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J. Phys. B: At. Mol. Opt. Phys. 33 (2000) 5045­5056. Printed in the UK

PII: S0953-4075(00)50295-8

On the origin of the Bloch correction in stopping
V A Khodyrev
Institute of Nuclear Physics, Moscow State University, Moscow 119899, Russia E-mail: khodyrev@anna19.npi.msu.su Received 7 March 2000, in final form 21 August 2000 Abstract. The energy loss in the collision of a moving charged projectile with a free electron is described in a rigorous approach. The collision is treated as stationary scattering of an electron in the projectile Coulomb field. In the laboratory frame, the picture can be represented as a spatial distribution of energy losses to the electron. It has been shown that the local rate of the energy gain can be presented as a product of the induced electron current and the projectile electric field. The analytical results and numerical calculations reveal a principal disagreement with the generally recognized condition for the classical description, = Z1 e2 /hv 1(Z1 e and v are, respectively, ¯ the charge and velocity of the projectile): for any value of , the quantum effects appear to be significant in the close vicinity of the projectile trajectory (small impact parameters) restricted by the distance = h/ mv . Essentially, the problem has been cleared in the qualitative analysis of ¯ collisions with electron wavepackets. The main results of the Bloch theory are reproduced in a simpler way. The clearer basis permits us to eliminate the ambiguity in the interpretation of the origin of the Bloch correction, which reflects in fact the evolution of the classical features in the quantum mechanical picture.

1. Introduction The famous paper by Bloch [1] has resolved the contradiction between the Bohr classical description of ion stopping [2] and the quantum perturbation result of Bethe [3]. It has been shown that, depending on the value of the parameter = Z1 e2 /hv , where Z1 is the atomic ¯ number of a bare ion moving with velocity v , the energy losses can be described classically if 1, and the quantum perturbation approach is applicable at the opposite limiting case of small . The result was presented as a correction to the stopping cross section determined in the first-order perturbation approach [3],
2 4Z1 Z2 e4 L0 (v , Z2 ), (1) mv 2 where Z2 is the target atomic number, m is the electron mass and L0 is the stopping number,

S=

2mv 2 . (2) I The mean ionization potential I = I(Z2 ) is expressed through the atomic dipole oscillator forces. The Bloch correction has a form L0 (v , Z2 ) = ln L() = -Re ( (1+i) - (1)), where is the logarithmic derivative of the projectile charge, L -1.2022 at small .
0953-4075/00/225045+12$30.00

(3)

-function. The correction is even over the 5045

© 2000 IOP Publishing Ltd


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V A Khodyrev

The correction (3) has been derived assuming that the nonperturbative effects in the ion­ electron interaction are mainly due to the close collisions where the electrons may be considered as free. Thus the expressions (1)­(3) present just a description of the Rutherford scattering in energy loss terms. However, as has been pointed out by Lindhard and Sorensen [4], paradoxical difficulties appear in attempting to reconcile this result with the description of the scattering cross section d/do. Really, the energy loss in a gas of free electrons is directly related to the transport cross section dtr = d/do(1 - cos ) ( is the angle of electron deflection in the projectile frame): in terms of the total stopping number L = L0 + L, L= m2 v 4 2 4Z1 e
4

dtr .

(4)

Since the Rutherford cross section is valid both classically, in quantum mechanics, and for perturbation, the only reason for the difference of energy loss could be the different treatment of the adiabatic effects in distant collisions (small ). However, if this is the case, the higher-order correction over Z1 should also be ascribed to the distant collisions. This strange conclusion was supported in [5] where the adiabatic effects in (4) were taken into account, though by a controversial method. On the other hand, expressing the integral in (4) as a sum over the electron angular momenta l [4, 6], L=
l =0

l +1 , (l +1)2 + 2

(5)

we recognize easily that the higher-order correction is mainly due to the collisions with small l . The contradictory conclusions suggest that the interrelation between the two descriptions, in terms of and l , should be analysed in detail. The rigorous description of Bloch illustrates, specifically, the role of quantum effects. Particularly, according to the asymptotic behaviour of (1+i) at large , L can be written as L = ln 2mv 3 h2 v 2 ¯ , - 2 2 CZ1 e 12Z1 e4 (6)

where = I/h is a characteristic frequency of atomic electrons, C = e and 0.577 ¯ is the Euler constant. The second term on the right-hand side of (6) presents the quantum correction to the Bohr classical result. The representation (5) shows that the quantum effects are also due to the close collisions; the motion with large l is quasiclassical as ordinarily. This conclusion seems contradictory to the Bohr criterion [7] for the classical description of Rutherford scattering. The criterion has the same form 1 and implies that the violation of the classical picture at smaller occurs simultaneously for all impact parameters (angular momenta). It would be useful to elucidate a cause of this inconsistency. The outlined problems concern mainly the interpretation of the basis of the description. Substantial difficulties emerge in attempting to incorporate the results for free electrons into the description of energy loss to atomic electrons. Especially, this concerns the description of the total higher-order correction, including the Barkas odd-order effect [8], and the impactparameter dependence of energy loss in an ion­atom collision E (b). The reason for these difficulties seems to be the different terms used in the treatment of the free electron model ( , l ) and in the standard procedure of the perturbation approach (see, e.g., [9, 10]). In this paper (section 3) a representation of energy loss is introduced where E (b) is expressed, in fact, through the distortion of the electron wavefunction during the collision. This provides the possibility of treating different models on a common basis. Particularly, using this formalism, the Bloch results for the energy loss to free electrons can be reproduced in a clear and simpler way. Also, the origin of the high-order correction and the quantum


On the origin of the Bloch correction in stopping

5047

effects can be analysed in detail. The structure of (3) shows that these two features are interdependent. In section 2, these features are analysed on the qualitative level. Using the introduced representation of energy loss, the Lindhard­Scharff model [11] for the stopping cross section (the local plasma frequency, LPF, approach) can be generalized [12] to describe E (b) in the linear response approach. The combination of this model with the exact description of energy loss to free electrons results in a general model [13] where both the Barkas and Bloch corrections to E (b) are described in a general scheme. Atomic units are used throughout, unless stated otherwise. 2. The qualitative analysis In the free electron model, the detailed analysis of nonlinear and quantum effects is feasible. First, in order to eliminate the inconsistencies in interpretation mentioned in the introduction and to facilitate further rigorous analysis, the main features of energy loss to a free electron can be analysed qualitatively. In the classical description, the energy loss to an electron in collision with the impact parameter s is determined as T(s) =
2 2 Z1 , v 2 (s 2 + a 2 )

(7)

where a = Z1 /v 2 is the collisional diameter. The higher-order effects are present here explicitly (at large v , the adiabatic limit in the integration over s , sad = v/ is not dependent on Z1 ). The condition of applicability of the classical approach can be derived considering the energy loss to an electron wavepacket. The uncertainty of the energy transfer is determined by the uncertainty of the impact parameter s according to the width of the wavepacket s . 2 At s a , when the classical impulse approach is applicable (T = 2Z1 /v 2 s 2 ), the resulting relative uncertainty T/ T s/s . On the other hand, one should take into account also the spread of the wavepacket in a time of collision s/v . The spread is determined by the dispersion of momentum in the wavepacket ps 1/s : sspr ps s/v s , which results in additional uncertainty of the energy transfer T/ T sspr /s 1/v s . The desired condition is a small summary uncertainty of T : s s
2

+

1 vs

2

1.
1/2

(8) which minimizes the left-hand side of (8), we (9)

Choosing for s the value smin = (s /v ) arrive at the condition vs = s/ 1,

where = 1/v is the de Broglie wavelength of an electron of velocity v . The criterion (9) will play an important role in the following analysis. It differs significantly from the Bohr criterion, 1, ensuring the quantum corrections to the angle of electron deflection 2Z1 /v 2 s to be insignificant. In the derivation of this condition [7], the initial uncertainty of the impact parameter s was taken into account analogously (without accounting for the wavepacket spread) as was the angular diffraction of the wavepacket directly contributing to the uncertainty of , diffr = ps /v 1/v s . It is clear, however, that the diffraction has no direct relationship to the energy loss. Thus, a small value of T can be realized simultaneously with a large . On the other hand, the inverse relation, also allowed by the two conditions (if vs is not large whereas 1), is certainly unreasonable. To obviate this difficulty we should take into account the wavepacket spread as a factor contributing also


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V A Khodyrev

to the uncertainty . The same procedure as used above results in the modification of the Bohr criterion: 1 1 + v 2 s 2 2 1. (10)

With the derived criteria (9) and (10), we can trace the transition from the quantum perturbation to the classical description of energy loss. For large projectile velocity (vsad = 1), the adiabatic effects may be treated classically. If this approach is applicable at v 2 / all impact parameters s , the stopping number L is determined by integration of (7): s ds v3 ln (11) 2 s +a Z1 0 (sad is assumed to be large relative to a ). The collisional diameter a set an effective lower limit in the integration (11). One may speculate, however, that at > a the lower limit is determined by the quantum effects. Thus L(v ) =
2 v/

v2 ds = ln , (12) s which coincides with Bethe's result. Such an interpretation of transition from the classical to quantum perturbation result is often suggested (see, e.g., [14]). The quantum effects at s are ordinarily associated with significant diffraction of an electron wavepacket of such size. However, such localization of quantum effects seems contradictory to the Bohr condition, 1, for the classical interpretation of electron scattering in the projectile field. This condition implies that the quantum effects, if significant, reveal themselves simultaneously at all s . With the condition (9), such an interpretation becomes plausible. The derivation of this condition shows that the classical picture is destroyed due to the spread of electron wavepackets during the collision. This results in effective averaging of the integrand in (11) over s in the range from 0 to s1 . As a result, at > a ( > 1), this region introduces just a small correction to the stopping number (11). As is seen from (11), the higher-order correction over Z1 has a classical origin and is mainly due to the close collisions. Alternatively, the appearance of the correction can be traced using the representation (4) of energy loss in terms of deflection angle . Here, the explicit usage of the classical picture can be avoided. In this representation, the stopping number is determined as 2 , (13) L = ln L(v )
min

v/

due to the adiabatic where min is taken as a lower cutoff in the integration over effect. Since this effect is represented naturally as a restriction of the interaction radius, s < sad , the relationship between min and sad should be defined. The classical estimation, 2 3 min 2Z1 /v sad = 2Z1 /v , results in the expression (11) for the stopping number. This estimate, however, loses its validity when the angle of diffraction of the electron wave, 2 diff 1/v sad = /v , has a larger value. In this case, min diff and L is given by the expression (12). Clearly, in the classical description, the fact that the dependence of L on Z1 is involved through min does not contradict the localization of the higher-order correction at small s ; the specifics of the variable transformation, tan /2 = a/s , should be taken into account. Also, at small , the diffraction broadening at the lower edge of the deflection angular profile cannot be ascribed only to distant collisions. For a clear representation, the consideration of energy loss at s < s1 can be given in terms of while the classical treatment is suitable


On the origin of the Bloch correction in stopping

5049

at s1 < s < sad . The spatial spread and the angular diffraction have the same origin: the dispersion of the momentum ps . These arguments ensure that descriptions in terms of s and are fully equivalent. 3. Energy loss to a free electron For the general situation of an ion­atom collision, the following exact representation of energy loss, summed over all atomic transitions, can be introduced: E (b) =
-

dt

d3 r E · j ,

(14)

where E (r ,t ) = -V (r , t )/ r is the electric field of the projectile and j (r ,t ) is the density of the current induced in the atomic shell. Considering the variation of the atomic energy, Eat , during collision, the expression (14) can be derived rigorously: d Eat d = dt dt 1 = 2i d3 r


H0

=

1 i +

d3 r




[H0 , H] = d3 r - V r
j , (15)

d3 r -

V r r

V r r

where H0 is the unperturbed Hamiltonian of the atom, the Hamiltonian H includes additionally the projectile­electron interaction, -V(r ,t ), and the summation over all atomic electrons is implied. With the current
j (r ,t ) =

1 2i

r

-



r

,

(16)

the energy loss (14) can be evaluated if the time evolution of the electron wavefunction (r ,t ) is known. For any approach, the corresponding expression for E (b) can be transformed to the form (14) with a specific description of j (r ,t ). The integrand in (14) has the obvious meaning of the rate of energy absorption as it is distributed within the atomic electron shell. Such a detailed visualization of the energy loss can be useful in analysis of the shortcomings of calculation approaches. It is important also that the current j (r ,t ) has a clear classical counterpart, thus the relation to the classical description can be also analysed in detail. Particularly, using the representation (14), the energy loss in the collision of a charged projectile with a free electron can be described rigorously if the unperturbed state of the electron is considered to be a plane wave with zero momentum. In this case, the wavefunction of the electron is presented as a Coulomb scattering wave (transformed to the laboratory frame),
+

(r ) = e

- 2

(1+i) (-i, 1; i(v r + vr )),

(17)

where (, ; z) is the confluent hypergeometric function. The normalization factor is defined so that the electron density in the impinging plane wave is equal to unity. (r ) is given for t = 0 when the ion moves through the origin. The parameter in (17) may have both signs. As a result, a united description is realized for both positive and negative projectiles (in what follows, the sign of = -Z1 /v is defined according to the electron affinity of the projectile charge). With the wavefunction (17), the current (16) is written as
j (r ,t ) = -

(e

2

2 1 ( - 1) 2



+



)

vr + r v , r

(18)


5050

V A Khodyrev

where, for the sake of brevity, the arguments of are omitted; means a derivative with respect to the last argument. In the case of an infinite wave, the integrals in (14) are divergent. However, we may interpret the integration over time as describing the stationary energy losses, E= dt v - dE dx (19)

(the x -axis is aligned with v ). With E = Z1 r /r 3 and j defined by (18), the specific energy loss can be written as Z1 2 vr + vr dE ( + ). (20) =- d3 r - dx 2v e2 - 1 r3 After the integration over longitudinal coordinate r , this equation acquires a familiar form - dE = dx d2 r E(r ), (21)

where E(r ) is the energy absorption at a given distance r from the ion trajectory. In this approach, the integral (21) represents energy loss analogously as in the classical integration of the energy transfer E(s ) over impact parameter s . Clearly E(r ) has identical meaning when the classical impulse approach is applicable. The divergence of the integral (21) at large r is to be eliminated by the adiabatic effects. Using the transformation (-i, 1, iu) = -i (1 - i, 2; iu) and the functional relations for the hypergeometric functions [15], we obtain


(22)

+

= u (1 - i, 2; iu) (1+i, 2;-iu).
2 2 1 2 e -

(23)

The change of variables in the integration (20) to u = vr + vr and r results in the expression E(r ) = 2Z 1
0

duu3 | (1 - i, 2; iu)|2 . 2 (u2 + v 2 r )2

(24)

At large r , one may expect convergence of E(r ) to the first-order perturbation result ( = 0). Hence, its subtraction from (24) should result in convergence of the integral over r . If the adiabatic effects take place at large distances, the correction to the first-order dE/dx can be presented as an unrestricted integral of the above-mentioned difference. The correction can be written in the standard form: - dE dx =
2 4Z1 L(), 2 v

(25)

where is the unperturbed electron density (taken before as being equal to unity). Using equation (24) we obtain L() = 1 2
0

duue

-u

2 | (1 - i, 2; iu)|2 -| (1, 2; iu)| e -1
2

2

.

(26)

The exponential factor with infinitesimal has been entered herein to provide the convergence in the integration of separate terms in the square brackets. These integrals can be taken by the method proposed in [16]: L() = 2 +i 1 2 ) e2 - 1 - i 2(1+ 1 -F 1, 1; 2; . 1+ 2
-i

F 1 - i, 1+i; 2;

1 1+

2

(27)


On the origin of the Bloch correction in stopping

5051

Straightforward use of the asymptotic form [15] of the hypergeometric function F (, ; + ; z) for z 1- , ( + ) F (, ; + ; z) [2(1) - ( ) - ( ) - ln(1 - z)], (28) () ( ) results in the expression (3) for the Bloch correction. The expression (24) permits us to perform a detailed analysis of the energy transfer to a free electron as a function of the `impact parameter' r . The hypergeometric function entering this equation can be expressed through the so-called Coulomb wavefunction of zero momentum F0 [15]: (1 - i, 2; iu) = 2 eiu/2 F C0 () u
0

,

u , 2

(29)

2 ¨ where C0 () = 2/(exp(2) - 1). The function F0 (, z) satisfies the radial Schrodinger equation with zero angular momentum,

2 F0 2 + 1- F0 = 0 , z2 z

(30)

for the boundary conditions F0 (, 0) = 0 and F0 (, z) = sin(z + ) at large z, where is the scattering phase. After substitution of (29) into (24), equation (21) can be rearranged as follows: 2Z 2 dE d2 r - Q(, ), (31) = 21 2 dx v r where Q(, ) = 4
0

dww F (1+ w 2 )2

2 0

,

w , 4

(32)

is a reduced transverse coordinate, = 2vr = 2r /. At Q = 1, equation (31) reproduces the classical impulse result, so the factor Q accounts for insufficiencies of the impulse approach and, simultaneously, for the quantum effects. The main result of this treatment, presented by (31) and (32), will be analysed in the next section. First, it would be useful to represent the perturbation and classical results in terms of the corrective factor Q(, ). In the former approach, the parameter in (32) should be taken to be zero. Accordingly, as evident from equation (30), F0 (0,z) = sin(z). Substituting this into (32), we present the first-order perturbation result in a form Q
pert
2

(33) 2 2

(, ) = Q(0, ) = e

-

Ei

- e 2 Ei -



,

(34)

where E i (z) is the exponential­integral function. The same result, though in a different form, was obtained in [17], where the perturbation approach was applied from the outset. The corresponding classical phenomenon, the stopping in a uniform electron gas, can be described using the quasiclassical solution of equation (30), 1 F0 (, z) = sin (z), (35) (1 - 2z )1/4 where (z) is the quasiclassical phase. Recall that > 0 corresponds to the repulsive projectile­electron interaction (a negatively charged projectile). In this case, the value of F0 (, z) at z < 2 (the under-barrier region) should be taken as zero. As usual, substituting (35)


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V A Khodyrev

Figure 1. The factor Q(, ) for the indicated values of . The solid curves show the exact quantum results (32); the classical results (36) are shown as dashed curves. The dot­dashed curves present the classical results in terms of the reduced impact parameter s , = 2vs (equation (37)).
2

into (32), we may replace sin Q(, ) is obtained: Qcl (, ) = 2

by its mean value. As a result the classical expression for


w

min

dww (1+ w 2 )

1
2

1-

8 w

,

(36)

where wmin = max(0, 8/ ). In fact, equation (36) defines Qcl (, ) as a function of one variable / = r /a . This is just a reflection of the invariance of classical motion in the Coulomb potential (the deflection angle is determined as = 2 arctan(a /s )). 4. Discussion Figure 1 illustrates how the factor Q(, ), with as a parameter, changes with . The classical results, equation (36), are also shown. It is seen from the figure that, at large , the classical description reproduces well the exact quantum results; the quantum effects become significant at 1. This confirms the conclusion made in the qualitative analysis (section 2). At large , all curves approach unity, demonstrating the applicability of the classical impulse approach for distant collisions. For comparison, we present also the results of the classical description in terms of the impact parameter s (equation (7)) (to underline the similarity of s and r , the same notation is used for the reduced variable = 2vs ). The corresponding corrective factor Q has the form 2 . (37) 2 +42 This representation reflects, in fact, the same classical picture as obtained in the quasiclassical approach (equation (36)). However, the data in figure 1 may cause confusion. At large , the factor Qcl (, ) converges rapidly to unity (Qcl 1 - 42 / 2 ). Seemingly, this implies that the electron displacement during collision is negligible (the condition for applicability of the impulse approach). When such is the case, also the variables s and r should become equivalent and Qcl (, ), likewise, should converge fast to unity. Yet, the factor Qcl reveals much slower convergence (Qcl 1 + 2/ ). This difference can be explained in the following way. The electron displacement reveals itself as a modification of the effective impact parameter (the transverse displacement) and also as a shorter effective time of collision (the longitudinal Qcl (, ) =


On the origin of the Bloch correction in stopping

5053

acceleration of electrons). In the case of the Coulomb interaction, these two effects almost cancel each other, so the deviation of Qcl from unity does not reflect the actual inaccuracy of the impulse approach. However, when the energy loss is treated in terms of the induced current, the picture looks different. The longitudinal acceleration results in corresponding modification of the current, which reproduces equivalently the effect for an individual electron. At the same time, the simultaneous transverse displacements of electrons do not produce an equivalent compensating effect. The compensation becomes effective when the integral over r is considered: the transverse displacements result in the concentration of electron gas behind the moving projectile. In figure 1 this is evident as the inversion at small of the relative vertical positions of curves Qcl (, ) for negative and positive projectiles. The applicability of the classical approach at r permits us to incorporate consistently the adiabatic effects. As a specific realization of the adiabatic effects, the screening of the ion field due to the polarization of electron gas [18] can be considered. The screening distance is estimated as v/p , where p = 4 is the plasma frequency of the electron gas. This effect results in an effective upper limit for r . A specific consequence of this cutoff is the appearance of the correction to the energy loss odd-order over the projectile charge. This follows immediately from the different behaviour of Q(, ) - Qpert (, ) for different signs of and from the fact that unrestricted integration in (31) should result in even total correction. In terms of the stopping number L, the total correction can be written as L= LBloch -
2v 2 /
p

d (Q - Q

pert

),

(38)

where, for simplicity, the adiabatic effect is taken into account as a sharp cutoff in the integral. If v/p is large, the integral in (38) can be estimated using the asymptotics of Q - Qpert Qcl - 1 2/ . As a result the total correction is represented as consisting of two terms: L = LBloch + Lodd , where Lodd Z1 p . v3 (39)

This consideration is consistent with the interpretation [14] of the Barkas correction. The representation (14) permits us to present [12] a comprehensive description of energy loss also for the uniform gas of interacting electrons. This is possible if the response of electrons on the projectile field is treated in the linear RPA approach. The current j (r ,t ) can be determined using the dielectric formalism [18]. The result is presented in the form of equation (31), which gives the possibility of combining it with the exact description for free electrons (section 3). The resulting calculation scheme takes into account both the higherorder correction and the effect of polarization of electron gas. Then, using the idea of the Lindhard­Scharff model [11], the impact-parameter dependence of energy loss in an ion­ atom collision, E (b), can be described in the local plasma frequency approach (the local current j (r ,t ) is calculated according to the electron density at a given point r of the atomic electron shell). This general model is presented in [13]. Here, for illustration, an example of the calculation is presented (figure 2). The dependence E (b) is shown for 0.5 MeV au-1 projectiles with Z1 = 1 and 6 experiencing collision with gold atoms. For comparison, the results obtained in the first-order perturbation approach and those for the equivalent negative projectiles are also presented. As is seen in the figure, the correction to E (b) can be fairly large and varies with b. These data, especially the curves for Z1 = 6 and -6, suggest the fact that, for positive projectiles, the Bloch and Barkas corrections have opposite signs and significantly compensate one another. For negative projectiles, in contrast, these corrections are both negative, which results in a much larger total effect. The discussion above shows,


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V A Khodyrev

Figure 2. The impact-parameter dependence of energy loss in collisions of hydrogen and carbon ions with gold atoms at the projectile energy E = 0.5 MeV au-1 . For comparison the first-order perturbation results (the dotted curves) and those for equivalent negative projectiles (the dashed curves) are also presented.

however, that the representation of the total correction as consisting of two parts may have no sense. According to the qualitative interpretation (section 2), the parameter = Z1 e2 /hv , ¯ completely determining the Bloch correction, characterizes the relation between the intensity of projectile­electron interaction and the quantum effects. In this respect, it is worth noting the additional misleading due to the customary representation of the relationship between the results of Bethe and Bloch. The classical limit is customarily associated with a small value of h ¯ while the perturbation approach seems to relate to a small value of Z1 . In fact, this illusion will disappear if one tries to draw the classical perturbation picture (the divergence in integration over s at small s cannot be eliminated) or to find the limit of the quantum perturbation result at h 0. However, some view of the interaction picture and its change between two limits ¯ would be more instructive. The consideration in section 2 demonstrates that the modification of collision when changes is suitably associated with the variation of quantum effects assuming the interaction potential to be unchanged. Being negligible in the classical case, these effects become dominant at small (¯ , the `ultra-quantum' limit). Alternatively, the evolution h with the increasing intensity of interaction, the variation of Z1 , should be traced, which seems too cumbersome. The fact that, at h , the treatment can be fulfilled using the perturbation ¯ ¨ approach is not accidental. This is seen from the Schrodinger equation written in the reduced

In the Bohmian quantum mechanics [19], this relation is presented as a competition of the quantum force with the actual Coulomb interaction.


On the origin of the Bloch correction in stopping variable = 2mv r /h: ¯

5055

1 2 1 h ¯ 1 ( ) = + U (40) ( ), 2 2 2 (2mv ) 2mv 8m ¯ where U(r ) is the scattering potential. An increase of h results effectively in narrowing of the interaction potential, which allows the usage of the perturbation approach [20]. - 5. Conclusion In summary, the energy loss by a moving charged particle has been represented as a spatial distribution of the energy gain in the target electron gas. For free electrons, this provides an exact solution of the problem. With these results, the higher-order correction over the projectile charge and the interplay of quantum and classical effects are analysed in detail. Particularly, the appearance of the higher-order correction within the Rutherford scattering picture is verified. It is shown that the origin of the correction is presented, in fact, in the classical picture of energy loss; the quantum effects result in its decrease. In this respect, the Bloch correction may be interpreted logically as describing a link between two limits of importance of the quantum effects. In the qualitative analysis of energy loss to an electron wavepacket, the condition for the classical description of energy loss is derived. This condition differs principally from the commonly used condition for the classical description of electron scattering in the projectile field. The introduced representation of energy loss through the density of the current, induced in the medium, provides a possibility of describing the impact-parameter dependence of energy loss in ion­atom collisions on a physical basis, equivalent to the Lindhard­Scharff model. The present results permit us to achieve an additional advancement by taking into account the higher-order corrections. The method proposed shows a way how the treatment of Bloch and Barkas corrections can be combined in a general scheme. Acknowledgments I have had fruitful discussions about some aspects of the considered problem with P Sigmund, J U Andersen, A H SÜrensen and D O Boerma. The author gratefully acknowledges the hospitality extended to him during his stays at the Groningen University. This stay was made possible by the Dutch Organization for Scientific Research (Nederlandse Organisatie voor Wetenschappelijk Onderzoek, NWO), grant 713-187. This work was partly supported by the Russian Foundation for Basic Research, grant 17-28-96, and by the Foundation for Russian Universities, grant 53-96-08. References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] Bloch F 1933 Ann. Phys., Lpz. 16 285 Bohr N 1913 Phil. Mag. 25 10 Bethe H A 1930 Ann. Phys., Lpz. 5 324 Lindhard J and SÜrensen A H 1996 Phys. Rev. A 53 2443 SÜrensen A H 1997 Phys. Rev. A 55 2896 Anderson V E, Ritchie R H, Sung C C and Eby P B 1985 Phys. Rev. A 31 2244 Bohr N 1948 K. Dansk. Vidensk. Selsk. Mat.-Fys. Meddr. 18 Barkas W H, Birnbaum W and Smith F M 1956 Phys. Rev. 101 778 Jackson J D and McCarthy R L 1972 Phys. Rev. B 6 4131 Mikkelsen H H and Sigmund P 1989 Phys. Rev. A 40 101 Lindhard J and Scharff M 1953 K. Dansk. Vidensk. Selsk. Mat.-Fys. Meddr. 27 Khodyrev V A Phys. Scr. submitted


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[13] [14] [15] [16] [17] [18] [19] [20]

V A Khodyrev

Khodyrev V A, Arnoldbik W M, Iferov G A and Boerma D O 2000 Nucl. Instrum. Methods B 164/165 191 Lindhard J 1976 Nucl. Instrum. Methods 132 1 Abramowitz M and Stegun I A 1972 Handbook of Mathematical Functions (New York: Dover) Nordsieck A 1954 Phys. Rev. 93 785 Khodyrev V A and Sirotinin E I 1983 Phys. Status Solidi b 116 659 Lindhard J 1954 K. Dansk. Vidensk. Selsk. Mat.-Fys. Meddr. 28 Bohm D 1952 Phys. Rev. 85 166 Landau L D and Lifshitz E M 1977 Quantum Mechanics (Oxford: Pergamon)