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Thermal properties of electron stopping in solids
RadiationMeasurements,Vol. 25, Nos I--4, pp. 105-106, 1995 Copyright O 1995 Elsevier Science Ltd Printed in Great Britain. All fights reserved 1350-4487/9559.50 + .00
Pergamon
1350-4487(95)00045.3
T H E R M A L PROPERTIES OF E L E C T R O N STOPPING IN SOLIDS
V. N. KONDRATYEV*t and V. A. SHCHEPUNOV$ ?Laboratorio Nazionale del Sud, INFN, V.S.Sofia 44, 95125 Catania, Italy; and :~High Energy Laboratory, JINR, Dubna, Russia
ABSTRACT Thermal effects in the stopping power of a solid target are analysed in the framework of the Bethe-Bloch theory. These effects due to the coupling of the electron dynamics with lattice phonons can be important in understanding the scatter of stopping cross section data. KEYWORDS Stopping power; solids; temperature; fluctuations; correlations. The electron stopping power of matter has been studied for several decades. However, some problems, for example, the scatter between the cross sections stopping data (Berger et al., 1993), are still under debate. This scatter may be reduced in systematic studies employing various techniques due care to the target preparations. Nevertheless, the different methods of measurements applied to the same target give also different results (Baket al., 1994). It is the temperature dependence of the electron stopping power that can be another possible reason of such a scatter. In a few previous theoretical works (Sabin et al., 1994 and references therein) some thermal features of the stopping phenomena have been discussed in the context of a large temperature scale of ten thousands Kelvin. We note, however, that the properties of the electron bulk excitation with a large momentum transfer (q2 ..~ e) can be modified significantly due to the coupling of the single particle electron dynamics with lattice degrees of freedom. Such a coupling results in a resonant thermal effect in the conduction electrons (c.e.) excitations that arises, in some cases, at the relatively small temperatures being of the order of thousand Kelvin (Kondratyev, 1993, 1994). The stopping power is characterized by the stopping(straggling) ($1(2)(v)) cross section. Within the Bethe-Bloch theory these values are given by (Lindhard and Winther, 1964):
S~(v)--
8~rZp2 / qdq ekde v2 (q2+rD2)2((q,c) ,
> I=, (1)
where r D is Debye radius, n~ are the occupation numbers of the single particle states [a > with energies e, in a target atom. We describe an excitation process in c.e. gas using the semiclassical approximation (Kondratyev, 1993, 1994) and obtain the strength (~) as: ~(q,e)
= e_Tr/ dt C(%,t) e-'a ; C = / ~
6(% -
h(~',f3) p(~*l,~'; t)exp{iq-~?-~l)} (2)
* p e r m a n e n t address: Institute for Nuclear Research, 47, Pr.Nauki, Kiev, 252098 Ukraine zs:l/~-J
105
106
V.N. KONDRATYEV and V. A. SHCHEPUNOV
with a probability p of finding an electron with initial conditions (~, p-') at the point ~'1 at the time t. % = k~/2 is the Fermi energy of c.e. gas. It is convenient to use for the cross sections Sk(v) the famous (Lindhard and Winther, 1964) results obtained in the independent-particle (e.g. Hartree: p = 5(~] - g(t)) at ~'(t) = ~'+ fit in eq. (2)) model. However, for the system at finite temperature (T) correlations reflecting the inexactness of the Hartree method become very important. These effects can be included in a phenomenological way using the Langevin equation of motion (Kondratyev, 1993, 1994). In the case of white noised Gaussian random forces the probability (p) is given by: p ~ (~rDt)312 . The spatial diffusion coefficient D is estimated as: D = y-iT, where 7/is a friction coefficient. Thus, using Eq. (2) we write the cross sections S/'k(V) due to close collisions with c.e. as: SAk(V) --
4'n'ZP2eok-lv2 Lak(n~,v);
Lak = 3
/
z~+ldz (1 +
'~A(Z,?2) = O(1 __ (Z/~)2 __ ?22) + 71"-1
[..[
[ so
du uk+lTa(z,u), 2Z/~
arctan \ [ z A ) 2 - 1 + u2/
(3)
'
where nc is the c.e. density, eo = vF/rD, A = 2D--g,X = 2kFr D. Note, that at zero temperature Eq.s (3) give the well-known (Lindhard and Winther, 1964) results in a high projectile velocity limit. q
Fig.1 Stopping number { dE1 =_ nc ~ l)'~ v s o
temperature and projectile velocity for a c.e. gas at normal density corresponding to the condition: % = 1 (see text).
From Fig. 1 we see that the thermal effects result in an anomalous temperature dependence of the c.e. stopping cross sections. Such thermal properties can be detected in the experiments where channeled ion energy losses are measured (Dygo et al., 1994). Furthermore, these effects are expected to he very important for the mesoscopic systems (e.g. Atomic Clusters) and heterogenetic materials, since it contains additional degrees of freedom associated with the surface and intersurfaces excitations. REFERENCES Bak, H.I., Y.D. Bae, C.S. Kim and M.S. Kim (1994). Nucl. Instr. and Meth., B93, 234. Berger, M. et al. (1993). Stopping Powers of Protons and Alpha Part., ICRU Report, 49. Dygo, A. et al. (1994). Nucl. Instr. and Meth., B93, 117. Kondratyev, V.N. (1993). Phys. Lett., A179, 209. ; (1994). Phys. Lett., A190, 465. Lindhard, J. and A. Winther (1964). Mat. Fys. Medd. Dan. Vid. Selsk., 34, no.4. Sabin, J.R., J. Oddershede and I. Paidarova (1994). Nucl. Instr. and Meth., B93, 161.
Pergamon
1350-4487(95)00045.3
T H E R M A L PROPERTIES OF E L E C T R O N STOPPING IN SOLIDS
V. N. KONDRATYEV*t and V. A. SHCHEPUNOV$ ?Laboratorio Nazionale del Sud, INFN, V.S.Sofia 44, 95125 Catania, Italy; and :~High Energy Laboratory, JINR, Dubna, Russia
ABSTRACT Thermal effects in the stopping power of a solid target are analysed in the framework of the Bethe-Bloch theory. These effects due to the coupling of the electron dynamics with lattice phonons can be important in understanding the scatter of stopping cross section data. KEYWORDS Stopping power; solids; temperature; fluctuations; correlations. The electron stopping power of matter has been studied for several decades. However, some problems, for example, the scatter between the cross sections stopping data (Berger et al., 1993), are still under debate. This scatter may be reduced in systematic studies employing various techniques due care to the target preparations. Nevertheless, the different methods of measurements applied to the same target give also different results (Baket al., 1994). It is the temperature dependence of the electron stopping power that can be another possible reason of such a scatter. In a few previous theoretical works (Sabin et al., 1994 and references therein) some thermal features of the stopping phenomena have been discussed in the context of a large temperature scale of ten thousands Kelvin. We note, however, that the properties of the electron bulk excitation with a large momentum transfer (q2 ..~ e) can be modified significantly due to the coupling of the single particle electron dynamics with lattice degrees of freedom. Such a coupling results in a resonant thermal effect in the conduction electrons (c.e.) excitations that arises, in some cases, at the relatively small temperatures being of the order of thousand Kelvin (Kondratyev, 1993, 1994). The stopping power is characterized by the stopping(straggling) ($1(2)(v)) cross section. Within the Bethe-Bloch theory these values are given by (Lindhard and Winther, 1964):
S~(v)--
8~rZp2 / qdq ekde v2 (q2+rD2)2((q,c) ,
> I=, (1)
where r D is Debye radius, n~ are the occupation numbers of the single particle states [a > with energies e, in a target atom. We describe an excitation process in c.e. gas using the semiclassical approximation (Kondratyev, 1993, 1994) and obtain the strength (~) as: ~(q,e)
= e_Tr/ dt C(%,t) e-'a ; C = / ~
6(% -
h(~',f3) p(~*l,~'; t)exp{iq-~?-~l)} (2)
* p e r m a n e n t address: Institute for Nuclear Research, 47, Pr.Nauki, Kiev, 252098 Ukraine zs:l/~-J
105
106
V.N. KONDRATYEV and V. A. SHCHEPUNOV
with a probability p of finding an electron with initial conditions (~, p-') at the point ~'1 at the time t. % = k~/2 is the Fermi energy of c.e. gas. It is convenient to use for the cross sections Sk(v) the famous (Lindhard and Winther, 1964) results obtained in the independent-particle (e.g. Hartree: p = 5(~] - g(t)) at ~'(t) = ~'+ fit in eq. (2)) model. However, for the system at finite temperature (T) correlations reflecting the inexactness of the Hartree method become very important. These effects can be included in a phenomenological way using the Langevin equation of motion (Kondratyev, 1993, 1994). In the case of white noised Gaussian random forces the probability (p) is given by: p ~ (~rDt)312 . The spatial diffusion coefficient D is estimated as: D = y-iT, where 7/is a friction coefficient. Thus, using Eq. (2) we write the cross sections S/'k(V) due to close collisions with c.e. as: SAk(V) --
4'n'ZP2eok-lv2 Lak(n~,v);
Lak = 3
/
z~+ldz (1 +
'~A(Z,?2) = O(1 __ (Z/~)2 __ ?22) + 71"-1
[..[
[ so
du uk+lTa(z,u), 2Z/~
arctan \ [ z A ) 2 - 1 + u2/
(3)
'
where nc is the c.e. density, eo = vF/rD, A = 2D--g,X = 2kFr D. Note, that at zero temperature Eq.s (3) give the well-known (Lindhard and Winther, 1964) results in a high projectile velocity limit. q
Fig.1 Stopping number { dE1 =_ nc ~ l)'~ v s o
temperature and projectile velocity for a c.e. gas at normal density corresponding to the condition: % = 1 (see text).
From Fig. 1 we see that the thermal effects result in an anomalous temperature dependence of the c.e. stopping cross sections. Such thermal properties can be detected in the experiments where channeled ion energy losses are measured (Dygo et al., 1994). Furthermore, these effects are expected to he very important for the mesoscopic systems (e.g. Atomic Clusters) and heterogenetic materials, since it contains additional degrees of freedom associated with the surface and intersurfaces excitations. REFERENCES Bak, H.I., Y.D. Bae, C.S. Kim and M.S. Kim (1994). Nucl. Instr. and Meth., B93, 234. Berger, M. et al. (1993). Stopping Powers of Protons and Alpha Part., ICRU Report, 49. Dygo, A. et al. (1994). Nucl. Instr. and Meth., B93, 117. Kondratyev, V.N. (1993). Phys. Lett., A179, 209. ; (1994). Phys. Lett., A190, 465. Lindhard, J. and A. Winther (1964). Mat. Fys. Medd. Dan. Vid. Selsk., 34, no.4. Sabin, J.R., J. Oddershede and I. Paidarova (1994). Nucl. Instr. and Meth., B93, 161.