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Effects of dislocations on the energy loss of charged particles in hyperchanneling
Volume 57A, number 5
PHYSICS LETTERS
12 July 1976
EFFECTS OF DISLOCATIONS ON THE ENERGY LOSS OF CHARGED PARTICLES IN HYPERCHANNELING A.P. PATHAK Section d’Etude des So/ides Irradths, Centre d’Etudes Nucleaires de Fontenay-aux-Roses, 92260 France Received 1 June 1976 A simple model to calculate the effects of dislocations on the energy loss of charged particles in proper axial channeling (hyperchanneling) is discussed. The cylindrically symmetric potential around the channel axis is obtained in the continuum approximation for (110> channel of diamond structure and (100> channels of fcc and bcc crystals. As in recent work on planar case, the channel curvature due to dislocations is approximated by an arc of constant radius which is determined from the displacement equations for the dislocation. An analytical expression for the change in the energy loss of initially well channeled particles is obtained for channels with small curvature.
We have recently studied the effects of dislocations on the energy loss in planar channeling [1]. For well channeled particles, the problem could be treated analytically under certain approximations and the experiments are in progress to test the predictions of the proposed model. The generalization of this treatment for those particles which are not well channeled initially, is also underway. A corresponding formalism for axial case is in general more complicated because firstly the geometry of different axial channels is widely different from each other and secondly the channeled particles wander from one acial channel to another(of the same family at least). However, it has been observed in transmission experiments for low velocity ions (protons through Si (110) channel [2] and iodine through Ag (100) and (110) channels [3,4]) that a special class of these axially channeled particles, having an order of magnitude smaller acceptance angle and exhibiting conspicuously lower energy loss, succeed in moving through a single axial channel throughout; the so called hyperchanneled particles [3,4]. In this communication, we present a simple analytical treatment for the effects of dislocations on the energy loss of these hyperchanneled particles and suggest that an experimental test (presently not feasible in our laboratory) of these results will be of great help to proceed further with the proposed model. Let us consider only those particles which move with a minimum of amplitude (well channeled partides) in a single axial channel and thus correspond to the edge of the hyperchanneling tail [2—4] in the energy spectrum. These particles move in the central
parts of the cylindrically symmetric potential which can be obtained from the geometry of the particular channel using Lindhards standard potential and continuum approximation [5]. For a symmetric channel, such as (100) in fcc and bce crystals and (110) in a diamond structure, we find [61 2 r2 ~ ~ ~ e2 C2 a2 V(r) = S 1 2 2 (1 + )E U 0 (1 +_~) d R0 R0 R0 (1) where r is distance measured from the channel axis, R0 is the distance from this axis to one of the strings, z1 and z2 are the atomic number of the incident ion and target atom respectively, C is Lindhards constant a is the Thomas-Fermi screening radius, d is the repeat distance along the strings and n~is the number of strings surrounding the channgel. For Si (110); ‘~= 6, d = A/~/~, R0 = A/2~./2;for bce (100); ‘~ = 4, d = A, R0 = A/2; and for fcc (100); ~ = d A, R0 = A/2../~,where A is the lattice constant of the corresponding crystal. In arriving at expression (I), the fourth and higher powers of r/R0 have been neglected so that it is valid in the central parts of the channel where the potential energy contours are circular [41for symmetric channels (for example, for r = R0/2, the maximum percentage error is about 6%). As previously [1] we consider the simpler case of screw dislocations, so that the radius of curvature R and the half length z (as shown in fig. 1) of the curved part of the dislocation effected channels, situated at a distance d0 from the dislocation of the Burger’s vector b, are given by [1] _~-~
(~r~r3),
467
Volume 57A, number S
PHYSICS LETTERS
12 July 1976
given by
2E R
~ dr
=0.
(4)
Ir—r
0
Using eq. (1) in eqs. (3) and (4), we get 2/2r ~=‘/~TmR”~./r r 0, where
(6)
0R.
Assuming, as in planar case [1,7] ,that the length of the curved part is not larger than the oscillation wavele1igth of the ion, we can take the average times to cross the first and second parts of the curved channel (having curvature and hence the centrifugal force in opposite
n 2 E
z
2/U
r0—ER
(a)
__
directions [1])as t1 =z\/~7~i~each,sothatthe position r1 and the transverse velocity v(r1), after crossing first half of the curved channel, are obtained
z
from eq. (5) as [6] (see fig. 1(b)), 2(t = 2 r0 cos 1 ~/KT7T~0
-~--~B (b)
12E!
~
(7)
and
Fig. 1. (a) A typical channel at some finite distance d0 from a dislocation. (b) The model channel replacing the actual channel of the part (a), and showing the co-ordinates used in the text. Here R0 is the distance between the axis of the channel and one of the strings, r0 is the equilibrium position about which the particle will oscillate and r1 and r2 are the positions at which the particle arrives after traversing the first and second parts of the channel, respectively (i.e. left and right to AB, respectively).
u (r1)
-~/~E r0/m R
=
sin
(t1 ~r2Elm R
2 d~lbcos3~, z = irdolcos (2) R = 21T where 4p is the angle between the channel axis and plane perpendicular to the dislocation axis [7] It is again implicit that the channel under consideration is outside the dechanneling cylinder [7] so that the ~,
.
ions do not get dechanneled but their state of motion is appreciably modified due to the curvature of the channels. Replacing these effects by the centrifugal force [1,7] 2E/R on the ion of average energy E as shown inwell fig. channeled 1 and solving the equation initially particles, we get of [6]motion for V(r)
+
2 E r/R
(3)
,
=
2 r0
dOS
(t1 ~./2E/m R r0) (8)
v(r2)
=
Now
2\/2E r0/m R sin (t1 ~,/2Elm R r0). the oscillation amplitude of the particle enter-
ing the perfect channel with initial conditions given by
eqs. (8), is given by [1,6,81
V(r~~) = ~mv2(r 2) + V(r2) (9) and using eqs. (1) and (8)in this, we get ~ = 2ER~/U0R.Since various dislocations, uniformly distributed in the target crystal, have different orientations with respect toits theaverage channelvalue under consideration, 3~pby 4/37r in the we replace for cos the curvature I IR (from eq. (2)), and expression finally get average oscillation amplitude as
The equilibrium distance r
3 d~U 0 from the channel axis is
468
r0)
Using these as initial conditions, in solving the equations of motion for second half of the curved channel, we get after some algebra [6] the position and the transverse velocity of the ion, after it has crossed whole of the curve part, as
and
,~ ~7~.\/V(0)
(5)
T
amp
=
4 R~F b/3ir
0 .
(10)
Volume 57A, number 5
PHYSICS LETTERS
To be consistent with eq (1), we use the charge density corresponding to the Lindhard’s standard potential [5] to derive an axially averaged charge density due to n 5 strings surrounding the channel, and get 2 2C2 a2 + 4 r2) p(r) = ~ z2 C~ ~ a R~ ~ ird R
(
2p 4 r 0 =p(O)+— R~
1
=
p(O) +
2
ir r p(r)dr
2 P0 r~~/R~
(16)
For 21 .6 MeV lions in Ag crystals [4] for ~ = 1000 a.u. ~530 A this gives z~S = 0.065 s0. It may be pointed out that L~Sis very sensitive to the concentration of dislocations. For example, for d~X= 500 a.u. s
(11)
Tamp
f
—a—-— IT ramp o
2A 12
s0E I~s= 41472 ir6d~ z~z~ e4a4(A2 48 a2)
0 because d0 appears in the denominator of eq. (16) as its fourth power.
so that the energy loss of well channeled particles (in hyperchanneling) in perfect crystal for low velocity case is given by [9], 2 Qd Q p(O) (12) s0 = p(0) m u where m is the electron mass, v is the incident ion velocity and Qd is the momentum transfer cross seetion which can be evaluated by solving radial part of Schrodinger equation for various phase shifts [9]. Alternatively the experimental value of s 0 may be used to obtain Q from eq. (12). After introducing a concentration n of screw dislocations, so that d8’~ ll~J~, the new edge in hyperchanneling tail of the energy spectrum will correspond to particles having oscillation amplitude given by eq. (10) with =~ The energy loss rate of these particles2QdEISQ is given by (13) Sj5mv where ~ is the density sampled by the oscillating ions and is given by,
=
12 July 1976
(14)
This should at least, provide an order of magnitude estimate of the effects of dislocations on the energy loss rate of the hyperchanneled particles and the tail observed [2 4] should correspondingly shift towards low energy side, or perhaps should disappear and be’ come part of the normal axially channeled particles. M experimental verification of these predictions should be very helpful in guiding the direction of more detailed (if required, numerical) calculations using effective density due to individual target shells, and corresponding detailed and accurate shell potential, instead of the statistical densities and potentials used here. Such calculations are presently in planning stage. The author is thankful to Dr. Y. Quéré for his interest in this work.
References [1] A.P. Pathak, Phys. Lett. 55A(1975) 104;Phys. Rev. B (in press). [2] F.H. Eisen, Phys. Lett. 23 (1966) 401. [3] B.R. Appleton, C.D. Moak, T.S. Noggle and J.H. Barrett. Phys. Rev. Lett. 28 (1972) 1307. [4] J.H. et a!., Atomic collisions solids, eds. Datz,Barrett B.R. Appleton and C.D. Moak in (Plenum, N.Y.S.1975) Vol. 2 p. 645. [5] J. Lindhard, K.Dan. Vidensk.Selsk., Mat.-Fys.Medd. 34 (1965) no. 14.
so that the change in the energy loss rate is given by
[6] A.P. Pathak, to be published. [7] Y. Quér~,Phys. Lett. 26A (1968) 578; Phys. Stat. Sol. 30 (1968) 713.
~S=S
[8] M.T. Robinson, Phys. Rev. 179 (1969) 327; Phys. Rev. B4 (1971)1461.
so=2Qpor~~/R~
(15)
which (for example) for (100) axial channels in fcc crystals gives
[9] J.S. Bri~sand A.P. Pathak, J. Phys. C 6 (1973) L153 J. Phys. C 7 (1974) 1929.
469
PHYSICS LETTERS
12 July 1976
EFFECTS OF DISLOCATIONS ON THE ENERGY LOSS OF CHARGED PARTICLES IN HYPERCHANNELING A.P. PATHAK Section d’Etude des So/ides Irradths, Centre d’Etudes Nucleaires de Fontenay-aux-Roses, 92260 France Received 1 June 1976 A simple model to calculate the effects of dislocations on the energy loss of charged particles in proper axial channeling (hyperchanneling) is discussed. The cylindrically symmetric potential around the channel axis is obtained in the continuum approximation for (110> channel of diamond structure and (100> channels of fcc and bcc crystals. As in recent work on planar case, the channel curvature due to dislocations is approximated by an arc of constant radius which is determined from the displacement equations for the dislocation. An analytical expression for the change in the energy loss of initially well channeled particles is obtained for channels with small curvature.
We have recently studied the effects of dislocations on the energy loss in planar channeling [1]. For well channeled particles, the problem could be treated analytically under certain approximations and the experiments are in progress to test the predictions of the proposed model. The generalization of this treatment for those particles which are not well channeled initially, is also underway. A corresponding formalism for axial case is in general more complicated because firstly the geometry of different axial channels is widely different from each other and secondly the channeled particles wander from one acial channel to another(of the same family at least). However, it has been observed in transmission experiments for low velocity ions (protons through Si (110) channel [2] and iodine through Ag (100) and (110) channels [3,4]) that a special class of these axially channeled particles, having an order of magnitude smaller acceptance angle and exhibiting conspicuously lower energy loss, succeed in moving through a single axial channel throughout; the so called hyperchanneled particles [3,4]. In this communication, we present a simple analytical treatment for the effects of dislocations on the energy loss of these hyperchanneled particles and suggest that an experimental test (presently not feasible in our laboratory) of these results will be of great help to proceed further with the proposed model. Let us consider only those particles which move with a minimum of amplitude (well channeled partides) in a single axial channel and thus correspond to the edge of the hyperchanneling tail [2—4] in the energy spectrum. These particles move in the central
parts of the cylindrically symmetric potential which can be obtained from the geometry of the particular channel using Lindhards standard potential and continuum approximation [5]. For a symmetric channel, such as (100) in fcc and bce crystals and (110) in a diamond structure, we find [61 2 r2 ~ ~ ~ e2 C2 a2 V(r) = S 1 2 2 (1 + )E U 0 (1 +_~) d R0 R0 R0 (1) where r is distance measured from the channel axis, R0 is the distance from this axis to one of the strings, z1 and z2 are the atomic number of the incident ion and target atom respectively, C is Lindhards constant a is the Thomas-Fermi screening radius, d is the repeat distance along the strings and n~is the number of strings surrounding the channgel. For Si (110); ‘~= 6, d = A/~/~, R0 = A/2~./2;for bce (100); ‘~ = 4, d = A, R0 = A/2; and for fcc (100); ~ = d A, R0 = A/2../~,where A is the lattice constant of the corresponding crystal. In arriving at expression (I), the fourth and higher powers of r/R0 have been neglected so that it is valid in the central parts of the channel where the potential energy contours are circular [41for symmetric channels (for example, for r = R0/2, the maximum percentage error is about 6%). As previously [1] we consider the simpler case of screw dislocations, so that the radius of curvature R and the half length z (as shown in fig. 1) of the curved part of the dislocation effected channels, situated at a distance d0 from the dislocation of the Burger’s vector b, are given by [1] _~-~
(~r~r3),
467
Volume 57A, number S
PHYSICS LETTERS
12 July 1976
given by
2E R
~ dr
=0.
(4)
Ir—r
0
Using eq. (1) in eqs. (3) and (4), we get 2/2r ~=‘/~TmR”~./r r 0, where
(6)
0R.
Assuming, as in planar case [1,7] ,that the length of the curved part is not larger than the oscillation wavele1igth of the ion, we can take the average times to cross the first and second parts of the curved channel (having curvature and hence the centrifugal force in opposite
n 2 E
z
2/U
r0—ER
(a)
__
directions [1])as t1 =z\/~7~i~each,sothatthe position r1 and the transverse velocity v(r1), after crossing first half of the curved channel, are obtained
z
from eq. (5) as [6] (see fig. 1(b)), 2(t = 2 r0 cos 1 ~/KT7T~0
-~--~B (b)
12E!
~
(7)
and
Fig. 1. (a) A typical channel at some finite distance d0 from a dislocation. (b) The model channel replacing the actual channel of the part (a), and showing the co-ordinates used in the text. Here R0 is the distance between the axis of the channel and one of the strings, r0 is the equilibrium position about which the particle will oscillate and r1 and r2 are the positions at which the particle arrives after traversing the first and second parts of the channel, respectively (i.e. left and right to AB, respectively).
u (r1)
-~/~E r0/m R
=
sin
(t1 ~r2Elm R
2 d~lbcos3~, z = irdolcos (2) R = 21T where 4p is the angle between the channel axis and plane perpendicular to the dislocation axis [7] It is again implicit that the channel under consideration is outside the dechanneling cylinder [7] so that the ~,
.
ions do not get dechanneled but their state of motion is appreciably modified due to the curvature of the channels. Replacing these effects by the centrifugal force [1,7] 2E/R on the ion of average energy E as shown inwell fig. channeled 1 and solving the equation initially particles, we get of [6]motion for V(r)
+
2 E r/R
(3)
,
=
2 r0
dOS
(t1 ~./2E/m R r0) (8)
v(r2)
=
Now
2\/2E r0/m R sin (t1 ~,/2Elm R r0). the oscillation amplitude of the particle enter-
ing the perfect channel with initial conditions given by
eqs. (8), is given by [1,6,81
V(r~~) = ~mv2(r 2) + V(r2) (9) and using eqs. (1) and (8)in this, we get ~ = 2ER~/U0R.Since various dislocations, uniformly distributed in the target crystal, have different orientations with respect toits theaverage channelvalue under consideration, 3~pby 4/37r in the we replace for cos the curvature I IR (from eq. (2)), and expression finally get average oscillation amplitude as
The equilibrium distance r
3 d~U 0 from the channel axis is
468
r0)
Using these as initial conditions, in solving the equations of motion for second half of the curved channel, we get after some algebra [6] the position and the transverse velocity of the ion, after it has crossed whole of the curve part, as
and
,~ ~7~.\/V(0)
(5)
T
amp
=
4 R~F b/3ir
0 .
(10)
Volume 57A, number 5
PHYSICS LETTERS
To be consistent with eq (1), we use the charge density corresponding to the Lindhard’s standard potential [5] to derive an axially averaged charge density due to n 5 strings surrounding the channel, and get 2 2C2 a2 + 4 r2) p(r) = ~ z2 C~ ~ a R~ ~ ird R
(
2p 4 r 0 =p(O)+— R~
1
=
p(O) +
2
ir r p(r)dr
2 P0 r~~/R~
(16)
For 21 .6 MeV lions in Ag crystals [4] for ~ = 1000 a.u. ~530 A this gives z~S = 0.065 s0. It may be pointed out that L~Sis very sensitive to the concentration of dislocations. For example, for d~X= 500 a.u. s
(11)
Tamp
f
—a—-— IT ramp o
2A 12
s0E I~s= 41472 ir6d~ z~z~ e4a4(A2 48 a2)
0 because d0 appears in the denominator of eq. (16) as its fourth power.
so that the energy loss of well channeled particles (in hyperchanneling) in perfect crystal for low velocity case is given by [9], 2 Qd Q p(O) (12) s0 = p(0) m u where m is the electron mass, v is the incident ion velocity and Qd is the momentum transfer cross seetion which can be evaluated by solving radial part of Schrodinger equation for various phase shifts [9]. Alternatively the experimental value of s 0 may be used to obtain Q from eq. (12). After introducing a concentration n of screw dislocations, so that d8’~ ll~J~, the new edge in hyperchanneling tail of the energy spectrum will correspond to particles having oscillation amplitude given by eq. (10) with =~ The energy loss rate of these particles2QdEISQ is given by (13) Sj5mv where ~ is the density sampled by the oscillating ions and is given by,
=
12 July 1976
(14)
This should at least, provide an order of magnitude estimate of the effects of dislocations on the energy loss rate of the hyperchanneled particles and the tail observed [2 4] should correspondingly shift towards low energy side, or perhaps should disappear and be’ come part of the normal axially channeled particles. M experimental verification of these predictions should be very helpful in guiding the direction of more detailed (if required, numerical) calculations using effective density due to individual target shells, and corresponding detailed and accurate shell potential, instead of the statistical densities and potentials used here. Such calculations are presently in planning stage. The author is thankful to Dr. Y. Quéré for his interest in this work.
References [1] A.P. Pathak, Phys. Lett. 55A(1975) 104;Phys. Rev. B (in press). [2] F.H. Eisen, Phys. Lett. 23 (1966) 401. [3] B.R. Appleton, C.D. Moak, T.S. Noggle and J.H. Barrett. Phys. Rev. Lett. 28 (1972) 1307. [4] J.H. et a!., Atomic collisions solids, eds. Datz,Barrett B.R. Appleton and C.D. Moak in (Plenum, N.Y.S.1975) Vol. 2 p. 645. [5] J. Lindhard, K.Dan. Vidensk.Selsk., Mat.-Fys.Medd. 34 (1965) no. 14.
so that the change in the energy loss rate is given by
[6] A.P. Pathak, to be published. [7] Y. Quér~,Phys. Lett. 26A (1968) 578; Phys. Stat. Sol. 30 (1968) 713.
~S=S
[8] M.T. Robinson, Phys. Rev. 179 (1969) 327; Phys. Rev. B4 (1971)1461.
so=2Qpor~~/R~
(15)
which (for example) for (100) axial channels in fcc crystals gives
[9] J.S. Bri~sand A.P. Pathak, J. Phys. C 6 (1973) L153 J. Phys. C 7 (1974) 1929.
469