UDC 537 : 535.3

PHYSICS

L. K. IZRAILEVA

ON THE THEORY OF THE SHORT-WAVELENGTH REGION OF X-RAY $K$-ABSORPTION SPECTRA OF SINGLE CRYSTALS AND POLYCRYSTALLINE SAMPLES

(Presented by Academician N. V. Belov on 10 IX 1965)

The fine structure of x-ray $K$-absorption spectra of molecules and polycrystalline metal samples in the region from about ten to several hundred electron volts from the edge is successfully interpreted in a number of works $^{(1–4)}$ on the basis of a one-electron model, the principal features of which are as follows:

1. Reduction of the problem to a centrally symmetric one by averaging the true potential of the system $V(\mathbf r)$ over the surface of a sphere centered at the chosen atom;

2. Division of the averaged potential $\overline V(r)$ into two parts $\overline V(r)=V_a(r)+V'(r)$, of which the potential $V_a(r)$, which gives the behavior of the true potential $V(\mathbf r)$ inside the atom in the system sufficiently well, is taken into account exactly, while the remaining part $V'(r)$ is treated within perturbation theory. The physical cause of the fluctuations of the absorption coefficient $\tau_k$ in this model is the scattering of the electron wave in the field $V'(r)$ and, as a consequence of this, the nonmonotonic dependence of the amplitude of the photoelectron wave function in the region of localization of the initial $1s$ state $^{(1–3)}$.

In work $^{(5)}$ the absorption coefficient of a single crystal was determined for the case of polarized radiation for a specific model of the crystalline field, in which an effective polyhedron charge is introduced. The potential of one polyhedron $V_a=Ze/|\mathbf r-\mathbf R_a|$ was taken into account exactly, as in $^{(1–3)}$, while the potentials of the others were treated as a perturbation, but, in contrast to $^{(1–3)}$, were not averaged over the surfaces of spheres. A shortcoming of approximation $^{(5)}$ is the neglect of the phase $\eta_l$ introduced by the field $V_a$, which, according to $^{(3)}$, substantially affects the dependence of the absorption coefficient on photon energy. Below, an expression will be obtained for the absorption coefficient of a single crystal $\tau_k$ in the short-wavelength region in the case of polarized or oriented radiation, taking the phase into account and suitable for any crystalline field. It will also be shown that $\tau_k$ for a polycrystalline sample with arbitrary orientations of the crystallites automatically turns out to depend only on the mean value of the crystal potential on the surface of a sphere centered at the chosen atom. The latter may be regarded as a justification of the model $^{(1–3)}$.

The absorption coefficient of the $1s$ band, taking into account the quasi-atomic character of the $1s$ states, their localization near the nuclei ($\chi r_{1s}\ll 1$, where $\chi$ is the wave vector of the photon, $r_{1s}$ is the radius of the $1s$ state), and the translational symmetry of the crystal, can be written in the form:

\[ \tau_k \simeq \sum_i \left| \int \psi_i(x,y,z)\,\mathbf E^0 \mathbf r \psi_{1s}(r)\,d\mathbf r \right|^2 , \tag{1} \]

where $\psi_{1s}$ is the atomic wave function of the $1s$ state; $\mathbf E^0$ is the direction of polarization of the radiation; the summation in (1) is carried out over the final states of the photoelectron belonging to the energy $\mathcal E_i=\mathcal E_{1s}+\hbar\omega$, and ...

the integration actually extends over the region \(|\mathbf r|\lesssim r_{1s}\) around the selected atom. Thus, the problem of calculating \(t_k(\omega)\) reduces to finding the wave function of the photoelectron \(\psi_i(x,y,z)\) in the region \(|\mathbf r|\lesssim r_{1s}\). For large photoelectron energies \(\mathcal E_i>0\) (\(\mathcal E_i\) is measured from the average potential of the crystal) we use perturbation theory. Let us represent the “true” potential of the crystal in the form of a sum \(V(\mathbf r)=V_a(r)+V'(\mathbf r)\), where for \(r<r_0\) \((r_0>r_{1s})\) \(V_a(r)\approx V(r)\), \(V'(\mathbf r)\ll V_a(r)\); as \(r\to\infty\), \(V_a(r)\to0\), \(V'(\mathbf r)\to V(\mathbf r)\). In the equation for the wave function of the photoelectron

\[ \Delta\psi+\frac{2m}{\hbar^2}\,[\mathcal E-V_a(r)-V'(\mathbf r)]\psi=0 \tag{2} \]

we shall take the potential \(V_a(r)\) of the “quasiatom” into account exactly, and \(V'(\mathbf r)\) as a perturbation. As solutions of the unperturbed equation \((V'(\mathbf r)\equiv0)\) for \(\mathcal E>0\), we choose a system of wave functions of the continuous spectrum characterized by a definite value of the photoelectron momentum \(\hbar \mathbf p\):

\[ \psi_{\mathbf p}^{0}(\mathbf r) = \frac{1}{4\pi p} \sum_{l=0}^{\infty} i^l(2l+1)e^{i\delta_l(p)}R_{pl}(r)P_l\!\left(\frac{\mathbf n\mathbf r}{r}\right). \tag{3} \]

Then, for those \(\mathcal E\) for which perturbation theory for nondegenerate states is applicable, the solution of (2), accurate to first-order terms, will have the form \({}^{6}\)

\[ \psi_{\mathbf p}(\mathbf r) = \psi_{\mathbf p}^{0}(\mathbf r)+\psi_{\mathbf p}'(\mathbf r) = \psi_{\mathbf p}^{0}(\mathbf r) - \frac{2m}{\hbar^2} \int G_{\mathcal E}(\mathbf r,\mathbf r')V'(\mathbf r')\psi_{\mathbf p}^{0}(\mathbf r')\,d\mathbf r'. \tag{4} \]

Here

\[ G_{\mathcal E}(\mathbf r,\mathbf r') = \sum_{n,l,m} \frac{\psi_{nlm}^{0}(\mathbf r)\psi_{nlm}^{0*}(\mathbf r')} {-p_n^2-p^2} + \int \frac{\psi_{\mathbf p'}^{0}(\mathbf r)\psi_{\mathbf p'}^{0*}(\mathbf r')\,d\mathbf p'} {p'^2-p^2-i0} \]

\[ \left( p^2=\frac{2m}{\hbar^2}\mathcal E,\quad p_n^2=-\frac{2m}{\hbar^2}\mathcal E_n \right) \tag{5} \]

is the Green’s function of equation (2) for \(V'(\mathbf r)\equiv0\). Note that \(\psi_{\mathbf p}(\mathbf r)\) is normalized in the same way as \(\psi_{\mathbf p}^{0}(\mathbf r)\), accurate to first-order small terms.

In the zeroth approximation the matrix element \(M_{1sp}^{0}\) of the probability of the transition \(1s-\mathbf p\) upon absorption of x-ray radiation with frequency \(\omega=(|\mathcal E-\mathcal E_{1s}|)\,1/\hbar\) and polarization \(\mathbf E^0\) is equal to

\[ M_{1sp}^{0} \sim \int \psi_{\mathbf p}^{0}(\mathbf r)\,\mathbf E^{0}\mathbf r\,\psi_{1s}(\mathbf r)\,d\mathbf r = \cos \hat{\mathbf p}\mathbf E^{0}\, \frac{i e^{i\delta_1(p)}}{p} M_{1s}(p), \tag{6} \]

where

\[ M_{1s}(p)=\int_{0}^{\infty}dr\,r^3R_{p1}\psi_{1s}. \tag{6'} \]

As is seen from (6), \(M_{1sp}^{0}\) is different from zero only for the term of series (3) with \(l=1\). Similarly, in the matrix element of first-order smallness

\[ M_{1sp}'=\int \psi_{\mathbf p}'\mathbf E^{0}\mathbf r\psi_{1s}\,d\mathbf r \]

there will be a contribution from the term in (5) with \(l=1,\ m=0\), if the direction of polarization \(\mathbf E^0\) is chosen as the quantization axis. Substituting series (3) into (5), carrying out the integration over the directions of \(\mathbf p'\), and extracting from (5) the term with \(l=1,\ m=0\), it will be equal to

\[ G_{p1}(r,r')Y_{10}(\theta,\varphi)Y_{10}^{*}(\theta',\varphi') = \]

\[ = \left[ \sum_n \frac{R_{n1}(r)R_{n1}(r')} {-p_n^2-p^2} + \int \frac{R_{p'1}(r)R_{p'1}(r')\,dp'} {p'^2-p^2-i0} \right] Y_{10}(\theta,\varphi)Y_{10}^{*}(\theta',\varphi') \tag{7} \]

\[ (\theta,\theta' \text{ are the angles between } \mathbf E^0 \text{ and } \mathbf r,\mathbf r'). \]

The expression in square brackets in (7) is, as is not difficult to see, the Green’s function \(G_{p1}(r,r')\) of the equation arising from (2) for \(V'(r)=0\) and \(l=1\):

\[ \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)+ \left[p^2-\frac{2}{r^2}-\frac{2m}{\hbar^2}V_a(r)\right]R=0. \tag{8} \]

The Green’s function (8) can also be written \({}^{(6,7)}\) as the product of two different particular solutions of (8), one of which, \(R_{p1}(r_<)\), is regular at zero, while the other, \(\Phi_{p1}(r_>)\), is not regular \((r_<,r_>\) are respectively the smaller and larger of \(r,r')\):

\[ G_{p1}(r,r')=\frac{\pi}{2p}R_{p1}(r_<)\Phi_{p1}(r_>). \tag{9} \]

The coefficient in the product in (9) is determined from the condition that \(G_{p1}(r,r')\) for \(V_a(r)\equiv0\) pass directly into the corresponding expression \(pj_1(pr_<)n_1(pr_>)\) for free motion with \(l=1\) \({}^{(6)}\).

Taking into account relations (7), (9), the matrix element \(M_{1sp}\) of the probability of transition to the state (3) is equal to

\[ M_{1sp}=M^0_{1sp}+M'_{1sp} =M^0_{1sp}-\frac{2m}{\hbar^2}\int d\mathbf r'\,\psi^0_{\mathbf p}(\mathbf r')V'(\mathbf r') \int d\mathbf r\,G_\varepsilon(\mathbf r,\mathbf r')\,\mathbf E^0\mathbf r\,\psi_{1s}(\mathbf r) \]
\[ = M^0_{1sp} -\frac{m\pi}{\hbar^2p}\int d\mathbf r'\,\psi^0_{\mathbf p}(\mathbf r')V'(\mathbf r')P_1\left(\frac{\mathbf E^0\mathbf r'}{r'}\right) \int_0^\infty dr\,r^3R_{p1}(r_<)\Phi_{p1}(r_>)\psi_{1s}(r). \tag{10} \]

In order to obtain the absorption coefficient \(\tau_k\) for the \(1s\)-band, it is necessary, according to (1), to sum the transition probability \(|M_{1sp}|^2\) over all states with energy \(\mathcal E=\hbar^2p^2/2m\), i.e., over all directions \(\mathbf p\). To accuracy including terms of first order in smallness, \(\tau_k\) will be equal to

\[ \tau_k=\int |M_{1sp}|^2\,d\Omega =\int |M^0_{1sp}|^2\,d\Omega +2\operatorname{Re}\int d\Omega\,M^{0*}_{1sp}M'_{1sp}. \tag{11} \]

The first term in (11) is the absorption coefficient of the “quasiatom” \(\tau_k^0\). In the second term of (11), the product
\(\cos \mathbf p\wedge\mathbf E^0 P_l(\mathbf n\mathbf r'/r')\) \((l=0,\ldots,\infty)\) depends, according to (10), (6), and (3), on the direction of the momentum \(\mathbf n=\mathbf p/p\); let us carry out in (11) the integration over \(\mathbf n\). The potential \(V'(\mathbf r')\sim0\) for \(|\mathbf r'|\leq r_{1s}\); therefore we replace
\(R_{p1}(r_<)\Phi_{p1}(r_>)\to R_{p1}(r)\Phi_{p1}(r')\); in this case (11) reduces to the form

\[ \tau_k=\tau_k^0\left\{1-\frac{2m\pi}{\hbar^2p} \int R_{p1}(r')\Phi_{p1}(r')V'(\mathbf r') \frac{3}{4\pi}\left[P_1\left(\frac{\mathbf r'\mathbf E^0}{r'}\right)\right]^2 \,d\mathbf r'\right\}. \tag{12} \]

Formula (12) is the final expression for the absorption coefficient \(\tau_k\) of the \(1s\)-band of a single crystal in the case of polarized radiation, suitable, as already noted, in the short-wavelength region.

Expanding \(V'(\mathbf r)\) in a series in spherical functions (\(\mathbf E^0\) is the quantization axis):

\[ V'(\mathbf r)=V'(r)+\sum_{l\ne0,\,m} V^{\mathbf E^0}_{l,m}(r)Y_{l,m}(\theta,\varphi) \tag{13} \]

and substituting (13) into (12), we obtain that \(\tau_k\) of a single crystal for the \(1s\)-band depends only on the components \(V'(r)\) and \(V^{\mathbf E^0}_{20}(r)\) of the crystal potential:

\[ \tau_k^{\mathbf E^0} =\tau_k^0\left\{1-\frac{2m\pi}{\hbar^2p} \int_{r_0}^{\infty} d r'\,r'^2 R_{p1}(r')\Phi_{p1}(r') \left[V'(r')-\frac{1}{\sqrt{5\pi}}V^{\mathbf E^0}_{20}(r')\right]\right\}. \tag{14} \]

The absorption coefficient of a single crystal \(\tau_k^{\vec x}\) in the case of unpolarized oriented radiation is obtained from (12) as the result of averaging the expression

\[ \left[P_1(\mathbf r'\mathbf E^0/r')\right]^2 =\cos^2 \mathbf r\wedge\mathbf E^0 =\sin^2\chi^{\vec x\hat{\ }}_{r'}\cos^2\varphi_{\mathbf E^0}. \tag{15} \]

in a plane perpendicular to $\vec{\varkappa}$, over $\varphi^{E^0}$. Substituting series (13) into (12), where the direction $\vec{\varkappa}$ is now taken as the quantization axis, we obtain, analogously to the preceding case, that $\tau_k^{\vec{\varkappa}}$ depends only on the components $V'(r)$ and $V_{20}^{\vec{\varkappa}}(r)$ of the crystal potential:

\[ \tau_k^{\vec{\varkappa}} = \frac{1}{2\pi}\int \tau_k^{E^0}\,d\varphi = \tau_k^0 \left\{ 1-\frac{2m\pi}{\hbar^2 p} \int_{r_0}^{\infty} dr'\,r'^2 R_{p1}(r')\Phi_{p1}(r') \left[ V'(r')+\frac{1}{2\sqrt{5}}V_{20}^{\vec{\varkappa}}(r') \right] \right\}. \tag{16} \]

Finally, on averaging (12) over all polarizations, we arrive at an expression for the absorption coefficient $\bar{\tau}_k$ of a polycrystalline specimen:

\[ \bar{\tau}_k = \frac{1}{4\pi}\int \tau_k^{E^0}\,d\Omega = \tau_k^0 \left\{ 1-\frac{2m\pi}{\hbar^2 p} \int_{r_0}^{\infty} dr'\,r'^2 R_{p1}(r')\Phi_{p1}(r')V'(r') \right\}. \tag{17} \]

As is seen from (17), $\bar{\tau}_k$ depends only on the mean value of the crystal potential

\[ V'(r)=\frac{1}{4\pi}\int V'(r)\,d\Omega \]

on the surface of a sphere centered at the selected atom; in other words, on the zero component of series (13). Thus, models$^{(1-3)}$ in which, from the very beginning, $V'(r)$ was averaged over the surface of spheres are entirely legitimate for polycrystalline specimens and also, obviously, for molecular gases. For a single crystal, formulas (14) and (16) differ from formula (17) by additional terms depending on another component of the crystal potential, $V_{20}(r)$. Information about it can be obtained from the $K$ absorption spectrum of a single crystal, if the dependence of the spectrum on the polarization or orientation of the incident beam is established experimentally.

In the energy region of photoelectrons $pr_0 \gg 1$, the wave functions $R_{p1}(r)$, $\Phi_{p1}(r)$ in formulas (14)—(17) may be replaced by their asymptotic expressions

\[ R_{p1}\sim \sqrt{\frac{2}{\pi}}\frac{\sin(pr-\pi/2+\eta_1)}{r}, \qquad \Phi_{p1}\sim \sqrt{\frac{2}{\pi}}\frac{\cos(pr-\pi/2+\eta_1)}{r}, \]

then formula (17) for a polycrystalline specimen passes into the known expression$^{(3)}$

\[ \bar{\tau}_k = \tau_k^0 \left\{ 1+\frac{2m}{\hbar^2p} \int_{r_0}^{\infty} dr\,\sin(2pr+2\eta_1)V'(r) \right\}, \tag{18} \]

while formulas (14) and (16) will have the form

\[ \tau_k^{E^0} = \tau_k^0 \left\{ 1+\frac{2m}{\hbar^2p} \int_{r_0}^{\infty} dr\,\sin(2pr+2\eta_1) \left[ V'(r)-\frac{1}{\sqrt{5\pi}}V_{20}^{E^0}(r) \right] \right\}, \tag{19} \]

\[ \tau_k^{\vec{\varkappa}} = \tau_k^0 \left\{ 1+\frac{2m}{\hbar^2p} \int_{r_0}^{\infty} dr\,\sin(2pr+2\eta_1) \left[ V'(r)+\frac{1}{2\sqrt{5\pi}}V_{20}^{\vec{\varkappa}}(r) \right] \right\}. \tag{20} \]

Institute of Geology of Ore Deposits,
Petrography, Mineralogy, and Geochemistry

Received
7 IX 1965

CITED LITERATURE

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