THE ROLE OF VALENCE-ELECTRON PLASMA IN THE STOPPING OF CHANNELED PARTICLES IN CRYSTALS

E. T. Shipatov

(Presented by the scientific seminar of the Faculty of Physics and Technology)

The observed decrease in the specific energy losses of protons with initial energies \((E_0)\) of 3–7 MeV, channeled in single crystals (approximately by a factor of two relative to the specific energy losses of protons in amorphous substances of similar chemical composition), is explained by a significant reduction in the contribution of close electronic collisions to their stopping [1–4]. According to the “equal participation” rule, at sufficiently high particle velocities the contributions of energy loss in close and distant (resonant) collisions of particles with atomic electrons to the total (ionization) energy losses are approximately the same [5].

Let us consider the stopping mechanism of channeled protons in ionic crystals, assuming that their energy losses are caused only by distant collisions with atomic electrons.

In distant electronic collisions, the passing particle excites electrons on atomic shells located at distances from the particle trajectory not exceeding the maximum impact parameter \(b_{\max}\),

\[ b_{\max}=\frac{v\cdot \hbar}{E_i}. \tag{1} \]

Here \(v\) is the particle velocity,

\(E_i\) is the binding energy of electrons on the \(i\)-shell of the atoms.

The impact parameter \(b_{\mathrm{exp}}\) in collisions of a channeled particle with electrons on the \(i\)-shell of an atom is determined by the known radius \(R_k\) of the selected channel and the shell radius \(\rho_i\):

\[ b_{\mathrm{exp}} = R_k - y - \rho_i, \tag{2} \]

where \(y\) is the deviation of the particle from the channel axis. For well-channeled particles \(y \approx 0\), and

\[ b_{\mathrm{exp}} \approx R_k - \rho_i. \]

Let us consider the stopping of protons with energy \(E_0 = 4.7\) MeV, slowed along the \(\langle 100\rangle\) direction in a KCl single crystal. The mechanism of particle channeling in ionic crystals is reported in [4]. Let us estimate \(b_{\max}\), required for the excitation of \(K\)- and \(L\)-electrons in potassium atoms. For the \(K\)-shell, \(E_k = 3606\) eV [6] and \(b_{\max}=0.055\) Å (for \(E_0=4.7\) MeV). For well-channeled protons in the direc-

in the \(\langle 100\rangle\) channel, \(b_{\text{exp}}\approx 0.5\,\text{Å} \gg b_{\max}\), and, consequently, we neglect excitation of electrons in the \(K\)-shell. For \(L\)-electrons \(E_i=341\) eV and \(b_{\max}\approx 0.6\,\text{Å}\). Excitation of \(L\)-electrons cannot be neglected. However, the energy losses of channeled particles are reduced by the factor

\[ \ln\frac{b_{\max}}{b_{\text{exp}}}\bigg/\ln\frac{2mv^2}{E_i}. \tag{3} \]

From this expression we find that the energy losses due to excitation of \(L\)-electrons in potassium atoms are reduced by 95%, and they likewise may be disregarded. In a similar way we find that, for channeled protons, the energy losses due to excitation of the \(K\)- and \(L\)-shells in chlorine atoms may be neglected. For the \(M\)-electrons of potassium atoms \(b_{\max}\approx 5\,\text{Å}\) \((E_M=41\ \text{eV})\). The energy losses due to excitation of the \(M\)-electrons of potassium atoms are reduced for channeled protons by 50% and make a substantial contribution to the total stopping of the protons.

The ensemble of valence electrons in a medium with uniformly distributed positive charges (ionic cores) is treated as a plasma [7, 8]. We neglect the influence of the crystal structure on the excitation of valence electrons. We divide all space into two parts: 1. The region directly adjacent to the particle (proton) trajectory—the region of “binary collisions,” where it is necessary to take into account the direct interaction of the particle with individual (valence) electrons. 2. At large distances from the particle trajectory \((b>n^{-1/3})\)—the “plasma” region, where the excitation of individual electrons produced by the particle is small and collective excitation of the electrons predominates (excitation of plasma oscillations with characteristic frequency

\[ \omega_p=\left(\frac{4\pi ne^2}{m}\right)^{1/2}, \]

where \(n\) is the density of valence electrons.

Let us calculate the total energy transfer to valence electrons per unit path length:

\[ \left(-\frac{dE}{dx}\right)_{\text{val}} = \left(\frac{dE}{dx}\right)_{\text{plas}} + \left(-\frac{dE}{dx}\right)_{\text{coll}}, \tag{4} \]

\[ \left(-\frac{dE}{dx}\right)_{\text{plas}} = \frac{4\pi Z_1^2 e^4}{mv^2}\,NZ_{\text{val}}\ln\frac{v}{v_F}, \tag{5} \]

\[ \left(-\frac{dE}{dx}\right)_{\text{coll}} = \frac{4\pi Z_1^2 e^4}{mv^2}\,NZ_{\text{val}}\ln\frac{2mv\cdot v_F}{\hbar\omega_p}, \tag{6} \]

\[ NZ_{\text{val}}=n \]

where \(NZ'_{\text{val}}\) is the local density of valence electrons in the channel, excited by the particle in binary collisions,

\[ \hbar\omega_p \]

is the plasmon energy,

\[ v_F \]

is the Fermi velocity for the valence electrons.

For a KCl single crystal (8 valence electrons per KCl molecule, \(\hbar\omega_p=13.5\) eV) and proton energy \(4.7\) MeV,

\[ \left(-\frac{dE}{dx}\right)_{\text{plas}} \approx 14\ \frac{\text{MeV}}{\text{cm}}. \]

To calculate the energy losses of a proton in binary collisions with valence electrons (the second term in expression (4)), it is necessary to know the number of valence electrons participating in the stopping...

…of motion of the channeled proton. The quantity \(Z'_{\mathrm{val}}\) is determined by the chemical composition of the single crystal, the type of channel selected, the amplitudes of oscillation of the particles in the channel, and the particle energy. For \(E_0=4.7\ \mathrm{MeV}\) it is reasonable to assume that all valence electrons take part in the stopping of protons channeled along the crystallographic axis \(\langle 100\rangle\) of a KCl single crystal \(\left(b_{\max}\simeq 7\ \text{\AA} > R_k,\ n=1.3\cdot 10^{23}\ \mathrm{el}/\mathrm{cm}^3\right)\). Then

\[ \left(-\frac{dE}{dx}\right)_{\mathrm{st}} \simeq 25\ \frac{\mathrm{MeV}}{\mathrm{cm}} . \]

The total energy losses of channeled protons \(\left(E_0=4.7\ \mathrm{MeV}\right)\) for the excitation of valence electrons amount to

\[ \left(-\frac{dE}{dx}\right)_{\mathrm{val}} \simeq 39\ \frac{\mathrm{MeV}}{\mathrm{cm}}, \]

which is smaller than the measured specific energy losses of channeled protons \(\left(E_0=4.7\ \mathrm{MeV}\right)\) in KCl,

\[ \left(-\frac{dE}{dx}\right)_{\mathrm{chan}} = 43\ \frac{\mathrm{MeV}}{\mathrm{cm}}\ [4]. \]

The quantity

\[ \left(-\frac{dE}{dx}\right) \simeq 4\ \mathrm{MeV}/\mathrm{cm} \]

is the energy of the protons expended on excitation of \(M\)-electrons in potassium atoms. From the known value of \(\left(-dE/dx\right)_M\) one can determine the number of \(M\)-electrons \(Z'_M\) participating in the stopping of channeled protons.

Thus, at sufficiently high velocities of channeled particles, distant collisions of the particles with electrons on all shells of the atoms composing the single crystals and bounding the channels under consideration contribute to their stopping. At a fixed particle energy, the decrease in the contribution to their stopping from electrons on inner shells is associated with a decrease in the amplitudes of particle oscillations in the lattice channels. A further decrease in the contribution of distant electronic collisions to the stopping of channeled particles is possible as the energy of the latter decreases.

At energies of channeled particles below the critical value, when

\[ v < v_k \simeq \frac{R_k E_{OC}}{\hbar}, \]

where \(E_{OC}\) is the binding energy of the outer electrons of the ionic core, the main contribution to the total energy losses is due to collective excitation of valence electrons. For example, for protons slowed in a KCl single crystal along the \(\langle 100\rangle\) direction, \(E_k \simeq 0.5\ \mathrm{MeV}\). At still lower energies of particles slowed in amorphous substances (for protons \(E_0\simeq 1\text{--}1.5\ \mathrm{keV}\)), a transition occurs from the inelastic (ionization) stopping mechanism to the mechanism of elastic stopping, in which particles lose energy in elastic collisions with the nuclei of atoms of the stopping medium. Since elastic nuclear collisions are absent for channeled particles, ionization stopping remains for them the main mechanism of energy loss until the end of the range.

In ionic crystals, where the electron density in the central regions of the channels is practically zero, anomalously large ranges are possible for channeled particles.

References

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