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

International Tables for Crystallography (2006). Vol. C, ch. 7.4, pp. 657-661

Section 7.4.3. Compton scattering

N. G. Alexandropoulosa and M. J. Cooperb

7.4.3. Compton scattering

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7.4.3.1. Introduction

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In many diffraction studies, it is necessary to correct the intensities of the Bragg peaks for a variety of inelastic scattering processes. Compton scattering is only one of the incoherent processes although the term is often used loosely to include plasmon, Raman, and resonant Raman scattering, all of which may occur in addition to the more familiar fluorescence radiation and thermal diffuse scattering. The various interactions are summarized schematically in Fig. 7.4.3.1[link], where the dominance of each interaction is characterized by the energy and momentum transfer and the relevant binding energy.

[Figure 7.4.3.1]

Figure 7.4.3.1 | top | pdf |

Schematic diagram of the inelastic scattering interactions, ΔE = E1E2 is the energy transferred from the photon and K the momentum transfer. The valence electrons are characterized by the Fermi energy, EF, and momentum, kF ([\hbar] being taken as unity). The core electrons are characterized by their binding energy EB. The dipole approximation is valid when |K|a < 1, where a is the orbital radius of the scattering electron.

With the exception of thermal diffuse scattering, which is known to peak at the reciprocal-lattice points, the incoherent background varies smoothly through reciprocal space. It can be removed with a linear interpolation under the sharp Bragg peaks and without any energy analysis. On the other hand, in non-crystalline material, the elastic scattering is also diffused throughout reciprocal space; the point-by-point correction is consequently larger and without energy analysis it cannot be made empirically; it must be calculated. These calculations are imprecise except in the situations where Compton scattering is the dominant process. For this to be the case, there must be an encounter, conserving energy and momentum, between the incoming photon and an individual target electron. This in turn will occur if the energy lost by the photon, ΔE = E1E1, clearly exceeds the one-electron binding energy, [E_B], of the target electron. Eisenberger & Platzman (1970[link]) have shown that this binary encounter model – alternatively known as the impulse approximation – fails as (EBE)2.

The likelihood of this failure can be predicted from the Compton shift formula, which for scattering through an angle [\varphi] can be written. [\Delta E=E_1-E_2={E^2_1(1-\cos\varphi)\over mc^2[1+(E_1/mc^2)(1-\cos \varphi)]}. \eqno (7.4.3.1)]

This energy transfer is given as a function of the scattering angle in Table 7.4.3.1[link] for a set of characteristic X-ray energies; it ranges from a few eV for Cr Kα X-radiation at small angles, up to ∼2 keV for backscattered Ag Kα X-radiation. Clearly, in the majority of typical experiments Compton scattering will be inhibited from all but the valence electrons.

Table 7.4.3.1| top | pdf |
The energy transfer, in eV, in the Compton scattering process for selected X-ray energies

Scattering angle ϕ (°)Cr KαCu KαMo KαAg Kα
5411 eV8040 eV17 443 eV22 104 eV
0 0 0 0 0
30 8 17 79 127
60 29 63 292 467
90 57 124 575 915
120 85 185 849 1344
150 105 229 1043 1648
180 112 245 1113 1757

Data calculated from equation (7.4.3.1)[link].

7.4.3.2. Non-relativistic calculations of the incoherent scattering cross section

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7.4.3.2.1. Semi-classical radiation theory

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For weak scattering, treated within the Born approximation, the incoherent scattering cross section, (dσ/dΩ)inc, can be factorized as follows: [\bigg ({{\rm d}\sigma \over {\rm d} \Omega}\bigg)_{\rm inc}=\bigg ({{\rm d}\sigma \over {\rm d} \Omega}\bigg)_{0}S(E_1,E_2,{\bf K},Z), \eqno (7.4.3.2)]where (dσ/dΩ)0 is the cross section characterizing the interaction, in this case it is the Thomson cross section, [(e^2/mc^2)^2\boldvarepsilon_1\cdot\boldvarepsilon_2]; [\boldvarepsilon_1] and [\boldvarepsilon_2] being the initial and final state photon polarization vectors. The dynamics of the target are contained in the incoherent scattering factor S(E1, E2, K, Z), which is usually a function of the energy transfer [\Delta E=E_1-E_2], the momentum transfer K, and the atomic number Z.

The electromagnetic wave perturbs the electronic system through the vector potential A in the Hamiltonian [H={e\over me}{\bf p}\cdot {\bf A}+{e^2\over 2me^2}{\bf A}\cdot {\bf A}. \eqno (7.4.3.3)]

It produces photoelectric absorption through the [{\bf p}\cdot{\bf A}] term taken in first order, Compton and Raman scattering through the [{\bf A}\cdot{\bf A}] term and resonant Raman scattering through the [{\bf p}\cdot{\bf A}] terms in second order.

If resonant scattering is neglected for the moment, the expression for the incoherent scattering cross section becomes [S=\textstyle\sum\limits_f (E_2/E_1)^2\bigg| \langle\psi_f| \textstyle\sum\limits_j\exp (i{\bf K}\cdot {\bf r}_j)|\psi_i\rangle\bigg|^2\delta(E_f-E_i-\Delta E),\eqno (7.4.3.4)]where the Born operator is summed over the j target electrons and the matrix element is summed over all final states accessible through energy conservation. In the high-energy limit of [\Delta E\gg E_B], S(E1, E2, K, Z) [\rightarrow] Z but as Table 7.4.3.1[link] shows this condition does not hold in the X-ray regime.

The evaluation of the matrix elements in equation (7.4.3.4)[link] was simplified by Waller & Hartree (1929[link]) who (i) set E2 = E1 and (ii) summed over all final states irrespective of energy conservation. Closure relationships were then invoked to reduce the incoherent scattering factor to an expression in terms of form factors [f_{jk}]: [S=\textstyle\sum\limits_j[1-|\,f_j({\bf K})|^2]-\textstyle\sum\limits_j^{\phantom{X}j}\textstyle\sum\limits^{\ne \,\,k\phantom{XX}}_k|\, f_{jk}({\bf K})|^2,\eqno (7.4.3.5)]where [f_j({\bf K})=\langle\psi_j|\!\exp (i{\bf K}\cdot {\bf r}_j)|\psi_j\rangle]and [f_{jk}=\langle\psi_k|\!\exp [i{\bf K}\cdot ({\bf r}_k-{\bf r}_j)]|\psi_j\rangle,]the latter term arising from exchange in the many-electron atom.

According to Currat, DeCicco & Weiss (1971[link]), equation (7.4.3.5)[link] can be improved by inserting the prefactor (E2/E1)2, where E2 is calculated from equation (7.4.3.1)[link]; the factor is an average for the factors inside the summation sign of equation (7.4.3.4)[link] that were neglected by Waller & Hartree. This term has been included in a few calculations of incoherent intensities [see, for example, Bloch & Mendelsohn (1974[link])]. The Waller–Hartree method remains the chosen basis for the most extensive compilations of incoherent scattering factors, including those tabulated here, which were calculated by Cromer & Mann (1967[link]) and Cromer (1969[link]) from non-relativistic Hartree–Fock self-consistent-field wavefunctions. Table 7.4.3.2[link] is taken from the compilation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975[link]).

Table 7.4.3.2| top | pdf |
The incoherent scattering function for elements up to Z = 55

Element(sin θ)/λ (Å−1)
0.100.200.300.400.500.600.700.800.901.001.502.00
1 H 0.343 0.769 0.937 0.983 0.995 0.998 0.994 0.999 1.000 1.000 1.000 1.000
2 He 0.296 0.881 1.362 1.657 1.817 1.902 1.947 1.970 1.983 1.990 1.999 2.000
3 Li 1.033 1.418 1.795 2.143 2.417 2.613 2.746 2.834 2.891 2.928 2.989 2.998
4 Be 1.170 2.121 2.471 2.744 3.005 3.237 3.429 3.579 3.693 3.777 3.954 3.989
5 B 1.147 2.531 3.190 3.499 3.732 3.948 4.146 4.320 4.469 4.590 4.895 4.973
6 C 1.039 2.604 3.643 4.184 4.478 4.690 4.878 5.051 5.208 5.348 5.781 5.930
7 N 1.08 2.858 4.097 4.792 5.182 5.437 5.635 5.809 5.968 6.113 6.630 6.860
8 O 0.977 2.799 4.293 5.257 5.828 6.175 6.411 6.596 6.755 6.901 7.462 7.764
9 F 0.880 2.691 4.347 5.552 6.339 6.832 7.151 7.376 7.552 7.703 8.288 8.648
10 Ne 0.812 2.547 4.269 5.644 6.640 7.320 7.774 8.085 8.312 8.490 9.113 9.517
11 Na 1.503 2.891 4.431 5.804 6.903 7.724 8.313 8.729 9.028 9.252 9.939 10.376
12 Mg 2.066 3.444 4.771 6.064 7.181 8.086 8.784 9.304 9.689 9.975 10.766 11.229
13 Al 2.264 4.047 5.250 6.435 7.523 8.459 9.225 9.830 10.296 10.652 11.592 12.083
14 Si 2.293 4.520 5.808 6.903 7.937 8.867 9.667 10.330 10.864 11.286 12.408 12.937
15 P 2.206 4.732 6.312 7.435 8.419 9.323 10.131 10.827 11.411 11.888 13.209 13.790
16 S 2.151 4.960 6.795 8.002 8.960 9.829 10.626 11.336 11.952 12.472 13.990 14.641
17 Cl 2.065 5.074 7.182 8.553 9.539 10.382 11.158 11.867 12.499 13.050 14.750 15.487
18 Ar 1.956 5.033 7.377 8.998 10.106 10.967 11.726 12.424 13.061 13.629 15.489 16.324
19 K 2.500 5.301 7.652 9.405 10.650 11.568 12.329 13.014 13.645 14.220 16.212 17.152
20 Ca 3.105 5.690 7.981 9.790 11.157 12.163 12.953 13.635 14.256 14.830 16.921 17.970
21 Sc 3.136 5.801 8.169 10.071 11.561 12.648 13.545 14.256 14.885 15.460 17.630 18.782
22 Ti 3.114 5.860 8.312 10.304 11.901 13.140 14.093 14.856 15.509 16.095 18.334 19.585
23 V 3.067 5.858 8.375 10.454 12.156 13.514 14.574 15.413 16.111 16.721 19.032 20.379
24 Cr 2.609 5.577 8.206 10.415 12.264 13.770 14.960 15.902 16.670 17.323 19.730 21.168
25 Mn 2.949 5.791 8.380 10.604 12.486 14.062 15.346 16.376 17.211 17.910 20.411 21.938
26 Fe 2.891 5.781 8.432 10.733 12.687 14.343 15.716 16.831 17.737 18.488 21.097 22.704
27 Co 2.832 5.764 8.469 10.844 12.867 14.596 16.050 17.249 18.229 19.039 21.777 23.462
28 Ni 2.772 5.726 8.461 10.894 12.980 14.780 16.317 17.602 18.664 19.543 22.445 24.211
29 Cu 2.348 5.455 8.310 10.778 12.942 14.847 16.494 17.885 19.043 20.002 23.107 24.957
30 Zn 2.654 5.631 8.388 10.901 13.094 15.020 16.709 18.163 19.395 20.427 23.745 25.683
31 Ga 2.791 5.939 8.599 11.082 13.290 15.233 16.947 18.445 19.734 20.831 24.370 26.400
32 Ge 2.839 6.229 8.912 11.338 13.536 15.486 17.215 18.741 20.074 21.224 24.983 27.109
33 As 2.793 6.365 9.236 11.658 13.828 15.775 17.511 19.056 20.420 21.612 25.583 27.810
34 Se 2.799 6.589 9.601 12.033 14.168 16.098 17.835 19.391 20.778 22.003 26.171 28.504
35 Br 2.771 6.748 9.940 12.440 14.552 16.456 18.185 19.747 21.149 22.399 26.747 29.190
36 Kr 2.703 6.760 10.157 12.828 14.969 16.849 18.562 20.123 21.535 22.804 27.313 29.870
37 Rb 3.225 7.062 10.431 13.206 15.410 17.282 18.974 20.526 21.940 23.221 27.871 30.543
38 Sr 3.831 7.464 10.746 13.576 15.860 17.745 19.420 20.956 22.367 23.654 28.423 31.210
39 Y 3.999 7.700 11.010 13.899 16.279 18.215 19.891 21.416 22.820 24.110 28.970 31.870
40 Zr 4.064 7.879 11.236 14.176 16.658 18.672 20.373 21.895 23.294 24.583 29.517 32.522
41 Nb 3.672 7.684 11.213 14.317 16.949 19.081 20.844 22.386 23.787 25.077 30.067 33.167
42 Mo 3.625 7.690 11.260 14.444 17.196 19.455 21.300 22.877 24.288 25.581 30.620 33.808
43 Tc 3.987 7.984 11.512 14.653 17.456 19.816 21.748 23.370 24.797 26.093 31.173 34.447
44 Ru 3.559 7.857 11.531 14.782 17.685 20.150 22.172 23.855 25.312 26.621 31.740 35.081
45 Rh 3.499 7.863 11.591 14.883 17.858 20.428 22.557 24.318 25.819 27.148 32.309 35.715
46 Pd 3.103 7.725 11.441 14.824 17.943 26.653 22.904 24.756 26.316 27.677 32.888 36.349
47 Ag 3.362 7.785 11.598 14.969 18.082 20.858 23.212 25.162 26.792 28.195 33.465 36.983
48 Cd 3.700 7.980 11.812 15.185 18.263 21.064 23.501 25.546 27.252 28.705 34.046 37.618
49 In 3.852 8.297 12.083 15.444 18.489 21.288 23.779 25.906 27.691 29.203 34.634 38.255
50 Sn 3.917 8.615 12.415 15.746 18.760 21.541 24.059 26.252 28.113 29.687 35.226 38.894
51 Sb 3.871 8.811 12.777 16.088 19.067 21.823 24.349 26.590 28.518 30.157 35.822 39.536
52 Te 3.097 9.076 13.171 16.466 19.407 22.134 25.655 26.927 28.912 30.613 36.422 40.181
53 I 3.903 9.287 13.564 16.876 19.227 22.471 24.980 27.269 29.298 31.056 37.024 40.827
54 Xe 3.841 9.340 13.892 17.307 20.175 22.833 25.324 27.619 29.680 31.488 37.628 41.477
55 Cs 4.320 9.615 14.217 17.753 20.612 23.228 25.691 27.981 30.064 31.914 38.232 42.129

7.4.3.2.2. Thomas–Fermi model

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This statistical model of the atomic charge density (Thomas, 1927[link]; Fermi, 1928[link]) considerably simplifies the calculation of coherent and incoherent scattering factors since both can be written as universal functions of K and Z. Numerical values were first calculated by Bewilogua (1931[link]); more recent calculations have been made by Brown (1966[link]) and Veigele (1967[link]). The method is less accurate than Waller–Hartree theory, but it is a much simpler computation.

7.4.3.2.3. Exact calculations

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The matrix elements of (7.4.3.4)[link] can be evaluated exactly for the hydrogen atom. If one-electron wavefunctions in many-electron atoms are modelled by hydrogenic orbitals [with a suitable choice of the orbital exponent; see, for example, Slater (1937[link])], an analytical approach can be used, as was originally proposed by Bloch (1934[link]).

Hydrogenic calculations have been shown to predict accurate K- and L-shell photoelectric cross sections (Pratt & Tseng, 1972[link]). The method has been applied in a limited number of cases to K-shell (Eisenberger & Platzman, 1970[link]) and L-shell (Bloch & Mendelsohn, 1974[link]) incoherent scattering factors, where it has served to highlight the deficiencies of the Waller–Hartree approach. In chromium, for example, at an incident energy of ∼17 keV and a Bragg angle of 85°, the L-shell Waller–Hartree cross section is higher than the `exact' calculation by ∼50%. A comparison of Waller–Hartree and exact results for 2s electrons, taken from Bloch & Mendelsohn (1974[link]), is given in Table 7.4.3.3[link] for illustration. The discrepancy is much reduced when all electrons are considered.

Table 7.4.3.3| top | pdf |
Compton scattering of Mo Kα X-radiation through 170° from 2s electrons

ElementSexactSimpSW–H
Li 0.879 0.878 0.877
B 0.879 0.878 0.877
O 0.878 0.877 0.876
Ne 0.875 0.875 0.875
Mg 0.863 0.863 0.872
Si 0.851 0.850 0.868
Ar 0.843 0.826 0.877
V 0.663 0.716 0.875
Cr 0.568 0.636 0.875

Sexact is the incoherent scattering factor calculated analytically from a hydrogenic atomic model. Simp is the incoherent scattering factor calculated by taking the Compton profile derived in the impulse approximation and truncating it for ΔE < EB. SW–H is the Waller–Hartree incoherent scattering factor. Data taken from Bloch & Mendelsohn (1974[link]).

In those instances where the exact method has been used as a yardstick, the comparison favours the `relativistic integrated impulse approximation' outlined below, rather than the Waller–Hartree method.

7.4.3.3. Relativistic treatment of incoherent scattering

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The Compton effect is a relativistic phenomenon and it is accordingly more satisfactory to start from this basis, i.e. the Klein & Nishina (1929[link]) theory and the Dirac equation (see Jauch & Rohrlich, 1976[link]). In second-order relativistic perturbation theory, there is no overt separation of [{\bf p} \cdot {\bf A}] and [{\bf A} \cdot {\bf A}] terms. The inclusion of electron spin produces additional terms in the Compton cross section that depend upon the polarization (Lipps & Tolhoek, 1954a[link],b[link]); they are generally small at X-ray energies. They are of increasing interest in synchrotron-based experiments where the brightness of the source and its polarization characteristics compensate for the small cross section (Blume & Gibbs, 1988[link]).

Somewhat surprisingly, it is the spectral distribution, d2σ/dΩ dE2, rather than the total intensity, dσ/dΩ, which is the better understood. This is a consequence of the exploitation of the Compton scattering technique to determine electron momentum density distributions through the Doppler broadening of the scattered radiation [see Cooper (1985[link]) and Williams (1977[link]) for reviews of the technique]. Manninen, Paakkari & Kajantie (1976[link]) and Ribberfors (1975a[link],b[link]) have shown that the Compton profile – the projection of the electron momentum density distribution onto the X-ray scattering vector – can be isolated from the relativistic differential scattering cross section within the impulse approximation. Several experimental and theoretical investigations have been concerned with understanding the changes in the spectral distribution when electron binding energies cannot be discounted. It has been found (e.g. Pattison & Schneider, 1979[link]; Bloch & Mendelsohn, 1974[link]) that, to a high degree of accuracy, the spectral distribution is merely truncated at energy transfers [E\le E_B].

This has led to the suggestion that the incoherent intensity can be obtained by integrating the spectral distributions, i.e. from [{{\rm d}\sigma \over {\rm d}\Omega}=\int\limits^\infty_{E_1-E_B}{{\rm d}^2\sigma \over {\rm d}\Omega\,{\rm d}E_2}\,{\rm d}E_2. \eqno (7.4.3.6)]

Unfortunately, this requires the Compton profile of each electron shell as input [Compton line shapes have been tabulated by Biggs, Mendelsohn & Mann (1975[link])] for all elements.

Ribberfors (1983[link]) and Ribberfors & Berggren (1982[link]) have shown that this calculation can be dramatically simplified, without loss of accuracy, by crudely approximating the Compton line shape. Fig. 7.4.3.2[link] shows the incoherent scattering from aluminium, modelled in this way, and compared with experiment, Waller–Hartree theory, and an exact integral of the truncated impulse Compton profile.

[Figure 7.4.3.2]

Figure 7.4.3.2 | top | pdf |

The incoherent scattering function, S(x, Z)/Z, per electron for aluminium shown as a function of x = (sin θ)/λ. The Waller–Hartree theory (dotted line) is compared with the truncated impulse approximation in the tabulated Compton profiles (Biggs, Mendelsohn & Mann, 1975[link]) cut off at E < EB for each electron group (solid line). The third curve (dashed line) shows the simplification introduced by Ribberfors (1983[link]) and Ribberfors & Berggren (1982[link]). The predictions are indistinguishable to within experimental error except at low [(\sin\theta/\lambda)]. Reference to the measurements can be found in Ribberfors & Berggren (1982[link]).

7.4.3.4. Plasmon, Raman, and resonant Raman scattering

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In typical X-ray experiments, as is evident from Table 7.4.3.1[link], the energy transfer may be so low that Compton scattering will be inhibited from all but the most loosely bound electrons. Indeed, in the situation in metals where K, the momentum transfer, is less than kF (the Fermi momentum), Compton scattering from the conduction electrons may be restricted by exclusion because of the lack of unoccupied final states [see Bushuev & Kuz'min (1977[link])]. Fortunately, in these uncertain circumstances, the incoherent intensities are low. In this regime, the electron gas may be excited into collective motion. For almost all solids, the plasmon excitation energy is 20–30 eV and, in the random phase approximation, the incoherent scattering factor becomes SE, K) ∝ (K2/wp)δ(ΔEhωp), where ωp is the plasma frequency.

At slightly higher energies [(\Delta E \ge E_B)], Compton scattering and Raman scattering can coexist, though the Raman component is only evident at low momentum transfer (Bushuev & Kuz'min, 1977[link]). The resultant spectrum is often referred to as the Compton–Raman band. In semi-classical radiation theory, Raman scattering is usually differentiated from Compton scattering by dropping the requirement for momentum conservation between the photon and the individual target electron, the recoil being absorbed by the atom. The Raman band corresponds to transitions into the lowest unoccupied levels and these can be calculated within the dipole approximation as long as |K|a [\lt1], where K is the momentum transfer and a the orbital radius of the core electron undergoing the transition. The transition probability in equation (7.4.3.4)[link] becomes [\sum_f|\langle \psi_f|{\bf r}|\psi_i\rangle|^2\delta (E_f-E_i-\Delta E), \eqno (7.4.3.7)]which implies that the near-edge structure is similar to the photoelectric absorption spectrum.

Whereas plasmon and Raman scattering are unlikely to make dramatic contributions to the total incoherent intensity, resonant Raman scattering (RRS) may, when [E_1\le E_B]. The excitation involves a virtual K-shell vacancy in the intermediate state and a vacancy in the L (or M or N) shell and an electron in the continuum in the final state. It has now been observed in a variety of materials [see, for example, Sparks (1974[link]), Eisenberger, Platzman & Winick (1976[link]), Schaupp et al. (1984[link])]. It was predicted by Gavrila & Tugulea (1975[link]) and the theory has been treated comprehensively by Åberg & Tulkki (1985[link]). The effect is the exact counterpart, in the inelastic spectrum, of anomalous scattering in the elastic spectrum. It is important because, as the resonance condition is approached, the intensity will exceed that due to Compton scattering and therefore play havoc with any corrections to total intensities based solely on the latter.

Although systematic tabulations of resonance Raman scattering do not exist, Fig. 7.4.3.3[link], which is based on the calculations of Bannett & Freund (1975[link]), shows how the intensity of RRS clearly exceeds that of the Compton scattering for incident energies just below the absorption edge. However, since the problems posed by anomalous scattering and X-ray fluorescence are generally appreciated, the energy range [0.9\lt E_1 /E_B \lt 1.1] is wisely avoided by crystallographers intent upon absolute intensity measurements.

[Figure 7.4.3.3]

Figure 7.4.3.3 | top | pdf |

The cross section for resonant Raman scattering (RRS) and fluorescence (F) as a function of the ratio of the incident energy, E, and the K-binding energy, EB. The units of dσ/dΩ are (e2/mc2)2 and the data are taken from Bannett & Freund (1975[link]). For comparison, the intensity of Compton scattering (C) from copper through an angle of 30° is also shown [data taken from Hubbell et al. (1975[link])].

7.4.3.5. Magnetic scattering

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Finally, and for completeness, it should be noted that the intensity of Compton scattering from a magnetic material with a net spin moment will, in principle, differ from that from a non-magnetic material. For unpolarized radiation, the effects are only discernible at photon energies greatly in excess of the electron rest mass energy, mc2 = 511 keV, but for circularly polarized radiation effects at the 1% level can be found in Compton scattering experiments carried out at [E_1\simeq1/10\,mc^2] on ferromagnets such as iron. See Lipps & Tolhoek (1954a[link],b[link]) for a comprehensive description of polarization phenomena in magnetic scattering and Lovesey (1993[link]) for an account of the scattering theory.

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