Atoms.Molecules and Clusters

Z. Phys. D - Atoms, Molecules and Clusters 13, 33-43 (1989)

ffir phySik D

© Springer-Verlag 1989

Anomalous density effect near the interface between two media V.A. Cheehin and V.K. Ermflova P.N. Lebedev Physical Institute, Academy of Sciences of the USSR, Leninsky prospect 53, Moscow 117924, USSR Received 14 September 1988; final version 18 December 1988

A theoretical investigation is presented for the local energy loss of a relativistic charged particle and the K-shell excitation cross section aK when the particle crosses a sharp interface between two different media. The logarithmic increase of the local energy loss and o-K with the particle energy is shown to differ essentially from the case of a homogeneous medium. The calculations of e~ for aluminum and copper foils are in good agreement with the experimental data. PACS: 34.90; 34.00

I. Introduction

In recent years detailed measurements of the K-shell excitation cross section ~K(7) were performed for a large spread in y-values, y,-~4+104 ( y = ( 1 - v 2 ) -1/2, v is the particle velocity in units of the velocity of light, c = 1) [1-9]. The unexpected result was the discovery that o.K(7)increases proportionally to the logarithm of the Lorentz-factor y for very high v-values, for which the density effect (DE) should have limited this increase. It is known [10, 11], that in condensed matter the excitation and ionization of the individual target electrons by a charged relativistic projectile are partially suppressed for large v-values because of the strong influence of target polarization. As a result, the logarithmic increase with 7 of the cross sections of these processes and of the associated quantities (ionization energy loss, primary ionization, etc.) saturates - comes onto the Fermi plateau - for y ~>o)~ff/o)p, where o)off denotes the mean effective energy transferred to atomic electrons in the process under consideration and cov is the plasma frequency (h = 1). In particular, the cross section a~(y) must reach the Fermi plateau for y > oJK/O~p,where coK is the minimum excitation energy of the K-shell electron. For e.g., copper, this estimation is 7>200, but Middleman [1] had observed the growth of o~r(y) until 7~-1800. The anomalous behaviour of ~K(Y) has repeatedly been observed for a wide range of targets with atomic numbers 20~
was paid to the necessity of taking into account transition electrodynamic effects in thin targets and the observed K-shell excitation yields were successfully described on the basis of a simple model. Independently at the same time theoretical studies were performed for the suppression of the density effect in the energy loss AE(y) experienced by a charged particle penetrating thin matter layers [ 1 4 4 6 ] and in the local energy loss d W(z, y)/dz near two media interface, i.e. for small enough penetrated depth z [17, 18]. Note that DE suppression in the energy loss AE was predicted by Garibian in 1959, but no convincing experimental proof of this phenomenon has been obtained as yet. The situation is quite different with the study of the cross section GK(y), where numerous experimental data have been obtained. The theoretical approach [17, 18] is the closest for the DE problem for o K. It was shown, that the local energy loss near the vacuum-medium boundary and in thin foils exhibit an unlimited logarithmic increase in y. To observe this increase for AE is rather difficult. At the same time the anomalously large increase of o.r(Y) in thin foils just confirms this theoretical result [i9]. The inverse effect is also possible, i.e. in a gas DE increases when a particle crosses the dense matter-gas interface. In the present paper the anomalous density effect for the local energy loss and for the K-shell excitation cross section is considered from a unified point of view. The general expressions for dW(z, y)/dz and aK(z, 7) and their qualitative consequences are presented in Sect. II. In the general case the calculations

34

according to these equations are too difficult, therefore in Sect. III we deal with the relativistic limit (7>> 1). In Sect.IV we give simple expressions for dW(z, 7)/dz for very small z, when the anomalous DE is most strongly pronounced. The calculations of aK(z, 7) are in close agreement with experimental data as well as with the theoretical results obtained in [13] (Sect. V). The inverse phenomenon - an amplification of DE in thin gas layers surrounded by a dense matter is discussed.

where P = ( 4 r c ) - l ( g - 1 ) E is the polarization vector equal to the dipole moment of unit volume. The electric field may be expressed in terms of its Fourier transform E(z, p, t) = ; E(z, p, co) exp(--

icot) do

-oo

= ; ~E(z,q, co)exp(--icot+iqp)dcodq,

(2)

co

then II. General expressions. Qualitative considerations

Let a particle with charge e move a constant velocity v along the z-axis in an polarizable medium characterized by a dielectric function e (co, z) which is piecewise constant in direct space where it depends on the zcoordinate only. For example, we imagine a situation where a particle intersects one or several plates, v~> 0. The electromagnetic field in such a problem can be obtained by sewing on the boundaries of the Maxwell equations solutions for a charge moving in a homogeneous medium. Further one can calculate the Poynting vector, the force acting upon the charge, the total energy loss of the particle, the field energy dissipated in the medium, the atomic ionization cross section, etc. In the transition radiation (TR) theory the Poynting vector is usually calculated for Izt ~ oo (but Ipl/ z = tg 0 = const, p is the two-dimensional vector in the plane z = 0) under the assumption of medium transparency, i.e. for zco Ime(co)~l. The corresponding flux of TR quanta just determines the quantum counting rate in the detector located far from the radiation region. We are concerned here with the "local" quantities specified by the electromagnetic field within the radiation region, in particular, with the local ionization energy loss dW(z)/dz and the cross section o-K(z). In the former case we deal with the energy dissipated in a medium, which shows up in charged particle detectors in the form of bubbles, developed grains, clusters, etc. The K-shell excitation is usually observed by the characteristic X-ray radiation. Let us calculate the energy d Wdissipated between the planes z and z + dz, which coincides with the work done by the electromagnetic field upon the charges ei of the medium" ~ ei ~ v~E(r~,

t)dt.

Summarizing

over all charges we obtain

dW(z)-sE(z,p, t) ~~dz

P(z, p, t) dp dt,

(1)

dW(z) d ~ = 4 ~ z 2 ; do) ~co Im e(co, z)lE(z, q, co)12dq o _=

de

o

co Ime(co, Z)-d-o-~codco,

(3)

where it is taken into account that E(z, - q , -co) = E* (z, q, co). The factor co Im e(co) describes the field energy absorption per unit length, the spectral intensity being equal to

dQ=4~2 SIE(z,q, co)12dq=SIE(z,p, co)I2dp.

do

(4)

The number of collisions with atoms per unit length dN(z)/dz can be obtained by inserting the factor 1/co under integral (3). To calculate o-r(z) it is necessary to single out in Im e(co) the contribution of the absorption on K-electrons only, then ~(~)

=

1 dNr n,(z) dz

__ 4re 2 ~ dcofIme(K)(co, z)lE(z,q,e})12dq ha(Z) o~x

-

1

~

K

dQ

j I m e ( )(co, z ) d e ) , no(z) ~,,

(5)

where n, denotes the number of atoms per unit volume. Since (3) and (5) are identical everything said about the energy loss dW(z)/dz holds also for the cross section er~(z). Equation (3) is true for any 7- If 7 >>1 it coincides with the energy absorbed at a distance d z and calculated according to the Weizsficker-Williams method of virtual photons [13]. The spectral intensity (4) in this method is known to diverge logarithmically as the impact parameter tends to zero, p--->0 (or q ~ oo). Therefore (3) can be used to describe the contribution of "distant" collisions (p > P0 or q < q0). The contribution of "close" collisions essentially may be obtained

35

from the differential cross section for collisions between free particles. In the presence of the interface between two media the divergence will be the same. If the spatial dispersion is taken into account, as e.g. in the Photoabsorption Ionization Model [11, 18, 20], then the integrals in (3) converge and (3) can be used to describe all collisions. Let us consider (3) and (5) qualitatively for 7 >>1. The field E (z, q, co) depends on the properties of the medium for any z'. However, for ~ >>1, disregarding the reflected waves, one may assume E(z, el, co) to be determined by the region z' 0) = e2(co), el, 2 are the dielectric functions of the two media) the field E(z, q, co) for z < 0 will be close to a " v a c u u m " one. This "vacuum" field is also preserved at a certain depth z2(co) in the medium e2, i.e. for 0 < z ~ z2 (co), because the basic for 7 >>1 tangential component of E is continuous on the boundary. 1 Consequently, dQ/dco in (4) for 0 0) and the plasma frequency in the rare medium

In a plate of thickness a,-~ Zz the density effect suppression will be partial as a result of averaging of d W(z)/ dz and aK(z) over the penetrated depth z. The dense medium-gas interface can be considered in a quite similar manner. In the general case the local energy loss and o-K reach the Fermi plateau for 0 • Z ~ Z2, (Dz > 0) when 7 > coeff (Z ~> O)/COp (Z < 0). The DE is also suppressed for the total energy AE(y) lost by the particle crossing a thin plate. According to [14] T

AE(v)=ev ~ E(z=vt;O;t)dt;

(T~o~).

-T

This quantity involves besides the local energy liberaa

tion W= ~(dl4]dz)dz, also the transition radiation o energy. In this case the effective frequency c%ff depends on 7 and the plate thickness a, which change drastically the character of DE suppression as compared with one considered above [15, 16].

IlL Relativistic approximation The qualitative conclusions of Sect. II can be rigorously grounded by using in (3) and (5) the expressions for E(z, q, co) known from the transition radiation theory. In the case of one interface between two media we have for z > 0 in the medium ~2 [14]: E(z, q, co) = 2-~e-~-2 v [Ez (q, co) exp (/~zff- ) + E~(q, co) exp (i 22 z)],

(6)

where the proper field is given by

(z < 0). For z >>z~ (co) the field E(z, q, co) acquires a stationary form typical for the medium e2, and therefore >. ~ ~ (2) for large enough z ~> z 2 ~ z2(c%f0 one can derive from (3) the Bethe-Bloch-Sternheimer formula with the density effect correction. In the relativistic case the above conclusions are true also for a plate. For instance, a logarithmic growth of cq¢ in a copper target of a thickness a ~ z2 surrounded by a gas under normal conditions must stop only for 7~>104 (cor-~11 KeV, cop~-i eV). This conclusion is in agreement with experiment (Sect. V).

~ (V292- 1) n--q E2 (q, co) =

l)

~2

- A2

(7)

and the transition field by

"t-£2221t)(14_A_1 /~1 ~1

Etr (q, 69)= ~1 (qZn-)'2

with the definitions ~j = (gj (.02 - - q 2 ) 1 / 2

1 The function Zz(~O) depends, generally speaking, on q2. We assume here averaging over the effective region q 2 ~qeff. z The estimation of zz(co) is given in Sect. III

(.O2

Aj-~-~2--}- q2 -- gj o,) 2,

( j = 1, 2),

~1 AzV21) ez co

(8)

36 n is the unit vector of the surface normal; I m 2 s > 0 for co>0. By insertion of (6)-(8), we may write (3)-(5) as a sum of three terms which have distinct physical meanings:

As is seen from (9), the influence of the two media interface ceases for (2) Im 52 (coeff) (2) ] - 1 = 2 2, , z >>(1/Ira 225 ~~ 2 [co~ef when d Q/d co ~ d Q 2/d o), d W(z)/d z ~ d W2/d z

and aM(z) ~- a~ ). dQ dQ2 dQ,, d--~ = d c o + ~ 4 : -

7Z/) 2

dQi, do)

By insertion of (6)-(8) (or similar expressions for several interfaces), the q and co integration in (3), (5) may be performed only numerically. Such calculations are simpler in the relativistic case ~> 1 ( v ~ l ) when q2/co2 ~t 1/I)2 --el ~ ~ - 2 4 1. In the region of the effective frequencies one can put es ~ 1 in (6)-(8) everywhere except in the dominators A s and exp(i22z). As a result to an accuracy of the terms O(7-2) and 0(11 -el) we obtain from (6)-(9):

'0,°{

IE2[ 2 + IEt~]2 e x p ( - 2 z Im 22)

+2 dW(z) = dW2 + dWt~+__dWi. dz dz dz dz ' (9)

o-K(z) = ~ + a,~(z) + %(z).

The contribution of the proper field can be written

as

--ieq E(z, q, co)_ 2n z fexp(izco/v) ~_(1 a

"1

dQ2

2 e 2 qeo q(q 2 +co 2 v 2

dco = rcV-~ oj

2e2 ~V2 Im52 ,~__ --

i~-~

Im ~o q(v 2 _ 1/g2) oJ

dQ(z) dco

lzzl 2 ) . aq A2

e2

n Im52 I m [ z 2 In q2° ] L52

o)2 g2]'

(10)

where zj = 1/V 2-eJ. For d Wz/dz we will derive the standard expression for the energy loss in the medium 5 2 [21]. The two last terms in (9) give the corrections to the local energy loss near the target surface. Let us consider dWt~/dz which describes the absorption of the transition field. Integrating over z > 0 up to z m , ~ o o and setting Ime2--*0 and sin20 = q2/52 o)2 we obtain the frequency-angular T R energy spectrum which in this case coincides with the energy spectrum absorbed in the medium 52 [22]. The interference term d W~,/d z includes the factor t / = e x p [ i z ( R e 2 2 - c o / v ) - z Im 22] typical of the T R theory. In order to calculate the radiation in the medium 52, one should set Im 52(co~a)~ O, (Ira 22 ~ 0) so that the main role will be played by the oscillating part of q, which determines "the formation zone" z I = 1/lRe22-co/vt. For the local quantities d W(z)/dz and a~c(z) essential is the strong absorption region (2) where Im 22 can be of order (Re 22-o)/v). o) ~ coorf, Therefore the interference term vanishes for ~ ,> < l / l & - c o / v l ) (2) I ~ i f ) (~):{-- 1/vl]-I = z2" -~ [2 o-hff

(11)

2e 2 rt

qo°[ 1 [1 "j A2+~7-~z)

dq

~__2)exp (i 22 z)t,

A2

1\

.

co

2

e X p [ t ( 2 2 - - v ) z] qadq •

(12)

Consequently, for z ~ z 2 when dQ(z)/dco,~dQ1/do), the density effect will be suppressed in d W(z)/dz and aM(z) (see Sect, II). It should be stressed that z2 and z~, corresponding to the quantities 1/122-co/vl and 1/Im 22 averaged over q and co are generally speaking distinct. However in the strong absorption region co . . .U-,ef . (2) t the z2 value is only several times less than r Z2 .

In the relativistic approximation one can consider the problem for several interfaces between media. Each interface with z-coordinate a, < 0 (n=0, - 1 , - 2 , ..., a._l 0 can be disregarded 9, makes a contribution to the field E(z, q, co) for z > 0 : ieq(1 2re 2 A,

1 ) A~-+I exp(i22z)

[m=o

(

. e x p i ~ (a,. + l -- a,.) 2 , . + 1 -

,

(13)

m=n

where 5 (a, < z < a, + 1) = 5. + 1. The formulas (11)-(13) are also applicable for oblique incidence of a particle onto the boundary because for 7 > 1 the field E depends weakly on the incidence angle up to small sliding angles 0H 7-1. Moreover the same relations can be used for a piecewise

37 constant dielectric function e(r). It is only necessary that the characteristic scale of variation of e(r) in the direction transverse to v exceed substantially Pelf 1/qeff ~'7/coeff [18].

Let us perform the q integration in (12). The contribution of the proper field is equal (instead of (10)): dQ2,~ dco-

e2 ( nImg2Im %ln

q~) .

(14)

To evaluate the contribution of the transition field we shall perform the q integration disregarding the dependence on q in Im 22. It is therefore adequate to put Im 22 ~-co Im ez/2. Since the integral converges on the upper limit we may assume qo ~ Go then the result is d ~ r ~ exp (-- z co Im

g2)

e 2 ~Im(z 1 In %) rr [ Im el

Consequently, besides damping due to absorption there also occurs a depletion of the spectrum according to the power law dQ~,/do~z -2 which is caused by the q integration of the rapidly oscillating function. One can readily make sure that for z~z2, when q)(z)~z In z, the sum of the expressions (14), (15) and (16) is equal to dQ1/dco.

Im(z 2 In "g2) Im e2

- 2 Re (% In z x - z * In z*)~ (~~ ~) J

IV. Local energy loss for z ~ z z

Choosing a realistic model for e 1(co) and e2(c0) in the spectra (14)-(16) and integrating over co in (3) and (5), we obtain the dependence of the energy liberation and of the number of K-vacancies on the coordinate z. For z,-~z2 such an integration is possible only numerically; for a~(z) it is discussed in Sect• V. Let us consider d W(z~ zz)/dz when the DE suppression is most pronounced. According to (12)

dQ(z ~ Zz)/d co ~ dQ1/dco = ( - eZ/rc Im el) •Im (zl ln(q~/co2 z 0), (15)

therefore

-

The factor at the exponent in (15) describes TR energy spectrum which takes into account, in particular, the Cherenkov radiation and the distinction from zero of the imaginary parts of ~ and e2. In the X-ray region it coincides with the spectrum [-14] and for e 2 ~ 1 and arbitrary e~ with the spectrum obtained in [23]. When calculating the interference term one should necessarily to take into account the first power of q2 in the expansion of 2 2 : co co 22 - - - ~ - co~f~2 v v ,,~ co [ 1

qZ 209 q2

Then for qo -~ Go we have dQi.~_2eZRe exp do) 7z .1}0(%)--~o(z~)

go(%)-- go(z*)_T~'

(16)

dW

e2

~zz (Z~Zz)~_~im ~

dQi, do)

8e 2 e x p ( - z c o Im82/2) 7r CO2Z2 ['C2I2

•t~e ['(ee-- el) exp [[izco ~ (Re e2 - ~ ) ~ .

(17)

e2

o Imel

(18) The most general method for computing e (co)-containing integrals is based on the use of the analytical properties of e(co) in the upper half-plane of the complex co [21]. In general this method cannot be applied to (18) because the function e*(co) contains in the upper half-plane numerous zeros and poles. An exception is the case when ea and e2 describe just the same substance with different density, i.e. Ime2/Imea = N2/N1 = const, where Nj is the electron density, and (18) reduces to the relativistic limit for the energy loss in the medium el. Deforming the integration contour in (18) into the upper half-plane we obtain as in [21-1: d wl ~ ( z ~ z2)= N2 N~ dz -

where q~(za)=z a exp(izcozj2) Ei(--izcozj2), Ei is the integral exponential function. For zco IzjI >>1

Im

2 2 2 e 2 cop2_f, q07 ~m~--i 2

41~

+~ )

"1

(19)

Here In (2~=(ln(co2+~,2)); averaging is carried out with the weight co Im e~ ~-,co Im e2; ~1 is the root of the equation 1/vZ=gl(i~l) if 1/v2 g1(0); c02j= 4 rcNj eZ/m, m is the electron mass. It should be emphasized that the parameter ~ ~=0 describing the density effect is determined in this case

38 not by the medium e2 in which the energy loss is observed but by the medium e~. The contribution of "close" collisions (e2co~j2)x ln(2mT/q~) does not change near the interface between the media. By adding (19) to the "close" contribution one obtains the local energy loss

dW r d ~ (z 4. zz) eZco2 { 2mTy 2 v~ In -

2

1+

~}

f2~

(20) '

where T = const denotes the maximum energy transferred to an atomic electron. If ~ = 0 , lnO~ = ( l n c o ) = l n I and (20) coincides with the BetheBloch formula for the medium /32 with no density effect correction for 7 >>1 (in this case 11 = I2 = I). For large enough 7 we have ¢1~com7, and the increase of d wr(z 4` az)/dz with In 7 will stop for corn~ >I; the corresponding limiting value for ~ >>l/com is given by

dW T

2 e 2 corn 2mT

(z 4 ` z 2 ) - " ~

in ~ "

(21)

If in the medium ~t the Cherenkov radiation condition /)2 R e a l _ l > > i m e l > 0 is fulfilled for the frequency co, we have 0 ~ - r e (assume that Im et--+0). The second term in the brackets of (23) describes the energy of Cherenkov radiation coming out from the medium el (from the absorption length 1/co Im el) and absorbed in unit length in the medium e2. The contribution of such frequencies is determined by the relation Imsz(co)/Ime1(co) and can be very large if Im al/Im/32 ~ 0. Suppose that in the entire frequency region, where Im a2 4=0, the Cherenkov radiation in the medium el is impossible (for example, a x=~ t). Then in (23) 0 ~ - Im al/(1 -/)2 Re a 0. Taking into account the contribution of "close" collisions it follows that dW r

dz (z4`z2) ~--2e2(@2 (

2m T

}

......~ln ~7222

(lnla--v2e,I)-- 1_.

(24)

Here In 12 = (In co), averaging is carried out with the o9

cop~

weight co Im ez ; 5 a) Im e2 dco=rtco~2/2. To evaluate

Let us compare (21) with the energy loss of the relativistic projectile in the homogeneous medium /32 for 7 >>I/cov~ (the region of the Fermi plateau)

(24) is possible provided that the characteristic absorption Dequencies in the medium/32 are much larger than that in the medium/31, i.e. 12>11. Then e I ~1 - co2vl/co2; (In [1 --/)2/;1 [) ~ in (7- 2 + c@jI~),

0

d W f _" e 2c o2P 2 1 2roT n__ dz

2

dWr

co~ "

It is readily seen that near the interface (z4`z2) for 7>>I/eom the local energy loss in the medium ez differs from the stationary value d Wf/dz for 7 >>I/cop= by the quantity A2 = e2 c02~ ln(copjcom).

(22)

For the gas-dense medium interface cop, 4`cow and A2 > 0, i.e. we will observe the decrease of the density effect in the medium e2. For instance, for corn= 1 eV, cow=20eV, (2mT)l/2=10SeV, the quantity ~2 =A2/(dWf/dz)~-0.35. For the dense medium-gas boundary 62 -~ -0.26, i.e. in a gas DE is amplified. Expressions similar to (19)-(22) can also be obtained in the general case without the assumption Im t31~Im/32. Write (18) in the form

dW , ez -dz ( z 4 ` z 2 ) - 7 o { q2 (1--/) 2 Re el) 0} co Im g2 de), (23) • In c0211_vZel I } imel where 0 = arg(1 - i)2 '~1)"

e2co22 (" 2mT72 ki2>>it)~-~e~m(iz+co~y2)

} 1._

(25)

Expression (25) demonstrates clearly an anomalous DE for d Wr(z 4` z2)/dz. Coinciding for y ~ lz/com with the Bethe-Bloch formula without DE, formula (25) describes the increased (decreased) logarithmic growth of the local energy loss if corn < cop~(corn > cop=)" For y>>Iz/cop, (25) passes over to (21) with an additional term ( - 1 ) in brackets. This small difference is explained by the fact that in (21.) as distinct from (25), the absorption of Cherenkov radiation is taken into account. For I1 >12 such an account becomes essential and therefore one should use the general formula (23). In accordance with above results, in the relativistic case equations (18)-(25) hold also for very thin layers

a 4`z2. As is seen from (22) the local energy loss near the interface between two media differs substantially from the ordinary energy loss, but the restrictive condition Z 4 ` Z 2 makes it very difficult to observe an anomalous DE in d W/d z. Indeed, z 2 ,-, z'2 ~ I2/co2~ be2 2 (2) ~ copJI2. co(2)~I For a dense cause elf 2 and Ime2(coeff) matter I2~cow~-20 eV and we obtain z2---10 -6 cm; for a gas under normal conditions com-~l eV, Ia

39

V. The cross section a ~ ( z , , ) j'/

/" -g

./

/

The results obtained in Sect. IV can be partially applied to the calculation of a~: for z ~ z 2 . For instance, using (5) and (23) we obtain

g

e2

N

--

>a

( T K ( Z <~ Z 2 ) ~ -

[

co o_(K)(cO y ~ )

j,,

..m

co

{ q2 (l--v2 Re el) 0} do), • In c02]l_v2el I + Imel I

t

{n'~z

In"G

[n'~

Fig. 1. The local energy loss as a function of In y near the gas-dense medium interface for z ~ z 2 and (ov,@e)v= (solid line). The dashdotted curve represents the local energy toss for o)m -.+0, the dashed curve describes d W r / d z. Yl = I2/(-opt ; Y2 = I2/(0p2

(26)

where a~r)(o))=o)Ime~2K)(o))/G is the photoelectric cross section for K-shell excitation. Assuming Cherenkov radiation in the medium el is absent we find for coK>>I1 an expression similar to (25)

ar (z < z2) 4roe4 ~l n

mo)rk E1

E2

}

1 .

(27)

E1

It is taken into account here that

\

g-%

q2' 2

(o)~+o)2 72)

do)

~r~K)(o)) -

t

O)K

o)

2

~- ~zo:~v2/na (o r Z = 4 n 2 e2/m O)K.

[

/ I

i

zl

(z~÷ a) Z

Fig. 2. The local energy loss as a function of the depth z for the projectiles penetrating the target z2 surrounded by a medium g~; c%~ ~cot, ~ ; 11 ~-I2~-I. Solid line: ~>>i/c%~ ; dashed line: I/oovz4~y
-~20 eV and z2-~4-10 -4 cm. Since the mean ionization potential is approximately proportional to the atomic number Z and o)p2 ~ Z, the above estimations of z z depends weakly on Z. The lack of the density effect in the energy loss of relativistic electrons has previously been demonstrated experimentally for very thin targets (10 - 4 - - 1 0 - 6 c m ) in [24]. A decrease of the ionization energy loss in a gas proportional counter was reported in [251 and an attempt was made to explain this phenomenon by the transition effects on the counter walls. The latter experiment is however beyond the scope of the effects discussed in the present paper, because the gas layer thickness was a ~ 10 cm >>z z . The dependence of d W / d z on 7 and z is schematically shown in Figs. 1, 2.

' a (K) In this case the characteristic length z 2 ~ Z ,2 ~'~ 2/n, s (o)K)~o)rZ/o922 is much larger than that for the local energy loss d W / d z . In particular, for copper z 2

~- 10 -3 cm and for aluminum z2 ~_ 10-4 cm. These estimations explain qualitatively the D E suppression in o K in experiments with thin targets [1-8]. F o r instance, the logarithmic growth of a K in the copper foil with the thickness a <~10-3 cm must stop, in accordance with (27) only f o r , ~>c%/o)m _~ 10" (if a target is surrounded by a gas under normal conditions, i.e. c%1 ~- 1 eV). For the numerical computations of o-K it is important to take an accurate account the contribution of "close" collisions (q =>qo). It should be noted that the integrals dQtr/do) and dQi~/dco converge as qo ~ co. Therefore it is necessary to find this contribution only to the proper field spectrum dQE/do) (10) or (14). Consequently, the problem is reduced to the case of a homogeneous medium ~2- The close collisions are usually considered to be interactions of the projectile with free electrons. In calculating the local energy loss the limiting parameter qo falls out (see (20)). This is not the case for o-K; one can put q 2 ~ - 2 m o ) r in the contribution due to distant collisions. Although physically justified, this assumption has an uncertain accuracy. More rigorous is the method based on an approximate account of spatial dispersion in e 2 (see Ap-

40

:f AL

At

l•g•"

zo 13

= 10s

//I

"t3

103

Eu

F1 2"O

1o

c,

I

5

I

I

I

x°2

10

"o~ 0.7

g

t~

l~\

°=

Cu

Cu

iI

"~

0.6

z=

:~ "~ ._N

"....... 3 ~

P

0.4

0.5 z=25

0.3

0.4

_

=

I

I

I

0

10

20

0.2F

.

1

/ 101

102

103

10~

l0 s

30

Z (llm)

Fig. 3. K-shell excitation cross section in a l u m i n u m and copper foils as a function of z for a projectile of unit charge. The dash-dotted lines correspond to the cross section ~r2(7~>coK/~%~), dashed curves to the sum of e2(7) and atr(Z , 7), while the solid curves represent ] ~ ) + O ' i n ( 2 , 7)" The curves are the cross section aK(z, 7 ) = O ' 2 ( 7 ) + O ' t r ( Z ,

calculated for 7= 103, 104, 105

pendix) [ i i , I8, 20]. This method is reduced to the substitution q2-~2mco in dQ2/dco and an addition of the term

Fig. 4. The K-sheU excitation cross section as a function of ~ for z = 0 , 1, 10 g m (aluminum) and z = 0 , 3, 25 g m (copper). The theoretical curves describe, respectively, a2 (7) (dash-dotted), the s u m of a 2 (7) and ~rtr(z, 7) (dashed), and crK(z, "~) (solid). The dotted lines represent the relative K-shell excitation yields calculated for a foil thickness

of 10 gm (aluminum) and 25 gm (copper), see text. The values of z are given in microns. The experimental data are taken from [1] (open triangles), [5] (closed circles) and [7, 8] (open circles). The vertical scale on the right is for the dotted lines and open circles

where (,0L denotes the ionization potential of the Lttl L

shell electron; 1:2(COL)= ~ f2 (CO')d co' ~- (Z-- 2)/Z is the d

' Q2 =

dco

o

e2

co'

z~coz Im e~K)(co)

h n e(z~)(co') dco',

(28)

(.o K

(the exchange and spin corrections are negligible small for large enough 7). Numerical calculations of o-K(z, 7) = a2 (7) + o'er(z, 7) + ai.(z, 7) were performed according to (5) with the use of dQ/dco from (14)(16) and (28). We apply the photoelectric cross section av(co)=co Im ~2(co)/n, as tabulated by Veigele [-26] and assume % = 1. The complex function s2(co) was determined from the Kramers-Kronig relation 2 o~ co' Im~2(co')dco' e2(co)--I"kTZ-JL

C0,2

CO2

it~CO

2

cop2 F2 (COL), 0.) 2

(5 ~ 1 eV),

(29)

sum of the atomic oscillator strength fa(CO) in the medium e2 for the frequencies co < coL. The absolute values of the K-shell excitation cross section o'K(z, 7) calculated according to the above method for 7 = 4 + 1 0 5 and z = 0 + 3 0 gm are given in Figs. 3, 4. The width of the transition region z 2 ~ 2 lam (A1) and z2-~ 9 ~tm (Cu) is close to above estimation (Fig. 3). The role of the interference term is on the whole not large but for z ~ z 2 it should necessarily be taken into account to obtain a physically correct result: the continuity of the spectrum dQ(z)/dco should be maintained at the target surface (see Sect. II). This was also stressed in [12, 13]. Figures 3, 4 imply that a~(z~z2) is not subject to the influence of the density effect. In Fig. 4 the curves represent the results of our calculations of a~:(z=const, 7)These curves describe the proper cross section a2(7) in the homogeneous medium e2, the sum of the proper

41 and "transition" cross sections, the s u m O ' 2 ( } ' ) - t - O ' t r ( Z , 7) + o-i,(z, 7)= °'K(z, 7). The dotted curves in Fig. 4 represent the increase of the K-shell excitation yields related to that for 5 GeV/c protons (see the right vertical scale) and correspond to the conditions of the experiments reported in [7, 8]. In producing the dotted curves we have integrated o-K(z, ~) (Fig. 3) over z (OO)/dz. Such a situation may arise when the substratum of an explored target faces the incident particle beam. In this case the additional terms (13) should be taken into account. The rote of the substratum is inessential if its thickness ao~z~"~(1/[2~ r.'(2)f X ] ~ - l / v ] . In the opposite U.,ef limiting case an anomalous DE in the target will be determined by the substratum matter.

sional transferred momentum k = q + no)Iv. If the spatial dispersion is taken into account then d W2/dz will be written in the form (we replace e2 by e): dW 2e2} ° ~dk co do) dz TrY2 o ,o/v k ~Im ,~It (o), k) m2 (v2 k 2 _ 6o2) Im .~±(o), k)) •~. 181112 4 jkZ_o)2e±l 2 o~ d (yrllacro -na I ~'o)

do),

(A.1)

0

where the first and second summands correspond to "longitudinal" (E(o), k ) ~ k ) and "transverse" (E(o), k)±k) collisions. The macroscopic formula (A.1) is in accordance with the quantum-mechanical expression for the differential cross section dami~°/do) of the atom excitation by a charged projectile in the polarizable medium [27] : da_4ne4 ; d/~_ k do) mv2o) o,/v effll (co, k)

o)2(v2k2-o)2)fl(o), k)~

(A.2)

where f(co, k) and f(o))=f(oJ, 0) are respectively generalized and dipole oscillator strength. An account of spatial dispersion in (A.1) corresponds to an account of the dependence on k in f(o), k). Since the "transverse" contribution converges as k ~ o% one can ignore the dependence, of ~±(o), k) on k and put el(o), k)~e(m) in (A.I) or f± (o), k)~_f(o)) in (A.2). Integrating the transverse contribution over k, we have

dz

o

Im

where

do)

gV2 Ime

Im

vz -

In

(1-7v2 e)

.

(A.3)

To calculate the longitudinal contribution one should use either Imetl(co, k) or f(o), k ) = ~ (v, -7 o)l ~ e x p ( - ikq)[O) 2 (amos~k2). Here 10) j=l

Appendix

Let us consider (10) for dQ2/do). Instead of a twodimensional vector q ± n we introduce a three-dimen-

and Iv, o)) are the ground and the excited states of the atom respectively, r~ is the coordinate ofj-th electron, v is the set of quantum numbers numerating the generated states of atom. There are numerous calculations of f(o), k), but we may restrict ourselves

42

to the simplest approximation because, in fact, we

where

employ only integrals of the form i ~ k-~f(co, k)dk.

dQir

e2

dco

TO/.)2

co/v

Let us make use of the semiphenomenologicaI expression for fll (co, k) which approximately describes the dependence off! I(co, k) on k [11, 18, 20]:

fl,(co, k)=f(co)O(co-~m)+F(o.~)6(co-£),

(A.4)

cO

where F(co)= ~ f(co')do)', 5(x) is the Dirac delta-func0

tion; 0(x)= 1 if x > 0 , 0(x)=0, if x<0. The choice of fll (co, k) in the form (A.4) is based on the analysis of the Bethe surface f(co, k) above the plane (co, k) [28]. The function f(co, k) noticeably differs from zero in the region co~ 1~rat,where on the Bethe surface there is a splash referred to as the "Bethe ridge" which becomes more pronounced with increasing k. For k>> 1~rat, when k2/2m>>I, the particle-atom collision reduces to the elastic scattering on free atomic electrons, and then f(co, k) ~,,_Za(co-U/2m). It is of importance to note that for f(co, k) of the form (A.4) there holds "the sum rule": oO

f(co, k) dco = Z,

•{ ~

(A.5)

ln2mv2 4 1 ~° co'Im ( e--~ o ' ) ) dco'} . co co2 Ime o (A.7)

The photoabsorption method for calculating ionization phenomena [-11, 18, 20] is based on (A.3) and (A.7) for dQ±/dco and dQH/dco respectively and on the expression d o-macro

- d~

1 im~(co){dQ±+dQil~

= n~

\--d-~-~

--d-~-~]

2my2 ]

-e2uv2n" {Im [(v2--1) In co(i ~

1 ~ co' Im ( - - ~ ( ~ ) dco'}, +•o

e)] (A.8)

that follows from these formulas. Making use the relation Im e = na %(co)/co, the tabulated photoabsorption cross section o-7 and the Kramers-Kronig relation for e(co), one can calculate do-/dco according to (A.8). Next, integrating numerically over co, one can find dW/dz, dN/dz, dW (col <09<0)2)/dz, etc. In the calculation of the cross section a K the form (A.4) should be employed for the oscillator strength fr~(co, k) corresponding to the K-shell excitation, ~ f K(co, k) dco~-2:

0

OK

if it is fulfilled for f(co). For a dense medium when the system of levels of a individual atom is deformed the role of generalized oscillator strength is played by the quantity 37(co, k) =(2Zco/nco2) x I m ( - l/e) for which there holds the sum rule (A.5). The insertion of Y(co, k) in the form (A.4) into (A.1) yields: dW/[ - e2 dz roy2

{fcoIme ~

2my 2

In

co

fK (co, k) ~-f~ (co) 0 (co- k2/2 m) + F~:(co)6 (co-- k2/2 m) where FK(co)= ~ fK(co')dd.

(A.9)

al K

do) (A.6)

For the spectrum dQ/dco one can derive the expressions (A.3) and (A.7); the last term in (A.7) should be written in the form

The second term in (A.6) describes the contribution of "close" collisions and therefore

e2 1 ~ co' Ime(m(co')dco'. (A.10) ~/)2 (I)2Im e(m(co)Jg(co)[o,,

,do

+ j ~ - ~ co' Im -0

dWtt-

~ co I m e o

do' .

0

do),

Formula (28) in the text is the relativistic approximation of this expression•

43

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