International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 4.2, pp. 191-258
https://doi.org/10.1107/97809553602060000592

Chapter 4.2. X-rays

U. W. Arndt,a D. C. Creagh,b R. D. Deslattes,c J. H. Hubbell,d P. Indelicato,e E. G. Kessler Jrf and E. Lindrothg

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England,bDivision of Health, Design, and Science, University of Canberra, Canberra, ACT 2601, Australia,cNational Institute of Standards and Technology, Gaithersburg, MD 20899, USA,dRoom C314, Radiation Physics Building, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA,eLaboratoire Kastler-Brossel, Case 74, Université Pierre et Marie Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France,fAtomic Physics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA, and gDepartment of Atomic Physics, Stockholm University, S-104 05 Stockholm, Sweden

The generation of X-rays is discussed in the first section of this chapter. X-rays are generated by (1) the bombardment of a target by electrons, (2) the decay of certain radio isotopes, (3) as part of the synchrotron-radiation spectrum and (4) in plasmas produced by bombarding targets with high-energy laser beams. The spectra produced by the first method consist of a line spectrum characteristic of the target material accompanied by a continuum of white radiation (Bremsstrahlung). The intensities and wavelengths of these components are discussed and the nomenclatures of the lines in the characteristic spectrum are compared. Synchrotron radiation and plasma generation of X-rays are discussed very briefly. X-ray wavelengths are discussed in the second section of the chapter. Tables of K- and L-series reference wavelengths and K- and L-emission lines and absorption edges are provided. X-ray absorption spectra are discussed in the third section of the chapter. A detailed discussion of the problems associated with the measurement of X-ray absorption and the requirements for the absolute measurement of the X-ray attenuation coefficients is followed by a description of the problems which are encountered in making measurements at X-ray absorption edges of atomic species in materials. The popular, and extremely valuable, experimental technique of X-ray absorption fine structure (XAFS) is discussed. This is a relative, rather than an absolute measurement of X-ray absorption, and the theoretical analysis of XAFS spectra depends on the deviation of the data points from a cubic spline fit to the data rather that the deviation from the extrapolation of the free-atom absorption. A brief description of the origin and use of X-ray absorption near edge structure (XANES) is given. In the fourth section of the chapter, X-ray absorption (or attenuation) coefficients are considered. A detailed description of the theoretical and experimental techniques for the determination of X-ray absorption (and attenuation coefficients) is given. The tables supercede the existing experimental and theoretical tables that were available up to 2001. The theoretical values given here are restricted to the characteristic radiations commonly available from laboratory X-ray sources (Ti Kβ to Ag Kα). Computational problems become significant for elements in the lanthanide and actinide series of the periodic table. These tables include a significant recalculation of the absorption and attenuation coefficients for the lanthanide series. Note that although many theory-based tables exist, few accurate experimental measurements have been made on atomic species with atomic numbers greater than 50, especially in the region of absorption edges. A comparison is given of the extent to which theoretical and experimental data agree. A recent tabulation discusses soft X-ray absorption in the XANES region. Filters and monochromators are discussed in the fifth section of the chapter. This section serves as an introduction to the use of filters and monochromators in experimental apparatus. It discusses, using synchrotron-radiation experimental configurations as examples, the use of X-ray reflectivity, X-ray refraction and X-ray Bragg (and Laue) or Fresnel scattering for the production of monochromatic or quasi- monochromatic (pink) beams from polychromatic sources. Amongst the techniques discussed are: bent and curved mirrors, capillaries, quasi-Bragg (multilayer) mirrors, multiple filters, crystal monochromators, polarization and polarization-producing systems, and focusing using a Bragg–Fresnel optical system. The development of monochromators for synchrotron-radiation research is a rapidly evolving field, but in practice all depend on either the processes of reflection, refraction or scattering, or combinations of these processes. In the final section of the chapter, X-ray dispersion corrections are considered. The various theoretical methods for the computation of the X-ray dispersion corrections and the experimental techniques used for their determination are described in some detail. The theoretical values given here are restricted to the characteristic radiations commonly available from laboratory X-ray sources (Ti Kβ to Ag Kα). An artificial distinction is made between the relativistic Dirac–Hartree–Fock–Slater (RDHFS) formalism used to calculate the data in these tables and the S-matrix method used by a number of theoreticians. Formally, the two techniques are equivalent: in terms of the computations the two approaches are different, and the outcomes are different. Comparison with experimental data shows that the RDHFS formalism gives a better fit to experimental results.

4.2.1. Generation of X-rays

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U. W. Arndta

X-rays are produced by the interaction of charged particles with an electromagnetic field. There are four sources of X-rays that are of interest to the crystallographer.

  • (1) The bombardment of a target by electrons produces a continuous (`white') X-ray spectrum, called Bremsstrahlung, which is accompanied by a number of discrete spectral lines characteristic of the target material. The high-vacuum, or Coolidge, X-ray tube is the most important X-ray source for crystallographic studies.

  • (2) The decay of natural or artificial radio isotopes is often accompanied by the emission of X-rays. Radioactive X-ray sources are often used for the calibration of X-ray detectors. Mössbauer sources have the narrowest known spectral bandwidth and are used in nuclear resonance scattering studies.

  • (3) Sources of synchrotron radiation produced by relativistic electrons in orbital motion are of growing importance.

  • (4) X-rays are also produced in plasmas generated by the bombardment of targets by high-energy laser beams, but to date the yield has been principally in the form of soft X-rays.

The classical text on the generation and properties of X-rays is that by Compton & Allison (1935[link]), which still summarizes much of the information required by crystallographers. There is a more recent comprehensive book by Dyson (1973[link]). X-ray physics has received a new impetus on the one hand through the development of X-ray microprobe analysis dealt with in a number of monographs (Reed, 1975[link]; Scott & Love, 1983[link]) and on the other hand through the increasing utilization of synchrotron-radiation sources (see Subsection 4.2.1.5[link]).

4.2.1.1. The characteristic line spectrum

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Characteristic X-ray emission originates from the radiative decay of electronically highly excited states of matter. We are concerned mostly with excitation by electron bombardment of a target that results in the emission of spectral lines characteristic of the target elements. The electronic states occurring as initial and final states of a process involving the absorption of emission of X-rays are called X-ray levels. Levels involving the removal of one electron from the configuration of the neutral ground state are called normal X-ray levels or diagram levels.

Table 4.2.1.1[link] shows the relation between diagram levels and electron configurations. The notation used here is the IUPAC notation (Jenkins, Manne, Robin & Senemaud, 1991[link]), which uses arabic instead of the former roman subscripts for the levels. The IUPAC recommendations are to refer to X-ray lines by writing the initial and final levels separated by a hyphen, e.g. Cu K-L3 and to abandon the Siegbahn (1925[link]) notation, e.g. Cu Kα1, which is based on the relative intensities of the lines. The correspondence between the two notations is shown in Table 4.2.1.2[link]. Because this substitution has not yet become common practice, however, the Siegbahn notation is retained in Section 4.2.2[link], in which the wavelengths of the characteristic emission lines and absorption edges are discussed.

Table 4.2.1.1| top | pdf |
Correspondence between X-ray diagram levels and electron configurations; from Jenkins, Manne, Robin & Senemaud (1991[link]), courtesy of IUPAC

LevelElectron configurationLevelElectron configurationLevelElectron configuration
K [1s^{-1}] N1 [4s^{-1}] [O_1] [5s^{-1}]
L1 [2s^{-1}] N2 [4p^{-1}_{1/2}] [O_2] [5p^{-1}_{1/2}]
L2 [2p^{-1}_{1/2}] N3 [4p^{-1}_{3/2}] [O_3] [5p^{-1}_{3/2}]
L3 [2p^{-1}_{3/2}] N4 [4d^{-1}_{3/2}] [O_4] [5d^{-1}_{3/2}]
M1 [3s^{-1}] N5 [4d^{-1}_{5/2}] [O_5] [5d^{-1}_{5/2}]
M2 [3p^{-1}_{1/2}] N6 [4f^{-1}_{5/2}] [O_6] [5f^{-1}_{5/2}]
M3 [3p^{-1}_{3/2}] N7 [4f^{-1}_{7/2}] [O_7] [5f^{-1}_{7/2}]
M4 [3d^{-1}_{3/2}]        
M5 [3d^{-1}_{5/2}]        

Table 4.2.1.2| top | pdf |
Correspondence between IUPAC and Siegbahn notations for X-ray diagram lines; from Jenkins, Manne, Robin & Senemaud (1991[link]), courtesy of IUPAC

SiegbahnIUPACSiegbahnIUPACSiegbahnIUPAC
[K\alpha_1] [K\hbox{-}L_3] [L\alpha_1] [L_3\hbox{-}M_5] [L\gamma_1] [L_2\hbox{-}N_4]
[K\alpha_2] [K\hbox{-}L_2] [L\alpha_2] [L_3\hbox{-}M_4] [L\gamma_2] [L_1\hbox{-}N_2]
[K\beta_1] [K\hbox{-}M_3] [L\beta_1] [L_2\hbox{-}M_4] [L\gamma_3] [L_1\hbox{-}N_3]
[K\beta^1_2] [K\hbox{-}N_3] [L\beta_2] [L_3\hbox{-}N_5] [L\gamma_4] [L_1\hbox{-}O_3]
[K\beta^{11}_2] [K\hbox{-}N_2] [L\beta_3] [L_1\hbox{-}M_3] [L\gamma'_4] [L_1\hbox{-}O_2]
[K\beta_3] [K\hbox{-}M_2] [L\beta_4] [L_1\hbox{-}M_2] [L\gamma_5] [L_2\hbox{-}N_1]
[K\beta^1_4] [K\hbox{-}N_5] [L\beta_5] [L_3\hbox{-}O_{4,5}] [L\gamma_6] [L_2\hbox{-}O_4]
[K\beta^{11}_4] [K\hbox{-}N_4] [L\beta_6] [L_3\hbox{-}N_1] [L\gamma_8] [L_2\hbox{-}O_1]
[K\beta_{4x}] [K\hbox{-}N_4] [L\beta_7] [L_3\hbox{-}O_1] [L\gamma'_8] [L_2\hbox{-}N_{6(7)}]
[K\beta^1_5] [K\hbox{-}M_5] [L\beta'_7] [L_3\hbox{-}N_{6,7}] [L_\eta] [L_2\hbox{-}M_1]
[K\beta^{11}_5] [K\hbox{-}M_4] [L\beta_9] [L_1\hbox{-}M_5] [Ll] [L_3\hbox{-}M_1]
    [L\beta_{10}] [L_1\hbox{-}M_4] [Ls] [L_3\hbox{-}M_3]
    [L\beta_{15}] [L_3\hbox{-}N_4] [Lt] [L_3\hbox{-}M_2]
    [L\beta_{17}] [L_2\hbox{-}M_3] [Lu] [L_3\hbox{-}N_{6,7}]
        [Lv] [L_2\hbox{-}N_{6(7)}]

SiegbahnIUPAC
[M\alpha_1] [M_5\hbox{-}N_7]
[M\alpha_2] [M_5\hbox{-}N_6]
[M\beta] [M_4\hbox{-}N_6]
[M\gamma] [M_3\hbox{-}N_5]
[M\zeta] [M_{4,5}\hbox{-}N_{2,3}]

In the case of unresolved lines, such as [K\hbox{-}L_2] and [K\hbox{-}L_3], the recommended IUPAC notation is [K\hbox{-}L_{2,3}].

4.2.1.1.1. The intensity of characteristic lines

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The efficiency of the production of characteristic radiation has been calculated by a number of authors (see, for example, Dyson, 1973[link], Chap. 3). For a particular line, it depends on the fluorescence yield, that is the probability that the decay of an excited state leads to the emission of a photon, on the statistical weights of the X-ray levels involved, on the effects of the penetration and slowing down of the bombarding electrons in the target, on the fraction of electrons back-scattered out of the target, and on the contribution caused by fluorescent X-rays produced indirectly by the continuous spectrum. The emerging X-ray intensity is further affected by the partial absorption of the generated X-rays in the target.

Dyson (1973[link]) has also reviewed calculations and measurements made of the relative intensities of different lines in the K spectrum. The ratio of the [K\alpha_2] to [K\alpha_3] intensities is very close to 0.5 for Z between 23 and 48. The ratio of [K\beta_3] to [K\alpha_2] rises fairly linearly with Z from 0.2 at Z = 20 to 0.4 at Z = 80 and that of [K\beta_1] to [K\alpha_2] is near zero at Z = 29 and rises linearly with Z to about 0.1 at Z = 80. Relative intensities of lines in the L spectrum are given by Goldberg (1961[link]).

Green & Cosslett (1968[link]) have made extensive measurements of the efficiency of the production of characteristic radiation for a number of targets and for a range of electron accelerating voltages. Their results can be expressed empirically in the form [N_K/4\pi=N_0/4\pi(E_0-E_K-1){}^{1.63},\eqno (4.2.1.1)]where [N_K/4\pi] is the generated number of Kα photons per steradian per incident electron, N0 is a function of the atomic number of the target, E0 is the electron energy in keV and [E_K] is the excitation potential in keV. It should be noted that [N_K/4\pi] decreases with increasing Z.

For a copper target, this expression becomes [N_K/4\pi=1.8\times10^{-6}\,(E_0 - 8.9){}^{1.63}\eqno (4.2.1.2)]or [N'_K/4\pi=1.1\times10^{10}\,(E_0 - 8.9){}^{1.63},\eqno (4.2.1.3)]where [N'_K/4\pi] is the number of Kα photons per steradian per second per milliampere of tube current.

These expressions are probably accurate to within a factor of 2 up to values of [E_0/E_K] of about 10. Guo & Wu (1985[link]) found a linear relationship for the emerging number of photons with electron energy in the range [2\lt E_0/E_K\lt 5].

To obtain the number of photons that emerge from the target, the above expressions have to be corrected for absorption of the generated radiation in the target. The number of photons emerging at an angle [\varphi] to the surface, for normal electron incidence, is usually written [N_\varphi/4\pi=f(\chi)N/4\pi,\eqno (4.2.1.4)]where [\chi=(\mu/\rho)\hbox{ cosec }\varphi] (Castaing & Descamps, 1955[link]). Green (1963[link]) gives experimental values of the correction factor f(χ) for a series of targets over a range of electron energies. His curves for a copper target are given in Fig. 4.2.1.1[link] . It will be noticed that the correction factor increases with increasing electron energy since the effective depth of X-ray generation increases with voltage. As a result, curves of [N_\varphi] as a function of [E_0] have a broad maximum that is displaced towards lower voltages as [\varphi] decreases, as shown in the experimental curves for copper K radiation due to Metchnik & Tomlin (1963[link]) (Fig. 4.2.1.2[link] ). For very small take-off angles, therefore, X-ray tubes should be operated at lower than customary voltages. Note that the values in Fig. 4.2.1.2[link] agree to within ∼40% with those of Green & Cosslett. f(χ) at constant [E_0/E_K] increases with increasing Z, thus partly compensating for the decrease in [N_K], especially at small values of [\varphi]. A recent re-examination of the characteristic X-ray flux from Cr, Cu, Mo, Ag and W targets has been carried out by Honkimaki, Sleight & Suortti (1990[link]).

[Figure 4.2.1.1]

Figure 4.2.1.1 | top | pdf |

f(χ) curves for Cu K-L3 at a series of different accelerating voltages (in kV). From Green (1963[link]).

[Figure 4.2.1.2]

Figure 4.2.1.2 | top | pdf |

Experimental measurements of [N_\varphi] for Cu K-L3 as functions of the accelerating voltage for different take-off angles. From Metchnik & Tomlin (1963[link]).

4.2.1.2. The continuous spectrum

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The shape of the continuous spectrum from a thick target is very simple: [I_\nu], the energy per unit frequency band in the spectrum, is given by the expression derived by Kramers (1923[link]): [I_\nu=AZ(\nu_0-\nu)+BZ^2,\eqno (4.2.1.5)]where Z is the atomic number of the target and A and B are constants independent of the applied voltage [E_0]. B/A is of the order of 0.0025 so that the term in [Z^2] can usually be neglected (Fig. 4.2.1.3[link] ). [\nu_0] is the maximum frequency in the spectrum, i.e. the Duane–Hunt limit at which the entire energy of the bombarding electrons is converted into the quantum energy of the emitted photon, where [H\nu_0=hc/\lambda_0=E_0.\eqno (4.2.1.6)]Using the latest adjusted values of the fundamental constants (Cohen & Taylor, 1987[link]): [\eqalign{hc&=1.23984244\pm0.00000037\times10^{-6}\ {\rm eV\ m}\cr&=12.3984244\pm0.0000037 {\rm\ keV}\ {\rm \AA}.}]Equation (4.2.1.5)[link] can be rewritten in a number of forms. If [{\rm d}N_E] is the number of photons of energy E per incident electron, [{\rm d}N_E=bZ(E_0/E-1)\, {\rm d}E,\eqno (4.2.1.7)]where [b\sim2\times10^{-9}] photons eV−1  electron−1, and is known as Kramer's constant.

[Figure 4.2.1.3]

Figure 4.2.1.3 | top | pdf |

Intensity per unit frequency interval versus frequency in the continuous spectrum from a thick target at different accelerating voltages. From Kuhlenkampff & Schmidt (1943[link]).

From (4.2.1.7)[link], it follows that the total energy in the continuous spectrum per electron is [\textstyle\int\limits^{E_0}_0E\, {\rm d}N_E=bZE^2_0/2.\eqno (4.2.1.8)]Since the energy of the bombarding electron is [E_0], the efficiency of production of the continuous radiation is [\eta_c=bZE_0/2.\eqno (4.2.1.9)]Crystallographers are more accustomed to thinking of the spectrum in terms of wavelength. Equation (4.2.1.7)[link] can be transformed into [{\rm d}N_\lambda=hcbZ(1/\lambda^2-1/\lambda\lambda_0)\, {\rm d}\lambda,\eqno (4.2.1.10)]which has a maximum at [\lambda=2\lambda_0]. In practice, the emerging spectrum is modified by target absorption, which is greatest for the longer wavelengths and moves the maximum more nearly to [1.5\lambda_0].

It is of interest to compare the X-ray flux in a narrow wavelength band selected by an appropriate monochromator with the flux in a characteristic spectral line, in order to examine the practicability of XAFS (X-ray absorption fine-structure spectroscopy) or optimized anomalous-dispersion diffractometry experiments. For these purposes, the maximum permissible wavelength band is about 10−3 Å. From equation (4.2.1.10)[link], we see that, for a tungsten-target X-ray tube operated at 80 kV, [{\rm d}N_{\lambda}] is about 1.1 × 10−5 photons with the Kα energy electron−1 steradian−1 (10−3 δλ/λ)−1 for an X-ray wavelength in the neighbourhood of 1.5 Å. By comparison, from equation (4.2.1.2)[link], a copper-target tube operated at 40 kV produces about 5 × 10−4 Kα photons electron−1 steradian−1. In spite of this shortcoming by a factor of about 45, laboratory XAFS experiments are sufficiently common to have merited at least one specialized conference (Stern, 1980[link]; see also Tohji, Udagawa, Kawasak & Masuda, 1983[link]; Sakurai, 1993[link]; Sakurai & Sakurai, 1994[link]).

The use of continuous radiation for diffraction experiments is complicated by the fact that the radiation is polarized. The degree of polarization may be defined as [p=(I_\|-I_\perp)/(I_\|+I_\perp),\eqno (4.2.1.11)]where [I_\|] and [I_\perp] are the intensities of radiation with the electric vector parallel and perpendicular to the plane containing the incident electrons and the direction of the emitted photons. For an angle of π/2 between the electrons and the emitted beam, p varies smoothly through the spectrum; it is negative for the softest radiation, approximately zero at [\nu/\nu_0\sim0.1] and reaches values between +0.7 and +0.9 near the Duane–Hunt limit (Kirkpatrick & Wiedmann, 1945[link]). Since practical use of white radiation is likely to be in the vicinity of [\nu/\nu_0\sim0.1], the effect is not a large one.

It should also be noted that the spatial distribution of the white spectrum, even after correction for absorption in the target, is not isotropic. The intensity has a maximum at about 50° to the electron beam and non-zero minima at 0 and 180° to that beam (Stephenson, 1957[link]).

4.2.1.3. X-ray tubes

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The commonest source of X-rays is the high-vacuum, or Coolidge, X-ray tube, which may be either demountable and pumped continuously when in operation or permanently sealed after evacuation. The vacuum tube contains an electron gun that incorporates a thermionic cathode, which produces a well defined electron beam that is accelerated towards the anode or target, formerly often called the anticathode. In most X-ray tubes intended for crystallographic purposes, the anode is massive, i.e. its thickness is large compared with the range of the electrons; it is usually water-cooled and its surface is normal to the incident electron beam. Usually, it is desirable for the X-ray source to be small (between 25 μm and 1 mm square) and for the X-ray intensity from the tube to be the maximum possible for the amount of power that can be dissipated in the target. These objectives are best achieved by designing the electron gun to produce a line focus, that is the electron focus on the target face is approximately rectangular with the small dimension equal to the desired effective source size and the large dimension about 10 to 20 times larger. The focus is viewed at an angle between about 2 and 5° to the anode surface to produce an approximately square foreshortened effective source; and the X-ray windows are so positioned as to make these take-off angles possible. For some purposes, very fine line sources are required and windows may be provided to allow the focus to be viewed so as to foreshorten the line width. Higher power dissipation is possible in X-ray tubes in which the anode rotates: the line focus is now usually on the cylindrical surface of the anode with its long dimension parallel to the axis of rotation.

For focal-spot sizes down to about 100 μm, an electrostatic gun is adequate; this consists of a fine helical filament and a Wehnelt cathode, which produces a demagnified electron image of the filament on the anode. For most purposes, the Wehnelt cathode can be at the same potential as the filament but cleaner foci and adjustment of the focal spot size are possible when this electrode is negatively biased with respect to the filament. The filament is nearly always directly heated and made of tungsten. Lower filament temperatures, and smaller heating currents, could be achieved with activated heaters but the vacuum in high-power devices like X-ray tubes is rarely hard enough to permit their use since they are easily poisoned. However, Yao (1992[link]) has reported successful operation of a hot-pressed polycrystalline lanthanum hexaboride cathode in an otherwise unmodified RU-1000 rotating-target X-ray generator.

Very fine focus tubes, with foci in the range between 25 and 1 μm, require magnetic lenses. At one time, the all-electrostatic X-ray tube of Ehrenberg & Spear (1951[link]), which achieved foci between 20 and 80 μm, was very popular.

Sealed-off X-ray tubes for crystallographic use are nowadays made in the form of inserts containing a target of one of a range of standard metals to produce the desired characteristic radiation. A series of nominal focal-spot sizes, shown in Table 4.2.1.3[link], is commonly available. The insert is mounted inside a standard shield that is radiation- and shock-proof and that is fitted with X-ray shutters and filters and often also with a standardized track for mounting X-ray cameras. The water-cooled anode is normally at ground potential and the negative high voltage for the cathode, together with the filament supply, is brought in through a shielded shock-proof cable. The high voltage is nowadays generally of the constant-voltage type, that is, it is full-wave rectified and smoothed by means of solid-state rectifiers and capacitors housed in the high-voltage transformer tank, which also contains the filament transformer. The high tension and the tube current are frequently stabilized. Only the simplest X-ray generators now employ an alternating high tension that is rectified by the self-rectifying property of the X-ray tube itself.

Table 4.2.1.3| top | pdf |
Copper-target X-ray tubes and their loading

X-ray tubeAnode diameter (mm)Speedf1 × f2
(mm) (mm)
μLoading (kW)Recommended specific loading (kW mm−2)
r min−1mm s−1calc.recommended
Standard insert 8 × 0.15 0.295 1.0 0.8 0.67
8 × 0.4 0.359 1.2 1.5 0.47
10 × 1.0 0.425 1.8 2.0 0.20
12 × 2.0 0.493 2.5 2.7 0.11
AEI-GX21 89 6000 28000 1 × 0.1 0.425 1.4 1.2 12.0
      2 × 0.2 0.425 3.95 3.2 8.0
      3 × 0.3 0.425 7.3 5.2 5.8
      5 × 0.5 0.425 15.6 15.0 6.0
AEI-GX13 457 4500 108000 1 × 0.1 0.425 2.7 2.7 27.0
Rigaku-RU200 99 6000 31000 1 × 0.1 0.425 1.5 1.2 12.0
      2 × 0.2 0.425 4.2 3.0 7.5
      3 × 0.3 0.425 7.6 5.4 6.0
Rigaku-RU500 400 1250 26200 10 × 0.5 0.359 26.8 30 6.0
Rigaku-RU1000 400 2500 52450 10 × 1 0.425 60 60 6.0
Rigaku-RU1500 250 10000 131000 10 × 1 0.425 96 90 9.0
KFA-Jülich 250 12000 157000 14 × 1.4 0.425 173 120 6.1

A demountable continuously pumped form of construction is nowadays adopted mainly for rotating-anode and other specialized X-ray tubes. The pumping system must be capable of maintaining a vacuum of better than 10−5 Torr: filament life is critically dependent upon the quality of the vacuum.

Rotating-anode tubes have been reviewed by Yoshimatsu & Kozaki (1977[link]). The first successful tube of this type that incorporated a vacuum shaft seal was described by Clay (1934[link]). Modern tubes mostly contain vacuum-oil-lubricated shaft seals of the type due to Wilson (1941[link]) and are based on, or are similar to, the rotating-anode tubes described by Taylor (1949[link], 1956[link]). In some tubes, successful use has been made of ferro-fluidic vacuum seals (see Bailey, 1978[link]). The main problems in the operation of rotating-anode tubes is the lifetime of the seals and of bearings that operate in vacuo. In successful tubes, e.g. those manufactured by Enraf, Rigaku-Denki, and Siemens, these lifetimes are about the same as the lifetime of the filament under good vacuum conditions, that is, of the order of 1000 h.

Phillips (1985[link]) has written a review article on stationary and rotating-anode X-ray tubes that contains many important practical details.

4.2.1.3.1. Power dissipation in the anode

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The allowable power loading of X-ray tube targets is determined by the temperature of the target surface, which must remain below the melting point. Müller (1927[link], 1929[link], 1931[link]) first calculated the maximum loading both for stationary and for rotating anodes. His calculations were refined by Oosterkamp (1948a[link],b[link],c[link]) who considered, in particular, targets of finite thickness, and who also treated pulsed operation of the tube. For normal conditions, Oosterkamp's conclusions and those of Ishimura, Shiraiwa & Sawada (1957[link]) do not greatly differ from those of Müller, which are in adequate agreement with experimental observations.

For an elliptical focal spot with axes [f_1] and [f_2], Müller's formula for the maximum power dissipation on a stationary anode, assumed to be a water-cooled block of dimensions large compared with the focal-spot dimensions, can be written [W_{\rm stat}=2.063(T_M-T_0)Kf_1\mu(\,f_1,f_2),\eqno (4.2.1.12)]where K is the specific thermal conductivity of the target material in W mm−1, [T_M] is the maximum temperature at the centre of the focal spot on the target, that is, a temperature well below the melting point of the target material, and [T_0] is the temperature of the cold surface of the target, that is, of the cooling water. The function μ is shown in Fig. 4.2.1.4[link] . For copper, K is 400 W m−1 and, with [T_M-T_0] = 500 K, [W_{\rm stat}=425\mu f_1.\eqno (4.2.1.13)]For f2/f1 = 0.1, and μ = 0.425, this equation becomes [W_{\rm stat}=180 \, f_1.\eqno (4.2.1.14)]In these last two equations, f1 is in mm.

[Figure 4.2.1.4]

Figure 4.2.1.4 | top | pdf |

The function μ in Müller's equation (equation 4.2.1.12[link]) as a function of the ratio of width to length of the focal spot.

For a rotating target, Müller found that the permissible power dissipation was given by [W_{\rm rot}=1.428\,K(T_M-T_0)\,f_1(\,f_2\rho Cv/2K)^{1/2},\eqno (4.2.1.15)]where f2 is the short dimension of the focus, assumed to be in the direction of motion of the target, v is the linear velocity, ρ is the density of the target material, and C is its specific heat.

For a copper target with f1 and f2 in mm and v in mm s−1, [W_{\rm rot}=26.4\,f_1(\,f_2v)^{1/2}.\eqno (4.2.1.16)]Equation (4.2.1.16)[link] shows that for very narrow focal spots rotating-anode tubes give useful improvements in permissible loading only if the surface speed is very high (see Table 4.2.1.3[link]). The reason is that with large foci on stationary anodes the isothermal surfaces in the target are planar; with fine foci, these surfaces become cylindrical and this already makes for very efficient cooling without the need for rotation. Rotating anodes are thus most useful for medium-size foci (200 to 500 μm) since for the larger focal spots it becomes very expensive to construct power supplies capable of supplying the permissible amount of power.

Table 4.2.1.3[link] shows the recommended loading for a number of commercially available X-ray tubes with copper targets, which will be seen to be in qualitative agreement with the calculations. Some of the discrepancy is due to the fact that the value of [K(T_M-T_0)] for the copper–chromium alloy targets used in actual X-ray tubes is appreciably lower than the value for pure copper used here. To a good approximation, the permissible loading for other targets can be derived by multiplying those in Table 4.2.1.3[link] by the factors shown in Table 4.2.1.4[link]. It is worth noting that the recommended loading of commercial stationary-target X-ray tubes has increased steadily in recent years. This is largely due to improvements in the water cooling of the back surface of the target by increasing the turbulence of the water and the effective surface area of the cooled surface.

Table 4.2.1.4| top | pdf |
Relative permissible loading for different target materials

CuCrFeCoMoAgW
1.0 0.9 0.6 0.9 1.2 1.0 1.2

In considering Table 4.2.1.3[link], it should be noted that the linear velocities of the highest-power X-ray-tube anode have already reached a speed that Yoshimatsu & Kozaki (1977[link]) consider the practical limit, which is set by the mechanical properties of engineering materials. It should also be noted that much higher specific loads can be achieved for true micro-focus tubes, e.g. 50 kW mm−2 for a 25 μm Ehrenberg & Spear tube and 1000 kW mm−2 for a tube with a 1 μm focus (Goldsztaub, 1947[link]; Cosslett & Nixon, 1951[link], 1960[link]).

Some tubes with focus spots of less than 10 μm utilize foil or needle targets. These targets and the heat dissipated in them have been discussed by Cosslett & Nixon (1960[link]). The dissipation is less than that in a massive target by a factor of about 3 for a foil and 10 for a needle, but, in view of the low absolute power, target movement and even water-cooling can be dispensed with.

4.2.1.4. Radioactive X-ray sources

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Radioactive sources of X-rays are mainly of interest to crystallographers for the calibration of X-ray detectors where they have the great advantage of being completely stable with time, or at least of having an accurately known decay rate. For some purposes, spectral purity of the radiation is important; radionuclides that decay wholly by electron capture are particularly useful as they produce little or no β or other radiation. In this type of decay, the atomic number of the daughter nucleus is one less than that of the decaying isotope, and the emitted X-rays are characteristic of the daughter nucleus. In some cases, the probability of electron capture taking place from some shell other than the K shell is very small and most of the photons emitted are K photons. The number of photons emitted into a solid angle of 4π, uncorrected for absorption, is given by the strength of the source in Curies (1 Curie = 3.7 × 1010 disintegrations s−1), since each disintegration produces one photon. A list of these nuclei (after Dyson, 1973[link]) is given in Table 4.2.1.5[link].

Table 4.2.1.5| top | pdf |
Radionuclides decaying wholly by electron capture, and yielding little or no γ-radiation

NuclideHalf-lifeX-raysRemarks
Element[K{\alpha_1}] (keV)
37Ar 35 d Cl 2.622
51Cr 27.8 d V 4.952 γ at 320 keV
55Fe 2.6 a Mn 5.898
71Ge 11.4 d Ga 9.251
103Pd 17 d Rh 20.214 Several γ's; all weak
109Cd 453 d Ag 22.16 γ at 88 keV
125I 60 d Te 27.47 γ at 35.4 keV
131Cs 10 d Xe 29.80
145Pm 17.7 a Nd 37.36 γ's at 67 and 72 keV
145Sm 340 d Pm 38.65 γ's at 61 keV; weak γ at 485 keV
179Ta 600 d Hf 55.76
181W 140 d Ta 57.52 γ at 6.5 keV; weak γ's at 136, 153 keV
205Pb 5 × 107 a Tl L only (Lα1 = 10.27 keV)

Useful radioactive sources are also made by mixing a pure β-emitter with a target material. These sources produce a continuous spectrum in addition to the characteristic line spectrum. The nuclide most commonly used for this purpose is tritium which emits β particles with an energy up to 18 keV and which has a half-life of 12.4 a.

Radioactive X-ray sources have been reviewed by Dyson (1973[link]).

4.2.1.5. Synchrotron-radiation sources

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The growing importance of synchrotron radiation is attested by a large number of monographs (Kunz, 1979[link]; Winick, 1980[link]; Stuhrmann, 1982[link]; Koch, 1983[link]) and review articles (Godwin, 1968[link]; Kulipanov & Skrinskii, 1977[link]; Lea, 1978[link]; Winick & Bienenstock, 1978[link]; Helliwell, 1984[link]; Buras, 1985[link]). Project studies for storage rings such as the European Synchrotron Radiation Facility, the ESRF (Farge & Duke, 1979[link]; Thompson & Poole, 1979[link]; Buras & Marr, 1979[link]; Buras & Tazzari, 1984[link]) are still worth consulting for the reasoning that lay behind the design; the ESRF has, in fact, achieved or even exceeded the design parameters (Laclare, 1994[link]).

A charged particle with energy E and mass m moving in a circular orbit of radius R at a constant speed v radiates a power P into a solid angle of 4π, where [P=2e^2c(v/c)^4(E/mc^2)^4/3R^2.\eqno (4.2.1.17)]The orbit of the particle can be maintained only if the energy lost in the form of electromagnetic radiation is constantly replenished. In an electron synchrotron or in a storage ring, the circulating particles are electrons or positrons maintained in a closed orbit by a magnetic field; their energy is supplied or restored by means of an oscillating radio-frequency (RF) electric field at one or more places in the orbit. In a synchrotron, designed for nuclear-physics experiments, the circulating particles are injected from a linear accelerator, accelerated up to full energy by the RF field and then deflected into a target with a cycle frequency of about 50 Hz. The synchrotron radiation is thus produced in the form of pulses of this frequency. A storage ring, on the other hand, is filled with electrons or positrons and after acceleration the particle energy is maintained by the RF field; the current ideally circulates for many hours and decays only as a result of collisions with remaining gas molecules. At present, only storage rings are used as sources of synchrotron radiation and many of these are dedicated entirely to the production of radiation: they are not used at all, or are used only for limited periods, for nuclear-physics collision experiments.

In equation (4.2.1.17)[link], we may substitute for the various constants and obtain for the radiated power [P=0.0885\,E^4I/R,\eqno (4.2.1.18)]where E is in GeV (109 eV), I is the circulating electron or positron current in milliamperes, and R is in metres. Thus, for example, at the Daresbury storage ring in England, R = 5.5 m and, for operation at 2 GeV and 200 mA, P = 51.5 kW. Storage rings with a total power of the order of 1 MW are planned.

For relativistic electrons, the electromagnetic radiation is compressed into a fan-shaped beam tangential to the orbit with a vertical opening angle [\psi\simeq mc^2/E], i.e. ~0.25 mrad for E = 2 GeV (Fig. 4.2.1.5[link] ). This fan rotates with circulating electrons: if the ring is filled with n bunches of electrons, a stationary observer will see n flashes of radiation every 2πR/c s, the duration of each flash being less than 1 ns.

[Figure 4.2.1.5]

Figure 4.2.1.5 | top | pdf |

Synchrotron radiation emitted by a relativistic electron travelling in a curved trajectory. B is the magnetic field perpendicular to the plane of the electron orbit; ψ is the natural opening angle in the vertical plane; P is the direction of polarization. The slit S defines the length of the arc of angle Δθ from which the radiation is taken. From Buras & Tazzari (1984[link]); courtesy of ESRP.

The spectral distribution of synchrotron radiation extends from the infrared to the X-ray region; Schwinger (1949[link]) gives the instantaneous power radiated by a monoenergetic electron in a circular motion per unit wavelength interval as a function of wavelength (Winick, 1980[link]). An important parameter specifying the distribution is the critical wavelength [\lambda_c]: half the total power radiated, but only ∼9% of the total number of photons, is at [\lambda \lt \lambda_c] (Fig. 4.2.1.6[link] ). [\lambda_c] is given by [\lambda_c=4\pi R/3(E/mc^2){}^3,\eqno (4.2.1.19)]from which it follows that [\lambda_c] in Å can be expressed as [\lambda_c=18.64/(BE^2),\eqno (4.2.1.20)]where B (= 3.34 E/R) is the magnetic bending field in T, E is in GeV, and R is in metres.

[Figure 4.2.1.6]

Figure 4.2.1.6 | top | pdf |

Synchrotron-radiation spectrum: percentage per unit wavelength interval (a) of power of total power and (b) of number of photons of total number of photons at wavelengths greater than λ versus λ/λc. Note that half the power but only 9% of the photons are radiated at wavelengths less than λc; courtesy of H. Winick.

Synchrotron radiation is highly polarized. In an ideal ring where all electrons are parallel to one another in a central orbit, the radiation in the orbital plane is linearly polarized with the electric vector lying in this plane. Outside the plane, the radiation is elliptically polarized.

In practice, the electron path in a storage ring is not a circle. The `ring' consists of an alternation of straight sections and bending magnets and beam lines are installed at these magnets. So-called insertion devices with a zero magnetic field integral, i.e. wigglers and undulators, may be inserted in the straight sections (Fig. 4.2.1.7[link] ). A wiggler consists of one or more dipole magnets with alternating magnetic field directions aligned transverse to the orbit. The critical wavelength can thus be shifted towards shorter values because the bending radius can be made small over a short section, especially when superconducting magnets are used. Such a device is called a wavelength shifter. If it has N dipoles, the radiation from the different poles is added to give an N-fold increase in intensity. Wigglers can be horizontal or vertical.

[Figure 4.2.1.7]

Figure 4.2.1.7 | top | pdf |

Main components of a dedicated electron storage-ring synchrotron-radiation source. For clarity, only one bending magnet is shown. From Buras & Tazzari (1984[link]); courtesy of ESRP.

In a wiggler, the maximum divergence 2α of the electron beam is much larger than ψ, the vertical aperture of the radiation cone in the spectral region of interest (Fig. 4.2.1.5[link]). If [2\alpha\ll\psi] and if, in addition, the magnet poles of a multipole device have a short period [\lambda_0], the device becomes an undulator: interference will take place between the radiation emitted at two points [\lambda_0] apart on the electron trajectory (Fig. 4.2.1.8[link] ). The spectrum at an angle [\varphi] to the axis observed through a pin-hole will consist of a single spectral line and its harmonics of wavelengths [\lambda_i=i^{-1}\lambda_0[(E/mc^2)^{-2}+\alpha^{2}/2+\theta^{2}]/2\eqno (4.2.1.21)](Hofmann, 1978[link]). Typically, the bandwidth of the lines, δλ/λ, will be ∼0.01 to 0.1 and the photon flux per unit band width from the undulator will be many orders of magnitude greater than that from a bending magnet. Existing undulators have been designed for photon energies below 2 keV; higher energies, because of the relatively weak magnetic fields necessitated by the need to keep λ0 small [equation (4.2.1.21)[link]], require a high electron energy: undulators with a fundamental wavelength in the neighbourhood of 0.86 Å are planned for the European storage ring (Buras & Tazzari, 1984[link]).

[Figure 4.2.1.8]

Figure 4.2.1.8 | top | pdf |

Electron trajectory within a multipole wiggler or undulator. λ0 is the spatial period, α the maximum deflection angle, and θ the observation angle. From Buras & Tazzari (1984[link]); courtesy of ESRP.

The wavelength spectra for a bending magnet, a wiggler and an undulator for the ESRF, are shown in Fig. 4.2.1.9[link] . A comparison of the spectra from an existing storage ring with the spectrum of a rotating-anode tube is shown in Fig. 4.2.1.10[link] .

[Figure 4.2.1.9]

Figure 4.2.1.9 | top | pdf |

Spectral distribution and critical wavelengths for (a) a dipole magnet, (b) a wavelength shifter, and (c) a multipole wiggler for the proposed ESRF. From Buras & Tazzari (1984[link]); courtesy of ESRP.

[Figure 4.2.1.10]

Figure 4.2.1.10 | top | pdf |

Comparison of the spectra from the storage ring SPEAR in photons s−1 mA−1 mrad−1 per 1% passband (1978 performance) and a rotating-anode X-ray generator. From Nagel (1980[link]); courtesy of K. O. Hodgson.

The important properties of synchrotron-radiation sources are:

  • (1) high intensity;

  • (2) very broad continuous spectral range;

  • (3) narrow angular collimation;

  • (4) small source size;

  • (5) high degree of polarization;

  • (6) regularly pulsed time structure;

  • (7) computability of properties.

Table 4.2.1.6[link] (after Buras & Tazzari, 1984[link]) compares the most important parameters of the European Synchrotron Radiation Facility (ESRF) with a number of other storage rings. In this table, BM and W signify bending-magnet and wiggler beam lines, respectively, [\sigma_x] and [\sigma'_z] is the source divergence; the flux is integrated in the vertical plane. The ESRF is seen to have a higher flux than other sources; even more impressive by virtue of the small dimensions of the source size and divergence are its improvements in spectral brightness (defined as the number of photons s−1 per unit solid angle per 0.1% bandwidth) and in spectral brilliance (defined as the number of photons s−1 per unit solid angle per unit area of the source per 0.1% bandwidth). In comparing different synchrotron-radiation sources with one another and with conventional sources (Fig. 4.2.1.10[link]), the relative quantity for comparison may be flux, brightness or brilliance, depending on the type of diffraction experiment and the type of collimation adopted. Table 4.2.1.7[link] (due to Farge & Duke, 1979[link]) attempts to compare intensity factors for a number of typical experiments. In general, a high brightness is important in experiments that do not embody focusing elements, such as mirrors or curved crystals, and a high brilliance in those experiments that do.

Table 4.2.1.6| top | pdf |
Comparison of storage-ring synchrotron-radiation sources; the parameters were correct in 1985 and, for some sources, may be substantially different from those at earlier or later periods; after Buras & Tazzari (1984[link]), courtesy of ESRP

Storage ringSource typeNo. of polesI (mA)E (GeV)R (m)[\sigma_x] (mm)[\sigma_z] (mm)[\sigma'_z] (mrad)[\lambda_c] (Å)Ec (keV)[{\rm Flux} \Big[{{\rm photons \, \, s}^{-1}\over {\rm mrad}\times 0.1\%\, \, {\rm BW}}\Big]]
at [\lambda^c]at 1.54 Å
(1) ESRF BM 100 5.0 20.0 0.092 0.100 0.008 0.9 14 8 × 1012 1 × 1013
(2) ESRF W 30 100 5.0 11.56 0.062 0.040 0.016 0.5 24 2.4 × 1014 3 × 1014
(3) ADONE (Frascati) BM 100 1.5 5.0 0.8 0.4 0.04 8.0 1.5 2.4 × 1012 5 × 1010
(4) ADONE (Frascati) W 6 100 1.5 2.6 1.4 0.24 0.08 4.3 3 1.4 × 1013 3.4 × 1012
(5) SRS (Daresbury) BM 300 2.0 5.56 2.7 0.23 0.05 4.0 3 1 × 1013 3 × 1012
(6) SRS (Daresbury) W 1 300 2.0 1.33 5.3 0.17 0.05 0.9 13 1 × 1013 1.2 × 1013
(7) DCI (Orsay) BM 250 1.8 3.82 2.72 1.06 0.06 3.6 3.4 7 × 1012 2.4 × 1012
(8) DORIS (Hamburg) BM 100 3.7 12.22 1.0 0.3 0.05 1.3 9.2 6 × 1012 6.4 × 1012
(9) DORIS (Hamburg) BM 40 5.0 12.22 1.3 0.65 0.065 0.55 23 3 × 1012 4.4 × 1013
(10) DORIS (Hamburg) W 32 100 3.7 20.57 1.5 0.4 0.033 2.3 5.5 1.9 × 1014 1.3 × 1014
(11) CESR (Cornell) BM 40 5.5 32.0 1.44 1.0 0.065 1.0 11.5 3.5 × 1012 4 × 1012
(12) CESR (Cornell) W 6 40 5.5 13.2 1.9 1.2 0.05 0.4 28 2 × 1013 3 × 1013
(13) NSLS X-ray (Brookhaven) BM 300 2.5 6.83 0.25 0.1 0.01 2.4 5 1 × 1013 8 × 1012
(14) SPEAR BM 100 3.0 12.7 2.0 0.28 0.05 2.7 5 5 × 1012 3 × 1012
(15) SPEAR W 8 100 3.0 5.57 3.2 0.15 0.03 1.0 10 3.8 × 1013 4.5 × 1013
(16) SPEAR W 54 100 3.0 8.36 3.2 0.15 0.03 1.7 7 2.6 × 1014 2.4 × 1014
(17) Photon Factory (Tsukuba) BM 150 2.5 8.66 2.2 0.6 0.14 3.0 4 6 × 1012 3 × 1012
(18) Photon Factory (Tsukuba) W 3 150 2.5 1.85 1.9 0.7 0.18 0.7 19 1.8 × 1013 2.5 × 1013
(19) VEPP-3 BM 100 2.2 6.15 6.15 0.08 0.02 3.0 4 3.5 × 1012 1.5 × 1012
One standard deviation of Gaussian distribution.

Table 4.2.1.7| top | pdf |
Intensity gain with storage rings over conventional sources; from Farge & Duke (1979[link]), courtesy of ESF

 GX6 rotating-anode tube 2.4 kW (Cu Kα emission)DCI 1.72 GeV and 240 mAESRF 5 GeV and 565 mA
[Scheme scheme1] Small-angle scattering with a double monochromator ×500 to 1000 ×15000 to 3000
Protein crystallography with a single-focus monochromator    
  1 mm3 samples ×50 to 160 ×900 to 1800
  Small samples ×30 to 60 ×650 to 1300
Diffuse scattering (wide angles, low resolution and large samples) with a curved graphite monochromator ×20 to 40 ×160 to 320
Non characteristic wavelength (continuous background) EXAFS experimental set-up with a 100 kW rotating anode ×104 ×105

Many surveys of existing and planned synchrotron-radiation sources have been published since the compilation of Table 4.2.1.6[link]. Fig. 4.2.1.11[link] , taken from a recent review (Suller, 1992[link]), is a graphical illustration of the growth and the distribution of these sources. An earlier census is due to Huke & Kobayakawa (1989[link]). Many detailed descriptions of beam lines for particular purposes, such as protein crystallography (e.g. Fourme, 1992[link]) or at individual storage rings (e.g. Kusev, Raiko & Skuratowski, 1992[link]) have appeared: these are too numerous to list here and can be located by reference to Synchrotron Radiation News.

[Figure 4.2.1.11]

Figure 4.2.1.11 | top | pdf |

The evolution of storage-ring synchrotron-radiation sources over the decades, as illustrated by their increasing number and range of machine energies (based on Suller, 1992[link]).

4.2.1.6. Plasma X-ray sources

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Plasma sources of hard X-rays are being investigated in many laboratories. Most of the material in this section is derived from publications from the Laboratory for Laser Energetics, University of Rochester, USA. Plasma sources of very soft X-rays have been reviewed by Byer, Kuhn, Reed & Trail (1983[link]).

The peak wavelength of emission from a black-body radiator falls into the ultraviolet at about 105 K and into the X-ray region between 106 and 107 K. At these temperatures, matter is in the form of a plasma that consists of highly ionized atoms and of electrons with energies of several keV. The only successful methods of heating plasmas to temperatures in excess of 106 K is by means of high-energy laser beams with intensities of 1012 W mm−2 or more. The duration of the laser pulse must be less than 1 ns so that the plasma cannot flow away from the pulse. When the plasmas are created from elements with 15 [\lt] Z [\lt] 25, they consist mainly of ions stripped to the K shell, that is of hydrogen- and helium-like ions. The X-ray spectrum (Fig. 4.2.1.12[link] ) then contains a main group of lines with a bandwidth for the group of about 1%; the band is situated slightly below the K-absorption edge of the target material. The intensity of the band drops with increasing atomic number. For diffraction studies, Forsyth & Frankel (1980[link], 1984[link]) and Frankel & Forsyth (1979[link], 1985[link]) used a multi-stage Nd3+:glass laser (Seka, Soures, Lewis, Bunkenburg, Brown, Jacobs, Mourou & Zimmermann, 1980[link]), which was able to deliver up to 220 J per pulse of width 700 ps. They obtained [6\times10^{14}] photons pulse−1 for a Cl15+ plasma with a mean wavelength of about 4.45 Å and about [3\times10^{13}] photons pulse−1 for a Fe24+ plasma at about 1.87 Å (Yaakobi, Bourke, Conturie, Delettrez, Forsyth, Frankel, Goldman, McCrory, Seka, Soures, Burek & Deslattes, 1981[link]). More recently, the laser was fitted with a frequency conversion system that shifts the peak power of the laser light from the infrared (1.054 µm) to the ultraviolet (0.351 µm) (Seka, Soures, Lund & Craxton 1981[link]). This led to a more efficient X-ray production, which permitted a more than twofold increase in X-ray flux, even though the maximum pulse energies had to be reduced to ∼50 J to prevent damage to the optical components (Yaakobi, Boehli, Bourke, Conturie, Craxton, Delettrez, Forsyth, Frankel, Goldman, McCrory, Richardson, Seka, Shvarts & Soures, 1981[link]). Forsyth & Frankel (1984[link]) used the plasma X-ray source for diffraction studies with 4.45 Å X-rays with a focusing collimation system that delivered up to 1010 photons pulse−1 to the specimen over an area approximately 150 µm in diameter. More recently, by special target design (Forsyth, 1986, unpublished), fluxes have been increased by factors of 2 to 3 without altering the laser output. Other plasma sources have been described by Collins, Davanloo & Bowen (1986[link]) and by Rudakov, Baigarin, Kalinin, Korolev & Kumachov (1991[link]).

[Figure 4.2.1.12]

Figure 4.2.1.12 | top | pdf |

X-ray emission from various laser-produced plasmas. From Forsyth & Frankel (1980[link]); courtesy of J. M. Forsyth.

The cost of plasma sources is about an order of magnitude greater than that of rotating-anode generators (Nagel, 1980[link]). Their use is at present confined to flash-diffraction experiments, since the duty cycle is a maximum of one flash every 30 min. Attempts are being made to increase the laser repetition rate; a substantial improvement could lead to a source that would rival storage-ring sources.

4.2.1.7. Other sources of X-rays

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Parametric X-ray generation can be described as the diffraction of virtual photons associated with the field of a relativistic charged particle passing through a crystal. These diffracted photons appear as real photons with an energy that satisfies Bragg's law for the reflecting crystal planes, so that the energy can be tuned between 5 and 45 keV by rotating the mosaic graphite crystal. Linear accelerators with an energy between 100 and 500 MeV produce the incident relativistic electron beam (Maruyama, Di Nova, Snyder, Piestrup, Li, Fiorito & Rule, 1993[link]; Fiorito, Rule, Piestrup, Li, Ho & Maruyama, 1993[link]).

Transition-radiation X-rays with peak energies between 10 and 30 keV are produced when electrons from 100 to 400 MeV strike a stack of thin foils (Piestrup, Moran, Boyers, Pincus, Kephart, Gearhart & Maruyama, 1991[link]). Quasi-monochromatic X-rays result from a selection of target foils with appropriate K-, L- or M-edge frequencies (Piestrup, Boyers, Pincus, Harris, Maruyama, Bergstrom, Caplan, Silzer & Skopik, 1991[link]).

Channelling radiation, resulting from the incidence of electrons with an energy of only about 5 MeV on appropriately aligned diamond or silicon crystals hold out the hope of producing a bright tunable X-ray source.

One or more of these methods may, in the future, be developed as X-ray sources that can compete with synchrotron-radiation sources.

4.2.2. X-ray wavelengths

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R. D. Deslattes,c E. G. Kessler Jr,f P. Indelicatoe and E. Lindrothg

4.2.2.1. Historical introduction

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Wavelength tables in previous editions of this volume (Rieck, 1962[link]; Arndt, 1992[link]) were mainly obtained from the compilations prepared in Paris under the general direction of Professor Y. Cauchois (Cauchois & Hulubei, 1947[link]; Cauchois & Senemaud, 1978[link]). A separate effort by the late Professor J. A. Bearden and his collaborators (Bearden, 1967[link]) has been widely used in other aggregations of tabular data and was made available for some time through the Standard Reference Data Program at the National Institute of Standards and Technology (NIST). For simplicity in the following discussion, we use the Bearden database as a frame of reference with respect to which our current, and rather different, approach can be compared. Although a detailed comparison of the historical databases may be of some interest, the result would have only very small influence on the outcome presented here. To specify this framework, we begin with a brief description of the procedures used in establishing this reference database.

Bearden and his collaborators remeasured a group of five X-ray lines (Bearden, Henins, Marzolf, Sauder & Thomsen, 1964[link]), with the remaining entries in the wavelength table coming from a critically reviewed, and re-scaled, subset of earlier measurements (Bearden, 1967[link]). Line locations were given in Å* units, a scale defined by setting the wavelength of W Kα1 = 0.2090100 Å*. It was Bearden's intention that, for all but the most demanding applications, one could simply assign Å*/Å = 1, with an uncertainty arising from the fundamental physical constants, particularly NA and hc/e, combined with uncertainties arising from the measurement technology (Bearden, 1965[link]). Not long after the publication of the final compilation (Bearden, 1967[link]), it became clear that the fundamental constants used in defining Å* needed significant revision (Cohen & Taylor, 1973[link]), and that there were some inconsistencies in the metrology (Kessler, Deslattes & Henins, 1979[link]).

4.2.2.2. Known problems

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Aside from the particular issues noted above, all previous wavelength tables had certain limitations arising from the procedures used in their generation. In particular, except for a small group of five [K\alpha ] spectra (Bearden, Thomsen et al., 1964[link]), the Bearden tables relied entirely on data previously reported in the literature. Both of the other tabulations also proceeded using only reported experimental values (Cauchois & Hulubei, 1947[link]; Cauchois & Senemaud, 1978[link]). In the Bearden compilation process, available data for each emission line were weighted according to claimed uncertainties, modified in certain cases by Bearden's detailed knowledge of the measurement practices of the major sources of experimental wavelength values. The complete documentation of this remarkable undertaking is, unfortunately, not widely accessible. Our evident need to understand the origin of the `recommended' values has been greatly aided by the availability of a copy of the full documentation (Bearden, Thomsen et al., 1964[link]).

The actual experimental data array from which the previous tables emerged is not complete, even for the prominent (`diagram') lines. In the cases where experimental data were not available [as can be seen only in the source documentation (Bearden, Thomsen et al., 1964[link])], the gaps were filled by interpolated values based on measurements available from nearby elements, plotted on a modified Moseley diagram in which the [Z^{2}] term dependence is taken into account (Burr, 1996[link]). In the end, such a smooth scaling with respect to nuclear charge suppresses the effects of the atomic shell structure, a practice that must be avoided in order to obtain the significant improvement in the database that we hope to provide. Also obscured in smooth Z scaling are detectable contributions arising from the fact that nuclear sizes do not change smoothly as a function of the nuclear charge, Z.

4.2.2.3. Alternative strategies

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There are several possible approaches to generating an improved, `all-Z' table of X-ray wavelengths. These range from the option of conducting a massive measurement campaign to populate more fully the currently available tabular array to a large computational endeavor that might purport to carry out multiconfiguration, relativistic wavefunction calculations for the entire Periodic Table. It seems evident to us that there is little interest in, and even less support for, mounting the large effort needed to realize an improved tabulation of X-ray wavelengths by purely experimental means, while the possibility of proceeding in an entirely theoretical mode is not consistent with the evident need that at least some wavelengths be reported with uncertainties that approach the limit of what can be obtained from the naturally occurring X-ray lines. The actual location of any useful feature of a line is influenced not only by the physical and chemical environment of the emitting atom but also by inevitable multi-electron excitation processes that perturb the entire spectral profile. Calculation of such complexities currently lies beyond the limits of practicality, eliminating the option of proceeding without strong coupling to experimental profile locations, at least for crystallographically important X-ray lines. Similar considerations apply a fortiori to those lines needed as reference wavelengths for exotic atom measurements, such as those leading to masses of elementary particles and tests of basic theory [see e.g. Beyer, Indelicato, Finlayson, Liesen & Deslattes (1991[link])].

In constructing the accompanying tables, we have chosen a new procedure that differs from those described above, and accordingly requires some detailed commentary. We begin with the presently available network of well documented experimental measurements, originally established to provide a test bed for the theoretical methods developing at that time (Deslattes & Kessler, 1985[link]). This modest network was the first compilation to make use of the, then newly available, connection between the X-ray region and the base unit of the International System of measurement (the SI) based on optical interferometric measurement of a lattice period as revealed by X-ray interferometry. Details of the generation of this network and its subsequent expansion will be given below. Using this network as a test set gave clearer suggestions as to specific limitations of the theoretical modelling than had been evident from using other, less selective, experimental reference compilations available at that time. Extensive theoretical developments before and, especially, after the appearance of this new experimental reference set have shown a steady convergence toward these critically evaluated data. Following this evolution further, our long-term plan is to use these new theoretical calculations to provide a more structured and accurate interpolation procedure for estimating the spectra of elements lying between those for which we have accurate measurements, or spectra well connected to a directly established reference wavelength. The present table provides experimental and theoretical values for some of the more prominent K and L series lines and is a subset of a larger effort for all K and L series lines connecting the n = 1 to n = 4 shells. The more complete table will be published elsewhere and be made available on the the NIST Physical Reference Data web site. In addition, experimental values for the K and L edges are provided. Although the reference data are inadequate in both low and high ranges of Z, the general consistency of theory and experiment through the region 20 [\lt] Z [\lt] 90 for the strong K-series and L-series lines suggests that, in the absence of good reference measurements, the uncorrected theoretical values should be considered for applications not requiring the highest accuracy.

4.2.2.4. The X-ray wavelength scales, old and new

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Historically, from the first realizations of refined spectroscopy in the X-ray region (ca 1915–1925) up to the period 1975–1985, the best measured X-ray wavelengths had to be expressed in some local unit, most often designated as the xu (x unit) or kxu (kilo x unit). Uncertainty in the conversion factor between the X-ray and optical scales was the dominant contributor to the total uncertainties in the wavelength values of the sharper X-ray emission lines, such as those most frequently used in crystallography. (For a discussion of the present values in relation to previously assigned numerical values on the various scales, see Subsection 4.2.2.14.[link]) This local unit was, for most of the time, `officially' defined by assigning a specific numerical value to the lattice period of a particular reflection from `the purest instance' of a particular crystal. Originally this was rocksalt; later it was calcite. In practice, most work used as de facto standards certain values for Cu [K\alpha _{1}] and Mo [K\alpha _{1}] whose inconsistency, though noted by some crystallographers earlier, was seriously addressed by Bearden and co-workers only in the 1960's (Bearden, Henins et al., 1964[link]). This early history was summarized in 1968 (Thomsen & Burr, 1968[link]). Connection of the X-ray wavelength scale to the primary realizations of the length (wavelength) unit in the `metric system' was primarily (at least after about 1930) through ruled grating measurements of longer-wavelength X-ray lines such as Al [K\alpha _{1,2}].

The remainder of the X-ray wavelength database was derived from relative measurements using crystal diffraction spectroscopy. Unfortunately, even the most refined among the ruled grating measurements did not give accuracies comparable to the precision accessible by relative wavelength measurements (Henins, 1971[link]). As noted above, in connection with establishing the previous wavelength table, Bearden introduced a new local unit, the Å*, based on an explicit value for the wavelength of W [K\alpha _{1}], chosen to give a conversion factor near unity. This transitional period will not be treated further in the present documentation since, to a substantial extent, developments described in the following paragraphs have effectively eliminated the need for a local scale for X-ray wavelength metrology.

Following the demonstration of crystal lattice interferometry in the X-ray region (Bonse & Hart, 1965a[link]), efforts to combine such an X-ray interferometer with various optical interferometers were undertaken in several (mostly national standards) laboratories. Although the earliest of these, carried out at the National Bureau of Standards (NBS) (now the National Institute of Standards and Technology, NIST) (Deslattes & Henins, 1973[link]) was, in the end, found to be burdened by a serious systematic error (1.8 × 10−6) in later work at the Physikalisch Technishe Bundesanstalt (PTB) (Becker et al., 1981[link]; Becker, Seyfried & Siegert, 1982[link]), it was clear that accuracy limitations associated with ruled grating measurements no longer dominated the metrology of X-ray wavelengths. The origin of the systematic error in the early NBS measurement was subsequently understood (Deslattes, Tanaka, Greene, Henins & Kessler, 1987[link]), and, more recently, excellent results were obtained in Italy (Basile, Bergamin, Cavagnero, Mana, Vittone & Zosi, 1994[link], 1995[link]) and Japan (Fujimoto, Fujii, Tanaka & Nakayama, 1997[link]). In all cases, the goal was to obtain an optical measurement of a crystal lattice period (thus far, only Si 220) and to use the calibrated crystals in diffraction spectrometry to establish optically based X-ray wavelengths. Such exercises have been undertaken for several X-ray lines, but the most detailed and well documented results to date were obtained in Jena (Härtwig, Hölzer, Wolf & Förster, 1993[link]; Hölzer, Fritsch, Deutsch, Härtwig & Förster, 1997[link]), where the Kα and Kβ spectra of the elements Cr to Cu were evaluated using silicon crystals well connected with the crystal spacings measured at the PTB.

4.2.2.5. K-series reference wavelengths

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In addition to the Jena measurements noted above, a number of characteristic X-ray lines were measured on the optically based scale at NBS/NIST. The set of directly measured reference wavelengths is given in Table 4.2.2.1[link] in bold type. Most of the originally published (NBS/NIST) values were burdened by the 1.8 × 10−6 error in the silicon lattice period, as noted above. These have been corrected in the numerical results summarized in the table. The directly measured elements and lines appearing in this table were often chosen to meet the need for specific reference values in locations near those of certain optical transitions in highly charged ion spectra or spectra from pionic atoms. In addition, early NBS measurements specifically addressed the lines most often used in crystallography and the W Kα transition. In response to the needs of electron spectroscopy, Al and Mg K spectra were also determined (Schweppe, Deslattes, Mooney & Powell, 1994[link]). In this case, the original lattice error had been previously recognized, so that no rescaling was required. The remaining directly measured entries were obtained as opportunities to do so emerged.

Table 4.2.2.1| top | pdf |
K-series reference wavelengths in Å; bold numbers indicate a directly measured line

Numbers in parentheses are standard uncertainties in the least-significant figures.

ZSymbolA[K\alpha _2][K\alpha _1][K\beta _3][K\beta _1]References
12 Mg   9.89153 (10) 9.889554 (88)     (a)
13 Al   8.341831 (58) 8.339514 (58)     (a)
14 Si   7.12801 (14) 7.125588 (78)     (b)
16 S   5.374960 (89) 5.372200 (78)     (b)
17 Cl   4.730693 (71) 4.727818 (71)     (b)
18 Ar   4.194939 (23) 4.191938 (23)     (c)
19 K   3.7443932 (68) 3.7412838 (56)     (d)
24 Cr   2.2936510 (30) 2.2897260 (30) 2.0848810 (40) 2.0848810 (40) (e)
25 Mn   2.1058220 (30) 2.1018540 (30) 1.9102160 (40) 1.9102160 (40) (e)
26 Fe   1.9399730 (30) 1.9360410 (30) 1.7566040 (40) 1.7566040 (40) (e)
27 Co   1.7928350 (10) 1.7889960 (10) 1.6208260 (30) 1.6208260 (30) (e)
28 Ni   1.6617560 (10) 1.6579300 (10) 1.5001520 (30) 1.5001520 (30) (e)
29 Cu   1.54442740 (50) 1.54059290 (50) 1.3922340 (60) 1.3922340 (60) (e)
31 Ga   1.3440260 (40) 1.3401270 (96) 1.208390 (75) 1.207930 (34) (b), (f)
33 As   1.108830 (31) 1.104780 (12) 0.992689 (79) 0.992189 (53) (b), (f)
34 Se   1.043836 (30) 1.039756 (30) 0.933284 (74) 0.932804 (30) (b), (f)
36 Kr   0.9843590 (44) 0.9802670 (40) 0.8790110 (70) 0.8785220 (50) (b)
40 Zr   0.7901790 (25) 0.7859579 (27) 0.7023554 (30) 0.7018008 (30) (b)
42 Mo   0.713607 (12) 0.70931715 (41) 0.632887 (13) 0.632303 (13) (d), (f)
44 Ru   0.6474205 (61) 0.6430994 (61) 0.5730816 (42) 0.5724966 (42) (d), (f)
45 Rh   0.6176458 (61) 0.6132937 (61) 0.5462139 (42) 0.5456189 (42) (d), (f)
46 Pd   0.5898351 (60) 0.5854639 (46) 0.5211363 (41) 0.5205333 (41) (d), (f)
47 Ag   0.5638131 (26) 0.55942178 (76) 0.4976977 (60) 0.4970817 (60) (d), (f)
48 Cd   0.5394358 (46) 0.5350147 (46) 0.4757401 (71) 0.4751181 (71) (d), (f)
49 In   0.5165572 (60) 0.5121251 (46) 0.4551966 (41) 0.4545616 (41) (d), (f)
50 Sn   0.4950646 (46) 0.4906115 (46) 0.4358821 (51) 0.4352421 (51) (d), (f)
51 Sb   0.4748391 (45) 0.4703700 (45) 0.4177477 (41) 0.4170966 (31) (d), (f)
54 Xe   0.42088103 (71) 0.4163508 (14) 0.3694051 (13) 0.3687346 (13) (d)
56 Ba   0.38968378 (74) 0.38512464 (84) 0.3415228 (11) 0.34082708 (75) (d)
60 Nd   0.3248079 (59) 0.3201648 (59) 0.283634 (59) 0.282904 (44) (d), (f)
62 Sm   0.31369830 (79) 0.30904506 (46) 0.273764 (30) 0.273014 (30) (d), (f)
67 Ho   0.26549088 (84) 0.2607608 (42) 0.230834 (30) 0.230124 (30) (f), (g)
68 Er   0.2571133 (11) 0.25237359 (62) 0.2234766 (14) 0.22269866 (72) (d)
69 Tm   0.24910095 (61) 0.24434486 (44) 0.216366 (30) 0.21559182 (57) (f), (h)
74 W   0.21383304 (50) 0.20901314 (18) 0.18518317 (70) 0.1843768 (30) (d), (f)
79 Au   0.18507664 (61) 0.18019780 (47) 0.1598249 (13) 0.15899527 (77) (d)
82 Pb   0.17029527 (56) 0.16537816 (38) 0.1468129 (10) 0.14596836 (58) (d)
83 Bi   0.1657183 (20) 0.1607903 (46) 0.142780 (11) 0.1419492 (54) (f), (g)
90 Th 230 0.13782600 (31) 0.13282021 (36) 0.11828686 (78) 0.11740759 (59) (d)
91 Pa 231 0.1343516 (29) 0.1293302 (27) 0.1152427 (21) 0.1143583 (21) (i)
92 U 238 0.13099111 (78) 0.12595977 (36) 0.11228858 (66) 0.11140132 (65) (d)
93 Np 237 0.1277287 (39) 0.1226882 (36) 0.1094230 (39) 0.1085265 (28) (i)
94 Pu 239 0.1245782 (15) 0.11952120 (69)     (h)
94 Pu 244 0.1245705 (25) 0.1195140 (23) 0.1066611 (18) 0.1057595 (18) (i)
95 Am 243 0.1215158 (24) 0.1164463 (33) 0.1039794 (17) 0.1030803 (17) (i)
96 Cm 248 0.1185427 (23) 0.1134635 (21) 0.1013753 (17) 0.1004708 (16) (i)
97 Bk 249 0.1156630 (54) 0.1105745 (49) 0.0988598 (55) 0.0979514 (54) (i)
98 Cf 250 0.1128799 (82) 0.1077793 (75)     (i)

References: (a) Schweppe et al. (1994[link]); (b) Mooney (1996[link]); (c) Schweppe (1995[link]); (d) Deslattes & Kessler (1985[link]); (e) Hölzer et al. (1997[link]); (f) Bearden (1967[link]); (g) Borchert, Hansen, Jonson, Ravn & Desclaux (1980[link]); (h) Borchert (1976[link]); (i) Barreau, Börner, Egidy & Hoff (1982[link]).

The optically based data set was expanded by noting that several groups of accurate (relative) measurements in the literature either contained one of the directly measured lines (bold type) in Table 4.2.2.1[link] or were explicitly connected to one of them. Most often this situation was realized in reports that indicated a specific reference value, i.e. where it was stated numerical values are based on a scale where, for example, the wavelength of Mo [K\alpha _{1}] was taken as 707.831 xu. In such cases, and where other indicators of good measurement quality are presented, it is easy to re-scale the data reported so that it is consistent with the optically based data. This procedure was followed for important groups of measurements from earlier work by Bearden and co-workers, and from the X-ray laboratory at Uppsala. The rescaled numerical results are included in Table 4.2.2.1[link] in normal type along with the specific literature citations. The indicated uncertainties are standard uncertainties as defined by ISO (Taylor & Kuyatt, 1994[link]; Schwarzenbach, Abrahams, Flack, Prince & Wilson, 1995[link]).

4.2.2.6. L-series reference wavelengths

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To date, only a very limited number of L-series emission lines have been directly measured on an optically based scale. Wavelengths for directly measured L-series lines are reported in Table 4.2.2.2[link] along with the literature citations. In the future, we hope to expand this limited data set by including other lines and elements that are well connected to these reference lines in the literature.

Table 4.2.2.2| top | pdf |
Directly measured L-series reference wavelengths in Å

Numbers in parentheses are standard uncertainties in the least significant figures.

ZSymbol[L\alpha _2][L\alpha _1][L\beta _1]References
36 Kr 7.82032(13) 7.82032(13) 7.574441(98) (a)
40 Zr 6.07710(48) 6.070250(79) 5.836214(76) (a)
54 Xe 3.025940(22) 3.016582(15) 2.806553(19) (b)
60 Nd 2.38079(52) 2.370526(16) 2.167008(19) (a)
62 Sm 2.210430(24) 2.199873(13) 1.998432(30) (a)
67 Ho 1.856472(15) 1.845092(17) 1.647484(32) (a)
68 Er 1.795701(45) 1.784481(20) 1.587466(86) (a)
69 Tm 1.738003(19) 1.7267720(70) 1.5302410(70) (a)

References: (a) Mooney (1996[link]); (b) Mooney et al. (1992[link]).

4.2.2.7. Absorption-edge locations

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Only a small number of absorption-edge locations have been directly measured to high accuracy using the currently acceptable protocols. Some of the available data were obtained in order to provide wavelength determinations for spectra from highly charged and/or exotic atoms (Bearden, 1960[link]; Lum et al., 1981[link]). This small group was, however, significantly expanded very recently by an important set of new measurements, extending to Z = 51, that are well coupled to the optical wavelength scale (Kraft, Stümpel, Becker & Kuetgens, 1996[link]) The resulting experimental database is summarized in Table 4.2.2.3[link]. The effort that would be needed to expand the experimental database in a systematic way is quite large. Thus, we make use of a procedure, not previously used for this purpose, that combines available electron binding energy data with emission-line locations from our expanded reference set of emission-line data and emission lines that have been rescaled to be consistent with the optically based scale. At the same time, calculation of the location of absorption thresholds within the theoretical framework (see below) has been undertaken and will be made available in the longer publication and on the web site.

Table 4.2.2.3| top | pdf |
Directly measured and emission + binding energies (see text) K-absorption edges in Å

Numbers in parentheses are standard uncertainties in the least significant figures.

ZSymbolDirectly measuredEmission + binding energiesReferences
23 V 2.269211(21) 2.26893(11) (a)
24 Cr 2.070193(14) 2.07014(17) (a)
25 Mn 1.8964592(58) 1.896457(42) (a)
26 Fe 1.7436170(49) 1.743589(98) (a)
27 Co 1.6083510(42) 1.60836(17) (a)
28 Ni 1.4881401(36) 1.48823(25) (a)
29 Cu 1.3805971(31) 1.38060(16) (a)
30 Zn 1.2833798(40) 1.28338(15) (a)
39 Y 0.7277514(21) 0.727750(23) (a)
40 Zr 0.6889591(31) 0.688946(30) (a)
41 Nb 0.6531341(14) 0.653112(29) (a)
42 Mo 0.61991006(62) 0.619906(64) (a)
45 Rh 0.5339086(69) 0.533951(10) (a)
46 Pd 0.5091212(42) 0.509156(11) (a)
47 Ag 0.4859155(57) 0.4859168(91) (a)
48 Cd 0.4641293(35) 0.464135(12) (a)
49 In 0.4437454(48) 0.443740(11) (a)
50 Sn 0.4245978(29) 0.424590(13) (a)
51 Sb 0.4066324(27) 0.406612(12) (a)
68 Er 0.2156801(75) 0.2156762(50) (b)
82 Pb 0.1408821(74) 0.1408836(11) (c)

References: (a) Kraft et al. (1996[link]); (b) Lum et al. (1981[link]); (c) Bearden (1960[link]).

The feature of absorption spectra customarily designated as `the absorption edge' has been variously associated with: the first inflection point of the absorption spectrum; the energy needed to produce a single inner vacancy with the photo-electron `at rest at infinity'; or the energy needed to remove an electron from an inner shell and place it in the lowest unoccupied energy level. A general discussion of this question has been given by Parratt (1959[link]). If we choose the second alternative, then it is easy to see that, with some care for symmetry restrictions, one can estimate the absorption-edge energy by combining the binding energy for any accessible outer shell with the energy of an emission line for which the transition terminus lies in the same outer shell. Of course, this procedure does not focus on the details of absorption thresholds, the locations of which are important for a number of structural applications. On the other hand, our choice gives greater regularity with respect to nuclear charge and facilitates use of electron binding energies, since they are referenced to the Fermi energy or the vacuum.

Electron binding energies have been tabulated for the principal electron shells of all the elements considered in the present table (Fuggle, Burr, Watson, Fabian & Lang, 1974[link]; Cardona & Ley, 1978[link]; Nyholm, Berndtsson & Mårtensson, 1980[link]; Nyholm & Mårtensson, 1980[link]; Lebugle, Axelsson, Nyholm & Mårtensson, 1981[link]; Powell, 1995[link]). The number of values available offers the possibility of consistency checking, since the K and L shells are connected by emission lines to several final hole states, each of which has (possibly) been evaluated by photoelectron spectroscopy. For each of the elements for which well qualified reference spectra are available, we evaluated edge location estimates using several alternative transition cycles and used the distribution of results to provide a measure of the uncertainty. Comparison of edge estimates obtained by this procedure with experimental data provides a quantitative test of the utility of the chosen approach to edge location estimation. In Table 4.2.2.3[link], the numerical results in the column labelled `Emission + binding energies' were obtained by combining emission energies and electron binding energies using all possible redundancies. The estimated uncertainties indicated were obtained from the distribution of the redundant routes. As can be seen, the results are in general agreement with the available directly measured values. Accordingly, we have used this protocol to obtain the edge locations listed in the summary tables below.

4.2.2.8. Outline of the theoretical procedures

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Only recently has it become possible to understand the relativistic many-body problem in atoms with sufficient detail to permit meaningful calculation of transition energies between hole states (Indelicato & Lindroth, 1992[link]; Mooney, Lindroth, Indelicato, Kessler & Deslattes, 1992[link]; Lindroth & Indelicato, 1993[link], 1994[link]; Indelicato & Lindroth, 1996[link]). To deal with those hole states for atomic numbers ranging from 10 to 100, one needs to consider five kinds of contributions, all of which must be calculated in a relativistic framework, and the relative influence of which can change strongly as a function of the atomic number:

  • (i) nuclear size;

  • (ii) relativistic effects (corrections to Coulomb energy, magnetic and retardation energy);

  • (iii) Coulomb and Breit correlation;

  • (iv) radiative (QED) corrections (one- and two-electron Lamb shift etc.);

  • (v) Auger shift.

Such an undertaking, although much more advanced than any other done in the past, still suffers from severe limitations that need to be understood fully to make the best use of the table. The main limitation is probably that most lines are emitted by atoms in an elemental solid or a compound, while the calculation at present deals only with atoms isolated in vacuum. (A purely experimental database would have a similar limitation.) The second limitation is that it is not possible at present to include the coupling between the hole and open outer shells. Coupling between a [j={1\over2}], [j={3\over2}] or [j={5\over2}] hole and an external 3d or 4f shell can generate hundreds of levels, with splitting that can reach an eV. One then should calculate all radiative and Auger transition probabilities between hundreds of initial and final states. (The Auger final state would have one extra vacancy, leading eventually to thousands of final states.) Such an approach would give not only the mean line energy but also its shape and would thus be very desirable, but is impossible to do with present day theoretical tools and computers. We have thus limited ourselves to an approach in which one computes the weighted average energy for each hole state, and ignores possible distortion of the line profile due to the coupling between inner vacancies and outer shells.

Since we want to have good predictions for both light and heavy atoms, we have to include relativity non-perturbatively. To get a result approaching 1 ×  10−6 for uranium Kα by applying perturbation theory to the Schrödinger equation, for example, one would need to go to order 22 in powers of Zα = v/c. The natural framework in this case is thus to do a calculation exact to all orders in Zα by using the Dirac equation. We thus have used many-body methods, based on the Dirac equation, in which the main contributions to the transition energy are evaluated using the Dirac–Fock method. We use the Breit operator for the electron–electron interaction, to include magnetic (spin–spin, spin–other orbit and orbit–orbit interactions in the lower orders in Zα and (v/c)2 retardation effects. Higher-order retardation effects are also included. Many-body effects are calculated by using relativistic many-body perturbation theory (RMBPT). Since inner vacancy levels are auto-ionizing, one must include shifts in their energy due to the coupling between the discrete levels and Auger decay continuua.

In the following subsections, we describe in more detail the calculation of the different contributions.

4.2.2.9. Evaluation of the uncorrelated energy with the Dirac–Fock method

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The first step in the calculation, following Indelicato and collaborators (Indelicato & Desclaux, 1990[link]; Indelicato & Lindroth, 1992[link]; Mooney et al., 1992[link]; Lindroth & Indelicato, 1993[link]; Indelicato & Lindroth, 1996[link]) consists in evaluating the best possible energy with relativistic corrections, within the independent electron approximation, for each hole state (here [1s_{{1\over2}}], [2p_{{1\over2}}], [2p_{{3\over2}}], [3p_{{1\over2}}], [3p_{{3\over2}}] for K, LII, LIII, MII, MIII, respectively). Such a calculation must provide a suitable starting point for adding all many-body and QED contributions. We have thus chosen the Dirac–Fock method in the implementation of Desclaux (1975[link], 1993[link]). This method, based on the Dirac equation, allows treatment of arbitrary atoms with arbitrary structure and has been widely used for this kind of calculation. We have used it with full exchange and relaxation (to account for inactive orbital rearrangement due to the hole presence). The electron–electron interaction used in this program contains all magnetic and retardation effects, which is very important to have good results at large Z. The magnetic interaction is treated on an equal footing with the Coulomb interaction, to account for higher-order effects in the wavefunction (which are also useful for evaluating radiative corrections to the electron–electron interaction). All these calculations must be done with proper nuclear charge models to account for finite-nuclear-size corrections to all contributions. For heavy nuclei, nuclear deformations must be accounted for (Blundell, Johnson & Sapirstein, 1990[link]; Indelicato, 1990[link]). For all elements for which experiments have been performed, we used experimental nuclear charge radii. For the others we used a formula from Johnson & Soff (1985[link]), corrected for nuclear deformations for Z [\gt] 90. Contribution of deformation to the r.m.s. radius (the only parameter of importance to the atomic calculation) is roughly constant (0.11 fm) for Z [\gt] 90. There is an unknown region, between Bi and Th (83 [\lt] Z [\lt] 90), where deformation effects start to be important, but for which they are not known. When experiments are done for a particular isotope, we calculated separately the energies for each isotope.

As mentioned in the introduction, there are special difficulties involved when dealing with atoms with open outer shells (obviously this is the most common case). Computing all energies [E_{{J}}] for total angular momentum J would be both impossible and useless. The Dirac–Fock method circumvents this difficulty. One can evaluate directly an average energy that corresponds to the barycentre of all [E_{{J}}] with weight ([2{J}+1]). There are still a few cases for which the average calculation cannot converge (when the open shells have identical symmetry). In that case, the outer electrons have been rearranged in an identical fashion for all hole states of the atom, to minimize possible shifts due to this procedure.

4.2.2.10. Correlation and Auger shifts

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Once the Dirac–Fock energy is obtained, many-body effects beyond Dirac–Fock relaxation must be taken into account. These include relaxation beyond the spherical average, correlation (due to both Coulomb and magnetic interaction), and corrections due to the autoionizing nature of hole states (Auger shift). Since the many-body generalization of the Dirac–Fock method, the so-called MCDF (multiconfiguration Dirac–Fock), is very inefficient for hole states, we turned to RMBPT to evaluate those quantities. These many-body effects contribute very significantly to the final value. Coulomb correlation is mostly constant along the Periodic Table (at the level of a few eV). Magnetic correlations are very strong at high Z. Auger shift is very important for p states. The interested reader will find more details of these complicated calculations in the original references (Indelicato & Lindroth, 1992[link]; Mooney et al., 1992[link]; Lindroth & Indelicato, 1993[link]; Indelicato & Lindroth, 1996[link]). As these calculations are very time consuming, they are performed only for selected Z and interpolated. Since the Auger shifts do not always have a smooth Z dependence, care has been taken to evaluate them at as many different Z's as practical to ensure a good reproduction of irregularities.

4.2.2.11. QED corrections

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The QED corrections originate in the quantum nature of both the electromagnetic and electron fields. They can be divided in two categories, radiative and non-radiative. The first one includes self-energy and vacuum polarization, which are the main contributions to the Lamb shift in one-electron atoms. These corrections scale as [{Z}^{4}/{n}^{3}] (n being the principal quantum number) and are thus very important for inner shells and high Z. The second category is composed of corrections to the electron–electron interaction that cannot be accounted for by RMBPT or MCDF. These corrections start at the two-photon interaction and include three-body effects. The two-photon, non-radiative QED contribution has been calculated recently only for the ground state of two-electron ions (Blundell, Mohr, Johnson & Sapirstein, 1993[link]; Lindgren, Persson, Salomonson & Labzowsky, 1995[link]) and cannot be evaluated in practice for atoms with more than two or three electrons.

The radiative corrections split up into two contributions. The first contribution is composed of one-electron radiative corrections (self-energy and vacuum polarization). For the self-energy and [{Z}\gt10], one must use all-order calculations (Mohr, 1974a,[link]b[link], 1975[link], 1982[link], 1992[link]; Mohr & Soff, 1993[link]). Vacuum polarization can be evaluated at the Uehling (1935[link]) and Wichmann & Kroll (1956[link]) level. Higher-order effects are much smaller than for the self-energy (Soff & Mohr, 1988[link]) and have been neglected. The second contribution is composed of radiative corrections to the electron–electron interaction, and scales as [{Z}^{3}/{n}^{3}]. Ab initio calculations have been performed only for few-electron ions (Indelicato & Mohr, 1990[link], 1991[link]). Here we use the Welton approximation which has been shown to reproduce very closely ab initio results in all examples that have been calculated (Indelicato, Gorceix & Desclaux 1987[link]; Indelicato & Desclaux 1990[link]; Kim, Baik, Indelicato & Desclaux, 1991[link]; Blundell, 1993a[link],b[link]).

4.2.2.12. Structure and format of the summary tables

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Table 4.2.2.4[link] summarizes the theoretical and experimental results for prominent K-series lines and the K-absorption edge. For the emission lines, the upper number (in italics) is the theoretical estimate for this line and the lower number is the experimentally measured value (1) from Table 4.2.2.1[link] or (2) from the Bearden database or a reference that appeared after the Bearden database corrected to the optically based scale. For the K absorption edge, the upper number (also in italics) was obtained by combining emission lines and photoelectron spectroscopy (see Subsection 4.2.2.7[link]), and the lower number is the experimentally measured value (1) from Table 4.2.2.3[link] or (2) from the Bearden database or a reference that appeared after the Bearden database corrected to an optically based scale. For the experimental emission and absorption entries, bold type is used for wavelengths directly measured on an optically based scale. The numerical values for wavelengths in angstrom units (1 Å = 0.1 nm) are given to a number of significant figures commensurate with their estimated uncertainties, which appear in parentheses after each theoretical and experimental value.

Table 4.2.2.4| top | pdf |
Wavelengths of K-emission lines and K-absorption edges in Å; see text for explanation of typefaces

Numbers in parentheses are standard uncertainties in the least significant figures.

ZSymbolA[K\alpha _2 ][K\alpha _1 ][K\beta _3 ][K\beta _1 ][K\beta _2 ^{\rm II} ][K\beta _2 ^{\rm I}]K abs. edge
10 Ne   14.6020(93) 14.6006(93)         14.2391(26)
  14.6102(44) 14.6102(44) 14.4522(74) 14.4522(74)     14.30201(15)
11 Na   11.9013(59) 11.8994(59)         11.4784(16)
  11.9103(13) 11.9103(13) 11.5752(30) 11.5752(30)     11.5692(15)
12 Mg   9.8860(39) 9.8840(39)         9.4479(10)
  9.89153(10) 9.889554(88) 9.5211(30) 9.5211(30)     9.51234(15)
13 Al   8.3372(27) 8.3349(26) 7.9412(49)       7.89928(67)
  8.341831(58) 8.339514(58) 7.9601(30) 7.9601(30)     7.948249(74)
14 Si   7.1269(19) 7.1208(19) 6.7317(26)       6.70091(46)
  7.12801(14) 7.125588(78) 6.7531(15) 6.7531(15)     6.7381(15)
15 P   6.1587(14) 6.1539(14) 5.7834(16) 5.7914(27)     5.75537(33)
  6.1601(15) 6.1571(15) 5.7961(30) 5.7961(30)     5.7841(15)
16 S   5.3742(10) 5.3701(10) 5.0202(12) 5.0246(15)     4.99591(24)
  5.374960(89) 5.372200(78)   5.03168(30)     5.01858(15)
17 Cl   4.72993(80) 4.72560(77) 4.39810(99) 4.40038(77)     4.37679(18)
  4.730693(71) 4.727818(71) 4.40347(44) 4.40347(44)     4.39717(15)
18 Ar   4.19448(62) 4.19162(60) 3.88506(71) 3.88486(70)     3.86552(14)
  4.194939(23) 4.191938(23) 3.88606(30) 3.88606(30)     3.870958(74)
19 K   3.74352(50) 3.74055(48) 3.45189(69) 3.45216(58)     3.42856(11)
  3.7443932(68) 3.7412838(56) 3.45395(30) 3.45395(30)     3.43655(15)
20 Ca   3.36223(39) 3.35911(38) 3.08855(45) 3.08827(45)     3.061828(87)
  3.361710(44) 3.358440(44) 3.08975(30) 3.08975(30)     3.07035(15)
21 Sc   3.03479(33) 3.03129(31) 2.77919(50) 2.77809(49)     2.754176(71)
  3.0344010(63) 3.030854(14) 2.77964(30) 2.77964(30)     2.7620(15)
22 Ti   2.75272(27) 2.74886(26) 2.51445(43) 2.51262(45)     2.490681(59)
  2.7521950(57) 2.7485471(57) 2.513960(30) 2.513960(30)     2.497377(74)
23 V   2.50798(23) 2.50383(21) 2.28567(37) 2.28332(40)     2.263194(49)
  2.507430(30) 2.503610(30) 2.284446(30) 2.284446(30)     2.269211(21)
24 Cr   2.29428(19) 2.29012(18) 2.08702(32) 2.08478(35)     2.067898(41)
  2.2936510(30) 2.2897260(30) 2.0848810(40) 2.0848810(40)     2.070193(14)
25 Mn   2.10635(16) 2.10210(15) 1.91175(28) 1.90960(31)     1.892275(36)
  2.1058220(30) 2.1018540(30) 1.9102160(40) 1.9102160(40)     1.8964592(58)
26 Fe   1.94043(14) 1.93631(13) 1.75784(25) 1.75617(27)     1.739918(31)
  1.9399730(30) 1.9360410(30) 1.7566040(40) 1.7566040(40)     1.7436170(49)
27 Co   1.79321(12) 1.78919(11) 1.62166(22) 1.62039(24)     1.605127(27)
  1.7928350(10) 1.7889960(10) 1.6208260(30) 1.6208260(30)     1.6083510(42)
28 Ni   1.66199(10) 1.658049(96) 1.50059(19) 1.49964(21)     1.485300(24)
  1.6617560(10) 1.6579300(10) 1.5001520(30) 1.5001520(30)     1.4881401(36)
29 Cu   1.544324(93) 1.540538(85) 1.39246(17) 1.39201(18)     1.379448(23)
  1.54442740(50) 1.54059290(50) 1.3922340(60) 1.3922340(60)     1.3805971(31)
30 Zn   1.438963(84) 1.435151(74) 1.29544(17) 1.29506(16)     1.282346(20)
  1.439029(12) 1.435184(12) 1.295276(30) 1.295276(30) 1.283739(30) 1.283739(30) 1.2833798(40)
31 Ga   1.343987(72) 1.340095(65) 1.20821(13) 1.20774(14) 1.195547(25)   1.194711(18)
  1.3440260(40) 1.3401270(96) 1.208390(75) 1.207930(34) 1.196018(30) 1.196018(30) 1.19582(15)
32 Ge   1.257998(65) 1.254054(58) 1.12924(13) 1.12877(13) 1.116387(37)   1.115585(16)
  1.258030(13) 1.254073(13) 1.12938(13) 1.128957(30) 1.116877(30) 1.116877(30) 1.116597(74)
33 As   1.179921(57) 1.175932(52) 1.05774(11) 1.05724(11) 1.044699(56) 1.044836(20) 1.043925(16)
  1.179959(17) 1.17595600(90) 1.057898(76) 1.057368(33) 1.045016(44) 1.045016(44) 1.04502(15)
34 Se   1.108801(52) 1.104778(47) 0.992646(96) 0.992152(95) 0.979618(57) 0.979716(26) 0.978818(15)
  1.108830(31) 1.104780(12) 0.992689(79) 0.992189(53) 0.979935(74) 0.979935(74) 0.979755(15)
35 Br   1.043841(47) 1.039785(42) 0.933275(87) 0.932768(84) 0.920344(49) 0.920390(28) 0.919501(13)
  1.043836(30) 1.039756(30) 0.933284(74) 0.932804(30) 0.920474(30) 0.920474(30) 0.92041(15)
36 Kr   0.984347(42) 0.980267(38) 0.878967(81) 0.878495(75) 0.866209(36) 0.866169(35) 0.865324(12)
  0.9843590(44) 0.9802670(40) 0.8790110(70) 0.8785220(50) 0.86611(15) 0.86611(15) 0.865533(15)
37 Rb   0.929713(39) 0.925597(35) 0.829174(71) 0.828681(67) 0.816459(33) 0.816408(33) 0.815270(12)
  0.929704(15) 0.925567(13) 0.829222(44) 0.828692(30) 0.816462(44) 0.816462(44) 0.815552(74)
38 Sr   0.879444(36) 0.875298(32) 0.783413(63) 0.782911(58) 0.770774(33) 0.770718(20) 0.769359(11)
  0.879443(15) 0.875273(15) 0.783462(44) 0.782932(30) 0.770822(44) 0.770822(44) 0.769742(74)
39 Y   0.833059(32) 0.828875(29) 0.741232(58) 0.740716(53) 0.728801(27) 0.728663(21) 0.727270(10)
  0.833063(15) 0.828852(15) 0.741271(44) 0.740731(30) 0.728651(59) 0.728651(59) 0.7277514(21)
40 Zr   0.790181(30) 0.785960(27) 0.702296(53) 0.701766(48) 0.690079(28) 0.689895(21) 0.6884893(99)
  0.7901790(25) 0.7859579(27) 0.7023554(30) 0.7018008(30) 0.689940(59) 0.689940(59) 0.6889591(31)
41 Nb   0.750448(28) 0.746189(25) 0.666266(49) 0.665721(44) 0.654328(31) 0.654078(22) 0.652890(93)
  0.750451(15) 0.746211(15) 0.666350(44) 0.665770(30) 0.654170(59) 0.654170(59) 0.6531341(14)
42 Mo   0.713612(25) 0.709328(22) 0.632900(44) 0.632345(38) 0.621162(35) 0.620941(21) 0.6196481(87)
  0.713607(12) 0.70931715(41) 0.632887(13) 0.632303(13) 0.620999(30) 0.620999(30) 0.61991006(62)
43 Tc   0.679318(24) 0.675017(21) 0.601881(40) 0.601318(35) 0.590423(40) 0.590231(22) 0.5889852(84)
  0.679330(44) 0.675030(44) 0.601889(59) 0.601309(59) 0.590249(74) 0.590249(74) 0.589069(15)
44 Ru   0.647415(22) 0.643088(20) 0.573053(37) 0.572478(32) 0.561748(44) 0.561587(22) 0.5603122(81)
  0.6474205(61) 0.6430994(61) 0.5730816(42) 0.5724966(42) 0.561668(44) 0.561668(44) 0.560518(15)
45 Rh   0.617652(21) 0.613305(18) 0.546191(34) 0.545606(29) 0.535110(48) 0.534977(22) 0.5337192(74)
  0.6176458(61) 0.6132937(61) 0.5462139(42) 0.5456189(42) 0.535038(30) 0.535038(30) 0.5339086(69)
46 Pd   0.589822(20) 0.585459(18) 0.521117(29) 0.520514(27) 0.510283(46) 0.510177(51) 0.5090158(75)
  0.5898351(60) 0.5854639(46) 0.5211363(41) 0.5205333(41) 0.5102357(59) 0.5102357(59) 0.5091212(42)
47 Ag   0.563804(18) 0.559420(17) 0.497673(29) 0.497069(25) 0.487060(55) 0.487019(38) 0.4857609(74)
  0.5638131(26) 0.55942178(76) 0.4976977(60) 0.4970817(60) 0.4870393(59) 0.4870393(59) 0.4859155(57)
48 Cd   0.539426(18) 0.535020(15) 0.475739(27) 0.475124(23) 0.465335(62) 0.465346(28) 0.4640026(71)
  0.5394358(46) 0.5350147(46) 0.4757401(71) 0.4751181(71) 0.465335(10) 0.465335(10) 0.4641293(35)
49 In   0.516551(17) 0.512124(15) 0.455178(25) 0.454552(22) 0.445014(58) 0.445011(27) 0.4435977(70)
  0.5165572(60) 0.5121251(46) 0.4551966(41) 0.4545616(41) 0.445007(15) 0.445007(15) 0.4437454(48)
50 Sn   0.495060(16) 0.490612(14) 0.435878(24) 0.435241(20) 0.426120(12) 0.425928(26) 0.4244611(68)
  0.4950646(46) 0.4906115(46) 0.4358821(51) 0.4352421(51) 0.425921(12) 0.425921(12) 0.4245978(29)
51 Sb   0.474840(15) 0.470373(13) 0.417736(22) 0.417089(19) 0.408017(57) 0.408004(25) 0.4064886(65)
  0.4748391(45) 0.4703700(45) 0.4177477(41) 0.4170966(31) 0.4079791(74) 0.4079791(74) 0.4066324(27)
52 Te   0.455795(14) 0.451310(13) 0.400664(21) 0.400008(18) 0.391161(56) 0.391135(27) 0.3895899(64)
  0.4557908(44) 0.4513018(44) 0.4006650(59) 0.4000010(44) 0.3911079(89) 0.3911079(89) 0.389746(15)
53 I   0.437834(13) 0.433330(12) 0.384576(20) 0.383910(17) 0.375286(54) 0.375234(29) 0.3736775(61)
  0.437836(10) 0.4333245(74) 0.3845698(59) 0.3839108(59) 0.375236(30) 0.375236(30) 0.373816(15)
54 Xe   0.420879(42) 0.416358(40) 0.369407(40) 0.368730(38) 0.360326(72) 0.36034(12) 0.358683(27)
  0.42088103(71) 0.4163508(14) 0.3694051(13) 0.3687346(13) 0.360265(44) 0.360265(44) 0.35841(74)
55 Cs   0.404848(13) 0.400310(11) 0.355067(17) 0.354385(16) 0.346197(49) 0.346102(37) 0.3444778(59)
  0.4048411(59) 0.4002960(59) 0.3550553(59) 0.354369(10) 0.346115(30) 0.346115(30) 0.344515(15)
56 Ba   0.389684(12) 0.385129(11) 0.341517(16) 0.340826(15) 0.33285(14) 0.332728(12) 0.3310639(56)
  0.38968378(74) 0.38512464(84) 0.3415228(11) 0.34082708(75) 0.332775(15) 0.332775(15) 0.331045(15)
57 La   0.375320(11) 0.370748(10) 0.328692(16) 0.327993(14) 0.32023(13) 0.320101(11) 0.3184025(55)
  0.3753186(30) 0.3707426(30) 0.3286909(59) 0.3279879(44) 0.320122(10) 0.320122(10) 0.318445(74)
58 Ce   0.361685(11) 0.3570964(97) 0.316507(15) 0.315795(14) 0.30827(12) 0.308131(10) 0.3065382(54)
  0.3616884(30) 0.3570974(30) 0.3165248(59) 0.3158207(30) 0.308165(15) 0.308165(15) 0.306485(74)
59 Pr   0.348755(10) 0.3441494(94) 0.304970(14) 0.304249(13) 0.29694(11) 0.2967952(99) 0.2952418(53)
  0.3487542(30) 0.3441452(30) 0.3049796(74) 0.3042656(59) 0.296794(30) 0.296794(30) 0.295184(74)
60 Nd   0.336473(10) 0.3318514(91) 0.294021(13) 0.293290(13) 0.28619(10) 0.2860408(94) 0.2845288(52)
  0.33647921(73) 0.33185689(62) 0.2940366(40) 0.2933086(40) 0.28610(15) 0.28610(15) 0.284534(74)
61 Pm   0.3247982(98) 0.3201607(88) 0.283620(13) 0.282880(12) 0.275992(98) 0.2758335(91) 0.2743634(53)
  0.3248079(59) 0.3201648(59) 0.283634(59) 0.282904(44) 0.27590(15) 0.27590(15) 0.274314(74)
62 Sm   0.3136913(94) 0.3090384(84) 0.273732(12) 0.272984(12) 0.266459(91) 0.2661277(87) 0.2647027(51)
  0.31369830(79) 0.30904506(46) 0.273764(30) 0.273014(30) 0.26620(15) 0.26620(15) 0.264644(74)
63 Eu   0.3031139(91) 0.2984457(81) 0.264322(12) 0.263567(11) 0.257069(85) 0.2569028(81) 0.2555123(51)
  0.3031225(30) 0.2984505(30) 0.2643360(74) 0.2635810(74) 0.256927(12) 0.256927(12) 0.255534(15)
64 Gd   0.2930400(89) 0.2883568(79) 0.255371(11) 0.254610(11) 0.248289(23) 0.2481186(76) 0.2467265(48)
  0.2930424(30) 0.2883573(30) 0.255344(30) 0.254604(30) 0.248164(44) 0.248164(44) 0.246814(15)
65 Tb   0.2834212(86) 0.2787234(76) 0.246818(11) 0.246054(11) 0.239916(21) 0.2397496(75) 0.2384335(49)
  0.2834273(30) 0.2787242(30) 0.246834(30) 0.246084(30) 0.23970(30) 0.23970(30) 0.238414(15)
66 Dy   0.2742462(84) 0.2695341(74) 0.238671(10) 0.237902(10) 0.231955(12) 0.2318190(53) 0.2304867(46)
  0.2742511(30) 0.2695370(30) 0.238624(30) 0.237884(30) 0.23170(30) 0.23170(30) 0.230483(15)
67 Ho   0.2654851(81) 0.2607589(72) 0.230896(10) 0.230122(10) 0.224320(18) 0.2241536(66) 0.2229099(45)
  0.26549088(84) 0.2607608(42) 0.230834(30) 0.230124(30) 0.22410(30) 0.22410(30) 0.222913(15)
68 Er   0.2571059(79) 0.2523659(71) 0.2234662(97) 0.2226875(98) 0.217046(16) 0.2168806(64) 0.2156719(45)
  0.2571133(11) 0.25237359(62) 0.2234766(14) 0.22269866(72) 0.21670(30) 0.21670(30) 0.2156801(75)
69 Tm   0.2490952(77) 0.2443415(68) 0.2163665(94) 0.2155833(95) 0.210099(15) 0.2099331(62) 0.2087587(44)
  0.24910095(61) 0.24434486(44) 0.216366(30) 0.21559182(57) 0.20980(30) 0.20980(30) 0.208803(74)
70 Yb   0.2414274(75) 0.2366603(67) 0.2095741(93) 0.2087863(95) 0.203456(14) 0.2032912(59) 0.2021481(43)
  0.2414276(30) 0.2366586(30) 0.20960(15) 0.208843(30) 0.20330(30) 0.20330(30) 0.202243(74)
71 Lu   0.2340857(73) 0.2293053(65) 0.2030802(88) 0.2022872(90) 0.191017(13) 0.1969329(58) 0.1957973(42)
  0.2340845(30) 0.2293014(30) 0.203093(59) 0.202313(44) 0.19690(30) 0.19690(30) 0.195853(74)
72 Hf   0.2270507(72) 0.2222572(64) 0.1968603(86) 0.1960622(88) 0.191017(13) 0.1908468(56) 0.1897176(42)
  0.2270274(44) 0.2222303(44) 0.196863(59) 0.196073(44) 0.19080(30) 0.19080(30) 0.189823(74)
73 Ta   0.2203039(70) 0.2154977(63) 0.1908986(83) 0.1900954(86) 0.185143(12) 0.1849702(54) 0.1838657(41)
  0.2203083(59) 0.2155002(59) 0.1908929(30) 0.1900919(59) 0.185191(13) 0.185014(12) 0.183943(15)
74 W   0.2138327(69) 0.2090134(61) 0.1851834(81) 0.1843751(83) 0.179595(12) 0.1794215(52) 0.1783098(41)
  0.21383304(50) 0.20901314(18) 0.18518317(70) 0.1843768(30) 0.179603(15) 0.179424(10) 0.178373(15)
75 Re   0.2076150(67) 0.2027835(60) 0.1796955(79) 0.1788824(81) 0.174234(11) 0.1740571(51) 0.1729509(40)
  0.2076141(15) 0.2027840(30) 0.1796997(44) 0.1788827(44) 0.174253(15) 0.1740566(89) 0.173023(15)
76 Os   0.2016443(66) 0.1968007(59) 0.1744279(77) 0.1736101(79) 0.169085(11) 0.1689066(50) 0.1678092(40)
  0.2016420(30) 0.1967970(30) 0.1744336(44) 0.1736136(44) 0.169103(15) 0.1689085(89) 0.167873(15)
77 Ir   0.1959045(65) 0.1910492(57) 0.1693667(75) 0.1685444(77) 0.164150(11) 0.1639697(51) 0.1628853(39)
  0.1959069(30) 0.1910499(30) 0.1693695(30) 0.1685445(30) 0.164152(15) 0.163958(10) 0.162922(15)
78 Pt   0.1903859(61) 0.1855187(55) 0.1645026(72) 0.1636756(74) 0.1593872(99) 0.1592048(46) 0.1581346(38)
  0.1903839(59) 0.1855138(59) 0.1645035(44) 0.1636775(44) 0.159392(15) 0.159202(15) 0.158182(15)
79 Au   0.1850702(64) 0.1801914(57) 0.1598202(73) 0.1589887(75) 0.1548206(99) 0.1546363(48) 0.1535699(40)
  0.18507664(61) 0.18019780(47) 0.1598249(13) 0.15899527(77) 0.154832(30) 0.154620(13) 0.1535953(74)
80 Hg   0.1799628(61) 0.1750720(54) 0.1553217(69) 0.1544857(72) 0.1504204(94) 0.1502334(46) 0.1491786(38)
  0.1799607(44) 0.1750706(44) 0.1553233(44) 0.1544893(44) 0.150402(30) 0.150202(30) 0.149182(15)
81 Tl   0.1750380(60) 0.1701355(53) 0.1509866(68) 0.1501462(70) 0.1461874(92) 0.1459989(77) 0.1449460(37)
  0.1750386(30) 0.1701386(30) 0.1509823(89) 0.1501443(74) 0.146142(15) 0.145952(15) 0.144952(15)
82 Pb   0.1702924(59) 0.1653781(53) 0.1468107(67) 0.1459663(68) 0.1421118(88) 0.1419201(75) 0.1408707(37)
  0.17029527(56) 0.16537816(38) 0.1468129(10) 0.14596836(58) 0.142122(30) 0.141912(15) 0.1408821(74)
83 Bi 209 0.1657170(58) 0.1607911(52) 0.1427865(65) 0.1419372(66) 0.1381841(87) 0.1379910(72) 0.1369439(37)
  0.1657183(20) 0.1607903(46) 0.142780(11) 0.1419492(54) 0.138172(15) 0.137972(15) 0.136942(15)
84 Po 209 0.1613031(58) 0.1563656(51) 0.1389056(63) 0.1380520(65) 0.1343966(85) 0.1342012(69) 0.1331589(36)
  0.161302(15) 0.156362(15) 0.138922(30) 0.138072(30) 0.134382(30) 0.134182(30)  
85 At 210 0.1570444(56) 0.1520953(50) 0.1351623(62) 0.1343044(63) 0.1307448(83) 0.1305470(67) 0.1295098(36)
  0.157052(30) 0.152102(30) 0.135172(59) 0.134322(59) 0.130722(59) 0.130522(59)  
86 Rn 222 0.1529334(56) 0.1479727(49) 0.1315499(61) 0.1306882(61) 0.1272218(79) 0.1270211(66) 0.1259898(35)
  0.152942(44) 0.147982(44) 0.131552(74) 0.130692(74) 0.127192(74) 0.126982(74)  
87 Fr 223 0.1489599(56) 0.1439878(50) 0.1280599(60) 0.1271937(61) 0.1238183(79) 0.1236157(63) 0.1225852(36)
  0.148962(44) 0.143992(44) 0.128072(74) 0.127192(74) 0.123792(74) 0.123582(74)  
88 Ra 226 0.1451209(54) 0.1401373(48) 0.1246890(58) 0.1238185(59) 0.1205312(77) 0.1203271(60) 0.1192985(35)
  0.145119(20) 0.140132(19) 0.124689(15) 0.123815(15) 0.120535(14) 0.120320(14)  
89 Ac 227 0.1414083(54) 0.1364131(47) 0.1214301(57) 0.1205554(58) 0.1173552(73) 0.1171477(59) 0.1161246(34)
  0.141412(30) 0.136419(12) 0.121432(30) 0.120552(30) 0.117322(30) 0.117112(30)  
90 Th 232 0.1378266(53) 0.1328194(47) 0.1182861(56) 0.1174071(56) 0.1142910(71) 0.1140810(57) 0.1130642(34)
  0.13782600(31) 0.13282021(36) 0.11828686(78) 0.11740759(59) 0.114262(15) 0.114042(13) 0.113072(15)
91 Pa 231 0.1343514(52) 0.1293324(46) 0.1152364(55) 0.1143530(55) 0.1113088(69) 0.1110964(56) 0.1101087(34)
  0.1343516(29) 0.1293302(27) 0.1152427(21) 0.1143583(21) 0.111292(30) 0.111072(30)  
92 U 238 0.1309879(52) 0.1259572(46) 0.1122860(53) 0.1113979(54) 0.1084449(67) 0.1082301(54) 0.1072452(33)
  0.13099111(78) 0.12595977(36) 0.11228858(66) 0.11140132(65) 0.108372(15) 0.108182(15) 0.107232(15)
93 Np 237 0.1277298(51) 0.1226871(45) 0.1094299(52) 0.1085378(53) 0.1056621(66) 0.1054450(53) 0.1044744(33)
  0.1277287(39) 0.1226882(36) 0.1094230(39) 0.1085265(28) 0.105670(31) 0.105457(31) 0.1044605(62)
94 Pu 244 0.1245763(50) 0.1195212(45) 0.1066627(51) 0.1057661(52) 0.1029688(64) 0.1027494(52) 0.1017982(33)
  0.1245705(25) 0.1195140(23) 0.1066611(18) 0.1057595(18) 0.1029724(26) 0.1027429(26)  
95 Am 243 0.1215172(50) 0.1164501(45) 0.1039811(51) 0.1030805(51) 0.1003579(63) 0.1001364(51) 0.0991999(33)
  0.1215158(24) 0.1164463(33) 0.1039794(17) 0.1030803(17) 0.1003537(24) 0.1001357(24)  
96 Cm 248 0.1185536(49) 0.1134742(44) 0.1013837(50) 0.1004790(50) 0.0978295(63) 0.0976059(50) 0.0966801(33)
  0.1185427(23) 0.1134635(21) 0.1013753(17) 0.1004708(16) 0.0978355(23) 0.0975952(15)  
97 Bk 249 0.1156777(49) 0.1105860(43) 0.0988636(48) 0.0979546(49) 0.0953724(61) 0.0951469(49) 0.0942405(32)
  0.1156630(54) 0.1105745(49) 0.0988598(55) 0.0979514(54)   0.0942501(50)  
98 Cf 250 0.1128873(48) 0.1077832(43) 0.0964130(47) 0.0955000(48) 0.0929867(61) 0.0927593(48) 0.0918695(32)
  0.1128799(82) 0.1077793(75) 0.0963915(83) 0.0954860(90) 0.0929715(82) 0.0927508(84) 0.091862(10)
99 Es 251 0.1101788(47) 0.1050620(43) 0.0940403(46) 0.0931231(47) 0.0906838(60) 0.0904543(47) 0.0895840(32)
  0.1102072(98) 0.1050554(89) 0.094036(14) 0.093090(14)     0.0895878(97)
100 Fm 254 0.1075497(47) 0.1024201(42) 0.0917379(45) 0.0908165(46) 0.0884443(59) 0.0882127(45) 0.0873575(32)
  0.107514(14) 0.102386(13) 0.091715(10) 0.0907943(98) 0.0884212(100) 0.0881872(99) 0.0873356(80)

Figure 4.2.2.1[link] shows plots of the relative deviation between theoretical and experimental values for the K-series lines and the K-absorption edge as a function of Z. The error bars shown in the figure are the experimental uncertainties. In general, these plots show good agreement between theory and experiment except in the low-Z and high-Z regions. At the low-Z end of the table, the particular calculational approach used is not optimum, and the experimental data are surprisingly weak. At the high end, experimental data have rather large uncertainties, and thus do not provide an accurate test of the theory.

[Figure 4.2.2.1]

Figure 4.2.2.1 | top | pdf |

Relative deviations between theoretical and experimental results for K-series spectra. The topmost data set concerns the K-edge location, while the other data sets, beginning at the bottom, refer to the Kα2, Kα1, Kβ3 and Kβ1, respectively. The ordinate scales have been displaced for clarity by the indicated multiples of 0.001.

Table 4.2.2.5[link] summarizes the theoretical and experimental results for prominent L-series lines and the L-absorption edges. The experimental database of high-accuracy emission data is much more limited than was the case for the K series, and there have been very few high-accuracy edge-location measurements. The format of this table is similar to that of Table 4.2.2.4[link]. For the emission lines, the upper number (in italics) is the theoretical estimate for this line, and the lower number is the experimentally measured value. Numbers in bold type were directly measured on the optical scale (see Table 4.2.2.2[link]), and numbers in normal type are from the Bearden database or a reference that appeared after the Bearden database corrected to an optically based scale. For the L-absorption edges, the upper number (also in italics) is obtained by combining emission lines and photoelectron spectroscopy (see Subsection 4.2.2.7[link]) and the lower number is the experimentally measured value. The numbers in bold type are recent measurements by Kraft, Stümpel, Becker & Kuetgens (1996[link]), and the numbers in normal type are from the Bearden database or a reference that appeared after the Bearden database corrected to an optically based scale. Fig. 4.2.2.2[link] shows relative deviations between the theoretical and experimental values for most of the tabulated data. The error bars shown in the figure are the experimental uncertainties.

Table 4.2.2.5| top | pdf |
Wavelengths of L-emission lines and L-absorption edges in Å; see text for explanation of typefaces

Numbers in parentheses are standard uncertainties in the least significant figures.

ZSymbolALα2Lα1Lβ1Lβ2LI abs. edgeLII abs. edgeLIII abs. edge
20 Ca           28.275(32) 35.384(40) 35.7704(68)
  36.331(30) 36.331(30) 35.941(30)     35.131(15) 35.491(15)
21 Sc   30.947(46)   30.587(47)   24.896(15) 30.718(17) 31.109(36)
  31.350(44) 31.350(44) 31.020(30)        
22 Ti   27.215(37)   26.843(37)   22.099(24) 26.953(14) 27.3105(36)
  27.420(30) 27.420(30) 27.050(30)     27.290(15) 27.290(15)
23 V   24.143(30)   23.764(30)   19.779(19) 23.8561(89) 24.206(10)
  24.250(44) 24.250(44) 23.880(59)        
24 Cr   21.640(24) 21.490(11) 21.276(24)   17.804(15) 21.246(18) 21.5867(49)
  21.640(44) 21.640(44) 21.270(15)   16.70(15) 17.90(15) 20.70(15)
25 Mn   19.390(20) 19.359(21) 19.036(20)   16.113(19) 19.0781(57) 19.4063(43)
  19.450(15) 19.450(15) 19.110(30)        
26 Fe   17.525(17) 17.503(17) 17.194(17)   14.611(34) 17.2248(92) 17.5402(35)
  17.590(30) 17.590(30) 17.260(15)     17.2023(74) 17.5253(74)
27 Co   15.922(14) 15.905(15) 15.610(14)   13.4000(86) 15.627(14) 15.9290(44)
  15.9722(89) 15.9722(89) 15.666(12)     15.6182(74) 15.9152(74)
28 Ni   14.532(12) 14.520(12) 14.236(12)   12.295(13) 14.251(23) 14.5396(57)
  14.5612(44) 14.5612(44) 14.2712(89)     14.2422(74) 14.5252(74)
29 Cu   13.341(10) 13.336(11) 13.063(10)   11.292(16) 13.016(14) 13.2934(64)
  13.3362(44) 13.3362(44) 13.0532(44)     13.0142(15) 13.2882(15)
30 Zn   12.2529(90) 12.2489(90) 11.9819(93)   10.361(12) 11.8652(66) 12.134(14)
  12.2542(44) 12.2542(44) 11.9832(44)   13.060(15) 11.8622(15) 12.1312(15)
31 Ga   11.2916(77) 11.2858(78) 11.0226(78)   9.518(11) 10.8414(29) 11.1040(29)
  11.2922(15) 11.2922(15) 11.0232(30)   9.5171(74) 10.8282(74) 11.1002(15)
32 Ge   10.4371(68) 10.4306(68) 10.1717(69)   8.775(12) 9.9340(27) 10.1849(46)
  10.4363(12) 10.4363(12) 10.1752(15)   8.7731(15) 9.9241(15) 10.1872(15)
33 As   9.6744(60) 9.6680(60) 9.4126(59)   8.092(13) 9.1182(17) 9.3649(29)
  9.6710(12) 9.6710(12) 9.4142(12)   8.1071(15) 9.1251(15) 9.3671(15)
34 Se   8.9914(52) 8.9852(52) 8.7335(52)   7.498(13) 8.4105(58) 8.64459(77)
  8.99013(74) 8.99013(74) 8.73593(74)   7.5031(15) 8.4071(15) 8.6461(15)
35 Br   8.3776(46) 8.3715(46) 8.1233(46)   6.958(14) 7.7669(35) 7.9991(30)
  8.37473(74) 8.37473(74) 8.12522(74)   6.9591(74) 7.7531(74) 7.9841(74)
36 Kr   7.8242(41) 7.8180(41) 7.5736(40)   6.4561(41) 7.1630(21) 7.3841(17)
  7.82032(13) 7.82032(13) 7.574441(98)   6.470(15) 7.1681(15) 7.3921(15)
37 Rb   7.3226(37) 7.3164(36) 7.0749(36)   6.0010(11) 6.6449(59) 6.8643(67)
  7.32521(44) 7.31841(30) 7.07601(44)   6.0081(74) 6.6441(15) 6.8621(15)
38 Sr   6.8674(33) 6.8610(32) 6.6224(33)   5.5945(16) 6.17624(70) 6.38937(84)
  6.86980(44) 6.86290(30) 6.62400(44)   5.5921(74) 6.1731(15) 6.3871(15)
39 Y   6.4539(30) 6.4466(29) 6.2110(29)   5.22968(53) 5.75742(82) 5.9658(15)
  6.45590(44) 6.44890(30) 6.21209(44)   5.2171(74) 5.7561(15) 5.9621(15)
40 Zr   6.0766(27) 6.0684(26) 5.8357(26)   4.89881(41) 5.3773(15) 5.5816(15)
  6.0766(27) 6.070250(79) 5.836214(76) 5.58638(44) 4.8791(74) 5.3781(15) 5.5791(15)
41 Nb   5.7326(24) 5.7226(23) 5.4931(23)   4.59975(43) 5.03480(63) 5.23529(98)
  5.73199(44) 5.72439(30) 5.49238(44) 5.23798(44) 4.5751(74) 5.0311(15) 5.2301(15)
42 Mo   5.4151(22) 5.4054(21) 5.1778(21) 4.91857(74) 4.32423(40) 4.72145(60) 4.9179(31)
  5.41445(12) 5.40663(12) 5.17716(12) 4.92327(30) 4.3041(74) 4.7191(15) 4.9131(15)
43 Tc   5.1228(20) 5.1139(19) 4.8880(19) 4.6341(13)   4.4368(13) 4.62991(94)
    5.11488(44) 4.8874(12)   4.0581(74) 4.4361(15) 4.6301(15)
44 Ru   4.8541(18) 4.8449(17) 4.6210(17) 4.3681(13) 3.8443(16) 4.17814(78) 4.36776(32)
  4.85388(10) 4.845823(74) 4.620649(44) 4.37187(30) 3.8351(74) 4.1801(15) 4.3691(15)
45 Rh   4.6055(16) 4.5966(16) 4.3744(16) 4.1277(12) 3.6334(17) 3.94053(55) 4.12730(50)
  4.60552(13) 4.59750(13) 4.374206(59) 4.13106(30) 3.6291(74) 3.94256(74) 4.12996(74)
46 Pd   4.3753(15) 4.3672(15) 4.1461(14) 3.9088(10) 3.43948(61) 3.72251(52) 3.90655(62)
  4.37595(10) 4.367736(74) 4.146282(74) 3.908929(59) 3.4371(15) 3.72286(15) 3.90746(15)
47 Ag   4.1623(14) 4.1541(13) 3.9347(13) 3.7034(10) 3.25639(29) 3.51704(26) 3.69817(53)
  4.163002(74) 4.154492(44) 3.934789(44) 3.703406(44) 3.25645(15) 3.51645(15) 3.69996(15)
48 Cd   3.9644(13) 3.9560(12) 3.7382(12) 3.51355(97) 3.08443(17) 3.32528(29) 3.50348(45)
  3.965020(89) 3.956409(59) 3.738286(59) 3.514133(59) 3.08495(15) 3.32575(15) 3.50475(15)
49 In   3.7802(12) 3.7716(11) 3.5553(11) 3.33796(83) 2.92533(19) 3.14784(47) 3.32322(42)
  3.780787(89) 3.771977(59) 3.555363(59) 3.338430(44) 2.92604(15) 3.14735(15) 3.32375(15)
50 Sn   3.6084(11) 3.5997(10) 3.38472(100) 3.17475(77) 2.776792(71) 2.98309(56) 3.15521(70)
  3.606964(59) 3.599994(44) 3.384921(44) 3.175098(44) 2.77694(15) 2.98234(15) 3.15575(15)
51 Sb   3.44794(99) 3.43913(93) 3.22551(92) 3.02325(67) 2.638437(69) 2.82990(51) 2.99986(66)
  3.448452(89) 3.439462(59) 3.225718(59) 3.023395(44) 2.63884(15) 2.82944(74) 3.00035(15)
52 Te   3.29788(92) 3.28894(86) 3.07663(85) 2.88209(61) 2.50998(50) 2.687685(87) 2.85523(35)
  3.29851(13) 3.289249(89) 3.076816(89) 2.88221(12) 2.50994(15) 2.68794(15) 2.85554(15)
53 I   3.15734(85) 3.14828(81) 2.93720(78) 2.75031(54) 2.38965(37) 2.55532(31) 2.72067(32)
  3.157957(89) 3.148647(89) 2.937484(89) 2.75057(12) 2.38804(74) 2.55424(74) 2.71964(74)
54 Xe   3.02568(78) 3.01640(76) 2.80659(69) 2.62740(47) 2.273869(70) 2.427862(95) 2.590303(89)
  3.025940(22) 3.016582(15) 2.806553(19)   2.27373(15) 2.42924(15) 2.59264(15)
55 Cs   2.90167(73) 2.89237(69) 2.68362(66) 2.51216(47) 2.1676(29) 2.3135(17) 2.47326(16)
  2.90204(30) 2.89244(30) 2.68374(30) 2.51184(30) 2.16733(74) 2.31393(15) 2.47404(15)
56 Ba   2.78522(68) 2.77580(64) 2.56812(61) 2.40421(26) 2.0697(15) 2.20482(12) 2.363082(97)
  2.785572(74) 2.775992(74) 2.568249(74) 2.404386(89) 2.06783(74) 2.20483(15) 2.36294(15)
57 La   2.67563(64) 2.66607(60) 2.45941(57) 2.30307(24) 1.97705(28) 2.10317(10) 2.25958(20)
  2.675383(60) 2.665740(74) 2.458947(74) 2.303312(98) 1.97803(74) 2.10533(74) 2.2610(15)
58 Ce   2.57122(59) 2.56108(56) 2.35598(53) 2.20843(21) 1.89320(71) 2.01084(14) 2.16586(39)
  2.57059(18) 2.56163(17) 2.35580(18) 2.20900(17) 1.89343(74) 2.01243(74) 2.1660(15)
59 Pr   2.47329(55) 2.46280(52) 2.25890(49) 2.11936(20) 1.81477(33) 1.92607(36) 2.07945(22)
  2.47294(44) 2.46304(30) 2.25883(44) 2.11943(59) 1.81413(74) 1.92553(74) 2.07913(74)
60 Nd   2.38081(51) 2.36999(48) 2.16724(45) 2.03554(18) 1.73904(18) 1.84373(16) 1.99616(19)
  2.38079(52) 2.370526(16) 2.167008(19) 2.035448(88) 1.73903(15) 1.84403(15) 1.99673(15)
61 Pm   2.29340(48) 2.28227(45) 2.08060(42) 1.95675(18)      
  2.29263(59) 2.28223(44) 2.07973(59) 1.95593(89) 1.66743(74) 1.76763(74) 1.91913(15)
62 Sm   2.21054(48) 2.19926(42) 1.99850(42) 1.88225(17) 1.60201(12) 1.69495(13) 1.84534(42)
  2.210430(24) 2.199873(13) 1.998432(30) 1.882206(41) 1.60022(15) 1.69533(15) 1.84573(15)
63 Eu   2.13214(42) 2.12081(40) 1.92080(37) 1.81237(16) 1.54065(17) 1.62830(21) 1.77767(16)
  2.13156(17) 2.120673(95) 1.92053(17) 1.81215(17) 1.53812(15) 1.62712(15) 1.77613(15)
64 Gd   2.05817(40) 2.04670(37) 1.84744(34) 1.74582(14) 1.47922(25) 1.56264(23) 1.71092(21)
  2.05783(30) 2.04683(30) 1.84683(30) 1.74553(30) 1.47842(15) 1.56322(15) 1.71173(15)
65 Tb   1.98699(37) 1.97586(35) 1.77701(32) 1.68377(14) 1.42285(98) 1.50195(80) 1.65023(44)
  1.98753(30) 1.97653(30) 1.77683(44) 1.68303(30) 1.42232(15) 1.50232(15) 1.64972(15)
66 Dy   1.91986(35) 1.90883(33) 1.71052(30) 1.62497(12) 1.37058(41) 1.44500(20) 1.59241(33)
  1.919939(44) 1.908839(44) 1.71065(10) 1.62371(10) 1.36922(15) 1.44452(15) 1.59162(15)
67 Ho   1.85606(33) 1.84511(31) 1.64732(28) 1.56818(11) 1.31957(28) 1.39091(27) 1.53614(34)
  1.856472(15) 1.845092(17) 1.647484(32) 1.567168(50) 1.31902(15) 1.39052(15) 1.53682(15)
68 Er   1.79537(31) 1.78449(29) 1.58720(26) 1.51486(10) 1.27145(14) 1.33792(26) 1.48318(27)
  1.795701(45) 1.784481(20) 1.587466(86) 1.51401(13) 1.27062(15) 1.33862(15) 1.48352(15)
69 Tm   1.73758(29) 1.72677(27) 1.52995(24) 1.464210(95) 1.22612(28) 1.28942(27) 1.43366(27)
  1.738003(19) 1.7267720(70) 1.5302410(70) 1.46402(30) 1.22502(15) 1.28922(15) 1.43342(15)
70 Yb   1.68248(29) 1.67177(26) 1.47538(24) 1.416041(89) 1.18266(60) 1.243391(70) 1.3858(10)
  1.682875(74) 1.671915(59) 1.475672(74) 1.415521(74) 1.18182(15) 1.24282(15) 1.38622(15)
71 Lu   1.63031(26) 1.61949(24) 1.42361(21) 1.370061(85) 1.14043(22) 1.197954(60) 1.341053(93)
  1.630314(74) 1.619534(44) 1.423611(44) 1.370141(44) 1.14022(15) 1.19852(15) 1.34052(15)
72 Hf   1.58049(25) 1.56959(23) 1.37419(20) 1.326241(78) 1.10009(24) 1.1550(10) 1.2972(14)
  1.580484(74) 1.569604(74) 1.374121(74) 1.326410(74) 1.1002640(49) 1.1548587(22) 1.2971383(68)
73 Ta   1.53290(23) 1.52194(22) 1.32697(19) 1.282314(74) 1.06152(30) 1.11368(14) 1.25506(34)
  1.532953(30) 1.521993(30) 1.327000(44) 1.284559(30) 1.06132(15) 1.11372(15) 1.25532(15)
74 W   1.48748(22) 1.47642(21) 1.28188(18) 1.244447(70) 0.91604(28) 1.07431(38) 1.21543(99)
  1.487452(30) 1.4763112(95) 1.281812(13) 1.2443048(98) 1.024685(74) 1.07452(15) 1.21552(15)
75 Re   1.44399(21) 1.43288(19) 1.23872(17) 1.206487(67) 0.98968(21) 1.03670(20) 1.17673(27)
  1.443982(74) 1.432922(59) 1.238599(30) 1.206618(59) 0.98941(15) 1.03712(15) 1.17732(15)
76 Os   1.40238(20) 1.39121(18) 1.19742(16) 1.170095(62) 0.95583(36) 1.000786(57) 1.14002(23)
  1.402361(74) 1.391231(74) 1.197288(74) 1.16981(12) 0.95581(15) 1.00142(15) 1.14082(15)
77 Ir   1.36252(19) 1.35130(19) 1.15786(15) 1.135812(72) 0.9240(12) 0.96675(18) 1.10535(22)
  1.362520(74) 1.351300(44) 1.157827(44) 1.135337(44) 0.92361(15) 0.96711(15) 1.10582(15)
78 Pt   1.32434(18) 1.31308(17) 1.11995(14) 1.102006(63) 0.8933(14) 0.93395(27) 1.07200(36)
  1.324340(30) 1.313060(44) 1.119917(30) 1.102017(44) 0.893213(19) 0.9341861(21) 1.0722721(19)
79 Au   1.28773(17) 1.27643(16) 1.08359(13) 1.070479(53) 0.86383(45) 0.90263(12) 1.04009(27)
  1.287739(44) 1.276419(44) 1.083546(44) 1.070236(44) 0.863683(30) 0.9027409(46) 1.0401625(52)
80 Hg   1.25261(16) 1.24126(15) 1.04869(13) 1.039584(51) 0.83546(43) 0.87238(26) 1.00919(30)
  1.25266(10) 1.241219(74) 1.048696(74) 1.03977(10) 0.83531(15) 0.87221(15) 1.00912(15)
81 Tl   1.21890(15) 1.20750(14) 1.01519(12) 1.01029(20) 0.80795(15) 0.843512(77) 0.97953(25)
  1.218768(44) 1.207408(59) 1.015145(59) 1.010325(44) 0.80811(15) 0.84341(15) 0.97931(15)
82 Pb   1.18651(15) 1.17507(14) 0.98298(11) 0.98221(19) 0.78172(24) 0.81575(18) 0.95113(22)
  1.186498(74) 1.175028(30) 0.982925(44) 0.98222(10) 0.7818404(49) 0.8157395(16) 0.9511590(22)
83 Bi 209 1.15540(14) 1.14390(13) 0.95205(11) 0.95526(18) 0.75649(58) 0.789102(88) 0.92387(11)
  1.155377(15) 1.143877(30) 0.951992(13) 0.955194(59) 0.75711(15) 0.78871(15) 0.92341(15)
84 Po 209 1.12549(13) 1.11393(12) 0.92228(10) 0.92932(18) 0.7332(13) 0.76325(13) 0.897554(85)
  1.125497(74) 1.113877(59) 0.92201(30) 0.929384(74)      
85 At 210 1.09670(13) 1.08510(12) 0.893639(96) 0.90444(17)   0.73868(13)  
  1.096726(74) 1.085016(74) 0.89350(13)        
86 Rn 222 1.06900(12) 1.05735(11) 0.866054(91) 0.88055(15)   0.71511(13)  
  1.069006(74) 1.057246(74) 0.86606(13)        
87 Fr 223 1.04232(11) 1.03063(11) 0.839482(86) 0.85751(15)   0.69240(13) 0.8251(27)
  1.042316(74) 1.030505(74) 0.83941(13) 0.8580(30)      
88 Ra 226 1.01662(11) 1.00489(10) 0.813866(82) 0.83533(16) 0.64449(15) 0.67077(12) 0.802768(44)
  1.016575(74) 1.004745(74) 0.813762(74) 0.835383(74) 0.64451(15) 0.67071(15) 0.80281(15)
89 Ac 227 0.99185(11) 0.980070(98) 0.789163(78) 0.81406(14)   0.64970(13)  
  0.991795(74) 0.979945(74) 0.78904(13)        
90 Th 232 0.96798(10) 0.956154(94) 0.765343(75) 0.79354(13) 0.60569(11) 0.62966(11) 0.760637(99)
  0.9679082(23) 0.9560826(15) 0.7652610(14) 0.7935516(15) 0.60591(15) 0.62991(15) 0.76071(15)
91 Pa 231 0.944896(96) 0.933002(90) 0.742301(71) 0.77321(12) 0.58759(12) 0.610354(92) 0.740958(97)
  0.944834(74) 0.932854(74) 0.742331(74) 0.77371(15)      
92 U 238 0.922622(93) 0.910674(86) 0.720056(68) 0.75462(12) 0.569885(39) 0.591930(66) 0.722319(52)
  0.922572(13) 0.910653(13) 0.719995(12) 0.754692(13) 0.56951(15) 0.59191(15) 0.72231(15)
93 Np 237 0.901230(88) 0.889223(83) 0.698624(65) 0.73623(11)      
  0.901059(13) 0.889141(13) 0.698488(13) 0.736241(13) 0.55239(34) 0.57368(37) 0.704136(20)
94 Pu 244 0.880355(85) 0.868290(79) 0.677776(60) 0.71848(11)      
          0.53651(15) 0.55721(15) 0.68671(15)
95 Am 243 0.860288(84) 0.848190(81) 0.657686(59) 0.70134(10)      
               
96 Cm 248 0.840918(80) 0.828776(78) 0.638265(56) 0.684815(98)      
               
97 Bk 249 0.822159(76) 0.809987(69) 0.619449(53) 0.668638(94)      
          0.49060(49) 0.50851(52) 0.63748(98)
98 Cf 250 0.803608(73) 0.791421(66) 0.601005(50) 0.652873(89)      
          0.476569(92) 0.493804(98) 0.62300(19)
99 Es 251 0.786043(70) 0.773837(63) 0.583354(49) 0.638227(82)      
               
100 Fm 254 0.769077(67) 0.756843(60) 0.566272(47) 0.623826(82)      
  0.76904(62) 0.75674(60) 0.56619(34) 0.62369(41) 0.44966(13) 0.46534(12) 0.59414(20)
These values are for the unresolved [L\beta _2] and [L\beta _{15}] emission lines.
[Figure 4.2.2.2]

Figure 4.2.2.2 | top | pdf |

Comparison of L-series data with experiment for the indicated range of Z. Indicated data, beginning at the bottom, refer to the Lα2, Lα1, and Lβ1 emission lines and the LIII, LII, and LI absorption edges. For clarity, the plots have been displaced vertically by multiples of 0.002 for the emission lines and 0.004 for the absorption edges.

4.2.2.13. Availability of a more complete X-ray wavelength table

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This article and the accompanying X-ray wavelength tables are an updated version of the contribution to the International Tables for Crystallography, Volume C, 2nd edition that was published in 1999. This article has been subject to more critical review and analysis and the data are consistent with the most recent adjustment of the fundamental physical constants (Mohr & Taylor, 2000[link]). We believe that these data represent a significant improvement in consistency, coverage and accuracy over previously available resources. The results presented here are a subset of a larger effort that includes all K- and L-series lines connecting the n = 1 to n = 4 shells. The more complete table has been submitted for archival publication and will be made available on the NIST Physical Reference Data web site. Electronic publishing of this resource will provide a convenient data resource to the scientific community that can be more easily up-dated and expanded.

4.2.2.14. Connection with scales used in previous literature

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In order to compare historical data for X-ray spectra with the results in the present tabulation, certain conversion factors are needed. As discussed in the introduction, the principal units found in the literature are the xu and the Å* unit. There is the additional complication that there were several different definitions in use at various times and at the same time in different laboratories. For the convenience of the reader, we summarize in Table 4.2.2.6[link] the main conversion factors needed. The numerical values for the wavelengths in Å can be converted to energies in electron volts by using the conversion factor 12398.41857 (49) eV Å (Mohr & Taylor, 2000[link]).

Table 4.2.2.6| top | pdf |
Wavelength conversion factors

Numbers in parentheses are standard uncertainties in the least-significant figures.

 Cu Kα1Mo Kα1W Kα1
λ (Å) 1.54059292(45) 0.70931713(41) 0.20901313(18)
λ (Å*) 1.540562(3) 0.709300(1) 0.2090100
l kxu 1.537400 0.707831  
Å*/Å 1.0000201(20) 1.0000242(22) 1.00001498(86)
kxu/Å 1.00207683(29) 1.00209955(58)  

4.2.3. X-ray absorption spectra

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D. C. Creaghb

4.2.3.1. Introduction

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4.2.3.1.1. Definitions

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This section deals with the manner in which the photon scattering and absorption cross sections of an atom varies with the energy of the incident photon. Further information concerning these cross sections and tables of the X-ray attenuation coefficients are given in Section 4.2.4[link]. Information concerning the anomalous-dispersion corrections is given in Section 4.2.6[link].

When a highly collimated beam of monoenergetic photons passes through a medium of thickness t, it suffers a decrease in intensity according to the relation [I=I_0\exp(-\mu_lt), \eqno (4.2.3.1)]where μl is the linear attenuation coefficient. Most tabulations express μl in c.g.s. units, μl having the units cm−1.

An alternative, often more convenient, way of expressing the decrease in intensity involves the measurement of the mass per unit area mA of the specimen rather than the specimen thickness, in which case equation (4.2.3.1)[link] takes the form [I=I_0\exp[-(\mu/\rho)m_A], \eqno (4.2.3.2)]where ρ is the density of the material and (μ/ρ) is the mass absorption coefficient. The linear attenuation coefficient of a medium comprising atoms of different types is related to the mass absorption coefficients by [\mu_l=\rho\textstyle\sum\limits_ig_i(\mu/\rho)_i, \eqno (4.2.3.3)]where [g_i] is the mass fraction of the atoms of the ith species for which the mass absorption coefficient is [(\mu/\rho)_i]. The summation extends over all the atoms comprising the medium. For a crystal having a unit-cell volume of [V_c], [\mu_l={1\over V_c}\sum\sigma_i, \eqno (4.2.3.4)]where [\sigma_i] is the photon scattering and absorption cross section. If [\sigma_i] is expressed in terms of barns/atom then [V_c] must be expressed in terms of Å3 and μl is in cm−1. (1 barn = 10−28 m2.)

The mass attenuation coefficient μ/ρ is related to the total photon–atom scattering cross section σ according to [\eqalignno{{\mu\over\rho}({\rm cm}^2/{\rm g}) &=(N_{\rm A}/M)\sigma ({\rm cm}^2/{\rm atom})\cr &=(N_{\rm A}/M)\times10^{-24}\sigma\,({\rm barns/atom}), &(4.2.3.5)}]where [N_{\rm A}=] Avogadro's number = 6.0221367 (36) × 1023 atoms/gram atom (Cohen & Taylor, 1987[link]) and M = atomic weight relative to M(12C) = 12.0000.

4.2.3.1.2. Variation of X-ray attenuation coefficients with photon energy

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When a photon interacts with an atom, a number of different absorption and scattering processes may occur. For an isolated atom at photon energies of less than 100 keV (the limit of most conventional X-ray generators), contributions to the total cross section come from the photo-effect, coherent (Rayleigh) scattering, and incoherent (Compton) scattering. [\sigma=\sigma_{\rm pe}+\sigma_R+\sigma_C. \eqno (4.2.3.6)]The relation between the photo-effect absorption cross section [\sigma_{\rm pe}] and the X-ray anomalous-dispersion corrections will be discussed in Section 4.2.6[link].

The magnitudes of these scattering cross sections depend on the type of atom involved in the interactions and on the energy of the photon with which it interacts. In Fig. 4.2.3.1[link] , the theoretical cross sections for the interaction of photons with a carbon atom are given. Values of [\sigma_{\rm pe}] are from calculations by Scofield (1973[link]), and those for Rayleigh and Compton scattering are from tabulations by Hubbell & Øverbø (1979[link]) and Hubbell (1969[link]), respectively. Note the sharp discontinuities that occur in the otherwise smooth curves. These correspond to photon energies that correspond to the energies of the K and [L_{\rm I}] [L_{\rm II}] [L_{\rm III}] shells of the carbon energies. Notice also that [\sigma_{\rm pe}] is the dominant interaction cross section, and that the Rayleigh scattering cross section remains relatively constant for a broad range of photon energies, whilst the Compton scattering peaks at a particular photon energy (∼100 keV). Other interaction mechanisms exist [e.g. Delbrück (Papatzacos & Mort, 1975[link]; Alvarez, Crawford & Stevenson, 1958[link]), pair production, nuclear Thompson], but these do not become significant interaction processes for photon energies less than 1 MeV. This section will not address the interaction of photons with atoms for which the photon energy exceeds 100 keV.

[Figure 4.2.3.1]

Figure 4.2.3.1 | top | pdf |

Theoretical cross sections for photon interactions with carbon showing the contributions of photoelectric, elastic (Rayleigh), inelastic (Compton), and pair-production cross sections to the total cross sections. Also shown are the experimental data (open circles). From Gerstenberg & Hubbell (1982[link]).

4.2.3.1.3. Normal attenuation, XAFS, and XANES

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The curves shown in Fig. 4.2.3.1[link] are the result of theoretical calculations of the interactions of an isolated atom with a single photon. Experiments are not usually performed on isolated atoms, however. When experiments are performed on ensembles of atoms, a number of points of difference emerge between the experimental data and the theoretical calculations. These effects arise because the presence of atoms in proximity with one another can influence the scattering process. In short: the total attenuation coefficient of the system is not the sum of all the individual attenuation coefficients of the atoms that comprise the system.

Perhaps the most obvious manifestation of this occurs when the photon energy is close to an absorption edge of an atom. In Fig. 4.2.3.2[link] , the mass attenuation of several germanium compounds is plotted as a function of photon energy. The energy scale measures the distance from the K-shell edge energy of germanium (11.104 keV). These curves are taken from Hubbell, McMaster, Del Grande & Mallett (1974[link]). Not only does the experimental curve depart significantly from the theoretically predicted curve, but there is a marked difference in the complexity of the curves between the various germanium compounds.

[Figure 4.2.3.2]

Figure 4.2.3.2 | top | pdf |

The dependence of the X-ray attenuation coefficient on energy for a range of germanium compounds, taken in the neighbourhood of the germanium absorption edge (from IT IV, 1974[link]).

Far from the absorption edge, the theoretical calculations and the experimental data are in reasonable agreement with what one might expect using the sum rule for the various scattering cross sections and one could say that this region is one in which normal attenuation coefficients may be found.

Closer to the edge, the almost periodic variation of the mass attenuation coefficient is called the extended X-ray absorption fine structure (XAFS). Very close to the edge, more complicated fluctuations occur. These are referred to as X-ray absorption near edge fine structure (XANES). The boundary of the XAFS and XANES regions is somewhat arbitrary, and the physical basis for making the distinction between the two will be outlined in Subsection 4.2.3.4[link].

Even in the region where normal attenuation may be thought to occur, cooperative effects can exist, which can affect both the Rayleigh and the Compton scattering contributions to the total attenuation cross section. The effect of cooperative Rayleigh scattering has been discussed by Gerward, Thuesen, Stibius-Jensen & Alstrup (1979[link]), Gerward (1981[link], 1982[link], 1983[link]), Creagh & Hubbell (1987[link]), and Creagh (1987a[link]). That the Compton scattering contribution depends on the physical state of the scattering medium has been discussed by Cooper (1985[link]).

Care must therefore be taken to consider the physical state of the system under investigation when estimates of the theoretical interaction cross sections are made.

4.2.3.2. Techniques for the measurement of X-ray attenuation coefficients

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4.2.3.2.1. Experimental configurations

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Experimental configurations that set out to determine the X-ray linear attenuation coefficient [\mu_l] or the corresponding mass absorption coefficients (μ/ρ) must have characteristics that reflect the underlying assumptions from which equation (4.2.3.1)[link] was derived, namely:

  • (i) the incident and transmitted beams are parallel and there is no divergence in the transmitted beam;

  • (ii) the photons in the incident and transmitted beams have the same energy;

  • (iii) the specimen is of sufficient thickness.

Because of the considerable discrepancies that often exist in X-ray attenuation measurements (see, for example, IT IV, 1974[link]), the IUCr Commission on Crystallographic Apparatus set up a project to determine which, if any, of the many techniques for the measurement of X-ray attenuation coefficients is most likely to yield correct results. In the project, a number of different experimental configurations were used. These are shown in Fig. 4.2.3.3[link] . The configurations used ranged in complexity from that of Fig. 4.2.3.3(a)[link], which uses a slit-collimated beam from a sealed tube and a β-filter to select its characteristic radiation, and a proportional counter and associated electronics to detect the transmitted-beam intensity, to that of Fig. 4.2.3.3[link](f), which uses a modification to a commercial X-ray-fluorescence analyser. Sources of X-rays included conventional sealed X-ray tubes, X-ray-fluorescence sources, radioisotope sources, and synchrotron-radiation sources. Detectors ranged from simple ionization chambers, which have no capacity for photon energy detection, to solid-state detectors, which provide a relatively high degree of energy discrimination. In a number of cases (Figs. 4.2.3.3c, d, e, and f[link]), monochromatization of the beam was effected using single Bragg reflection from silicon single crystals. In Fig. 4.2.3.3(i)[link], the incident-beam monochromator is using reflections from two Bragg reflectors tuned so as to eliminate harmonic radiation from the source.

[Figure 4.2.3.3]

Figure 4.2.3.3 | top | pdf |

Schematic representations of experimental apparatus used in the IUCr X-ray Attenuation Project (Creagh & Hubbell, 1987[link]; Creagh, 1985[link]). X: characteristic line from sealed X-ray tube; b: Bremsstrahlung from a sealed X-ray tube; r: radioactive source; s: synchrotron-radiation source; β: β-filter for characteristic X-rays; S: collimating slits; M: monochromator.

The performance of these systems was evaluated for a range of materials that included:

  • (i) highly perfect silicon single crystals (Creagh & Hubbell, 1987[link]);

  • (ii) polycrystalline copper foils that exhibited a high degree of preferred orientation; and

  • (iii) pyrolytic graphite that contained a high density of regular voids.

The results of this study are outlined in Section 4.2.3.2.3[link].

4.2.3.2.2. Specimen selection

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Although the most important component in the experiment is the specimen itself, examination of the data files held at the US National Institute of Standards and Technology (Gerstenberg & Hubbell, 1982[link]; Saloman & Hubbell, 1986[link]; Hubbell, Gerstenberg & Saloman, 1986[link]) has shown that, in general, insufficient care has been taken in the past to select an experimental device with characteristics that are appropriate to the specimen chosen. Nor has sufficient care been taken in the determination of the dimensions, homogeneity, and defect structure of the specimens. To achieve the best results, the following procedures should be followed.

  • (i) The dimensions of the specimen should be determined using at least two different techniques, and sample thicknesses should be chosen such that the Nordfors (1960[link]) criterion, later confirmed by Sears (1983[link]), that the condition [2\le\ln(I_0/I)\le4\eqno (4.2.3.7)]be satisfied. This enables the best compromise between achieving good counting statistics and avoiding multiple photon scattering within the sample.

    Wherever possible, different sample thicknesses should be chosen to enable a test of equation (4.2.3.1)[link] to be made. If deviations from equation (4.2.3.1)[link] exist, either the sample material or the experimental configuration, or both, are not appropriate for the measurement of [\mu_l]. If the attenuation of the material under test falls outside the limits set by the Nordfors criterion and the material is in the form of a powder, the mixing of this powder with one with low attenuation and no absorption edge in the region of interest can be used to bring the total attenuation of the sample within the Nordfors range.

  • (ii) The sample should be examined by as many means as possible to ascertain its regularity, homogeneity, defect structure, and, especially for very thin specimens, freedom from pinholes and cracks. Where a diluent has been used to reduce the attenuation so that the Nordfors criterion is satisfied, care must be taken to ensure intimate mixing of the two materials and the absence of voids.

    Since the theory upon which equation (4.2.3.1)[link] is based envisages that each atom scatters as an individual, it is necessary to be aware of whether such cooperative effects as Laue–Bragg scattering (which may become significant in single-crystal specimens) and small-angle X-ray scattering (SAXS) (which may occur if a distribution of small voids or inclusions exists) occur in polycrystalline and amorphous specimens. Knowledge that cooperative scattering may occur influences the choice of collimation of the beam.

  • (iii) The sample should be mounted normal to the beam.

4.2.3.2.3. Requirements for the absolute measurement of μl or (μ/ρ)

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The following prescription should be followed if accurate, absolute measurements of [\mu_l] and [(\mu/\rho)] are to be obtained.

  • (i) X-ray source and X-ray monochromatization. The energy of the incident photons should be measured directly using reflections from a single-crystal silicon monochromator, and the energy spread of the beam should be measured. Measurements should be made of the state of polarization, since X-ray-polarization effects are known to be significant in some measurements (Templeton & Templeton, 1982[link], 1985a[link], 1986[link]). The results of a survey on X-ray polarization were given by Jennings (1984[link]). If a single-crystal monochromator is employed, it should be placed between the sample and the detector.

  • (ii) Collimation. It is of some advantage if both the incident-beam- and the transmitted-beam-defining slits can be varied in width.

    Should it be necessary to combat the effects of Laue–Bragg scattering in a single-crystal specimen, an incident beam with a high degree of collimation is required (Gerward, 1981[link]).

    To counter the effects of small-angle X-ray scattering, it may be necessary to widen the detector aperture (Chipman, 1969[link]). That these effects can be marked has been shown by Parratt, Porteus, Schnopper & Watanabe (1959[link]), who investigated the influence collimator and monochromator configurations have on X-ray-attenuation measurements.

  • (iii) Detection. Detectors that give some degree of energy discrimination should be used. Compromise may be necessary between sensitivity and energy resolution, however, and these factors should be taken into account when a choice is being made between proportional and solid-state detectors.

Whichever detection system is chosen, it is essential that the system dead-time be determined experimentally. For descriptions of techniques for the determination of system dead-time, see, for example, Bertin (1975[link]).

4.2.3.3. Normal attenuation coefficients

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Fig. 4.2.3.1[link] shows that the X-ray attenuation coefficients are a smooth function of photon energy over a relatively large range of photon energies, and that discontinuities occur whenever the photon energy corresponds to a resonance in the electron cloud surrounding the nucleus. In Fig. 4.2.3.2[link], the effect of the interaction of the ejected photoelectron with the electron's neighbouring atoms is shown. Such edge effects (XAFS) can extend 1000 eV from the edge.

It is conventional, however, to extrapolate the smooth curve to the edge value, and a curve of normal attenuation coefficients results. These are taken to be the attenuation coefficients of the individual atoms. Tables of these normal attenuation coefficients are given in Section 4.2.4[link].

4.2.3.4. Attenuation coefficients in the neighbourhood of an absorption edge

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4.2.3.4.1. XAFS

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Although the existence of XAFS has been known for more than 60 years following experiments by Fricke (1920[link]) and Hertz (1920[link]), it is only in the last decade that a proper theoretical description has been developed. Kronig (1932a[link]) suggested a long-range-order theory based on quantum-mechanical precepts, although later (Kronig, 1932b[link]) he applied a short-range-order (SRO) theory to explain the existence of XAFS in molecular spectra. As time progressed, important suggestions were made by others, notably Kostarev (1941[link], 1949[link]), who applied this SRO theory to condensed matter, Sawada, Tsutsumi, Shiraiwa, Ishimura & Obashi (1959[link]), who accounted for the lifetime of the excited photoelectron and the core-hole state in terms of a mean free path, and Schmidt (1961a[link],b[link], 1963[link]), who showed the influence atomic vibrations have on the phase of the back-scattered waves.

Nevertheless, neither the experimental data nor the theories were sufficiently good to enable Azaroff & Pease (1974[link]) to decide which theory was the correct one to use. However, Sayers, Lytle & Stern (1970[link]) produced a theoretical approach based on SRO theory, later extended by Lytle, Stern & Sayers (1975[link]), and this is the foundation upon which all modern work has been built. Since 1970, a great deal of theoretical effort has been expended to improve the theory because of the need to interpret the wealth of data that became available through the increasing use of synchrotron-radiation sources in XAFS experiments.

A number of major reviews of XAFS theory and its use for the resolution of experimental data have been published. Contributions have been made by Stern, Sayers & Lytle (1975[link]), Lee, Citrin, Eisenberger & Kincaid (1981[link]), Lee (1981[link]), and Teo (1981[link]). The rapid growth of the use of synchrotron-radiation sources has led to the development of the use of XAFS in a wide variety of research fields. The XAFS community has met regularly at conferences, producing conference proceedings that demonstrate the maturation of the technique. The reader is directed to the proceedings edited by Mustre de Leon, Stern, Sayers, Ma & Rehr (1988[link]), Hasnain (1990[link]), and Kuroda, Ohta, Murata, Udagawa & Nomura (1992[link]), and to the papers contained therein. In the following section, a brief, simplified, description will be given of the theory of XAFS and of the application of that theory to the interpretation of XAFS data.

4.2.3.4.1.1. Theory

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The theory that will be outlined here has evolved through the efforts of many workers over the past decade. The oscillatory part of the X-ray attenuation relative to the `background' absorption may be written as [\chi(E)={\mu_l(E)-\mu_{l0}(E)\over\mu_{l0}(E)}, \eqno (4.2.3.8)]where [\mu_l(E)] is the measured value of the linear attenuation coefficient at a photon energy E and [\mu_{l0}(E)] is the `background' linear attenuation coefficient. This is sometimes the extrapolation of the normal attenuation curve to the edge energy, although it is usually found necessary to modify this extrapolation somewhat to improve the matching of the higher-energy data with the XAFS data (Dreier, Rabe, Malzfeldt & Niemann, 1984[link]). In most computer programs, the normal attenuation curve is fitted to the data using cubic spline fitting routines.

The origin of XAFS lies in the interaction of the ejected photoelectron with electrons in its immediate vicinity. The wavelength of a photoelectron ejected when a photon is absorbed is given by λ = 2π/k, where [k=[(2m/\hbar^2)(E-E_0)]{}^{1/2}. \eqno (4.2.3.9)]

This outgoing spherical wave can be back-scattered by the electron clouds of neighbouring atoms. This back-scattered wave interferes with the outgoing wave, resulting in the oscillation of the absorption rate that is observed experimentally and called XAFS. Equation (4.2.3.8)[link] was written with the assumption that the absorption rate was directly proportional to the linear absorption coefficient.

It is conventional to express [\chi(E)] in terms of the momentum of the ejected electron, and the usual form of the theoretical expression for χ(k) is [\chi({\bf k})=\textstyle\sum\limits_i(N_i/kr^2_j)|\,f_i(k)|\exp(\sigma^2_i k^2-r_i/\rho)\sin[2kr_i + \varphi_i(k)]. \eqno (4.2.3.10)]Here the summation extends over the shells of atoms that surround the absorbing atom, [N_i] representing the number of atoms in the ith shell, which is situated a distance [r_i] from the absorbing atom. The back-scattering amplitude from this shell is [f_i(k)] for which the associated phase is [\varphi_i(k)]. Deviations due to thermal motions of the electrons are incorporated through a Debye–Waller factor, [\exp(-\sigma^2_ik^2)], and ρ is the mean free path of the electron.

The amplitude function [f_i(k)] depends only on the type of back-scattering atom. The phase, however, contains contributions from both the absorber and the back-scatterer: [\varphi^l_i(k)=\varphi^l_j(k)+\varphi_i(k)-l\pi,\eqno (4.2.3.11)]where l = 1 for K and [L_{\rm I}] edges, and l = 2 or 0 for [L_{\rm II}] and [L_{\rm III}] edges. The phase is sensitive to variations in the energy threshold, the magnitude of the effect being larger for small electron energies than for electrons with considerable kinetic energy, i.e. the effect is more marked in the neighbourhood of the absorption edge. Since the position of the edge varies somewhat for different compounds (Azaroff & Pease, 1974[link]), some impediment to the analysis of experimental data might occur, since the determination of the interatomic distance [r_i] depends upon the precise knowledge of the value of [\varphi_i(k)].

In fitting the experimental data based on an empirical value of threshold energy using theoretically determined phase shifts, the difference between the theoretical and the experimental threshold energies [\Delta E_0] cannot produce a good fit at an arbitrarily chosen distance [r_i], since the effect will be seen primarily at low k values [(\sim0.3r\Delta E_0/k)], whereas changing [r_i] affects [\varphi_i(k)] at high k values [(\sim2k\Delta r)]. This was first demonstrated by Lee & Beni (1977[link]).

The significance of the Debye–Waller factor [\exp(-\sigma^2_ik^2)] should not be underestimated in this type of investigation. In XAFS studies, one is seeking to determine information regarding such properties of the system as nearest- and next-nearest-neighbour distances and the number of nearest and next-nearest neighbours. The theory is a short-range-order theory, hence deviations of atoms from their expected positions will influence the analysis significantly. Thus, it is often of value, experimentally, to work at liquid-nitrogen temperatures to reduce the effect of atomic vibrations.

Two distinct types of disorder are observed: vibrational, where the atom vibrates about a mean position in the structure, and static, where the atom occupies a position not expected theoretically. These terms can be separated from one another if the variation of XAFS spectra with temperature is studied, because the two have different temperature dependences. A discussion of the effect of a thermally activated disorder that is large compared with the static order has been given by Sevillano, Meuth & Rehr (1978[link]). For systems with large static disorders, e.g. liquids and amorphous solids, equation (4.2.3.10)[link] has to be modified somewhat. The XAFS equation has to be averaged over the pair distribution function g(r) for the system: [\chi(k)={ F(k)\over k}\int\limits^\propto_0g(r)\exp(-2r/\rho){ \sin(2kr+\varphi_k)\over r^2}{\,\rm d}r. \eqno (4.2.3.12)]

Other factors that must be taken into account in XAFS analyses include: inelastic scattering (due to multiple scattering in the absorbing atom and excitations of the atoms surrounding the atom from which the photoelectron was ejected) and multiple scattering of the photoelectron. Should multiple scattering be significant, the simple model given in equation (4.2.3.10)[link] is inappropriate, and more complex models such as those proposed by Pendry (1983[link]), Durham (1983[link]), Gurman (1988[link], 1995[link]), Natoli (1990[link]), and Rehr & Albers (1990[link]) should be used. Several computer programs are now available commercially for use in personal computers (EXCURVE, FEFF5, MSCALC). Readers are referred to scientific journals to find how best to contact the suppliers of these programs.

4.2.3.4.1.2. Techniques of data analysis

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Three assumptions must be made if XAFS data are to be used to provide accurate structural and chemical information:

  • (i) XAFS occurs through the interaction of waves singly scattered by neighbouring atoms;

  • (ii) the amplitude function of the atoms is insensitive to the type of chemical bond (the postulate of transferability), which implies that one can use the same amplitude function for a given atom in problems involving compounds of that atom, whatever the nature of its neighbours or the nature of the bond; and

  • (iii) the phase function can be transferred for each pair of absorber–back-scatterer atoms.

Of these three assumptions, (ii) is of the most questionable validity. See, for example, Stern, Bunker & Heald (1981[link]).

It is usual, when analysing XAFS data, to search the literature for, or make sufficient measurements of, [\mu_{l0}] remote from the absorption edge to produce a curve of [\mu_{l0}(E)] versus E that can be extrapolated to the position of the edge. From equation (4.2.3.8)[link], it is possible to produce a curve of [\chi(E)] versus E from which the variation of [\chi(k)] with k can be deduced using equation (4.2.3.9)[link].

It is also customary to multiply [\chi(k)] by some power of k to compensate for the damping of the XAFS amplitudes with increasing k. The power chosen is somewhat arbitrary but [k^3] is a commonly used weighting function.

Two different techniques may be used to analyse the new data set, the Fourier-transform technique or the curve-fitting technique.

In the Fourier-transform technique (FF), the Fourier transform of the [k^n\chi(k)] is determined for that region of momentum space from the smallest, [k_1], to the largest, [k_2], wavevectors of the photoelectron, yielding the radical distribution function [\rho_n(r')] in coordinate [(r')] space. [\rho_n(r')={ 1\over(2\pi)^{1/2}}\int\limits^{k_2}_{k_1}k^n\chi(k)\exp(i2kr'){\, \rm d}k. \eqno (4.2.3.13)]

The Fourier spectrum contains peaks indicating that the nearest-neighbour, next-nearest-neighbour, etc. distances will differ from the true spacings by between 0.2 to 0.5 Å depending on the elements involved. These position shifts are determined for model systems and then transferred to the unknown systems to predict interatomic spacings. Fig. 4.2.3.4[link] illustrates the various steps in the Fourier-transform analysis of XAFS data.

[Figure 4.2.3.4]

Figure 4.2.3.4 | top | pdf |

Steps in the reduction of data from an XAFS experiment using the Fourier transform technique: (a) after the removal of background χ(k) versus k; (b) after multiplication by a weighting function (in this case k3); (c) after Fourier transformation to determine r′.

The technique works best for systems having well separated peaks. Its primary weakness as a technique lies in the fact that the phase functions are not linear functions of k, and the spacing shift will depend on [E_0], the other factors including the weighting of data before the Fourier transforms are made, the range of k space transformed, and the Debye–Waller factors of the system.

In the curve-fitting technique (CF), least-squares refinement is used to fit the spectra in k space using some structural model for the system. Such techniques, however, can only indicate which of several possible choices is more likely to be correct, and do not prove that that structure is the correct structure.

It is possible to combine the FF and CF techniques to simplify the data analysis. Also, for data containing single-scatter peaks, the phase and amplitude components can be separated and analysed separately using either theory or model compounds (Stern, Sayers & Lytle, 1975[link]).

Each XAFS data set depends on two sets of strongly correlated variables: [\{F(k),\sigma,\rho,N\}] and [\{\varphi(k),E_0,r\}]. The elements of each set are not independent of one another. To determine N and σ, one must know F(k) well. To determine r, [\varphi(k)] must be known accurately.

Attempts have been made by Teo & Lee (1979[link]) to calculate F(k) and [\varphi(k)] from first principles using an electron–atom scattering model. Parametrized versions have been given by Teo, Lee, Simons, Eisenberger & Kincaid (1977[link]) and Lee et al. (1981[link]). Claimed accuracies for r, σ, and N in XAFS determinations are 0.5, 10, and 20%, respectively.

Acceptable methods for data analysis must conform to a number of basic criteria to have any validity. Amongst these are the following:

  • (i) the data analysis must not give rise to systematic error in the sense that it must provide unbiased estimates of parameters;

  • (ii) the assumed (hypothetical) model must be able to describe the data adequately;

  • (iii) the number of parameters used to describe the best fit of data must not exceed the number of independent data points;

  • (iv) where multiple solutions exist, supplementary information or assumptions used to resolve the ambiguity must conform to the philosophy of choice of the model structure.

The techniques for estimation of the parameters must always be given, including all known sources of uncertainty.

A complete list of criteria for the correct analysis and presentation of XAFS data is given in the reports of the International Workshops on Standards and Criteria in XAFS (Lytle, Sayers & Stern, 1989[link]; Bunker, Hasnain & Sayers, 1990[link]).

4.2.3.4.1.3. XAFS experiments

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The variety and number of experiments in which XAFS experiments have been used is so large that it is not possible here to give a comprehensive list. By consulting the papers given in such texts as those edited by Winick & Doniach (1980[link]), Teo & Joy (1981[link]), Bianconi, Incoccia & Stipcich (1983[link]), Mustre de Leon et al. (1988[link]), Hasnain (1990[link]), and Kuroda et al. (1992[link]), the reader may find references to a wide variety of experiments in fields of research ranging from archaeology to zoology.

In crystallography, XAFS experiments have been used to assist in the solution of crystal structures; the large variations in the atomic scattering factors can be used to help solve the phase problem. Helliwell (1984[link]) reviewed the use of these techniques in protein crystallography. A further discussion of the use of these anomalous-dispersion techniques in crystallography has been given by Creagh (1987b[link]). The relation that exists between the attenuation (related to the imaginary part of the dispersion correction, f′′) and intensity (related to the atomic form factor and the real part of the dispersion correction, f′) is discussed by Creagh in Section 4.2.6[link]. Specifically, modulations occur in the observed diffracted intensities from a specimen as the incident photon energy is scanned through the absorption edge of an atomic species present in the specimen. This technique, referred to as diffraction anomalous fine structure (DAFS) is complementary to XAFS. Because of the dependence of intensity on the geometrical structure factor, and the fact that the structure factor itself depends on the positional coordinates of the absorbing atom, it is possible to discriminate, in some favourable cases, between the anomalous scattering between atoms occupying different sites in the unit cell (Sorenson et al., 1994[link]).

In many systems of biological interest, the arrangement of radicals surrounding an active site must be found in order that the role of that site in biochemical processes may be assessed. A study of the XAFS spectrum of the active atom yields structural information that is specific to that site. Normal crystallographic techniques yield more general information concerning the crystal structure. An example of the use of XAFS in biological systems is the study of iron–sulfur proteins undertaken by Shulman, Weisenberger, Teo, Kincaid & Brown (1978[link]). Other, more recent, studies of biological systems include the characterization of the Mn site in the photosynthetic oxygen evolving complexes including hydroxylamine and hydroquinone (Riggs, Mei, Yocum & Penner-Hahn, 1993[link]) and an XAFS study with an in situ electrochemical cell on manganese Schiff-base complexes as a model of a photosystem (Yamaguchi et al., 1993[link]).

It must be stressed that the theoretical expression (equation 4.2.3.10[link]) does not take into account the state of polarization of the incident photon. Templeton & Templeton (1986[link]) have shown that polarization effects may be observed in some materials, e.g. sodium bromate. Given that most XAFS experiments are undertaken using the highly polarized radiation from synchrotron-radiation sources, it is of some importance to be aware of the possibility that dichroic effects may occur in some specimens.

Because XAFS is a short-range-order phenomenon, it is particularly useful for the structural study of such disordered systems as liquid metals and amorphous solids. The analysis of such disordered systems can be complicated, particularly in those cases where excluded-volume effects occur. Techniques for analysis for these cases have been suggested by Crozier & Seary (1980[link]). Fuoss, Eisenberger, Warburton & Bienenstock (1981[link]) suggested a technique for the investigation of amorphous solids, which they call the differential anomalous X-ray scattering (DAS) technique. This method has some advantages when compared with conventional XAFS methods because it makes more effective use of low-k information, and it does not depend on a knowledge of either the electron phase shifts or the mean free paths.

Both the conventional XAFS and DAS techniques may be used for studies of surface effects and catalytic processes such as those investigated by Sinfelt, Via & Lytle (1980[link]), Hida et al. (1985[link]), and Caballero, Villain, Dexpert, Le Peltier & Lynch (1993[link]).

It must be stressed that in all the foregoing discussion it has been assumed that the detection of XAFS has been by measurement of the linear attenuation coefficient of the specimen. However, the process of photon absorption followed by the ejection of a photoelectron has as its consequence both X-ray fluorescence and surface XAFS (SEXAFS) and Auger electron emission. All of these techniques are extremely useful in the analysis of dilute systems.

SEXAFS techniques are extremely sensitive to surface conditions since the mean free path of electrons is only about 20 Å. Discussions of the use of SEXAFS techniques have been given by Citrin, Eisenberger & Hewitt (1978[link]) and Stohr, Denley & Perfettii (1978[link]). A major review of the topic is given in Lee et al. (1981[link]). SEXAFS has the capacity of sensing thin films deposited on the surface of substrates, and has applications in experiments involving epitaxic growth and absorption by catalysts.

Fluorescence techniques are important in those systems for which the absorption of the specimen under investigation contributes only very slightly to the total attenuation coefficient since it detects the fluorescence of the absorbing atom directly. Experiments by Hastings, Eisenberger, Lengeler & Perlman (1975[link]) and Marcus, Powers, Storm, Kincaid & Chance (1980[link]) proved the importance of this technique in analysing dilute alloy and biological specimens. Materlik, Bedzyk & Frahm (1984[link]) have demonstrated its use in determining the location of bromine atoms absorbed on single-crystal silicon substrates. Oyanagi, Matsushita, Tanoue, Ishiguro & Kohra (1985[link]) and Oyanagi, Takeda, Matsushita, Ishiguro & Sasaki (1986[link]) have also used fluorescence XAFS techniques for the characterization of very thin films. More recently, Oyanagi et al. (1987[link]) have applied the technique to the study of short-range order in high-temperature superconductors. Oyanagi, Martini, Saito & Haga (1995[link]) have studied in detail the performance of a 19-element high-purity Ge solid-state detector array for fluorescence X-ray absorption fine structure studies.

A less-sensitive technique, but one that can be usefully employed for thin-film studies, is that in which XAFS modulations are detected in the beam reflected from a sample surface. This technique, ReflEXAFS, has been used by Martens & Rabe (1980[link]) to investigate superficial regions of copper oxide films by means of reflection of the X-rays close to the critical angle for total reflection.

If a thin film is examined in a transmission electron microscope, the electron beam loses some of its kinetic energy in interactions between the electron beam and the electrons within the film. If the resultant energy loss is analysed using a magnetic analyser, XAFS-like modulations are observed in the electron energy spectrum. These modulations, electron-energy-loss fine structure (EELS), which were first observed in a conventional transmission electron microscope by Leapman & Cosslett (1976[link]), are now used extensively for microanalyses of light elements incorporated into heavy-element matrices. Most major manufacturers of transmission electron microscopes supply electron-energy-loss spectrometers for their machines. There are more problems in analysing electron-energy-loss spectra than there are for XAFS spectra. Some of the difficulties encountered in producing reliable techniques for the routine analysis of EELS have been outlined by Joy & Maher (1985[link]). This matter is discussed more fully in §4.3.4.4.2[link] .

A more recent development has been the observation of topographic XAFS (Bowen, Stock, Davies, Pantos, Birnbaum & Chen, 1984[link]). This fine structure is observed in white-beam topographs taken using synchrotron-radiation sources. The technique provides the means of simultaneously determining spatially resolved microstructural and spectroscopic information for the specimen under investigation.

In all the preceding discussion, however, the electron was assumed to undergo only single-scattering processes. If multiple scattering occurs, then the theory has to be changed somewhat. §4.2.3.4.2[link] discusses the effect of multiple scattering.

4.2.3.4.2. X-ray absorption near edge structure (XANES)

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In Fig. 4.2.3.2(c)[link], there appears to be one cycle of strong oscillation in the neighbourhood of the absorption edge before the quasi-periodic variation of the XAFS commences. The electrons that cause this strong modulation of the photoelectric scattering cross section have low k values, and the electron is strongly scattered by neighbouring atoms. It was mentioned in §4.2.3.4.1[link] that conventional XAFS theory assumes a weak, single-scattering interaction between the ejected photoelectron and its environment. A schematic diagram illustrating the difference between single- and multiple-scattering processes is given in Fig. 4.2.3.5[link] . Evidently, the multiple-scattering process is very complicated and a discussion of the theory of XANES is too complex to be given here. The reader is directed to papers by Pendry (1983[link]), Lee (1981[link]), and Durham (1983[link]). A more recent review of the study of fine structure in ionization cross sections and their use in surface science has been given by Woodruff (1986[link]).

[Figure 4.2.3.5]

Figure 4.2.3.5 | top | pdf |

Schematic representations of the scattering processes undergone by the ejected photoelectron in the single-scattering (XAFS) case and the full multiple-scattering regime (XANES).

The data from XANES experiments can be analysed to determine structural information such as coordination geometry, the symmetry of unoccupied valence electronic states, and the effective charge on the absorbing atom (Natoli, Misemer, Doniach & Kutzler, 1980[link]; Kutzler, Natoli, Misemer, Doniach & Hodgson, 1981[link]). XANES experiments have been performed to resolve many problems, inter alia: the origin of white lines (Lengeler, Materlik & Müller, 1983[link]); absorption of gases on metal surfaces (Norman, Durham & Pendry, 1983[link]); the effect of local symmetry in 3d elements (Petiau & Calas, 1983[link]); and the determination of valence states in materials (Lereboures, Dürr, d'Huysser, Bonelle & Lenglet, 1980[link]).

4.2.3.5. Comments

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For reliable experiments using XAFS and XANES to be undertaken, intense-radiation sources must be used. Synchrotron-radiation sources are such a source of highly intense X-rays. Their ready availability to experimenters and the comparative simplicity of the equipment required to perform the experiments have made experiments involving XAFS and XANES very much easier to perform than has hitherto been the case.

At some synchrotron-radiation sources, database and program libraries for the storage and analysis of XAFS and XANES data exist. These are usually part of the general computing facilities (Pantos, 1982[link]).

Crystallographers seeking information concerning the nature and extent of these computer facilities can find such information by contacting the computer centre at one of the synchrotron-radiation establishments listed in Table 4.2.3.1[link].

Table 4.2.3.1| top | pdf |
Some synchrotron-radiation facilities providing XAFS databases and analysis utilities

CountrySynchrotron sourceAddress
France LURE Université Paris-Sud, LURE, 91405 Orsay, France
Italy Frascati Laboratori Nationali di Frascati, CP 13, 00044 Frascati, Italy
Japan Photon Factory Photon Factory, National Laboratory for High Energy Physics, 1-1 Oho, Tsukuba-gun, Ibaraki 305, Japan
Germany DESY DESY, Notkestrasse 85, 2000 Hamburg 52, Germany
United Kingdom SRC/Daresbury Daresbury Laboratory, Daresbury, Warrington WA4 4AD, England
USA CHESS CHESS, Cornell University, Ithaca, New York 14853, USA
NSLS NSLS, Brookhaven National Laboratory, Upton, New York 11973, USA
SPEAR SSRL, Stanford University, Bin 69, PO Box 4349, Stanford, California 94305, USA

4.2.4. X-ray absorption (or attenuation) coefficients

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D. C. Creaghb and J. H. Hubbelld

4.2.4.1. Introduction

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This data set is intended to supersede those data sets given in International Tables for X-ray Crystallography, Vols. III (Koch, MacGillavry & Milledge, 1962[link]) and IV (Hubbell, McMaster, Del Grande & Mallett, 1974[link]).

It is not intended here to give a detailed bibliography of experimental data that have been obtained in the past 90 years. This has been the subject of a number of publications, e.g. Saloman & Hubbell (1987[link]), Hubbell, Gerstenberg & Saloman (1986[link]), Saloman & Hubbell (1986[link]), and Saloman, Hubbell & Scofield (1988[link]). Further commentary on the validity and the quality of the experimental data in existing tabulations has been given by Creagh & Hubbell (1987[link]) and Creagh (1987a[link]).

Existing tabulations of X-ray attenuation (or absorption) cross sections fall into three distinct categories: purely theoretical, purely experimental, and an evaluated mixture of theoretical and experimental data.

Compilations of the purely theoretically derived data exist for:

  • photo-effect absorption cross sections (Storm & Israel, 1970[link]; Cromer & Liberman, 1970[link]; Scofield, 1973[link]; Hubbell, Veigele, Briggs, Brown, Cromer & Howerton, 1975[link]; Band, Kharitonov & Trzhaskovskaya, 1979[link]; Yeh & Lindau, 1985[link]);

  • Compton scattering cross sections (Hubbell et al., 1975[link]);

  • Rayleigh scattering cross sections (Hubbell et al., 1975[link]; Hubbell & Øverbø, 1979[link]; Schaupp, Schumacher, Smend, Rullhusen & Hubbell, 1983[link]).

Many purely experimental compilations exist, and the cross-section data given in computer programs used in the analysis of results in X-ray-fluorescence spectroscopy, electron-probe microanalysis, and X-ray diffraction are usually (evaluated) compilations of several of the following compilations: Allen (1935[link], 1969[link]), Victoreen (1949[link]), Liebhafsky, Pfeiffer, Winslow & Zemany (1960[link]), Koch et al. (1962[link]), Heinrich (1966[link]), Theisen & Vollath (1967[link]), Veigele (1973[link]), Leroux & Thinh (1977[link]), Montenegro, Baptista & Duarte (1978[link]), and Plechaty, Cullen & Howerton (1981[link]). If a comparison is made between these data sets, significant discrepancies are found, and questions must be asked concerning the reliability of the data sets that are compared. Jackson & Hawkes (1981[link]) and Gerward (1986[link]) have produced sets of parametric tables to simplify the application of X-ray attenuation data for the solution of problems in computer-aided tomography and X-ray-fluorescence analysis.

Compilations by Henke, Lee, Tanaka, Shimambukuro & Fujikawa (1982[link]) and the earlier tables of McMaster, Del Grande, Mallett & Hubbell (1969/1970[link]) are examples of the judicious application of both theoretical and experimental data to produce a comprehensive data set of X-ray interaction cross sections.

Because of the discrepancies that appear to exist between experimental data sets, the IUCr Commission on Crystallographic Apparatus set up a project to establish which, if any, of the existing methods for measuring X-ray interaction cross sections (X-ray attenuation coefficients) and which theoretical calculations could be considered to be the most reliable. A discussion of some of the major results of this project is given in Section 4.2.3[link]. A more detailed description of this project has been given by Creagh & Hubbell (1987[link], 1990[link]).

In this section, tabulations of the total X-ray interaction cross sections σ and the mass absorption coefficient [\mu_m] are given for a range of characteristic X-ray wavelengths [Ti Kα 2.7440 Å (or 4.509 keV) to Ag Kβ 0.4470 Å (or 24.942 keV)]. The interaction cross sections are expressed in units of barns/atom (1 barn = 10−28 m2) whilst the mass absorption coefficient is given in cm2 g−1. Table 4.2.4.1[link] sets out the wavelengths of the characteristic wavelengths used in Tables 4.2.4.2[link] and 4.2.4.3[link], which list values of σ and [\mu_m], respectively.

Table 4.2.4.1| top | pdf |
Table of wavelengths and energies for the characteristic radiations used in Tables 4.2.4.2[link] and 4.2.4.3[link]

Radiationλ (Å)E (keV)
Ag [K \bar \alpha] 0.5608 22.103
Ag [K \beta _1] 0.4970 24.942
Pd [K \bar \alpha] 0.5869 21.125
Pd [K \beta _1] 0.5205 23.819
Rh [K \bar \alpha] 0.6147 20.169
Rh [K \beta _1] 0.5456 22.724
Mo [K \bar \alpha] 0.7107 17.444
Mo [K \beta _1] 0.6323 19.608
Zn [K \bar \alpha] 1.4364 8.631
Zn [K \beta _1] 1.2952 9.572
Cu [K \bar \alpha] 1.5418 8.041
Cu [K \beta _1] 1.3922 8.905
Ni [K \bar \alpha] 1.6591 7.472
Ni [K \beta _1] 1.5001 8.265
Co [K \bar \alpha] 1.7905 6.925
Co [K \beta _1] 1.6208 7.629
Fe [K \bar \alpha] 1.9373 6.400
Fe [K \beta _1] 1.7565 7.038
Mn [K \bar \alpha] 2.1031 5.895
Mn [K \beta _1] 1.9102 6.490
Cr [K \bar \alpha] 2.2909 5.412
Cr [K \beta _1] 2.0848 5.947
Ti [K \bar \alpha] 2.7496 4.509
Ti [K \beta _1] 2.5138 4.932

Table 4.2.4.2| top | pdf |
Total phonon interaction cross section (barns/atom)

RadiationEnergy (MeV)12345678
HydrogenHeliumLithiumBerylliumBoronCarbonNitrogenOxygen
Ag K[\beta _1] 2.494E−02 6.10E−01 1.26E+00 2.01E+00 2.97E+00 4.40E+00 6.59E+00 1.00E+01 1.52E+01
Pd K[\beta _1] 2.382E−02 6.10E−01 1.26E+00 2.01E+00 2.99E+00 4.44E+00 6.68E+00 1.02E+01 1.55E+01
Rh K[\beta _1] 2.272E−02 6.12E−01 1.27E+00 2.04E+00 3.06E+00 4.47E+00 6.78E+00 1.05E+01 1.62E+01
Ag K[\bar {\alpha }] 2.210E−02 6.14E−01 1.28E+00 2.06E+00 3.13E+00 4.79E+00 7.45E+00 1.17E+01 1.82E+01
Pd K[\bar {\alpha }] 2.112E−02 6.16E−01 1.29E+00 2.09E+00 3.23E+00 5.05E+00 8.02E+00 1.28E+01 2.02E+01
Rh K[\bar {\alpha }] 2.017E−02 6.18E−01 1.30E+00 2.13E+00 3.35E+00 5.35E+00 8.68E+00 1.41E+01 2.25E+01
Mo K[\beta _1] 1.961E−02 6.19E−01 1.31E+00 2.16E+00 3.42E+00 5.56E+00 9.14E+00 1.50E+01 2.41E+01
Mo K[\bar {\alpha }] 1.744E−02 6.24E−01 1.34E+00 2.28E+00 3.83E+00 6.61E+00 1.15E+01 1.96E+01 3.25E+01
Zn K[\beta _1] 9.572E−03 6.47E−01 1.69E+00 4.19E+00 1.07E+01 2.54E+01 5.37E+01 1.03E+02 1.80E+02
Cu K[\beta _1] 8.905E−03 6.50E−01 1.78E+00 4.74E+00 1.28E+01 3.10E+01 6.64E+01 1.27E+02 2.24E+02
Zn K[\bar {\alpha }] 8.631E−03 6.51E−01 1.82E+00 5.02E+00 1.38E+01 3.39E+01 7.28E+01 1.40E+02 2.46E+02
Ni K[\beta _1] 8.265E−03 6.53E−01 1.89E+00 5.46E+00 1.54E+01 3.83E+01 8.28E+01 1.59E+02 2.80E+02
Cu K[\bar {\alpha}] 8.041E−03 6.55E−01 1.94E+00 5.76E+00 1.66E+01 4.15E+01 8.99E+01 1.73E+02 3.04E+02
Co K[\beta _1] 7.649E−03 6.58E−01 2.04E+00 6.40E+00 1.90E+01 4.80E+01 1.04E+02 2.01E+02 3.54E+02
Ni K[\bar {\alpha}] 7.472E−03 6.59E−01 2.09E+00 6.73E+00 2.02E+01 5.14E+01 1.12E+02 2.16E+02 3.80E+02
Fe K[\beta _1] 7.058E−03 6.63E−01 2.23E+00 7.65E+00 2.37E+01 6.09E+01 1.33E+02 2.57E+02 4.51E+02
Co K[\bar {\alpha}] 6.925E−03 6.64E−01 2.28E+00 7.99E+00 2.50E+01 6.45E+01 1.41E+02 2.72E+02 4.78E+02
Mn K[\beta _1] 6.490E−03 6.69E−01 2.48E+00 9.34E+00 3.01E+01 7.84E+01 1.72E+02 3.31E+02 5.81E+02
Fe K[\bar {\alpha}] 6.400E−03 6.70E−01 2.53E+00 9.67E+00 3.13E+01 8.18E+01 1.79E+02 3.46E+02 6.06E+02
Cr K[\beta _1] 5.947E−03 6.77E−01 2.83E+00 1.16E+01 3.88E+01 1.02E+02 2.24E+02 4.32E+02 7.56E+02
Mn K[\bar {\alpha}] 5.895E−03 6.78E−01 2.87E+00 1.19E+01 3.99E+01 1.05E+02 2.30E+02 4.44E+02 7.76E+02
Cr K[\bar {\alpha}] 5.412E−03 6.89E−01 3.31E+00 1.50E+01 5.14E+01 1.36E+02 2.99E+02 5.75E+02 1.00E+03
Ti K[\beta _1] 4.932E−03 7.04E−01 3.94E+00 1.94E+01 6.82E+01 1.81E+02 3.98E+02 7.62E+02 1.33E+03
Ti K[\bar {\alpha}] 4.509E−03 7.24E−01 4.73E+00 2.51E+01 8.97E+01 2.39E+02 5.23E+02 1.00E+03 1.73E+03

RadiationEnergy (MeV)910111213141516
FluorineNeonSodiumMagnesiumAluminiumSiliconPhosphorusSulfur
Ag K[\beta _1] 2.494E−02 2.27E+01 3.33E+01 4.77E+01 6.68E+01 9.16E+01 1.23E+02 1.62E+02 2.10E+02
Pd K[\beta _1] 2.382E−02 2.32E+01 3.40E+01 4.88E+01 6.85E+01 9.40E+01 1.26E+02 1.67E+02 2.16E+02
Rh K[\beta _1] 2.272E−02 2.50E+01 3.62E+01 5.07E+01 8.26E+01 1.05E+02 1.40E+02 1.85E+02 2.02E+02
Ag K[\bar {\alpha }] 2.210E−02 2.77E+01 4.12E+01 5.96E+01 8.42E+01 1.16E+02 1.56E+02 2.06E+02 2.67E+02
Pd K[\bar {\alpha }] 2.112E−02 3.11E+01 4.65E+01 6.75E+01 9.55E+01 1.32E+02 1.78E+02 2.35E+02 3.05E+02
Rh K[\bar {\alpha }] 2.017E−02 3.50E+01 5.26E+01 7.67E+01 1.09E+02 1.51E+02 2.03E+02 2.69E+02 3.49E+02
Mo K[\beta _1] 1.961E−02 3.76E+01 5.68E+01 5.30E+01 1.18E+02 1.63E+02 2.20E+02 2.92E+02 3.78E+02
Mo K[\bar {\alpha }] 1.744E−02 5.15E+01 7.86E+01 1.16E+02 1.65E+02 2.29E+02 3.10E+02 4.10E+02 5.32E+02
Zn K[\beta _1] 9.572E−03 2.95E+02 4.57E+02 6.77E+02 9.67E+02 1.34E+03 1.79E+03 2.36E+03 3.03E+03
Cu K[\beta _1] 8.905E−03 3.66E+02 5.67E+02 8.39E+02 1.20E+03 1.65E+03 2.21E+03 2.90E+03 3.72E+03
Zn K[\bar {\alpha }] 8.631E−03 4.02E+02 6.22E+02 9.20E+02 1.31E+03 1.81E+03 2.42E+03 3.17E+03 4.06E+03
Ni K[\beta _1] 8.265E−03 4.58E+02 7.08E+02 1.05E+03 1.49E+03 2.05E+03 2.75E+03 3.59E+03 4.60E+03
Cu K[\bar {\alpha }] 8.041E−03 4.98E+02 7.68E+02 1.14E+03 1.61E+03 2.22E+03 2.97E+03 3.88E+03 4.97E+03
Co K