EFFECT OF THE CRYSTALLINE STRUCTURE OF MATTER ON ENERGY LOSSES OF FAST PROTONS IN SINGLE CRYSTALS OF ALKALI METAL HALIDES

E. T. SHIPATOV, B. A. KONONOV

The paper considers the question of channeling of fast protons in ionic crystals possessing a simple cubic structure. Cases are considered of the passage of protons in the \([100]\) direction and between the \((100)\) planes of single crystals. Potentials for the atomic planes and atomic rows of crystals have been calculated, and the limiting channeling angles have been determined for sodium and potassium halide compounds.

The energy spectra of protons that have passed through single-crystal targets of various compounds have been studied, and a correlation has been found between the shape of the energy spectrum and the ratio of the ionic radii in the compound, and between the fraction of channeled particles and the ordinal number of the halide in the series.

The channeling of fast protons in simple ionic crystals of cubic structure is considered in this paper.

The penetration of protons along the \([100]\) directions and between the \((100)\) planes of monocrystals is considered.

The potentials of the atomic planes and the atomic rows of crystals are calculated and the critical angles for channeling for sodium and potassium halogenide combinations are defined.

The energy spectra of protons having passed through the monocrystals of various compositions are examined and the correlation between the shape of the energy spectrum and the ratio of radii in composition, between the share of particles channelling and the ordinal number of halide in rows is discovered.

In the theory of the passage of particles through matter, the principal approximation is the replacement of the actual medium through which the particle passes by an isotropic, homogeneous medium composed of individual atoms distributed randomly throughout the entire volume. The processes of scattering of particles by individual atoms are independent of one another and are characterized by a differential transverse scattering cross section. The stopping of particles in such an amorphous medium does not depend on the direction of particle motion, and the distribution of energy losses and the angular distribution of particles depend only on the number of atoms per unit volume or on the thickness of the layer traversed by the particles.

However, for the most part, real solids possess a quite definite crystalline structure, that is, they are representatives of another type of medium—a heterogeneous, anisotropic, and ordered medium. The development of radiation physics at the present time sharply raises the question of the role of crystalline structure in all processes of interaction of particles with matter.

The first studies of the processes of interaction of particles with single-crystalline solids were connected with cathode sputtering

substances and led to the discovery by Vener in 1954 of the so-called focusing collisions of atoms [1].

The theoretical and experimental studies carried out in recent years on the passage of fast charged particles through single-crystalline substances have shown that the crystal structure has a strong influence also on the character of particle motion. When the directions of a particle beam and of some principal crystallographic axis of a single crystal coincide, anomalously low energy losses are observed both for protons [2–4] and for heavy ions [5]; the yields of atomic [6] and nuclear [7–8] processes decrease; the rate of formation of radiation defects [9] is substantially reduced; and anomalies are also observed in the scattering of particles by single-crystal targets [10]. All these anisotropy phenomena are associated with the motion of particles in relatively free spaces of the lattice—“channels”—formed by sets of neighboring, mutually parallel atomic rows or atomic planes lying in definite crystallographic directions.

The capture of particles into channels is effected by successive glancing collisions of the particles with atoms of the rows or planes forming the channel walls. This phenomenon has been called “channeling” of particles in crystal lattices.

It should be noted that there is also another group of phenomena associated with the influence of the crystal structure on the motion of particles formed inside the crystals themselves in nuclear reactions [11] or scattered through large angles [12]. For such particles, the principal crystallographic directions, in contrast to particles channeled through crystals, prove to be closed. This phenomenon has been called “blocking” or the “shadow effect” [13].

Table 1

Here

\(a\)—lattice constant [14],
\(r_{\mathrm{a}}\)—anion radius in the compound [14],
\(r_{\mathrm{k}}\)—cation radius in the compound [14],
\(q\)—packing coefficient of the structure [15],
\(T_{\mathrm{m}}\)—melting temperature [16],
\(\Theta_{\mathrm{D}}\)—characteristic Debye temperature [17].

Let us note further that all studies of particle-channeling effects up to the present time have been carried out on a narrow range of simple substances;

such as Si, Ge, Au, Cu, which have different crystal structures, and at different particle energies. Therefore, the results of these studies are difficult to systematize, and it is impossible to establish a connection between changes in the properties of the crystalline medium and its influence on the motion of particles. From this point of view it is expedient to investigate channeling effects in single-crystalline chemical compounds that have the same crystal lattice but different lattice parameters. The study of chemical compounds is all the more interesting because it makes it possible to trace the influence of the crystal structure in a series of substances, provided that one component of the compound remains constant while the others are varied.

Proceeding from these considerations, we selected as objects of investigation single crystals of alkali-metal halides: NaCl, NaBr, KCl, KBr, KI. All these compounds belong to the type of ionic crystals and have a simple cubic lattice.

Some characteristics of these compounds are presented in Table 1.

The most open directions in a simple cubic lattice are the directions [100] and [110].

Figure 1 shows the structure of the [100] channel in a NaCl single crystal and schematically shows the distribution of electron density in the (100) plane.

Fig. 1. Structure of the [100] channel in a simple cubic lattice.
\(\Psi\) — angle of entry of the proton into the channel,
\(y_0\) — amplitude of the oscillation of the proton trajectory,
\(R\) — channel radius,
\(d\) — distance between neighboring ions in a row,
\(\lambda\) — wavelength of the oscillation of the proton trajectory,
\(Z_2'\) and \(Z_2''\) — ordinal numbers of the ions.

It may be noted that the space relatively free of electrons is asymmetric with respect to the channel axis, owing to the different ordinal numbers of the ions located at the lattice sites.

Channeling mechanism

Analysis of the motion of particles along the lattice channels in the classical approximation, valid for protons [18], shows that when a particle enters a crystal at a sufficiently small angle to the channel axis, it

over a relatively long time will move along an oscillating trajectory with amplitude \(y_0\) in a certain average potential field \(V(\rho)\), produced by all the atoms in a row or plane. In this case it is convenient to consider only the transverse motion of a particle with energy \(E_\perp = E\Psi^2\), where \(E\) is the energy of the particle and \(\Psi\) is the angle between the direction of the channel axis and the direction of the particle beam.

The potentials of atomic rows and atomic planes are easily obtained if the interaction potentials \(U(r)\) of the particle with individual atoms are known.

It is known that, in the scattering of protons with energies in the range from 1 keV to 10 MeV through small angles, allowance for the screening of nuclei by atomic electrons becomes especially important.

Bohr [19], in order to take screening into account, introduced an interaction potential between a particle and an atom of the form:

\[ U(r)=\frac{Z_1Z_2e^2}{r}\exp\left(-\frac{r}{a_{\mathrm{B}}}\right). \tag{1} \]

Here and below we use the following notation:

\[ \begin{aligned} Z_1e&\text{ — charge of the proton,}\\ Z_2e&\text{ — charge of the scattering atom,}\\ r&\text{ — distance from the center of the atom to the incident particle,}\\ a_{\mathrm{B}}&\text{ — screening radius.} \end{aligned} \]

\[ a_{\mathrm{B}}=a_0\left(Z_1^{2/3}+Z_2^{2/3}\right)^{-1/2}, \tag{2} \]

where \(a_0=0.529\cdot10^{-8}\ \mathrm{cm}\) is the Bohr radius of the hydrogen atom.

In most cases a more accurate potential, especially for heavy atoms and at small distances, is the Thomas–Fermi potential, written in the form [20]

\[ U(r)=\frac{Z_1Z_2e^2}{r}\,\chi\left(\frac{r}{a_{\mathrm{TF}}}\right). \tag{3} \]

Here \(\chi\left(\frac{r}{a_{\mathrm{TF}}}\right)\) is the Thomas–Fermi screening function for an individual atom.

\(a_{\mathrm{TF}}\) is the Thomas–Fermi screening radius.

\[ a_{\mathrm{TF}}=0.8853\,a_0\left(Z_1^{2/3}+Z_2^{2/3}\right)^{-1/2}. \tag{4} \]

In what follows, since \(Z_1<Z_2\), \(a_{\mathrm{TF}}=0.468 Z^{-1/3}\). For values of \(r\) comparable with the screening radius, the function \(\chi\left(\frac{r}{a_{\mathrm{TF}}}\right)\) may be approximated by the expression

\[ \chi\left(\frac{r}{a_{\mathrm{TF}}}\right)\sim \frac{a_{\mathrm{TF}}}{2r} \tag{5} \]

and the potential is written in the form

\[ U(r)=\frac{Z_1Z_2e^2a_{\mathrm{TF}}}{2r^2}. \tag{6} \]

For large values of \(r\), namely for \(0\le r\le 6a_{\mathrm{TF}}\), the potential obtained by Molière [21] is more accurate:

\[ U(r)=\frac{Z_1Z_2e^2}{r}\left[ 0.1\exp\left(-\frac{6r}{a_{\mathrm{TF}}}\right) +0.55\exp\left(-\frac{1.2r}{a_{\mathrm{TF}}}\right) +0.35\exp\left(-\frac{0.3r}{a_{\mathrm{TF}}}\right) \right]. \tag{7} \]

We obtained expressions for the potentials of atomic rows and atomic planes in single crystals of two-component chemical compounds composed of ions with different atomic numbers \(Z_2'\) and \(Z_2''\).

For atomic rows the Bohr (1), Molière (7), and Nielsen (6) potentials have the form:

\[ \overline{V}(\rho)=\frac{Z_1e^2}{a}\left[Z_2'K_0\left(\frac{\rho}{a_B'}\right)+Z_2''K_0\left(\frac{\rho}{a_B''}\right)\right]. \tag{8} \]

\[ \begin{aligned} \overline{V}(\rho)=\frac{Z_1e^2}{a}\Bigg\{& Z_2'\left[0.1K_0\left(\frac{6\rho}{a_{\mathrm{TF}}'}\right) +0.55K_0\left(\frac{1.2\rho}{a_{\mathrm{TF}}'}\right) +0.35K_0\left(\frac{0.3\rho}{a_{\mathrm{TF}}'}\right)\right]\\ &+Z_2''\left[0.1K_0\left(\frac{6\rho}{a_{\mathrm{TF}}''}\right) +0.55K_0\left(\frac{0.3\rho}{a_{\mathrm{TF}}''}\right) +0.35K_0\left(\frac{0.3\rho}{a_{\mathrm{TF}}''}\right)\right]\Bigg\}. \end{aligned} \tag{9} \]

\[ \overline{V}(\rho)=\frac{\pi Z_1e^2}{4\rho\cdot a}\left[Z_2'a_{\mathrm{TF}}'+Z_2''a_{\mathrm{TF}}''\right]. \tag{10} \]

Here

\[ K_0(z) \]

is the Macdonald function [22], and \(\rho\) is the radial distance from the center of the atomic row.

For atomic planes, the following expressions were obtained for the Bohr, Molière, and Nielsen potentials:

\[ \overline{V}(\rho)_{\mathrm{pl.}}=\frac{\pi Z_1e^2}{a^2}\left[Z_2'a_B'\exp\left(-\frac{\rho}{a_B'}\right)+Z_2''a_B''\exp\left(-\rho/a_B''\right)\right]. \tag{11} \]

\[ \begin{aligned} \overline{V}(\rho)_{\mathrm{pl}}=\frac{\pi Z_1e^2}{a^2}\Bigg\{& Z_2'a_{\mathrm{TF}}'\frac{0.1}{6}\exp\left(-\frac{6\rho}{a_{\mathrm{TF}}'}\right)\\ &+\left[\frac{0.55}{1.2}\exp\left(-\frac{1.2\rho}{a_{\mathrm{TF}}'}\right) +\frac{0.35}{0.3}\exp\left(-\frac{0.3\rho}{a_{\mathrm{TF}}'}\right)\right]\\ &+Z_2''a_{\mathrm{TF}}''\times\\ &\times\left[\frac{0.1}{6}\exp\left(-\frac{6\rho}{a_{\mathrm{TF}}''}\right) +\frac{0.55}{1.2}\exp\left(-\frac{1.2\rho}{a_{\mathrm{TF}}''}\right) +\frac{0.35}{0.3}\exp\left(-\frac{0.3\rho}{a_{\mathrm{TF}}''}\right)\right]\Bigg\}. \end{aligned} \tag{12} \]

\[ \overline{V}(\rho)_{\mathrm{pl}}\simeq\frac{\pi Z_1e^2}{2a^2}\left[Z_2'a_{\mathrm{TF}}'+Z_2'a_{\mathrm{TF}}''\right]. \tag{13} \]

The calculated dependences of the potentials \(\overline{V}(k)\) for the atomic rows [100] and the atomic planes (100) in KBr are presented in Figs. 2 and 3.

Let us compare the potentials of rows and planes with one another. The figures show that the potentials of rows and planes are small up to certain values \(\rho \simeq a_{\mathrm{TF}}\), and then a sharp increase is observed. It should be noted that the row potentials decrease with distance more rapidly than the plane potentials; the two potentials become equal at a distance \(\sim d/2\), whence row effects must be more sensitive to lattice disturbances. Motion in the interplanar space, moreover, is two-dimensional. Note that the row potentials depend only on the charge per unit length of the row, \(Z_2e/d\), and do not depend on the energy. Thus, channels situated along the principal crystallographic directions have not only larger field-

of the cross section, but are also more effective from the standpoint of influencing the character of the motion of the particles.

In the series of compounds considered, the magnitudes of the potentials increase with increasing atomic numbers.

Fig. 2. Average potentials \(V(\rho)\) for rows [100] in KBr.
Here and in the following figure the following notation is used:
1 — Bohr potential,
2 — Nelson potential,
3 — Molière potential.

The approximation of continuous rows is valid at distances \(\rho\) greater than the screening radius of the atoms.

For this approximation to be valid, it is necessary to require that the scattering of the particle by the row be determined simultaneously by many atoms in the row. This condition is evidently better fulfilled at high particle energies. It can be shown that channeling becomes possible if the angle of incidence of the particle into the channel \(\Psi\) is smaller than a certain limiting angle \(\Psi_{\max}\) [23]

\[ \Psi < \Psi_{\max} = \sqrt{\frac{2b}{d}}, \tag{14} \]

where \(b = \dfrac{Z_1 Z_2 e^2}{E}\) is the collision diameter in the laboratory coordinate system.

The magnitude of the limiting channeling angle can be found if the minimum distance of approach of the particle to the row is known, at which the continuous description of rows and planes is still valid. According to Lindhard [23], for stable channeling \(\rho_{\min} \simeq a_{\mathrm{TF}}\). On the other hand, if the particle trajectory lies in the plane of an atomic row, the minimum distance can be found from the solution of the equation

Fig. 3. Average potentials \(\overline{V}(\rho)_{\mathrm{pl}}\) for the (100) planes in KBr.

\[ \overline{V}(\rho) = E\Psi^{2}. \tag{15} \]

Hence

\[ \Psi = \left[\frac{\overline{V}(\rho)}{E}\right]^{1/2}. \tag{16} \]

If one now finds the value of the potential \(\overline{V}(\rho_{\min})\), one can obtain the value of the maximum channeling angle for the given particle energy.

Particles moving along the channels of the lattice in the average field \(\overline{V}(\rho)\), with transverse energy \(E\Psi^{2}\), when approaching rows or planes to a distance \(\rho_{\min}\), encounter, as it were, a potential barrier. At high particle velocities the potential barriers are, in order of magnitude, equal to:

for rows

\[ \sim \frac{4 Z_1 Z_2 e^2}{d} \simeq Z_1 Z_2 \cdot 20\ \text{eV} \tag{17} \]

at \(d \sim 5a_0\);

for planes

\[ \sim 5 Z_1 Z_2 e^2 \cdot N \cdot d_{\text{p}} a_{\text{TF}} \simeq 5 Z_1 Z_2^{2/3}\ \text{eV}, \tag{18} \]

where \(d_{\text{p}}\) is the distance between planes,
\(N \cdot d_{\text{p}}\) is the average number of atoms per unit area of the plane.

The potential barriers for planes are lower by a factor \(b \sim 4Z_2^{1/2}\) than the barriers for rows, and hence the critical channeling angles for planes must be lower by a factor \(\sim 2Z_2^{1/6}\) than the critical angles for rows.

If expression (16) is used to calculate the amplitude of oscillation of the particle trajectory in the channel, \(y\), as a function of the angle \(\Psi\) at different particle energies \(E\), one can obtain the curves shown in Figs. 4 and 5. The calculation was performed for protons with energies of 1 MeV, 5 MeV, and 10 MeV for the \([100]\) and \([100]\) channels in KBr and for the Bohr potential.

Fig. 4. Amplitude of oscillations of the proton trajectory in the \([100]\) channel of KBr as a function of the angle of entry of the proton into the channel at different proton energy values.

It is evident from the figures that, for different energies, the curves are located close to one another and, consequently, the amplitudes \(y\) depend only weakly on the angle \(\Psi\). The convergence of the curves as the amplitude approaches the value of the channel radius \(R_k\), and their saturation, are due to the sharp increase of the potentials beginning from the values

\[ \rho_{\min} = R_k - y_0 \simeq a_{\text{TF}} . \tag{19} \]

Taking this value for \(\rho_{\min}\), one can find the critical channeling angles of protons in single crystals. In the case of KBr, for pro-

protons with an energy of 6.72 MeV, the angles for channeling along the \([100]\) direction and between the \((100)\) planes are equal to

\[ \Psi_{\max[100]} \simeq 0.49^\circ, \]

\[ \Psi_{\max(100)} \simeq 0.17^\circ . \tag{20} \]

The time of one oscillation of the particle trajectory is approximately equal to [18]

\[ \frac{4y}{\Psi v}, \tag{21} \]

Fig. 5. Amplitude of oscillations of the proton trajectory between the \((100)\) planes of KBr as a function of the angle of incidence for different values of the proton energy

and, consequently, the wavelength of the trajectory \(\lambda\) is equal to

\[ \lambda \simeq \frac{4y}{\Psi}. \tag{22} \]

The relation between \(y\) and \(\Psi\), shown in Figs. 4 and 5, indicates that \(\lambda\) decreases with increasing \(\Psi\), as was to be expected. From Figs. 4 and 5 one can find that, for protons with an energy of 6.72 MeV in a KBr single crystal,

\[ \lambda_{[100]} \simeq 2 \cdot 10^{-5}\ \text{cm}, \]

\[ \lambda_{(100)} \simeq 3.9 \cdot 10^{-5}\ \text{cm}. \tag{23} \]

The relation between the amplitude \(y\) and the angle \(\Psi\) in a series of compounds makes it possible to conclude that, with increasing atomic numbers of the components, the wavelength increases only slightly.

Let us now consider qualitatively the behavior of a beam of particles passing through a crystal lattice. At angles of incidence of the particles on the crystal, relative to the principal directions or planes, smaller than the limiting-

...angle \(\Psi < \Psi_{\max}\), the particles are captured into the lattice channels. Their approach to the atomic rows to distances smaller than the screening radius is hindered, and the angular distribution of such particles will differ from the angular distribution of particles in an amorphous medium. Such a beam has been called the central beam of particles. If \(\Psi > \Psi_{\max}\), then the particles, passing through the lattice, undergo ordinary scattering by individual atoms and in most cases their motion is analogous to the motion of particles in an amorphous medium. Such a beam is called normal or chaotic. It is interesting to note that these two beams are not connected with one another, and the transition of particles from one beam to the other in a perfect crystal is impossible. Indeed, in order for a particle to be able to pass from the central beam into the chaotic one, it must undergo very strong scattering by an individual atom; and for this it must approach the atom to a distance \(\rho < a_{\mathrm{T}}\Phi\), which is impossible for central particles. For the same reason, the transition of particles from the normal beam into the central one is also hindered. The region of angles \(\Psi \simeq \Psi_{\max}\) is intermediate, and consideration of the behavior of such particles is extremely difficult because many effects are superposed.

Up to now we have considered the model of a rigid lattice with atoms fixed immovably at their sites. Let us estimate the influence of thermal vibrations of the lattice atoms on the magnitudes of the potentials.

Consider harmonic vibrations of the lattice from the classical point of view. Such a treatment gives a good approximation for temperatures greater than the Debye temperature. We assume that the displacements of atoms obey a Gaussian distribution with mean-square displacement \(\bar{s}^{2}\):

\[ \omega(s)\,ds = \frac{1}{\sqrt{2\pi \bar{s}^{2}}} \exp\left(-\frac{s^{2}}{2\bar{s}^{2}}\right). \tag{24} \]

For such a distribution of atomic displacements, a potential of the form

\[ \bar{V}(\rho)=A\exp(-\rho/a), \]

where \(A\) and \(a\) are constants, is transformed into a potential of the form [24]

\[ \bar{V}'(\rho)=\bar{V}(\rho)\exp\left(\frac{\bar{s}_{\perp}^{2}}{2a^{2}}\right), \tag{25} \]

where \(\bar{s}_{\perp}^{2}\) is the mean-square displacement in the direction perpendicular to the trajectory of the particle in the channel. According to the dynamical theory of crystal lattices, for the deviation of an atom from its equilibrium position one may write [25]

\[ \bar{s}_{\perp}^{2} = \frac{0.5}{T_{\mathrm{m}}}\cdot 10^{-2} \begin{cases} T, & \text{for } T \geq \Theta_{\mathrm{D}},\\[4pt] \dfrac{\Theta_{\mathrm{D}}}{4}, & \text{for } T < \Theta_{\mathrm{D}}. \end{cases} \tag{26} \]

This expression is valid only for simple cubic lattices, where the condition \(\bar{s}_{\perp}^{2}=\dfrac{1}{3}\bar{s}^{2}\) is satisfied.

To estimate the influence of thermal vibrations on the magnitude \(V(\rho)\), let us consider the change of the potential in a KCl single crystal with a change in temperature \(T\). For the estimate we use the Molière potential (14) for the atomic plane (100). The results of the calculation are given in Table 2.

It may be noted that thermal vibrations of the atoms lead to an insignificant increase of the potential, and this increase causes a corresponding and still weaker increase of the angle \(\Psi_{\max}\). On the other hand,

thermal vibrations of atoms lead to a decrease in the dimensions of the channel \(R_k\) by the amount of the mean vibration amplitude \(S_\perp\). Now, for stable channeling, the distance of closest approach to the channel wall \(\rho_{min}\) will be determined by the expression [26].

\[ \rho_{\min} \geq cS_\perp, \tag{27} \]

where \(c\) is a small constant quantity. From simple considerations one should expect that \(c\) will be a function of the length of the channeled trajectory of the proton \(t\), which is approximately equal to the thickness of the single crystal.

Table 2

Molière potential for atomic planes (100) in a KCl single crystal, taking into account thermal vibrations of the ions

The dependence of \(c\) on \(t\) means that \(\Psi_{max}\) will decrease with increasing \(t\), and, consequently, the influence of temperature for thick specimens is more substantial than for thin ones. It can be shown that the dependence of \(\Psi_{max}\) on temperature is given by the expression

\[ \Psi_{max} \sim T^{- \frac{1}{2}} . \tag{28} \]

Thus, an increase in the temperature of the specimen will lead to a decrease in the intensity of the channeled proton beam for a given specimen thickness \(t\).

Energy losses of protons in single crystals of compounds

For heavy charged particles passing through matter, energy is lost in two processes: in inelastic collisions with atomic electrons (electronic stopping) and in elastic collisions with atomic nuclei (nuclear stopping). At high particle energies, electronic stopping predominates, and the particle energy is spent on excitation or ejection of electrons in atoms. In a chaotic system, nuclear stopping predominates over electronic stopping at \(\varepsilon < 10\), where \(\varepsilon\) is a dimensionless quantity characterizing the energy, introduced by Thomas–Fermi.

\[ \varepsilon = \frac{Q_{\mathrm{TF}} \cdot E}{Z_1 Z_2 e^2} \cdot \frac{M_2}{M_1 + M_2}, \tag{29} \]

where

\[ \begin{aligned} M_1 &\text{ is the mass of the incident particle (in our case, the proton),}\\ M_2 &\text{ is the mass of the nucleus of an atom of the medium.} \end{aligned} \]

Since we are interested in the passage of fast protons through matter, where nuclear stopping is insignificant and amounts to only \(\sim 0.1\%\) of

of the total energy losses, we shall dwell on the ionization energy losses of the particles.

In considering energy losses it is convenient to separate the energy losses into close collisions, with a large transfer of momentum to the electrons, and distant ones, with a small transfer of momentum to the electrons. Such a division is due to the fact that in close collisions quantum-mechanical phenomena must be taken into account, whereas in distant collisions they play no role in determining the energy losses. This separation is also convenient because channeled particles move in regions where the electron density is considerably lower than the average and, consequently, the role of close collisions in channels is small, whereas distant collisions are little sensitive to the structure of the substance.

At high energies and small atomic numbers \(Z_1\) of the particles, for calculating the average energy losses the Bethe–Bloch formula is valid, having the form [23]

\[ \frac{dE}{dx}=S_{\mathrm{el}}\cdot N\cdot Z_2 = \frac{4\pi Z_1^2 e^4}{mv^2}\cdot N\cdot B . \tag{30} \]

Here

\[ N \]
is the number of atoms per unit volume,

\[ S_{\mathrm{el}} \]
is the cross section for electronic stopping for an amorphous substance,

\[ B \]
is the stopping coefficient. \(B=Z_2\ln \dfrac{2mv^2}{I}\), \(I\) is the mean ionization potential of an atom with atomic number \(z\).

For complex compounds the quantity \(\overline{I}\) is found from Bragg’s rule [27]:

\[ \ln \overline{I}= \frac{\sum_i N_i Z_i \ln I_i}{\sum_i N_i Z_i}, \tag{31} \]

where \(N_i\) is the number of atoms per unit volume with atomic number \(Z_i\), and \(I_i\) is the mean ionization potential of the atoms of the element.

We have calculated the energy losses of protons with energy 6.72 MeV in all the substances studied, taking into account corrections for the binding effect of \(K\)-electrons (Table 3). Corrections for the density effect of the medium and for the

Table 3

Mass stopping power of compounds

binding effect of \(L\)-electrons in atoms at this proton energy may be neglected [20]. Bethe [28], in deriving formula (31) for the case of an electron gas, established that at high particle velocities the energy losses in electronic collisions asymptotically consist of

equal contributions from distant and close collisions. Recently Lindhard [29] confirmed this rule for the case of atomic systems. On the basis of these remarks, it is convenient to divide the energy losses into distant and close ones [23]:

\[ \frac{dE}{dx}(r)=S_{\rm el}\left[\alpha N\cdot Z_2+(1-\alpha)\rho(\vec r)\right], \tag{32} \]

where \(\rho(\vec r)\) is the electron density at a certain point in space through which the particle passes. The second term in the expression determines the energy losses in close collisions with electrons. The coefficient \(\alpha \lesssim \frac{1}{2}\), and with increasing energy approaches the value \(1/2\). This expression shows that, for fast particles passing along lattice channels, the energy losses should be reduced by a factor of two, since the probability of close collisions for the central beam is insignificant. The losses will almost completely be determined by distant collisions of the particles with the electrons of the medium. Since distant collisions are not sensitive to the exact location of the proton trajectory in the channel, a change in the temperature \(T\) should not noticeably affect the energy losses of protons in single crystals.

One can introduce an effective atomic number \(Z_2^*(E_\perp)\), which determines the number of electrons per atom located at distances greater than \(\rho_{\min}\) from the row, when

\[ \overline{V}(\rho_{\min})=E_\perp . \]

Then the transverse cross section of electronic stopping can be rewritten in the form [23]

\[ S_{\rm el}(E_\perp)=S_{\rm el}\left[(1-\alpha)+\alpha\,\frac{Z_2^*(E_\perp)}{Z_2}\right]. \tag{33} \]

It can be shown that the ratio \(\dfrac{Z_2^*(E_\perp)}{Z_2}\) is given by the expression

\[ \frac{Z_2^*(E_\perp)}{Z_2} = \frac{d\cdot \rho_{\min}\,\overline{V}'(\rho_{\min})}{2Z_1Z_2e^2} \tag{34} \]

and is approximately equal to

\[ \frac{\gamma E_\perp}{E\Psi_{\max}^2}, \]

if the potential varies as \(\rho^{-\gamma}\).

Energy loss is a statistical process, and therefore formula (31) determines only the mean energy loss per unit path. For heavy charged particles, when they pass through sufficiently large thicknesses, the distribution function of the energy losses about the mean value has the form of a Gaussian distribution. Fluctuations of the energy loss per unit path of the particle are due only to close collisions and, consequently, we have the right to replace the mean electron density \(Z_2N\) by \(\rho(\vec r)\).

Then

\[ \left[\frac{d(\Delta E)^2}{dx}\right]_{\rm el} = 4\pi Z_1^2 e^4 \rho(\vec r) \tag{35} \]

or, using the effective atomic number,

\[ \left[\frac{d(\Delta E)^2}{dx}\right]_{\rm el} = 4\pi Z_1^2 e^4 Z_2^*(E_\perp). \tag{36} \]

Thus, the half-width of the anomalous peak should be narrower than that of the normal one.

Let us note that at \(\Psi \simeq \Psi_{\max}\), for physical reasons, energy losses higher than for a normal beam of particles should be observed, owing to blocking on atomic rows or planes. If the angular distribution of channeled particles should have a maximum at \(\Psi = 0\) and then rapidly decrease as the angle increases, then the angular distribution of blocked particles should have a maximum at angles \(\Psi \simeq \Psi_{\max}\). Such behavior of particles was observed in a Si single crystal [4].

Experimental Results

The experiments on the investigation of energy losses and channeling of protons in single crystals of alkali-metal halides were carried out by us on an experimental setup whose scheme is shown in

Fig. 6. Scheme of the experimental setup

Fig. 6. Protons 1 from the cyclotron of the Tomsk Polytechnic Institute enter a scattering chamber, where they impinge on a thin platinum target 2 with a thickness of about \(3 \text{ mg/cm}^2\), installed at the center of the chamber. The proton beam scattered at an angle of \(30^\circ\) from the target was collimated by thin tantalum diaphragms and entered the goniometer chamber, where the single-crystal sample under study 3 was placed. The spot diameter on the sample was \(2\) mm, the energy of the proton beam was \(6.72\) MeV, and the angular divergence of the beam was less than \(\pm 0.1^\circ\).

Thin single-crystal samples of various compounds, with thicknesses from \(15 \text{ mg/cm}^2\) to \(50 \text{ mg/cm}^2\), preliminarily oriented with the \([100]\) axis along the beam direction, were mounted on the rotating device of the goniometer, which had adjustment in the horizontal and vertical planes with angle-setting accuracies of \(\pm 0.05\) and \(\pm 0.1\) degrees, respectively. The energy of the protons after passing through the samples was determined with a scintillation spectrometer 4. The energy resolution of the spectrometer for protons with an energy of \(6.72\) MeV is \(2.8\%\).

The energy distribution of protons that had passed through single-crystal samples of NaCl with a thickness of \(17.3 \text{ mg/cm}^3\) and KBr with a thickness of \(41.2 \text{ mg/cm}^2\) is presented in Figs. 7 and 8. In the right-hand part of the figures the energy spectrum of protons incident on the samples is shown.

The specimens were rotated in the horizontal plane about the \([001]\) axis so that the proton beam remained parallel to the \((010)\) planes. Curves 2, 3, 4, and 5 represent the energy spectra of protons at rotation angles of 0, 0.6, 1.2, and 6 degrees, respectively. All spectra are normalized to a single maximum value. The slight shift of the spectra toward lower energy values and their broadening are due to the increase in the specimen thickness upon its rotation.

Fig. 7. Energy distribution of protons with energy 6.72 MeV after passage through an NaCl single crystal at different angles relative to the \([100]\) direction: \(0^\circ\)—2, \(0.6^\circ\)—3, \(1.2^\circ\)—4, \(6^\circ\)—5.

It is evident from the figures that protons incident on single crystals in a direction sufficiently far from the \([100]\) direction have the energy distribution expected for polycrystalline and amorphous substances. When the \([100]\) direction of the single crystals coincides with the beam direction, the spectrum is distorted and a “tail” appears in the high-energy part of the spectrum, indicating that some fraction of the particles loses less energy than the particles in the normal spectrum. As the rotation angle is increased, the “tail” gradually disappears. The number of particles in the central beam can be determined by simply subtracting the normal spectrum from the normalized one.

Table 4 gives the mean energy losses of protons in NaCl and KBr, calculated by the Bethe–Bloch formula and found experimentally, as well as the value of the anisotropy coefficient, which is the ratio of the number of particles in the anomalous part of the spectrum to the total number of particles in the spectrum.

It may be noted that the greater part of the protons is in the normal beam, and its energy losses agree fairly well with the calculated values. The mean energy losses of the proton beam, found from the energy spectra, prove to be somewhat lower because of the fraction of particles that channel through the crystals. The fraction of particles in the channel-

[[unclear: beginning of word]] beam at zero angle is 0.36 for KBr and 0.28 for NaCl. As the angle of rotation of the crystal is increased, the fraction of channeled particles decreases, and the mean proton losses approach the calculated values more and more.

Fig. 8. Energy distribution of protons with energy 6.72 MeV after passing through a KBr single crystal at various angles relative to the [100] direction: 0°—2, 0.6°—3, 1.2°—4, 6°—5.

Figure labels: vertical axis, “number of protons”; horizontal axis, “proton energy, MeV.”

Figure markers: 1, 2, 3, 4, 5.

Figure tick labels visible: 2000, 1500, 1000, 500; 6.0, 6.72.

In Fig. 9 the dependence of the anisotropy coefficient on the angle of incidence of the protons into the [100] channel is presented for NaCl and KBr. For both crystals—

Table 4

Energy losses of protons in single crystals of compounds

the course of the curves is analogous. The angular half-widths of the curves for NaCl and KBr are equal to ∼0.68° and 0.8°, respectively. According to the theoretical estimates (20), the half-widths of the curves should be ∼0.25°

for KBr and \(\sim 0.02^\circ\) for NaCl. The fact that the “tails” disappear significantly more slowly than the theory predicts confirms the conclusion that, at high proton energies, orientation effects are due mainly to atomic planes and not to atomic rows of single crystals.

Fig. 9. Dependence of the fraction of channeled protons in KBr (1) and NaCl (2) single crystals on the angle of incidence of the protons into the \([100]\) channel.

Below we discuss several factors that may affect the change in the shape of the spectra. Since the energy losses of protons are determined substantially by the distribution of electron density in the space between the \((010)\) planes, or in the \([100]\) direction, we may suppose that changes in the electron-density distribution in the series of compounds from NaCl to NaI and from KCl to KI will also lead to a change in the shape of the energy spectrum. For clarity, let us turn to Fig. 1. We have already noted that the electron-free space in ionic crystals is asymmetric with respect to the \([100]\) axis, which is due to the difference in the atomic numbers of the ions at the lattice sites. The asymmetry of the electron-density distribution increases in the sodium and potassium halide series with increasing atomic numbers of the halides. As a particle moves along the \([100]\) direction, the pattern of the electron-density distribution will rotate through an angle of \(90^\circ\) upon passing from one atomic layer to another, owing to the alternation of ions at the lattice sites. Sta-

it becomes obvious that the entire space enclosed between the atomic rows \([100]\) is conveniently divided into three parts.

A particle whose trajectory can be enclosed in a cylinder with its axis along the \([100]\) direction and with base radius \(\rho_1\) moves in a region where it loses energy predominantly in distant collisions with atomic electrons. The energy losses of such particles amount to approximately one half of the energy losses of a normal particle beam. At larger angles the particle trajectory may fall within the region bounded by a cylinder of radius \(\rho_2\), where, in addition to distant collisions, close collisions with electrons belonging to negatively charged ions with larger atomic numbers also begin to play a role in the energy losses. The energy losses of such particles are somewhat greater than the energy losses of the first group of particles, but less than those of particles in a normal beam. Finally, at angles of incidence into the crystal \(\Psi > \Psi_{\max}\), we are dealing with a normal beam of particles. The relationship among these three regions in the interionic space determines the form of the energy spectrum of the protons. The greater asymmetry in the distribution of electron density in a NaCl single crystal than in a KBr single crystal explains the fact that the anomalous part of the spectrum in the case of KBr is expressed more distinctly than in the case of NaCl.

Conclusion

1. The influence of the crystal structure of alkali-metal halide compounds having a simple cubic structure on the passage of fast protons has been considered.

2. For the compounds NaCl, NaBr, NaI, KCl, KBr, and KI, the potentials of atomic rows and atomic planes have been found. With an increase in the atomic number of the halide in the series, the potentials increase.

3. The limiting angles for channeling of fast protons in ionic crystals along the principal directions \([100]\) and \([110]\) and between the planes \((100)\) and \((110)\) have been found. In the series of compounds, with an increase in the atomic number of the halide, the limiting angle increases. Capture of protons into lattice channels is most effective in compounds with larger atomic numbers.

4. The energy losses of protons with an initial energy of 6.72 MeV have been experimentally investigated in a series of compounds. When the \([100]\) axis of the single crystals coincides with the direction of the proton beam, a fraction of the protons channels through the single crystal, losing half as much energy in comparison with an unchanneled beam.

5. The fraction of channeled protons increases in the series of compounds with an increase in the atomic number of the halide, which confirms the theoretical estimates.

6. A conclusion has been drawn concerning the determining influence of the atomic planes of crystals on the channeling of fast protons.

7. Channeling in the rows of sodium and potassium halides is most pronounced in NaCl and KCl single crystals and decreases in the series of compounds as the atomic number of the halide increases, with a change in the ratio between the ionic radii in the compound.

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