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High-energy approximations to nuclear scattering
ANNALS
OF PHYSICS
84, 147-164 (1974)
High-Energy
Approximations
to Nuclear
Scattering
W. E. FRAHN AND B. SCHBRMANN Physics
Department,
University
of Cape
Town,
Rondebosch
(Cape),
South
Africa
Received April 13, 1973
Starting from Glauber’s eikonal approximation, we develop systematic approximation procedures for high-energy nuclear scattering based on expansions of the free-space propagator. We distinguish two different expansions, the “eikonal expansion” and the “Fresnel expansion,” and interpret their physical meaning in optical terms. In Fresnel approximation we derive, by means of a unitary transformation, a closed representation of the scattering amplitude which is formally similar to Glauber’s expression. In the present paper these approximation methods are formulated for potential scattering; application to multiple scattering in nuclei will be described in a subsequent article.
1. INTRODUCTION Nuclear scattering at high energies has been very successfully described by means of Glauber’s eikonal approximation [l]. This approximation rests on the assumption that (i) each scatterer (nucleon) stays at a fixed position during a collision, (ii) the propagation of the projectile between two successive collisions proceeds according to geometrical optics, and (iii) the total phase shift of the scattered wave is composed additively out of the phase shifts imparted by the individual collisions. Glauber’s theory has been most successfully applied to multiple scattering of nucleons (and other hadrons) by complex nuclei at energies in the GeV region [2], where it allows to calculate nucleon-nucleus collisions in terms of the parameters of free nucleon-nucleon scattering. The crucial assumption of Glauber’s approximation is the second of those enumerated above; propagation according to geometrical optics amounts to replacing the full free-space Green function by a propagator of eikonal type. In this version, Glauber’s theory can also be applied to potential scattering, which was in fact the first application considered in Ref. [l]. The GIauber approximation is the better the higher the energy. If the theory is applied at energies below the multi-GeV region, various corrections to its basic assumptions may become significant. If we adopt the viewpoint that the essential assumption is the one concerning the propagator, we should seek to improve the 147 Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
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FRAHN AND SCHijRMANN
theory for lower energies by taking account of deviations from straight-line propagation. Over the past few years there have been many efforts in this direction [3-121. Sugar and Blankenbecler [5] were the first to study corrections to the Glauber model in a systematic way by expanding the full free-space Green propagator in terms of an eikonal-type propagator. A relativistic eikonal expansion has been proposed by Abarbanel and Itzykson [6]. Its nonrelativistic version has been studied by Kujawski [lo]. Recently Wallace has discussed in great detail an eikonal expansion with the Glauber expression as the leading term [8, 121. Based on Glaubcr’s original method of deriving a first-order eikonal approximation [I], Baker has obtained corrections to the Glauber model of first order in the reciprocal wave number k-l [ 111. The relationship between certain corrections to the Glauber theory and Fresnel diffraction effects was first pointed out by Gottfried in a study of high-energy multiple scattering from deuterium [9]. In the present paper we develop systematic approximation procedures by suitable expansions of the full free-space propagator, similar to the method used by Sugar and Blankenbecler. These are first formulated in momentum representation; by transformation to configuration space we obtain expressions for the various approximate Green functions which allow an interpretation of their physical meaning in wave-optical terms, much in the spirit of Gottfried’s lucid article [9]. Both Gottfried’s “Fresnel approximation” and Baker’s “second-order eikonal approximation” are obtained very simply as first-order corrections from our systematic expansions. For simplicity, our approximation methods will be formulated for potential scattering, even though the potential concept has little significance at the energies for which they are valid. Furthermore, we confine ourselves to nonrelativistic dynamics. Application to multiple scattering of nucleons in complex nuclei will be presented in a subsequent paper [13].
2. FIRST-ORDER EIKONAL APPROXIMATION
The wave function for nonrelativistic scattering of a particle of mass m and wave number k by a local potential V(r) satisfies the integral equation
I)(‘) = eik*-
I dr’ G(r, r’) V(f) fir’),
where eiklr-r’l
G(r, r’) = -!!2dP 1r - r’ I
(2)
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is the free-space Green function momentum representation G(r, r’) = $
j f$
z-e 2k hv
149
SCA’ITERING
for outgoing-wave
boundary
conditions.
In
.&~(r-r’)(~Z - ,Z$ _ is)-1 d3A &-‘)(A2
ik(z-z’)
s cw3
(3)
+ 2kA - +-I,
where we have introduced A = p - k, taking k in z-direction, and the particle speed v = fiklm. Now we split all momenta into transverse (index JJ and longitudinal (index 11) components with respect to the z-direction; a similar split in coordinate space introduces the two-dimensional impact parameter vectors b. Then we may write G(b - b’, z - z’) = z1 eikh-2’)
j
;f;
1
@$
eiA.(k-k’)ei~,(z-r’)e(Al
, A ,,),
(4)
where (?(A, , A ,,) = (h + A ,, - i+’
(5)
and where we have defined h = h, + h,, = AL2/2k + A,,2/2k.
(6)
If in Eq. (5) we neglect h compared with A ,, , we obtain the “eikonal propagator” CT!) = (A,, - ie)-l.
(7)
The index (1) indicates that we shall regard this as the “first-order eikonal approximation,” as distinct from higher-order eikonal approximations to be defined in Section 5. In coordinate space the eikonal Green function has the form G:‘(b - b’, z - z’) = (i/h)
6(2)(b - b’) e(z - z’) eik(‘-“),
(8)
where O(z) is the unit step function. The delta function of impact parameters means that the propagating wave suffers no transverse displacement; thus, the eikonal approximation corresponds to propagation according to geometrical optics. To solve the integral equation (I), the wave function is first written in the form of the incoming plane wave multiplied by a modulating function 4(r), #(r) = eikr$(r),
(9)
150
FRAHN
AND
SCHijRMANN
where 4(r) satisfies d(r) = 1 - 1 d3r’ evikcr-*‘)G(r, r’) V(r’) +(r’). By inserting (8) in place of G(r, r’) we have @(b, z) = 1 - ;
J-1 dz’ O(z - z’) V(b, z’) @‘(II, z’),
(11)
which has the solution $F’(b, z) = exp [ - i
1-L e(z - z’) V(b, z’) dz’].
(12)
It means that the only effect of the interaction is a modulation of the incoming plane wave, by a phase shift in case of a real potential, and by a reduction in amplitude as well, in case of a complex potential. The scattering amplitude has the general form s =--
e”q*V(r) $(r) d3r
k & 2n-h s
(13) eW
s
dz eiouzV(b, z) $00, z),
where fiq = h(k - k’) is the momentum transfer. To obtain the first-order eikonal approximation 4:“. In addition, since
to Eq. (13) we replace q5 by
(14)
is of the order k-l for fixed momentum transfer, we may set 4 ,, = 0 in the eikonal limit k -+ co. This amounts to the assumption of forward scattering as described by the eikonal propagator (8). Equation (13) then becomes fe(qJ = - &
j d2b eiqLb 1 dz V(b, z) &b,
z)
with the profile function F,(b) = 1 - S(b) = 1 - eixfb).
(16)
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SCATTERING
151
Here $x(b) plays the role of the phase shift in impact parameter representation and is given by
x(b) = - & j-;00dz v(b,z).
(17)
Equations (15)-(17) are the well known results of Glauber.
3. SECOND-ORDER EIKONAL
APPROXIMATION
An alternative derivation of the eikonal result (12), which emphasizes the highenergy character of the eikonal approximation, proceeds through partial integration in Eq. (10) in spherical coordinates over the cosine of the scattering angle, and dropping the resulting terms of order @a)-l and higher, where (z is a characteristic dimension of the scattering potential. This shows that the first-order eikonal approximation holds in the high-energy limit ka + co. To obtain corrections for lower energies it seems natural to carry the same procedure to first order in (ka)-l. This has been done recently by Baker [l I] who, after a somewhat lengthy calculation, arrives at a result which in our notation may be written in the form
+ &
J-1 dz’ (z - z’) O(z - z’) V,2V(‘b, z’) @‘@, z’),
(18)
where Vh2 is the two-dimensional Laplacian in impact parameter space. In Section 5 we shall rederive Baker’s equation (18) in a very simple fashion as the second order of a systematic approximation method, and also give a physical interpretation of it in “optical” terms.
4. FRESNEL APPROXIMATION In view of the fact that the (first-order) eikonal Green function (8) represents propagation according to geometrical optics, one may look for corrections which describe wave-optical modifications of the scattered wave. This has been done by Gottfried [9] for multiple scattering and applied to high-energy scattering by deuterium. Let us forget about potential scattering for a moment and assume that the scatterer (nucleon within a nucleus) may be regarded as perfectly absorbing. The eikonal Green function (8) then describes the formation of a cylindrical
152
FRAHN AND SCHiiRMANN
“shadow” extending to infinity behind the scatterer. In actual fact, diffraction at the scatterer causes a distortion of the geometric-optical wave field which destroys the shadow at distances of the order of and larger than the Rayleigh distance dR = kc?. Thus the eikonal Green function (8) approximates the propagation only at distances z < dR , and the Glauber approximation for multiple scattering will only be valid if the Rayleigh distance is large compared with the mean free path of the projectile in the target nucleus. Since dR is proportional to k, this condition can always be satisfied if the energy is made high enough. At lower energies, however, the diffractive distortion of the geometric-optical shadow may become appreciable. Gottfried takes these effects into account by employing the Fresnel-Kirchhoff formulation of Huygens’ principle in representing the scattered wave function. This amounts to replacing the eikonal Green function (8) by the “Fresnel-Green function” Gj;l’(b - b’, z - z’) = Go
e(z _ z,) eik(z.+‘) exp[i &k(bz--b’L,“/(z - z’)] . (19)
The index (1) again indicates that we shall regard this as the first order of a systematic “Fresnel approximation” method to be defined in the following section. In momentum representation the Fresnel propagator has the form G!) = (h I + d I - ie)-’ ,
(20)
i.e., it arises from the full propagator (5) by neglecting h,, . The relation between (19) and (20) was noticed by Gottfried (see footnote 9 of Ref. [9]).
5. SYSTEMATIC APPROXIMATION
METHODS
We shall now develop systematic approximation methods for improving the eikonal propagator (7) which proceed in different directions indicated by the and the “first-order Fresnel approxi“second-order eikonal approximation” mation,” respectively. We start with the full propagator (5) in momentum representation, e
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SCATTERING
Alternatively, we may regard h ,, as a “perturbation” of the Fresnel term h, + d H, and write the relation between C? and C?j.?’in the form G = ($!’ - ($‘h ue: *
(22)
Finally, we have a relation between C$) and ekl), ($’
= ($1) _ ,‘$l’, ($1, e e I F-
(23)
Our approximation methods now consist in iterating in Eqs. (21) and (22) with respect to @) and e$), respectively, e = @) _ ,p),,!’
+ ($)h@‘h@’
7 .. .
(24)
and G = ej$ - @‘h,@’
+ @‘h,@‘h,($!’
F -.. .
(25)
Equations (24) and (25) define the “eikonal expansion” and the “Fresnel expansion” of the free-space propagator, respectively. Going beyond the first-order eikonal approximation e M CA’) to the nexthigher order, we obtain
The physical meaning of this second-order eikonal approximation in coordinate representation. The result is
is best discussed
G$‘(b - b’, z - z’) = eikb-z’) .-& S(Z)@- j,‘) e(z - z’) [ - Au
V,26(2)(b - b’)(z - z’) e(z - z’)
- &
8(2@ - b’) 6(z - z’)].
(27)
Inserting (27) in place of G(r - r’) in Eq. (10) yields precisely Baker’s equation (18). It may be useful to indicate briefly how Eq. (27) is derived. When G(A, , LI ,,) in Eq. (4) is replaced by GL2) we obtain three contributions. The first, from @), yields GLl)(b - b’, z - z’). The second, from the term with h, , involves the integral
= - g z&z).
(28)
154
FRAHN
The third contribution, * f
eiAl
AND
SCHiiRMANN
from the term with h,, , involves Z(-$‘,
I,
27r
1
G(l) e
=-
.&
2k
eiA1/“A ,,‘(A ,, - kp2
2~
s
(2% = $6(z).
Going beyond the first-order higher order, we have
Fresnel approximation
G w Gg) to the next-
I$ M I$!’ - C$‘h ,,C$’ c cl-“‘.
(30)
In coordinate representation we find Gt’(b - b’, z - z’) =
1
_
1.
@
-
b’)’
_
2 (z - z’)2
j
1
k
4
cb
-
b’)4
1
($)(,,
(z - 233
+Ava(z _z')emu--l')
_
b’,
z
exp[i &k(b - b’)2/(z - z’)] z - z’
_
zy
(31)
Again we give a brief outline of how (31) is derived. When G(n, , d ,,) in Eq. (4) (2), the first contribution, from I?$), yields G$)(h - b’, z - z’). is replaced by C?, The second contribution, from the term with h,, , is (2) GF
-
Gj71'=
-
__ '
eik(z-z’)
j
;f;
j
;$
eiAL(b-b’)eiA,(z-z’)
2fikv (32)
Transforming yields GF’
from d ,, to d ,,’ = AL2/2k + A,, and thereafter dropping the prime ik(Z-2”
-G$)=-Le
aAlfb-b’)e-i(d,a/2k)(z-z’)
2fikv . 1 $$
eiA~(z-z’)[A12 - (A12/k) A,, + A14/4k2](A,, -
i~)-~. (33)
With [A ,,2 - (A12/k) A ,, + AL4/4k21(A ,, - i~)-~ = 1 - (A,s/k)(A ,, - ir)-l + (A14/k2)(A ,, - ie)-“,
(34)
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SCATTERING
the integration over d ,, can be carried out with the result
&)-G&--leik(Z-2’) s
iAl(b-b’)e-i(d,2/2k)(z-z’)
2?ikv
- [6(z - z’) - i(AL2/k) O(z - z’) - (dL4/4k2)(z - z’) O(z - z’)]. (35) The integration over LIP is then straightforward.
6. PARTIAL
SUMMATION
OF THE EIKONAL
EXPANSION
After writing Eq. (21) in the form
the eikonal expansion (24) may be regarded as composed of three contributions,
e = G f e2f G ,
(37)
where Q;‘, contains @) plus the terms involving h, only, e, those involving h,, only, and e, the “mixed” terms involving products of h, and h,, . Since all the factors in the expansion are ordinary functions and commute, we have t$ = @) - (@‘)2 h, + (($‘)3 hL2 F ... = @) f
(-l)n
(@)h,)n
= f$‘,
?Z=O
(38) where the last equality follows from Eq. (23). Thus the sum total of the terms involving h, in the eikonal expansion is just the first-order Fresnel approximation. Hence the coordinate representation G,@ - b’, z - z’) is given by Eq. (19). The second contribution can also be summed in closed form, C?, = -(GF’)2 h,, + (@‘)3 h,,2 T ... = ,e(,’ f (-l)“(@h,)“. n=l
(39)
With @)h ,, = d ,,/2k and (el’))” h ,, = (2k)-l we have e, = - &
f TZ=O
(-1)”
(d,,/2k)“.
(40)
156
FRAHN
AND
SCHiiRMANN
In coordinate representation this yields Gz(b - b’, z - z’)
zxz
-
kv
8(z)@
-
eikcz+ ) (1 + 2kl -g-J a -l 6(z - 2’).
b’)
Since we have closed expressions for G, = G$!) and Gz, and of course, for the full propagator G, the mixed contribution G, may also be regarded as known in closed form G3 = G - G!’ - G2.
(42)
7. PHYSICAL INTERPRETATION The two expansions of the full free-space propagator defined in Section 5, Eqs. (24) and (25), yield successive approximations subject to the conditions 4 +h,
9
eikonal approximation,
(43)
Fresnel approximation.
WI
Thus the Fresnel approximation only requires that h ,, = d ,,2/2k is small compared with the eikonal term d I , whereas in the eikonal approximation the sum h = (AL2 + d ,,“)/k is small compared to d ,, . Indeed, as we have shown in the preceding section, the first-order Fresnel approximation @$ already represents a partial sum of the eikonal expansion: it comprises to all orders the terms involving h, only. Thus the two expansions are physically distinct, emphasizing different aspects of the scattering process. The Fresnel expansion concerns the wave-optical (difsractive) deviations, in the transverse direction, from geometric-optical propagation. As we have discussed earlier, following Gottfried’s interpretation, the first-order Fresnel approximation accounts for the diffractive distortion of the geometrical shadow beyond the Rayleigh distance ka2. This is caused by a distortion of the scattered wave front due to diffraction at the boundary of the scatterer. The Fresnel propagator G$) describes this distortion to first-order as a transverse curvature of the wave front, by retaining the term proportional to 00 - b’)2 in the phase of the Green function.
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This is best seen by an alternative derivation of G,(‘) from the full free-space Green function in the form G(b - b’, z -
z’) eik[(b-b~)a+(z--z')2]1'e
k =
7&i
[(b
-
by
+
(z
-
k z')2]1/2
eik(z-z~)u+8P'~
= 27Tfiu (2 - z’)(l + /3)1/S .
(45)
Expansion in powers of fi = (b - b’)2/(z - z’)~ and retaining the leading term in the phase only, yields Eq. (19) except for the factor 0(z - z’) which restricts propagation to the forward direction. The integral equation (1) or (10) may be regarded as an exact formulation of (the quanta1 analog of) Huygens’ principle: the scattered wave at observation point r is formed by interference of freely propagating spherical waves G(r - r’) originating with amplitude V# from each source point r’ in the interaction region. The firstorder Fresnel approximation represents an approximate form of Huygens’ principle (corresponding to the Fresnel-Kirchhoff approximation in physical optics), in which only the leading distortion of the wave front in transverse direction (i.e., over the b-plane) is retained and the longitudinal (z) propagation is restricted to the forward direction. Higher-order terms in the Fresnel expansion correspond to higher powers of /3 in the expansion of both amplitude and phase of G in Eq. (45). This may be seen from the form (31) of the second-order Fresnel propagator. Each term of the Fresnel expansion contains contributions of all orders in k-l. If such a term is further expanded in powers of k-l, the transverse curvature of the wave front is represented by powers of the transverse Laplacian Vb2. For instance, the expansion of the Fresnel propagator G$) with respect to k-l is G$‘(b
=
b’ 3 z - z’) ew2k)(s-z')v
Ir2G, (1) (b -
= e(z - z’) &M-Z’) k
b’, z -
[ia(
z’)
- b’) - LIZ&
V,28(2)(b _ b’) + ***I,
(46)
which is, of course, nothing else than the coordinate representation of the expansion obtained by iteration of Eq. (23), (47) Comparison with Eq. (24) shows that this forms part (e,) of the eikonal expansion which we consider next. The eikonal expansion represents a systematic expansion in increasing orders of k-l, or rather in powers of (ka)-l where a is the characteristic dimension of the scatterer. It therefore concerns the deviations from the asymptotic energy limit
FRAHN AND SCHiiRMANN
158
ka -+ co. In each order beyond G,(l) it collects contributions describing both longitudinal (eikonal type) and transverse (Fresnel type) distortions of the geometric-optical pattern represented by G,(I). For instance, in the second-order eikonal approximation Gi2) these are given by the third and second terms, respectively, in Eq. (27). The partial summations given in Section 6 show that the transverse distortions, represented by the derivatives of 8(2)(b - b’), result mainly from the contribution G, = G$) determined by h, , and that the longitudinal distortions, described by the derivatives of e(z - z’), are mainly due to the contribution G, determined by h,, . However, the presence of the “mixed” contribution G3 shows that there is no clear-cut separation between transverse and longitudinal distortions in the eikonal expansion.
8. FFCESNJXL APPROXIMATION
IN EIKONAL
REPRESENTATION
In the preceding section we have seen that the (first-order) Fresnel and eikonal Green functions are related by a unitary operator, ,‘$‘(,, _ ,.,‘, z _ z’) = e-(il’iQ)~~(Z-Z’)G~)(b _ ,,‘, z _ z’),
(48)
where R, = -(ii2/2m)
Vb2
(49)
is the transverse kinetic energy operator. This remarkable connection allows us to represent the Fresnel approximation of the scattering amplitude by a closed expression which is formally analogous to the eikonal (Glauber) amplitude. This is achieved by means of a unitary transformation of the Fresnel solution which is a precise formal analog of the transformation from the Schrbdinger picture to the interaction picture in time-dependent perturbation theory. Starting from the SchrGdinger equation for the modulating function 4(r) = +(b, z) defined by Eq. (9), (50) where (51) is the full kinetic energy operator, the Fresnel and eikonal wave functions +F and c#@ satisfy the wave equations (ifiWW and
- KJ Mh
4 = Wh z> 4dh z>
(52)
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respectively. These equations are formally analogous to the time-dependent Schrodinger equation with a “time parameter” T = z/v and a time-dependent interaction Hamiltonian. (According to Ref. [14], this analogy for the eikonal equation (53) was previously noticed by Reading.) By a unitary transformation from the “Fresnel picture” 1c$~) to an “interaction picture” ] 4,) with respect to the interaction V(b, z), $F(b, z) = e-(i’fiv)kLz~I(b, z),
Eq. (52) transforms to an equation of “eikonal”
(54)
type,
ifiv@lW +100, 4 = Q, 4 4dh 4,
(55)
with an interaction operator (56) The “z-translation
operator”
&(z, z,J defined by
Mb, 4 = r3,k 4 4,@, 4
(57)
satisfies the same equation (55); in integrated form it$,(z, z,,) = 1 - ;
1’ dz’ p(b, z’) it&(z’, zJ. 10
Equation (58) is formally solved by a Dyson-type perturbation &(z, zo) = 1 + f
n=l
(-
k)n
$1’
-0
dz, ..a s:, dz, Z[@,
(58) expansion [15]
z,) ..a p(b, z,)],
(59)
where the z-ordering operator 2 arranges the noncommuting interaction operators P in the order of increasing z-values from right to left. Writing symbolically O,,(z, zo) = Z exp [ - k jZl dz’ @b, zf)], the modulating
(60)
function in the “eikonal interaction picture” becomes Mb, z> = %,
-a>
Mb, -a>
= (r3,(z, -a)>,
(61)
where (o,,(z, -co)) denotes the expectation value, in the constant state &(b, -co) = 1, of th e operator 4 . The asymptotic form of Eq. (61) for z + co, ;+%
$I@,
z> =
SF(b)
(62)
160
FRAHN
AND
SCHtiRMANN
defines the scattering function S,(b) = (&,). The scattering operator $, is given by 9, = Q(co, -co)
=
Z[&b)
]9
(63)
where in analogy to Eq. (17) j(b) = - & j-:
m
w
dz p(b, z)
is the “phase shift operator” in the interaction picture. In order to obtain the Fresnel approximation to the scattering amplitude we replace the wave function 4 in Eq. (13) by $F of Eq. (54). In addition we must use an approximation for q,, which is compatible with the Fresnel propagator (20). Since the Fresnel approximation is supposed to be valid for larger scattering angles than the Glauber approximation, the forward scattering restriction q,, = 0 is now inadequate and has to be replaced by
which is obtained by neglecting the term quadratic in q,, in Eq. (14). Using this approximation for q,, in Eq. (13) and noting that exp[i(qL2/2k) z] eiaLb = exp[(z@) KLz] eiaLb,
(66)
a partial integration over b yields hh)
= -
&
= &
I d2b eiaLb 1 dz I%,
z) &(b,
l d2b eiaLbTF(b)
z)
(67)
with the profile function T,(b) = 1 - S,(b) = 1 - eixptb).
(68)
The second equality in Eq. (67) has been obtained by using fl(b, .+@&,
- co)) = ifiV g <&(z, - CO)).
(6%
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Thus, the Fresnel approximation has been cast into a form which resembles Glauber’s expressions (15)-(17) for the eikonal approximation to potential scattering. It is shown in Ref. [13] that Eq. (67) is identical with the amplitude obtained in the “optical limit”
of the Fresnel approximation
to multiple scattering.
Although Eq. (67) shows that Fresnel diffraction in high-energy scattering can be described by a closed “Glauber-type” expression of the scattering amplitude, this result may appear to be merely of formal interest and of no immediate practical use. Of course, there would be no point in trying to evaluate the perturbation expansion in powers of the potential operator P; thereby one would lose the decisive advantage of Glauber theory (over, say, the Born approximation) of incorporating the effects of the interaction potential to all orders. However, the result (67) is useful as a starting point of a different approximation method, by expanding the interaction operator P (defined by Eq. (56)) in powers of to k-l is the (z/h) I?, = -(2/2k) Vb2, which because of its proportionality appropriate expansion operator at high energy. In fact, this approximation procedure is equivalent to the expansion (46) or (47) of the Fresnel propagator in powers of the eikonal propagator. To first order in this expansion, Ti = V, and Eq. (67) reduces to the Glauber formula. To second order, v = v + i(z/2k)[ V, V,2]. Inserting (70) in (63) and retaining only contributions the perturbation series yields
(70) of order k-1 in each term of
St’(b) = S(b)[l - (i/k) d,(b)].
(71)
Here S(h) is the Glauber scattering function of Eq. (16) and d,(b) is defined as h(b)=
-&va2jw
--m
dzzV(b,z)
For spherically symmetrical potentials this reduces to d,(b) = (+-)’
b $ JOmdz [V(b, z)]“.
(73)
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FRAHN AND SCHtiRMANN
The form of the function d,(b) indicates that its main contribution comes from the surface region of the potential V, in accordance with the physical picture of Fresnel diffraction. Details of the derivation of Eqs. (72) and (73), and the lowest-order Fresnel correction to the Glauber approximation for nuclear multiple scattering, are described in Ref. [16]. The second-order eikonal approximation, which contains ail terms to the order k-l, differs from the first-order Fresnel approximation by the contribution arising from the longitudinal kinetic energy operator R,, = -(fi/2m) P/az2. In this approximation the scattering function is given by [16]
$%9 = W)[l - (d&h@) + G-N,
(74)
where d,(b) = $(l/fi#
jrn dz [V(b, z)]“. --m
(75)
For a spherically symmetrical potential, Eq. (74) agrees with the expression given by Wallace [12]. It disagrees, however, with Baker’s result [ll]. The reason presumably is that in calculating the scattering amplitude from 4:“) of Eq. (18), Baker has used the forward scattering restriction exp(iq,,z) M 1, whereas the correct approximation is exp(iq ,,z) m 1 + i(qL2/2k) z, in accordance with Eq. (65). Finally, we mention that for nuclear multiple scattering there are additional first-order eikonal-type corrections which arise from rescattering on the same nucleon [16]. These terms largely cancel the contribution corresponding to Eq. (75), leaving the Fresnel correction as the dominant term. This underlines the significance of the Fresnel approximation in nuclear multiple scattering.
9. RELATION
TO OTHER EXPANSIONS
In this section we briefly discuss how our expansions are related to those mentioned in the introduction. The approaches by Abarbanel and Itzykson [6], Kujawski [lo], and Wallace [8, 121 differ from ours in that the intermediate momentum p is expanded about the average momentum & = &(k + k’) rather than about the initial momentum k. Whereas the leading term in Wallace’s series is the Glauber expression, it has a slightly different form in the series of AbarbanelItzykson. This is because in the former the projectile speed in the direction i; is u = +ik/m, while in the latter it is u’ = ZJcos $9, where 0 is the scattering angle. Taking the z-axis in the direction of & in both cases automatically yields qn = 0 so that no additional assumption about q ,, has to be made. This is often considered to be an advantage over expansions about the initial momentum k.
HIGH-ENERGY
SCATTERING
163
On the other hand, as indicated at the end of Section 8, taking the effect of the longitudinal momentum transfer properly into account to first order in k-l, yields a result equivalent to Wallace’s It is obvious that with the appropriate assumptions about q,, this equivalence holds true for any given order of k-l. (It should be mentioned that the importance of the longitudinal momentum transfer in connection with corrections to the Glauber model has previously been pointed out in Refs. [17] and [9].) Because of q ,, = 0 in his approach, Wallace immediately obtains an impact parameter representation for the scattering amplitude. However, this is at the cost of an expansion for the Green propagator more complicated than ours. In particular, there seems to be no simple way of extracting a Fresnel propagator from Wallace’s expansion. Since the main aim of the present paper is the development of systematic high-energy scattering approximations which have an immediate physical interpretation in wave-optical terms, the expansion about the initial momentum is the more appropriate choice.
10. SUMMARY AND CONCLUSION We have developed systematic approximation methods for high-energy nuclear scattering which improve upon the Glauber approximation. These methods are based on expansions of the free-space propagator which emphasize different aspects of the scattering process. Since the zeroth order of these expansions, the eikonal propagator, represents at the same time the high-energy limit ka -+ co and the geometric-optical limit of propagation, we have distinguished two different approximation procedures. The eikonal expansion emphasizes the deviations from the asymptotic energy limit and proceeds in powers of (ka)-l. The Fresnel expansion emphasizes the deviations from geometric-optical propagation caused by d@ractive distortion of the propagating wave front in transverse direction. To first order in these expansions we recover expressions derived by Gottfried [9] and by Baker [l I] and Wallace [12], respectively. By partial summation of the eikonal expansion we establish the connection between the two approximation procedures and give a physical interpretation in “optical” terms. By means of a unitary transformation analogous to the one that leads from the Schrodinger picture to the interaction picture in time-dependent perturbation theory, we have derived a closed expression for the scattering amplitude in Fresnel approximation which is formally similar to the Glauber formula. From this expression we calculate, to first order in k-l, the Fresnel correction to the Glauber amplitude. Although these methods have been formulated for potential scattering, they are concerned only with the propagators and can therefore readily be applied to multiple scattering. The Fresnel approximation to multiple scattering in complex nuclei will be dealt with in a subsequent paper [13].
164
FRAHN AND SCHiiRMANN REFERENCES
“Lectures in Theoretical Physics,” Vol. 1 (W. E. Brittin and L. G. Dunham, Eds.), Interscience, New York, 1959. R. J. GLAUBER, “High-Energy Physics and Nuclear Structure” (S. Devons, Ed.), Plenum Press, New York, 1970, and references given therein. D. S. SAXON AND L. I. SCHIFF, Nuovo Cimento 6 (1957), 614. T. T. WV, Phys. Rev. 108 (1957), 466. R. L. SUGAR AND R. BLANKENEIECLER, Phys. Rev. 183 (1969), 1387. H. D. I. ABARBANEL AND C. ITZYKSON, Phy.s. Rev. Lett. 23 (1969), 53. R. J. MOORE, Phys. Rev. 02 (1970), 313. S. J. WALLACE, Phys. Rev. Lett. 27 (1971), 622. K. GOTTFRIED,Ann. Phys. (N. Y.) 66 (1971), 868; “Optical Concepts in High-Energy Physics,” CERN lectures 1971, CERN 72-20, Geneva, 1972. E. KUJAWSKI, Phys. Rev. D 4 (1971), 2573; Ann. Phys. (N. Y.) 74 (1972), 567. A. BAKER, Phys. Rev. D 6 (1972), 3462. S. J. WALLACE, Ann. Phys. (N. Y.) 78 (1973), 190. This paper appeared after the original version of the present article had been submitted. B. SCH~~RMANNAND W. E. FRAHN, Nucl. Phys. B 62 (1973), 365. H. FESHBACH AND J. H-R, Ann. Phys. (N. Y.) 56 (1970), 268. F. J. DYSON, Phys. Rev. 75 (1949), 486. B. SCH&~MANN, Nucl. Phys. B 67 (1973), 425. W. H. BASICHIS, H. FESHBACH, AND J. F. READING, Ann. Phys. (N. Y.) 68 (1971), 462.
1. R. J. GLAUBER, 2.
3. 4. 5. 6. 7. 8.
9. 10.
11. 12. 13. 14. 15. 16. 17.
OF PHYSICS
84, 147-164 (1974)
High-Energy
Approximations
to Nuclear
Scattering
W. E. FRAHN AND B. SCHBRMANN Physics
Department,
University
of Cape
Town,
Rondebosch
(Cape),
South
Africa
Received April 13, 1973
Starting from Glauber’s eikonal approximation, we develop systematic approximation procedures for high-energy nuclear scattering based on expansions of the free-space propagator. We distinguish two different expansions, the “eikonal expansion” and the “Fresnel expansion,” and interpret their physical meaning in optical terms. In Fresnel approximation we derive, by means of a unitary transformation, a closed representation of the scattering amplitude which is formally similar to Glauber’s expression. In the present paper these approximation methods are formulated for potential scattering; application to multiple scattering in nuclei will be described in a subsequent article.
1. INTRODUCTION Nuclear scattering at high energies has been very successfully described by means of Glauber’s eikonal approximation [l]. This approximation rests on the assumption that (i) each scatterer (nucleon) stays at a fixed position during a collision, (ii) the propagation of the projectile between two successive collisions proceeds according to geometrical optics, and (iii) the total phase shift of the scattered wave is composed additively out of the phase shifts imparted by the individual collisions. Glauber’s theory has been most successfully applied to multiple scattering of nucleons (and other hadrons) by complex nuclei at energies in the GeV region [2], where it allows to calculate nucleon-nucleus collisions in terms of the parameters of free nucleon-nucleon scattering. The crucial assumption of Glauber’s approximation is the second of those enumerated above; propagation according to geometrical optics amounts to replacing the full free-space Green function by a propagator of eikonal type. In this version, Glauber’s theory can also be applied to potential scattering, which was in fact the first application considered in Ref. [l]. The GIauber approximation is the better the higher the energy. If the theory is applied at energies below the multi-GeV region, various corrections to its basic assumptions may become significant. If we adopt the viewpoint that the essential assumption is the one concerning the propagator, we should seek to improve the 147 Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
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FRAHN AND SCHijRMANN
theory for lower energies by taking account of deviations from straight-line propagation. Over the past few years there have been many efforts in this direction [3-121. Sugar and Blankenbecler [5] were the first to study corrections to the Glauber model in a systematic way by expanding the full free-space Green propagator in terms of an eikonal-type propagator. A relativistic eikonal expansion has been proposed by Abarbanel and Itzykson [6]. Its nonrelativistic version has been studied by Kujawski [lo]. Recently Wallace has discussed in great detail an eikonal expansion with the Glauber expression as the leading term [8, 121. Based on Glaubcr’s original method of deriving a first-order eikonal approximation [I], Baker has obtained corrections to the Glauber model of first order in the reciprocal wave number k-l [ 111. The relationship between certain corrections to the Glauber theory and Fresnel diffraction effects was first pointed out by Gottfried in a study of high-energy multiple scattering from deuterium [9]. In the present paper we develop systematic approximation procedures by suitable expansions of the full free-space propagator, similar to the method used by Sugar and Blankenbecler. These are first formulated in momentum representation; by transformation to configuration space we obtain expressions for the various approximate Green functions which allow an interpretation of their physical meaning in wave-optical terms, much in the spirit of Gottfried’s lucid article [9]. Both Gottfried’s “Fresnel approximation” and Baker’s “second-order eikonal approximation” are obtained very simply as first-order corrections from our systematic expansions. For simplicity, our approximation methods will be formulated for potential scattering, even though the potential concept has little significance at the energies for which they are valid. Furthermore, we confine ourselves to nonrelativistic dynamics. Application to multiple scattering of nucleons in complex nuclei will be presented in a subsequent paper [13].
2. FIRST-ORDER EIKONAL APPROXIMATION
The wave function for nonrelativistic scattering of a particle of mass m and wave number k by a local potential V(r) satisfies the integral equation
I)(‘) = eik*-
I dr’ G(r, r’) V(f) fir’),
where eiklr-r’l
G(r, r’) = -!!2dP 1r - r’ I
(2)
HIGH-ENERGY
is the free-space Green function momentum representation G(r, r’) = $
j f$
z-e 2k hv
149
SCA’ITERING
for outgoing-wave
boundary
conditions.
In
.&~(r-r’)(~Z - ,Z$ _ is)-1 d3A &-‘)(A2
ik(z-z’)
s cw3
(3)
+ 2kA - +-I,
where we have introduced A = p - k, taking k in z-direction, and the particle speed v = fiklm. Now we split all momenta into transverse (index JJ and longitudinal (index 11) components with respect to the z-direction; a similar split in coordinate space introduces the two-dimensional impact parameter vectors b. Then we may write G(b - b’, z - z’) = z1 eikh-2’)
j
;f;
1
@$
eiA.(k-k’)ei~,(z-r’)e(Al
, A ,,),
(4)
where (?(A, , A ,,) = (h + A ,, - i+’
(5)
and where we have defined h = h, + h,, = AL2/2k + A,,2/2k.
(6)
If in Eq. (5) we neglect h compared with A ,, , we obtain the “eikonal propagator” CT!) = (A,, - ie)-l.
(7)
The index (1) indicates that we shall regard this as the “first-order eikonal approximation,” as distinct from higher-order eikonal approximations to be defined in Section 5. In coordinate space the eikonal Green function has the form G:‘(b - b’, z - z’) = (i/h)
6(2)(b - b’) e(z - z’) eik(‘-“),
(8)
where O(z) is the unit step function. The delta function of impact parameters means that the propagating wave suffers no transverse displacement; thus, the eikonal approximation corresponds to propagation according to geometrical optics. To solve the integral equation (I), the wave function is first written in the form of the incoming plane wave multiplied by a modulating function 4(r), #(r) = eikr$(r),
(9)
150
FRAHN
AND
SCHijRMANN
where 4(r) satisfies d(r) = 1 - 1 d3r’ evikcr-*‘)G(r, r’) V(r’) +(r’). By inserting (8) in place of G(r, r’) we have @(b, z) = 1 - ;
J-1 dz’ O(z - z’) V(b, z’) @‘(II, z’),
(11)
which has the solution $F’(b, z) = exp [ - i
1-L e(z - z’) V(b, z’) dz’].
(12)
It means that the only effect of the interaction is a modulation of the incoming plane wave, by a phase shift in case of a real potential, and by a reduction in amplitude as well, in case of a complex potential. The scattering amplitude has the general form s =--
e”q*V(r) $(r) d3r
k & 2n-h s
(13) eW
s
dz eiouzV(b, z) $00, z),
where fiq = h(k - k’) is the momentum transfer. To obtain the first-order eikonal approximation 4:“. In addition, since
to Eq. (13) we replace q5 by
(14)
is of the order k-l for fixed momentum transfer, we may set 4 ,, = 0 in the eikonal limit k -+ co. This amounts to the assumption of forward scattering as described by the eikonal propagator (8). Equation (13) then becomes fe(qJ = - &
j d2b eiqLb 1 dz V(b, z) &b,
z)
with the profile function F,(b) = 1 - S(b) = 1 - eixfb).
(16)
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SCATTERING
151
Here $x(b) plays the role of the phase shift in impact parameter representation and is given by
x(b) = - & j-;00dz v(b,z).
(17)
Equations (15)-(17) are the well known results of Glauber.
3. SECOND-ORDER EIKONAL
APPROXIMATION
An alternative derivation of the eikonal result (12), which emphasizes the highenergy character of the eikonal approximation, proceeds through partial integration in Eq. (10) in spherical coordinates over the cosine of the scattering angle, and dropping the resulting terms of order @a)-l and higher, where (z is a characteristic dimension of the scattering potential. This shows that the first-order eikonal approximation holds in the high-energy limit ka + co. To obtain corrections for lower energies it seems natural to carry the same procedure to first order in (ka)-l. This has been done recently by Baker [l I] who, after a somewhat lengthy calculation, arrives at a result which in our notation may be written in the form
+ &
J-1 dz’ (z - z’) O(z - z’) V,2V(‘b, z’) @‘@, z’),
(18)
where Vh2 is the two-dimensional Laplacian in impact parameter space. In Section 5 we shall rederive Baker’s equation (18) in a very simple fashion as the second order of a systematic approximation method, and also give a physical interpretation of it in “optical” terms.
4. FRESNEL APPROXIMATION In view of the fact that the (first-order) eikonal Green function (8) represents propagation according to geometrical optics, one may look for corrections which describe wave-optical modifications of the scattered wave. This has been done by Gottfried [9] for multiple scattering and applied to high-energy scattering by deuterium. Let us forget about potential scattering for a moment and assume that the scatterer (nucleon within a nucleus) may be regarded as perfectly absorbing. The eikonal Green function (8) then describes the formation of a cylindrical
152
FRAHN AND SCHiiRMANN
“shadow” extending to infinity behind the scatterer. In actual fact, diffraction at the scatterer causes a distortion of the geometric-optical wave field which destroys the shadow at distances of the order of and larger than the Rayleigh distance dR = kc?. Thus the eikonal Green function (8) approximates the propagation only at distances z < dR , and the Glauber approximation for multiple scattering will only be valid if the Rayleigh distance is large compared with the mean free path of the projectile in the target nucleus. Since dR is proportional to k, this condition can always be satisfied if the energy is made high enough. At lower energies, however, the diffractive distortion of the geometric-optical shadow may become appreciable. Gottfried takes these effects into account by employing the Fresnel-Kirchhoff formulation of Huygens’ principle in representing the scattered wave function. This amounts to replacing the eikonal Green function (8) by the “Fresnel-Green function” Gj;l’(b - b’, z - z’) = Go
e(z _ z,) eik(z.+‘) exp[i &k(bz--b’L,“/(z - z’)] . (19)
The index (1) again indicates that we shall regard this as the first order of a systematic “Fresnel approximation” method to be defined in the following section. In momentum representation the Fresnel propagator has the form G!) = (h I + d I - ie)-’ ,
(20)
i.e., it arises from the full propagator (5) by neglecting h,, . The relation between (19) and (20) was noticed by Gottfried (see footnote 9 of Ref. [9]).
5. SYSTEMATIC APPROXIMATION
METHODS
We shall now develop systematic approximation methods for improving the eikonal propagator (7) which proceed in different directions indicated by the and the “first-order Fresnel approxi“second-order eikonal approximation” mation,” respectively. We start with the full propagator (5) in momentum representation, e
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153
SCATTERING
Alternatively, we may regard h ,, as a “perturbation” of the Fresnel term h, + d H, and write the relation between C? and C?j.?’in the form G = ($!’ - ($‘h ue: *
(22)
Finally, we have a relation between C$) and ekl), ($’
= ($1) _ ,‘$l’, ($1, e e I F-
(23)
Our approximation methods now consist in iterating in Eqs. (21) and (22) with respect to @) and e$), respectively, e = @) _ ,p),,!’
+ ($)h@‘h@’
7 .. .
(24)
and G = ej$ - @‘h,@’
+ @‘h,@‘h,($!’
F -.. .
(25)
Equations (24) and (25) define the “eikonal expansion” and the “Fresnel expansion” of the free-space propagator, respectively. Going beyond the first-order eikonal approximation e M CA’) to the nexthigher order, we obtain
The physical meaning of this second-order eikonal approximation in coordinate representation. The result is
is best discussed
G$‘(b - b’, z - z’) = eikb-z’) .-& S(Z)@- j,‘) e(z - z’) [ - Au
V,26(2)(b - b’)(z - z’) e(z - z’)
- &
8(2@ - b’) 6(z - z’)].
(27)
Inserting (27) in place of G(r - r’) in Eq. (10) yields precisely Baker’s equation (18). It may be useful to indicate briefly how Eq. (27) is derived. When G(A, , LI ,,) in Eq. (4) is replaced by GL2) we obtain three contributions. The first, from @), yields GLl)(b - b’, z - z’). The second, from the term with h, , involves the integral
= - g z&z).
(28)
154
FRAHN
The third contribution, * f
eiAl
AND
SCHiiRMANN
from the term with h,, , involves Z(-$‘,
I,
27r
1
G(l) e
=-
.&
2k
eiA1/“A ,,‘(A ,, - kp2
2~
s
(2% = $6(z).
Going beyond the first-order higher order, we have
Fresnel approximation
G w Gg) to the next-
I$ M I$!’ - C$‘h ,,C$’ c cl-“‘.
(30)
In coordinate representation we find Gt’(b - b’, z - z’) =
1
_
1.
@
-
b’)’
_
2 (z - z’)2
j
1
k
4
cb
-
b’)4
1
($)(,,
(z - 233
+Ava(z _z')emu--l')
_
b’,
z
exp[i &k(b - b’)2/(z - z’)] z - z’
_
zy
(31)
Again we give a brief outline of how (31) is derived. When G(n, , d ,,) in Eq. (4) (2), the first contribution, from I?$), yields G$)(h - b’, z - z’). is replaced by C?, The second contribution, from the term with h,, , is (2) GF
-
Gj71'=
-
__ '
eik(z-z’)
j
;f;
j
;$
eiAL(b-b’)eiA,(z-z’)
2fikv (32)
Transforming yields GF’
from d ,, to d ,,’ = AL2/2k + A,, and thereafter dropping the prime ik(Z-2”
-G$)=-Le
aAlfb-b’)e-i(d,a/2k)(z-z’)
2fikv . 1 $$
eiA~(z-z’)[A12 - (A12/k) A,, + A14/4k2](A,, -
i~)-~. (33)
With [A ,,2 - (A12/k) A ,, + AL4/4k21(A ,, - i~)-~ = 1 - (A,s/k)(A ,, - ir)-l + (A14/k2)(A ,, - ie)-“,
(34)
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SCATTERING
the integration over d ,, can be carried out with the result
&)-G&--leik(Z-2’) s
iAl(b-b’)e-i(d,2/2k)(z-z’)
2?ikv
- [6(z - z’) - i(AL2/k) O(z - z’) - (dL4/4k2)(z - z’) O(z - z’)]. (35) The integration over LIP is then straightforward.
6. PARTIAL
SUMMATION
OF THE EIKONAL
EXPANSION
After writing Eq. (21) in the form
the eikonal expansion (24) may be regarded as composed of three contributions,
e = G f e2f G ,
(37)
where Q;‘, contains @) plus the terms involving h, only, e, those involving h,, only, and e, the “mixed” terms involving products of h, and h,, . Since all the factors in the expansion are ordinary functions and commute, we have t$ = @) - (@‘)2 h, + (($‘)3 hL2 F ... = @) f
(-l)n
(@)h,)n
= f$‘,
?Z=O
(38) where the last equality follows from Eq. (23). Thus the sum total of the terms involving h, in the eikonal expansion is just the first-order Fresnel approximation. Hence the coordinate representation G,@ - b’, z - z’) is given by Eq. (19). The second contribution can also be summed in closed form, C?, = -(GF’)2 h,, + (@‘)3 h,,2 T ... = ,e(,’ f (-l)“(@h,)“. n=l
(39)
With @)h ,, = d ,,/2k and (el’))” h ,, = (2k)-l we have e, = - &
f TZ=O
(-1)”
(d,,/2k)“.
(40)
156
FRAHN
AND
SCHiiRMANN
In coordinate representation this yields Gz(b - b’, z - z’)
zxz
-
kv
8(z)@
-
eikcz+ ) (1 + 2kl -g-J a -l 6(z - 2’).
b’)
Since we have closed expressions for G, = G$!) and Gz, and of course, for the full propagator G, the mixed contribution G, may also be regarded as known in closed form G3 = G - G!’ - G2.
(42)
7. PHYSICAL INTERPRETATION The two expansions of the full free-space propagator defined in Section 5, Eqs. (24) and (25), yield successive approximations subject to the conditions 4 +h,
9
eikonal approximation,
(43)
Fresnel approximation.
WI
Thus the Fresnel approximation only requires that h ,, = d ,,2/2k is small compared with the eikonal term d I , whereas in the eikonal approximation the sum h = (AL2 + d ,,“)/k is small compared to d ,, . Indeed, as we have shown in the preceding section, the first-order Fresnel approximation @$ already represents a partial sum of the eikonal expansion: it comprises to all orders the terms involving h, only. Thus the two expansions are physically distinct, emphasizing different aspects of the scattering process. The Fresnel expansion concerns the wave-optical (difsractive) deviations, in the transverse direction, from geometric-optical propagation. As we have discussed earlier, following Gottfried’s interpretation, the first-order Fresnel approximation accounts for the diffractive distortion of the geometrical shadow beyond the Rayleigh distance ka2. This is caused by a distortion of the scattered wave front due to diffraction at the boundary of the scatterer. The Fresnel propagator G$) describes this distortion to first-order as a transverse curvature of the wave front, by retaining the term proportional to 00 - b’)2 in the phase of the Green function.
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SCATTERING
This is best seen by an alternative derivation of G,(‘) from the full free-space Green function in the form G(b - b’, z -
z’) eik[(b-b~)a+(z--z')2]1'e
k =
7&i
[(b
-
by
+
(z
-
k z')2]1/2
eik(z-z~)u+8P'~
= 27Tfiu (2 - z’)(l + /3)1/S .
(45)
Expansion in powers of fi = (b - b’)2/(z - z’)~ and retaining the leading term in the phase only, yields Eq. (19) except for the factor 0(z - z’) which restricts propagation to the forward direction. The integral equation (1) or (10) may be regarded as an exact formulation of (the quanta1 analog of) Huygens’ principle: the scattered wave at observation point r is formed by interference of freely propagating spherical waves G(r - r’) originating with amplitude V# from each source point r’ in the interaction region. The firstorder Fresnel approximation represents an approximate form of Huygens’ principle (corresponding to the Fresnel-Kirchhoff approximation in physical optics), in which only the leading distortion of the wave front in transverse direction (i.e., over the b-plane) is retained and the longitudinal (z) propagation is restricted to the forward direction. Higher-order terms in the Fresnel expansion correspond to higher powers of /3 in the expansion of both amplitude and phase of G in Eq. (45). This may be seen from the form (31) of the second-order Fresnel propagator. Each term of the Fresnel expansion contains contributions of all orders in k-l. If such a term is further expanded in powers of k-l, the transverse curvature of the wave front is represented by powers of the transverse Laplacian Vb2. For instance, the expansion of the Fresnel propagator G$) with respect to k-l is G$‘(b
=
b’ 3 z - z’) ew2k)(s-z')v
Ir2G, (1) (b -
= e(z - z’) &M-Z’) k
b’, z -
[ia(
z’)
- b’) - LIZ&
V,28(2)(b _ b’) + ***I,
(46)
which is, of course, nothing else than the coordinate representation of the expansion obtained by iteration of Eq. (23), (47) Comparison with Eq. (24) shows that this forms part (e,) of the eikonal expansion which we consider next. The eikonal expansion represents a systematic expansion in increasing orders of k-l, or rather in powers of (ka)-l where a is the characteristic dimension of the scatterer. It therefore concerns the deviations from the asymptotic energy limit
FRAHN AND SCHiiRMANN
158
ka -+ co. In each order beyond G,(l) it collects contributions describing both longitudinal (eikonal type) and transverse (Fresnel type) distortions of the geometric-optical pattern represented by G,(I). For instance, in the second-order eikonal approximation Gi2) these are given by the third and second terms, respectively, in Eq. (27). The partial summations given in Section 6 show that the transverse distortions, represented by the derivatives of 8(2)(b - b’), result mainly from the contribution G, = G$) determined by h, , and that the longitudinal distortions, described by the derivatives of e(z - z’), are mainly due to the contribution G, determined by h,, . However, the presence of the “mixed” contribution G3 shows that there is no clear-cut separation between transverse and longitudinal distortions in the eikonal expansion.
8. FFCESNJXL APPROXIMATION
IN EIKONAL
REPRESENTATION
In the preceding section we have seen that the (first-order) Fresnel and eikonal Green functions are related by a unitary operator, ,‘$‘(,, _ ,.,‘, z _ z’) = e-(il’iQ)~~(Z-Z’)G~)(b _ ,,‘, z _ z’),
(48)
where R, = -(ii2/2m)
Vb2
(49)
is the transverse kinetic energy operator. This remarkable connection allows us to represent the Fresnel approximation of the scattering amplitude by a closed expression which is formally analogous to the eikonal (Glauber) amplitude. This is achieved by means of a unitary transformation of the Fresnel solution which is a precise formal analog of the transformation from the Schrbdinger picture to the interaction picture in time-dependent perturbation theory. Starting from the SchrGdinger equation for the modulating function 4(r) = +(b, z) defined by Eq. (9), (50) where (51) is the full kinetic energy operator, the Fresnel and eikonal wave functions +F and c#@ satisfy the wave equations (ifiWW and
- KJ Mh
4 = Wh z> 4dh z>
(52)
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SCATTERING
respectively. These equations are formally analogous to the time-dependent Schrodinger equation with a “time parameter” T = z/v and a time-dependent interaction Hamiltonian. (According to Ref. [14], this analogy for the eikonal equation (53) was previously noticed by Reading.) By a unitary transformation from the “Fresnel picture” 1c$~) to an “interaction picture” ] 4,) with respect to the interaction V(b, z), $F(b, z) = e-(i’fiv)kLz~I(b, z),
Eq. (52) transforms to an equation of “eikonal”
(54)
type,
ifiv@lW +100, 4 = Q, 4 4dh 4,
(55)
with an interaction operator (56) The “z-translation
operator”
&(z, z,J defined by
Mb, 4 = r3,k 4 4,@, 4
(57)
satisfies the same equation (55); in integrated form it$,(z, z,,) = 1 - ;
1’ dz’ p(b, z’) it&(z’, zJ. 10
Equation (58) is formally solved by a Dyson-type perturbation &(z, zo) = 1 + f
n=l
(-
k)n
$1’
-0
dz, ..a s:, dz, Z[@,
(58) expansion [15]
z,) ..a p(b, z,)],
(59)
where the z-ordering operator 2 arranges the noncommuting interaction operators P in the order of increasing z-values from right to left. Writing symbolically O,,(z, zo) = Z exp [ - k jZl dz’ @b, zf)], the modulating
(60)
function in the “eikonal interaction picture” becomes Mb, z> = %,
-a>
Mb, -a>
= (r3,(z, -a)>,
(61)
where (o,,(z, -co)) denotes the expectation value, in the constant state &(b, -co) = 1, of th e operator 4 . The asymptotic form of Eq. (61) for z + co, ;+%
$I@,
z> =
SF(b)
(62)
160
FRAHN
AND
SCHtiRMANN
defines the scattering function S,(b) = (&,). The scattering operator $, is given by 9, = Q(co, -co)
=
Z[&b)
]9
(63)
where in analogy to Eq. (17) j(b) = - & j-:
m
w
dz p(b, z)
is the “phase shift operator” in the interaction picture. In order to obtain the Fresnel approximation to the scattering amplitude we replace the wave function 4 in Eq. (13) by $F of Eq. (54). In addition we must use an approximation for q,, which is compatible with the Fresnel propagator (20). Since the Fresnel approximation is supposed to be valid for larger scattering angles than the Glauber approximation, the forward scattering restriction q,, = 0 is now inadequate and has to be replaced by
which is obtained by neglecting the term quadratic in q,, in Eq. (14). Using this approximation for q,, in Eq. (13) and noting that exp[i(qL2/2k) z] eiaLb = exp[(z@) KLz] eiaLb,
(66)
a partial integration over b yields hh)
= -
&
= &
I d2b eiaLb 1 dz I%,
z) &(b,
l d2b eiaLbTF(b)
z)
(67)
with the profile function T,(b) = 1 - S,(b) = 1 - eixptb).
(68)
The second equality in Eq. (67) has been obtained by using fl(b, .+@&,
- co)) = ifiV g <&(z, - CO)).
(6%
HIGH-ENERGY
161
SCATTERING
Thus, the Fresnel approximation has been cast into a form which resembles Glauber’s expressions (15)-(17) for the eikonal approximation to potential scattering. It is shown in Ref. [13] that Eq. (67) is identical with the amplitude obtained in the “optical limit”
of the Fresnel approximation
to multiple scattering.
Although Eq. (67) shows that Fresnel diffraction in high-energy scattering can be described by a closed “Glauber-type” expression of the scattering amplitude, this result may appear to be merely of formal interest and of no immediate practical use. Of course, there would be no point in trying to evaluate the perturbation expansion in powers of the potential operator P; thereby one would lose the decisive advantage of Glauber theory (over, say, the Born approximation) of incorporating the effects of the interaction potential to all orders. However, the result (67) is useful as a starting point of a different approximation method, by expanding the interaction operator P (defined by Eq. (56)) in powers of to k-l is the (z/h) I?, = -(2/2k) Vb2, which because of its proportionality appropriate expansion operator at high energy. In fact, this approximation procedure is equivalent to the expansion (46) or (47) of the Fresnel propagator in powers of the eikonal propagator. To first order in this expansion, Ti = V, and Eq. (67) reduces to the Glauber formula. To second order, v = v + i(z/2k)[ V, V,2]. Inserting (70) in (63) and retaining only contributions the perturbation series yields
(70) of order k-1 in each term of
St’(b) = S(b)[l - (i/k) d,(b)].
(71)
Here S(h) is the Glauber scattering function of Eq. (16) and d,(b) is defined as h(b)=
-&va2jw
--m
dzzV(b,z)
For spherically symmetrical potentials this reduces to d,(b) = (+-)’
b $ JOmdz [V(b, z)]“.
(73)
162
FRAHN AND SCHtiRMANN
The form of the function d,(b) indicates that its main contribution comes from the surface region of the potential V, in accordance with the physical picture of Fresnel diffraction. Details of the derivation of Eqs. (72) and (73), and the lowest-order Fresnel correction to the Glauber approximation for nuclear multiple scattering, are described in Ref. [16]. The second-order eikonal approximation, which contains ail terms to the order k-l, differs from the first-order Fresnel approximation by the contribution arising from the longitudinal kinetic energy operator R,, = -(fi/2m) P/az2. In this approximation the scattering function is given by [16]
$%9 = W)[l - (d&h@) + G-N,
(74)
where d,(b) = $(l/fi#
jrn dz [V(b, z)]“. --m
(75)
For a spherically symmetrical potential, Eq. (74) agrees with the expression given by Wallace [12]. It disagrees, however, with Baker’s result [ll]. The reason presumably is that in calculating the scattering amplitude from 4:“) of Eq. (18), Baker has used the forward scattering restriction exp(iq,,z) M 1, whereas the correct approximation is exp(iq ,,z) m 1 + i(qL2/2k) z, in accordance with Eq. (65). Finally, we mention that for nuclear multiple scattering there are additional first-order eikonal-type corrections which arise from rescattering on the same nucleon [16]. These terms largely cancel the contribution corresponding to Eq. (75), leaving the Fresnel correction as the dominant term. This underlines the significance of the Fresnel approximation in nuclear multiple scattering.
9. RELATION
TO OTHER EXPANSIONS
In this section we briefly discuss how our expansions are related to those mentioned in the introduction. The approaches by Abarbanel and Itzykson [6], Kujawski [lo], and Wallace [8, 121 differ from ours in that the intermediate momentum p is expanded about the average momentum & = &(k + k’) rather than about the initial momentum k. Whereas the leading term in Wallace’s series is the Glauber expression, it has a slightly different form in the series of AbarbanelItzykson. This is because in the former the projectile speed in the direction i; is u = +ik/m, while in the latter it is u’ = ZJcos $9, where 0 is the scattering angle. Taking the z-axis in the direction of & in both cases automatically yields qn = 0 so that no additional assumption about q ,, has to be made. This is often considered to be an advantage over expansions about the initial momentum k.
HIGH-ENERGY
SCATTERING
163
On the other hand, as indicated at the end of Section 8, taking the effect of the longitudinal momentum transfer properly into account to first order in k-l, yields a result equivalent to Wallace’s It is obvious that with the appropriate assumptions about q,, this equivalence holds true for any given order of k-l. (It should be mentioned that the importance of the longitudinal momentum transfer in connection with corrections to the Glauber model has previously been pointed out in Refs. [17] and [9].) Because of q ,, = 0 in his approach, Wallace immediately obtains an impact parameter representation for the scattering amplitude. However, this is at the cost of an expansion for the Green propagator more complicated than ours. In particular, there seems to be no simple way of extracting a Fresnel propagator from Wallace’s expansion. Since the main aim of the present paper is the development of systematic high-energy scattering approximations which have an immediate physical interpretation in wave-optical terms, the expansion about the initial momentum is the more appropriate choice.
10. SUMMARY AND CONCLUSION We have developed systematic approximation methods for high-energy nuclear scattering which improve upon the Glauber approximation. These methods are based on expansions of the free-space propagator which emphasize different aspects of the scattering process. Since the zeroth order of these expansions, the eikonal propagator, represents at the same time the high-energy limit ka -+ co and the geometric-optical limit of propagation, we have distinguished two different approximation procedures. The eikonal expansion emphasizes the deviations from the asymptotic energy limit and proceeds in powers of (ka)-l. The Fresnel expansion emphasizes the deviations from geometric-optical propagation caused by d@ractive distortion of the propagating wave front in transverse direction. To first order in these expansions we recover expressions derived by Gottfried [9] and by Baker [l I] and Wallace [12], respectively. By partial summation of the eikonal expansion we establish the connection between the two approximation procedures and give a physical interpretation in “optical” terms. By means of a unitary transformation analogous to the one that leads from the Schrodinger picture to the interaction picture in time-dependent perturbation theory, we have derived a closed expression for the scattering amplitude in Fresnel approximation which is formally similar to the Glauber formula. From this expression we calculate, to first order in k-l, the Fresnel correction to the Glauber amplitude. Although these methods have been formulated for potential scattering, they are concerned only with the propagators and can therefore readily be applied to multiple scattering. The Fresnel approximation to multiple scattering in complex nuclei will be dealt with in a subsequent paper [13].
164
FRAHN AND SCHiiRMANN REFERENCES
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