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

International Tables for Crystallography (2006). Vol. C, ch. 4.4, pp. 430-487
https://doi.org/10.1107/97809553602060000594

Chapter 4.4. Neutron techniques

I. S. Anderson,a P. J. Brown,a J. M. Carpenter,b G. Lander,c R. Pynn,d J. M. Rowe,e O. Schärpf,f V. F. Searsg and B. T. M. Willish

aInstitut Laue–Langevin, Avenue des Martyrs, BP 156X, F-38042 Grenoble CEDEX, France,bIntense Pulsed Neutron Source, Building 360, Argonne National Laboratory, Argonne, IL 60439, USA,cITU, European Commission, Postfach 2340, D-76125 Karlsruhe, Germany,dLANSCE, MS H805, Los Alamos National Laboratory, PO Box 1663, Los Alamos, NM 87545, USA,eNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA,fPhysik-Department E13, TU München, James-Franck-Strasse 1, D-85748 Garching, Germany,gAtomic Energy of Canada Limited, Chalk River Laboratories, Chalk River, Ontario, Canada K0J 1J0, and hChemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England

The first section of this chapter briefly describes the fission and spallation mechanisms for producing neutrons. The general properties of the slow neutron spectra that they produce are discussed and the methods of using beams from steady (fission reactor) sources and pulsed (accelerator-driven) sources are compared. The second section provides an overview of the different kinds of beam-definition devices. Resolution functions are then described in the third section. Scattering lengths for neutrons are tabulated in the fourth section, and scattering and absorption cross sections, isotope effects and correction for electromagnetic interactions are discussed. The fifth section gives tables of the coefficients in analytic approximations to the form factors used in the calculation of the cross sections for magnetic scattering of neutrons. The coefficients for atoms and ions in the 3d and 4d transition series are derived from wavefunctions obtained using Hartree–Fock theory and the coefficients for the rare-earth and actinide ions are obtained by fitting the analytic forms to published form factors calculated from Dirac–Fock wavefunctions. In the final section, the absorption cross sections and 1/e penetration depths of the elements are tabulated for neutrons of 1.8 Å wavelength.

4.4.1. Production of neutrons

| top | pdf |
J. M. Carpenterb and G. Landerc

The production of neutrons of sufficient intensity for scattering experiments is a `big-machine' operation; there is no analogue to the small laboratory X-ray unit. The most common sources of neutrons, and those responsible for the great bulk of today's successful neutron scattering programs, are the nuclear reactors. These are based on the continuous, self-sustaining fission reaction. Research-reactor design emphasizes power density, that is the highest power within a small `leaky' volume, whereas power reactors generate large amounts of power over a large core volume. In research reactors, fuel rods are of highly enriched 235U. Neutrons produced are distributed in a fission spectrum centred about 1 MeV: Most of the neutrons within the reactor are moderated (i.e. slowed down) by collisions in the cooling liquid, normally D2O or H2O, and are absorbed in fuel to propagate the reaction. As large a fraction as possible is allowed to leak out as fast neutrons into the surrounding moderator (D2O and Be are best) and to slow down to equilibrium with this moderator. The neutron spectrum is Maxwellian with a mean energy of ∼300 K (= 25 meV), which for neutrons corresponds to 1.8 Å since [E_n \ ({\rm meV})=81.8/\lambda^2\ ({\rm \AA}^2).]Neutrons are extracted in beams through holes that penetrate the moderator.

There are two points to remember: (a) neutrons are neutral so that we cannot focus the beams and (b) the spectrum is broad and continuous; there is no analogy to the characteristic wavelength found with X-ray tubes, or to the high directionality of synchrotron-radiation sources.

Neutron production and versatility in reactors reached a new level with the construction of the High-Flux Reactor at the French–German–English Institut Laue–Langevin (ILL) in Grenoble, France. An overview of the reactor and beam-tube assembly is shown in Fig. 4.4.1.1[link] . To shift the spectrum in energy, both a cold source (25 l of liquid deuterium at 25 K) and a hot source (graphite at 2400 K) have been inserted into the D2O moderator. Special beam tubes view these sources allowing a range of wavelengths from ∼0.3 to ∼17 Å to be used. Over 30 instruments are in operation at the ILL, which started in 1972.

[Figure 4.4.1.1]

Figure 4.4.1.1 | top | pdf |

A plane view of the installation at the Institut Laue–Langevin, Grenoble. Note especially the guide tubes exiting from the reactor that transport the neutron beams to a variety of instruments; these guide tubes are made of nickel-coated glass from which the neutrons are totally internally reflected.

The second method of producing neutrons, which historically predates the discovery of fission, is with charged particles (α particles, protons, etc.) striking a target nuclei. The most powerful source of neutrons of this type uses proton beams. These are accelerated in short bursts ([\lt] 1 µs) to 500–1000 MeV, and after striking the target produce an instantaneous supply of high-energy `evaporation' neutrons. These extend up in energy close to that of the incident proton beam. Shielding for spallation sources tends to be even more massive than that for reactors. The targets, usually tungsten or uranium and typically much smaller than a reactor core, are surrounded by hydrogenous moderators such as polyethylene (often at different temperatures) to produce the `slow' neutrons (En < 10 eV) used in scattering experiments. The moderators are very different from those of reactors; they are designed to slow down neutrons rapidly and to let them leak out, rather than to store them for a long time. If the accelerated particle pulse is short enough, the duration of the moderated neutron pulses is roughly inversely proportional to the neutron speed.

These accelerator-driven pulsed sources are pulsed at frequencies of between 10 and 100 Hz.

There are two fundamental differences between a reactor and a pulsed source.

  • (1) All experiments at a pulsed source must be performed with time-of-flight techniques. The pulsed source produces neutrons in bursts of 1 to 50 µs duration, depending on the energy, spaced about 10 to 100 ms apart, so that the duty cycle is low but there is very high neutron intensity within each pulse. The time-of-flight technique makes it possible to exploit that high intensity. With the de Broglie relationship, for neutrons [\lambda\, ({\rm \AA})=0.3966t\,({\mu}{\rm s})/L\,({\rm cm}),]where t is the flight time in µs and L is the total flight path in cm.

  • (2) The spectral characteristics of pulsed sources are somewhat different from reactors in that they have a much larger component of higher-energy (above 100 meV) neutrons than the thermal spectrum at reactors. The exploitation of this new energy regime accompanied by the short pulse duration is one of the great opportunities presented by spallation sources.

Fig. 4.4.1.2[link] illustrates the essential difference between experiments at a steady-state source (left panel) and a pulsed source (right panel). We confine the discussion here to diffraction. If the time over which useful information is gathered is equivalent to the full period of the source Δt (the case suggested by the lower-right figure), the peak flux of the pulsed source is the effective parameter to compare with the flux of the steady-state source. Often this is not the case, so one makes a comparison in terms of time-averaged flux (centre panel). For the pulsed source, this is lowered from the peak flux by the duty cycle, but with the time-of-flight method one uses a large interval of the spectrum (shaded area). For the steady-state source, the time-averaged flux is high, but only a small wavelength slice (stippled area) is used in the experiment. It is the integrals of the two areas which must be compared; for the pulsed sources now being designed, the integral is generally favourable compared with present-day reactors. Finally, one can see from the central panel that high-energy neutrons (100–1000 meV) are especially plentiful at the pulsed sources. These various features can be exploited in the design of different kinds of experiments at pulsed sources.

[Figure 4.4.1.2]

Figure 4.4.1.2 | top | pdf |

Schematic diagram for performing diffraction experiments at steady-state and pulsed neutron sources. On the left we see the familiar monochromator crystal allowing a constant (in time) beam to fall on the sample (centre left), which then diffracts the beam through an angle 2θs into the detector. The signal in the latter is also constant in time (lower left). On the right, the pulsed source allows a wide spectrum of neutrons to fall on the sample in sharp pulses separated by Δt (centre right). The neutrons are then diffracted by the sample through 2θs and their time of arrival in the detector is analysed (lower right). The centre figure shows the time-averaged flux at the source. At a reactor, we make use of a narrow band of neutrons (heavy shading), here chosen with λ = 1.5 Å. At a pulsed source, we use a wide spectral band, here chosen from 0.4 to 3 Å and each one is identified by its time-of-flight. For the experimentalist, an important parameter is the integrated area of the two-shaded areas. Here they have been made identical.

4.4.2. Beam-definition devices

| top | pdf |
I. S. Andersona and O. Schärpff

4.4.2.1. Introduction

| top | pdf |

Neutron scattering, when compared with X-ray scattering techniques developed on modern synchrotron sources, is flux limited, but the method remains unique in the resolution and range of energy and momentum space that can be covered. Furthermore, the neutron magnetic moment allows details of microscopic magnetism to be examined, and polarized neutrons can be exploited through their interaction with both nuclear and electron spins.

Owing to the low primary flux of neutrons, the beam definition devices that play the role of defining the beam conditions (direction, divergence, energy, polarization, etc.) have to be highly efficient. Progress in the development of such devices not only results in higher-intensity beams but also allows new techniques to be implemented.

The following sections give a (non-exhaustive) review of commonly used beam-definition devices. The reader should keep in mind the fact that neutron scattering experiments are typically carried out with large beams (1 to 50 cm2) and divergences between 5 and 30 mrad.

4.4.2.2. Collimators

| top | pdf |

A collimator is perhaps the simplest neutron optical device and is used to define the direction and divergence of a neutron beam. The most rudimentary collimator consists of two slits or pinholes cut into an absorbing material and placed one at the beginning and one at the end of a collimating distance L. The maximum beam divergence that is transmitted with this configuration is [\alpha _{\rm max }=(a_{1}+a_{2})/L, \eqno (4.4.2.1)]where [a_{1}] and [a_{2}] are the widths of the slits or pinholes.

Such a device is normally used for small-angle scattering and reflectometry. In order to avoid parasitic scattering by reflection from slit edges, very thin sheets of a highly absorbing material, e.g. gadolinium foils, are used as the slit material. Sometimes wedge-formed cadmium plates are sufficient. In cases where a very precise edge is required, cleaved single-crystalline absorbers such as gallium gadolinium garnet (GGG) can be employed.

To avoid high intensity losses when the distances are large, sections of neutron guide can be introduced between the collimators, as in, for example, small-angle scattering instruments with variable collimation. In this case, for maximum intensity at a given resolution (divergence), the collimator length should be equal to the camera length, i.e. the sample–detector distance (Schmatz, Springer, Schelten & Ibel, 1974[link]).

As can be seen from (4.4.2.1)[link], the beam divergence from a simple slit or pinhole collimator depends on the aperture size. In order to collimate (in one dimension) a beam of large cross section within a reasonable distance L, Soller collimators, composed of a number of equidistant neutron absorbing blades, are used. To avoid losses, the blades must be as thin and as flat as possible. If their surfaces do not reflect neutrons, which can be achieved by using blades with rough surfaces or materials with a negative scattering length, such as foils of hydrogen-containing polymers (Mylar is commonly used) or paper coated with neutron-absorbing paint containing boron or gadolinium (Meister & Weckerman, 1973[link]; Carlile, Hey & Mack, 1977[link]), the angular dependence of the transmission function is close to the ideal triangular form, and transmissions of 96% of the theoretical value can be obtained with 10′ collimation. If the blades of the Soller collimator are coated with a material whose critical angle of reflection is equal to [\alpha _{\rm max }/2] (for one particular wavelength), then a square angular transmission function is obtained instead of the normal triangular function, thus doubling the theoretical transmission (Meardon & Wroe, 1977[link]).

Soller collimators are often used in combination with single-crystal monochromators to define the wavelength resolution of an instrument but the Soller geometry is only useful for one-dimensional collimation. For small-angle scattering applications, where two-dimensional collimation is required, a converging `pepper pot' collimator can be used (Nunes, 1974[link]; Glinka, Rowe & LaRock, 1986[link]).

Cylindrical collimators with radial blades are sometimes used to reduce background scattering from the sample environment. This type of collimator is particularly useful with position-sensitive detectors and may be oscillated about the cylinder axis to reduce the shadowing effect of the blades (Wright, Berneron & Heathman, 1981[link]).

4.4.2.3. Crystal monochromators

| top | pdf |

Bragg reflection from crystals is the most widely used method for selecting a well defined wavelength band from a white neutron beam. In order to obtain reasonable reflected intensities and to match the typical neutron beam divergences, crystals that reflect over an angular range of 0.2 to 0.5° are typically employed. Traditionally, mosaic crystals have been used in preference to perfect crystals, although reflection from a mosaic crystal gives rise to an increase in beam divergence with a concomitant broadening of the selected wavelength band. Thus, collimators are often used together with mosaic monochromators to define the initial and final divergences and therefore the wavelength spread.

Because of the beam broadening produced by mosaic crystals, it was soon recognised that elastically deformed perfect crystals and crystals with gradients in lattice spacings would be more suitable candidates for focusing applications since the deformation can be modified to optimize focusing for different experimental conditions (Maier-Leibnitz, 1969[link]).

Perfect crystals are used commonly in high-energy-resolution backscattering instruments, interferometry and Bonse–Hart cameras for ultra-small-angle scattering (Bonse & Hart, 1965[link]).

An ideal mosaic crystal is assumed to comprise an agglomerate of independently scattering domains or mosaic blocks that are more or less perfect, but small enough that primary extinction does not come into play, and the intensity reflected by each block may be calculated using the kinematic theory (Zachariasen, 1945[link]; Sears, 1997[link]). The orientation of the mosaic blocks is distributed inside a finite angle, called the mosaic spread, following a distribution that is normally assumed to be Gaussian. The ideal neutron mosaic monochromator is not an ideal mosaic crystal but rather a mosaic crystal that is sufficiently thick to obtain a high reflectivity. As the crystal thickness increases, however, secondary extinction becomes important and must be accounted for in the calculation of the reflectivity. The model normally used is that developed by Bacon & Lowde (1948[link]), which takes into account strong secondary extinction and a correction factor for primary extinction (Freund, 1985[link]). In this case, the mosaic spread (usually defined by neutron scatterers as the full width at half maximum of the reflectivity curve) is not an intrinsic crystal property, but increases with wavelength and crystal thickness and can become quite appreciable at longer wavelengths.

Ideal monochromator materials should have a large scattering-length density, low absorption, incoherent and inelastic cross sections, and should be available as large single crystals with a suitable defect concentration. Relevant parameters for some typical neutron monochromator crystals are given in Table 4.4.2.1[link].

Table 4.4.2.1| top | pdf |
Some important properties of materials used for neutron monochromator crystals (in order of increasing unit-cell volume)

MaterialStructureLattice constant(s) at 300 K
a, c (Å)
Unit-cell volume
V0 (10−24 cm3)
Coherent scattering length
b (10−12 cm)
Square of scattering-length density
(10−21cm−4)
Ratio of incoherent to total scattering cross section
σincs
Absorption cross section
σabs (barns) (at λ = 1.8 Å)
Atomic mass ADebye temperature
[\theta_D] (K)
[A\theta^2_D] (106  K2)
Beryllium h.c.p. a : 2.2856
c : 3.5832
16.2 0.779 (1) 9.25 [6.5\times10^{-4}] 0.0076 (8) 9.013 1188 12.7
Iron b.c.c. a : 2.8664 23.5 0.954 (6) 6.59 0.033 2.56 (3) 55.85 411 9.4
Zinc h.c.p. a : 2.6649
c : 4.9468
30.4 0.5680 (5) 1.50 0.019 1.11 (2) 65.38 253 4.2
Pyrolytic graphite layer hexag. a : 2.461
c : 6.708
35.2 0.66484 (13) 5.71 [>2\times10^{-4}] 0.00350 (7) 12.01 800 7.7
Niobium b.c.c. 3.3006 35.9 0.7054 (3) 1.54 [4\times10^{-4}] 1.15 (5) 92.91 284 7.5
Nickel (58Ni) f.c.c. 3.5241 43.8 1.44 (1) 17.3 0 4.6 (3) 58.71 417 9.9
Copper f.c.c. 3.6147 47.2 0.7718 (4) 4.28 0.065 3.78 (2) 63.54 307 6.0
Aluminium f.c.c. 4.0495 66.4 0.3449 (5) 0.43 [5.6\times10^{-3}] 0.231 (3) 26.98 402 4.4
Lead f.c.c. 4.9502 121 0.94003 (14) 0.97 [2.7\times10^{-4}] 0.171 (2) 207.21 87 1.6
Silicon diamond 5.4309 160 0.41491 (10) 0.43 [6.9\times10^{-3}] 0.171 (3) 28.09 543 8.3
Germanium diamond 5.6575 181 0.81929 (7) 1.31 0.020 2.3 (2) 72.60 290 6.1
1 barn = 10−28 m2.

In principle, higher reflectivities can be obtained in neutron monochromators that are designed to operate in reflection geometry, but, because reflection crystals must be very large when takeoff angles are small, transmission geometry may be used. In that case, the optimization of crystal thickness can only be achieved for a small wavelength range.

Nickel has the highest scattering-length density, but, since natural nickel comprises several isotopes, the incoherent cross section is quite high. Thus, isotopic 58Ni crystals have been grown as neutron monochromators despite their expense. Beryllium, owing to its large scattering-length density and low incoherent and absorption cross sections, is also an excellent candidate for neutron monochromators, but the mosaic structure of beryllium is difficult to modify, and the availability of good-quality single crystals is limited (Mücklich & Petzow, 1993[link]). These limitations may be overcome in the near future, however, by building composite monochromators from thin beryllium blades that have been plastically deformed (May, Klimanek & Magerl, 1995[link]).

Pyrolytic graphite is a highly efficient neutron monochromator if only a medium resolution is required (the minimum mosaic spread is of the order of 0.4°), owing to high reflectivities, which may exceed 90% (Shapiro & Chesser, 1972[link]), but its use is limited to wavelengths above 1.5 Å, owing to the rather large d spacing of the 002 reflection. Whenever better resolution at shorter wavelengths is required, copper (220 and 200) or germanium (311 and 511) monochromators are frequently used. The advantage of copper is that the mosaic structure can be easily modified by plastic deformation at high temperature. As with most face-centred cubic crystals, it is the (111) slip planes that are functional in generating the dislocation density needed for the desired mosaic spread, and, depending on the required orientation, either isotropic or anisotropic mosaics can be produced (Freund, 1976[link]). The latter is interesting for vertical focusing applications, where a narrow vertical mosaic is required regardless of the resolution conditions.

Although both germanium and silicon are attractive as monochromators, owing to the absence of second-order neutrons for odd-index reflections, it is difficult to produce a controlled uniform mosaic spread in bulk samples by plastic deformation at high temperature because of the difficulty in introducing a spatially homogenous microstructure in large single crystals (Freund, 1975[link]). Recently this difficulty has been overcome by building up composite monochromators from a stack of thin wafers, as originally proposed by Maier-Leibnitz (1967[link]; Frey, 1974[link]).

In practice, an artificial mosaic monochromator can be built up in two ways. In the first approach, illustrated in Fig. 4.4.2.1(a)[link] , the monochromator comprises a stack of crystalline wafers, each of which has a mosaic spread close to the global value required for the entire stack. Each wafer in the stack must be plastically deformed (usually by alternated bending) to produce the correct mosaic spread. For certain crystal orientations, the plastic deformation may result in an anisotropic mosaic spread. This method has been developed in several laboratories to construct germanium monochromators (Vogt, Passell, Cheung & Axe, 1994[link]; Schefer et al., 1996[link]).

[Figure 4.4.2.1]

Figure 4.4.2.1 | top | pdf |

Two methods by which artificial mosaic monochromators can be constructed: (a) out of a stack of crystalline wafers, each with a mosaicity close to the global value. The increase in divergence due to the mosaicity is the same in the horizontal (left picture) and the vertical (right picture) directions; (b) out of several stacked thin crystalline wafers each with a rather narrow mosaic but slightly misoriented in a perfectly controlled way. This allows the shape of the reflectivity curve to be rectangular, Gaussian, Lorentzian, etc., and highly anisotropic, i.e. vertically narrow (right picture) and horizontally broad (left figure).

In the second approach, shown in Fig. 4.4.2.1(b)[link], the global reflectivity distribution is obtained from the contributions of several stacked thin crystalline wafers, each with a rather narrow mosaic spread compared with the composite value but slightly misoriented with respect to the other wafers in the stack. If the misorientation of each wafer can be correctly controlled, this technique has the major advantage of producing monochromators with a highly anisotropic mosaic structure. The shape of the reflectivity curve can be chosen at will (Gaussian, Lorentzian, rectangular), if required. Moreover, because the initial mosaicity required is small, it is not necessary to use mosaic wafers and therefore for each wafer to undergo a long and tedious plastic deformation process. Recently, this method has been applied successfully to construct copper monochromators (Hamelin, Anderson, Berneron, Escoffier, Foltyn & Hehn, 1997[link]), in which individual copper wafers were cut in a cylindrical form and then slid across one another to produce the required mosaic spread in the scattering plane. This technique looks very promising for the production of anisotropic mosaic monochromators.

The reflection from a mosaic crystal is visualized in Fig. 4.4.2.2(b)[link] . An incident beam with small divergence is transformed into a broad exit beam. The range of k vectors, Δk, selected in this process depends on the mosaic spread, η, and the incoming and outgoing beam divergences, [\alpha _{1}] and [\alpha _{2}.] [ \Delta k/k=\Delta\tau /\tau +\alpha \cot \theta, \eqno (4.4.2.2)]where τ is the magnitude of the crystal reciprocal-lattice vector (τ = 2π/d) and α is given by [\alpha =\sqrt {\;{\alpha {}\/_{1}^{2}\alpha {}\/_{2}^{2}\;+\;\alpha {}\/_{1}^{2}\eta ^{2}\;+\;\alpha {}\/_{2}^{2}\eta ^{2}\;}\over{\alpha {}\/_{1}^{2}\;+\;\alpha {}\/_{2}^{2}\;+\;4\eta ^{2}}}. \eqno (4.4.2.3)]The resolution can therefore be defined by collimators, and the highest resolution is obtained in backscattering, where the wavevector spread depends only on the intrinsic Δd/d of the crystal.

[Figure 4.4.2.2]

Figure 4.4.2.2 | top | pdf |

Reciprocal-lattice representation of the effect of a monochromator with reciprocal-lattice vector τ on the reciprocal-space element of a beam with divergence α. (a) For an ideal crystal with a lattice constant width Δτ; (b) for a mosaic crystal with mosaicity η, showing that a beam with small divergence, α, is transformed into a broad exit beam with divergence 2η + α; (c) for a gradient crystal with interplanar lattice spacing changing over Δτ, showing that the divergence is not changed in this case.

In some applications, the beam broadening produced by mosaic crystals can be detrimental to the instrument performance. An interesting alternative is a gradient crystal, i.e. a single crystal with a smooth variation of the interplanar lattice spacing along a defined crystallographic direction. As shown in Fig. 4.4.2.2(c)[link], the diffracted phase-space element has a different shape from that obtained from a mosaic crystal. Gradients in d spacing can be produced in various ways, including thermal gradients (Alefeld, 1972[link]), vibrating crystals by piezoelectric excitation (Hock, Vogt, Kulda, Mursic, Fuess & Magerl, 1993[link]), and mixed crystals with concentration gradients, e.g. Cu–Ge (Freund, Guinet, Maréschal, Rustichelli & Vanoni, 1972[link]) and Si–Ge (Maier-Leibnitz & Rustichelli, 1968[link]; Magerl, Liss, Doll, Madar & Steichele, 1994[link]).

Both vertically and horizontally focusing assemblies of mosaic crystals are employed to make better use of the neutron flux when making measurements on small samples. Vertical focusing can lead to intensity gain factors of between two and five without affecting resolution (real-space focusing) (Riste, 1970[link]; Currat, 1973[link]). Horizontal focusing changes the k-space volume that is selected by the monochromator through the variation in Bragg angle across the monochromator surface (k-space focusing) (Scherm, Dolling, Ritter, Schedler, Teuchert & Wagner, 1977[link]). The orientation of the diffracted k-space volume can be modified by variation of the horizontal curvature, so that the resolution of the monochromator may be optimized with respect to a particular sample or experiment without loss of illumination. Monochromatic focusing can be achieved. Furthermore, asymmetrically cut crystals may be used, allowing focusing effects in real space and k space to be decoupled (Scherm & Kruger, 1994[link]).

Traditionally, focusing monochromators consist of rectangular crystal plates mounted on an assembly that allows the orientation of each crystal to be varied in a correlated manner (Bührer, 1994[link]). More recently, elastically deformed perfect crystals (in particular silicon) have been exploited as focusing elements for monochromators and analysers (Magerl & Wagner, 1994[link]).

Since thermal neutrons have velocities that are of the order of km s−1, their wavelengths can be Doppler shifted by diffraction from moving crystals. The k-space representation of the diffraction from a crystal moving perpendicular to its lattice planes is shown in Fig. 4.4.2.3(a)[link] . This effect is most commonly used in backscattering instruments on steady-state sources to vary the energy of the incident beam. Crystal velocities of 9–10 m s−1 are practically achievable, corresponding to energy variations of the order of ±60 µeV.

[Figure 4.4.2.3]

Figure 4.4.2.3 | top | pdf |

Momentum-space representation of Bragg scattering from a crystal moving (a) perpendicular and (b) parallel to the diffracting planes with a velocity vk. The vectors kL and [{\bf k}^{'}_{L}] refer to the incident and reflected wavevectors in the laboratory frame of reference. In (a), depending on the direction of vk, the reflected wavevector is larger or smaller than the incident wavevector, kL. In (b), a larger incident reciprocal-space volume, ΔvL, is selected by the moving crystal than would have been selected by the crystal at rest. The reflected reciprocal-space element, [\Delta{\bf v}^{'}_{L}], has a large divergence, but can be arranged to be normal to [{\bf k}^{'}_{L}], hence improving the resolution [\Delta{\bf k}^{'}_{L}].

The Doppler shift is also important in determining the resolution of the rotating-crystal time-of-flight (TOF) spectrometer, first conceived by Brockhouse (1958[link]). A pulse of monochromatic neutrons is obtained when the reciprocal-lattice vector of a rotating crystal bisects the angle between two collimators. Effectively, the neutron k vector is changed in both direction and magnitude, depending on whether the crystal is moving towards or away from the neutron. For the rotating crystal, both of these situations occur simultaneously for different halves of the crystal, so that the net effect over the beam cross section is that a wider energy band is reflected than from the crystal at rest, and that, depending on the sense of rotation, the beam is either focused or defocused in time (Meister & Weckerman, 1972[link]).

The Bragg reflection of neutrons from a crystal moving parallel to its lattice planes is illustrated in Fig. 4.4.2.3(b)[link]. It can be seen that the moving crystal selects a larger Δk than the crystal at rest, so that the reflected intensity is higher. Furthermore, it is possible under certain conditions to orientate the diffracted phase-space volume orthogonal to the diffraction vector. In this way, a monochromatic divergent beam can be obtained from a collimated beam with a larger energy spread. This provides an elegant means of producing a divergent beam with a sufficiently wide momentum spread to be scanned by the Doppler crystal of a backscattering instrument (Schelten & Alefeld, 1984[link]).

Finally, an alternative method of scanning the energy of a monochromator in backscattering is to apply a steady but uniform temperature variation. The monochromator crystal must have a reasonable thermal expansion coefficient, and care has to be taken to ensure a uniform temperature across the crystal.

4.4.2.4. Mirror reflection devices

| top | pdf |

The refractive index, n, for neutrons of wavelength λ propagating in a nonmagnetic material of atomic density N is given by the expression [ n^{2}=1 - {{\lambda ^{2}Nb_{\rm {coh}}}\over{\pi }}, \eqno (4.4.2.4)]where [b_{\rm {coh}}] is the mean coherent scattering length. Values of the scattering-length density [Nb_{\rm {coh}}] for some common materials are listed in Table 4.4.2.2[link], from which it can be seen that the refractive index for most materials is slightly less than unity, so that total external reflection can take place. Thus, neutrons can be reflected from a smooth surface, but the critical angle of reflection, [\gamma _{c},] given by [\gamma _{c}=\lambda \sqrt {{Nb_{\rm {coh}}}\over{\pi }}, \eqno (4.4.2.5)]is small, so that reflection can only take place at grazing incidence. The critical angle for nickel, for example, is 0.1° Å−1.

Table 4.4.2.2| top | pdf |
Neutron scattering-length densities, Nbcoh, for some commonly used materials

Material Nb (10−6 Å−2)
58Ni 13.31
Diamond 11.71
Nickel 9.40
Quartz 3.64
Germanium 3.62
Silver 3.50
Aluminium 2.08
Silicon 2.08
Vanadium −0.27
Titanium −1.95
Manganese −2.95

Because of the shallowness of the critical angle, reflective optics are traditionally bulky, and focusing devices tend to have long focal lengths. In some cases, however, depending on the beam divergence, a long mirror can be replaced by an equivalent stack of shorter mirrors.

4.4.2.4.1. Neutron guides

| top | pdf |

The principle of mirror reflection is the basis of neutron guides, which are used to transmit neutron beams to instruments that may be situated up to 100 m away from the source (Christ & Springer, 1962[link]; Maier-Leibnitz & Springer, 1963[link]). A standard neutron guide is constructed from boron glass plates assembled to form a rectangular tube, the dimensions of which may be up to 200 mm high by 50 mm wide. The inner surface of the guide is coated with approximately 1200 Å of either nickel, 58Ni ([\gamma_c] = 0.12° Å−1), or a `supermirror' (described below). The guide is usually evacuated to reduce losses due to absorption and scattering of neutrons in air.

Theoretically, a neutron guide that is fully illuminated by the source will transmit a beam with a square divergence of full width [2\gamma _{c}] in both the horizontal and vertical directions, so that the transmitted solid angle is proportional to [\lambda^{2}]. In practice, owing to imperfections in the assembly of the guide system, the divergence profile is closer to Gaussian than square at the end of a long guide. Since the neutrons may undergo a large number of reflections in the guide, it is important to achieve a high reflectivity. The specular reflectivity is determined by the surface roughness, and typically values in the range 98.5 to 99% are achieved. Further transmission losses occur due to imperfections in the alignment of the sections that make up the guide.

The great advantage of neutron guides, in addition to the transport of neutrons to areas of low background, is that they can be multiplexed, i.e. one guide can serve many instruments. This is achieved either by deflecting only a part of the total cross section to a given instrument or by selecting a small wavelength range from the guide spectrum. In the latter case, the selection device (usually a crystal monochromator) must have a high transmission at other wavelengths.

If the neutron guide is curved, the transmission becomes wavelength dependent, as illustrated in Fig. 4.4.2.4[link] . In this case, one can define a characteristic wavelength, [\lambda ^{*}], given by the relation [\theta ^{*}=\sqrt {2a/\rho }], so that [\lambda ^{*}=\sqrt { {\vphantom2}{\pi }\over{Nb_{\rm coh}\vphantom{\rho}}} \sqrt { {2a}\over{\rho }} \eqno (4.4.2.6)](where a is the guide width and ρ the radius of curvature), for which the theoretical transmission drops to 67%. For wavelengths less than [\lambda ^{*}], neutrons can only be transmitted by `garland' reflections along the concave wall of the curved guide. Thus, the guide acts as a low-pass energy filter as long as its length is longer than the direct line-of-sight length [L_{1}=\sqrt {8a\rho }]. For example, a 3 cm wide nickel-coated guide whose characteristic wavelength is 4 Å (radius of curvature 1300 m) must be at least 18 m long to act as a filter. The line-of-sight length can be reduced by subdividing the guide into a number of narrower channels, each of which acts as a miniguide. The resulting device, often referred to as a neutron bender, since deviation of the beam is achieved more rapidly, is used in beam deviators (Alefeld et al., 1988[link]) or polarizers (Hayter, Penfold & Williams, 1978[link]). A microbender was devised by Marx (1971[link]) in which the channels were made by evaporating alternate layers of aluminium (transmission layer) and nickel (mirror layer) onto a flexible smooth substrate.

[Figure 4.4.2.4]

Figure 4.4.2.4 | top | pdf |

In a curved neutron guide, the transmission becomes λ dependent: (a) the possible types of reflection (garland and zig-zag), the direct line-of-sight length, the critical angle θ*, which is related to the characteristic wavelength [\lambda^*=\theta^*{\sqrt{\pi/Nb_{\rm coh}}}]; (b) transmission across the exit of the guide for different wavelengths, normalized to unity at the outside edge; (c) total transmission of the guide as a function of λ.

Tapered guides can be used to reduce the beam size in one or two dimensions (Rossbach et al., 1988[link]), although, since mirror reflection obeys Liouville's theorem, focusing in real space is achieved at the expense of an increase in divergence. This fact can be used to calculate analytically the expected gain in neutron flux at the end of a tapered guide (Anderson, 1988[link]). Alternatively, focusing can be achieved in one dimension using a bender in which the individual channel lengths are adjusted to create a focus (Freund & Forsyth, 1979[link]).

4.4.2.4.2. Focusing mirrors

| top | pdf |

Optical imaging of neutrons can be achieved using ellipsoidal or torroidal mirrors, but, owing to the small critical angle of reflection, the dimensions of the mirrors themselves and the radii of curvature must be large. For example, a 4 m long toroidal mirror has been installed at the IN15 neutron spin echo spectrometer at the Institut Laue–Langevin, Grenoble (Hayes et al., 1996[link]), to focus neutrons with wavelengths greater than 15 Å. The mirror has an in-plane radius of curvature of 408.75 m, and the sagittal radius is 280 mm. A coating of 65Cu is used to obtain a high critical angle of reflection while maintaining a low surface roughness. Slope errors of less than 2.5 × 10−5 rad (r.m.s.) combined with a surface roughness of less than 3 Å allow a minimum resolvable scattering vector of about 5 × 10−4 Å−1 to be reached.

For best results, the slope errors and the surface roughness must be low, in particular in small-angle scattering applications, since diffuse scattering from surface roughness gives rise to a halo around the image point. Owing to its low thermal expansion coefficient, highly polished Zerodur is often chosen as substrate.

4.4.2.4.3. Multilayers

| top | pdf |

Schoenborn, Caspar & Kammerer (1974[link]) first pointed out that multibilayers, comprising alternating thin films of different scattering-length densities ([Nb_{\rm {coh}}]) act like two-dimensional crystals with a d spacing given by the bilayer period. With modern deposition techniques (usually sputtering), uniform films of thickness ranging from about twenty to a few hundred ångströms can be deposited over large surface areas of the order of 1 m2. Owing to the rather large d spacings involved, the Bragg reflection from multilayers is generally at grazing incidence, so that long devices are required to cover a typical beam width, or a stacked device must be used. However, with judicious choice of the scattering-length contrast, the surface and interface roughness, and the number of layers, reflectivities close to 100% can be reached.

Fig. 4.4.2.5[link] illustrates how variation in the bilayer period can be used to produce a monochromator (the minimum Δλ/λ that can be achieved is of the order of 0.5%), a broad-band device, or a `supermirror', so called because it is composed of a particular sequence of bilayer thicknesses that in effect extends the region of total mirror reflection beyond the ordinary critical angle (Turchin, 1967[link]; Mezei, 1976[link]; Hayter & Mook, 1989[link]). Supermirrors have been produced that extend the critical angle of nickel by a factor, m, of between three and four with reflectivities better than 90%. Such high reflectivities enable supermirror neutron guides to be constructed with flux gains, compared with nickel guides, close to the theoretical value of m2.

[Figure 4.4.2.5]

Figure 4.4.2.5 | top | pdf |

Illustration of how a variation in the bilayer period can be used to produce a monochromator, a broad-band device, or a supermirror.

The choice of the layer pairs depends on the application. For non-polarizing supermirrors and broad-band devices (Høghøj, Anderson, Ebisawa & Takeda, 1996[link]), the Ni/Ti pair is commonly used, either pure or with some additions to relieve strain and stabilize interfaces (Elsenhans et al., 1994[link]) or alter the magnetism (Anderson & Høghøj, 1996[link]), owing to the high contrast in scattering density, while for narrow-band monochromators a low contrast pair such as W/Si is more suitable.

4.4.2.4.4. Capillary optics

| top | pdf |

Capillary neutron optics, in which hollow glass capillaries act as waveguides, are also based on the concept of total external reflection of neutrons from a smooth surface. The advantage of capillaries, compared with neutron guides, is that the channel sizes are of the order of a few tens of micrometres, so that the radius of curvature can be significantly decreased for a given characteristic wavelength [see equation (4.4.2.6)[link]]. Thus, neutrons can be efficiently deflected through large angles, and the device can be more compact.

Two basic types of capillary optics exist, and the choice depends on the beam characteristics required. Polycapillary fibres are manufactured from hollow glass tubes several centimetres in diameter, which are heated, fused and drawn multiple times until bundles of thousands of micrometre-sized channels are formed having an open area of up to 70% of the cross section. Fibre outer diameters range from 300 to 600 µm and contain hundreds or thousands of individual channels with inner diameters between 3 and 50 µm. The channel cross section is usually hexagonal, though square channels have been produced, and the inner channel wall surface roughness is typically less than 10 Å r.m.s., giving rise to very high reflectivities. The principal limitations on transmission efficiency are the open area, the acceptable divergence (note that the critical angle for glass is 1 mrad Å−1) and reflection losses due to absorption and scattering. A typical optical device will comprise hundreds or thousands of fibres threaded through thin screens to produce the required shape.

Fig. 4.4.2.6[link] shows typical applications of polycapillary devices. In Fig. 4.4.2.6[link](a), a polycapillary lens is used to refocus neutrons collected from a divergent source. The half lens depicted in Fig. 4.4.2.6[link](b) can be used either to produce a nearly parallel (divergence = [2\gamma _{c}]) beam from a divergent source or (in the reverse sense) to focus a nearly parallel beam, e.g. from a neutron guide. The size of the focal point depends on the channel size, the beam divergence, and the focal length of the lens. For example, a polycapillary lens used in a prompt γ-activation analysis instrument at the National Institute of Standards and Technology to focus a cold neutron beam from a neutron guide results in a current density gain of 80 averaged over the focused beam size of 0.53 mm (Chen et al., 1995[link]).

[Figure 4.4.2.6]

Figure 4.4.2.6 | top | pdf |

Typical applications of polycapillary devices: (a) lens used to refocus a divergent beam; (b) half-lens to produce a nearly parallel beam or to focus a nearly parallel beam; (c) a compact bender.

Fig. 4.4.2.6(c)[link] shows another simple application of polycapillaries as a compact beam bender. In this case, such a bender may be more compact than an equivalent multichannel guide bender, although the accepted divergence will be less. Furthermore, as with curved neutron guides, owing to the wavelength dependence of the critical angle the capillary curvature can be used to filter out thermal or high-energy neutrons.

It should be emphasized that the applications depicted in Fig. 4.4.2.6[link] obey Liouville's theorem, in that the density of neutrons in phase space is not changed, but the shape of the phase-space volume is altered to meet the requirements of the experiment, i.e. there is a simple trade off between beam dimension and divergence.

The second type of capillary optic is a monolithic configuration. The individual capillaries in monolithic optics are tapered and fused together, so that no external frame assembly is necessary (Chen-Mayer et al., 1996[link]). Unlike the multifibre devices, the inner diameters of the channels that make up the monolithic optics vary along the length of the component, resulting in a smaller more compact design.

Further applications of capillary optics include small-angle scattering (Mildner, 1994[link]) and lenses for high-spatial-resolution area detection.

4.4.2.5. Filters

| top | pdf |

Neutron filters are used to remove unwanted radiation from the beam while maintaining as high a transmission as possible for the neutrons of the required energy. Two major applications can be identified: removal of fast neutrons and γ-rays from the primary beam and reduction of higher-order contributions (λ/n) in the secondary beam reflected from crystal monochromators. In this section, we deal with non-polarizing filters, i.e. those whose transmission and removal cross sections are independent of the neutron spin. Polarizing filters are discussed in the section concerning polarizers.

Filters rely on a strong variation of the neutron cross section with energy, usually either the wavelength-dependent scattering cross section of polycrystals or a resonant absorption cross section. Following Freund (1983[link]), the total cross section determining the attenuation of neutrons by a crystalline solid can be written as a sum of three terms, [\sigma =\sigma _{\rm {abs}}+\sigma _{\rm {tds}}+\sigma _{\rm {Bragg}}. \eqno (4.4.2.7)]Here, [\sigma _{\rm {abs}}] is the true absorption cross section, which, at low energy, away from resonances, is proportional to [E^{-1/2}]. The temperature-dependent thermal diffuse cross section, [\sigma _{\rm {tds}}], describing the attenuation due to inelastic processes, can be split into two parts depending on the neutron energy. At low energy, [E\ll k_{b}\Theta _{{D}}], where [k_{b}] is Boltzmann's constant and [\Theta _{{D}}] is the characteristic Debye temperature, single-phonon processes dominate, giving rise to a cross section, [\sigma _{\rm {sph}}], which is also proportional to [E^{-1/2}]. The single-phonon cross section is proportional to [T^{7/2}] at low temperatures and to T at higher temperatures. At higher energies, [E\ge k_{b}\Theta_D], multiphonon and multiple-scattering processes come into play, leading to a cross section, [\sigma _{\rm {mph}}], that increases with energy and temperature. The third contribution, [\sigma _{\rm {Bragg}}], arises due to Bragg scattering in single- or polycrystalline material. At low energies, below the Bragg cut-off (λ [\gt] [2{d}_{\rm {max}}]), [\sigma _{\rm {Bragg}}] is zero. In polycrystalline materials, the cross section rises steeply above the Bragg cut-off and oscillates with increasing energy as more reflections come into play. At still higher energies, [\sigma _{\rm {Bragg}}] decreases to zero.

In single-crystalline material above the Bragg cut-off, [\sigma _{\rm {Bragg}}] is characterized by a discrete spectrum of peaks whose heights and widths depend on the beam collimation, energy resolution, and the perfection and orientation of the crystal. Hence a monocrystalline filter has to be tuned by careful orientation.

The resulting attenuation cross section for beryllium is shown in Fig. 4.4.2.7[link] . Cooled polycrystalline beryllium is frequently used as a filter for neutrons with energies less than 5 meV, since there is an increase of nearly two orders of magnitude in the attenuation cross section for higher energies. BeO, with a Bragg cut-off at approximately 4 meV, is also commonly used.

[Figure 4.4.2.7]

Figure 4.4.2.7 | top | pdf |

Total cross section for beryllium in the energy range where it can be used as a filter for neutrons with energy below 5 meV (Freund, 1983[link]).

Pyrolytic graphite, being a layered material with good crystalline properties along the c direction but random orientation perpendicular to it, lies somewhere between a polycrystal and a single crystal as far as its attenuation cross section is concerned. The energy-dependent cross section for a neutron beam incident along the c axis of a pyrolytic graphite filter is shown in Fig. 4.4.2.8[link] , where the attenuation peaks due to the 00ξ reflections can be seen. Pyrolytic graphite serves as an efficient second- or third-order filter (Shapiro & Chesser, 1972[link]) and can be `tuned' by slight misorientation away from the c axis.

[Figure 4.4.2.8]

Figure 4.4.2.8 | top | pdf |

Energy-dependent cross section for a neutron beam incident along the c axis of a pyrolytic graphite filter. The attenuation peaks due to the 00ξ reflections can be seen.

Further examples of typical filter materials (e.g. silicon, lead, bismuth, sapphire) can be found in the paper by Freund (1983[link]).

Resonant absorption filters show a large increase in their attenuation cross sections at the resonant energy and are therefore used as selective filters for that energy. A list of typical filter materials and their resonance energies is given in Table 4.4.2.3[link].

Table 4.4.2.3| top | pdf |
Characteristics of some typical elements and isotopes used as neutron filters

Element or isotopeResonance (eV)σs (resonance) (barns)λ (Å)σs(λ) (barns)[\displaystyle{\sigma _{s}( \lambda /2) }\over {\sigma _{s}( \lambda) }]
In 1.45 30000 0.48 94 319
Rh 1.27 4500 0.51 76 59.2
Hf 1.10 5000 0.55 58 86.2
240Pu 1.06 115000 9.55 145 793
Ir 0.66 4950 0.70 183 27.0
229Th 0.61 6200 0.73 < 100 [\gt] 62.0
Er 0.58 1500 0.75 127 11.8
Er 0.46 2300 0.84 125 18.4
Eu 0.46 10100 0.84 1050 9.6
231Pa 0.39 4900 0.92 116 42.2
239Pu 0.29 5200 1.06 700 7.4

1 barn = 10−28 m2.

4.4.2.6. Polarizers

| top | pdf |

Methods used to polarize a neutron beam are many and varied, and the choice of the best technique depends on the instrument and the experiment to be performed. The main parameter that has to be considered when describing the effectiveness of a given polarizer is the polarizing efficiency, defined as [ P=(N_{+}-N_{-})/(N_{+}+N_{-}), \eqno (4.4.2.8)]where [N_{+}] and [N_{-}] are the numbers of neutrons with spin parallel (+) or antiparallel (−) to the guide field in the outgoing beam. The second important factor, the transmission of the wanted spin state, depends on various factors, such as acceptance angles, reflection, and absorption.

4.4.2.6.1. Single-crystal polarizers

| top | pdf |

The principle by which ferromagnetic single crystals are used to polarize and monochromate a neutron beam simultaneously is shown in Fig. 4.4.2.9[link] . A field B, applied perpendicular to the scattering vector [\boldkappa], saturates the atomic moments [{\bf M}_{\nu}] along the field direction. The cross section for Bragg reflection in this geometry is [({\rm d}\sigma /{\rm d}\Omega)=F_{N}({\boldkappa})^{2}+2F_{N}({\boldkappa})F_{M}({\boldkappa})({\bf {}P}\cdot {\boldmu})+F_{M}({\boldkappa}){}^{2}, \eqno (4.4.2.9)]where [F_{N}({\boldkappa})] is the nuclear structure factor and [F_{M}({\boldkappa}) = [({\gamma }/{2})r_{0}]\sum _{\nu }M_{\nu }\,f(hkl)\exp [2\pi (hx+ky+lz)]] is the magnetic structure factor, with f(hkl) the magnetic form factor of the magnetic atom at the position (x, y, z) in the unit cell. The vector P describes the polarization of the incoming neutron with respect to B; P =1 for + spins and P = −1 for − spins and [{\boldmu}] is a unit vector in the direction of the atomic magnetic moments. Hence, for neutrons polarized parallel to B [({\bf P}\cdot {\boldmu}=1)], the diffracted intensity is proportional to [[F_{N}({\boldkappa})+F_{M}({\boldkappa})]{}^{2}], while, for neutrons polarized antiparallel to B [({\bf P}\cdot {\boldmu}=-1)], the diffracted intensity is proportional to [[F_{N}({\boldkappa})-F_{M}({\boldkappa})]{}^{2}]. The polarizing efficiency of the diffracted beam is then [ P=\pm 2F_{N}({\boldkappa})F_{M}({\boldkappa})/[F_{N}({\boldkappa})^{2}+F_{M}({\boldkappa}){}^{2}], \eqno (4.4.2.10)]which can be either positive or negative and has a maximum value for [|F_{N}({\boldkappa})|= |F_{M}({\boldkappa})|]. Thus, a good single-crystal polarizer, in addition to possessing a crystallographic structure in which [F_{N}] and [F_{M}] are matched, must be ferromagnetic at room temperature and should contain atoms with large magnetic moments. Furthermore, large single crystals with `controllable' mosaic should be available. Finally, the structure factor for the required reflection should be high, while those for higher-order reflections should be low.

[Figure 4.4.2.9]

Figure 4.4.2.9 | top | pdf |

Geometry of a polarizing monochromator showing the lattice planes (hkl) with |FN| = |FM|, the direction of P and [\boldmu], the expected spin direction and intensity.

None of the three naturally occurring ferromagnetic elements (iron, cobalt, nickel) makes efficient single-crystal polarizers. Cobalt is strongly absorbing and the nuclear scattering lengths of iron and nickel are too large to be balanced by their weak magnetic moments. An exception is 57Fe, which has a rather low nuclear scattering length, and structure-factor matching can be achieved by mixing 57Fe with Fe and 3% Si (Reed, Bolling & Harmon, 1973[link]).

In general, in order to facilitate structure-factor matching, alloys rather than elements are used. The characteristics of some alloys used as polarizing monochromators are presented in Table 4.4.2.4[link]. At short wavelengths, the 200 reflection of Co0.92Fe0.08 is used to give a positively polarized beam [[F_{N}({\boldkappa})] and [F_{M}({\boldkappa})] both positive], but the absorption due to cobalt is high. At longer wavelengths, the 111 reflection of the Heusler alloy Cu2MnAl (Delapalme, Schweizer, Couderchon & Perrier de la Bathie, 1971[link]; Freund, Pynn, Stirling & Zeyen, 1983[link]) is commonly used, since it has a higher reflectivity and a larger d spacing than Co0.92Fe0.08. Since for the 111 reflection [F_{N}\approx -F_{M}], the diffracted beam is negatively polarized. Unfortunately, the structure factor of the 222 reflection is higher than that of the 111 reflection, leading to significant higher-order contamination of the beam.

Table 4.4.2.4| top | pdf |
Properties of polarizing crystal monochromators (Williams, 1988[link])

 Co0.92Fe0.08Cu2MnAlFe3Si57Fe:FeHoFe2
Matched reflection [|F_{{N}}|\sim |F_{\rm {M}}|] 200 111 111 110 620
d spacing (Å) 1.76 3.43 3.27 2.03 1.16
Take-off angle [2\theta _{{B}}] at 1 Å (°) 33.1 16.7 17.6 28.6 50.9
Cut-off wavelength, λmax (Å) 3.5 6.9 6.5 4.1 2.3

Other alloys that have been proposed as neutron polarizers are Fe3−xMnxSi, 7Li0.5Fe2.5O4 (Bednarski, Dobrzynski & Steinsvoll, 1980[link]), Fe3Si (Hines et al., 1976[link]), Fe3Al (Pickart & Nathans, 1961[link]), and HoFe2 (Freund & Forsyth, 1979[link]).

4.4.2.6.2. Polarizing mirrors

| top | pdf |

For a ferromagnetic material, the neutron refractive index is given by [n_{\pm }^{2}=1-\lambda ^{2}N(b_{\rm {coh}}\pm p)/\pi, \eqno (4.4.2.11)]where the magnetic scattering length, p, is defined by [ p=2\mu (B-H)m\pi /h^{2}N. \eqno (4.4.2.12)]Here, m and μ are the neutron mass and magnetic moment, B is the magnetic induction in an applied field H, and h is Planck's constant.

The − and + signs refer, respectively, to neutrons whose moments are aligned parallel and antiparallel to B. The refractive index depends on the orientation of the neutron spin with respect to the film magnetization, thus giving rise to two critical angles of total reflection, γ and γ+. Thus, reflection in an angular range between these two critical angles gives rise to polarized beams in reflection and in transmission. The polarization efficiency, P, is defined in terms of the reflectivity [r_{+}] and [r_{-}] of the two spin states, [ P=(r_{+}-r_{-})/(r_{+}+r_{-}). \eqno (4.4.2.13)]The first polarizers using this principle were simple cobalt mirrors (Hughes & Burgy, 1950[link]), while Schaerpf (1975[link]) used FeCo sheets to build a polarizing guide. It is more common these days to use thin films of ferromagnetic material deposited onto a substrate of low surface roughness (e.g. float glass or polished silicon). In this case, the reflection from the substrate can be eliminated by including an antireflecting layer made from, for example, Gd–Ti alloys (Drabkin et al., 1976[link]). The major limitation of these polarizers is that grazing-incidence angles must be used and the angular range of polarization is small. This limitation can be partially overcome by using multilayers, as described above, in which one of the layer materials is ferromagnetic. In this case, the refractive index of the ferromagnetic material is matched for one spin state to that of the non-magnetic material, so that reflection does not occur. A polarizing supermirror made in this way has an extended angular range of polarization, as indicated in Fig. 4.4.2.10[link] . It should be noted that modern deposition techniques allow the refractive index to be adjusted readily, so that matching is easily achieved. The scattering-length densities of some commonly used layer pairs are given in Table 4.4.2.5[link]

Table 4.4.2.5| top | pdf |
Scattering-length densities for some typical materials used for polarizing multilayers

Magnetic layerN(b + p) (10−6 Å−2)N(bp) (10−6 Å−2)Nb (10−6 Å−2)Nonmagnetic layer
Fe 13.04 3.08 3.64 Ge
3.50 Ag
3.02 W
2.08 Si
2.08 Al
Fe:Co (50:50) 10.98 −0.52 −0.27 V
−1.95 Ti
Ni 10.86 7.94    
Fe:Co:V (49:49:2) 10.75 −0.63 −0.27 V
−1.95 Ti
Fe:Co:V (50:48:2) 10.66 −0.64 −0.27 V
−1.95 Ti
Fe:Ni (50:50) 10.53 6.65    
Co 6.65 −2.00 −1.95 Ti
Fe:Co:V (52:38:10) 6.27 2.12 2.08 Si
2.08 Al

For the non-magnetic layer we have only listed the simple elements that give a close match to the N(bp) value of the corresponding magnetic layer. In practice excellent matching can be achieved by using alloys (e.g. TixZry alloys allow Nb values between −1.95 and 3.03 × 10−6 Å−2 to be selected) or reactive sputtering (e.g. TiNx)
[Figure 4.4.2.10]

Figure 4.4.2.10 | top | pdf |

Measured reflectivity curve of an FeCoV/TiZr polarizing supermirror with an extended angular range of polarization of three times that of γc(Ni) for neutrons without spin flip, ↑↑, and with spin flip, ↑↓.

Polarizing multilayers are also used in monochromators and broad-band devices. Depending on the application, various layer pairs have been used: Co/Ti, Fe/Ag, Fe/Si, Fe/Ge, Fe/W, FeCoV/TiN, FeCoV/TiZr, 63Ni0.6654Fe0.34/V and the range of fields used to achieve saturation varies from about 100 to 500 Gs.

Polarizing mirrors can be used in reflection or transmission with polarization efficiencies reaching 97%, although, owing to the low incidence angles, their use is generally restricted to wavelengths above 2 Å.

Various devices have been constructed that use mirror polarizers, including simple reflecting mirrors, V-shaped transmission polarizers (Majkrzak, Nunez, Copley, Ankner & Greene, 1992[link]), cavity polarizers (Mezei, 1988[link]), and benders (Hayter, Penfold & Williams, 1978[link]; Schaerpf, 1989[link]). Perhaps the best known device is the polarizing bender developed by Schärpf. The device consists of 0.2 mm thick glass blades coated on both sides with a Co/Ti supermirror on top of an antireflecting Gd/Ti coating designed to reduce the scattering of the unwanted spin state from the substrate to a very low Q value. The device is quite compact (typically 30 cm long for a beam cross section up to 6 × 5 cm) and transmits over 40% of an unpolarized beam with the collimation from a nickel-coated guide for wavelengths above 4.5 Å. Polarization efficiencies of over 96% can be achieved with these benders.

4.4.2.6.3. Polarizing filters

| top | pdf |

Polarizing filters operate by selectively removing one of the neutron spin states from an incident beam, allowing the other spin state to be transmitted with only moderate attenuation. The spin selection is obtained by preferential absorption or scattering, so the polarizing efficiency usually increases with the thickness of the filter, whereas the transmission decreases. A compromise must therefore be made between polarization, P, and transmission, T. The `quality factor' often used is [P\sqrt {T}] (Tasset & Resouche, 1995[link]).

The total cross sections for a generalized filter may be written as [ \sigma _{\pm }=\sigma _{0}\pm \sigma _{p}, \eqno (4.4.2.14)]where [\sigma _{0}] is a spin-independent cross section and σp = (σ+ + σ)/2 is the polarization cross section. It can be shown (Williams, 1988[link]) that the ratio [\sigma _{p}/\sigma _{0}] must be [\ge] 0.65 to achieve |P| [\gt] 0.95 and T [\gt] 0.2.

Magnetized iron was the first polarizing filter to be used (Alvarez & Bloch, 1940[link]). The method relies on the spin-dependent Bragg scattering from a magnetized polycrystalline block, for which [\sigma _{p}] approaches 10 barns near the Fe cut-off at 4 Å (Steinberger & Wick, 1949[link]). Thus, for wavelengths in the range 3.6 to 4 Å, the ratio [\sigma _{p}/\sigma _{0}\simeq 0.59,] resulting in a theoretical polarizing efficiency of 0.8 for a transmittance of [\sim 0.3]. In practice, however, since iron cannot be fully saturated, depolarization occurs, and values of [P\simeq 0.5] with [T\sim 0.25] are more typical.

Resonance absorption polarization filters rely on the spin dependence of the absorption cross section of polarized nuclei at their nuclear resonance energy and can produce efficient polarization over a wide energy range. The nuclear polarization is normally achieved by cooling in a magnetic field, and filters based on 149Sm (Er = 0.097 eV) (Freeman & Williams, 1978[link]) and 151Eu (Er = 0.32 and 0.46 eV) have been successfully tested. The 149Sm filter has a polarizing efficiency close to 1 within a small wavelength range (0.85 to 1.1 Å), while the transmittance is about 0.15. Furthermore, since the filter must be operated at temperatures of the order of 15 mK, it is very sensitive to heating by γ-rays.

Broad-band polarizing filters, based on spin-dependent scattering or absorption, provide an interesting alternative to polarizing mirrors or monochromators, owing to the wider range of energy and scattering angle that can be accepted. The most promising such filter is polarized 3He, which operates through the huge spin-dependent neutron capture cross section that is totally dominated by the resonance capture of neutrons with antiparallel spin. The polarization efficiency of an 3He neutron spin filter of length l can be written as [ P_{n}(\lambda)=\tanh [{\cal O}(\lambda) P_{\rm {He}}], \eqno (4.4.2.15)]where [P_{\rm {He}}] is the 3He polarization, and [{\cal O}(\lambda)=[^{3}{\rm He}]l\sigma _{0}(\lambda)] is the dimensionless effective absorption coefficient, also called the opacity (Surkau et al., 1997[link]). For gaseous 3He, the opacity can be written in more convenient units as [ {\cal O}^{\prime }=p[{\rm bar}]\times l\,{[{\rm {}cm}}]\times \lambda[{\rm \AA }], \eqno (4.4.2.16)]where p is the 3He pressure (1 bar = 105 Pa) and [{\cal O}=] [7.33\times 10^{-2}{\cal O}^{\prime }]. Similarly, the residual transmission of the spin filter is given by [ T_{n}(\lambda)=\exp [-{\cal O}(\lambda)]\cosh [{\cal O}(\lambda) P_{\rm {He}}]. \eqno (4.4.2.17)]It can be seen that, even at low 3He polarization, full neutron polarization can be achieved in the limit of large absorption at the cost of the transmission.

3He can be polarized either by spin exchange with optically pumped rubidium (Bouchiat, Carver & Varnum, 1960[link]; Chupp, Coulter, Hwang, Smith & Welsh, 1996[link]; Wagshul & Chupp, 1994[link]) or by pumping of metastable 3He* atoms followed by metastable exchange collisions (Colegrove, Schearer & Walters, 1963[link]). In the former method, the 3He gas is polarized at the required high pressure, whereas 3He* pumping takes place at a pressure of about 1 mbar, followed by a polarization conserving compression by a factor of nearly 10 000. Although the polarization time constant for Rb pumping is of the order of several hours compared with fractions of a second for 3He* pumping, the latter requires several `fills' of the filter cell to achieve the required pressure.

An alternative broad-band spin filter is the polarized proton filter, which utilizes the spin dependence of nuclear scattering. The spin-dependent cross section can be written as (Lushchikov, Taran & Shapiro, 1969[link]) [\sigma _{\pm }=\sigma _{1}+\sigma _{2}P_{\rm {H}}^{2}\mp \sigma _{3}P_{\rm {H}}, \eqno (4.4.2.18)]where [\sigma _{1}], [\sigma _{2}], and [\sigma _{3}] are empirical constants. The viability of the method relies on achieving a high nuclear polarization [P_{\rm {H}}]. A polarization PH = 0.7 gives [\sigma _{p}/\sigma _{0}\approx 0.56] in the cold-neutron region. Proton polarizations of the order of 0.8 are required for a useful filter (Schaerpf & Stuesser, 1989[link]). Polarized proton filters can polarize very high energy neutrons even in the eV range.

4.4.2.6.4. Zeeman polarizer

| top | pdf |

The reflection width of perfect silicon crystals for thermal neutrons and the Zeeman splitting (ΔE = 2μB) of a field of about 10 kGs are comparable and therefore can be used to polarize a neutron beam. For a monochromatic beam (energy [E_{0}]) in a strong magnetic field region, the result of the Zeeman splitting will be a separation into two polarized subbeams, one polarized along B with energy E0 + μB, and the other polarized antiparallel to B with energy E0 − μB. The two polarized beams can be selected by rocking a perfect crystal in the field region B (Forte & Zeyen, 1989[link]).

4.4.2.7. Spin-orientation devices

| top | pdf |

Polarization is the state of spin orientation of an assembly of particles in a target or beam. The beam polarization vector P is defined as the vector average of this spin state and is often described by the density matrix ρ = [1\over2](1 + σP). The polarization is then defined as P = Tr(ρσ). If the polarization vector is inclined to the field direction in a homogenous magnetic field, B, the polarization vector will precess with the classical Larmor frequency [\omega _{{L}}=|\gamma |{B}]. This results in a precessing spin polarization. For most experiments, it is sufficient to consider the linear polarization vector in the direction of an applied magnetic field. If, however, the magnetic field direction changes along the path of the neutron, it is also possible that the direction of P will change. If the frequency, [\Omega], with which the magnetic field changes is such that [\Omega={\rm d}({\bf {}B}/|{\bf {}B}|)/{\rm d} t\ll \omega _{{L}}, \eqno (4.4.2.19)]then the polarization vector follows the field rotation adiabatically. Alternatively, when [\Omega\gg \omega _{{L}}], the magnetic field changes so rapidly that P cannot follow, and the condition is known as non-adiabatic fast passage. All spin-orientation devices are based on these concepts.

4.4.2.7.1. Maintaining the direction of polarization

| top | pdf |

A polarized beam will tend to become depolarized during passage through a region of zero field, since the field direction is ill defined over the beam cross section. Thus, in order to keep the polarization direction aligned along a defined quantization axis, special precautions must be taken.

The simplest way of maintaining the polarization of neutrons is to use a guide field to produce a well defined field B over the whole flight path of the beam. If the field changes direction, it has to fulfil the adiabatic condition [\Omega\ll \omega _{L}], i.e. the field changes must take place over a time interval that is long compared with the Larmor period. In this case, the polarization follows the field direction adiabatically with an angle of deviation [\Delta\theta \le 2\arctan (\Omega/\omega _{L})] (Schärpf, 1980[link]).

Alternatively, some instruments (e.g. zero-field spin-echo spectrometers and polarimeters) use polarized neutron beams in regions of zero field. The spin orientation remains constant in a zero-field region, but the passage of the neutron beam into and out of the zero-field region must be well controlled. In order to provide a well defined region of transition from a guide-field region to a zero-field region, a non-adiabatic fast passage through the windings of a rectangular input solenoid can be used, either with a toroidal closure of the outside field or with a μ-metal closure frame. The latter serves as a mirror for the coil ends, with the effect of producing the field homogeneity of a long coil but avoiding the field divergence at the end of the coil.

4.4.2.7.2. Rotation of the polarization direction

| top | pdf |

The polarization direction can be changed by the adiabatic change of the guide-field direction so that the direction of the polarization follows it. Such a rotation is performed by a spin turner or spin rotator (Schärpf & Capellmann, 1993[link]; Williams, 1988[link]).

Alternatively, the direction of polarization can be rotated relative to the guide field by using the property of precession described above. If a polarized beam enters a region where the field is inclined to the polarization axis, then the polarization vector P will precess about the new field direction. The precession angle will depend on the magnitude of the field and the time spent in the field region. By adjustment of these two parameters together with the field direction, a defined, though wavelength-dependent, rotation of P can be achieved. A simple device uses the non-adiabatic fast passage through the windings of two rectangular solenoids, wound orthogonally one on top of the other. In this way, the direction of the precession field axis is determined by the ratio of the currents in the two coils, and the sizes of the fields determine the angle [\varphi] of the precession. The orientation of the polarization vector can therefore be defined in any direction.

In order to produce a continuous rotation of the polarization, i.e. a well defined precession, as required in neutron spin-echo (NSE) applications, precession coils are used. In the simplest case, these are long solenoids where the change of the field integral over the cross section can be corrected by Fresnel coils (Mezei, 1972[link]). More recently, Zeyen & Rem (1996[link]) have developed and implemented optimal field-shape (OFS) coils. The field in these coils follows a cosine squared shape that results from the optimization of the line integral homogeneity. The OFS coils can be wound over a very small diameter, thereby reducing stray fields drastically.

4.4.2.7.3. Flipping of the polarization direction

| top | pdf |

The term `flipping' was originally applied to the situation where the beam polarization direction is reversed with respect to a guide field, i.e. it describes a transition of the polarization direction from parallel to antiparallel to the guide field and vice versa. A device that produces this 180° rotation is called a π flipper. A π/2 flipper, as the name suggests, produces a 90° rotation and is normally used to initiate precession by turning the polarization at 90° to the guide field.

The most direct wavelength-independent way of producing such a transition is again a non-adiabatic fast passage from the region of one field direction to the region of the other field direction. This can be realized by a current sheet like the Dabbs foil (Dabbs, Roberts & Bernstein, 1955[link]), a Kjeller eight (Abrahams, Steinsvoll, Bongaarts & De Lange, 1962[link]) or a cryoflipper (Forsyth, 1979[link]).

Alternatively, a spin flip can be produced using a precession coil, as described above, in which the polarization direction makes a precession of just π about a direction orthogonal to the guide field direction (Mezei, 1972[link]). Normally, two orthogonally wound coils are used, where the second, correction, coil serves to compensate the guide field in the interior of the precession coil. Such a flipper is wavelength dependent and can be easily tuned by varying the currents in the coils.

Another group of flippers uses the non-adiabatic transition through a well defined region of zero field. Examples of this type of flipper are the two-coil flipper of Drabkin, Zabidarov, Kasman & Okorokov (1969[link]) and the line-shape flipper of Korneev & Kudriashov (1981[link]).

Historically, the first flippers used were radio-frequency coils set in a homogeneous magnetic field. These devices are wavelength dependent, but may be rendered wavelength independent by replacing the homogeneous magnetic field with a gradient field (Egorov, Lobashov, Nazarento, Porsev & Serebrov, 1974[link]).

In some devices, the flipping action can be combined with another selection function. The wavelength-dependent magnetic wiggler flipper proposed by Agamalyan, Drabkin & Sbitnev (1988[link]) in combination with a polarizer can be used as a polarizing monochromator (Majkrzak & Shirane, 1982[link]). Badurek & Rauch (1978[link]) have used flippers as choppers to pulse a polarized beam.

In neutron resonance spin echo (NRSE) (Gähler & Golub, 1987[link]), the precession coil of the conventional spin-echo configuration is replaced by two resonance spin flippers separated by a large zero-field region. The radio-frequency field of amplitude [B_{1}] is arranged orthogonal to the DC field, [B_{0}], with a frequency [\omega = \omega _{L}], and an amplitude defined by the relation [\omega _{1}\tau] = π, where τ is the flight time in the flipper coil and [\omega _{1} = \gamma {B}_{1}]. In this configuration, the neutron spin precesses through an angle π about the resonance field in each coil and leaves the coil with a phase angle [\varphi]. The total phase angle after passing through both coils, [\varphi =2\omega L/v], depends on the velocity v of the neutron and the separation L between the two coils. Thus, compared with conventional NSE, where the phase angle comes from the precession of the neutron spin in a strong magnetic field compared with a static flipper field, in NRSE the neutron spin does not precess, but the flipper field rotates. Effectively, the NRSE phase angle [\varphi] is a factor of two larger than the NSE phase angle for the same DC field [{B}_{0}]. Furthermore, the resolution is determined by the precision of the RF frequencies and the zero-field flight path L rather than the homogeneity of the line integral of the field in the NSE precession coil.

4.4.2.8. Mechanical choppers and selectors

| top | pdf |

Thermal neutrons have relatively low velocities (a 4 Å neutron has a reciprocal velocity of approximately 1000 µs m−1), so that mechanical selection devices and simple flight-time measurements can be used to make accurate neutron energy determinations.

Disc choppers rotating at speeds up to 20 000 revolutions per minute about an axis that is parallel to the neutron beam are used to produce a well defined pulse of neutrons. The discs are made from absorbing material (at least where the beam passes) and comprise one or more neutron-transparent apertures or slits. For polarized neutrons, these transparent slits should not be metallic, as the eddy currents in the metal moving in even a weak guide field will strongly depolarize the beam. The pulse frequency is determined by the number of apertures and the rotation frequency, while the duty cycle is given by the ratio of open time to closed time in one rotation. Two such choppers rotating in phase can be used to monochromate and pulse a beam simultaneously (Egelstaff, Cocking & Alexander, 1961[link]). In practice, more than two choppers are generally used to avoid frame overlap of the incident and scattered beams. The time resolution of disc choppers (and hence the energy resolution of the instrument) is determined by the beam size, the aperture size and the rotation speed. For a realistic beam size, the rotation speed limits the resolution. Therefore, in modern instruments, it is normal to replace a single chopper with two counter-rotating choppers (Hautecler et al., 1985[link]; Copley, 1991[link]). The low duty cycle of a simple disc chopper can be improved by replacing the single slit with a series of slits either in a regular sequence (Fourier chopper) (Colwell, Miller & Whittemore, 1968[link]; Hiismäki, 1997[link]) or a pseudostatistical sequence (pseudostatistical chopper) (Hossfeld, Amadori & Scherm, 1970[link]), with duty cycles of 50 and 30%, respectively.

The Fermi chopper is an alternative form of neutron chopper that simultaneously pulses and monochromates the incoming beam. It consists of a slit package, essentially a collimator, rotating about an axis that is perpendicular to the beam direction (Turchin, 1965[link]). For optimum transmission at the required wavelength, the slits are usually curved to provide a straight collimator in the neutron frame of reference. The curvature also eliminates the `reverse burst', i.e. a pulse of neutrons that passes when the chopper has rotated by 180°.

A Fermi chopper with straight slits in combination with a monochromator assembly of wide horizontal divergence can be used to time focus a polychromatic beam, thus maintaining the energy resolution while improving the intensity (Blanc, 1983[link]).

Velocity selectors are used when a continuous beam is required with coarse energy resolution. They exist in either multiple disc configurations or helical channels rotating about an axis parallel to the beam direction (Dash & Sommers, 1953[link]). Modern helical channel selectors are made up of light-weight absorbing blades slotted into helical grooves on the rotation axis (Wagner, Friedrich & Wille, 1992[link]). At higher energies where no suitable absorbing material is available, highly scattering polymers [poly(methyl methacrylate)] can be used for the blades, although in this case adequate shielding must be provided. The neutron wavelength is determined by the rotation speed, and resolutions, Δλ/λ, ranging from 5% to practically 100% (λ/2 filter) can be achieved. The resolution is fixed by the geometry of the device, but can be slightly improved by tilting the rotation axis or relaxed by rotating in the reverse direction for shorter wavelengths. Transmissions of up to 94% are typical.

4.4.3. Resolution functions

| top | pdf |
R. Pynnd and J. M. Rowee

In a Gedanken neutron scattering experiment, neutrons of wavevector [{\bf k}_I] impinge on a sample and the wavevector, [{\bf k}_F], of the scattered neutrons is determined. A number of different types of spectrometer are used to achieve this goal (cf. Pynn, 1984[link]). In each case, finite instrumental resolution is a result of uncertainties in the definition of [{\bf k}_I] and [{\bf k}_F]. Propagation directions for neutrons are generally defined by Soller collimators for which the transmission as a function of divergence angle generally has a triangular shape. Neutron monochromatization may be achieved either by Bragg reflection from a (usually) mosaic crystal or by a time-of-flight method. In the former case, the mosaic leads to a spread of [|k_I|] while, in the latter, pulse length and uncertainty in the lengths of flight paths (including sample size and detector thickness) produce a similar effect. Calculations of instrumental resolution are generally lengthy and lack of space prohibits their detailed presentation here. In the following paragraphs, the concepts involved are indicated and references to original articles are provided.

In resolution calculations for neutron spectrometers, it is usually assumed that the uncertainty of the neutron wavevector does not vary spatially across the neutron beam, although this reasoning may not apply to the case of small samples and compact spectrometers. To calculate the resolution of the spectrometer in the large-beam approximation, one writes the measured intensity I as [I\propto\textstyle\int{\rm d}^3k_i\int {\rm d}^3k_f\, P_i({\bf k}_i)S({\bf k}_i\rightarrow{\bf k}_f)P_f({\bf k}_f), \eqno (4.4.3.1)]where [P_i({\bf k}_i)] is the probability that a neutron of wavevector [{\bf k}_i] is incident on the sample, [P_f({\bf k}_f)] is the probability that a neutron of wavevector [{\bf k}_f] is transmitted by the analyser system and [S({\bf k}_i\rightarrow{\bf k}_f)] is the probability that the sample scatters a neutron from [{\bf k}_i] to [{\bf k}_f]. The fluctuation spectrum of the sample, [S({\bf k}_i\rightarrow{\bf k}_f)], does not depend separately on [{\bf k}_i] and [{\bf k}_f] but rather on the scattering vector Q and energy transfer [\hbar\omega] defined by the conservation equations [{\bf Q}={\bf k}_i-{\bf k}_f; \quad \hbar\omega={\hbar^2\over 2m}(k^2_i-k^2_f), \eqno (4.4.3.2)]where m is the neutron mass.

A number of methods of calculating the distribution functions [P_i({\bf k}_i)] and [P_f({\bf k}_f)] have been proposed. The method of independent distributions was used implicitly by Stedman (1968[link]) and in more detail by Bjerrum Møller & Nielson (MN) (Nielsen & Bjerrum Møller, 1969[link]; Bjerrum Møller & Nielsen, 1970[link]) for three-axis spectrometers. Subsequently, the method has been extended to perfect-crystal monochromators (Pynn, Fujii & Shirane, 1983[link]) and to time-of-flight spectrometers (Steinsvoll, 1973[link]; Robinson, Pynn & Eckert, 1985[link]). The method involves separating [P_i] and [P_f] into a product of independent distribution functions each of which can be convolved separately with the fluctuation spectrum S(Q, ω) [cf. equation (4.4.3.1)[link]]. Extremely simple results are obtained for the widths of scans through a phonon dispersion surface for spectrometers where the energy of scattered neutrons is analysed (Nielson & Bjerrum Møller, 1969[link]). For diffractometers, the width of a scan through a Bragg peak may also be obtained (Pynn et al., 1983[link]), yielding a result equivalent to that given by Caglioti, Paoletti & Ricci (1960[link]). In this case, however, the singular nature of the Bragg scattering process introduces a correlation between the distribution functions that contribute to [P_i] and [P_f] and the calculation is less transparent than it is for phonons.

A somewhat different approach, which does not explicitly separate the various contributions to the resolution, was proposed by Cooper & Nathans (CN) (Cooper & Nathans, 1967[link], 1968[link]; Cooper, 1968[link]). Minor errors were corrected by several authors (Werner & Pynn, 1971[link]; Chesser & Axe, 1973[link]). The CN method calculates the instrumental resolution function R(QQ0, ω − ω0) as [R(\Delta{\bf Q},\Delta\omega)=R_0\exp-{\textstyle{1\over2}} \,\textstyle\sum\limits_{\alpha,\beta}\,M_{\alpha \beta}\,X_\alpha\, X_\beta, \eqno (4.4.3.3)]where [X_1], [X_2], and [X_3] are the three components of ΔQ, X4 = Δω, and Q0 and ω0 are obtained from (4.4.3.2)[link] by replacing [{\bf k}_i] and [{\bf k}_f] by [{\bf k}_I] and [{\bf k}_F], respectively. The matrix M is given in explicit form by several authors (Cooper & Nathans, 1967[link], 1968[link]; Cooper, 1968[link]; Werner & Pynn, 1971[link]; Chesser & Axe, 1973[link]) and the normalization [R_0] has been discussed in detail by Dorner (1972[link]). [A refutation (Tindle, 1984[link]) of Dorner's work is incorrect.] Equation (4.4.3.3)[link] implies that contours of constant transmission for the spectrometer [RQ, Δω) = constant] are ellipsoids in the four-dimensional Q–ω space. Optimum resolution (focusing) is achieved by a scan that causes the resolution function to intersect the feature of interest in S(Q, ω) (e.g. Bragg peak or phonon dispersion surface) for the minimum scan interval. The optimization of scans for a diffractometer has been considered by Werner (1971[link]).

The MN and CN methods are equivalent. Using the MN formalism, it can be shown that [{\bi M} = ({\bi A})^{-1}\quad{\rm with}\quad A_{\alpha\beta}=\textstyle\sum\limits_j \chi_{j\alpha}\chi _{j\beta}, \eqno (4.4.3.4)]where the [\chi_{j\alpha}] are the components of the standard deviations of independent distributions (labelled by index j) defined by Bjerrum Møller & Nielsen (1970[link]). In the limit [Q\rightarrow] 0, the matrices M and A are of rank three and other methods must be used to calculate the resolution ellipsoid (Mitchell, Cowley & Higgins, 1984[link]). Nevertheless, the MN method may be used even in this case to calculate widths of scans.

To obtain the resolution function of a diffractometer (in which there is no analysis of scattered neutron energy) from the CN form for M, it is sufficient to set to zero those contributions that arise from the mosaic of the analyser crystal. For elastic Bragg scattering, the problem is further simplified because [X_4] [cf. equation (4.4.3.3)[link]] is zero. The spectrometer resolution function is then an ellipsoid in Q space.

For the measurement of integrated intensities (of Bragg peaks for example), the normalization [R_0] in (4.4.3.3)[link] is required in order to obtain the Lorentz factor. The latter has been calculated for an arbitrary scan of a three-axis spectrometer (Pynn, 1975[link]) and the results may be modified for a diffractometer as described in the preceding paragraph.

4.4.4. Scattering lengths for neutrons

| top | pdf |
V. F. Searsg

The use of neutron diffraction for crystal-structure determinations requires a knowledge of the scattering lengths and the corresponding scattering and absorption cross sections of the elements and, in some cases, of individual isotopes. This information is needed to calculate unit-cell structure factors and to correct for effects such as absorption, self-shielding, extinction, thermal diffuse scattering, and detector efficiency (Bacon, 1975[link]; Sears, 1989[link]). Table 4.4.4.1[link] lists the best values of the neutron scattering lengths and cross sections that are available at the time of writing (January 1995). We begin by summarizing the basic relationships between the scattering lengths and cross sections of the elements and their isotopes that have been used in the compilation of this table. More background information can be found in, for example, the book by Sears (1989[link]).

Table 4.4.4.1| top | pdf |   interactive version
Bound scattering lengths, b, in fm and cross sections, σ, in barns (1 barn = 100 fm2) of the elements and their isotopes

Z: atomic number; A: mass number; I(π): spin (parity) of the nuclear ground state; c: % natural abundance (for radioisotopes, the half-life is given instead in annums); bc: bound coherent scattering length; bi: bound incoherent scattering length; σc: bound coherent scattering cross section; σi: bound incoherent scattering cross section; σs: total bound scattering cross section; σa: absorption cross section for 2200 m s−1 neutrons (E = 25.30 meV, k = 3.494 Å−1, λ = 1.798 Å); i = [{\sqrt-1}].

Element[Z][A][I(\pi)][c][b_c][b_i][\sigma_c][\sigma_i][\sigma_s][\sigma_a]
H 1       −3.7390(11)   1.7568(10) 80.26(6) 82.02(6) 0.3326(7)
    1 1/2(+) 99.985 −3.7406(11) 25.274(9) 1.7583(10) 80.27(6) 82.03(6) 0.3326(7)
    2 1(+) 0.015 6.671(4) 4.04(3) 5.592(7) 2.05(3) 7.64(3) 0.000519(7)
    3 1/2(+) (12.32a) 4.792(27) −1.04(17) 2.89(3) 0.14(4) 3.03(5) 0
                     
He 2       3.26(3)   1.34(2) 0.00 1.34(2) 0.00747(1)
    3 1/2(+) 0.00014 5.74(7) −2.5(6) 4.42(10) 1.6(4) 6.0(4) 5333.(7.)
          −1.483(2)[i] +2.568(3)[i]        
    4 0(+) 99.99986 3.26(3) 0 1.34(2) 0 1.34(2) 0
                     
Li 3       −1.90(2)   0.454(14) 0.92(3) 1.37(3) 70.5(3)
    6 1(+) 7.5 2.00(11) −1.89(5) 0.51(5) 0.46(2) 0.97(7) 940.(4.)
          −0.261(1)[i] 0.257(11)[i]        
    7 3/2(−) 92.5 −2.22(2) −2.49(5) 0.619(11) 0.78(3) 1.40(3) 0.0454(3)
                     
Be 4 9 3/2(−) 100 7.79(1) 0.12(3) 7.63(2) 0.0018(9) 7.63(2) 0.0076(8)
                     
B 5       5.30(4)   3.54(5) 1.70(12) 5.24(11) 767.(8.)
          0.213(2)[i]          
    10 3(+) 20.0 −0.1(3) −4.7(3) 0.144(8) 3.0(4) 3.1(4) 3835.(9.)
          1.066(3)[i] 1.231(3)[i]        
    11 3/2(−) 80.0 6.65(4) −1.3(2) 5.56(7) 0.22(6) 5.78(9) 0.0055(33)
                     
C 6       6.6460(12)   5.550(2) 0.001(4) 5.551(3) 0.00350(7)
    12 0(+) 98.90 6.6511(16) 0 5.559(3) 0 5.559(3) 0.00353(7)
    13 1/2(−) 1.10 6.19(9) −0.52(9) 4.81(14) 0.034(12) 4.84(14) 0.00137(4)
                     
N 7       9.36(2)   11.01(5) 0.50(12) 11.51(11) 1.90(3)
    14 1(+) 99.63 9.37(2) 2.0(2) 11.03(5) 0.5(1) 11.53(11) 1.91(3)
    15 1/2(−) 0.37 6.44(3) −0.02(2) 5.21(5) 0.00005(10) 5.21(5) 0.000024(8)
                     
O 8       5.803(4)   4.232(6) 0.000(8) 4.232(6) 0.00019(2)
    16 0(+) 99.762 5.803(4) 0 4.232(6) 0 4.232(6) 0.00010(2)
    17 5/2(+) 0.038 5.78(12) 0.18(6) 4.20(22) 0.004(3) 4.20(22) 0.236(10)
    18 0(+) 0.200 5.84(7) 0 4.29(10) 0 4.29(10) 0.00016(1)
                     
F 9 19 1/2(+) 100 5.654(10) −0.082(9) 4.017(17) 0.0008(2) 4.018(14) 0.0096(5)
                     
Ne 10       4.566(6)   2.620(7) 0.008(9) 2.628(6) 0.039(4)
    20 0(+) 90.51 4.631(6) 0 2.695(7) 0 2.695(7) 0.036(4)
    21 3/2(+) 0.27 6.66(19) [\pm]0.6(1) 5.6(3) 0.05(2) 5.7(3) 0.67(11)
    22 0(+) 9.22 3.87(1) 0 1.88(1) 0 1.88(1) 0.046(6)
                     
Na 11 23 3/2(+) 100 3.63(2) 3.59(3) 1.66(2) 1.62(3) 3.28(4) 0.530(5)
                     
Mg 12       5.375(4)   3.631(5) 0.08(6) 3.71(4) 0.063(3)
    24 0(+) 78.99 5.66(3) 0 4.03(4) 0 4.03(4) 0.050(5)
    25 5/2(+) 10.00 3.62(14) 1.48(10) 1.65(13) 0.28(4) 1.93(14) 0.19(3)
    26 0(+) 11.01 4.89(15) 0 3.00(18) 0 3.00(18) 0.0382(8)
                     
Al 13 27 5/2(+) 100 3.449(5) 0.256(10) 1.495(4) 0.0082(7) 1.503(4) 0.231(3)
                     
Si 14       4.1491(10)   2.1633(10) 0.004(8) 2.167(8) 0.171(3)
    28 0(+) 92.23 4.107(6) 0 2.120(6) 0 2.120(6) 0.177(3)
    29 1/2(+) 4.67 4.70(10) 0.09(9) 2.78(12) 0.001(2) 2.78(12) 0.101(14)
    30 0(+) 3.10 4.58(8) 0 2.64(9) 0 2.64(9) 0.107(2)
                     
P 15 31 1/2(+) 100 5.13(1) 0.2(2) 3.307(13) 0.005(10) 3.312(16) 0.172(6)
                     
S 16       2.847(1)   1.0186(7) 0.007(5) 1.026(5) 0.53(1)
    32 0(+) 95.02 2.804(2) 0 0.9880(14) 0 0.9880(14) 0.54(4)
    33 3/2(+) 0.75 4.74(19) 1.5(1.5) 2.8(2) 0.3(6) 3.1(6) 0.54(4)
    34 0(+) 4.21 3.48(3) 0 1.52(3) 0 1.52(3) 0.227(5)
    36 0(+) 0.02 3.(1.) E 0 1.1(8) 0 1.1(8) 0.15(3)
                     
Cl 17       9.5770(8)   11.526(2) 5.3(5) 16.8(5) 33.5(3)
    35 3/2(+) 75.77 11.65(2) 6.1(4) 17.06(6) 4.7(6) 21.8(6) 44.1(4)
    37 3/2(+) 24.23 3.08(6) 0.1(1) 1.19(5) 0.001(3) 1.19(5) 0.433(6)
                     
Ar 18       1.909(6)   0.458(3) 0.22(2) 0.683(4) 0.675(9)
    36 0(+) 0.337 24.90(7) 0 77.9(4) 0 77.9(4) 5.2(5)
    38 0(+) 0.063 3.5(3.5) 0 1.5(3.1) 0 1.5(3.1) 0.8(2)
    40 0(+) 99.600 1.830(6) 0 0.421(3) 0 0.421(3) 0.660(9)
                     
K 19       3.67(2)   1.69(2) 0.27(11) 1.96(11) 2.1(1)
    39 3/2(+) 93.258 3.74(2) 1.4(3) 1.76(2) 0.25(11) 2.01(11) 2.1(1)
    40 4(−) 0.012 3.(1.) E   1.1(8) 0.5(5) 1.6(9) 35.(8.)
    41 3/2(+) 6.730 2.69(8) 1.5(1.5) 0.91(5) 0.3(6) 1.2(6) 1.46(3)
                     
Ca 20       4.70(2)   2.78(2) 0.05(3) 2.83(2) 0.43(2)
    40 0(+) 96.941 4.80(2) 0 2.90(2) 0 2.90(2) 0.41(2)
    42 0(+) 0.647 3.36(10) 0 1.42(8) 0 1.42(8) 0.68(7)
    43 7/2(−) 0.135 −1.56(9) 0.31(4) 0.5(5) E   0.8(5) 6.2(6)
    44 0(+) 2.086 1.42(6) 0 0.25(2) 0 0.25(2) 0.88(5)
    46 0(+) 0.004 3.6(2) 0 1.6(2) 0 1.6(2) 0.74(7)
    48 0(+) 0.187 0.39(9) 0 0.019(9) 0 0.019(9) 1.09(14)
                     
Sc 21 45 7/2(−) 100 12.29(11) −6.0(3) 19.0(3) 4.5(5) 23.5(6) 27.5(2)
                     
Ti 22       −3.370(13)   1.427(11) 2.63(3) 4.06(3) 6.43(6)
    46 0(+) 8.2 4.725(5) 0 2.80(6) 0 2.80(6) 0.59(18)
    47 5/2(−) 7.4 3.53(7) −3.5(2) 1.57(6) 1.5(2) 3.1(2) 1.7(2)
    48 0(+) 73.8 −5.86(2) 0 4.32(3) 0 4.32(3) 8.30(9)
    49 7/2(−) 5.4 0.98(5) 5.1(2) 0.12(1) 3.3(3) 3.4(3) 2.2(3)
    50 0(+) 5.2 5.88(10) 0 4.34(15) 0 4.34(15) 0.179(3)
                     
V 23       −0.3824(12)   0.01838(12) 5.08(6) 5.10(6) 5.08(2)
    50 6(+) 0.250 7.6(6)   7.3(1.1) 0.5(5) E 7.8(1.0) 60.(40.)
    51 7/2(−) 99.750 −0.402(2) 6.435(4) 0.0203(2) 5.07(6) 5.09(6) 4.9(1)
                     
Cr 24       3.635(7)   1.660(6) 1.83(2) 3.49(2) 3.05(8)
    50 0(+) 4.35 −4.50(5) 0 2.54(6) 0 2.54(6) 15.8(2)
    52 0(+) 83.79 4.920(10) 0 3.042(12) 0 3.042(12) 0.76(6)
    53 3/2(−) 9.50 −4.20(3) 6.87(10) 2.22(3) 5.93(17) 8.15(17) 18.1(1.5)
    54 0(+) 2.36 4.55(10) 0 2.60(11) 0 2.60(11) 0.36(4)
                     
Mn 25 55 5/2(−) 100 −3.750(18) 1.79(4) 1.77(2) 0.40(2) 2.17(3) 13.3(2)
                     
Fe 26       9.45(2)   11.22(5) 0.40(11) 11.62(10) 2.56(3)
    54 0(+) 5.8 4.2(1) 0 2.2(1) 0 2.2(1) 2.25(18)
    56 0(+) 91.7 9.94(3) 0 12.42(7) 0 12.42(7) 2.59(14)
    57 1/2(−) 2.2 2.3(1) 0.66(6)   0.3(3) E 1.0(3) 2.48(30)
    58 0(+) 0.3 15.(7.) 0 28.(26.) 0 28.(26.) 1.28(5)
                     
Co 27 59 7/2(−) 100 2.49(2) −6.2(2) 0.779(13) 4.8(3) 5.6(3) 37.18(6)
                     
Ni 28       10.3(1)   13.3(3) 5.2(4) 18.5(3) 4.49(16)
    58 0(+) 68.27 14.4(1) 0 26.1(4) 0 26.1(4) 4.6(3)
    60 0(+) 26.10 2.8(1) 0 0.99(7) 0 0.99(7) 2.9(2)
    61 3/2(−) 1.13 7.60(6) [\pm]3.9(3) 7.26(11) 1.9(3) 9.2(3) 2.5(8)
    62 0(+) 3.59 −8.7(2) 0 9.5(4) 0 9.5(4) 14.5(3)
    64 0(+) 0.91 −0.37(7) 0 0.017(7) 0 0.017(7) 1.52(3)
                     
Cu 29       7.718(4)   7.485(8) 0.55(3) 8.03(3) 3.78(2)
    63 3/2(−) 69.17 6.43(15) 0.22(2) 5.2(2) 0.006(1) 5.2(2) 4.50(2)
    65 3/2(−) 30.83 10.61(19) 1.79(10) 14.1(5) 0.40(4) 14.5(5) 2.17(3)
                     
Zn 30       5.60(5)   4.054(7) 0.077(7) 4.131(10) 1.11(2)
    64 0(+) 48.6 5.22(4) 0 3.42(5) 0 3.42(5) 0.93(9)
    66 0(+) 27.9 5.97(5) 0 4.48(8) 0 4.48(8) 0.62(6)
    67 5/2(−) 4.1 7.56(8) −1.50(7) 7.18(15) 0.28(3) 7.46(15) 6.8(8)
    68 0(+) 18.8 6.03(3) 0 4.57(5) 0 4.57(5) 1.1(1)
    70 0(+) 0.6 6.(1.) E 0 4.5(1.5) 0 4.5(1.5) 0.092(5)
                     
Ga 31       7.288(2)   6.675(4) 0.16(3) 6.83(3) 2.75(3)
    69 3/2(−) 60.1 7.88(2) −0.85(5) 7.80(4) 0.091(11) 7.89(4) 2.18(5)
    71 3/2(−) 39.9 6.40(3) −0.82(4) 5.15(5) 0.084(8) 5.23(5) 3.61(10)
                     
Ge 32       8.185(20)   8.42(4) 0.18(7) 8.60(6) 2.20(4)
    70 0(+) 20.5 10.0(1) 0 12.6(3) 0 12.6(3) 3.0(2)
    72 0(+) 27.4 8.51(10) 0 9.1(2) 0 9.1(2) 0.8(2)
    73 9/2(+) 7.8 5.02(4) 3.4(3) 3.17(5) 1.5(3) 4.7(3) 15.1(4)
    74 0(+) 36.5 7.58(10) 0 7.2(2) 0 7.2(2) 0.4(2)
    76 0(+) 7.8 8.21(1.5) 0 8.(3.) 0 8.(3.) 0.16(2)
                     
As 33 75 3/2(−) 100 6.58(1) −0.69(5) 5.44(2) 0.060(10) 5.50(2) 4.5(1)
                     
Se 34       7.970(9)   7.98(2) 0.33(6) 8.30(6) 11.7(2)
    74 0(+) 0.9 0.8(3.0) 0 0.1(6) 0 0.1(6) 51.8(1.2)
    76 0(+) 9.0 12.2(1) 0 18.7(3) 0 18.7(3) 85.(7.)
    77 1/2(−) 7.6 8.25(8) [\pm]0.6(1.6) 8.6(2) 0.05(26) 8.65(16) 42.(4.)
    78 0(+) 23.5 8.24(9) 0 8.5(2) 0 8.5(2) 0.43(2)
    80 0(+) 49.6 7.48(3) 0 7.03(6) 0 7.03(6) 0.61(5)
    82 0(+) 9.4 6.34(8) 0 5.05(13) 0 5.05(13) 0.044(3)
                     
Br 35       6.795(15)   5.80(3) 0.10(9) 5.90(9) 6.9(2)
    79 3/2(−) 50.69 6.80(7) −1.1(2) 5.81(12) 0.15(6) 5.96(13) 11.0(7)
    81 3/2(−) 49.31 6.79(7) 0.6(1) 5.79(12) 0.05(2) 5.84(12) 2.7(2)
                     
Kr 36       7.81(2)   7.67(4) 0.01(14) 7.68(13) 25.(1.)
    78 0(+) 0.35   0   0   6.4(9)
    80 0(+) 2.25   0   0   11.8(5)
    82 0(+) 11.6   0   0   29.(20.)
    83 9/2(+) 11.5   185(30.)        
    84 0(+) 57.0   0   0   0.113(15)
    86 0(+) 17.3 8.1(2) 0 8.2(4) 0 8.2(4) 0.003(2)
                     
Rb 37       7.09(2)   6.32(4) 0.5(4) 6.8(4) 0.38(4)
    85 5/2(−) 72.17 7.03(10) 6.2(2) 0.5(5) E 6.7(5) 0.48(1)
    87 3/2(−) 27.83 7.23(12) 6.6(2) 0.5(5) E 7.1(5) 0.12(3)
                     
Sr 38       7.02(2)   6.19(4) 0.06(11) 6.25(10) 1.28(6)
    84 0(+) 0.56 7.(1.) E 0 6.(2.) 0 6.(2.) 0.87(7)
    86 0(+) 9.86 5.67(5) 0 4.04(7) 0 4.04(7) 1.04(7)
    87 9/2(+) 7.00 7.40(7) 6.88(13) 0.5(5) E 7.4(5) 16.(3.)
    88 0(+) 82.58 7.15(6) 0 6.42(11) 0 6.42(11) 0.058(4)
                     
Y 39 89 1/2(−) 100 7.75(2) 1.1(3) 7.55(4) 0.15(8) 7.70(9) 1.28(2)
                     
Zr 40       7.16(3)   6.44(5) 0.02(15) 6.46(14) 0.185(3)
    90 0(+) 51.45 6.4(1) 0 5.1(2) 0 5.1(2) 0.011(5)
    91 5/2(+) 11.32 8.7(1) −1.08(15) 9.5(2) 0.15(4) 9.7(2) 1.17(10)
    92 0(+) 17.19 7.4(2) 0 6.9(4) 0 6.9(4) 0.22(6)
    94 0(+) 17.28 8.2(2) 0 8.4(4) 0 8.4(4) 0.0499(24)
    96 0(+) 2.76 5.5(1) 0 3.8(1) 0 3.8(1) 0.0229(10)
                     
Nb 41 93 9/2(+) 100 7.054(3) −0.139(10) 6.253(5) 0.0024(3) 6.255(5) 1.15(5)
                     
Mo 42       6.715(2)   5.67(3) 0.04(5) 5.71(4) 2.48(4)
    92 0(+) 14.84 6.91(8) 0 6.00(14) 0 6.00(14) 0.019(2)
    94 0(+) 9.25 6.80(7) 0 5.81(12) 0 5.81(12) 0.015(2)
    95 5/2(+) 15.92 6.91(6) 6.00(10) 0.5(5) E 6.5(5) 13.1(3)
    96 0(+) 16.68 6.20(6) 0 4.83(9) 0 4.83(9) 0.5(2)
    97 5/2(+) 9.55 7.24(8) 6.59(15) 0.5(5) E 7.1(5) 2.5(2)
    98 0(+) 24.13 6.58(7) 0 5.44(12) 0 5.44(12) 0.127(6)
    100 0(+) 9.63 6.73(7) 0 5.69(12) 0 5.69(12) 0.4(2)
                     
Tc 43                  
    99 9/2(+) (2.13×105a) 6.8(3) 5.8(5) 0.5(5) E 6.3(7) 20.(1.)
                     
Ru 44       7.03(3)   6.21(5) 0.4(1) 6.6(1) 2.56(13)
    96 0(+) 5.5 0 0 0.28(2)      
    98 0(+) 1.9 0 0 < 8.0      
    99 5/2(+) 12.7 6.9(1.0)          
    100 0(+) 12.6 0 0 4.8(6)      
    101 5/2(+) 17.0 3.3(9)          
    102 0(+) 31.6 0 0 1.17(7)      
    104 0(+) 18.7 0 0 0.31(2)      
                     
Rh 45 103 1/2(−) 100 5.88(4) 4.34(6) 0.3(3) E 4.6(3) 144.8(7)
                     
Pd 46       5.91(6)   4.39(9) 0.093(9) 4.48(9) 6.9(4)
    102 0(+) 1.02 7.7(7) E 0 7.5(1.4) 0 7.5(1.4) 3.4(3)
    104 0(+) 11.14 7.7(7) E 0 7.5(1.4) 0 7.5(1.4) 0.6(3)
    105 5/2(+) 22.33 5.5(3) −2.6(1.6) 3.8(4) 0.8(1.0) 4.6(1.1) 20.(3.)
    106 0(+) 27.33 6.4(4) 0 5.1(6) 0 5.1(6) 0.304(29)
    108 0(+) 26.46 4.1(3) 0 2.1(3) 0 2.1(3) 8.5(5)
    110 0(+) 11.72 7.7(7)E 0 7.5(1.4) 0 7.5(1.4) 0.226(31)
                     
Ag 47       5.922(7)   4.407(10) 0.58(3) 4.99(3) 63.3(4)
    107 1/2(−) 51.839 7.555(11) 1.00(13) 7.17(2) 0.13(3) 7.30(4) 37.6(1.2)
    109 1/2(−) 48.161 4.165(11) −1.60(13) 2.18(1) 0.32(5) 2.50(5) 91.0(1.0)
                     
Cd 48       4.87(5)   3.04(6) 3.46(13) 6.50(12) 2520.(50.)
          −0.70(1)[i]          
    106 0(+) 1.25 5.(2.) E 0 3.1(2.5) 0 3.1(2.5) 1.
    108 0(+) 0.89 5.4(1) 0 3.7(1) 0 3.7(1) 1.1(3)
    110 0(+) 12.51 5.9(1) 0 4.4(1) 0 4.4(1) 11.(1.)
    111 1/2(+) 12.81 6.5(1) 5.3(2) 0.3(3) E 5.6(4) 24(3.)
    112 0(+) 24.13 6.4(1) 0 5.1(2) 0 5.1(2) 2.2(5)
    *113 1/2(+) 12.22 −8.0(2) 12.1(4) 0.3(3) E 12.4(5)   20600(400.)
          −5.73(11)[i]          
    114 0(+) 28.72 7.5(1) 0 7.1(2) 0 7.1(2) 0.34(2)
    116 0(+) 7.47 6.3(1) 0 5.0(2) 0 5.0(2) 0.075(13)
                     
In 49     4.065(20) 2.08(2) 0.54(11) 2.62(11) 193.8(1.5)    
          −0.0539(4)[i]          
    113 9/2(+) 43 5.39(6) [\pm]0.017(1) 3.65(8) 0.000037(5) 3.65(8) 12.0(1.1)
    115 9/2(+) 957 4.01(2) −2.1(2) 2.02(2) 0.55(11) 2.57(11) 202(2.)
          −0.0562(6)[i]          
                     
Sn 50       6.225(2)   4.870(3) 0.022(5) 4.892(6) 0.626(9)
    112 0(+) 1.0 6.1(1.) E 0 4.5(1.5) 0 4.5(1.5) 1.01(11)
    114 0(+) 0.7 6.2(3) 0 4.8(5) 0 4.8(5) 0.114(30)
    115 1/2(+) 0.4 6.(1.) E 4.5(1.5) 0.3(3) E 4.8(1.5) 30(7.)  
    116 0(+) 14.7 5.93(5) 0 4.42(7) 0 4.42(7) 0.14(3)
    117 1/2(+) 7.7 6.48(5) 5.28(8) 0.3(3) E 5.6(3) 2.3(5)  
    118 0(+) 24.3 6.07(5) 0 4.63(8) 0 4.63(8) 0.22(5)
    119 1/2(+) 8.6 6.12(5) 4.71(8) 0.3(3) E 5.0(3) 2.2(5)  
    120 0(+) 32.4 6.49(5) 0 5.29(8) 0 5.29(8) 0.14(3)
    122 0(+) 4.6 5.74(5) 0 4.14(7) 0 4.14(7) 0.18(2)
    124 0(+) 5.6 5.97(5) 0 4.48(8) 0 4.48(8) 0.133(5)
                     
Sb 51       5.57(3)   3.90(4) 0.00(7) 3.90(6) 4.91(5)
    121 7/2(+) 57.3 5.71(6) −0.05(15) 4.10(9) 0.0003(19) 4.10(9) 5.75(12)
    123 5/2(+) 42.7 5.38(7) −0.10(15) 3.64(9) 0.001(4) 3.64(9) 3.8(2)
                     
Te 52       5.80(3)   4.23(4) 0.09(1) 4.32(4) 4.05(5)
    120 0(+) 0.096 5.3(5) 0 3.5(7) 0 3.4(7) 2.3(3)
    122 0(+) 2.60 3.8(2) 0 1.8(2) 0 1.8(2) 3.4(5)
    123 1/2(+) 0.908 −0.05(25) −2.04(9) 0.002(3) 0.52(5) 0.52(5) 418(30.)
          −0.116(8)[i]          
    124 0(+) 4.816 7.96(10) 0 8.0(2) 0 8.0(2) 6.8(1.3)
    125 1/2(+) 7.14 5.02(8) −0.26(13) 3.17(10) 0.008(8) 3.18(10) 1.55(16)
    126 0(+) 18.95 5.56(7) 0 3.88(10) 0 3.88(10) 1.04(15)
    128 0(+) 31.69 5.89(7) 0 4.36(10) 0 4.36(10) 0.215(8)
    130 0(+) 33.80 6.02(7) 0 4.55(11) 0 4.55(11) 0.29(6)
                     
I 53 127 5/2(+) 100 5.28(2) 1.58(15) 3.50(3) 0.31(6) 3.81(7) 6.15(6)
                     
Xe 54       4.92(3)   3.04(4)     23.9(1.2)
    124 0(+) 0.10   0   0   165.(20.)
    126 0(+) 0.09   0   0   3.5(8)
    128 0(+) 1.91   0   0   < 8.
    129 1/2(+) 26.4           21.(5.)
    130 0(+) 4.1   0   0   < 26.
    131 3/2(+) 21.2           85.(10.)
    132 0(+) 26.9   0   0   0.45(6)
    134 0(+) 10.4   0   0   0.265(20)
    136 0(+) 8.9   0   0   0.26(2)
                     
Cs 55 133 7/2(+) 100 5.42(2) 1.29(15) 3.69(3) 0.21(5) 3.90(6) 29.0(1.5)
                     
Ba 56       5.07(3)   3.23(4) 0.15(11) 3.38(10) 1.1(1)
    130 0(+) 0.11 −3.6(6) 0 1.6(5) 0 1.6(5) 30(5.)
    132 0(+) 0.10 7.8(3) 0 7.6(6) 0 7.6(6) 7.0(8)
    134 0(+) 2.42 5.7(1) 0 4.08(14) 0 4.08(14) 2.0(1.6)
    135 3/2(+) 6.59 4.67(10)   2.74(12) 0.5(5) E 3.2(5) 5.8(9)
    136 0(+) 7.85 4.91(8) 0 3.03(10) 0 3.03(10) 0.68(17)
    137 3/2(+) 11.23 6.83(10)   5.86(17) 0.5(5) E 6.4(5) 3.6(2)
    138 0(+) 71.70 4.84(8) 0 2.94(10) 0 2.94(10) 0.27(14)
                     
La 57       8.24(4)   8.53(8) 1.13(19) 9.66(17) 8.97(5)
    138 5(+) 0.09 8.(2.) E 8.(4.) 0.5(5) E 8.5(4.0) 57.(6.)  
    139 7/2(+) 99.91 8.24(4) 3.0(2) 8.53(8) 1.13(15) 9.66(17) 8.93(4)
                     
Ce 58       4.84(2)   2.94(2) 0.00(10) 2.94(10) 0.63(4)
    136 0(+) 0.19 5.80(9) 0 4.23(13) 0 4.23(13) 7.3(1.5)
    138 0(+) 0.25 6.70(9) 0 5.64(15) 0 5.64(15) 1.1(3)
    140 0(+) 88.48 4.84(9) 0 2.94(11) 0 2.94(11) 0.57(4)
    142 0(+) 11.08 4.75(9) 0 2.84(11) 0 2.84(11) 0.95(5)
                     
Pr 59 141 5/2(+) 100 4.58(5) −0.35(3) 2.64(6) 0.015(3) 2.66(6) 11.5(3)
                     
Nd 60       7.69(5)   7.43(10) 9.2(8) 16.6(8) 50.5(1.2)
    142 0(+) 27.16 7.7(3) 0 7.5(6) 0 7.5(6) 18.7(7)
    143 7/2(−) 12.18 14.2(5) E [\pm]21.1(6) 25.(7.) 55.(7.) 80.(2.) 334.(10.)
    144 0(+) 23.80 2.8(3) 0 1.0(2) 0 1.0(2) 3.6(3)
    145 7/2(−) 8.29 14.2(5) E 25.(7.) 5.(5.) E 30.(9.) 42.(2.)
    146 0(+) 17.19 8.7(2) 0 9.5(4) 0 9.5(4) 1.4(1)
    148 0(+) 5.75 5.7(3) 0 4.1(4) 0 4.1(4) 2.5(2)
    150 0(+) 5.63 5.3(2) 0 3.5(3) 0 3.5(3) 1.2(2)
                     
Pm 61                  
    147 7/2(+) (2.62a) 12.6(4) [\pm]3.2(2.5) 20.0(1.3) 1.3(2.0) 21.3(1.5) 168.4(3.5)
                     
Sm 62       0.80(2)   0.422(9) 39.(3.) 39.(3.) 5922.(56.)
          −1.65(2)[i]          
    144 0(+) 3.1 −3.(4.) E 0 1.(3.) 0 1.(3.) 0.7(3)
    147 7/2(−) 15.1 14(3.) [\pm]11.(7.) 25.(11.) 14.(19.) 39(16.) 57(3.)
    148 0(+) 11.3 −3.(4.) E 0 1.(3.) 0 1.(3.) 2.4(6)
    *149 7/2(−) 13.9 −19.2(1) [\pm]31.4(6) 63.5(6) 137.(5.) 200.(5.) 42080.(400.)
          −11.7(1)[i] −10.3(1)[i]        
    150 0(+) 7.4 14(3.) 0 25(11.) 0 25(11.) 104(4.)
    152 0(+) 26.6 −5.0(6) 0 3.1(8) 0 3.1(8) 206.(6.)
    154 0(+) 22.6 9.3(1.0) 0 11.(2.) 0 11.(2.) 8.4(5)
                     
Eu 63       7.22(2)   6.75(4) 2.5(4) 9.2(4) 4530.(40.)
          −1.26(1)[i]          
    *151 5/2(+) 47.8 6.13(14) [\pm]4.5(4) 5.5(2) 3.1(4) 8.4(4) 9100(100.)
          −2.53(3)[i] −2.14(2)[i]        
    153 5/2(+) 52.2 8.22(12) [\pm]3.2(9) 8.5(2) 1.3(7) 9.8(7) 312.(7.)
                     
Gd 64       6.5(5)   29.3(8) 151.(2.) 180.(2.) 49700.(125.)
          −13.82(3)[i]          
    152 0(+) 0.2 10.(3.) E 0 13.(8.) 0 13.(8.) 735.(20.)
    154 0(+) 2.1 10.(3.) E 0 13.(8.) 0 13.(8.) 85.(12.)
    *155 3/2(−) 14.8 6.0(1) [\pm]5.(5.) E 40.8(4.) 25.(6.) 66.(6.) 61100.(400.)
          −17.0(1)[i] −13.16(9)[i]        
    156 0(+) 20.6 6.3(4) 0 5.0(6) 0 5.0(6) 1.5(1.2)
    *157 3/2(−) 15.7 −1.14(2) [\pm]5.(5.) E 650(4.) 394.(7.) 1044.(8.) 259000.(700.)
          −71.9(2)[i] −55.8(2)[i]        
    158 0(+) 24.8 9.(2.) 0 10.(5.) 0 10.(5.) 2.2(2)
    160 0(+) 21.8 9.15(5) 0 10.52(11) 0 10.52(11) 0.77(2)
                     
Tb 65 159 3/2(+) 100 7.38(3) −0.17(7) 6.84(6) 0.004(3) 6.84(6) 23.4(4)
                     
Dy 66       16.9(2)   35.9(8) 54.4(1.2) 90.3(9) 994.(13.)
          −0.276(4)[i]          
    156 0(+) 0.06 6.1(5) 0 4.7(8) 0 4.7(8) 33.(3.)
    158 0(+) 0.10 6.(4.) E 0 5.(6.) 0 5.(6.) 43.(6.)
    160 0(+) 2.34 6.7(4) 0 5.6(7) 0 5.6(7) 56.(5.)
    161 5/2(+) 19.0 10.3(4) [\pm]4.9(8) 13.3(1.0) 3.(1.) 16.(1.) 600.(25.)
    162 0(+) 25.5 −1.4(5) 0 0.25(18) 0 0.25(18) 194.(10.)
    163 5/2(−) 24.9 5.0(4) 1.3(3) 3.1(5) 0.21(10) 3.3(5) 124.(7.)
    164 0(+) 28.1 49.4(5) 0 307.(3.) 0 307.(3.) 2840.(40.)
          −0.79(1)[i]          
                     
Ho 67 165 7/2(−) 100 8.01(8) −1.70(8) 8.06(16) 0.36(3) 8.42(16) 64.7(1.2)
                     
Er 68       7.79(2)   7.63(4) 1.1(3) 8.7(3) 159.(4.)
    162 0(+) 0.14 8.8(2) 0 9.7(4) 0 9.7(4) 19.(2.)
    164 0(+) 1.56 8.2(2) 0 8.4(4) 0 8.4(4) 13.(2.)
    166 0(+) 33.4 10.6(2) 0 14.1(5) 0 14.1(5) 19.6(1.5)
    167 7/2(+) 22.9 3.0(3) 1.0(3) 1.1(2) 0.13(8) 1.2(2) 659.(16.)
    168 0(+) 27.1 7.4(4) 0 6.8(7) 0 6.9(7) 2.74(8)
    170 0(+) 14.9 9.6(5) 0 11.6(1.2) 0 11.6(1.2) 5.8(3)
                     
Tm 69 169 1/2(+) 100 7.07(3) 0.9(3) 6.28(5) 0.10(7) 6.38(9) 100.(2.)
                     
Yb 70       12.43(3)   19.42(9) 4.0(2) 23.05(18) 34.8(8)
    168 0(+) 0.14 −4.07(2) 0 2.13(2) 0 2.13(2) 2230.(40.)
          −0.62(1)[i]          
    170 0(+) 3.06 6.77(10) 0 5.8(2) 0 5.8(2) 11.4(1.0)
    171 1/2(−) 143 9.66(10) −5.59(17) 11.7(2) 3.9(2) 15.6(3) 48.6(2.5)
    172 0(+) 21.9 9.43(10) 0 11.2(2) 0 11.2(2) 0.8(4)
    173 5/2(−) 16.1 9.56(7) −5.3(2) 11.5(2) 3.5(3) 15.0(4) 17.1(1.3)
    174 0(+) 31.8 19.3(1) 0 46.8(5) 0 46.8(5) 69.4(5.0)
    176 0(+) 12.7 8.72(10) 0 9.6(2) 0 9.6(2) 2.85(5)
                     
Lu 71       7.21(3)   6.53(5) 0.7(4) 7.2(4) 74.(2.)
    175 7/2(+) 97.39 7.24(3) [\pm]2.2(7) 6.59(5) 0.6(4) 7.2(4) 21.(3.)
    *176 7(−) 2.61 6.1(1) [\pm]3.0(4) 4.7(2) 1.2(3) 5.9(4) 2065.(35.)
          −0.57(1)[i] +0.61(1)[i]        
                     
Hf 72       7.77(14)   7.6(3) 2.6(5) 10.2(4) 104.1(0.5)
    174 0(+) 0.2 10.9(1.1) 0 15.(3.) 0 15.(3.) 561.(35.)
    176 0(+) 5.2 6.61(18) 0 5.5(3) 0 5.5(3) 23.5(3.1)
    177 7/2(−) 18.6 0.8(1.0) E [\pm]0.9(1.3) 0.1(2) 0.1(3) 0.2(2) 373.(10.)
    178 0(+) 27.1 5.9(2) 0 4.4(3) 0 4.4(3) 84.(4.)
    179 9/2(+) 13.7 7.46(16) [\pm]1.06(8) 7.0(3) 0.14(2) 7.1(3) 41.(3.)
    180 0(+) 35.2 13.2(3) 0 21.9(1.0) 0 21.9(1.0) 13.04(7)
                     
Ta 73       6.91(7)   6.00(12) 0.01(17) 6.01(12) 20.6(5)
    *180 9(−) 0.012 7.(2.) E 6.2(3.5) 0.5(5) E 7.(4.) 563.(60.)
    181 7/2(+) 99.988 6.91(7) −0.29(3) 6.00(12) 0.011(2) 6.01(12) 20.5(5)
                     
W 74       4.86(2)   2.97(2) 1.63(6) 4.60(6) 18.3(2)
    180 0(+) 0.1 5.(3.) E 0 3.(4.) 0 3.(4.) 30.(20.)
    182 0(+) 26.3 6.97(14) 0 6.10(7) 0 6.10(7) 20.7(5)
    183 1/2(−) 14.3 6.53(4)   5.36(7) 0.3(3) E 5.7(3) 10.1(3)
    184 0(+) 30.7 7.48(6) 0 7.03(11) 0 7.03(11) 1.7(1)
    186 0(+) 28.6 −0.72(4) 0 0.065(7) 0 0.065(7) 37.9(0.6)
                     
Re 75       9.2(2)   10.6(5) 0.9(6) 11.5(3) 89.7(1.0)
    185 5/2(+) 37.40 9.0(3) [\pm]2.0(1.8) 10.2(7) 0.5(9) 10.7(6) 112.(2.)
    187 5/2(+) 62.60 9.3(3) [\pm]2.8(1.1) 10.9(7) 1.0(8) 11.9(4) 76.4(1.0)
                     
Os 76       10.7(2)   14.4(5) 0.3(8) 14.7(6) 16.0(4)
    184 0(+) 0.02 10.(2.) E 0 13.(5.) 0 13.(5.) 3000.(150.)
    186 0(+) 1.58 11.6(1.7) 0 17.(5.) 0 17.(5.) 80.(13.)
    187 1/2(−) 1.6 10.(2.) E 13.(5.) 0.3(3) E 13.(5.) 320(10.)
    188 0(+) 13.3 7.6(3) 0 7.3(6) 0 7.3(6) 4.7(5)
    189 3/2(−) 16.1 10.7(3)   14.4(8) 0.5(5) E 14.9(9) 25(4.)
    190 0(+) 26.4 11.0(3) 0 15.2(9) 0 15.2(8) 13.1(3)
    192 0(+) 41.0 11.5(4) 0 16.6(1.2) 0 16.6(1.2) 2.0(1)
                     
Ir 77       10.6(3)   14.1(8) 0.(3.) 14.(3.) 425.3(2.4)
    191 3/2(+) 37.3           954.(10.)
    193 3/2(+) 62.7           111.(5.)
                     
Pt 78       9.60(1)   11.58(2) 0.13(11) 11.71(11) 10.3(3)
    190 0(+) 0.01 9.0(1.0) 0 10.(2.) 0 10.(2.) 152.(4.)
    192 0(+) 0.79 9.9(5) 0 12.3(1.2) 0 12.3(1.2) 10.0(2.5)
    194 0(+) 32.9 10.55(8) 0 14.0(2) 0 14.0(2) 1.44(19)
    195 1/2(−) 33.8 8.83(9) −1.00(17) 9.8(2) 0.13(4) 9.9(2) 27.5(1.2)
    196 0(+) 25.3 9.89(8) 0 12.3(2) 0 12.3(2) 0.72(4)
    198 0(+) 7.2 7.8(1) 0 7.7(2) 0 7.6(2) 3.66(19)
                     
Au 79 197 3/2(+) 100 7.63(6) −1.84(10) 7.32(12) 0.43(5) 7.75(13) 98.65(9)
                     
Hg 80       12.692(15)   20.24(5) 6.6(1) 26.8(1) 372.3(4.0)
    196 0(+) 0.2 30.3(1.0) 0 115(8.) 0 115(8.) 3080(180.)
    198 0(+) 10.1   0   0   2.0(3)
    199 1/2(−) 17.0 16.9(4) [\pm]15.5(8) 36.(2.) 30.(3.) 66.(2.) 2150.(48.)
    200 0(+) 23.1   0   0   < 60.
    201 3/2(−) 13.2     7.8(2.0)      
    202 0(+) 29.6   0   0   4.89(5)
    204 0(+) 6.8   0   0   0.43(10)
                     
Tl 81       8.776(5)   9.678(11) 0.21(15) 9.89(15) 3.43(6)
    203 1/2(+) 29.524 6.99(16) 1.06(14) 6.14(28) 0.14(4) 6.28(28) 11.4(2)
    205 1/2(+) 70.476 9.52(7) −0.242(17) 11.39(17) 0.007(1) 11.40(17) 0.104(17)
                     
Pb 82       9.405(3)   11.115(7) 0.0030(7) 11.118(7) 0.171(2)
    204 0(+) 1.4 9.90(10) 0 12.3(2) 0 12.3(2) 0.65(7)
    206 0(+) 24.1 9.22(5) 0 10.68(12) 0 10.68(12) 0.0300(8)
    207 1/2(−) 22.1 9.28(4) 0.14(6) 10.82(9) 0.002(2) 10.82(9) 0.699(10)
    208 0(+) 52.4 9.50(2) 0 11.34(5) 0 11.34(5) 0.00048(3)
                     
Bi 83 209 9/2(−) 100 8.532(2) 0.259(15) 9.148(4) 0.0084(10) 9.156(4) 0.0338(7)
                     
Po 84                  
                     
At 85                  
                     
Rn 86                  
                     
Fr 87                  
                     
Ra 88                  
    226 0(+) (1.60×103a) 10.0(1.0) 0 13.(3.) 0 13.(3.) 12.8(1.5)
                     
Ac 89                  
                     
Th 90 232 0(+) 100 10.31(3) 0 13.36(8) 0 13.36(8) 7.37(6)
                     
Pa 91                  
    231 3/2(−) (3.28×104a) 9.1(3) 1 0.4(7) 0.1(3.3) 10.5(3.2) 200.6(2.3)
                     
U 92       8.417(5)   8.903(11) 0.005(16) 8.908(11) 7.57(2)
    233 5/2(+) (1.59×105a) 10.1(2) [\pm]1.(3.) 12.8(5) 0.1(6) 12.9(3) 574.7(1.0)
    234 0(+) 0.005 12.4(3) 0 19.3(9) 0 19.3(9) 100.1(1.3)
    235 7/2(−) 0.720 10.47(3) [\pm]1.3(6) 13.78(11) 0.2(2) 14.0(2) 680.9(1.1)
    238 0(+) 99.275 8.402(5) 0 8.871(11) 0 8.871(11) 2.68(2)
                     
Np 93                  
    237 5/2(+) (2.14×106a) 10.55(10)   14.0(3) 0.5(5)E 14.5(6) 175.9(2.9)
                     
Pu 94                  
    238 0(+) (87.74a) 14.1(5) 0 25.0(1.8) 0 25.0(1.8) 558.(7.)
    239 1/2(+) (2.41×104a) 7.7(1) [\pm]1.3(1.9) 7.5(2) 0.2(6) 7.7(6) 1017.3(2.1)
    240 0(+) (6.56×103a) 3.5(1) 0 1.54(9) 0 1.54(9) 289.6(1.4)
    242 0(+) (3.76×105a) 8.1(1) 0 8.2(2) 0 8.2(2) 18.5(5)
                     
Am 95                  
    243 5/2(−) (7.37×103a) 8.3(2) [\pm]2.(7.) 8.7(4) 0.3(2.6) 9.0(2.6) 75.3(1.8)
                     
Cm 96                  
    244 0(+) (18.10a) 9.5(3) 0 11.3(7) 0 11.3(7) 16.2(1.2)
    246 0(+) (4.7×103a) 9.3(2) 0 10.9(5) 0 10.9(5) 1.36(17)
    248 0(+) (3.5×105a) 7.7(2) 0 7.5(4) 0 7.5(4) 3.00(26)

4.4.4.1. Scattering lengths

| top | pdf |

The scattering of a neutron by a single bound nucleus is described within the Born approximation by the Fermi pseudopotential, [V({\bf r}) =\left({2\pi\hbar^2\over m}\right) b\delta({\bf r}), \eqno (4.4.4.1)]in which r is the position of the neutron relative to the nucleus, m the neutron's mass, and b the bound scattering length. The neutron has spin s and the nucleus spin I so that, if [I\neq0], the Fermi pseudopotential and, hence, the bound scattering length will be spin dependent. Since s = 1/2, the most general rotationally invariant expression for b is [b=b_c+{2b_i\over\sqrt{I(I+1)}}\,{\bf s}\cdot {\bf I}, \eqno (4.4.4.2)]in which the coefficients [b_c] and [b_i] are called the bound coherent and incoherent scattering lengths. If I = 0, then bi = 0 by convention.

4.4.4.2. Scattering and absorption cross sections

| top | pdf |

When a thermal neutron collides with a nucleus, it may be either scattered or absorbed. By absorption, we mean reactions such as [(n,\gamma)], (n, p), or (n, α), in which there is no neutron in the final state. The effect of absorption can be included by allowing the bound scattering length to be complex, [b=b'-ib''. \eqno (4.4.4.3)]The total bound scattering cross section is then given by [\sigma_s=4\pi\left\langle|b|{}^2\right\rangle, \eqno (4.4.4.4)]in which [\langle\,\rangle] denotes a statistical average over the neutron and nuclear spins and the absorption cross section is given by [\sigma_a={4\pi\over k}\langle b''\rangle, \eqno (4.4.4.5)]where k = 2π/λ is the wavevector of the incident neutron and λ is the wavelength.

If the neutron and/or the nucleus is unpolarized, then the total bound scattering cross section is of the form [\sigma_s=\sigma_c+\sigma_i, \eqno (4.4.4.6)]in which [\sigma_c] and [\sigma_i] are called the bound coherent and incoherent scattering cross sections and are given by [\sigma_c=4\pi|b_c|{}^2, \quad \sigma_i=4\pi|b_i|{}^2. \eqno (4.4.4.7)]Also, [b_c=\langle b\rangle, \eqno (4.4.4.8)]so that the absorption cross section is given by [\sigma_a={4\pi \over k}\, b''_c. \eqno (4.4.4.9)]The absorption cross section is therefore uniquely determined by the imaginary part of the bound coherent scattering length. It is only when the neutron and the nucleus are both polarized that the imaginary part of the bound incoherent scattering length contributes to the value of [\sigma_a].

For most nuclides, the scattering lengths and, hence, the scattering cross sections are constant in the thermal-neutron region, and the absorption cross sections are inversely proportional to k. Since k is proportional to the neutron velocity v, the absorption is said to obey a 1/v law. By convention, absorption cross sections are tabulated for a velocity v = 2200 m s−1, which corresponds to a wavevector k = 3.494 Å−1, a wavelength λ = 1.798 Å, or an energy E = 25.30 meV.

The only major deviations from the 1/v law are for a few heavy nuclides (specifically, 113Cd, 149Sm, 151Eu, 155Gd, 157Gd, 176Lu, and 180Ta), which have an (n, γ) resonance at thermal-neutron energies. For these nuclides (which are indicated by the symbol * in Table 4.4.4.1[link]), the scattering lengths and cross sections are strongly energy dependent. The scattering lengths of the resonant rare-earth nuclides have been tabulated as a function of energy by Lynn & Seeger (1990[link]).

4.4.4.3. Isotope effects

| top | pdf |

The coefficients [b_c] and [b_i] in (4.4.4.2)[link] for the bound scattering length depend on the particular isotope under consideration, and this provides an additional source of incoherence in the scattering of neutrons by a mixture of isotopes. If [\langle\,\rangle] is now taken to denote an average over both the spin and isotope distributions, then the expressions (4.4.4.8)[link] for [b_c], (4.4.4.4)[link] for [\sigma_s], and (4.4.4.5)[link] for [\sigma_a] also apply to a mixture of isotopes. Hence, if [c_l] denotes the mole fraction of isotopes of type l, so that [\textstyle\sum\limits_l\,c_l=1, \eqno (4.4.4.10)]then, for an isotopic mixture, [b_c=\textstyle\sum\limits_l\, c_l\,b_{cl}, \eqno (4.4.4.11)] [\sigma_s=\textstyle\sum\limits_l\, c_l\,\sigma_{sl}, \eqno (4.4.4.12)]and [\sigma_a=\textstyle\sum\limits_l\,c_l\,\sigma_{al}. \eqno (4.4.4.13)]The bound coherent scattering cross section of the mixture is given, as before, by [\sigma_c=4\pi|b_c|{}^2, \eqno (4.4.4.14)]while the bound incoherent scattering cross section is defined as [\sigma_i=\sigma_s-\sigma_c. \eqno (4.4.4.15)]Hence, it follows that [\sigma_i\equiv 4\pi|b_i|^2=\sigma_i({\rm spin})+\sigma_i({\rm isotope}), \eqno (4.4.4.16)]in which the contribution from spin incoherence is given by [\sigma_i({\rm spin})=\textstyle\sum\limits_l\, c_l\sigma_{il} = 4\pi\sum\limits_l\, c_l|b_{il}|{}^2, \eqno (4.4.4.17)]and that from isotope incoherence is given by [\sigma_i({\rm isotope})= 4\pi\textstyle\sum\limits_{l\lt l'} \,c_l c_{l'}|b_{cl}-b_{cl'}|{}^2. \eqno (4.4.4.18)]Note that for a mixture of isotopes only the magnitude of [b_i] is defined by (4.4.4.16)[link], and its sign is arbitrary. However, for the individual isotopes, both the magnitude and sign (or complex phase) of [b_i] are defined in (4.4.4.2)[link].

4.4.4.4. Correction for electromagnetic interactions

| top | pdf |

The effective bound coherent scattering length that describes the interaction of a neutron with an atom includes additional contributions from electromagnetic interactions (Bacon, 1975[link]; Sears, 1986a[link], 1996[link]). For a neutral atom with atomic number Z, this quantity is of the form [b_c(q)=b_c(0)-b_e[Z-f(q)], \eqno (4.4.4.19)]where q is the wavevector transfer in the collision, [b_c(0)] and [b_e] are constants, and f(q) is the atomic scattering factor (Section 6.1.1[link] ). The latter quantity is the Fourier transform of the electron number density and is normalized such that f(0) = Z.

The main contribution to [b_c(0)] is from the nuclear interaction between the neutron and the nucleus but there is also a small electrostatic contribution ([\le] 0.5%) arising from the neutron electric polarizability. The coefficient [b_e] is called the neutron–electron scattering length and has the value −1.32 (4) × 10−3 fm (Koester, Waschkowski & Meier, 1988[link]). This quantity is due mainly to the Foldy interaction with a small additional contribution (∼10%) from the intrinsic charge distribution of the neutron.

The correction of the bound coherent scattering length for electromagnetic interactions requires a knowledge of the atomic scattering factor f(q). Tables 6.1.1.1[link] and 6.1.1.3[link] provide accurate values of f(q) obtained from relativistic Hartree–Fock calculations for all the atoms and chemically important ions in the Periodic Table. Alternatively, since the correction is small (∼1%), one can often use the approximate analytical expression (Sears, 1986a[link], 1996[link]) [f(q)={Z \over\sqrt{1+3(q/q_0)^2}} \eqno (4.4.4.20)]with [q_0=\gamma Z^{1/3}]. The value γ = 1.90 ± 0.07 Å−1 provides a good fit to the Hartree–Fock results in Table 6.1.1.1[link] for [Z\ge20].

4.4.4.5. Measurement of scattering lengths

| top | pdf |

The development of modern neutron-optical techniques during the past 25 years has produced a dramatic increase in the accuracy with which scattering lengths can be measured (Koester, 1977[link]; Klein & Werner, 1983[link]; Werner & Klein, 1986[link]; Sears, 1989[link]; Koester, Rauch & Seymann, 1991[link]). The measurements employ a number of effects – mirror reflection, prism refraction, gravity refractometry, Christiansen filter, and interferometry – all of which are based on the fact that the neutron index of refraction, n, is uniquely determined by [b_c(0)] through the relation [n^2 =1-{4\pi\over k^2}\,\rho\,b_c(0), \eqno (4.4.4.21)]in which ρ is the number of atoms per unit volume. Apart from a small (≤0.01%) local-field correction (Sears, 1985[link], 1989[link]), this expression is exact.

In methods based on diffraction, such as Bragg reflection by powders or dynamical diffraction by perfect crystals, the measured quantity is the unit-cell structure factor [|F_{hkl}|]. This quantity depends on [b_c(q)] in which q is equal to the magnitude of the reciprocal-lattice vector corresponding to the relevant Bragg planes, i.e. [q=2k\sin\theta_{hkl}, \eqno (4.4.4.22)]where [\theta_{hkl}] is the Bragg angle. In dynamical diffraction measurements, it is usual for the authors to correct their results for electromagnetic interactions so that the published quantity is again [b_c(0)]. In the past, this correction has not usually been made for the scattering lengths obtained from Bragg reflection by powders. However, these latter measurements are accurate only to ±2 or 3% so that the correction is then relatively unimportant.

The essential point is that all the bound coherent scattering lengths in Table 4.4.4.1[link] with the experimental uncertainties less than 1% represent [b_c(0)] and should therefore be corrected for electromagnetic interactions before being used in the interpretation of neutron diffraction experiments. Failure to make this correction will introduce systematic errors of 0.5 to 2% in the unit-cell structure factors at large q, and corresponding errors of 1 to 4% in the calculated intensities.

Expression (4.4.4.21)[link] assumed that the neutrons and/or the nuclei are unpolarized. If the neutrons and the nuclei are both polarized then [b_c(0)] is replaced by [\langle b(0)\rangle], which depends on both the coherent and incoherent scattering lengths. If the coherent scattering length is known, neutron-optical experiments with polarized neutrons and nuclei can then be used to determine the incoherent scattering length (Glättli & Goldman, 1987[link]).

4.4.4.6. Compilation of scattering lengths and cross sections

| top | pdf |

The bound scattering lengths and cross sections of almost all the elements in the Periodic Table, as well as those of the individual isotopes, are listed in Table 4.4.4.1[link]. As in earlier versions of this table (Sears, 1984[link], 1986b[link], 1992a[link],b[link]), our primary aim, has been to take the best current values of the bound coherent and incoherent neutron scattering lengths and to compute from them a consistent set of bound scattering cross sections. In the present version, we have used the values of the coherent and incoherent scattering lengths recommended by Koester, Rauch & Seymann (1991[link]), supplemented with a few more recently measured values, and have computed from them the corresponding scattering cross sections. The trailing digits in parentheses give the standard errors calculated from the errors in the input data using the statistical theory of error propagation (Young, 1962[link]). The imaginary parts of the scattering lengths, which are appreciable only for strongly absorbing nuclides, were calculated from the measured absorption cross sections (Mughabghab, Divadeenam & Holden, 1981[link]; Mughabghab, 1984[link]) and are listed beneath the real parts of Table 4.4.4.1[link].

In a few cases, where the scattering lengths have not yet been measured directly, the available scattering cross-section data (Mughabghab, Divadeenam & Holden, 1981[link]; Mughabghab, 1984[link]) were used to obtain the scattering lengths. Equations (4.4.4.11)[link], (4.4.4.12)[link], and (4.4.4.13)[link] were used, where necessary, to fill gaps in Table 4.4.4.1[link]. For some elements, these relations indicated inconsistencies in the data. In such cases, appropriate adjustments in the values of some of the quantities were made. In almost all cases, such adjustments were comparable with the stated errors. Finally, for some elements, it was necessary to estimate arbitrarily the scattering lengths of one or two isotopes in order to be able to complete the table. Such estimates are indicated by the letter `E' and were usually made only for isotopes of low natural abundance where the estimated values have only a marginal effect on the final results. Apart from the inclusion of new data for Ti and Mn, the values listed in Table 4.4.4.1[link] are the same as in Sears (1992b[link]).

4.4.5. Magnetic form factors

| top | pdf |
P. J. Browna

The form factors used in the calculations of the cross sections for magnetic scattering of neutrons are defined in Subsection 6.1.2.3[link] as [\langle\, j_l(k)\rangle =\textstyle \int\limits^\infty_0\,U^2(r)\, j_l(kr)4\pi r^2\,{\rm d} r, \eqno (4.4.5.1)]in which U(r) is the radial wavefunction for the unpaired electrons in the atom, k is the length of the scattering vector, and [j_l(kr)] is the lth-order spherical Bessel function.

Tables 4.4.5.1[link][link][link][link][link][link][link]–4.4.5.8[link] give the coefficients in an analytical approximation to the [\langle \,j_0\rangle] magnetic form factors for the 3d and 4d transition series, the 4f electrons of rare-earth ions, and the 5f electrons of some actinide ions. The approximation has the form used by Forsyth & Wells (1959[link]) but allowing three instead of two exponential terms: [\eqalignno{ \langle\, j_0(s)\rangle &=A\exp(-as^2)+B\exp(-bs^2) \cr &\quad +C\exp(-cs^2)+D, &(4.4.5.2)}]where s is the value of [(\sin\theta)/\lambda] in Å−1.

Table 4.4.5.1| top | pdf |
<j0> form factors for 3d transition elements and their ions

Atom or ionAaBbCcDe
Sc 0.2512 90.030 0.3290 39.402 0.4235 14.322 −0.0043 0.2029
Sc+ 0.4889 51.160 0.5203 14.076 −0.0286 0.179 0.0185 0.1217
Sc2+ 0.5048 31.403 0.5186 10.990 −0.0241 1.183 0.0000 0.0578
Ti 0.4657 33.590 0.5490 9.879 −0.0291 0.323 0.0123 0.1088
Ti+ 0.5093 36.703 0.5032 10.371 −0.0263 0.311 0.0116 0.1125
Ti2+ 0.5091 24.976 0.5162 8.757 −0.0281 0.916 0.0015 0.0589
Ti3+ 0.3571 22.841 0.6688 8.931 −0.0354 0.483 0.0099 0.0575
V 0.4086 28.811 0.6077 8.544 −0.0295 0.277 0.0123 0.0970
V+ 0.4444 32.648 0.5683 9.097 −0.2285 0.022 0.2150 0.1111
V2+ 0.4085 23.853 0.6091 8.246 −0.1676 0.041 0.1496 0.0593
V3+ 0.3598 19.336 0.6632 7.617 −0.3064 0.030 0.2835 0.0515
V4+ 0.3106 16.816 0.7198 7.049 −0.0521 0.302 0.0221 0.0433
Cr 0.1135 45.199 0.3481 19.493 0.5477 7.354 −0.0092 0.1975
Cr+ −0.0977 0.047 0.4544 26.005 0.5579 7.489 0.0831 0.1114
Cr2+ 1.2024 −0.005 0.4158 20.548 0.6032 6.956 −1.2218 0.0572
Cr3+ −0.3094 0.027 0.3680 17.035 0.6559 6.524 0.2856 0.0436
Cr4+ −0.2320 0.043 0.3101 14.952 0.7182 6.173 0.2042 0.0419
Mn 0.2438 24.963 0.1472 15.673 0.6189 6.540 −0.0105 0.1748
Mn+ −0.0138 0.421 0.4231 24.668 0.5905 6.655 −0.0010 0.1242
Mn2+ 0.4220 17.684 0.5948 6.0050 0.0043 −0.609 −0.0219 0.0589
Mn3+ 0.4198 14.283 0.6054 5.469 0.9241 −0.009 −0.9498 0.0392
Mn4+ 0.3760 12.566 0.6602 5.133 −0.0372 0.563 0.0011 0.0393
Fe 0.0706 35.008 0.3589 15.358 0.5819 5.561 −0.0114 0.1398
Fe+ 0.1251 34.963 0.3629 15.514 0.5223 5.591 −0.0105 0.1301
Fe2+ 0.0263 34.960 0.3668 15.943 0.6188 5.594 −0.0119 0.1437
Fe3+ 0.3972 13.244 0.6295 4.903 −0.0314 0.350 0.0044 0.0441
Fe4+ 0.3782 11.380 0.6556 4.592 −0.0346 0.483 0.0005 0.0362
Co 0.4139 16.162 0.6013 4.780 −0.1518 0.021 0.1345 0.1033
Co+ 0.0990 33.125 0.3645 15.177 0.5470 5.008 −0.0109 0.0983
Co2+ 0.4332 14.355 0.5857 4.608 −0.0382 0.134 0.0179 0.0711
Co3+ 0.3902 12.508 0.6324 4.457 −0.1500 0.034 0.1272 0.0515
Co4+ 0.3515 10.778 0.6778 4.234 −0.0389 0.241 0.0098 0.0390
Ni −0.0172 35.739 0.3174 14.269 0.7136 4.566 −0.0143 0.1072
Ni+ 0.0705 35.856 0.3984 13.804 0.5427 4.397 −0.0118 0.0738
Ni2+ 0.0163 35.883 0.3916 13.223 0.6052 4.339 −0.0133 0.0817
Ni3+ 0.0012 35.000 0.3468 11.987 0.6667 4.252 −0.0148 0.0883
Ni4+ −0.0090 35.861 0.2776 11.790 0.7474 4.201 −0.0163 0.0966
Cu 0.0909 34.984 0.4088 11.443 0.5128 3.825 −0.0124 0.0513
Cu+ 0.0749 34.966 0.4147 11.764 0.5238 3.850 −0.0127 0.0591
Cu2+ 0.0232 34.969 0.4023 11.564 0.5882 3.843 −0.0137 0.0532
Cu3+ 0.0031 34.907 0.3582 10.914 0.6531 3.828 −0.0147 0.0665
Cu4+ −0.0132 30.682 0.2801 11.163 0.7490 3.817 −0.0165 0.0767

Table 4.4.5.2| top | pdf |
<j0> form factors for 4d atoms and their ions

Atom or ionAaBbCcDe
Y 0.5915 67.608 1.5123 17.900 −1.1130 14.136 0.0080 0.3272
Zr 0.4106 59.996 1.0543 18.648 −0.4751 10.540 0.0106 0.3667
Zr+ 0.4532 59.595 0.7834 21.436 −0.2451 9.036 0.0098 0.3639
Nb 0.3946 49.230 1.3197 14.822 −0.7269 9.616 0.0129 0.3659
Nb+ 0.4572 49.918 1.0274 15.726 −0.4962 9.157 0.0118 0.3403
Mo 0.1806 49.057 1.2306 14.786 −0.4268 6.987 0.0171 0.4135
Mo+ 0.3500 48.035 1.0305 15.060 −0.3929 7.479 0.0139 0.3510
Tc 0.1298 49.661 1.1656 14.131 −0.3134 5.513 0.0195 0.3869
Tc+ 0.2674 48.957 0.9569 15.141 −0.2387 5.458 0.0160 0.3412
Ru 0.1069 49.424 1.1912 12.742 −0.3176 4.912 0.0213 0.3597
Ru+ 0.4410 33.309 1.4775 9.553 −0.9361 6.722 0.0176 0.2608
Rh 0.0976 49.882 1.1601 11.831 −0.2789 4.127 0.0234 0.3263
Rh+ 0.3342 29.756 1.2209 9.438 −0.5755 5.332 0.0210 0.2574
Pd 0.2003 29.363 1.1446 9.599 −0.3689 4.042 0.0251 0.2453
Pd+ 0.5033 24.504 1.9982 6.908 −1.5240 5.513 0.0213 0.1909

Table 4.4.5.3| top | pdf |
<j0> form factors for rare-earth ions

IonAaBbCcDe
Ce2+ 0.2953 17.685 0.2923 6.733 0.4313 5.383 −0.0194 0.0845
Nd2+ 0.1645 25.045 0.2522 11.978 0.6012 4.946 −0.0180 0.0668
Nd3+ 0.0540 25.029 0.3101 12.102 0.6575 4.722 −0.0216 0.0478
Sm2+ 0.0909 25.203 0.3037 11.856 0.6250 4.237 −0.0200 0.0408
Sm3+ 0.0288 25.207 0.2973 11.831 0.6954 4.212 −0.0213 0.0510
Eu2+ 0.0755 25.296 0.3001 11.599 0.6438 4.025 −0.0196 0.0488
Eu3+ 0.0204 25.308 0.3010 11.474 0.7005 3.942 −0.0220 0.0356
Gd2+ 0.0636 25.382 0.3033 11.212 0.6528 3.788 −0.0199 0.0486
Gd3+ 0.0186 25.387 0.2895 11.142 0.7135 3.752 −0.0217 0.0489
Tb2+ 0.0547 25.509 0.3171 10.591 0.6490 3.517 −0.0212 0.0342
Tb3+ 0.0177 25.510 0.2921 10.577 0.7133 3.512 −0.0231 0.0512
Dy2+ 0.1308 18.316 0.3118 7.665 0.5795 3.147 −0.0226 0.0315
Dy3+ 0.1157 15.073 0.3270 6.799 0.5821 3.020 −0.0249 0.0146
Ho2+ 0.0995 18.176 0.3305 7.856 0.5921 2.980 −0.0230 0.1240
Ho3+ 0.0566 18.318 0.3365 7.688 0.6317 2.943 −0.0248 0.0068
Er2+ 0.1122 18.122 0.3462 6.911 0.5649 2.761 −0.0235 0.0207
Er3+ 0.0586 17.980 0.3540 7.096 0.6126 2.748 −0.0251 0.0171
Tm2+ 0.0983 18.324 0.3380 6.918 0.5875 2.662 −0.0241 0.0404
Tm3+ 0.0581 15.092 0.2787 7.801 0.6854 2.793 −0.0224 0.0351
Yb2+ 0.0855 18.512 0.2943 7.373 0.6412 2.678 −0.0213 0.0421
Yb3+ 0.0416 16.095 0.2849 7.834 0.6961 2.672 −0.0229 0.0344

Table 4.4.5.4| top | pdf |
<j0> form factors for actinide ions

IonAaBbCcDe
U3+ 0.5058 23.288 1.3464 7.003 −0.8724 4.868 0.0192 0.1507
U4+ 0.3291 23.548 1.0836 8.454 −0.4340 4.120 0.0214 0.1757
U5+ 0.3650 19.804 3.2199 6.282 −2.6077 5.301 0.0233 0.1750
Np3+ 0.5157 20.865 2.2784 5.893 −1.8163 4.846 0.0211 0.1378
Np4+ 0.4206 19.805 2.8004 5.978 −2.2436 4.985 0.0228 0.1408
Np5+ 0.3692 18.190 3.1510 5.850 −2.5446 4.916 0.0248 0.1515
Np6+ 0.2929 17.561 3.4866 5.785 −2.8066 4.871 0.0267 0.1698
Pu3+ 0.3840 16.679 3.1049 5.421 −2.5148 4.551 0.0263 0.1280
Pu4+ 0.4934 16.836 1.6394 5.638 −1.1581 4.140 0.0248 0.1242
Pu5+ 0.3888 16.559 2.0362 5.657 −1.4515 4.255 0.0267 0.1287
Pu6+ 0.3172 16.051 3.4654 5.351 −2.8102 4.513 0.0281 0.1382
Am2+ 0.4743 21.776 1.5800 5.690 −1.0779 4.145 0.0218 0.1253
Am3+ 0.4239 19.574 1.4573 5.872 −0.9052 3.968 0.0238 0.1054
Am4+ 0.3737 17.862 1.3521 6.043 −0.7514 3.720 0.0258 0.1113
Am5+ 0.2956 17.372 1.4525 6.073 −0.7755 3.662 0.0277 0.1202
Am6+ 0.2302 16.953 1.4864 6.116 −0.7457 3.543 0.0294 0.1323
Am7+ 0.3601 12.730 1.9640 5.120 −1.3560 3.714 0.0316 0.1232

Table 4.4.5.5| top | pdf |
<j2> form factors for 3d transition elements and their ions

Atom or ionAaBbCcDe
Sc 10.8172 54.327 4.7353 14.847 0.6071 4.218 0.0011 0.1212
Sc+ 8.5021 34.285 3.2116 10.994 0.4244 3.605 0.0009 0.1037
Sc2+ 4.3683 28.654 3.7231 10.823 0.6074 3.668 0.0014 0.0681
Ti 4.3583 36.056 3.8230 11.133 0.6855 3.469 0.0020 0.0967
Ti+ 6.1567 27.275 2.6833 8.983 0.4070 3.052 0.0011 0.0902
Ti2+ 4.3107 18.348 2.0960 6.797 0.2984 2.548 0.0007 0.0640
Ti3+ 3.3717 14.444 1.8258 5.713 0.2470 2.265 0.0005 0.0491
V 3.7600 21.831 2.4026 7.546 0.4464 2.663 0.0017 0.0556
V+ 4.7474 23.323 2.3609 7.808 0.4105 2.706 0.0014 0.0800
V2+ 3.4386 16.530 1.9638 6.141 0.2997 2.267 0.0009 0.0565
V3+ 2.3005 14.682 2.0364 6.130 0.4099 2.382 0.0014 0.0252
V4+ 1.8377 12.267 1.8247 5.458 0.3979 2.248 0.0012 0.0399
Cr 3.4085 20.127 2.1006 6.802 0.4266 2.394 0.0019 0.0662
Cr+ 3.7768 20.346 2.1028 6.893 0.4010 2.411 0.0017 0.0686
Cr2+ 2.6422 16.060 1.9198 6.253 0.4446 2.372 0.0020 0.0480
Cr3+ 1.6262 15.066 2.0618 6.284 0.5281 2.368 0.0023 0.0263
Cr4+ 1.0293 13.950 1.9933 6.059 0.5974 2.346 0.0027 0.0366
Mn 2.6681 16.060 1.7561 5.640 0.3675 2.049 0.0017 0.0595
Mn+ 3.2953 18.695 1.8792 6.240 0.3927 2.201 0.0022 0.0659
Mn2+ 2.0515 15.556 1.8841 6.063 0.4787 2.232 0.0027 0.0306
Mn3+ 1.2427 14.997 1.9567 6.118 0.5732 2.258 0.0031 0.0336
Mn4+ 0.7879 13.886 1.8717 5.743 0.5981 2.182 0.0034 0.0434
Fe 1.9405 18.473 1.9566 6.323 0.5166 2.161 0.0036 0.0394
Fe+ 2.6290 18.660 1.8704 6.331 0.4690 2.163 0.0031 0.0491
Fe2+ 1.6490 16.559 1.9064 6.133 0.5206 2.137 0.0035 0.0335
Fe3+ 1.3602 11.998 1.5188 5.003 0.4705 1.991 0.0038 0.0374
Fe4+ 1.5582 8.275 1.1863 3.279 0.1366 1.107 −0.0022 0.0327
Co 1.9678 14.170 1.4911 4.948 0.3844 1.797 0.0027 0.0452
Co+ 2.4097 16.161 1.5780 5.460 0.4095 1.914 0.0031 0.0581
Co2+ 1.9049 11.644 1.3159 4.357 0.3146 1.645 0.0017 0.0459
Co3+ 1.7058 8.859 1.1409 3.309 0.1474 1.090 −0.0025 0.0462
Co4+ 1.3110 8.025 1.1551 3.179 0.1608 1.130 −0.0011 0.0374
Ni 1.0302 12.252 1.4669 4.745 0.4521 1.744 0.0036 0.0338
Ni+ 2.1040 14.866 1.4302 5.071 0.4031 1.778 0.0034 0.0561
Ni2+ 1.7080 11.016 1.2147 4.103 0.3150 1.533 0.0018 0.0446
Ni3+ 1.4683 8.671 0.1794 1.106 1.1068 3.257 −0.0023 0.0373
Ni4+ 1.1612 7.700 1.0027 3.263 0.2719 1.378 0.0025 0.0326
Cu 1.9182 14.490 1.3329 4.730 0.3842 1.639 0.0035 0.0617
Cu+ 1.8814 13.433 1.2809 4.545 0.3646 1.602 0.0033 0.0590
Cu2+ 1.5189 10.478 1.1512 3.813 0.2918 1.398 0.0017 0.0429
Cu3+ 1.2797 8.450 1.0315 3.280 0.2401 1.250 0.0015 0.0389
Cu4+ 0.9568 7.448 0.9099 3.396 0.3729 1.494 0.0049 0.0330

Table 4.4.5.6| top | pdf |
<j2> form factors for 4d atoms and their ions

Atom or ionAaBbCcDe
Y 14.4084 44.658 5.1045 14.904 −0.0535 3.319 0.0028 0.1093
Zr 10.1378 35.337 4.7734 12.545 −0.0489 2.672 0.0036 0.0912
Zr+ 11.8722 34.920 4.0502 12.127 −0.0632 2.828 0.0034 0.0737
Nb 7.4796 33.179 5.0884 11.571 −0.0281 1.564 0.0047 0.0944
Nb+ 8.7735 33.285 4.6556 11.605 −0.0268 1.539 0.0044 0.0855
Mo 5.1180 23.422 4.1809 9.208 −0.0505 1.743 0.0053 0.0655
Mo+ 7.2367 28.128 4.0705 9.923 −0.0317 1.455 0.0049 0.0798
Tc 4.2441 21.397 3.9439 8.375 −0.0371 1.187 0.0066 0.0645
Tc+ 6.4056 24.824 3.5400 8.611 −0.0366 1.485 0.0044 0.0806
Ru 3.7445 18.613 3.4749 7.420 −0.0363 1.007 0.0073 0.0533
Ru+ 5.2826 23.683 3.5813 8.152 −0.0257 0.426 0.0131 0.0830
Rh 3.3651 17.344 3.2121 6.804 −0.0350 0.503 0.0146 0.0545
Rh+ 4.0260 18.950 3.1663 7.000 −0.0296 0.486 0.0127 0.0629
Pd 3.3105 14.726 2.6332 5.862 −0.0437 1.130 0.0053 0.0492
Pd+ 4.2749 17.900 2.7021 6.354 −0.0258 0.700 0.0071 0.0768

Table 4.4.5.7| top | pdf |
<j2> form factors for rare-earth ions

IonAaBbCcDe
Ce2+ 0.9809 18.063 1.8413 7.769 0.9905 2.845 0.0120 0.0448
Nd2+ 1.4530 18.340 1.6196 7.285 0.8752 2.622 0.0126 0.0461
Nd3+ 0.6751 18.342 1.6272 7.260 0.9644 2.602 0.0150 0.0450
Sm2+ 1.0360 18.425 1.4769 7.032 0.8810 2.437 0.0152 0.0345
Sm3+ 0.4707 18.430 1.4261 7.034 0.9574 2.439 0.0182 0.0510
Eu2+ 0.8970 18.443 1.3769 7.005 0.9060 2.421 0.0190 0.0511
Eu3+ 0.3985 18.451 1.3307 6.956 0.9603 2.378 0.0197 0.0447
Gd2+ 0.7756 18.469 1.3124 6.899 0.8956 2.338 0.0199 0.0441
Gd3+ 0.3347 18.476 1.2465 6.877 0.9537 2.318 0.0217 0.0484
Tb2+ 0.6688 18.491 1.2487 6.822 0.8888 2.275 0.0215 0.0439
Tb3+ 0.2892 18.497 1.1678 6.797 0.9437 2.257 0.0232 0.0458
Dy2+ 0.5917 18.511 1.1828 6.747 0.8801 2.214 0.0229 0.0439
Dy3+ 0.2523 18.517 1.0914 6.736 0.9345 2.208 0.0250 0.0476
Ho2+ 0.5094 18.515 1.1234 6.706 0.8727 2.159 0.0242 0.0560
Ho3+ 0.2188 18.516 1.0240 6.707 0.9251 2.161 0.0268 0.0503
Er2+ 0.4693 18.528 1.0545 6.649 0.8679 2.120 0.0261 0.0413
Er3+ 0.1710 18.534 0.9879 6.625 0.9044 2.100 0.0278 0.0489
Tm2+ 0.4198 18.542 0.9959 6.600 0.8593 2.082 0.0284 0.0457
Tm3+ 0.1760 18.542 0.9105 6.579 0.8970 2.062 0.0294 0.0468
Yb2+ 0.3852 18.550 0.9415 6.551 0.8492 2.043 0.0301 0.0478
Yb3+ 0.1570 18.555 0.8484 6.540 0.8880 2.037 0.0318 0.0498

Table 4.4.5.8| top | pdf |
<j2> form factors for actinide ions

IonAaBbCcDe
U3+ 4.1582 16.534 2.4675 5.952 −0.0252 0.765 0.0057 0.0822
U4+ 3.7449 13.894 2.6453 4.863 −0.5218 3.192 0.0009 0.0928
U5+ 3.0724 12.546 2.3076 5.231 −0.0644 1.474 0.0035 0.0477
Np3+ 3.7170 15.133 2.3216 5.503 −0.0275 0.800 0.0052 0.0948
Np4+ 2.9203 14.646 2.5979 5.559 −0.0301 0.367 0.0141 0.0532
Np5+ 2.3308 13.654 2.7219 5.494 −0.1357 0.049 0.1224 0.0553
Np6+ 1.8245 13.180 2.8508 5.407 −0.1579