ORIENTATION EFFECTS FOR ELECTRONS AND POSITRONS IN SINGLE CRYSTALS

S. A. VOROB’EV

Recently, considerable attention has been devoted to studies of the motion of charged particles in single crystals, which demonstrates interesting features absent when particles pass through amorphous targets. As a result of these studies for heavy positively charged particles it has been shown that, when such a particle passes through a crystal parallel to a low-index axis or plane, a sharp decrease in the specific energy losses is observed, as well as a decrease in the yield of processes requiring a small collision parameter of the particle with an atom. The channeling of positively charged particles, that is, motion in a lattice channel with a reduced atomic density, leads to a decrease in the yield of particles scattered through large angles, in the yield of nuclear reactions and characteristic x-ray radiation, and to an increase in the ranges of charged particles in single crystals. The channeling effect is described quite well with the aid of classical mechanics.

Motion of particles in lattice crystals

Theoretical studies of the channeling effect [1] have shown that there exists a certain critical angle \(\psi_k\) between the direction of motion of a particle and the axis of the channel, which makes it possible conditionally to divide the flux of particles incident on the crystal into channeled and chaotic beams. Channeled particles undergo successive small-angle collisions with the atoms of the lattice rows and cannot approach an atom more closely than the value of the Thomas–Fermi screening parameter, without leaving the channel. The trajectory of a channeled particle in this case is approximated by an oscillating curve with maximum oscillation amplitude \((R_k-a)\), where \(R_k\) is the radius of the channel under study, and \(a\) is the Thomas–Fermi screening parameter lying in the region of the effective potential of the row \(V(y)\), where \(y\) is the transverse displacement of the particle from the channel axis. This oscillating trajectory is characterized by the transverse displacement of the particle \(y\) and the angle \(\psi_k\) at which the particle crosses the channel axis. The characteristics indicated above are related by the condition of stable channeling, according to which the maximum transverse energy \(\psi_k^2 E\) must correspond to the potential energy at the maximum of the trajectory amplitude \(\psi_k^2 E \leq \bar V(y_{\max})\). If the poten-

…to choose the potential of a row of atoms in the form of a Coulomb distribution, the critical channeling angle has the form

\[ \Psi_k=\left(\frac{2Z_1Z_2e^2}{Ed}\right)^{1/2} \]

provided that \(\Psi_k<a/d\).

In [1] an expression was also obtained for determining the yield of charged particles scattered by lattice atoms at a sufficiently small depth near the crystal surface in the case of a well-channeled beam,

\[ X_{\min}=\pi Nd\,[a^2+\rho^2(T^\circ)] \]

in the approximation that the root-mean-square amplitude \(\rho(T^\circ)\) of oscillations of lattice atoms perpendicular to the axis of the chain is comparable with the Thomas–Fermi screening parameter \(a\). The quantity \(N\) determines the density of atoms in a row or chain.

Numerous and varied experiments with heavy charged particles [2] confirm the conclusions of Lindhard’s theory [1], which is based on the concepts of classical mechanics. As for light charged particles, electrons and positrons, their motion through a crystal must have features of a nonclassical character [3]; this applies especially to negatively charged electrons, for which the phenomenon of wave diffraction is well known. At present there are experimental data confirming both the wave and the classical mechanisms of channeling of charged particles.

Emission of beta particles from single crystals

In the first study [4] of the angular dependence, with a detector resolution of \(1.5^\circ\), of the emission of electrons emitted by radioactive Xe\(^{133}\) ions from a single-crystal tantalum source, a complex spatial structure of the flux of scattered electrons was found. Radioactive ions implanted into the single crystal to an average depth of approximately 260 atomic layers occupied positions at the lattice sites, and the emergence of conversion \(K\)-electrons with an energy of 44.8 keV and \(L\)-electrons with an energy of 75.8 keV was shadowed in the direction of the crystal axis by a chain of lattice atoms. As a result of this shielding action of the atoms of the chain, when the crystal axis \(\langle100\rangle\) coincided with the detector axis, a minimum in the electron yield was observed; on both sides of it, at several larger values of the crystal tilt angle, two symmetric narrow maxima in the yield of emitted electrons were found (Fig. 1).

Fig. 1. Dependence of the emission yield on the tilt angle of the single crystal [4]

The observed pattern of the angular distribution was the same for conversion electrons from both investigated shells of the Xe\(^{133}\) ion, and the position of the maxima and their width at half height do not fit within the framework of the simple kinematic theory of diffraction.

The study of the emission of electrons and positrons emitted by radioactive \( \mathrm{Cu}^{64} \) ions, localized at an average depth of about 200 Å in the lattice sites of a copper single crystal [5, 6], confirmed the fact that the crystal lattice exerts an orienting influence on the yield of beta particles from crystals. In the cited works, the detection system using a magnetic spectrometer made it possible, simultaneously, with angular resolution down to \(0.5^\circ\), to separate and record a beam of electrons with energy \((150 \div 250)\) keV and a beam of positrons with energy \((200 \div 300)\) keV. The measurement results showed that, when the detector axis coincides with the crystallographic axis of the emitter, a decrease in the positron yield is observed in comparison with a disoriented crystal, and an increase in the electron yield. Figure 2 shows the dependence of the emission yield

Fig. 2. Dependence of the emission yield of electrons and positrons on the tilt angle of a single crystal [5, 6]

of positrons and electrons of different energies on the angle between the direction of particle emission and the \(\langle 110\rangle\) axis of a copper single crystal. The authors attribute the difference obtained in these works in the emission patterns for electrons, as compared with [4], to possible interference effects in [4], where the electrons were of comparatively low energy, especially since in Fig. 2 some oscillations of the electron yield are also noticeable, occurring on the sides of the main maximum. The measured minimum of the positron yield has a width somewhat greater than the width of the electron maximum and is very similar to the corresponding minimum in the yield distribution of backscattered heavy positively charged particles. Calculation of the critical channeling angles by Lindhard’s relativistic formula gives theoretical angle values approximately coinciding with the experimental values, and this agreement improves as the positron energy increases.

Characteristic X-ray Radiation

The excitation of characteristic X-ray radiation requires a sufficiently small interaction parameter of the incident particle with the atoms of the target and therefore should likewise be sensitive to the orientation of the target relative to the incident beam. In [7, 8] it was shown experimentally that characteristic X-ray radiation is excited more weakly by channeled positrons, whereas for electrons, on the contrary, an enhancement of the yield of X-ray quanta takes place.

When measuring the yield of characteristic X-ray radiation generated in a silver single crystal of thickness up to \(2000\ \text{Å}\) by electrons with an energy of \(350\ \text{keV}\) (the angular divergence of the electron beam was about \(10^{-3}\ \text{rad}\)), an increase in the yield of X-ray quanta was found when the \(\langle 100\rangle\) axis of the crystal coincided with the beam axis [7]. The width of the peak at half its height was about \(3^\circ\), which is in good agreement with the results of [5, 6]. A fine structure of the maximum was also observed; however, the authors do not relate it to diffraction structures, but explain it by a distortion of the orientation effect due to the intersection of crystallographic planes by the electron beam during rotation of the crystal. An interference splitting of the maximum of the X-ray yield into two narrow peaks, similar to that observed in [4], was not found in this experiment.

In [8] a similar experiment was performed using a radioactive positron source \(^{58}\mathrm{Co}\), having a continuous spectrum with a mean energy of \(220\ \text{keV}\), but owing to the non-monoenergetic character of the positron source, as well as the large statistical error of the experiment, no deviations from the classical behavior of orientation effects were found in measuring the yield of characteristic X-ray radiation from an RbBr single crystal.

Scattering of Beta Particles by Crystals

A detailed study of orientation phenomena for the scattering of electrons by single crystals was carried out in [9] with the aim of clarifying the nature of the observed effects. Using a Van de Graaff accelerator, Rutherford scattering of electrons with an energy of \(1\ \text{MeV}\), incident on single-crystal gold films of thickness less than \(5000\ \text{Å}\), was measured. The angular divergence of the incident beam was \(0.1^\circ\), and the angular resolution of the semiconductor detector was \(10^\circ\). Figure 3 shows typical results for the dependence of the yield of Rutherford scattering on the tilt angle of the crystal when electrons are incident along the crystallographic axis \(\langle 111\rangle\) of the target. When the axis of the incident beam coincided with this axis of the crystal, maxima were obtained in the yield of electrons scattered through large angles, with a width agreeing with Lindhard’s formula for the classical channeling angle. The small “humps” that distort the observed maxima of the particle yield are attributed by the authors, analogously to [7], to the influence of planar effects manifested during rotation of the target. In confirmation of the assumption made, measurement of the electron yield along crystallographic planes revealed no distortions of the orientation maxima. The numerous experiments performed in [9], including measurements of the angular distribution of multiply scattered electrons, testify to the classical behavior of electrons in single crystals and to a certain “focusing” of electrons on chains of atoms. In accordance with Lindhard’s reciprocity rule [1]

orientation effects in the scattering of electrons through large angles should be analogous to the emission of electrons from lattice sites.

Consequently, the results obtained in [9] can be compared with the data of [7], and a conclusion can be drawn in favor of the possibility of a classical description of axial orientation effects.

As was shown in [10], the behavior of charged particles can be regarded as classical, independent of mass, in the case when the number of bound states in transverse phase space is sufficiently large. According to this condition, in [9] an approximate expression was obtained for the number of bound states of an electron moving in an axial channel

Fig. 3. Dependence of the yield of scattered electrons on the orientation of a single crystal [9].

\[ n_k \simeq \frac{4a_0 Z^{1/3}}{d}\,(1-\beta^2)^{-1/2} \]

and for interplanar channeling

\[ n_p \simeq \frac{2a_0}{d_p}(1-\beta^2)^{-1/4}, \]

where \(\beta=v/c\), \(v\) is the velocity of the incident particle, \(c\) is the speed of light in vacuum, \(Z\) is the atomic number of the crystal, \(a_0\) is the classical Bohr radius, \(d\) is the distance between atoms in a chain, and \(d_p\) is the interplanar spacing. For electrons with an energy of \(1\) MeV in a gold single crystal, \(n_k \simeq 10\), while \(n_p < 1\), which explains the possibility of applying the apparatus of classical mechanics in the description of axial channeling. Interplanar channeling of electrons, according to the criterion obtained, should be described by quantum-mechanical theory; however, the experimental results of the works analyzed did not reveal a wave pattern of interplanar orientation patterns for electrons.

Further confirmation of the possibility of a classical description of axial orientation phenomena for electrons was obtained from the temperature dependence of the efficiency of anomalous backscattering of electrons [11]. The yield of electrons from a radioactive source \((\mathrm{Sr}+\mathrm{Y})^{90}\) was studied for backscattering of electrons incident along the \(\langle 100\rangle\) axis of alkali-halide single crystals KCl and KBr in the temperature range \((300 \div 700)^\circ\mathrm{K}\). It was found that, upon heating the single-crystal target, the efficiency of the orientation phenomena decreases—

decrease with increasing temperature approximately linearly (Fig. 4), and at a certain critical temperature \(T_{\mathrm{cr}}^{0} \simeq 4\Theta_D\) (\(\Theta_D\) is the characteristic Debye temperature) they practically cease to be observed, which agrees with the temperature dependence for the channeling efficiency of positively charged heavy particles and positrons. In [11], similarly to the results of [5, 6], classical maxima of backscattered electrons and a minimum for positrons were also observed when the \(<100>\) axis of single crystals was oriented along the axis of the beam of charged particles. The half-width of the peaks obtained for electrons and positrons agrees with the calculated values according to Lindhard’s theory.

It is of definite interest that the contribution of interference effects to the classical picture of orientation phenomena for light charged particles was first established rather clearly in an experiment with positrons [12], under experimental conditions less stringent than in [9]. The angular divergence of the incident beam of positrons with energy \((1.00 \pm 0.15)\) MeV was \((0.14 \div 0.30)^\circ\), and the angular resolution of the detector was \(6^\circ\). For a more correct determination of the interference contribution to the channeling mechanism, the authors compared the minima in the yield of positrons scattered through large angles in the case of channeling of positrons and protons of the corresponding energy. Figure 5 shows the positron and proton minima observed for the \(<110>\) axis in a gold single-crystal film of thickness \(1200\ \text{Å}\). The minimum yield and the half-width of the minima coincide well; however, in the compensating peaks at angles somewhat larger than the critical angle there is a noticeable difference. The authors attribute this difference to the effect of the inaccuracy of the orientation of the single crystal relative to the beam, but the fact that in the “shoulders” of the positron minimum there are small oscillations, absent in the case of protons, apparently indicates a contribution of interference effects, though not very significant for the case of axial channeling.

More strongly, quantum-mechanical effects manifest themselves in the case of planar channeling of positrons.

Fig. 4. Temperature dependence of the yield of electrons scattered through an angle of \(150^\circ\) [11]

Fig. 5. Comparison of the orientation dependences of the yield of positrons and protons with energy 1 MeV scattered through large angles [12]

Figure 6 shows the measured dependence of the yield of single scattering through large angles of positrons channeling in the \(\{110\}\) planes on the angle of inclination of the crystal relative to the direction of the incident beam. An interference fine structure is clearly manifested both in the minimum of the positron yield and in the compensating peak. For comparison, the same figure shows an analogous dependence calculated by the 10-beam theory of dynamic diffraction, which do-

agrees quite well with the experimentally measured curve for positrons.

The criterion for the applicability of classical mechanics is written, for axial channeling of positrons, by means of the representation of the number of quantum states in the minimum of the potential of an atomic chain in the form [12]

\[ n \simeq Z\gamma/\pi N d_2 a_0, \]

where \(\gamma m_0\) is the relativistic mass of the positron. For the case illustrated in Fig. 6, \(n \simeq 100\), and, consequently, Lindhard’s theory is fully applicable. In planar channeling, however, \(n\) turns out to be close to unity, and interference effects begin to manifest themselves; for certain planes these effects, apparently, may become dominant.

Fig. 6. Comparison of the measured and calculated orientational dependences for the \(\{110\}\) plane of a single crystal [12].

Thus, at present there is very contradictory information on the nature of orientational effects for light charged particles. The solution of this question, outlined in [12], will make it possible to identify the domains in which orientational effects of beta particles are preferably used in practice.

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