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

International Tables for Crystallography (2006). Vol. C, ch. 4.2, pp. 192-193

Section The continuous spectrum

U. W. Arndta The continuous spectrum

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


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

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

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

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

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


Cohen, E. R. & Taylor, B. N. (1987). The 1986 adjustment of the fundamental physical constants. Rev. Mod. Phys. 89, 1121–1148.
Kirkpatrick, P. & Wiedmann, L. (1945). Theoretical continuous X-ray energy and polarization. Phys. Rev. 67, 321–339.
Kramers, H. A. (1923). On the theory of X-ray absorption and of the continuous X-ray spectrum. Philos. Mag. 46, 836–871.
Kulenkampff, H. & Schmidt, L. (1943). Die Energieverteilung im Spektrum der Röntgen Bremsstrahlung. Ann. Phys. (Leipzig), 43, 494–512.
Sakurai, K. (1993). High-intensity X-ray line-focal spot for laboratory EXAFS measurements. Rev. Sci. Instrum. 64, 267–268.
Sakurai, K. & Sakurai, H. (1994). Comment on high intensity low tube-voltage X-ray source for laboratory EXAFS measurements. Rev. Sci. Instrum. 65, 2417–2418.
Stephenson, S. T. (1957). The continuous X-ray spectrum. Handbuch der Physik. XXX: X-rays, edited by S. Flügge, pp. 337–370. Berlin: Springer.
Stern, E. A. (1980). Editor. Laboratory EXAFS facilities 1980. AIP Conf. Proc. No. 64.
Tohji, K., Udagawa, Y., Kawasaki, T. & Masuda, K. (1983). Laboratory EXAFS spectrometer with a bent-crystal, a solid-state detector, and a fast detection system. Rev. Sci. Instrum. 54, 1482–1487.

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