ON THE POSSIBILITY OF DIRECT DESTRUCTION OF \(F\)-CENTERS BY X-RAYS

M. M. ARINSHTEIN

Irradiation of alkali-halide crystals with X-rays leads both to the generation and to the destruction of \(F\)-centers [1].

In the present work an attempt is made to estimate the role of the effect of destruction of \(F\)-centers due to direct interactions of ionizing-radiation quanta with the electron of an \(F\)-center. Such processes may be the photoelectric effect and Compton scattering by the electron of an \(F\)-center.

As a first approximation, a hydrogen-like wave function may be adopted for the electron of an \(F\)-center [2]. The effective cross section for the photoelectric effect for such a state has been calculated [3] and, in the case of interest to us, may be obtained by replacing the orbital radius \(a_0\) by the quantity \(\dfrac{a}{\xi}\) from [2], where \(a\) is the lattice constant and \(\xi\) is the variational screening parameter.

In considering the Compton effect it is usually assumed that the electron may be regarded as free [3]. This is valid for hard quanta with energies of the order of hundreds of keV and higher. However, at lower quantum energies the binding forces of the electron may manifest themselves. The presence of binding will lead to a violation of the law of conservation of momentum in the process of Compton scattering. Then the number of final states of the electron increases and the effective scattering cross section changes.

This effect may be of interest for the theory of the interaction of X-radiation with matter. In the present work the Compton effect on a weakly bound electron is considered, i.e., when

\[ |E_{\mathrm{b}}| \ll h\omega, \]

where \(E_{\mathrm{b}}\) is the binding energy of the electron. The Hamiltonian of the interaction of radiation with the electron is taken in the form

\[ \hat{H}=-e_0(\boldsymbol{\alpha}\hat{\mathbf{A}}), \tag{1} \]

where \(\boldsymbol{\alpha}, \beta\) are the Dirac matrices, and \(\hat{\mathbf{A}}\) is the vector potential of the electromagnetic field.

The initial state of the bound electron may be modeled by a hydrogen-like wave function

\[ \psi_e=\frac{u_0}{\sqrt{\pi}}a_0^{-3/2}\exp\left(-\frac{r}{a_0}\right), \tag{2} \]

where \(u_0\) is the spinor corresponding to states with positive energy, for which the last two components are zero. This spinor satisfies the equation: \(\beta u_0=u_0\); \(a_0\) is the radius of the electron orbit in the bound state. As \(E_{\mathrm{b}}\) decreases, \(a_0\) increases.

We shall take the final state of the electron in the form of a plane spinor wave normalized in the volume \(V\). We expand the vector potential in plane waves, also normalized in the volume \(V\):

\[ \hat{\mathbf A} = c\sqrt{\frac{2\pi\hbar}{V}} \sum \frac{\mathbf e_\kappa}{\sqrt{\omega_\kappa}} e^{i\mathbf k\mathbf r} \left(\hat a_\kappa+\hat a^+_{-\kappa}\right), \]

where \(\hat a_\kappa,\ \hat a^+_{-\kappa}\) are the operators of absorption and creation of quanta with frequency \(\omega_\kappa\) and polarization \(\mathbf e_\kappa\).

In the second order of perturbation theory we obtain the transition matrix element in the form

\[ H_{10} = -\left( \hat H \frac{1}{\hat H_0-E_0} \hat H \right)_{10} = -\frac{e_0^2\,2\pi\hbar c^2}{V^{3/2}\sqrt{\omega_0\omega_1}} \{I_1+I_2\}, \tag{3} \]

where

\[ I_1= \int e^{i\mathbf k_0\mathbf r-i\mathbf k_e\mathbf r} u_1^+(\mathbf e_0\boldsymbol\alpha) \frac{1}{\hat H_e+\hbar\omega_1-E_e^0} e^{-i\mathbf k_1\mathbf r} (\mathbf e_1\boldsymbol\alpha)\psi_e\,d\mathbf r, \]

\[ \hat H_e=\hat H_D+W,\qquad \hat H_D=c(\boldsymbol\alpha,\hat{\mathbf p})+\beta mc^2, \]

\[ E_e^0=mc^2+E_{\mathrm{b}}, \]

\(W\) is the potential energy;

\(\omega_0,\ \mathbf k_0,\ \mathbf e_0\) are the frequency, wave vector, and unit polarization vector for the quantum in the initial state;

\(\omega_1,\ \mathbf k_1,\ \mathbf e_1\) are the same quantities for the final state of the quantum; \(u_1\) is the spinor describing the final states of the electron;

\(\mathbf k_e\) is the electron wave vector.

\(I_2\) may be obtained from \(I_1\) by replacing \(\omega_0\) by \(-\omega_1\), \(\mathbf k_0\) by \(-\mathbf k_1\), and \(\mathbf k_1\) by \(-\mathbf k_0\). Therefore it is sufficient to calculate one of the integrals. In view of the fact that the binding energy is small, the potential energy may be treated as a perturbation. Then

\[ \frac{1}{\hat H_e-\hbar\omega_0-E_e^0} = \frac{1}{\hat H_D-\hbar\omega_0-mc^2} - \frac{1}{\hat H_D-\hbar\omega_0-mc^2} \left(W-E_{\mathrm{b}}\right) \]

\[ \times \frac{1}{\hat H_D-\hbar\omega_0-mc^2} +\ldots \tag{4} \]

Expanding the function

\[ \frac{1}{\sqrt{\pi a_0^3}}\exp\left(-\frac{r}{a_0}\right) \]

in a Fourier integral and restricting ourselves in formula (4) to one term, we obtain

\[ I_1= \frac{8\sqrt{\pi}\,a_0^{3/2}}{(1+a_0^2K^2)^2} \frac{ u_1^+(\mathbf e_0\boldsymbol\alpha) \left[\hbar\omega_1-E_{\mathrm{b}}+\hbar c(\mathbf k_0-\mathbf k_e;\boldsymbol\alpha)\right] (\mathbf e_1,\boldsymbol\alpha)u_0 }{ 2mc^2\hbar\omega_1 +c^2\hbar^2\left[(\mathbf k_0-\mathbf k_e)^2-\mathbf k_1^2\right] } +\ldots, \tag{5} \]

where \(\hbar \mathbf{k} = (\mathbf{k}_0-\mathbf{k}_1-\mathbf{k}_e)\hbar\) is the magnitude of the nonconservation of momentum.

The operator projecting the spinor \(u_1\) onto the state with positive energy is equal to

\[ \frac{c\hbar}{2E_e} \left[ (\boldsymbol{\alpha}\mathbf{k}_e)+\frac{mc}{\hbar}\beta+\frac{E_e}{c\hbar} \right], \]

and the spinor

\[ u_0=\frac{1+\beta}{2}. \]

Introducing these operators, averaging over the initial spin states of the electron and summing over the final spin states, which upon introduction of projection operators reduces to taking the trace, we obtain

\[ \begin{aligned} 2\overline{H_{01}^{+}H_{10}} &= \frac{64\pi^3 e_0^4\hbar^2 a_0^3} {V^3 m k_0 k_1 E_e(1+a_0^2 k^2)^4} \Bigg\{ \frac{\omega_0}{\omega_1}+\frac{\omega_1}{\omega_0}-2+4(\mathbf{e}_0\mathbf{e}_1)^2 \\ &\quad +\frac{\hbar}{mc}\frac{k_0-k_1}{k_0k_1} \left[ (k_0+k_1)\left(\frac{\mathbf{k}_0}{k_0}+\frac{\mathbf{k}_1}{k_1};\mathbf{k}\right) +(\mathbf{k}_e\mathbf{k}) +\frac{(k_0+k_1)^2+k_0k_1}{2k_0k_1}k^2 \right] \\ &\quad +\frac{2\hbar}{mc}(\mathbf{e}_1\mathbf{e}_1) \left[ \frac{1}{k_1}(\mathbf{e}_0\mathbf{k}_1)(\mathbf{e}_1\mathbf{k}_1) -\frac{1}{k_0}(\mathbf{e}_1\mathbf{k}_0)(\mathbf{e}_0\mathbf{k}) \right] \\ &\quad +\frac{2\hbar}{mc}(\mathbf{e}_0\mathbf{e}_1)^2 \left( \frac{\mathbf{k}_0}{k_0}+\frac{\mathbf{k}_1}{k_1} -2\frac{k_0-k_1}{k_0k_1}\mathbf{k}_e;\mathbf{k} \right) \Bigg\} \\ &= \frac{64\pi^3 e_0^4\hbar^2 a_0^3} {V^3 m k_0 k_1 E_e(1+a_0^2 k^2)^4} \left\{ \frac{\omega_0}{\omega_1}+\frac{\omega_1}{\omega_0}-2+4(\mathbf{e}_0\mathbf{e}_1)^2+\Delta \right\}, \end{aligned} \tag{6} \]

where \(E_e\) is the energy of the electron in the final state.

From this we obtain the differential effective cross section of the Compton effect

\[ d\sigma = \frac{V}{c}\frac{2\pi}{\hbar} \delta(E_{\mathrm{fin.}}-E_{\mathrm{init.}}) 2\overline{H_{01}^{+}H_{10}} \frac{V^3}{(2\pi)^6} k_e^2\,dk_e\,d\Omega_e\,k_1^2\,dk_1\,d\Omega_1, \tag{7} \]

where the \(\delta\)-function expresses the law of conservation of energy.

Since the energy of the electron is completely determined by the energy of the initial and final states of the quantum, expression (7) can be simplified by integrating it over \(dk_e\). Since

\[ c^2\hbar^2 k_e^2+m^2c^4=E_e^2, \quad\text{then}\quad \frac{dk_e}{E_e}=\frac{dE_e}{c^2\hbar^2 k_e}, \]

integration over \(dE_e\) gives

\[ d\sigma = \frac{2r_0^2 mca_0^3 k_e k_1\,dk_1\,d\Omega_e\,d\Omega_1} {\hbar\pi^2 k_0(1+a_0^2k^2)^4} \left\{ \frac{\omega_0}{\omega_1}+\frac{\omega_1}{\omega_0}-2+4(\mathbf{e}_0\mathbf{e}_1)^2+\Delta \right\}, \tag{8} \]

where

\[ r_0=\frac{e_0^2}{mc^2} \]

is the classical electron radius.

Formula (8) gives the distribution of the scattered quanta over energies and directions and the distribution of the Compton electrons over directions. Since the function \((1+a_0^2k^2)^{-4}\) has a sufficiently sharp maximum at \(k=0\), the transfer of binding momentum cannot exceed the value \(\hbar/a_0\). For sufficiently large \(a_0\) this value is very small, and the corrections to the Klein–Nishina formula, given by \(\Delta\sim k\), are sufficiently small. To obtain the distribution of quanta, we integrate

formula (8) over the directions of emission of the electrons, taking the magnitude of the electron momentum to be prescribed:

\[ k^{2}=(k_{0}-k_{1})^{2}+2(k_{0}-k_{1})\frac{mc}{\hbar}. \]

As is directly seen from formula (8), the width of the distribution of secondary quanta with respect to momenta will be \(\sim \hbar/a\). The maximum of the distribution falls at the momentum determined by the laws of conservation of energy and momentum in the usual theory of the Compton effect,

\[ \frac{1}{k_{1}}=\frac{1}{k_{0}}+\frac{\hbar}{2mc}(1-\cos\vartheta). \]

Integrating this distribution near the maximum, we obtain the differential effective scattering cross section into a given angle:

\[ \begin{aligned} d\sigma &=\frac{r_{0}^{4}}{4}\frac{\omega_{1}^{2}}{\omega_{0}^{2}} \Biggl\{ \frac{\omega_{0}}{\omega_{1}}+\frac{\omega_{0}}{\omega_{1}}-2 +4(\mathbf e_{0}\mathbf e_{1})^{2} +\frac{\hbar}{ma_{0}^{2}}\frac{\omega_{0}-\omega_{1}}{\omega_{0}\omega_{1}} \\ &\quad \times \Biggl[ \frac{(\omega_{0}+\omega_{1})^{2}}{2\omega_{0}\omega_{1}} +2(\mathbf e_{0}\mathbf e_{1}) +\frac{\frac{1}{2}(\omega_{0}^{2}-\omega_{1}^{2})(1+\cos\vartheta)} {\omega_{0}^{2}+\omega_{1}^{2}-2\omega_{0}\omega_{1}\cos\vartheta} \\ &\qquad +\frac{(\mathbf e_{0}\mathbf e_{1})^{2}\omega_{0}\omega_{1}(1+\cos\vartheta)} {\omega_{0}^{2}+\omega_{1}^{2}-2\omega_{0}\omega_{1}\cos\vartheta} +\frac{(\mathbf e_{0}\mathbf e_{1})(\mathbf e_{1}\mathbf k_{0})(\mathbf e_{0}\mathbf k_{1})} {(\mathbf k_{0}-\mathbf k_{1})^{2}} \Biggr] \Biggr\}d\Omega_{1}=d\sigma_{0}+d\sigma'. \tag{9} \end{aligned} \]

In the case of unpolarized radiation it is necessary to average over the polarizations of the initial quantum and to sum over the states of the final quantum. Under this averaging \((\mathbf e_{0}\mathbf e_{1})^{2}\) gives

\[ \frac{1+\cos^{2}\vartheta}{2}, \]

\[ (\mathbf e_{0}\mathbf e_{1})(\mathbf e_{1}\mathbf k_{0})(\mathbf e_{0}\mathbf k_{1}) \]

gives

\[ -\frac{1}{2}\sin^{2}\vartheta\cos\vartheta\, k_{0}k_{1}. \]

In the limiting cases, nonrelativistic and extremely relativistic, the formulas obtained are simplified.

a. Nonrelativistic case:

\[ \alpha=\frac{\hbar\omega_{0}}{mc^{2}}=\frac{\lambda_{k}}{\lambda_{0}}\ll 1. \]

Then the correction to the differential effective cross section will be

\[ d\sigma'=\frac{r_{0}^{2}}{32}\left(\frac{\lambda_{k}}{2\pi a_{0}}\right)^{2} [13-12\cos\vartheta+5\cos^{2}\vartheta-2\cos^{3}\vartheta]\,d\Omega. \tag{10} \]

The correction to the integral cross section is

\[ \sigma'=\frac{11}{6}\pi r_{0}^{2}\left(\frac{\lambda_{k}}{2\pi a_{0}}\right)^{2}, \tag{11} \]

whereas the Klein—Nishina formula in the nonrelativistic case gives the Thomson cross section

\[ \sigma_{0}=\frac{8}{3}\pi r_{0}^{2}. \]

6. Extremely relativistic case:

\[ \alpha \gg 1. \]

The corrections to the differential and integral cross sections will be

\[ d\sigma'=\frac{r_0^2}{8\alpha}\left(\frac{\lambda_k}{2\pi a_0}\right)^2 d\Omega \tag{12} \]

and

\[ \sigma'=\frac{\pi r_0^2}{2\alpha}\left(\frac{\lambda_k}{2\pi a_0}\right)^2 . \tag{13} \]

In this case the Klein—Nishina formula gives

\[ \sigma_0=\frac{\pi r_0^2}{\alpha}\ln 2\alpha . \]

Thus, we see that the corrections to the Klein—Nishina formula acquire an appreciable magnitude when the effective dimensions of the wave function \(a_0\) are sufficiently small (not much greater than the Compton wavelength), which is possible only in the case of not very weak binding.

Under our conditions, when \(a_0 \sim 10^{-8}\ \text{cm}\), the relative magnitude of the correction

\[ \frac{\sigma'}{\sigma_0}\sim 10^{-5}, \]

that is, the Klein—Nishina formula gives sufficient accuracy.

Upon irradiation with X-rays with \(\lambda_{\mathrm{char}}=1.537\ \text{Å}\) (X-ray tube with a copper anticathode) and \(\lambda_{\mathrm{eff}}=0.385\ \text{Å}\) (X-ray tube with a tungsten anticathode), the value of the cross section for direct processes has the following values:

Table 1

From Table 1 it is clear that, under our conditions, the direct effects have a very small cross section.

Usually, in studying the kinetics of accumulation of \(F\)-centers under X-ray irradiation, fluxes of \(10^{11}\ \text{quanta}/\text{cm}^2\) in \(1\ \text{sec}\) are used.

If \(\sigma_0 \sim 10^{-20}\), then the probability of destruction of one \(F\)-center upon incidence of \(10^{11}\ \text{quanta}/\text{cm}^2\) in \(1\ \text{sec}\) is equal to \(10^{-9}\ \text{sec}^{-1}\), i.e., in \(1\ \text{sec}\) one out of \(10^9\) \(F\)-centers is destroyed.

REFERENCES

1. P. V. Mitchell, D. A. Wiegand, and R. Smoluchowski, Phys. Rev., 121, 484, 1961.
2. B. Gurari and F. Adrian. In the collection Color Centers. IL, 1958.
3. W. Heitler. The Quantum Theory of Radiation. IL, 1956.