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Effect of correlations of lattice thermal vibrations on particle dechanneling rate
Volume 88A, number 4
PHYSICS LETTERS
8 March 1982
EFFECT OF CORRELATIONS OF LATTICE THERMAL VIBRATIONS ON PARTICLE DECHANNELING RATE ~ E.I. SIROTININ, A.F. TULINOV, V.A. KHODYREV and V.A. KHRUSHCHEV Institute of Nuclear Physics, Moscow State University, Moscow, USSR Received 21 July 1981
Revised manuscript received 4 January 1982
The dechanneling mechanism due to thermal vibrations of crystal atoms has been investigated. The fluctuations of the particle position in the transverse plane, due to thermal atomic displacements, and correlations of the lattice thermal vibrations are shown to be necessary for explaining the experimentally observable dechanneling rate.
Dechanneling of charged particles in crystals is currently a topic of much theoretical and experimental investigation. Theoretical investigations of dechanneling and an interpretation of the experimental data are usually done in the framework of the model of a steady increase in transverse energy [11, the diffusive approach [1,2] or using the Fokker—Planck equation [3,4] . Despite the profusion of the works dedicated to this problem, a theoretical description of the experimentally observed regularities of dechanneling is rather difficult (see, for example, refs. [3,4]). At the same time, it is felt that the mechanisms determining the dechanneling rate of particles are not sufficiently well explored. In particular it has been shown [5] that correlations of the lattice thermal vibrations can affect markedly the dechanneling rate of particles in crystal lattices with a complex basis. The present paper reports preliminary results of the experimental investigation and calculations, aiming at elucidating the effect of correlations of the lattice thermal vibrations on the dechanneling rate in the main axial channels of a W crystal. The experimental results are analyzed in terms of the model of a steady increase in transverse energy which is good approximation at small depths where the integral dechanneling
function x(x) ~ 0.2 (see, for example, ref. [3] p. 16). The energy spectra of channeled proton backscattering for the four most open axial channels of a W crystal were measured with an electrostatic accelerator at an energy of the proton beam E
0 = 1.0 MeV. The measured backscattering spectra were used to determine the integral dechanneling function by the method analogous to the one employed in ref. [61. Fig. 1 presents the results obtained in the variables~’ = x!(d/d0) and ~ = x(d/d0) where d is the interatomic spacing along the strings and d0 is the lattice constant. It is easy to show [5] from the similarity considerations that the dependence ~‘(~) must be the same for all channels if correlations of the lattice thermal vibrations are neglected. It can be seen from the figure that
0,05
— —
2
~l[Mv~
Fig. 1. Experimental values of~(~) for the axial channels ~ This paper was first presented at the 11th Soviet National Conference on Physics of charged particle interaction with single crystals, Moscow, May 25—27, 1981.
0 031-9163/82/0000—0000/s 02.75 © 1982 North-Holland
(111) (—), (———), <110> (—.—), and <113>(—I—) in a W crystal. The shaded areas indicate the position of the curves
shown in fig. 3.
199
Volume 88A, number 4
PHYSICS LETTERS
the curves ~(~) corresponding to the (111), (100),
ofp
(110), (113) channels approach each other. In order to explain these results we have attempted to consider the dechanneling mechanism associated with the lattice thermal displacements making as full allowance as possible for correlations of the lattice thermal vibrations. When determining the change in transverse energy of a particle following its collision with an atom of the string, it was taken into account that the thermal atomic displacements did not only lead to the change in particle transverse momentum !3p1 [1] but also to the displacement of a particle trajectory in the transverse plane ~r1 2/2M bE1 = (p1 + bp1) 1 —p~/2M1 +
‘
[aU(r )/&r ]br I
I
i
experimental data, The effect of correlations of the lattice thermal vibrations on bE1 can be taken into account if the value
/
-~
~
10 ,
/
0,5
0,5
/
2
~
Fig. 2. The dependence2); of ~d(c)/d~on = 2irZ the 2xd/E. reduced 1 Contribution transverse energythermal from e = E1d/(Z1Z2e vibrations of lattice. 1Z2e 2 Contribution from the multiple scattering by electrons. 3 The fraction introduced by the last term of eq. (1) to the total rate of variations in <~> (the scale for this curve is shown to the left of the figure).
200
1 in eq. (1) is taken as nearly equal to p10 ÷~.1öp11 where p1~is the value ofp1 obtained with the neglect of the displacements of! atoms which pre-
ceded the collision in question. If the number of penods where the correlations are important, 1, is assumed to be ! ~ N where N is the total number of collisions in a single scattering by the string, we obtain, after simple transformations 2
(~p1)/ (öE1) =
1 + 0<
\ ~lU(r1) ii ~ i K~)+ (~r1)~ (2) --~--—
where the angular brackets stand for the averaging over thermal atomic displacements, K~= p1.
-‘
where U(r1) is the continuum transverse potential, M1 is the particle mass and p1 is the transverse compo. nent of the momentum of a particle following the scattering by the undisplaced atom. The last term in eq. (1), just as the sum of the first two terms, is inversely proportional to the particle energy. Its relative value is in. dicated in fig. 2. As seen from the fig. 2 the last term should be included in a quantitative description of the
/ / / / /
8 March 1982
The averaging was done numerically. The result of
the averaging proved to be well described by the sum of two terms of the expansion in powers of u1(cr u~ and ut). This approximation was, therefore, used to calculate x(x). In calculating K1 at small E1, we use only the first order terms in the expansion of op1 in powers of u1. Then K~ k~where k~= (u1u1~)/(u~) are the correlation coefficients of the lattice thermal vibrations. This equality was assumed to hold for all values of E1. The values of k~were determined in the Debye approximation [71.In this case, the value of the coefficient in parenthesis in eq. (2) proved to be nearly 2.5 for the most open channel and 1.3 for the least open channel considered. This implies that the thermal vibration correlations lead to a considerable increase of the dechanneling rate. It will be noted that the replacement of K~by k~ can be illegitimate at small r1 where the expansion in powers of u1 is not applicable. However, it is clear that the result of the calculation of x(x) is weakly depen. dent of d(E1)/dx at E1 close to the critical transverse energy E±~ because in this case E1 undergoes changes at distances which are small compared to the experimentally explored depth intervals. The contribution of multiple scattering by electrons to d(E1)/clx was calculated in the electron local density approximation using the expression for a uniform gas of free electrons [8]
plotted The results in fig. 3. of Differences calculationsinofthe thevalues function of the x(x)coefare Ek~]in various channels which lead to a mismatch of the curves ~(x), are compensated, to a great extent, by different contributions of multiple ficients [1 +
Volume 88A, number 4
PHYSICS LETTERS
-
8 March 1982
refs. [3,12]). The approach is employed to consider the correlations of thermal atomic vibrations and the displacement of a particle trajectory due to the lattice
thermal vibrations. The obtained results indicate that correlations of the lattice thermal vibrations play a sig-
a~5
nificant role in the process of particle dechanneling. The Debye approximation used to estimate the correlation coefficients does not enable us to speak of a
quantitative agreement with experiment. Nevertheless, 1
2
~1MgM]
Fig. 3. The results of the calculationsof the dependence ~(x) in W made including and disregarding the correlations of thermal vibrations of the lattice (the upper and lower sets of curves respectively). The notation is the same as in fig. 1.
the results are important in studying the dechanneling mechanisms. Moreover, the present results permit to state that the experimental data on dechanneling can be used to extract new information on the character of thermal atomic vibrations in a crystal.
References scattering by electrons. For a comparison with the experimental data, fig. 1 includes the data of fig. 3 (shading). As is seen, taking account of the correlations improves the agreement with experiment. Deviations of the calculations from the experimental data at small depth [in particular, the values of~(0)] can be due to inapplicability of the model of statistical equilibrium at these depths. The value of~(0)calculated using the results of Barrett [9] ‘is the same as the measured one. It will be noted that there are methods for studying multiple scattering of channeled particles that enable us, in principle, to take a strict account of inelastic scattering with the~excitationof degrees of freedom of the crystal (see, for example, refs. [10,111). In particular, this approach is used [111 to obtain the expression for (bE 1) which is a good approximation at largeE1. Up to now, the possibilities of the approach have not yet been realized to full advantage. In the present paper, the uncertainties in Lindhard’s approach [1] at high E1 are eliminated by using the resuits of the numerical averaging of (bp~>over the distribution in thermal atomic displacements (see, also,
[11 J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (1965) No. 14.
[21 N. Matsunami and L.M. Howe, Radiat. Eff. 51(1980) [3] M.A. Kumakhov and G. Shirmer, Atomic collisions in crystals (in Russian) (Atomizdat, Moscow, 1980). [4] T. Oshiyama and M. Mannaini, Phys. Lett. 81A (1981) [5] Sh.Z. Izmallov, E.l. Sirotinin, A.F. Tulinov and V.A. Khodyrev, Fiz. Tverd. Tela 23 (1981) 1933. [6] V.0. Roslyakov, A.S. Rudnev, E.I. Sirotinin,
A.F. Tulinov and V.A. Khodyrev, Phys. Stat. Sol. (a)43 (1977) 59. [71 K.J. Glauber, Phys. Rev. 98 (1955) 1692. [81 J. Lindhard, K. Dan. Vidensk. Seisk. Mat. Fys. Medd. 28 (1954) No. 8. [9] J.H. Barrett, Phys. Rev. B3 (1971) 1527. [101 Yu. Kagan and Yu.V. Kononetz, Zh. Eksp. Teor. Fiz. 64 (1973) 1042, [11] M. Kitagawa and Y.H. Ohtsuki, Phys. Rev. B8 (1973) 3117. [121 H.E. Shi~ltt,EAtomic . Bonderup, J.U. Andersen andS. Datz, H. Esbensen, collisions in solids, ed. B,A. Appleton and C.D. Moak (Plenum, New York, 1975) p. 843.
201
PHYSICS LETTERS
8 March 1982
EFFECT OF CORRELATIONS OF LATTICE THERMAL VIBRATIONS ON PARTICLE DECHANNELING RATE ~ E.I. SIROTININ, A.F. TULINOV, V.A. KHODYREV and V.A. KHRUSHCHEV Institute of Nuclear Physics, Moscow State University, Moscow, USSR Received 21 July 1981
Revised manuscript received 4 January 1982
The dechanneling mechanism due to thermal vibrations of crystal atoms has been investigated. The fluctuations of the particle position in the transverse plane, due to thermal atomic displacements, and correlations of the lattice thermal vibrations are shown to be necessary for explaining the experimentally observable dechanneling rate.
Dechanneling of charged particles in crystals is currently a topic of much theoretical and experimental investigation. Theoretical investigations of dechanneling and an interpretation of the experimental data are usually done in the framework of the model of a steady increase in transverse energy [11, the diffusive approach [1,2] or using the Fokker—Planck equation [3,4] . Despite the profusion of the works dedicated to this problem, a theoretical description of the experimentally observed regularities of dechanneling is rather difficult (see, for example, refs. [3,4]). At the same time, it is felt that the mechanisms determining the dechanneling rate of particles are not sufficiently well explored. In particular it has been shown [5] that correlations of the lattice thermal vibrations can affect markedly the dechanneling rate of particles in crystal lattices with a complex basis. The present paper reports preliminary results of the experimental investigation and calculations, aiming at elucidating the effect of correlations of the lattice thermal vibrations on the dechanneling rate in the main axial channels of a W crystal. The experimental results are analyzed in terms of the model of a steady increase in transverse energy which is good approximation at small depths where the integral dechanneling
function x(x) ~ 0.2 (see, for example, ref. [3] p. 16). The energy spectra of channeled proton backscattering for the four most open axial channels of a W crystal were measured with an electrostatic accelerator at an energy of the proton beam E
0 = 1.0 MeV. The measured backscattering spectra were used to determine the integral dechanneling function by the method analogous to the one employed in ref. [61. Fig. 1 presents the results obtained in the variables~’ = x!(d/d0) and ~ = x(d/d0) where d is the interatomic spacing along the strings and d0 is the lattice constant. It is easy to show [5] from the similarity considerations that the dependence ~‘(~) must be the same for all channels if correlations of the lattice thermal vibrations are neglected. It can be seen from the figure that
0,05
— —
2
~l[Mv~
Fig. 1. Experimental values of~(~) for the axial channels ~ This paper was first presented at the 11th Soviet National Conference on Physics of charged particle interaction with single crystals, Moscow, May 25—27, 1981.
0 031-9163/82/0000—0000/s 02.75 © 1982 North-Holland
(111) (—),
shown in fig. 3.
199
Volume 88A, number 4
PHYSICS LETTERS
the curves ~(~) corresponding to the (111), (100),
ofp
(110), (113) channels approach each other. In order to explain these results we have attempted to consider the dechanneling mechanism associated with the lattice thermal displacements making as full allowance as possible for correlations of the lattice thermal vibrations. When determining the change in transverse energy of a particle following its collision with an atom of the string, it was taken into account that the thermal atomic displacements did not only lead to the change in particle transverse momentum !3p1 [1] but also to the displacement of a particle trajectory in the transverse plane ~r1 2/2M bE1 = (p1 + bp1) 1 —p~/2M1 +
‘
[aU(r )/&r ]br I
I
i
experimental data, The effect of correlations of the lattice thermal vibrations on bE1 can be taken into account if the value
/
-~
~
10 ,
/
0,5
0,5
/
2
~
Fig. 2. The dependence2); of ~d(c)/d~on = 2irZ the 2xd/E. reduced 1 Contribution transverse energythermal from e = E1d/(Z1Z2e vibrations of lattice. 1Z2e 2 Contribution from the multiple scattering by electrons. 3 The fraction introduced by the last term of eq. (1) to the total rate of variations in <~> (the scale for this curve is shown to the left of the figure).
200
1 in eq. (1) is taken as nearly equal to p10 ÷~.1öp11 where p1~is the value ofp1 obtained with the neglect of the displacements of! atoms which pre-
ceded the collision in question. If the number of penods where the correlations are important, 1, is assumed to be ! ~ N where N is the total number of collisions in a single scattering by the string, we obtain, after simple transformations 2
(~p1)/ (öE1) =
1 + 0<
\ ~lU(r1) ii ~ i K~)+ (~r1)~ (2) --~--—
where the angular brackets stand for the averaging over thermal atomic displacements, K~= p1.
-‘
where U(r1) is the continuum transverse potential, M1 is the particle mass and p1 is the transverse compo. nent of the momentum of a particle following the scattering by the undisplaced atom. The last term in eq. (1), just as the sum of the first two terms, is inversely proportional to the particle energy. Its relative value is in. dicated in fig. 2. As seen from the fig. 2 the last term should be included in a quantitative description of the
/ / / / /
8 March 1982
The averaging was done numerically. The result of
the averaging proved to be well described by the sum of two terms of the expansion in powers of u1(cr u~ and ut). This approximation was, therefore, used to calculate x(x). In calculating K1 at small E1, we use only the first order terms in the expansion of op1 in powers of u1. Then K~ k~where k~= (u1u1~)/(u~) are the correlation coefficients of the lattice thermal vibrations. This equality was assumed to hold for all values of E1. The values of k~were determined in the Debye approximation [71.In this case, the value of the coefficient in parenthesis in eq. (2) proved to be nearly 2.5 for the most open channel and 1.3 for the least open channel considered. This implies that the thermal vibration correlations lead to a considerable increase of the dechanneling rate. It will be noted that the replacement of K~by k~ can be illegitimate at small r1 where the expansion in powers of u1 is not applicable. However, it is clear that the result of the calculation of x(x) is weakly depen. dent of d(E1)/dx at E1 close to the critical transverse energy E±~ because in this case E1 undergoes changes at distances which are small compared to the experimentally explored depth intervals. The contribution of multiple scattering by electrons to d(E1)/clx was calculated in the electron local density approximation using the expression for a uniform gas of free electrons [8]
plotted The results in fig. 3. of Differences calculationsinofthe thevalues function of the x(x)coefare Ek~]in various channels which lead to a mismatch of the curves ~(x), are compensated, to a great extent, by different contributions of multiple ficients [1 +
Volume 88A, number 4
PHYSICS LETTERS
-
8 March 1982
refs. [3,12]). The approach is employed to consider the correlations of thermal atomic vibrations and the displacement of a particle trajectory due to the lattice
thermal vibrations. The obtained results indicate that correlations of the lattice thermal vibrations play a sig-
a~5
nificant role in the process of particle dechanneling. The Debye approximation used to estimate the correlation coefficients does not enable us to speak of a
quantitative agreement with experiment. Nevertheless, 1
2
~1MgM]
Fig. 3. The results of the calculationsof the dependence ~(x) in W made including and disregarding the correlations of thermal vibrations of the lattice (the upper and lower sets of curves respectively). The notation is the same as in fig. 1.
the results are important in studying the dechanneling mechanisms. Moreover, the present results permit to state that the experimental data on dechanneling can be used to extract new information on the character of thermal atomic vibrations in a crystal.
References scattering by electrons. For a comparison with the experimental data, fig. 1 includes the data of fig. 3 (shading). As is seen, taking account of the correlations improves the agreement with experiment. Deviations of the calculations from the experimental data at small depth [in particular, the values of~(0)] can be due to inapplicability of the model of statistical equilibrium at these depths. The value of~(0)calculated using the results of Barrett [9] ‘is the same as the measured one. It will be noted that there are methods for studying multiple scattering of channeled particles that enable us, in principle, to take a strict account of inelastic scattering with the~excitationof degrees of freedom of the crystal (see, for example, refs. [10,111). In particular, this approach is used [111 to obtain the expression for (bE 1) which is a good approximation at largeE1. Up to now, the possibilities of the approach have not yet been realized to full advantage. In the present paper, the uncertainties in Lindhard’s approach [1] at high E1 are eliminated by using the resuits of the numerical averaging of (bp~>over the distribution in thermal atomic displacements (see, also,
[11 J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (1965) No. 14.
[21 N. Matsunami and L.M. Howe, Radiat. Eff. 51(1980) [3] M.A. Kumakhov and G. Shirmer, Atomic collisions in crystals (in Russian) (Atomizdat, Moscow, 1980). [4] T. Oshiyama and M. Mannaini, Phys. Lett. 81A (1981) [5] Sh.Z. Izmallov, E.l. Sirotinin, A.F. Tulinov and V.A. Khodyrev, Fiz. Tverd. Tela 23 (1981) 1933. [6] V.0. Roslyakov, A.S. Rudnev, E.I. Sirotinin,
A.F. Tulinov and V.A. Khodyrev, Phys. Stat. Sol. (a)43 (1977) 59. [71 K.J. Glauber, Phys. Rev. 98 (1955) 1692. [81 J. Lindhard, K. Dan. Vidensk. Seisk. Mat. Fys. Medd. 28 (1954) No. 8. [9] J.H. Barrett, Phys. Rev. B3 (1971) 1527. [101 Yu. Kagan and Yu.V. Kononetz, Zh. Eksp. Teor. Fiz. 64 (1973) 1042, [11] M. Kitagawa and Y.H. Ohtsuki, Phys. Rev. B8 (1973) 3117. [121 H.E. Shi~ltt,EAtomic . Bonderup, J.U. Andersen andS. Datz, H. Esbensen, collisions in solids, ed. B,A. Appleton and C.D. Moak (Plenum, New York, 1975) p. 843.
201