Planar channelling and diffraction of relativistic electrons in thin crystals

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Planar channelling and diffraction of relativistic electrons in thin crystals

S.B. Nurmagambetov, S.A. Vorobiev

To cite this version:

S.B. Nurmagambetov, S.A. Vorobiev. Planar channelling and diffraction of relativistic electrons in thin crystals. Journal de Physique, 1986, 47 (7), pp.1227-1232. �10.1051/jphys:019860047070122700�.

�jpa-00210310�

Planar channelling ^and diffraction of relativistic electrons in thin crystals

S. B. Nurmagambetov and S. A. Vorobiev

Nuclear Physics Institute, ^Tomsk Polytechnical Institute, 634050 Tomsk, ^U.S.S.R.

(Reçu le 12 dgcembre 1984, révisé le 4 février ^1986, accepti ^{le 21} fgvrier ¹⁹⁸⁶⁾

Résumé.

On a étudié l’interaction des électrons relativistes avec les cristaux minces dans l’approximation ^du potentiel continu des plans cristallographiques. ^{Nous avons formulé une méthode} numérique de résolution de

l’équation ^de Schrödinger ^{à une} dimension pour ^un potentiel périodique. ^Les formes des distributions angulaires

pour les électrons, trouvées dans les différentes bandes du mouvement transversal, ^sont déterminées sur la base de calculs numériques. ^Les résultats des calculs ont été appliqués ^à l’analyse des données obtenues expérimentalement

en mesurant les distributions angulaires des électrons de 5,1 ^{MeV lancés aux} petits angles d’incidence sur les

plans cristallographiques (110) de Si. Nos investigations expérimentales ^et théoriques ^ont montré d’une part ^la possibilité ^de peuplement préférentiel ^{des états ftxés du mouvement} transversal et, ^d’autre part, ^ont permis ^de séparer les effets de canalisation des effets de diffraction d’électrons.

Abstract

The interaction of relativistic electrons with thin crystals ^{has been} investigated ^{in the} approximation of a continuum potential ^{of a} system ^of crystallographic planes. ^A numerical method of solution of the one-dimen- sional Schrödinger equation has been worked out The shapes ^of angular distribution of the electrons which occupy the different bands of transverse motion have been determined on the basis of numerical calculations. The calculated results have been used to analyse ^the experimental ^{data obtained} by measuring ^the angular distributions of 5.1 MeV electrons incident along ^the (110) planes ^{of Si} crystal. ^{On the one} hand, experimental ^and theoretical investigations

indicate a possibility ^of preferential population ^{of the} transverse motion states, and, ^{on the other} hand, ^{make it} possible ^to separate ^the channelling effects from those of electron diffraction.

Classification Physics ^Abstracts

07.85 - 29.90

1. Introductiom

Transmission of the relativistic charged particles along

the crystallographic planes ^is governed by ^{the corre-}

lated collisions with the atoms of the crystal lattice,

so that it can be described as a motion in an effective

averaged potential ^{of the} crystallographic planes [1].

The results, ^{obtained in this} approximation, ^{made a} great contribution to the physics ^{of the fast} particle

interactions with the crystals, ^{which is based on the}

well-known theory ^of the electron wave diffraction in

crystalline ^matter [2].

Using channelling phenomenon ^{one can} speak ^about

some prospects ^of designing, ^on the basis of a crystal target, ^new devices for producing ^the electromagnetic

radiation in ^a wide region ^{of spectrum} and also for the electron beam manipulation. Moreover, the channel-

ling ^effect gives ^a possibility ^to prove experimentally

the Doppler ^anomalous effect, predicted by

I. M. Frank [3]. ^Other practical aspects ^are pos- sible [4], but in all cases the most important question

is in what quantum ^{states of the} transverse motion an

electron will be localized and what is the dependence

of probability of the electron _occupancy of some states on the incidence angle. Further, instead of the proba- bility of the electron localization we will use the term of probability ^{of the} quantum ^state population. ^The

most suitable and direct experimental procedure allowing ^to determine the population ^{of some state}

of a channelled electron is the measurement of the

angular distribution of the electrons transmitted

through ^a crystal target, ^since every quantum ^{state of}

transverse motion has an extremely characteristic

momentum distribution.

In this _paper the detailed experimental ^{and theo-}

retical investigation ^{of the} orientational dependence

of angular distribution of MeV _energy electrons transmitted through ^{Si thin} crystal ^{has been} per- formed. An analogous investigation has been made

earlier 10], ^{but we} concentrate our attention to the

properties ^{of the} sub-barrier (bound) ^{state of chan-} nelling ^electrons.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047070122700

1228

2. Experimental technique and the results of measu- rement.

Investigations ^of angular distributions of the electrons transmitted through ^the crystal target ^{have been} carried out with the use of equipment (Fig. 1), ^created

on the basis of microtron. A pulse magnitude ^of

electron beam extracted from the microtron was

about 25 mA with the _energy spread ^{of 0.3} % ^and angular divergence about 0.1 °. The electron beam extracted from the microtron [1], ⁱⁿ passing through ^a

deflection magnet [2], system of collimators [3] ^was

formed _up to the angular divergence of 0.02° and fell

on a crystal ^{fixed in a vacuum chamber} [4] ^{in a triaxial} goniometer [5], ^which provides ^the accuracy of the orientation angles establishing of 0.01 °. A preliminary crystal orientation was carried out with the accuracy of about o.32° with the help of the laser beam reflection

by ^a crystal ^surface.

The crystal targets ^{have a thickness from 1 to} 2 gm and have been made from a dislocationless silicon with an impurity concentration less than 1015 cm- 3 by the method of dynamic etching ^{in the} polishing

solution. A recording of the electrons transmitted

through ^a crystal target has been done with a photo- technique ^film FT-lOlp placed ^{in a holder} [6] ^{at a}

distance of 1 250 mm behind the crystal. ^The efficiency

of the electron beam recording ^was (80 - 90) % ^at

rather low sensitivity ^{to a} background X-ray ^radia-

tion. The obtained electronogramms have been treated at a microphotometer ^{MF-4. The} exposure times were

of some minutes.

Figure ² represents ^the photos of the electrons

angular distributions and curves of their intensities,

obtained by photometric measurement of electro- nogramms for different angles ^{0 of a} beam incidence relative to ( 110) ^Si crystal planes. ^{In the case 0 = 0°} (a) ^a three-peak ^{structure of} angular distribution with the

intensity ^{maxima at S =} 0° and 0.1 ° is observed. At 0 ⁼ 0.05°(b) ^{there are four} intensity ^{maxima at the} angles ^{different from the} corresponding positions

obtained in the previous ^{case. For the} crystal ^orien-

tation 0 ⁼ 0.12°(c) ^a picture of three maxima in the

angular distribution similar to orientation 0 ⁼ 0° is observed again, but it is somewhat asymmetric. ^The

case 0 ⁼ 0.18°(d) demonstrates a four-peak ^{curve of} intensity but with the maxima in the positions ^different

from 0 ⁼ 0.05° orientation. The electron angular

distribution is also asymmetric ^{relative to a} (110) plane direction. At least, ^{at 0 =} 0.29°(a) ^{an electron}

flux in the directions close to 8 ⁼ 0° is not observed.

In this case the angular distribution is _very asymmetric

and similar to that under the random irradiation of

crystal.

At the planar channelling of electrons a set of the bound states (bands) ^{with the} crystallographic planes

occurs in the transverse motion of electrons [1]. ^{It is}

reasonable to suggest ^{that the observed effect of}

« change-over» ^{of the} angular distributions of a

scattered beam is determined by changing the charac-

Fig. ^1.

^{Scheme of} experimental equipment : ^{1- electron} accelerator; ²

deflection magnet; ³

collimators sys- tem ; ⁴

scattering chamber; ⁵

goniometer device with

a target; ⁶

recording photoplate.

Fig. ^2.

Photos of the angular distribations ^of scattered electrons and the results of photometric measurements (8 ^is

the incidence angle of electron beam at the crystallographic plane).

ter of fast electrons interactions with a crystal. ^{In this}

case a preferential population ^of the individual _quan- tum states of a quasidiscrete spectrum ^{of the bands of}

transverse motion for the channelled particules ^occurs.

Under channelling of heavy particles ^or high relativistic electrons such ^« change-over >> effect would not be observed due to a great density of the levels in a

potential well, ^{in other} words, ^a quasiclassical ^charac-

ter of the transverse motion.

3. Discussion of the experimental ^results.

A theoretical consideration of the present experiments

was based on the use of a continuum approximation

for the crystal potential [1]. ^To describe the transverse motion of a particle in the continuum potential ^the equation has been resolved

where Elcl is the relativistic mass of the electron,

’Pjk(X) is the Bloch _wave, corresponding to j-band ^and

k is a magnitude ^{of a wave vector.} The method of

resolving ^the equation (1), which has been used by us, is presented ^{in the} paper [5]. ^{An effective} potential ^has

been defined in the Moli6re approximation taking ^into

account two neighbour ^atomic planes [6]. ^{The cal-}

culated results of transverse motion for a kinetic _energy of electrons Ek ^{= 5.1 MeV are shown in} figure ^{3a, b.}

Figure ^{3a shows a} periodical chain of the potential

CtLp(x) ^{for (110) Si planes and the density of particle}

distribution in the planar channel I ’Fjo(x) 11 for the

initial three bands of the transverse motion. Figure ^3b

shows a band spectrum ^of energies ^{of the transverse}

motion of electron. Only ^two sub-barrier bands are seen to be realized at the given ^kinetic energy of electrons. A transverse energy of the particle ^{for these}

bands almost does not depend ^{on the wave vector.}

The Bloch _waves, corresponding ^to these bands also

weakly depend ^{on the wave vector and describe the}

states of electrons, ^{localized at} any place ^{in channel}

Fig. ^3.

Calculation results for transverse motion of electrons with the total energy E ⁼ 5.61 MeV in a (110) planar ^{channel of Si} crystal. a) ^{Periodic chain of} potential

’11p(x) ^and the electron distribution in channel for first three bands (k ⁼ 0). b) ^Band spectrum ^{of electron trans-}

verse motion. Sign ( + ) ^and ( - ) ^{show the} parity of the Bloch

waves at the edges ^{of the} energetic ^bands.

(Fig. 3a). Thus, the sub-barrier bands describe the states of the channelled electron bound in channel. As for the above-barrier bands, ^{here the} quantum ^states describe the channelled electron scattering by ^a system ^{of the} crystallographic planes. ^{In this case the}

distribution of the particle density ^{in a} cross-section of

planar channel is almost uniform, except ^{for a centre} and the edges ^{of a} Brillouin band (k ^{= 0 and k =}

± nl2 d), where the Bloch wave has a certain _sym- metry ^{at the interval} of periodicity ^{relative to an}

inversion of a system of coordinates. Odd and ^even

parity of the Bloch wave on the boundaries of the

energetic bands is shown in figure ^3b by ( - ) ^{and ( + ),} respectively. ^{Let us} note, ^{that at} the boundaries of the Brillouin band k ⁼ ± n/2 d, ^{where the} Bragg ^reflection

of the electrons from the crystallographic planes ^takes place, ^the parity of the Bloch waves for the adjacent energetic ^{band is} different. This corresponds ^{to S-}

waves and P-waves, ^{formed at} the electron diffraction.

Their density ^{maxima are on the} crystallographic planes (for S-waves) ^or between them (for P-waves) [2].

Also, ^{let us note} that, ^{as a} rule, ^the parity ^{on the bot-}

tom and the top ^{of a} sub-barrier bands is different,

but it is identical for ^an above-barrier one. The devia- tion from this regularity ^takes place when the above- barrier band crosses the top ^{of the} potential ^barrier

due to its shift with the increase of the _energy of incident electrons [5]. ^{As a} result, ^the parity ^{of the near-}

barrier bands changes similarly ^to the sub-barrier ^ones while transmitting ^from the band bottom to the top.

Then, ^for higher-lying ^{bands a} regular variation of the

parity alternation restores. For the high-lying ^above-

barrier bands the failures in the parity alternation connected with vanishing ^the coefficients of transmis- sion or reflection for a periodic potential ^{chain may}

occur [7].

We notice, ^{that the} phenomena connected with the failure of the regular parity alternation can really

manifest themselves in experiment, that will be demonstrated later. Being interested in the angular

distributions of channelled electrons and the popula-

tions of the bound states with respect ^{to the beam} incidence angle ^{at the} crystal target, ^we have calculated the coefficients of expansion ^{in Fourier} series of a

periodic part of the Bloch wave

where j ^{is a} band number J ⁼ 1, 2, 3...). Then, ^initial population ^{of some} quantum ^state in j ^{band at the}

incidence of a monoenergetic ^{beam at the} angle ⁰

on a system ^of crystallographic planes has the form

The angular distribution of electrons determined by.

some state in j band, ^{will be}

1230

Let Fjk(O) ^{denote a resulting} population ^{of k state in}

band with the incidence of an initial beam at angle ^{0 to}

the crystallographic planes, ^then neglecting ^{the inter-}

ferential effects the total angular distribution has a

form

In our case the surfaces of the crystal targets ^{were not} perfectly ^smooth (the ^{thickness variation was AL -}

0.7 J.1m). ^{Then in} (5) interference effect connected with

a crystal ^{thickness was} neglected.

The calculated populations Pik(o) for the different bands are represented ⁱⁿ figure ^{4. It is seen,} that for the sub-barrier bands the populations ^differ sufficiently

from zero in a wide interval of the incidence angle ^of

an initial beam, resulting ^{from a} strong localization of particles ^for the sub-barrier states. The populations

of the above-barrier bands are represented ^well by

the step-like (functions, which differ from zero for an

interval of angles U - 1) OB 0 JOB’ where j ^is

a band number and OB ^{is an} Bragg angle. ^For ( 110) ^Si planes and electrons with the kinetic _energy Ek ⁼

5.1 MeV 0B ⁼ 0.0370. It _means, that the Bloch _waves,

corresponding ^{to the} above-barrier bands are the

plane ^{waves. We} note, ^that population into these bands ^occurs at the incidence angles ⁰ > 03B8L, ^where 0L

is a critical Lindhard’s angle (in ^{our case} OL ⁼ 0.0870).

In this case one says, that transition from ^a channelling

to quasi-channelling motion takes place [8].

To our mind the quasi-channelling ^states represent the usual diffraction of electrons by ^a system of crystal- lographic planes. ^{So, there is no} evidence for introduc-

ing ^{a new} type ^{of motion} calling ^{it «} quasi-channel- ling ^». Something analogous ^takes place, when instead of the coherent bremsstrahlung ^one says about the

quasi-channelling ^radiation (see [9]). Specificity ^of

above-barrier states of the channelled particle ^consists

in application ^of approximation ^{of a continuum} potential, ^{that makes it} possible ^{to obtain a more}

accurate results, ⁱⁿ comparison ^{with the two-wave} approximation ^{in the} theory of electron diffraction [2].

Fig. ^4.

Orientational dependence ^{of the states} population

in the bands at the plane ^{wave incidence at the} crystal (Oø ⁼ 0.037°).

In their turn, the channelling ^{states differ} basically

from the above-barrier _ones, taking ^{into account both}

the band spectrum ^{and the wave} functions and _popu- lations.

As it was shown, ^the angular distribution of the channelled electrons behind the crystal ^{is a linear}

combination of the angular distributions of each quantum ^state in energy bands Wjk(S), ^{the weight}

of every individual distribution Wjk(S) being resulting population ^{of the states} Pjk(O). ^{It is evident, that the}

values Fjk(o) ^may be defined by resolving ^{the corres-} ponding equations, describing ^the redistribution of

particles ^{in different states in} planar ^{channel due to}

different _processes [11]. However, ^{if we} choose rather thin crystal target ^{we can} neglect ^{these processes and}

consider j5jk(O) ⁼ Pik(o) ^to be initial population.

The calculated angular distributions of electrons with respect ^{to the} angle ^{of incidence} according ^{to the} equation (5) ^are represented ⁱⁿ figure ^{5. In} calculation

one considers that the angular divergence ^{of the}

incident beam is A0 ⁼ 0.25 OB. ^{As it is seen from} figure ⁵ for the incidence angles ^{0 3} 03B8B - OL a picture ^of angular distribution has a complicated

form : with the changing ^{of the} angle ^{0 from 0 to 3} OB ^a

number of maxima in angular distribution is varied from 3 to 6. If we take into account only ^the explicit maxima, i.e., ^{with a} sufficiently high intensity, ^{a scheme}

of the alternation of these maxima is the following :

three maxima (0 03B8 03B8B, four maxima (8B 8

2 OB), three maxima (2 OB 1 ^{0 3} OB)

^{and four}

maxima (3 OB 1 6 ^3.5 0p). With the further change

of the incidence angle the variation of the picture ^of

Fig. ^5.

Angular distributions of electrons in dependence

on the observation angle S. Arrow shows the incidence

angle ^0.

angular distribution showed another regularity :

at the beginning ^{there is one} explicit maximum, ^then

at 0 ⁼ 4 OB ^{there are two} symmetrically placed ^maxi-

ma, then again only ^one maximum and at 0 ⁼ 5 BB

two symmetric explicit ^{maxima. It is seen} that for the

angles ^{of incidence} larger ^{than 3} OB ^the angular

distribution is explained easily ^on the basis of the

Bragg theory. ^That is, ^{with the} angles of incidence

being multiple ^of Bragg angle, the reflection from the

planes occurs, that stipulates ^the existence of two

symmetrically placed maxima in the angular ^dis- tribution, ^formed by the transmitted and reflected beams. At other values of the angles ^{0 we have} only ^the

transmitted beam. At the angles smaller than 3 OB ^the

observed picture ^{is more} complicated : 1) ^{the number}

of the transmitted and reflected beams is larger ^than

two and varies with the change of the incidence angle;

2) reflection takes place ^{at the} angles ^{which are not} multiple ^{of the} Bragg angle, the reflection angle being

not equal ^to the incidence angle. This is connected with the existence of the sub-barrier states, ^which

population ^with respect ^to the incidence angle ^differs

from zero in a wide interval of these angles (see Fig. 4)

in contrast to the above-barrier states. In the latter

case the states in one or two bands are populated ^at

the multiplicity of the incidence angle ^{to the} Bragg angle.

Thus, study ^{of the} orientational dependence ^allows

to separate ^the region, in which the channelling ^effects (the existence of the sub-barrier states) ^{are of} impor-

tance, from the region of electron diffraction. The

angular interval, ^where channelling ^{effects are of} importance, ^is connected with a number of the sub- barrier bands. In our case it is equal ^to (2 ⁺ 1) OB,

where we have added the first above-barrier band,

which is like to the sub-barrier ones by their charac- teristics.

So, ^the present experimental ^{data on the} angular

distributions of the channelled particles ^can give ^some

information on the existence of the sub-barrier and near-barrier bands and their number. Estimation of the latter value will be the multiplicity ^{of a maximum} angle of incidence to the Bragg angle, ^{at which we}

obtain a difference from simple Bragg reflection.

The number of the sub-barrier bands depends ^{on the}

energy of the electrons incident at the crystal, ^{due to the} change ^{of an} electron relativistic mass [12]. ^{As men-}

tioned above, ^the parity of the Bloch ^{waves at} the

edges ^{of the} energetic bands is different for the sub- barrier and above-barrier bands. For example, ^for

the electron _energy 100 keV, 0.85 MeV and 1.30 MeV the parity ^{of the} edges ^{of the} above-barrier ^{bands is} identical and the sub-barrier one is different. It can be

seen from the shapes ^{of the} angular distributions of particles ^for every band (Fig. 6). At Ek ^{= 100 keV}

the angular distributions have almost the step-like form, because of the Bloch wave proximity ^{to the} plane

wave, and are located in angular interval ± E(i - 1) OB, JOBI, where j ^{is a} band number. Because of the ^even

Fig. ^6.

Variation of the angular distributions of electrons

Wjk(S) ⁱⁿ dependence upon the electron kinetic energy for the first three bands of transverse motion ( j ^{is a band} number).

parity of the Bloch wave at the top of the second band the probability of electron _emergence from the second band at zero incidence angle ^is different from zero.

With the increase of the electron _energy the broa-

dening ^{of the} angular distribution from the first band takes place ^{due to the} increasing localization of the particles ^{near the} crystallographic planes, ^and

in the angular distribution from the second band the growth of the central maximum and rise of its relative height ^takes place. With the further increase of electron _energy the second band crosses the tops of potential barriers and alternation of the parity

will take place. ^{As a} result, the central maximum in the angular distribution from the second band changes by the central maximum from the third band. This

parity alternation leads to the « glimmer >> ^{effect of}

the central maximum of the angular distributions, obtained with the rather thick single crystals, ^where only the near-barrier band forms the observed picture

of the angular distribution. Similar experimental

results have been obtained and discussed on the basis of the Kronig-Penny ^model [ 13].

4. Conclusion.

The performed experimental investigations ^{and theo-}

retical analysis ^{of the} orientational dependences ^{of the} angular distributions of the electrons, transmitted

through ^a crystal along ^the crystallographic planes,

show an influence of several bound states at small

angles of orientation 0 0L and almost one state at 0 > 03B8L. ^The comparison ^of experimental ^electrono-

gramms with the theoretical angular distributions shows a reasonable agreement. The results demonstrate the effect of a preferential population ^{of the} definite

bands of transverse energy for the channelled electrons at some orientation angles. ^Our experimental ^results

show that at small angles ^{of a} crystal orientation e OL ^{to form an} angular distribution ^{two near-} barrier bands are essential. If at 0 ⁼ 00 the shape ^of angular distribution is determined mainly by ^the

first above-barrier band, ^{then at 0 =} 0.0450 contri-

1232

bution of the second band, being ^{a last} sub-barrier

one, is dominant. The phenomenon ^{of a} preferential population ^{of quantum states} of channelled electrons may be used to manipulate the electron beam by ^the crystal, because of different localization of electrons

in channel for different states. By ^a crystal tilting ^we

can direct the electron beam near or far from the

crystal lattice atom, that defines the preferential yield

of the corresponding processes producing by ^the

electron beam in a crystalline ^materials.

References

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Medd., ³⁴ (1965) ^N14.

[2] HIRSCH, ^P. B., HOWIE, A., NICHOLSON, ^R. B., PASHLEY,

D. W., WHELAN, ^M. J., ^Electron Microscopy of

Thin Crystals (London, Butterworths) ^1965.

[3] FRANK, ^I. M., Izv. Akad. Nauk SSSR Fiz. 6 (1942) ^3.

[4] BELOSHITSKY, ^V. V., KOMAROV, ^F. F., Phys. Rep. ⁹³ (1982) 117.

[5] NURMAGAMBETOV, ^S. B., Preprint ^{VTNITI N2688-83} DEP, ^1983.

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[11] BELOSHITSKY, ^V. V., KUMAKHOV, ^M. A., ^Zh. Eksper.

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