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Theory of dechannelling of electrons in single crystals
Volume 89A, number 6
PHYSICS LETTERS
24 May 1982
THEORY OF DECHANNELLING OF ELECTRONS IN SINGLE CRYSTALS K. KIMURA and M. MANNAMI Department of EngineeringScience,Kyoto University,Kyoto 606, Japan Received 6 January 1982 Revised manuscript received 18 March 1982
The probability of dechannelling of axially channelled electrons due to lattice vibrations and the discreteness of the atomic row is calculated. It is shown that the deehanneUing due to lattice vibrations is the dominant process except for very low energy electrons. The calculated results are in good agreement with the observed linewidths of channelling radiation.
Radiation from transitions between bound states of transverse motion for channelled electrons was predicted by Kumakhov [ 1]. Recently, this "r-radiation from axiaUy channelled electrons has been observed [ 2 - 5 ] . A1 though the enhancement of ~/-radiation for channelling condition is observed, the "pray spectra from high energy electrons ( 2 8 - 5 6 MeV) are structureless [2]. On the other hand, low energy electrons (1.5--4 MeV) show interesting 7-ray spectra which have several peaks corresponding to individual transitions between bound states and the energies of these peaks are in good agreement with calculated results [ 3 - 5 ] . The linewidths of these peaks were also measured, from which the widths of the energy levels of the bound states can be deduced. There are several mechanisms of line broadening; (1) the lifetime of the bound state due to dechannelling, (2) the lifetime due to the finite crystal thickness, (3) radiative lifetime and (4) the band structure of the energy levels of the bound states due to the periodicity of the crystal perpendicular to the direction of the axis. The line broadening corresponding to mechanism (3) is very small compared with experimental results [6], and that of (4) is also small except for highly excited levels [7]. Thus the observed linewidth is considered to be mainly caused by dechannelling of the electrons and the finite crystal thickness. The line broadening due to the finite crystal thickness was observed in recent investigations of MeV electrons with thin silicon crystals [5,8]. As for the dechannelling of electrons, there is only one theoretical treatment of line broadening due to scattering by the thermal displacements of lattice atoms based on the sudden collision approximation [5]. In the present letter, a theoretical model for calculating the dechannelling probability due to lattice vibrations and the discreteness of the atomic row for axially channelled electrons is proposed. The calculated line broadening due to dechannelling is compared with experimental results. The potential of a thermally vibrating periodic chain of atoms, whose direction is taken to be the z-axis, is written approximately
zq
+
+ n~0 Z
ox ( ZTz ) ,
where
gn(rt)
=
(
)
Vatom(r)ex p - i --j- z dz,
V~'(rt) is the continuum potential taking account of the lattice vibration, r t the distance of the electron from the atomic row, Uq the amplitude of lattice vibration of mode q, qz the z-component of q, d the mean interatomic 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
299
Volume 89A, n u m b e r 6
PHYSICS LETTERS
24 May 1982
spacing and Vatom(r ) the atomic potential. The second term of eq. (1) is due to the lattice vibration and the third term is due to the discreteness of the atomic row. Eq. (1), which is the potential in the laboratory frame, can be transformed into the "rest" frame of the electrons,
U(r) = "),VoT(rt) - D "y{UqVgo(rt) exp(iqzW/t) + U~VVo(rt) exp(-iqzvTt)} + ~ q
[. 27rn
7Vn(rt) exp[1---j- vTt
)
,
(2)
nv~O
where v is the electron velocity parallel to the atomic row in the laboratory frame and 7 = [1 - (v/c) 2 ] -1/2. The second and the third term of eq. (2) are neglected in the usual channelling theory. We calculate wave functions of the bound states of the transverse motion with the first term of eq. (2) and treat the second and third term as a perturbation. These terms are periodic with time and cause transitions with AE t = h(27rn/d)v7 and l~qzVT. Using time dependent perturbation theory, the transition probability from a bound state to a continuum state, which corresponds to dechannelling, can be calculated. The dechannelling probability of the ith bound state per unit time due to the second term and that due to the third term of eq. (2), which are referred to as thermal dechannelling and resonance dechannelling respectively, hereafter, are written respectively
2.f ](,f(rt)l,,/U~ OVo(rt)/3xl~)i(rt))lZp(Ef)f(Ef
w t = 4 ~q h
w[ = n~= = l 27r
_ Ei _ ~qzUT) dEf
(3)
(4)
[(dpf(rt)l'),Vn(rt)lc~i(rt))12p(l:})8(Ef - E i - h(27rn/d)v'),) d E f ,
where q~i(rt) and ~bf(rt) are the wave functions of the initial ith bound state and the final continuum state respectively, El, Ef the energies of the initial and final states respectively and p(Ef) the density of the final state. The dechannelling rate of the ith state per unit length along the atomic row is given by Pi = ( wt + wr)/v7 in the laboratory frame and the partial width of the energy level of the ith state corresponding to the dechannelling is given by I~i = h(w/ + W[) in the "rest" frame of the electrons. The linewidth of the channelling radiation emitted along the beam direction from the transition i ~ j becomes [ ' i ~ / ~ 27(Fi + F/) in the laboratory frame because of the Doppler shift. If the wave function of the initial bound state is calculated with a harmonic approximation to the bottom of the potential vT(rt), i.e. VT(rt) ~ U 0 + ½mwor2t, and that of the final continuum state is approximated to be a plane wave in the mean inner potential Vin [the mean value of U(r) in the crystal], and Vo(rt) is approximated to be (2Ze2/d) In rt, i.e. neglecting the screening of Fatom(r), eq. (3) for the ls state becomes
W~s
-
2 e2
1
kBT c
n Pic Naod2 MV2s v
Z2COO~'3/2 f
qD
{1 -- exp[--(q z - ql)V71/2/CO0]} 2
ln(qD/qz ) --
qmin
qz - ql
-
-dqz ,
(5)
where N is the number of crystal atoms per unit volume, a 0 the Bohr radius, M the mass of the crystal atom, Z the atomic number, v s the sonic velocity in the crystal, ql = (IEil - Vin)/17°')', qmin = ([/fil - Ec)fflO'g, E c the critical transverse energy of channelling, qD the Debye wave number, and the Debye model of thermal vibrations is employed. The neglect of atomic screening in eq. (5) has little effect because the wave function of the 1 s state is localized in the vicinity of the atomic row, e.g. the radius of the ls state of MeV electrons in silicon crystal is ~< 10 -2 nm and the screening radius is ~ 2 × 10 - 2 nm. The integral in eq. (5) increases slightly with increasing 7 and is roughly approximated to be ln(ttqDV/Uo) ln(qDo'y1/2/COO). Using the same approximation for initial and final states as used in deriving eq. (5), eq. (4) for the ls state becomes
W~s-
300
16n~oZ2@/2 ~ Ifexp(m°~o'I/2r2~ aZod2 n:l I 2h -]Ko(rt[(2rrn/d)2
+(1/aB)2]l/2)Jo(ktrt)rt drt 2 ,
(6)
Volume 89A, number 6
0
PHYSICS LETTERS
ENERGY OF ELECTRON (MeV) I 10 , " " '
I },o ll
]
Si
'
,
'
'
I
[110] ls state
24 May 1982
0 ENERGY OF ELECTRON10 (MeV) • Si [111] ls state I
102
I0~~"~ ~:"~
,
1102
,,,,
[
,
,
.
,
[
lO2
10
r e s o ~
\ ~ 1c
1 '
=
101
. . . . -''2
°I 1 1
'
'
'
'
101
,
,
,
,
--
Fig. 1. Calculated dechannelling rates for the 1s state in the laboratory system as a function o f ~, = [ 1 - (o/c) 2 ]-1/2, (a) for Si [ 110] and (b) for Si [ 111 ]. The scale o f the linewidth in the "rest" system o f electrons, r , and the scale o f the electron energy are also shown.
where k t = (2m [E i + ti (27rn/d)tr)" + Vin ] }l/2h, a B the screening radius for the Bohr potential, which is employed for the calculation of Vn(rt). In the ultrarelativistic limit, the integration can be performed analytically and then eq. (6) becomes oo
WIs = ~
7r-1 (e2 /ptc)2 Z 2 ~o Tl/2 /n2 = ~ 7r(e2/hc)2Z 2 600 71[2 .
(7)
n=l
The energy dependence of the dechannelling rate and the width of the energy level of the 1s state for silicon [110] and [111] calculated with the use of eqs. (5) and (6) are shown in fig. 1. The resonance dechannelling rate decreases monotonically with increasing energy. The thermal dechanneUing rate also decreases with increasing energy in the low energy region, but in the high energy region it increases with increasing energy and becomes dominant The widths of energy levels of excited states can also be calculated with eq~. (3) and (4). However, the calculation is complicated becuase the harmonic approximation to the potential V(~(rt) is not adequate for excited states. A rough estimation showed that the width of the 2p state is very small and about one eigth of that of the Is state [4], as the amplitude of 2p wave functions is zero at the atomic row. Thus we can compare the observed linewidths for the transition from the 2p state to the ls state with the calculated width of the ls state. The calculated and the observed linewidths are shown in table 1. The calculated linewidths are in good agreement with the experimental results of 3.81 MeV electrons for [ 110] and 4 MeV for [ 111 ]. The thermal dechannelling is dominant for [ 110], but the resonance dechannelling is comparable to the thermal dechannelling for [ 111 ] at 4 MeV. The calculation for higher energy electrons gives very large linewidths. These are also consistent with experimental spectra, in which individual transitions between bound states are not resolved due to the superposition of broad peaks. In the present calculation, the dechannelling due to electronic scattering is neglected. It plays an important role in the theory of dechannelling of positively charged particles. They pass through the channel centre and have 301
Volume 89A, number 6
PHYSICS LETTERS
24 May 1982
Table 1 Comparison of the calculated linewidths of channelling radiation with the experimental results for the 2 p - I s transition in Si single crystals [3,4]. F t s and l?[ s are the partial widths of the 1s state corresponding to thermal dechannelling and resonance dechannelling respectively in the "rest" system of the electrons. The experimental results are also the linewidths in the "rest" system, which are calculated from the observed linewidths. The individual transitions between bound states were not resolved in the experimental spectra for 28 and 56 MeV electrons [2]. Channel
Energy (MeV)
Linewidths (eV) calculation
rts
experiment
rts
l'~s+ r~
r~p-ls
[110]
3.81 28.0 56.0
40 1024 3110
7 18 26
47 1042 3136
47.2 [4] [2] [21
[111]
4.O 28.0 56.0
27 610 1945
54 138 194
8l 748 2139
53
a very large probability o f e n c o u n t e r s with electrons c o m p a r e d with nuclei. As for pass around the a t o m i c row and their e n c o u n t e r probability w i t h electrons is only nuclei. So the dechannelling probability due to the electronic scattering is roughly due to the nuclei because o f the Z 2 d e p e n d e n c e o f W~s and W~s. The details o f the derivation o f the formulas and the application o f the present will be published elsewhere.
[3] [2] [2]
the channelled electrons, they Z times as large as that w i t h e s t i m a t e d to be 1/Z times that m o d e l to planar channelling
References [1] [2] [3] [4] [5]
M.A. Kumakhov, Zh. Eksp. Teor. Fiz. 72 (1977) 1489 [Sov. Phys. JETP 45 (1977) 781]. R.L. Swent et al., Phys. Rev. Lett. 43 (1979) 1723. J.U. Andersen and E. Laegsgaard, Phys. Rev. Lett. 44 (1980) 1079. N. Cue et al., Phys. Lett. 80A (1980) 26. J.U. Andersen, E. Bonderup, E. Laegsgaard, B.B. Marsh and A.H. Sorensen, Proc. 9th Intern. Conf. on Atomic collisions in solids, Lyon, France (July 6 - 1 0 , 1981), to be published. [6] R.H. Pantell and M.J. Atguard, J. Appl. Phys. 50 (1979) 798. [ 7 ] K. Komaki and F. Fujimoto, Phys. Lett. 82A (1981 ) 51. [8] J.U. Andersen, K.R. Eriksen and E. Laegsgaard, Phys. Scr. 24 (1981) 588.
302
PHYSICS LETTERS
24 May 1982
THEORY OF DECHANNELLING OF ELECTRONS IN SINGLE CRYSTALS K. KIMURA and M. MANNAMI Department of EngineeringScience,Kyoto University,Kyoto 606, Japan Received 6 January 1982 Revised manuscript received 18 March 1982
The probability of dechannelling of axially channelled electrons due to lattice vibrations and the discreteness of the atomic row is calculated. It is shown that the deehanneUing due to lattice vibrations is the dominant process except for very low energy electrons. The calculated results are in good agreement with the observed linewidths of channelling radiation.
Radiation from transitions between bound states of transverse motion for channelled electrons was predicted by Kumakhov [ 1]. Recently, this "r-radiation from axiaUy channelled electrons has been observed [ 2 - 5 ] . A1 though the enhancement of ~/-radiation for channelling condition is observed, the "pray spectra from high energy electrons ( 2 8 - 5 6 MeV) are structureless [2]. On the other hand, low energy electrons (1.5--4 MeV) show interesting 7-ray spectra which have several peaks corresponding to individual transitions between bound states and the energies of these peaks are in good agreement with calculated results [ 3 - 5 ] . The linewidths of these peaks were also measured, from which the widths of the energy levels of the bound states can be deduced. There are several mechanisms of line broadening; (1) the lifetime of the bound state due to dechannelling, (2) the lifetime due to the finite crystal thickness, (3) radiative lifetime and (4) the band structure of the energy levels of the bound states due to the periodicity of the crystal perpendicular to the direction of the axis. The line broadening corresponding to mechanism (3) is very small compared with experimental results [6], and that of (4) is also small except for highly excited levels [7]. Thus the observed linewidth is considered to be mainly caused by dechannelling of the electrons and the finite crystal thickness. The line broadening due to the finite crystal thickness was observed in recent investigations of MeV electrons with thin silicon crystals [5,8]. As for the dechannelling of electrons, there is only one theoretical treatment of line broadening due to scattering by the thermal displacements of lattice atoms based on the sudden collision approximation [5]. In the present letter, a theoretical model for calculating the dechannelling probability due to lattice vibrations and the discreteness of the atomic row for axially channelled electrons is proposed. The calculated line broadening due to dechannelling is compared with experimental results. The potential of a thermally vibrating periodic chain of atoms, whose direction is taken to be the z-axis, is written approximately
zq
+
+ n~0 Z
ox ( ZTz ) ,
where
gn(rt)
=
(
)
Vatom(r)ex p - i --j- z dz,
V~'(rt) is the continuum potential taking account of the lattice vibration, r t the distance of the electron from the atomic row, Uq the amplitude of lattice vibration of mode q, qz the z-component of q, d the mean interatomic 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
299
Volume 89A, n u m b e r 6
PHYSICS LETTERS
24 May 1982
spacing and Vatom(r ) the atomic potential. The second term of eq. (1) is due to the lattice vibration and the third term is due to the discreteness of the atomic row. Eq. (1), which is the potential in the laboratory frame, can be transformed into the "rest" frame of the electrons,
U(r) = "),VoT(rt) - D "y{UqVgo(rt) exp(iqzW/t) + U~VVo(rt) exp(-iqzvTt)} + ~ q
[. 27rn
7Vn(rt) exp[1---j- vTt
)
,
(2)
nv~O
where v is the electron velocity parallel to the atomic row in the laboratory frame and 7 = [1 - (v/c) 2 ] -1/2. The second and the third term of eq. (2) are neglected in the usual channelling theory. We calculate wave functions of the bound states of the transverse motion with the first term of eq. (2) and treat the second and third term as a perturbation. These terms are periodic with time and cause transitions with AE t = h(27rn/d)v7 and l~qzVT. Using time dependent perturbation theory, the transition probability from a bound state to a continuum state, which corresponds to dechannelling, can be calculated. The dechannelling probability of the ith bound state per unit time due to the second term and that due to the third term of eq. (2), which are referred to as thermal dechannelling and resonance dechannelling respectively, hereafter, are written respectively
2.f ](,f(rt)l,,/U~ OVo(rt)/3xl~)i(rt))lZp(Ef)f(Ef
w t = 4 ~q h
w[ = n~= = l 27r
_ Ei _ ~qzUT) dEf
(3)
(4)
[(dpf(rt)l'),Vn(rt)lc~i(rt))12p(l:})8(Ef - E i - h(27rn/d)v'),) d E f ,
where q~i(rt) and ~bf(rt) are the wave functions of the initial ith bound state and the final continuum state respectively, El, Ef the energies of the initial and final states respectively and p(Ef) the density of the final state. The dechannelling rate of the ith state per unit length along the atomic row is given by Pi = ( wt + wr)/v7 in the laboratory frame and the partial width of the energy level of the ith state corresponding to the dechannelling is given by I~i = h(w/ + W[) in the "rest" frame of the electrons. The linewidth of the channelling radiation emitted along the beam direction from the transition i ~ j becomes [ ' i ~ / ~ 27(Fi + F/) in the laboratory frame because of the Doppler shift. If the wave function of the initial bound state is calculated with a harmonic approximation to the bottom of the potential vT(rt), i.e. VT(rt) ~ U 0 + ½mwor2t, and that of the final continuum state is approximated to be a plane wave in the mean inner potential Vin [the mean value of U(r) in the crystal], and Vo(rt) is approximated to be (2Ze2/d) In rt, i.e. neglecting the screening of Fatom(r), eq. (3) for the ls state becomes
W~s
-
2 e2
1
kBT c
n Pic Naod2 MV2s v
Z2COO~'3/2 f
qD
{1 -- exp[--(q z - ql)V71/2/CO0]} 2
ln(qD/qz ) --
qmin
qz - ql
-
-dqz ,
(5)
where N is the number of crystal atoms per unit volume, a 0 the Bohr radius, M the mass of the crystal atom, Z the atomic number, v s the sonic velocity in the crystal, ql = (IEil - Vin)/17°')', qmin = ([/fil - Ec)fflO'g, E c the critical transverse energy of channelling, qD the Debye wave number, and the Debye model of thermal vibrations is employed. The neglect of atomic screening in eq. (5) has little effect because the wave function of the 1 s state is localized in the vicinity of the atomic row, e.g. the radius of the ls state of MeV electrons in silicon crystal is ~< 10 -2 nm and the screening radius is ~ 2 × 10 - 2 nm. The integral in eq. (5) increases slightly with increasing 7 and is roughly approximated to be ln(ttqDV/Uo) ln(qDo'y1/2/COO). Using the same approximation for initial and final states as used in deriving eq. (5), eq. (4) for the ls state becomes
W~s-
300
16n~oZ2@/2 ~ Ifexp(m°~o'I/2r2~ aZod2 n:l I 2h -]Ko(rt[(2rrn/d)2
+(1/aB)2]l/2)Jo(ktrt)rt drt 2 ,
(6)
Volume 89A, number 6
0
PHYSICS LETTERS
ENERGY OF ELECTRON (MeV) I 10 , " " '
I },o ll
]
Si
'
,
'
'
I
[110] ls state
24 May 1982
0 ENERGY OF ELECTRON10 (MeV) • Si [111] ls state I
102
I0~~"~ ~:"~
,
1102
,,,,
[
,
,
.
,
[
lO2
10
r e s o ~
\ ~ 1c
1 '
=
101
. . . . -''2
°I 1 1
'
'
'
'
101
,
,
,
,
--
Fig. 1. Calculated dechannelling rates for the 1s state in the laboratory system as a function o f ~, = [ 1 - (o/c) 2 ]-1/2, (a) for Si [ 110] and (b) for Si [ 111 ]. The scale o f the linewidth in the "rest" system o f electrons, r , and the scale o f the electron energy are also shown.
where k t = (2m [E i + ti (27rn/d)tr)" + Vin ] }l/2h, a B the screening radius for the Bohr potential, which is employed for the calculation of Vn(rt). In the ultrarelativistic limit, the integration can be performed analytically and then eq. (6) becomes oo
WIs = ~
7r-1 (e2 /ptc)2 Z 2 ~o Tl/2 /n2 = ~ 7r(e2/hc)2Z 2 600 71[2 .
(7)
n=l
The energy dependence of the dechannelling rate and the width of the energy level of the 1s state for silicon [110] and [111] calculated with the use of eqs. (5) and (6) are shown in fig. 1. The resonance dechannelling rate decreases monotonically with increasing energy. The thermal dechanneUing rate also decreases with increasing energy in the low energy region, but in the high energy region it increases with increasing energy and becomes dominant The widths of energy levels of excited states can also be calculated with eq~. (3) and (4). However, the calculation is complicated becuase the harmonic approximation to the potential V(~(rt) is not adequate for excited states. A rough estimation showed that the width of the 2p state is very small and about one eigth of that of the Is state [4], as the amplitude of 2p wave functions is zero at the atomic row. Thus we can compare the observed linewidths for the transition from the 2p state to the ls state with the calculated width of the ls state. The calculated and the observed linewidths are shown in table 1. The calculated linewidths are in good agreement with the experimental results of 3.81 MeV electrons for [ 110] and 4 MeV for [ 111 ]. The thermal dechannelling is dominant for [ 110], but the resonance dechannelling is comparable to the thermal dechannelling for [ 111 ] at 4 MeV. The calculation for higher energy electrons gives very large linewidths. These are also consistent with experimental spectra, in which individual transitions between bound states are not resolved due to the superposition of broad peaks. In the present calculation, the dechannelling due to electronic scattering is neglected. It plays an important role in the theory of dechannelling of positively charged particles. They pass through the channel centre and have 301
Volume 89A, number 6
PHYSICS LETTERS
24 May 1982
Table 1 Comparison of the calculated linewidths of channelling radiation with the experimental results for the 2 p - I s transition in Si single crystals [3,4]. F t s and l?[ s are the partial widths of the 1s state corresponding to thermal dechannelling and resonance dechannelling respectively in the "rest" system of the electrons. The experimental results are also the linewidths in the "rest" system, which are calculated from the observed linewidths. The individual transitions between bound states were not resolved in the experimental spectra for 28 and 56 MeV electrons [2]. Channel
Energy (MeV)
Linewidths (eV) calculation
rts
experiment
rts
l'~s+ r~
r~p-ls
[110]
3.81 28.0 56.0
40 1024 3110
7 18 26
47 1042 3136
47.2 [4] [2] [21
[111]
4.O 28.0 56.0
27 610 1945
54 138 194
8l 748 2139
53
a very large probability o f e n c o u n t e r s with electrons c o m p a r e d with nuclei. As for pass around the a t o m i c row and their e n c o u n t e r probability w i t h electrons is only nuclei. So the dechannelling probability due to the electronic scattering is roughly due to the nuclei because o f the Z 2 d e p e n d e n c e o f W~s and W~s. The details o f the derivation o f the formulas and the application o f the present will be published elsewhere.
[3] [2] [2]
the channelled electrons, they Z times as large as that w i t h e s t i m a t e d to be 1/Z times that m o d e l to planar channelling
References [1] [2] [3] [4] [5]
M.A. Kumakhov, Zh. Eksp. Teor. Fiz. 72 (1977) 1489 [Sov. Phys. JETP 45 (1977) 781]. R.L. Swent et al., Phys. Rev. Lett. 43 (1979) 1723. J.U. Andersen and E. Laegsgaard, Phys. Rev. Lett. 44 (1980) 1079. N. Cue et al., Phys. Lett. 80A (1980) 26. J.U. Andersen, E. Bonderup, E. Laegsgaard, B.B. Marsh and A.H. Sorensen, Proc. 9th Intern. Conf. on Atomic collisions in solids, Lyon, France (July 6 - 1 0 , 1981), to be published. [6] R.H. Pantell and M.J. Atguard, J. Appl. Phys. 50 (1979) 798. [ 7 ] K. Komaki and F. Fujimoto, Phys. Lett. 82A (1981 ) 51. [8] J.U. Andersen, K.R. Eriksen and E. Laegsgaard, Phys. Scr. 24 (1981) 588.
302