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UDC 537:535.3
PHYSICS
L. K. IZRAILEVA
LIMITS OF APPLICABILITY OF THE THEORY OF $K$-ABSORPTION SPECTRA OF POLYCRYSTALLINE SAMPLES
(Presented by Academician N. V. Belov on 10 IX 1965)
For calculations of the $K$-absorption spectra of molecules in the short-wavelength region, a method was proposed in ($^1$), which was subsequently developed and refined as applied to the $K$-absorption spectra of metals ($^2$, $^3$). A feature of the method ($^1$) is that, when it is used, the potential of the system $V(\mathbf r)$ is averaged over the surface of a sphere centered at the absorbing atom. In other words, only the zero component is retained in the expansion of $V(\mathbf r)$ in a series in spherical functions $V_0(r)$:
\[ V(\mathbf r)=V_0(r)+\sum_{l\ne 0,\ m} V_{l,m}(r)Y_{l,m}(\theta,\varphi), \tag{1} \]
\[ V_0(r)=\frac{1}{4\pi}\iint V(\mathbf r)d\Omega. \tag{2} \]
Further, in finding the wave function of the final state of the photoelectron, the part $V_0(r)$—the potential inside the absorbing atom in the system $V_a(r)$—was taken into account exactly, while the difference $V'(r)=V_0(r)-V_a(r)$, or the “potential of the surrounding atoms” ($^2$, $^3$), was treated as a perturbation. In ($^4$) it was shown that the sphericity of the initial $1s$ state and the averaging of the absorption coefficient $\tau_k$ over the polarization of the radiation, which in fact occurs in experiments with molecules and polycrystalline samples, leads to the result that for these objects $\tau_k$ does indeed depend only on the component $V_0(r)$ of the potential, but only for those photoelectron energies for which perturbation theory for nondegenerate states is applicable ($^5$). This condition is violated in the case of a periodic $V(\mathbf r)$ for electrons with quasimomenta $\mathbf k$ satisfying the Wulff–Bragg condition $(\mathbf k-2\pi\mathbf g)^2=k^2$ ($\mathbf g$ is a reciprocal-lattice vector) ($^6$). Let us show how this fact is reflected in the results of ($^2$, $^3$).
For simplicity we restrict ourselves to the case of a crystal with a center of symmetry and expand $V(\mathbf r)$ in a Fourier series
\[ V(\mathbf r)=\sum_g V_g\cos(2\pi\mathbf g\mathbf r). \tag{3} \]
Let us average (3) over $\theta,\varphi$ in order to obtain $V_0(r)$ (2). Passing in the integration to variables $\chi,\psi$, where $\chi=\hat{\mathbf g}\mathbf r$, and $\psi$ is the angle in the plane normal to $\mathbf g$, we obtain:
\[ V_0(r)=\frac{1}{4\pi}\iint V(\mathbf r)d\Omega =V_{\mathrm{cp}}+\sum_{g\ne 0} V_g\frac{1}{4\pi} \int_0^\pi\int_0^{2\pi}\cos(2\pi gr\cos\chi)\sin\chi\,d\chi\,d\psi = \]
\[ =V_{\mathrm{cp}}+\sum_{g\ne 0} V_g\frac{\sin 2\pi gr}{2\pi gr}, \tag{4} \]
where $V_{\mathrm{cp}}$ is the mean potential of the crystal. The formula for the relative absorption coefficient (with ionization of the $1s$ shell) in the model ($^3$) has
form:
\[ \tau_k/\tau_0=1+2\left(I'\cos 2\eta_1+I''\sin 2\eta_1\right), \tag{5} \]
where \(\eta_1(k)\) is the phase of the electron wave in the field \(V_a(r)\) for \(l=1\),
\[ I'=\frac{m}{\hbar^2 k}\int_{r_0}^{\infty} V'(\xi)\sin 2k\xi\,d\xi, \]
\[ I''=\frac{m}{\hbar^2 k}\int_{r_0}^{\infty} V'(\xi)\cos 2k\xi\,d\xi, \]
\[ V'(r)=V_0(r)-V_{\mathrm{av}}\quad \text{for } r>r_0. \]
We have not restricted the region of integration in \(I'\) and \(I''\) to some value \(R\), but have extended the integration to the entire crystal, i.e., formally to \(\infty\). Substituting series (4) into the integrands \(I'\) and \(I''\) and assuming that \(V'(\xi)\) is measured in units of \(e^2/a\) (\(a\) is a certain characteristic lattice parameter), \(\xi, r_0\) in units of \(a\), and \(k\) in units of \(1/a\), after elementary transformations we obtain:
\[ I'=\frac{a/a_0}{ka}\sum_{g\ne 0}\frac{V_g}{4\pi g} \left[\operatorname{ci}2(k+\pi g)r_0-\operatorname{ci}2|k-\pi g|r_0\right], \tag{6} \]
\[ I''=\frac{a/a_0}{ka}\sum_{g\ne 0}\frac{V_g}{4\pi g} \left[\frac{\pi}{2}-\operatorname{si}2(k+\pi g)r_0 +\operatorname{sign}(\pi g-k)\left(\frac{\pi}{2}-\operatorname{si}2r_0|k-\pi g|\right)\right], \tag{7} \]
where
\[ \operatorname{si}(x)=\int_0^x \frac{\sin t}{t}\,dt \quad \text{and} \quad \operatorname{ci}(x)=-\int_x^\infty \frac{\cos t}{t}\,dt \]
are the sine and cosine integrals, and \(a_0\) is the Bohr radius.
From expression (6) it is immediately evident that, since \(\operatorname{ci}x\sim C+\ln x\) as \(x\to 0\), \(I'\) tends to \(\infty\) as \(k\to \pi g\), and near such a \(k\), \(I'\sim \ln|k-\pi g|\). Together with \(I'\), the first-order correction to the photoelectron wave function defined in (3) also diverges. This means that the model \((^{2,3})\), based on perturbation theory for nondegenerate states, is inapplicable for polycrystalline samples at photoelectron energies
\[ \mathcal{E}=\frac{\hbar^2 k^2}{2m}=\frac{\hbar^2}{2m}(\pi g)^2. \]
For \(k\) close to \(\pi g\), in the nearly-free-electron approximation one obtains a fluctuation of the density of states \((^{6,7})\). We shall show that in the model \((^{2,3})\) a fluctuation of the absorption coefficient may be observed in this region. Let \(\zeta=|k-\pi g|>0\) (“resonance deficiency” \((^6)\)). Assuming that there exist \(\zeta\ll \pi g_1-\pi g_2\) (\(g_1, g_2\) are successive reciprocal-lattice vectors) for which the theory \((^{2,3})\) is still applicable, and neglecting \(\zeta\) in comparison with \(\pi g\), we obtain from (6) and (7):
a) for \(k=\pi g_1-\zeta\)
\[ I'\approx \frac{a/a_0}{ka}\sum_{|g|\ne 0} \left[\operatorname{ci}2(\pi g_1+\pi g)r_0-\operatorname{ci}2\zeta r_0\right] \frac{1}{4\pi g} \sum_{|g|=\mathrm{const}} V_g, \tag{8} \]
\[ I''\approx \frac{a/a_0}{ka} \left\{ \sum_{|g|\ne 0,g_1} \left[\pi-\operatorname{si}2(\pi g_1+\pi g)r_0-\operatorname{si}2|\pi g_1-\pi g|r_0\right] \frac{1}{4\pi g} \sum_{|g|=\mathrm{const}} V_g +\right. \]
\[ \left. +\left[\pi-\operatorname{si}(4\pi g_1 r_0)-\operatorname{si}(2\eta r_0)\right] \frac{1}{4\pi g_1} \sum_{|g|=g_1} V_g \right\}; \tag{9} \]
b) for \(k=\pi g_1+\zeta\)
\[ I''\simeq \frac{a/a_0}{ka}\left\{ \sum_{|g|\ne 0,g_1} \left[\pi-\operatorname{si}2(\pi g_1+\pi g)r_0-\operatorname{si}2|\pi g_1-\pi g|r_0\right] \frac{1}{4\pi g}\sum_{|g|=\mathrm{const}} V_g + \left[-\operatorname{si}(4\pi g_1 r_0)+\operatorname{si}(2\zeta r_0)\right] \frac{1}{4\pi g_1}\sum_{|g|=g_1} V_g \right\}. \tag{10} \]
\(I'\) does not change in comparison with (8).
According to (9), (10), upon passing through the value \(k=\pi g_1\), \(I''\) changes by a finite amount:
\[ I''(\pi g_1-\zeta)-I''(\pi g_1+\zeta)\simeq \frac{a/a_0}{ka}\, \frac{\sum_{|g|=g_1} V_g}{4g_1}. \tag{11} \]
The change in \(I''\) will lead, in accordance with (5), to a finite change \(\Delta\tau\) in the relative absorption coefficient:
\[ \Delta\tau=\tau(\pi g_1-\zeta)-\tau(\pi g_1+\zeta) = 2\frac{a/a_0}{ka}\, \frac{\sum_{|g|=g_1} V_g}{4g_1}\, \sin 2\eta_1(\pi g_1) = \frac{\dfrac{e^2}{a}N_{g_1}V_{g_1}} {\dfrac{\hbar^2}{2ma^2}(\pi g_1)^2}\, \frac{\pi}{4}\sin 2\eta_1(\pi g_1) \tag{12} \]
(in the last expression (12), for simplicity, we have put \(V_g=V_{|g|}\), \(N_{g_1}\) is the number of reciprocal-lattice points on the sphere of radius \(g_1\)).
Thus, just as in the approximation \((^{6,7})\), the magnitude of the fluctuation of the absorption coefficient near \(k=\pi g\) is proportional to
\[ \frac{N_g V_g}{E_g} \left(E_g=\frac{\hbar^2}{2ma^2}(\pi g)^2\right), \]
but, in addition, it also depends on the phase \(\eta_1(\pi g_1)\) of the field of the absorbing atom in the crystal.
Let us estimate the energy interval \(\Delta E\) near \(E_g\), outside which the theory \((^{2,3})\), and hence formulas (5)—(12), are still valid. For this, for \(E<E_g-\Delta E\), \(E>E_g+\Delta E\), the condition
\[ I'\sim \frac{\dfrac{e^2}{a}N_{g_1}V_{g_1}} {\dfrac{\hbar^2}{2ma^2}(\pi g_1)^2}\, \frac{1}{8}\ln 2\zeta r_0<1, \tag{13} \]
must be satisfied, where \(\zeta=|k-\pi g|\), and \(\Delta E\) is related to \(\zeta\) as follows:
\[ \Delta E=E-E_{g_1} = \frac{\hbar^2}{2ma^2}(\pi g_1+\zeta)^2 - \frac{\hbar^2}{2ma^2}(\pi g_1)^2 \sim \frac{E_{g_1}}{\pi g_1}\,2\zeta. \tag{14} \]
For a KCl crystal in the region from \(E_g=11.7\) eV \((g=\sqrt{12}/2)\) to \(E_g=31.2\) eV \((g=\sqrt{32}/2)\), the interval \(\Delta E\) is largest for \(E_{g_1}=19.5\) eV \((g_1=\sqrt{20}/2)\) and is \(\sim 0.5\) eV. The estimate was made using formulas (13), (14); \(N_{g_1}=24\), \(r_0=0.5a\), \(a=5.9a_0\); \(V_{g_1}=-0.57e^2/a\) was calculated from known atomic factors of the ions K\(^+\) and Cl\(^-\) \((^8)\). Since the region \(2\Delta E\lesssim 1\) eV, where the theory \((^{2,3})\) is not valid for KCl, proves comparable with the width of the \(K\)-level itself (\(\gamma_K\sim 0.65\) eV for K and \(\gamma_K\sim 0.60\) eV for Cl \((^9)\)), and for a number of values of \(g\) is considerably smaller, it may be assumed that in many cases its presence is insignificant and does not show up when comparing the results \((^{2,3})\) with experiment. However, in each specific-
in that case a special estimate of the quantity \(\Delta E\) is necessary for different \(g\).
I express my deep gratitude to K. I. Narbutt for his attention to and interest in this work.
Institute of Geology of Ore Deposits,
Petrography, Mineralogy, and Geochemistry
Received
7 IX 1965
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