- Email: [email protected]
Electronic stopping of ions in the low velocity limit
Nuclear Instruments
and Methods in Physics Research B 96 (1995) 626-632
Beam Interactbn~ with YIlterials& Atoms ELSEVIER
Electronic stopping of ions in the low velocity limit James W. Dufty
*, Mikhail Berkovsky
Department of Physics, University of Florida, Gadnesuille, FL 32611, USA
Abstract The change in kinetic energy of an ion moving in an equilibrium electron gas is considered as a model for electronic stopping power. The exact small velocity limit is obtained for an ion of infinite mass. It is shown to be linear in the ion speed with a coefficient determined from the autocorrelation function for the force on the ion at rest in the equilibrium electron gas. The result has no limitations with respect to the ion charge, the electron coupling parameter, and the electron degeneracy parameter. Therefore it provides a firm basis for testing various theoretical approximations. Some comments are offered on the semiclassical limit, to provide a model appropriate for study via molecular dynamics simulation.
1. Introduction Stopping power is a measure of a particle’s energy degradation on passing through a target. In general it can depend on the geometry, surface effects, and internal structure of the target’s constituent particles. Here, we idealize the problem to consider only free electronic stopping in bulk materials. The electron gas has been the historical prototype for electronic stopping, and we consider it a valuable example for “benchmark” tests of approximate theories to be applied in more realistic contexts and for tests against idealized computer “experiments”. We do not propose it is an appropriate model for comparison with real experiments, although it becomes more practical for the case of plasma stopping (as contrasted to electronic stopping by bound and free electrons in crystals). Indeed, new experiments and theory for plasma stopping of heavy ions is a primary motivation for this work. Entirely different state conditions are being sampled in attempts to use heavy ions as drivers for inertial confinement fusion, and in experiments directed at heavy-ionpumped X-ray lasers. The present problems in this field are reviewed rather completely in Ref. [l]. A point (structureless) ion of mass m, and charge Z,e is at equilibrium with an electron gas (free electrons in a uniform neutralizing background); subsequently, it is given an initial velocity, 6. The stopping power is defined as the (negative) average kinetic energy change divided by the average path length for a time interval T, Y= E(r)
-{E(7)
_E(O)}/L(T),
= ($rr,vf(r>>s,
* Corresponding
author.
L(r)
(I) =irdt
(u,(t))s,
(2)
where u,, is the ion velocity operator and ( . )B denotes a trace over the Gibbs ensemble for the ion and electron gas, constrained such that the initial ion velocity is 6 (see below). This is perhaps the most direct and specific definition of the stopping function; more familiar forms obtained from linear response, binary scattering, force due to induced potential, can be obtained from Eq. (1) in the appropriate limits. Here, our objective is precision so that there are no ambiguities in the defined quantities or the limiting process considered. For analysis of this problem it is useful first to consider the appropriate dimensionless form. All velocities are scaled relative to the electron thermal velocity, U, = (kgT/me)‘/‘, and the time is scaled to the inverse plasma frequency, oP = (47rn e*/m,)‘/‘; the corresponding length scale becomes the Debye length, A, = (k,T/47rrn e * ) ‘I2 . In these units the dimensionless electron coupling parameter is y= e*/(k,TA,), and the electron degeneracy parameter is 17= A,/& where A, = (fi*/m,k,Z’)‘~* is proportional to the thermal de Broglie wavelength. Finally, the electron/ion mass ratio is denoted by LY= m,/m,. In dimensionless form, the stopping power is a function of many parameters, P*(a, 6, Z, T, y, 77). Clearly, this encompasses complex phenomena in a wide range of different physical conditions. The analysis below identifies criteria to determine the asymptotic behavior for 6-+ 0. In many experiments and approximate theories [2-41, a linear dependence on the velocity is observed for 6 < 1 (in the above units). As will be seen below, this is expected when the mass ratio of the target particle to incident particle is small. However, when this mass ratio is of the order of unity (e.g., protons on He or H) deviations from linearity are observed [s]. For the case of heavy ion stopping by plasmas the former case is relevant, and we restrict attention here to the small mass ratio limit. A direct expansion of P* for finite me describes physics quite
0168-583X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)00234-0
J. W. Dufty, M. Berkocsky /Nucl. Instr. and Meth. in Phys. Res. B 96 (1995) 626-632
unrelated to the stopping problem, since the ion dynamics is diffusive for speeds of the order or smaller than the ion thermal velocity. For a systematic expansion in 6, therefore, it is essential to take first the limit of infinite mass ((Y -+ 0, fixed m,). In this way the ion thermal velocity is driven first to zero, so that the subsequent Taylor series expansion in 6 about 0 still corresponds to speeds above the diffusive limit. A related simplification occurs for the choice of observation interval T. In most approximate theories stopping is related to the force autocorrelation function, whose correlation time is assumed short compared to the observation time T. The latter is then extended formally to infinity. This is justified only in the context of certain approximations (e.g., Born-RPA approximation [2,3]), since this limit applied to the exact theory gives .Y* + 0 (thermalization of the ion by the target). In practice this does not occur because of the thin sample size used. The infinite mass limit avoids this problem by driving the crossover time for diffusion and thermalization to infinity. The main results presented here are (1) an exact asymptotic linear velocity dependence of 9*, and (2) a slope determined by the force autocorrelation function. These results are formally exact for cr + 0, 6 -+ 0, and r-f a, taken in that order. The coefficient of this linear velocity dependence, A@, y, q), applies for arbitrary Z, y, and 7. This linear velocity domain is of current interest for both experiment [6,7] and simulation [%-lo], and provides the controlled basis for a corresponding theoretical investigation of this very rich parameter space. The low velocity Born-RPA approximation results are encompassed by this analysis (see Section 31, but our primary interest is in conditions outside the limitations of this approximation. For example, both simulations and experiments suggest an unexplained Z3/’ dependence of A(Z, y, 7) at strong coupling, yw 1 (i.e., the Born series gives integer powers of Z, nonlinear Vlasov gives Z5/‘, see Ref. [l]). In the next section, the results of the above limit are given and discussed briefly. Their derivation is outlined in the Appendices. In Section 3, the limit of small Z is considered to make contact with the more familiar linear response results. The classical limit is considered in Section 4, to provide an expression suitable for study via molecular dynamics simulation methods. The complication due to the singular ion-electron interaction in the case of positive ions is noted, and a suitable classical model is indicated. Finally, the results are summarized briefly in the last section.
the following it will be understood units described above are used), H=a-‘HO+H,+
yZV,
V=
627
that the dimensionless
CV( UI
I’a-‘,,I), (3)
where H, is the Hamiltonian (kinetic energy only) for the ion, H, is the Hamiltonian for the electrons (including electron-electron interactions and the positive background), and V is the ion-electron interaction potential energy. The functional form of these interactions is not required at this point (e.g., either Coulomb or screened Coulomb). It is necessary to specify the precise meaning of the brackets in Eq. (2), i.e. the meaning of the average. The initial state is an ion with sharply defined velocity in an equilibrium electron gas. The corresponding density matrix (ensemble) for the system is an equilibrium Gibbs state, constrained such that the probability for the ion velocity is peaked at a chosen eigenvalue 8 of the operator ~‘(r, p= i{e-“, Z(6)
6(6-
0,)}/Z(S),
=Tr e +%(rY-
Q).
(4)
The brackets (,} denote an anticommutator, and Tr denotes a trace over the ion and electron Hilbert spaces. It is easily verified that diagonal matrix elements of p with respect to ion velocity states (the probability density for a selected velocity) are proportional to a delta function at 6, as required. The presence of V in Eq. (4) implies that the electron distribution around the ion is initially distorted by the ion in the same way as at equilibrium. This is relevant for real stopping conditions only at sufficiently low velocities and/or after remaining in the target sufficiently long. For the case of bulk stopping in the low velocity limit considered here this is an appropriate representation. The average kinetic energy of the ion at time t is given by Eq. (2) E(t)
= (Y-’ Tr &u:(t).
(5)
During the interval (0,~) the change in kinetic energy is, E(r)
-E(O)
=~‘dt$E(r)
= (ZY/2)JoT where F,(t) = -aIf/&, to all other charges,
dt({ uai( t)> Fi( r)} )o 1
(6)
is the force acting on the ion due
2. Large mass, small velocity limits The system consists of N, electrons in a uniform neutralizing background, and one ion (positive or negative) of charge +Z,e. The Hamiltonian has the form (in all of
Eq. (6) shows the energy change in terms of the rate at which work is done by the electrons. A similar expression for L(T) gives the path length as a straight I%- plus corrections due to the impulse along the path. In the
VI. ELECTRONIC STOPPING POWER
J. W. Dufty, M. Berkovsky / Nucl. lnstr. and Meth. in Phys. Res. B 96 (1995) 626432
623
is the final thermal energy, and Y* -+ 0). The details of the small 6 limit are given in Appendix B with the result,
following, only the limit (Y+ 0 will be considered so that L -+ &. The reduction of the expectation value in Eq. (6) for the large mass limit is described in Appendix A, where the stopping power in this limit is found to be,
P*(
o = 0, -9, Z, 00, Y, 7) + aA(Z,
p’( a=o,
A(Z,
Y, n) = (Zy)‘$
C(t)
= <5W-(t)),.
6, Z, 7, Y, 77)
+ -.(Z~/r)/rdt
Trt,, ~e(*)~(~;
t).
(8)
-Cz
Y, v),
(IO)
dt c(r)> (11)
0
The trace now extends only over the electron Hilbert space, where p,(6) represents the initial state for the electrons subject to an external potential due to a classica ion at the origin moving with velocity 6. This is perhaps a more familiar expression for the stopping function Y: the average force exerted on the particle opposite to the direction of motion [I]. It is determined from Eqs. (A.17) and (A.8) of Appendix A. The time dependent force along the initial direction of motion, F(6; t), has the initial condition Y(6; 0) = 8. F (where & is a unit vector) and evolves according to, J&Y;
t) = 6. %5r(l9;
t) + +?(r),
F(l9;
t)], (9)
where z(r) = H, + V,,(r) is the Hamiltonian for the electrons interacting with a classical ion at r. The vectors r and r9 are not operators, but rather eigenvalues of the corresponding ion position and velocity operators. The general properties of p. and Y(t) will not be considered further here beyond a few observations. The first is that although the infinite ion mass implies a vanishing thermal de Broglie wavelength, p. is not the Gibbs ensemble for electrons in the potential for a classical ion. The reason is that there are contributions of order 776. In most applications, all relevant velocities are determined by the Gibbs density operator to be of the order of the thermal velocity and @+ no. Here, however, the velocity ti is held fixed as (Y+ 0 and residual quantum effects of the ion occur in p,(6). Of course, for the limit 6 + 0 considered below the Gibbs density operator is recovered to leading order. Regarding the equation for F(t), the first term on the right side of Eq. (9) generates the straight line motion for the ion with a constant velocity, as expected in this limit. The second term generates the Heisenberg equations of motion for the electrons and is still valid for arbitrary degeneracy. These two terms do not commute, due to the ion-electron interaction potential so the classical and quantum dynamics are coupled. As discussed in the Introduction, it is now possible to consider the limit of small 6, without the complications of remaining greater than the ion thermal speed. In addition, the limit T-+ ~0 can be taken with the limitation only that the observation time is large compared to the correlation time for the electrons (for finite me it is easily seen from Eq. (6) that the E(T) - E(O) -+ (E), - E(O) where (E),
The coefficient of 6 is simply determined by the time integral of the force autocorrelation function. The brackets ( . . ), now denote a Gibbs density operator for the Hamiltonian 2, and s(t) is determined from the corresponding Heisenberg equations, (FY(
t)), = TrCe,eezYg( = e iXt/qy
F(t)
t)/Trt,,
emdy,
e-i.Tt/7j (12)
This is our primary result.
3. Weak coupling To confirm the above analysis we illustrate the result for the case of weak coupling between the ion and electrons. The weak coupling limit is defined by Zy -=z 1. More precisely, we consider A(Zy)-’ evaluated at Zy = 0 and constant n. In this case the Hamiltonian in Eq. (12) simplifies to X?(r) + H,. The force autocorrelation function in this limit is conveniently represented in terms of a Fourier representation, F(t)
= (2rr)-‘/
n(k,
r)=
z
dkfi(k)n(k,
eiP,r,(r)
t),
,
(13)
a=1
where I’(k) is the Fourier transformed ion-electron interaction, and n(k) is the Fourier transformed electron number density around the ion. Use of this in Eq. (12) gives, A(Z,
Y> n)
=+(ZY)~(~*)-‘/
dk]@(k)12S(k,
where S( k, w) is the equilibrium for the electrons, S(k,
w) = lrn dt e’“‘(n(k, --m
w=O),
dynamic structure factor
t)n( -k)&.
(15)
A more familiar form in terms of the dielectric E(k, w) is obtained from the relationship, S(k,
w)=2~(1-e
-q”)-’
(14)
Im(?(k)E(k,
function
0,)-i, (16)
J. W. Dufty, M. Berkocsky / Nucl. Ins@. and Meth. in Phys. Res. B 96 (1995) 626-632
?((k)
8 is Fourier transformed pair potential, Im denotes imagiSubstitution of (16) into (14) gives,
nary
A(&
~3
77)
= (ZY)~(~IT~)-‘&--
X
[
;Irn
~(k,
dkk4f(k)l+,
w)
O)/-’
. 1OJ=O
(17)
This is the small velocity limit for the stopping power in the Born approximation [2,3]. The dielectric function in Eq. (17) is still exact for the electron subsystem. More realistically, the condition for weak coupling of the ion with electrons, Zy < 1, also implies weak electron-electron coupling, y < 1, in which case ~(k, w) is given by the usual random phase approximation. The most interesting and relevant case is the fully coupled electron and ion conditions, 1 < Z < 100 and y _ 1. This represents strong coupling for all charges and is outside the Born-RPA limit. Detailed many-body analysis is required to describe the collective dynamics of the electron subsystem at strong coupling, and some improvements within the Born approximation are described in Ref. [3]. Approximate corrections to the Born approximation, e.g. the non-linear distortion of the charge distribution near the ion, are considered in Ref. [1] (nonlinear Vlasov equation) and Ref. [4] (density functional methods).
4. Classical limit Our primary results from Eqs. (10) and (11) apply without restriction to the state conditions of the electron gas - both weak and strong degeneracy limits are included. In this section we consider the semi-classical limit of Eq. (11) to obtain an expression that allows the study of the strongly coupled conditions via molecular dynamics simulation. For negative ions this limit is straightforward to obtain, with the force autocorrelation function becoming a classical phase space average over electron degrees of freedom, and the dynamics of s(t) determined from Hamilton’s equations of motion. Then standard methods of molecular dynamics simulation can be applied directly in this case to provide benchmark tests of theoretical approximation schemes [ll]. In this way a controlled assessment of these schemes is possible before proceeding to more realistic systems of experimental interest. Strong coupling conditions place no restrictions on the molecular dynamics, and in fact the simulations become more accurate at stronger coupling. The more interesting case of positive ions poses inherent difficulties, due to the singular nature of the attractive
629
ion-electron interaction. The classical limit of Eq. (11) does not exist for Coulomb interactions. However, the expression is certainly well-behaved for n ==K1 where the electron system is clearly in its classical limit. Under these conditions, the quantum features of the problem persist only within a sphere of radius _ n around the ion (i.e inside a thermal de Broglie wavelength) [12]. Within this sphere the Heisenberg uncertainty principle prevents attraction of the electron to the origin with infinite energy. To obtain a practical classical expression corresponding to these conditions, the Coulomb potential for the ion-electron interaction, Vii,, is modified at these short distances. Of course, there is no unique prescription for this modification but several choices have been used in related contexts. One such pair potential is [13],
V;(r)
= -
?(I - emr/v).
Alternatively, it can be required that the charge distribution around the ion calculated classically from Vk and quantum mechanically from Vi, agree in the respective linear response approximation. This leads to the result,
where n, is the average electron density and x0(k) is the static polarizability for the free electron gas. The two choices (Eqs. (18) and (19)) are quite similar, and it is largely a matter of taste and convenience as to which is used. A classical limit is now associated with Eq. (11) by replacing the ion-electron potential with Vi: or Vie and setting 77= 0, holding the potential fixed. The result is a purely classical expression as in the case of negative ions described above, but still parameterized by 77 through the effective potential. This system can now be studied via molecular dynamics simulation in the usual way. One new aspect of the simulations and theory in this classical limit for positive ions is bound or long-lived orbiting trajectories. This is only a technical problem in the simulations, but can provide significant difficulties in the theory. To illustrate the latter, consider the simplest calculation of the initial condition for the force autocorrelation function in the classical limit. A straightforward calculation gives [14],
C(t=O)
= (6.1”)-*im
dkk41i,‘,(k)&(k).
Here i,(k) is the Fourier transform of gi,(r> - 1, where gi,(r) is the radial distribution function for electrons at a distance r from the ion, using the pair potential y:. For negative ions the distribution function can be calculated for all fluid phase values of the coupling y to a very good approximation from the HNC (hypernetted chain) integral
VI. ELECTRONIC STOPPING POWER
630
J. W. Du&, M. Berkousky / Nucl. Instr. and Meth. in Phys. Res. B 96 (1995) 626-632
equation [IS]. These are approximate equations that have been applied and tested extensively for the one and two components plasmas for both thermodynamics and structural properties. However, for positive ions the quality of this approximation degenerates with increasing values of Zy/v (N pair potential energy for r - 7). In fact, we find from numerical studies that no solutions to the HNC equations exist for Z-r/v > 1. This may be due to a failure of the HNC approximation when bound states appear. A more sophisticated theory is required for this portion of the parameter space.
velocity are denoted by ( r) and 1u), respectively. Also, to simplify the notation let X(t) = (1/2)(u&), Fi(t)). Then the expectation value in Eq. (6) can be written, Tr p(O)X(r) = Z-‘(6)
Trt,tTrtij i{e-“,
=Z-‘(8)0-1(2n~7(y)-3 X
/
a(#--
v,,)}X(t)
Re Trt,)
dr (rlemH(8)(81X(t)lr).
(A.1)
Here 0 is the volume of the system and Re denotes the real part. The expression for Z(8) in Eq. (4) also becomes, 5. Summary Z(9) The analysis presented here has isolated a precise expression for theoretical study of electronic stopping at low velocities. The result in Eq. (10) is exact in the limit of small velocities and therefore serves as a suitable starting point for application of fundamental (e.g., quantum Monte Carlo) as well as approximate many-body techniques. The simplifications were obtained first by the infinite ion mass limit; this in turn made possible the expansion to first order in the initial ion velocity. The coefficient A(Z, y, 7) is simply proportional to the area under the force autocorrelation function; the infinite mass limit plays an essential role in establishing this identification. The derivation of Eq. (11) does not limit the values of Z, y, or n so the degree of coupling between ions and electrons is arbitrary, as well as the degree of electron degeneracy. To guide and test theoretical methods it is useful to have exact “experiments” via molecular dynamics simulations. The appropriate way to assign a classical limit for such simulations has been noted in Section 4, together with some interesting theoretical challenges to be faced.
This research was supported by National Science Foundation grant PHY 9312723.
Appendix A. Large mass limit In terms of the dimensionless variables introduced in Section 1, the thermal de Broglie wavelength of the ion is (~77. Thus for fixed temperature and electron mass, it is expected that the ion degrees of freedom in Eqs. (6) and (7) can be treated as classical variables in the large mass limit (a -+ 0). To identify this limit the trace in Eq. (6) is performed over product states for the ion and electron degrees of freedom. Eigenstates for the ion position and
Trte, ( dr (rj eeH IS>(S
1r). (A.21
The Hamiltonian Eq. (3) has a singular dependence on cr through the ion kinetic energy. This can be removed in Eq. (A.l) by defining the function Y(r, 6, x) by, (r~e-XHI~)~(r~~)e-xiY*‘2aY(r,6,x).
(A.3)
The matrix element required in Eqs. (A.l) and (A.2) is determined from Y(r, 6, x = 1) = Y(r, 6). This form is suggested by the expected classical limit, for which the distribution function has a Maxwellian distribution of velocities. For similar reasons we define, (6 I X(t) I r) = (6 I r)X(r,
6; t).
(A.4)
Then Eq. (A.1) becomes, Tr p(O)X(r) =2-l(6)
x(6) Acknowledgements
= W’(2a77~))~
Re/
dr Tr(,)Y(r,
= Re / dr Trt,,Y(r,
Cf)X(r,
6; f),
8).
(AS)
(A.6)
The quantities Y and X(t) are functions of the ion position and velocity eigenvalues (r, 6), but are still operators in the electron Hilbert space. This representation is still exact. An equation for Y is obtained by differentiating Eq. (A.3) with respect to x,
a - - $78. ax ( = 0,
V, + H, +ZyVei - $q2
Vr2 Y(r,
1
9, x) (A.7)
with the boundary condition Y(r, 6, 0) = 1. Consequently, the desired result for a = 0 is determined from,
a - --$#. ax
q+H,
+ZyV,,(r)
.I.W. Dufry,M. Be&o&y/
with Ya(r, 6, 0) = 1. Next consider (19 1 X(t)
1 r) = (6 I eiH’/“X(0)
Nucl. Instr.andMeth. in Phys. Res. B 96 (I 995) 626-632
(6 1X(t) 1r>,
e-iH’/q
I r).
cated in Eq. (A.1.5) can be taken explicitly, result Tr p(O)X(r)
6; t) =L?X(r,
L?X(r,
6; t)
6; t),
(A.lO)
+ 6 Trt,) pa(*)g(a;
where 9(6, PO(a)
6; t) + i[He,
+ +Z-y
X(r,
- scar’)
= [ Ya(0, 6) + Y,‘(O, a>]
-, -(Z-~/T)/~
( XX(r,
v; t)
-X(r,
6; t)&(r)
.
(A.ll)
1 The initial condition,
X(r,
6; 01, is calculated
directly to
be, X(r,
6; 0) = 6.F(r)
- (in(r/2)V,*F.
is
6) + Yo*(0,
~11.
(A.17)
The stopping power in the large mass limit is given by Eqs. (l), (6), and (A.16),
6; t)]
dr’ du ei’(‘-“)&(r
(27re3/
(~.16)
f),
t) = F(O, 6, t) and the initial ensemble
/TrI,I[Y,(O,
= 19. yX(r,
leading to the
(A.9)
Differentiation with respect to time leads, together with the definition in Eq. (A.4) to the equation, iX(r,
631
(A.12)
The (Y= 0 limit is easily identified now from Eqs. (A.lO)(A.12),
0
dr TrceI pa(-9)9(
(A.18)
5; r).
Appendix B. Small 6 limit The small obtained from and (A.13) to which becomes yo(r,
6)
+
6 limit for the stopping power can be expansions of the solutions to Eqs. (A.8) first order in 6. Consider first Yo(r, 6) to this order,
e-x(r)
’
+ $7
e(x-1)x(r)8.
dx
V,
eexzcr),
/0
(B.1)
(A.13) = H, + Vei( r),
Z’(r)
(B.2)
The derivative in Eq. (B.l) is given by _9. y e-xx”(r) %‘(r)=He+Zzy&,
(A.14)
with the initial condition X,&r, 6, 0) = 6. F(r). The Hamiltonian 2?(r) is that for the electron system interacting with a point source at the position r, where the latter is no longer an operator. The expectation value in Eq. (AS) becomes in this limit,
=.-.
+ 8x;‘(6) Xy(r,
dx’ e(+x)x’(r)[+
vrz(r)]
e-x’r(r), (B.3)
To linear order in 6, therefore, ~~(6)
Tr p(O)X(t)
* /0
--f p,{l + order a’},
p,(6)
becomes,
pe = e -z(o) / Tr lel e-“(o)
Re / dr Tree, Yo(r, S) 6; f),
(B.4)
(A.15)
where ST = 8. F is the component of the force along the initial direction of motion. The time dependence of F(t) is determined from Eq. (A.15). To simplify further, note that the trace over electrons leads to a result that is independent of r. This is easily verified by introducing a unitary transformation under the trace to generate an overall translation of the electron subsystem relative to r. Consequently, the integral over r can be performed to yield a volume factor and the initial position of the ion can be chosen to be at the origin. Finally, the real part indi-
(B.5) where use has been made of the fact that the normalization function, z(6) in Eq. (A.15) is 2(O) + order I?‘. The expansion of S(r, 6, t) can be done in an analogous fashion from Eq. (A.13), F(r,
6, t) =9(r, +
0, t)
/0
’ dt’
ei~(rxr-r’)/rl[+
&F(,.,
0,
f)]
Xe-iX(rXt-t’)/q (B.6)
VI. ELECTRONIC STOPPING POWER
632
J. W. Dufry, M. Berkousky/Nucl.
The correlation order,
function,
Tree) 00(fi)F(6;
Eq. (A.17),
t) + Tree) PX(O)
VF(r,
becomes
Instr. and Meth. in Phys. Res. B 96 (1995) 626-632
to this
+ I,’ dt’ Tree) ~,a
0, t’),
(B.7)
The first term vanishes from spherical symmetry; the last term can be simplified by integrating the gradient by parts and using Eq. (B.3), Trc,) p,$.
VF(0;
= -ZYi’
Y’*((Y=O,
+ 8(Zy)‘flm
= B(Zy)‘$
-ivx)F(r,
0, t’),
it
--oc
dt/o’ dx C(t + ivx)
dt C(t). --CL
(B.14) in the text.
References (B.8)
the correlation
in t. Consequently
6, Z, 03, Y, 77)
This is the result discussed
t’)
dx Tree, p,F(O;
Consequently,
time integrand is real and symmetric has the equivalent form,
function is linear in 6,
[l] T. Peter and J. Meyer ter Vehn, Phys. Rev. A 43 (1991) 201.5.
La N.R. Arista and W. Brandt, Phys. Rev. A 23 (1981) 1898. Trte, p,,(8),8(6;
t) + -Zy6
/0
’ dt’
/0
’ dx C(t’+
iTx), (B-9)
where C(t) is the force autocorrelation C(t)
function, (B.lO)
= TrteI QWt).
The stopping power in this limit is obtained from Eq. (MS), Y(cr=O,
6, Z, 7, Y, 77)
-+ u(ZY)2kTdt
1-
i
f
i’
dx C(t + i?x).
(B.ll)
1
This is the most general result for the small velocity limit. Assume now that the correlation time for C(t) is much smaller than T. Then the limit ~4 00 is justified and Eq. (B.12) simplifies to, y”( cr = 0, 6, Z, 00, Y, ?1) -+ ALLThe correlation
dtl’
dx C(t + ivx).
function has the following
(B.12) symmetries:
C*(t+iqx)=C(-t+ivx)=C(t+iv[l-xl), (B.13) The last equality can be used in Eq. (B.121, together with a change of variables in the x-integration, to show that the
131 S. Ichimaru and D. Utsumi, Phys. Rev. B 24 (1981) 32; C. Deutsch, in Strongly Coupled Plasma Physics, eds. F. Rogers and H. de Witt (Plenum, New York, 1986). [41 P.M. Echenique, A. Arnau and M. Penalba, Nucl. Instr. and Meth. B 56 (1991) 345. [51 R. Golser and D. Semrad, Phys. Rev. Lett. 66 (1991) 1831; M. Kimura, Phys. Rev. A 47 (1993) 2393. b1 M. Steck, K. Beckert, F. Bosch., H. Eickhoff, B. Franzke, 0. Klepper, R. Mooshammer, F. Nolden, H. Reich and P. Spadtke, Proc. Beam Cooling and Related Topics, Montreux, Switzerland, October 1993 (to be published). [71 A. Wolf, C. Ellert, M. Grieser, D. Habs, B. Hchadel, R. Repnow and D. Schwalm, Proc. Beam Cooling and Related Topics, Montreux, Switzerland, October 1993 (to be published). [81G. Zwicknagel, D. Klakow, P.-G. Reinhard and C. Toepffer, Contrib. Plasma Phys. 33 (1993) 395. [91 G. Zwicknagel, C. Toepffer and P.-G. Reinhard, Nuovo Cimento A 107 (1993). [lOI G. Zwicknagel, C. Toepffer and P.-G. Reinhard, Proc. Beam Cooling and Related Topics, Montreux, Switzerland, October 1993 (to be published). ml J.-P. Hansen and McDonald, Physics of Simple Liquids (Academic Press, New York, 1976) p. 45. r121 T. Morita, Prog. Theor. Phys. 22 (1959) 757. 1131H. Minoo, M. Gombert and C. Deutsch, Phys. Rev. A 23 (1981) 924. 1141J.-P. Hansen and L. Sjogren, Phys. Fluids 25 (1982) 617. 1151J.-P. Hansen and McDonald, Physics of Simple Liquids (Academic Press, New York, 1976) p. 112.
and Methods in Physics Research B 96 (1995) 626-632
Beam Interactbn~ with YIlterials& Atoms ELSEVIER
Electronic stopping of ions in the low velocity limit James W. Dufty
*, Mikhail Berkovsky
Department of Physics, University of Florida, Gadnesuille, FL 32611, USA
Abstract The change in kinetic energy of an ion moving in an equilibrium electron gas is considered as a model for electronic stopping power. The exact small velocity limit is obtained for an ion of infinite mass. It is shown to be linear in the ion speed with a coefficient determined from the autocorrelation function for the force on the ion at rest in the equilibrium electron gas. The result has no limitations with respect to the ion charge, the electron coupling parameter, and the electron degeneracy parameter. Therefore it provides a firm basis for testing various theoretical approximations. Some comments are offered on the semiclassical limit, to provide a model appropriate for study via molecular dynamics simulation.
1. Introduction Stopping power is a measure of a particle’s energy degradation on passing through a target. In general it can depend on the geometry, surface effects, and internal structure of the target’s constituent particles. Here, we idealize the problem to consider only free electronic stopping in bulk materials. The electron gas has been the historical prototype for electronic stopping, and we consider it a valuable example for “benchmark” tests of approximate theories to be applied in more realistic contexts and for tests against idealized computer “experiments”. We do not propose it is an appropriate model for comparison with real experiments, although it becomes more practical for the case of plasma stopping (as contrasted to electronic stopping by bound and free electrons in crystals). Indeed, new experiments and theory for plasma stopping of heavy ions is a primary motivation for this work. Entirely different state conditions are being sampled in attempts to use heavy ions as drivers for inertial confinement fusion, and in experiments directed at heavy-ionpumped X-ray lasers. The present problems in this field are reviewed rather completely in Ref. [l]. A point (structureless) ion of mass m, and charge Z,e is at equilibrium with an electron gas (free electrons in a uniform neutralizing background); subsequently, it is given an initial velocity, 6. The stopping power is defined as the (negative) average kinetic energy change divided by the average path length for a time interval T, Y= E(r)
-{E(7)
_E(O)}/L(T),
= ($rr,vf(r>>s,
* Corresponding
author.
L(r)
(I) =irdt
(u,(t))s,
(2)
where u,, is the ion velocity operator and ( . )B denotes a trace over the Gibbs ensemble for the ion and electron gas, constrained such that the initial ion velocity is 6 (see below). This is perhaps the most direct and specific definition of the stopping function; more familiar forms obtained from linear response, binary scattering, force due to induced potential, can be obtained from Eq. (1) in the appropriate limits. Here, our objective is precision so that there are no ambiguities in the defined quantities or the limiting process considered. For analysis of this problem it is useful first to consider the appropriate dimensionless form. All velocities are scaled relative to the electron thermal velocity, U, = (kgT/me)‘/‘, and the time is scaled to the inverse plasma frequency, oP = (47rn e*/m,)‘/‘; the corresponding length scale becomes the Debye length, A, = (k,T/47rrn e * ) ‘I2 . In these units the dimensionless electron coupling parameter is y= e*/(k,TA,), and the electron degeneracy parameter is 17= A,/& where A, = (fi*/m,k,Z’)‘~* is proportional to the thermal de Broglie wavelength. Finally, the electron/ion mass ratio is denoted by LY= m,/m,. In dimensionless form, the stopping power is a function of many parameters, P*(a, 6, Z, T, y, 77). Clearly, this encompasses complex phenomena in a wide range of different physical conditions. The analysis below identifies criteria to determine the asymptotic behavior for 6-+ 0. In many experiments and approximate theories [2-41, a linear dependence on the velocity is observed for 6 < 1 (in the above units). As will be seen below, this is expected when the mass ratio of the target particle to incident particle is small. However, when this mass ratio is of the order of unity (e.g., protons on He or H) deviations from linearity are observed [s]. For the case of heavy ion stopping by plasmas the former case is relevant, and we restrict attention here to the small mass ratio limit. A direct expansion of P* for finite me describes physics quite
0168-583X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)00234-0
J. W. Dufty, M. Berkocsky /Nucl. Instr. and Meth. in Phys. Res. B 96 (1995) 626-632
unrelated to the stopping problem, since the ion dynamics is diffusive for speeds of the order or smaller than the ion thermal velocity. For a systematic expansion in 6, therefore, it is essential to take first the limit of infinite mass ((Y -+ 0, fixed m,). In this way the ion thermal velocity is driven first to zero, so that the subsequent Taylor series expansion in 6 about 0 still corresponds to speeds above the diffusive limit. A related simplification occurs for the choice of observation interval T. In most approximate theories stopping is related to the force autocorrelation function, whose correlation time is assumed short compared to the observation time T. The latter is then extended formally to infinity. This is justified only in the context of certain approximations (e.g., Born-RPA approximation [2,3]), since this limit applied to the exact theory gives .Y* + 0 (thermalization of the ion by the target). In practice this does not occur because of the thin sample size used. The infinite mass limit avoids this problem by driving the crossover time for diffusion and thermalization to infinity. The main results presented here are (1) an exact asymptotic linear velocity dependence of 9*, and (2) a slope determined by the force autocorrelation function. These results are formally exact for cr + 0, 6 -+ 0, and r-f a, taken in that order. The coefficient of this linear velocity dependence, A@, y, q), applies for arbitrary Z, y, and 7. This linear velocity domain is of current interest for both experiment [6,7] and simulation [%-lo], and provides the controlled basis for a corresponding theoretical investigation of this very rich parameter space. The low velocity Born-RPA approximation results are encompassed by this analysis (see Section 31, but our primary interest is in conditions outside the limitations of this approximation. For example, both simulations and experiments suggest an unexplained Z3/’ dependence of A(Z, y, 7) at strong coupling, yw 1 (i.e., the Born series gives integer powers of Z, nonlinear Vlasov gives Z5/‘, see Ref. [l]). In the next section, the results of the above limit are given and discussed briefly. Their derivation is outlined in the Appendices. In Section 3, the limit of small Z is considered to make contact with the more familiar linear response results. The classical limit is considered in Section 4, to provide an expression suitable for study via molecular dynamics simulation methods. The complication due to the singular ion-electron interaction in the case of positive ions is noted, and a suitable classical model is indicated. Finally, the results are summarized briefly in the last section.
the following it will be understood units described above are used), H=a-‘HO+H,+
yZV,
V=
627
that the dimensionless
CV( UI
I’a-‘,,I), (3)
where H, is the Hamiltonian (kinetic energy only) for the ion, H, is the Hamiltonian for the electrons (including electron-electron interactions and the positive background), and V is the ion-electron interaction potential energy. The functional form of these interactions is not required at this point (e.g., either Coulomb or screened Coulomb). It is necessary to specify the precise meaning of the brackets in Eq. (2), i.e. the meaning of the average. The initial state is an ion with sharply defined velocity in an equilibrium electron gas. The corresponding density matrix (ensemble) for the system is an equilibrium Gibbs state, constrained such that the probability for the ion velocity is peaked at a chosen eigenvalue 8 of the operator ~‘(r, p= i{e-“, Z(6)
6(6-
0,)}/Z(S),
=Tr e +%(rY-
Q).
(4)
The brackets (,} denote an anticommutator, and Tr denotes a trace over the ion and electron Hilbert spaces. It is easily verified that diagonal matrix elements of p with respect to ion velocity states (the probability density for a selected velocity) are proportional to a delta function at 6, as required. The presence of V in Eq. (4) implies that the electron distribution around the ion is initially distorted by the ion in the same way as at equilibrium. This is relevant for real stopping conditions only at sufficiently low velocities and/or after remaining in the target sufficiently long. For the case of bulk stopping in the low velocity limit considered here this is an appropriate representation. The average kinetic energy of the ion at time t is given by Eq. (2) E(t)
= (Y-’ Tr &u:(t).
(5)
During the interval (0,~) the change in kinetic energy is, E(r)
-E(O)
=~‘dt$E(r)
= (ZY/2)JoT where F,(t) = -aIf/&, to all other charges,
dt({ uai( t)> Fi( r)} )o 1
(6)
is the force acting on the ion due
2. Large mass, small velocity limits The system consists of N, electrons in a uniform neutralizing background, and one ion (positive or negative) of charge +Z,e. The Hamiltonian has the form (in all of
Eq. (6) shows the energy change in terms of the rate at which work is done by the electrons. A similar expression for L(T) gives the path length as a straight I%- plus corrections due to the impulse along the path. In the
VI. ELECTRONIC STOPPING POWER
J. W. Dufty, M. Berkovsky / Nucl. lnstr. and Meth. in Phys. Res. B 96 (1995) 626432
623
is the final thermal energy, and Y* -+ 0). The details of the small 6 limit are given in Appendix B with the result,
following, only the limit (Y+ 0 will be considered so that L -+ &. The reduction of the expectation value in Eq. (6) for the large mass limit is described in Appendix A, where the stopping power in this limit is found to be,
P*(
o = 0, -9, Z, 00, Y, 7) + aA(Z,
p’( a=o,
A(Z,
Y, n) = (Zy)‘$
C(t)
= <5W-(t)),.
6, Z, 7, Y, 77)
+ -.(Z~/r)/rdt
Trt,, ~e(*)~(~;
t).
(8)
-Cz
Y, v),
(IO)
dt c(r)> (11)
0
The trace now extends only over the electron Hilbert space, where p,(6) represents the initial state for the electrons subject to an external potential due to a classica ion at the origin moving with velocity 6. This is perhaps a more familiar expression for the stopping function Y: the average force exerted on the particle opposite to the direction of motion [I]. It is determined from Eqs. (A.17) and (A.8) of Appendix A. The time dependent force along the initial direction of motion, F(6; t), has the initial condition Y(6; 0) = 8. F (where & is a unit vector) and evolves according to, J&Y;
t) = 6. %5r(l9;
t) + +?(r),
F(l9;
t)], (9)
where z(r) = H, + V,,(r) is the Hamiltonian for the electrons interacting with a classical ion at r. The vectors r and r9 are not operators, but rather eigenvalues of the corresponding ion position and velocity operators. The general properties of p. and Y(t) will not be considered further here beyond a few observations. The first is that although the infinite ion mass implies a vanishing thermal de Broglie wavelength, p. is not the Gibbs ensemble for electrons in the potential for a classical ion. The reason is that there are contributions of order 776. In most applications, all relevant velocities are determined by the Gibbs density operator to be of the order of the thermal velocity and @+ no. Here, however, the velocity ti is held fixed as (Y+ 0 and residual quantum effects of the ion occur in p,(6). Of course, for the limit 6 + 0 considered below the Gibbs density operator is recovered to leading order. Regarding the equation for F(t), the first term on the right side of Eq. (9) generates the straight line motion for the ion with a constant velocity, as expected in this limit. The second term generates the Heisenberg equations of motion for the electrons and is still valid for arbitrary degeneracy. These two terms do not commute, due to the ion-electron interaction potential so the classical and quantum dynamics are coupled. As discussed in the Introduction, it is now possible to consider the limit of small 6, without the complications of remaining greater than the ion thermal speed. In addition, the limit T-+ ~0 can be taken with the limitation only that the observation time is large compared to the correlation time for the electrons (for finite me it is easily seen from Eq. (6) that the E(T) - E(O) -+ (E), - E(O) where (E),
The coefficient of 6 is simply determined by the time integral of the force autocorrelation function. The brackets ( . . ), now denote a Gibbs density operator for the Hamiltonian 2, and s(t) is determined from the corresponding Heisenberg equations, (FY(
t)), = TrCe,eezYg( = e iXt/qy
F(t)
t)/Trt,,
emdy,
e-i.Tt/7j (12)
This is our primary result.
3. Weak coupling To confirm the above analysis we illustrate the result for the case of weak coupling between the ion and electrons. The weak coupling limit is defined by Zy -=z 1. More precisely, we consider A(Zy)-’ evaluated at Zy = 0 and constant n. In this case the Hamiltonian in Eq. (12) simplifies to X?(r) + H,. The force autocorrelation function in this limit is conveniently represented in terms of a Fourier representation, F(t)
= (2rr)-‘/
n(k,
r)=
z
dkfi(k)n(k,
eiP,r,(r)
t),
,
(13)
a=1
where I’(k) is the Fourier transformed ion-electron interaction, and n(k) is the Fourier transformed electron number density around the ion. Use of this in Eq. (12) gives, A(Z,
Y> n)
=+(ZY)~(~*)-‘/
dk]@(k)12S(k,
where S( k, w) is the equilibrium for the electrons, S(k,
w) = lrn dt e’“‘(n(k, --m
w=O),
dynamic structure factor
t)n( -k)&.
(15)
A more familiar form in terms of the dielectric E(k, w) is obtained from the relationship, S(k,
w)=2~(1-e
-q”)-’
(14)
Im(?(k)E(k,
function
0,)-i, (16)
J. W. Dufty, M. Berkocsky / Nucl. Ins@. and Meth. in Phys. Res. B 96 (1995) 626-632
?((k)
8 is Fourier transformed pair potential, Im denotes imagiSubstitution of (16) into (14) gives,
nary
A(&
~3
77)
= (ZY)~(~IT~)-‘&--
X
[
;Irn
~(k,
dkk4f(k)l+,
w)
O)/-’
. 1OJ=O
(17)
This is the small velocity limit for the stopping power in the Born approximation [2,3]. The dielectric function in Eq. (17) is still exact for the electron subsystem. More realistically, the condition for weak coupling of the ion with electrons, Zy < 1, also implies weak electron-electron coupling, y < 1, in which case ~(k, w) is given by the usual random phase approximation. The most interesting and relevant case is the fully coupled electron and ion conditions, 1 < Z < 100 and y _ 1. This represents strong coupling for all charges and is outside the Born-RPA limit. Detailed many-body analysis is required to describe the collective dynamics of the electron subsystem at strong coupling, and some improvements within the Born approximation are described in Ref. [3]. Approximate corrections to the Born approximation, e.g. the non-linear distortion of the charge distribution near the ion, are considered in Ref. [1] (nonlinear Vlasov equation) and Ref. [4] (density functional methods).
4. Classical limit Our primary results from Eqs. (10) and (11) apply without restriction to the state conditions of the electron gas - both weak and strong degeneracy limits are included. In this section we consider the semi-classical limit of Eq. (11) to obtain an expression that allows the study of the strongly coupled conditions via molecular dynamics simulation. For negative ions this limit is straightforward to obtain, with the force autocorrelation function becoming a classical phase space average over electron degrees of freedom, and the dynamics of s(t) determined from Hamilton’s equations of motion. Then standard methods of molecular dynamics simulation can be applied directly in this case to provide benchmark tests of theoretical approximation schemes [ll]. In this way a controlled assessment of these schemes is possible before proceeding to more realistic systems of experimental interest. Strong coupling conditions place no restrictions on the molecular dynamics, and in fact the simulations become more accurate at stronger coupling. The more interesting case of positive ions poses inherent difficulties, due to the singular nature of the attractive
629
ion-electron interaction. The classical limit of Eq. (11) does not exist for Coulomb interactions. However, the expression is certainly well-behaved for n ==K1 where the electron system is clearly in its classical limit. Under these conditions, the quantum features of the problem persist only within a sphere of radius _ n around the ion (i.e inside a thermal de Broglie wavelength) [12]. Within this sphere the Heisenberg uncertainty principle prevents attraction of the electron to the origin with infinite energy. To obtain a practical classical expression corresponding to these conditions, the Coulomb potential for the ion-electron interaction, Vii,, is modified at these short distances. Of course, there is no unique prescription for this modification but several choices have been used in related contexts. One such pair potential is [13],
V;(r)
= -
?(I - emr/v).
Alternatively, it can be required that the charge distribution around the ion calculated classically from Vk and quantum mechanically from Vi, agree in the respective linear response approximation. This leads to the result,
where n, is the average electron density and x0(k) is the static polarizability for the free electron gas. The two choices (Eqs. (18) and (19)) are quite similar, and it is largely a matter of taste and convenience as to which is used. A classical limit is now associated with Eq. (11) by replacing the ion-electron potential with Vi: or Vie and setting 77= 0, holding the potential fixed. The result is a purely classical expression as in the case of negative ions described above, but still parameterized by 77 through the effective potential. This system can now be studied via molecular dynamics simulation in the usual way. One new aspect of the simulations and theory in this classical limit for positive ions is bound or long-lived orbiting trajectories. This is only a technical problem in the simulations, but can provide significant difficulties in the theory. To illustrate the latter, consider the simplest calculation of the initial condition for the force autocorrelation function in the classical limit. A straightforward calculation gives [14],
C(t=O)
= (6.1”)-*im
dkk41i,‘,(k)&(k).
Here i,(k) is the Fourier transform of gi,(r> - 1, where gi,(r) is the radial distribution function for electrons at a distance r from the ion, using the pair potential y:. For negative ions the distribution function can be calculated for all fluid phase values of the coupling y to a very good approximation from the HNC (hypernetted chain) integral
VI. ELECTRONIC STOPPING POWER
630
J. W. Du&, M. Berkousky / Nucl. Instr. and Meth. in Phys. Res. B 96 (1995) 626-632
equation [IS]. These are approximate equations that have been applied and tested extensively for the one and two components plasmas for both thermodynamics and structural properties. However, for positive ions the quality of this approximation degenerates with increasing values of Zy/v (N pair potential energy for r - 7). In fact, we find from numerical studies that no solutions to the HNC equations exist for Z-r/v > 1. This may be due to a failure of the HNC approximation when bound states appear. A more sophisticated theory is required for this portion of the parameter space.
velocity are denoted by ( r) and 1u), respectively. Also, to simplify the notation let X(t) = (1/2)(u&), Fi(t)). Then the expectation value in Eq. (6) can be written, Tr p(O)X(r) = Z-‘(6)
Trt,tTrtij i{e-“,
=Z-‘(8)0-1(2n~7(y)-3 X
/
a(#--
v,,)}X(t)
Re Trt,)
dr (rlemH(8)(81X(t)lr).
(A.1)
Here 0 is the volume of the system and Re denotes the real part. The expression for Z(8) in Eq. (4) also becomes, 5. Summary Z(9) The analysis presented here has isolated a precise expression for theoretical study of electronic stopping at low velocities. The result in Eq. (10) is exact in the limit of small velocities and therefore serves as a suitable starting point for application of fundamental (e.g., quantum Monte Carlo) as well as approximate many-body techniques. The simplifications were obtained first by the infinite ion mass limit; this in turn made possible the expansion to first order in the initial ion velocity. The coefficient A(Z, y, 7) is simply proportional to the area under the force autocorrelation function; the infinite mass limit plays an essential role in establishing this identification. The derivation of Eq. (11) does not limit the values of Z, y, or n so the degree of coupling between ions and electrons is arbitrary, as well as the degree of electron degeneracy. To guide and test theoretical methods it is useful to have exact “experiments” via molecular dynamics simulations. The appropriate way to assign a classical limit for such simulations has been noted in Section 4, together with some interesting theoretical challenges to be faced.
This research was supported by National Science Foundation grant PHY 9312723.
Appendix A. Large mass limit In terms of the dimensionless variables introduced in Section 1, the thermal de Broglie wavelength of the ion is (~77. Thus for fixed temperature and electron mass, it is expected that the ion degrees of freedom in Eqs. (6) and (7) can be treated as classical variables in the large mass limit (a -+ 0). To identify this limit the trace in Eq. (6) is performed over product states for the ion and electron degrees of freedom. Eigenstates for the ion position and
Trte, ( dr (rj eeH IS>(S
1r). (A.21
The Hamiltonian Eq. (3) has a singular dependence on cr through the ion kinetic energy. This can be removed in Eq. (A.l) by defining the function Y(r, 6, x) by, (r~e-XHI~)~(r~~)e-xiY*‘2aY(r,6,x).
(A.3)
The matrix element required in Eqs. (A.l) and (A.2) is determined from Y(r, 6, x = 1) = Y(r, 6). This form is suggested by the expected classical limit, for which the distribution function has a Maxwellian distribution of velocities. For similar reasons we define, (6 I X(t) I r) = (6 I r)X(r,
6; t).
(A.4)
Then Eq. (A.1) becomes, Tr p(O)X(r) =2-l(6)
x(6) Acknowledgements
= W’(2a77~))~
Re/
dr Tr(,)Y(r,
= Re / dr Trt,,Y(r,
Cf)X(r,
6; f),
8).
(AS)
(A.6)
The quantities Y and X(t) are functions of the ion position and velocity eigenvalues (r, 6), but are still operators in the electron Hilbert space. This representation is still exact. An equation for Y is obtained by differentiating Eq. (A.3) with respect to x,
a - - $78. ax ( = 0,
V, + H, +ZyVei - $q2
Vr2 Y(r,
1
9, x) (A.7)
with the boundary condition Y(r, 6, 0) = 1. Consequently, the desired result for a = 0 is determined from,
a - --$#. ax
q+H,
+ZyV,,(r)
.I.W. Dufry,M. Be&o&y/
with Ya(r, 6, 0) = 1. Next consider (19 1 X(t)
1 r) = (6 I eiH’/“X(0)
Nucl. Instr.andMeth. in Phys. Res. B 96 (I 995) 626-632
(6 1X(t) 1r>,
e-iH’/q
I r).
cated in Eq. (A.1.5) can be taken explicitly, result Tr p(O)X(r)
6; t) =L?X(r,
L?X(r,
6; t)
6; t),
(A.lO)
+ 6 Trt,) pa(*)g(a;
where 9(6, PO(a)
6; t) + i[He,
+ +Z-y
X(r,
- scar’)
= [ Ya(0, 6) + Y,‘(O, a>]
-, -(Z-~/T)/~
( XX(r,
v; t)
-X(r,
6; t)&(r)
.
(A.ll)
1 The initial condition,
X(r,
6; 01, is calculated
directly to
be, X(r,
6; 0) = 6.F(r)
- (in(r/2)V,*F.
is
6) + Yo*(0,
~11.
(A.17)
The stopping power in the large mass limit is given by Eqs. (l), (6), and (A.16),
6; t)]
dr’ du ei’(‘-“)&(r
(27re3/
(~.16)
f),
t) = F(O, 6, t) and the initial ensemble
/TrI,I[Y,(O,
= 19. yX(r,
leading to the
(A.9)
Differentiation with respect to time leads, together with the definition in Eq. (A.4) to the equation, iX(r,
631
(A.12)
The (Y= 0 limit is easily identified now from Eqs. (A.lO)(A.12),
0
dr TrceI pa(-9)9(
(A.18)
5; r).
Appendix B. Small 6 limit The small obtained from and (A.13) to which becomes yo(r,
6)
+
6 limit for the stopping power can be expansions of the solutions to Eqs. (A.8) first order in 6. Consider first Yo(r, 6) to this order,
e-x(r)
’
+ $7
e(x-1)x(r)8.
dx
V,
eexzcr),
/0
(B.1)
(A.13) = H, + Vei( r),
Z’(r)
(B.2)
The derivative in Eq. (B.l) is given by _9. y e-xx”(r) %‘(r)=He+Zzy&,
(A.14)
with the initial condition X,&r, 6, 0) = 6. F(r). The Hamiltonian 2?(r) is that for the electron system interacting with a point source at the position r, where the latter is no longer an operator. The expectation value in Eq. (AS) becomes in this limit,
=.-.
+ 8x;‘(6) Xy(r,
dx’ e(+x)x’(r)[+
vrz(r)]
e-x’r(r), (B.3)
To linear order in 6, therefore, ~~(6)
Tr p(O)X(t)
* /0
--f p,{l + order a’},
p,(6)
becomes,
pe = e -z(o) / Tr lel e-“(o)
Re / dr Tree, Yo(r, S) 6; f),
(B.4)
(A.15)
where ST = 8. F is the component of the force along the initial direction of motion. The time dependence of F(t) is determined from Eq. (A.15). To simplify further, note that the trace over electrons leads to a result that is independent of r. This is easily verified by introducing a unitary transformation under the trace to generate an overall translation of the electron subsystem relative to r. Consequently, the integral over r can be performed to yield a volume factor and the initial position of the ion can be chosen to be at the origin. Finally, the real part indi-
(B.5) where use has been made of the fact that the normalization function, z(6) in Eq. (A.15) is 2(O) + order I?‘. The expansion of S(r, 6, t) can be done in an analogous fashion from Eq. (A.13), F(r,
6, t) =9(r, +
0, t)
/0
’ dt’
ei~(rxr-r’)/rl[+
&F(,.,
0,
f)]
Xe-iX(rXt-t’)/q (B.6)
VI. ELECTRONIC STOPPING POWER
632
J. W. Dufry, M. Berkousky/Nucl.
The correlation order,
function,
Tree) 00(fi)F(6;
Eq. (A.17),
t) + Tree) PX(O)
VF(r,
becomes
Instr. and Meth. in Phys. Res. B 96 (1995) 626-632
to this
+ I,’ dt’ Tree) ~,a
0, t’),
(B.7)
The first term vanishes from spherical symmetry; the last term can be simplified by integrating the gradient by parts and using Eq. (B.3), Trc,) p,$.
VF(0;
= -ZYi’
Y’*((Y=O,
+ 8(Zy)‘flm
= B(Zy)‘$
-ivx)F(r,
0, t’),
it
--oc
dt/o’ dx C(t + ivx)
dt C(t). --CL
(B.14) in the text.
References (B.8)
the correlation
in t. Consequently
6, Z, 03, Y, 77)
This is the result discussed
t’)
dx Tree, p,F(O;
Consequently,
time integrand is real and symmetric has the equivalent form,
function is linear in 6,
[l] T. Peter and J. Meyer ter Vehn, Phys. Rev. A 43 (1991) 201.5.
La N.R. Arista and W. Brandt, Phys. Rev. A 23 (1981) 1898. Trte, p,,(8),8(6;
t) + -Zy6
/0
’ dt’
/0
’ dx C(t’+
iTx), (B-9)
where C(t) is the force autocorrelation C(t)
function, (B.lO)
= TrteI QWt).
The stopping power in this limit is obtained from Eq. (MS), Y(cr=O,
6, Z, 7, Y, 77)
-+ u(ZY)2kTdt
1-
i
f
i’
dx C(t + i?x).
(B.ll)
1
This is the most general result for the small velocity limit. Assume now that the correlation time for C(t) is much smaller than T. Then the limit ~4 00 is justified and Eq. (B.12) simplifies to, y”( cr = 0, 6, Z, 00, Y, ?1) -+ ALLThe correlation
dtl’
dx C(t + ivx).
function has the following
(B.12) symmetries:
C*(t+iqx)=C(-t+ivx)=C(t+iv[l-xl), (B.13) The last equality can be used in Eq. (B.121, together with a change of variables in the x-integration, to show that the
131 S. Ichimaru and D. Utsumi, Phys. Rev. B 24 (1981) 32; C. Deutsch, in Strongly Coupled Plasma Physics, eds. F. Rogers and H. de Witt (Plenum, New York, 1986). [41 P.M. Echenique, A. Arnau and M. Penalba, Nucl. Instr. and Meth. B 56 (1991) 345. [51 R. Golser and D. Semrad, Phys. Rev. Lett. 66 (1991) 1831; M. Kimura, Phys. Rev. A 47 (1993) 2393. b1 M. Steck, K. Beckert, F. Bosch., H. Eickhoff, B. Franzke, 0. Klepper, R. Mooshammer, F. Nolden, H. Reich and P. Spadtke, Proc. Beam Cooling and Related Topics, Montreux, Switzerland, October 1993 (to be published). [71 A. Wolf, C. Ellert, M. Grieser, D. Habs, B. Hchadel, R. Repnow and D. Schwalm, Proc. Beam Cooling and Related Topics, Montreux, Switzerland, October 1993 (to be published). [81G. Zwicknagel, D. Klakow, P.-G. Reinhard and C. Toepffer, Contrib. Plasma Phys. 33 (1993) 395. [91 G. Zwicknagel, C. Toepffer and P.-G. Reinhard, Nuovo Cimento A 107 (1993). [lOI G. Zwicknagel, C. Toepffer and P.-G. Reinhard, Proc. Beam Cooling and Related Topics, Montreux, Switzerland, October 1993 (to be published). ml J.-P. Hansen and McDonald, Physics of Simple Liquids (Academic Press, New York, 1976) p. 45. r121 T. Morita, Prog. Theor. Phys. 22 (1959) 757. 1131H. Minoo, M. Gombert and C. Deutsch, Phys. Rev. A 23 (1981) 924. 1141J.-P. Hansen and L. Sjogren, Phys. Fluids 25 (1982) 617. 1151J.-P. Hansen and McDonald, Physics of Simple Liquids (Academic Press, New York, 1976) p. 112.