Explicit evaluation of the amplitude for elastic heavy ion scattering

ANNALS

OF PHYSICS

Explicit

101,

Evaluation

520473

(1976)

of the Amplitude

for Elastic

Heavy

Ion Scattering

W. E. FRAHN Physics Department, University of Cape Town, Rondebosch (Cape) 7700, South Africa AND

D. H. E. GROSS Sektor Kerphysik,

Hahn-Meitner

Institut ftir Kernforschung, I Berlin 39, Germany

Received Sanuary 26, 1976

The partial-wave expansion of the amplitude for elastic heavy ion scattering is explicitly evaluated for realistic quanta1 scattering functions. Closed expressions are derived for general forms of the absorptive shape function in the limits of weak and strong absorption. By assuming a Gaussian model of the absorptive shape function, the explicit evaluation can be carried out by a systematic asymptotic expansion valid over the whole range between the weak and strong absorption limits. The result enables us to give a detailed analytic discussion of possible scattering conditions and to identify the physical mechanisms that operate in different limiting situations. It is shown that the transition between the quanta1 (diffractive) and semiclassical (nondiffractive) regimes is controlled by a characteristic parameter 7~ = &P 1 @‘(ll) j. The explicit formalism provides a systematic description of the main physical processes in elastic heavy ion scattering as well as quantitative criteria for their regimes of validity.

1. MOTIVATION An accurate description and valid physical interpretation of elastic scattering is necessary for the understanding of all heavy ion interactions. Because in fully quanta1 descriptions such as the optical model the scattering amplitude can be calculated only in numerical form, there have been many attempts to gain physical insight by studying simple approximations in which explicit expressions can be obtained. In recent years the development of the semiclassical theories [l] has been based on the fact that heavy ions behave under certain conditions almost like classical bodies moving along well-defined trajectories. These theories highlight the particle aspects of the heavy ion interaction and give a simple and accurate description of potential scattering and of the refractive phenomena associated with 520 Copyright All rights

0 1976 by Academic Press, Inc. of reproduction in any form reserved.

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521

the real part of the interaction potential, such as rainbow and glory scattering, orbiting, etc. However, modifications are required to incorporate wave-mechanical features into the classical framework [2]. Quanta1 phase factors multiplying the classical amplitudes take account of the interference between contributions from different trajectories that emerge at the same scattering angle; attenuation coefficients describe weak absorption along the trajectories. More drastic generalizations are necessary to account for such essentially nonclassical effects as quantum diffraction associated with strong absorption, and reflections from the surface of the real and imaginary parts of the optical potential; these may be described by introducing complex turning points and complex trajectories [3,4]. Much earlier, a different approach towards a simplified description was developed, which starts as it were from the opposite extreme to potential scattering. It emphasizes the strong absorption properties of the interaction of complex nuclei above the Coulomb barrier and is based on an exact wave-mechanical expression (the partial-wave series) of the elastic scattering amplitude. The simplification is achieved by taking advantage of the simple form of the elastic scattering function (S-matrix) in angular momentum space under conditions of strong absorption. The first of these descriptions was Blair’s “sharp-cutoff model” [5]. Subsequently it was shown [6] that the Blair model amplitude can be evaluated explicitly in the limits of weak and strong Coulomb interaction; at the same time the closed formalism was generalized for arbitrary “smooth-cutoff” reflection functions and for the presence of (small) nuclear phase shifts. While in the neutral limit the formulas described Fraunhofer diffraction scattering, the angular pattern that results from strong absorption in the presence of strong Coulomb interaction was identified as quanta1 diffraction of Fresnel type [7]. The relation between the type of diffraction pattern and the strength of the Coulomb field was given a simple physical interpretation and provided the basis for a general classification of nuclear scattering processes [8]. Although the strong absorption model of Ref. [6] has been used quite successfully in analyses of heavy ion scattering data, it is unduly restrictive in the way it accounts for the effects of the nuclear phase shifts, which have to be assumed small compared with the phase shift for Rutherford scattering. Recently, a generalized formalism has been developed [9] which includes realistic nuclear phase shifts and gives a systematic expansion of the elastic scattering amplitude in the strong absorption limit. In the present paper we develop an alternative procedure for explicit evaluation of the elastic scattering amplitude, and put both methods on a firm mathematical basis. The basic approximations and the assumptions about the general form of the elastic scattering function are discussed in Section 2. We then describe in Section 3 the evaluation methods by which we derive explicit expressions for the elastic amplitude in the limits of weak and strong absorption. The physical inter-

522

FRAHN AND GROSS

pretation of the results is discussed in Section 4. To gain insight into the mathematical nature of our methods and to establish a connection between our results for the weak and strong absorption limits, we study in Section 5 a specific (Gaussian) form of the absorptive shape function. In this case the explicit evaluation can be carried out quite generally; the resulting expression covers the entire range between the limits of strong and weak absorption. We thereby provide, firstly, a justification of the procedures described in Section 3 and quantitative conditions for their validity. Secondly we show that the transition between the quanta1 (diffractive) and semiclassical (nondiffractive) regimes of scattering is controlled by a characteristic parameter 7A . In Section 6 we discuss the effects of nonmonotonic deviations from the average quanta1 deflection function. Finally, in Section 7, we complete the physical interpretation of our results and define, by means of quantitative criteria, the regimes of validity of the various physical mechanisms that operate in heavy ion scattering.

2. THE ELASTIC SCATTERING FUNCTION

We start with the partial-wave (for spin-0 particlesl),

expansion of the elastic scattering amplitude

f(e) = + 5 (1+ Nl - S,) P&OS6

(2.1)

1=0

where (2.2)

& = 77p=1

are the elements of the elastic scattering matrix in angular momentum representation. Without further discussion2 we make the usual approximations that are appropriate for heavy ion scattering because the wavelengths are short compared to the interaction radius: (i) (ii)

replace the summation over I by an integration over h = If use the asymptotic form of the Legendre polynomials, i(AB-(l/dh) Pdcos

0) =

@TX

s;n

fp

Ee

+

e-

ih9-Wahr)

4, and

13

1 An extension of the present formalism to nonzero spin has been developed, and explicit expressions for the amplitudes and the vector and tensor polarizations of spin-4 and spin-l particles have been derived, which will be given in another publication [lo]. 9 A detailed justification of these approximations, with special regard to the expansion methods developed here, will be given elsewhere [ll].

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SCATTERING

valid for h-l ;5 0 ,z5r - X-l. Furthermore we assume that the (real) reflection coefficients Q and the phase shifts 6, can be interpolated by smooth (continuously differentiable) functions T(A) and 6(h), respectively, so that SI may be replaced by the “elastic scattering (S-) function” S(A) = q(h) eiz6tA).

(2.4)

The amplitude (2.1) may then be written as a sum of two contribution9 from the two branches of the asymptotic Legendre polynomials (2.3), f(@ = f”‘(@

+ f’-w,

arising (2.5)

where f’*‘(e)

= - + (2rr ,i’, e)l,2 J= dA P,(A)

e@n,@)

112

with +,(A, e) = 28(x) T

(xe -

+I.

(2.7)

For our present purposes we suppose that the quantities Q and & are given, at each energy E, by a fully quanta1 calculation, for instance generated by solving the Schrbdinger equation for an optical potential. Our aim is to evaluate fhe integrals in Eq. (2.6) for such general forms of $A) and 6(h) that are typical of optical modelfits to actual heavy ion scattering data. It will be seen that the properties of f(6) are mainly determined by the form of the derivatives of 6(h) and q(A) in

X-space,

=2p ) @(A) =ep ) D(X)

(2.8)

which we call the “quanta1 deflection function”4 and the “absorptive shape function,” respectively. What can be said about the general form of these functions in heavy ion scattering? Firstly these forms change with energy. Well below the Coulomb barrier we have practically pure Rutherford scattering for which the phase shifts are given by (2.9)

8The superscripts(+) and (-) indicate that the two contributions correspond, in a semiclassical picture, to scattering through positive and negative angles, respectively. * The function O(A), which is determined by a smooth interpolation of the phase shift differences W&,1- a,), must be distinguished from the classical deflection function S&A) defined as the derivative of the WKB phase shifts with respect to angular momentum X = I + 4. Contrary to &r(A), @(A) in general does not represent the relation between deflection angle and orbital angular momentum as determined by the classical equations of motion.

524

FRAHN

AND

GROSS

where

II = MZ2e2 A2k

(2.10)

is the Sommerfeld parameter. Heavy-ion collisions are characterized by large values of n and X, so that we may approximate the Rutherford phase shift function by the asymptotic expression 26,(h) = 2S&)

- 2n ln(sin $9) + in9 cot 38 - $77 + O([n2 + X2]-1/2),

(2.11)

where 9 = 2 arc tan(n/h).

(2.12)

Thus, well below the Coulomb barrier, the deflection function is given by O(h) m O,(h) = d26,@)/dh = 6 = 77 - 2 arc tan(h/n).

(2.13)

Because there is practically no absorption we have q(X) = 1. At energies near and above the Coulomb barrier the nuclear interaction comes into play: The attractive nuclear force causes O(h) to deviate from O,(X) mainly in the region of X-space that corresponds to the nuclear surface. In addition, absorption and reflections from the nuclear surface make q(h) < 1 for low-l partial waves, so that D(X) has a broad distribution only weakly peaked in the “surface” region of X-space. A typical example of “weak absorption” forms of O(h) and D(h) is shown in Fig. 1. As the energy increases further the absorption becomes stronger, and D(h) shrinks to a narrow peak at the critical angular momentum ./I defined by +o

= ii.

(2.14)

The quanta1 deflection function may show, especially for deep real nuclear potentials, nonmonotonic deviations from the average smooth decrease with h; these deviations are, however, confined to a relatively narrow X-range around the critical value /I. A typical example of “strong absorption” forms of O(h) and D(A) is shown in Fig. 2. It should be emphasized that the quanta1 functions O(X) and q(h) are usually quite different from the deflection and reflection functions calculated semiclassically, as in first-order WKB approximation. The classical deflection function O,i(b) often describes “rainbow scattering” and “orbiting” at points b, and b, where Ob(b,.) = 0 and O&b,) = - co, respectively [14], while the classical reflection function v&b) for complex potentials represents almost complete absorption for all impact parameters below the orbiting value b,, . These features

525

HEAVY ION SCATTERING

‘*Oc @l(h) (dcqrees)

15 0 + 58Ni E ,o,.60MeV \

120

v,=2 r,:l.65fm a”=06

O(h) ,

O,(h)

MeV

w,.5 MeV rI I 1.25 fm a.=10 fm

fm

i

90

TO

30

-I "

- ‘12

10

20

t

30

40

t A

30

40

50

O.l~

‘12 10

20

50 A = I+‘/2

FIG. 1. Quanta1 deflection function @(X) (a), solid curve) and absorptive shape function D(X) (b), calculated with a “shallow” optical potential given in Ref. 1121. The broken curve in (a) represents the deflection function for Rutherford scattering, @R(X) = m - 2 arctan(l\/n).

are drastically modified by quanta1 effects: the “orbiting” singularity in f&(b) is removed and the sharp cutoff in yCl(b) for b < b,, is smoothed out. It has been shown [15, 41 that these modifications are due to the (essentially nonclassical) reflection from the surface of the optical potential. Thus the quanta1 deflection functions derived from most optical potentials that fit actual heavy ion scattering data are smooth functions of A, except for possible minor structure within a relatively narrow range near h = A. We shall assume in most of what follows that O(A) is a smooth monotonically decreasingfunction of A.

526

FRAHN

AND

GROSS

8 (Al v, .791

(degrees) 120

eh=eR

uev

w,.50

MeV

-

90 -

60 -

30 -

D A)

0.05

-

@ 0 'I2

10

20

t *

30

I 40

I 50 h.l+'h

FIG. 2. Quanta1 deflection function @(A) (a), solid curve) and absorptive shape function D(A) (b), calculated with a “deep” optical potential given in Ref. [13]. The broken curve in (a) represents the deflection function for Rutherford scattering, O,(A) = T - 2 arctan(h/n).

The effect of small deviations from monotonic behaviour on the elastic angular distribution will be discussed in Section 6. For such deflection functions there is a well-deGned angle en = O(A),

(2.15)

the “critical angle” associated with the critical (or “cutoff”) angular momentum (1, which separates two angular regions, 0 ,( 8 & Bn and eA < 8 < rr. The latter,

HEAW

527

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because it receives contributions from partial waves that are strongly absorbed, is called the “shadow” region; the former is the “illuminated” region (see Figs. la and 2a). The angle 8, is usually close to (but distinct from) the critical angle for Rutherford scattering defined by OR = @,(A) = 2 arc tan(n/Ll),

(2.16)

which is also indicated in Figs. la and 2a.

3. EXPLICIT

EVALUATION OF THE ELASTIC SCATTERING AMPLITUDE OF WEAK AND STRONG ABSORPTION

IN THE LIMITS

In this Section we derive closed-form expressions of f(8) for functions O(X) and D(h) that have the general properties described in Section 2, but are otherwise unspecified. We use different methods of evaluation in the weak and strong absorption limits. 3.1. Weak Absorption

This situation corresponds to energies near the Coulomb barrier, where q(h) varies slowly compared with the oscillatory variation of the integrands of f(*)(0) in Eq. (2.6). Then the main contributions to the integrals come from the vicinity of possible points of stationary phase. According to our assumptions on O(X) there is only one stationary point X, , defined by 0(x,) =

8,

(3.1)

in the positive (or “stationary”) branch f(+)(e). In stationary phase approximation (SPA) the slowly varying function r(x) may be taken out of the integral, and we obtain [9] f(e) = f(+)(e) = h(e) +u, (3.2) where

m---

i h? 1/Z k ( --O’(h,) sin B 1

ei12S(“/3)-A#31

(3.3)

is the amplitude forpotentialscattering in SPA. (Note that @‘(A,) < 0.) The negative branch f(-)(e) gives no contribution in this approximation. Thus in this case the effect of absorption is to multiply the potential scattering amplitude by the attenuation factor I, which is a function of the scattering angle 0 by virtue of the relation (3.1).

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FRAHN

AND

GROSS

3.2 Strong Absorption In this case the reflection function q(h) varies rapidly in the vicinity of h = /l; thus D(h) is localized within a small h-range of width d < (1, with a narrow peak at R. Then both the positive and negative branches of j(0) contribute. Let us first evaluate f(+)(e). By introducing an integral representation of q(h) [6, 91, (3.4) in terms of the Fourier transform of the absorptive shape function D(h), F(Az) = jm dh D(A) dA-“)z, -02

(3.5)

f(+)(e) can be written in the form

This expression shows clearly how the amplitude for potential scattering is modified by absorption: Aside from the pole near z = 0, the total amplitude is the convolution off0 with the Fourier transform F, that is, the potential scattering pattern is “smeared out” to a greater or lesser extent according whether F(Az) is broad or narrow. If F(Az) is sharply peaked at z = 0 (weak absorption), the features of fO(0) are essentially preserved; on the other hand, if F(Az) is broad (strong absorption), the angular distribution of potential scattering is drastically changed by quanta1 diffraction at the sharp “edge” of r(h) at the critical angular momentum (1. The reason for introducing the representation (3.4) is that in the case of strong absorption where D(h) varies rapidly, its Fourier transform F(Az) varies slowly, so that the resulting angle integration may again be carried out in SPA. If we now insert in Eq. (3.6) for fO(e + z) the expression obtained in SPA (see Eq. (3.3)), the result can be written as f(+)(e)

= 1

2rk

e-iAs

?ida ” (sin e)l12 !:?I Yi 0 c -@‘(A,)

] l”

FM@ - @I eir2s~+(+m i?--8fir

(3.;)

where h, is the stationary point pertaining to the angle 8 = z + 6. The range of the integration over 8 is restricted to physical angles because the SPA expression (3.3) for&(e) is defined only in this range and zero outside.

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SCATTERING

The oscillatory part of the integrand in (3.7) has a stationary point 8, given by Aas= A,

or

6, = @(A) = e/j )

(3.8)

i.e., by the “critical angle” defined by Eq. (2.15). Thus the integral will receive its main contributions from the vicinity of the stationary point On and from that of the pole near 6 = 0, which coincide at 8 = On. In order to separate these contributions we follow an approximate procedure which has been used earlier in a similar form for the h-space representation off(+)(O) [6,9]. If the main contribution to the integrand in (3.7) came from the stationary point, we could take the slowly varying function F from under the integral, leaving us with the positive branch of the “sharp-cutoff” amplitude +4e f:;;(e)

= J-

2rk

(sin @l/2 %9

* d9

O

ho 1 -@‘(A,)

l/Z

1

ei[2s(Aa)-(na-n)al 8-6+ic

’

(3.9)

which corresponds to A = 0, or

and Dsco(4 = w - 4,

F,,,(Az) = F(0) = 1.

(3.11)

We therefore write

f’+‘(e) = F[A(O, - 6>lf~~(6) + R(6’- k, e,,),

(3.12)

and show in Appendix III that the term R(B - ie, fl,), given by Eq. (3.15) below, is small at least of order AF’(O)LI-~/~ compared to the first term, in the “shadow region” e 3 8, . In the “illuminated region” 6’ < Oil we first rewrite Eq. (3.7) by using the symbolic relation lim ’ r-+0 6 - e + iE = -i27r8(8

- 0) + El0 6 _ i _ in .

(3.13)

The first term on the right-hand side of (3.13) yields the SPA expression (3.3) of the potential scattering amplitudef,(O); the contribution from the second term can be rewritten by using Eq. (3.13) again, with the result

f’+‘(@ = A@>- F[A(eA- @lIfo(R - f%(@l + No + k, OJ, (3.14)

530

FRAHN

AND

GROSS

where +4e

R(0 f ie, f3,) = --!2nk (sin e)1/z !%

xmv -8 e)l - m(eA - e)iea[2s(ns)-(Ae-n)ele - (e k k)

c3 15J

In Appendix III we show that R(B + ie, 0,) is small, at least of order &(0)/l-1/2 compared to the first two terms in Eq. (3.14), in the region 0 < en . Neglecting R(B & ie, 0,) we obtain f(+)(e) = .h(e) - [h(e) - kke)i f(+)(e) = f%(e) m(e,

ww,

- m

ea,

- e)i,

(3.16)

ea,

or f(+)(e) = ~b(e)(i - F+I 4e, - 0) + f&e)

F+ ,

(3.17)

where T(X) is the unit step function defined in Eq. (3.10), and F+ = F[d(O, - e)]. It remains to evaluate the sharp-cutoff amplitudef&!J(@ given by Eq. (3.9). For the negative branch f’-)(e) we follow a similar procedure. By using the integral representation (3.4) we can write

f(-)(e)- 27r1 -+O Jims--m mdzF(~z) einr sin(z-e) I 1’2fo(z _ e), z + ie [ sine

(3.18)

and by inserting the SPA expression (3.3), eine j--ye)

=

_

i

2nk (sin @l/2 !%

“d8

s ,,

[

A’ -@‘(A,)

] 1’2

mw

+

e)i

ei[28(na)-(Ae-n)el

a+e+ie

(3.19)

where h, is the stationary point pertaining to the angle 8 = z - 8. Because the pole near a0 = -0 is now outside the integration interval, the main contribution to the integral in (3.19) comes from the neighborhood of the stationary point 8, , which is again given by Eq. (3.8). Since F(dz) is slowly varying in the strong absorption limit, we may take it out of the integral at the stationary point Z, = 8, + 0 = en + 0 with the result (3.20)

f(-)(e) = a4o(e, + e)i fkW, where f;;:(e)

is the negative branch of the sharp-cutoff amplitude edne

fj$9)

=

-

i

A8 ] .s ,,mdt9 1 -@‘(A*) 2rk (sin e)1/2 !A%

l/2

ei[2s(AaL(As-n)al

8 + 8+ k

*(3.21)

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SCATTERING

3.3 Explicit Expressions for the Sharp-cutofl

Amplitude

can be evaluated in the form of an asymptotic The sharp-cutoff amplitudej&(8) expansion with respect to a large parameter /3 which for heavy ion scattering is either the critical angular momentum (1 or the Sommerfeld parameter n. This method will be described in detail in Section 5 for a Gaussian absorptive shape function. From the results derived there we obtain explicit expressions forfsco(8) as a special case in the limit d = 0. From Eqs. (5.39), (5.40) and (5.43) with p = (1 we find (3.22)

A -O’(fl) 112m -(2*) + eim+(n,e) k [ 2r sin e] r=O 2 c+ (3.23)

(3.24)

where fO(0) is given by the SPA expression (3.3), and erfc(u,) is the complementary error function defked as erfc(d

= p

2

m e-T2 d7, s%I

(3.25)

with the argument u. = ei(114h

The coefficients ey’(&, by Eqs. (5.48), (5.53)

e - en

[-20’((1)]1/2

*

(3.26)

F 8, 0,) are the 2rth derivatives of the functions defLi.ned

c+(e, - 8, en) = cl(e,l - e) ~[--4--]1’2 27) -0 (rl) - [ p&Ae)]1’2 eiY(YpeA’j.(3 c-(e, + 8, 0,) =

n(e,l+e)k&J”2~

(3.28)

where #(6J - 0,) is the correction of third and higher order in an expansion of the phase shift function in powers of 8 - en , 26(X,) - (he- n)e

= 26(A) + /T2;$&

+ 9x0 - u.

(3.29)

532

FRAHN

AND

GROSS

Because for heavy-ion scattering the parameter /I (i.e., n or n or both) is large compared to unity, we shall retain only the leading contributions to fsC0(8) of orders 8” and /3-l12. The discussion in Section 5.3 will show that these come only from the terms with r = 0 in Eqs. (3.23) (3.24), except at the point 8 = Bn, where there is an additional contribution of order /W2 from the term with r = 1 in Eq. (2.23) forf~~$3). Thus we obtain for B + 8, , f%(e)

= fo(e) i erfc(za,)

+e fkite)

&$+(/I$) fl -O’(fl) l/2 7F [ 2~ sin e I c+tb - 0, 0,) + owl),

= ei6-(nse) $ [ ;Y$j T

]1’2

c-(e, + e, e,) + o(p),

with the functions c+ , e- given by Eqs. (3.27), (3.28). The SPA expression (3.3) for the potential scattering amplitude

(3.30) (3.31)

may be written

fo(e) = - k(sitejl,2 ei”+(“~e)ei(1’4)~+u~eC+(e, e,), in terms of the function

c+(e, e,) = i [ -iiAe)]“’ introduced form f&e)

ei”(‘-‘“),

in Section 5, Eq. (5.34). Then we can write Eqs. (3.30), (3.31) in the = f,(e) 1; erfc(u,) _

e-i(l/4)n-%8

[ - T2A)] l” c+(cB;B fje”)1, + 2 A

(3.34)

i(zne+(l/4h-t4,*

f&9

-f(e) - 0

(3.35)

p-fl)~/ye, + 8) c+(e, e,) 9

in which fo(0) is an overall factor offs&@. At the point 0 = tin, a calculation carried out in Appendix IV shows that the function C, has the value (3.36) where

cd-4 = - ; [,(_‘,J +

0; (-On’)2

I’

(3.37)

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SCATTERING

-(ml = O(m)(A). However, this is not the only and where we use the abbreviation 0, contribution to order /?- 1/2; there is an additional contribution of the same order coming from the term with r = 1 in the expansion (3.23). It is also shown in Appendix IV that C, at 8 = Bn is to be replaced by

C+(O, e,>- i ; O,‘Cf’(O, e/l>= (~~;~~)l,2 ,

(3.38)

where

&Q = - ; [/q&)

ol; + ; (-@,‘)Z *

1

Thus we obtain from Eqs. (3.34), (3.35) hco(en>

=j@,)

1;

+

e-i(1'4)T

(*yr[-co(A)

+

is]

+

qp-l)/,

(3.40)

with

fo@,) = - ; ( -@A+sineny2

(p44@Ala

(3.41)

The difference between the leading contributions at the point 0 = 011and those for lJ # tin is a “Gibbs phenomenon” that occurs in the asymptotic expansion (3.23) for f;:;(e). This mathematical feature is of no consequence in physical applications, for which filA(e,) may be replaced by the limit (3.42)

so that Eqs. (3.30), (3.31) can be used in practice for all angles. 3.4 Alternative

Method

In this Section we consider another method of evaluating j&,(0), which was developed previously [9] and is based on an expansion in X-space rather than in &space. By using the step function (3.10) in Eq. (2.6) the sharp-cutoff amplitudes become

f!%(e)=

- $

(2n ,II, e)l,2 s,r dA x1’3ei~*(~,0).

Again we deal first with the positive branch. By expanding the phase shift function about L’I explicitly to second order in h - Al, 26(h) = 26(A) + (A - A)

e, +

+p(A)(h

- ny + x3(x - n),

(3.44)

534

F’RAHN AND GROSS

where X&i - A) = 5 @ ;y p=3

W-l’(A),

(3.45)

fA$(@ can be written as

fgg9) = - ; (&J

ei~+(“%l>)(e, - e>,

(3.46)

where6

p(zJ = frn evW- 4 z,+ (l/2) @‘VW - A>2 +x4(h - A)]] ‘4dh(A/A)l/” = smdpA+(P) (3.47) ei(z.~+(l/2)o‘q’2) 0

with

A+(p) = (1 + (p/~))"" &“)

(3.48)

and Z, = en - 8, p = A - A. If the main contribution to the integral (3.47) comes from the vicinity of A w A, we may expand A+(p) in a Taylor series about ,u = 0, A+(P) = f

W/m!)

A?%),

(3.49)

Wl=O

and obtain

Ll>'(zJ = f A:!'(O) L,(zJ,

(3.50)

Vl=O

where &(zJ

= (l/m 1)fom dp ~nzei(psu+(1/2)‘~‘e2).

(3.51)

The integrand of (3.51) has a point of stationary phase at

LkV=$SJ=

8, - e -i-&q-*

(3.52)

5 The integrands in Eqs. (3.47) and (3.57) below are supposed to contain a convergence factor exp(-cp), which amounts to replacing z, by z, + ie and z, - k, respectively, with the limit l -+ +0 taken after integration.

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SCATTERING

Thus the main contribution comes from p % 0 if pS -=c0, i.e., in the shadow region 0 > 19~. In this region the integral (3.51) can be expressed in terms of parabolic cylinder functions, using formula 3.462.1 of Ref. [24],

=e

111!

-i(1/4)x(m+1)

i(~,a/S’u)~-~-~

(2a)u/2)(m+l,

[ei(“4b7

;2;;;/2

1,

e

(m = 0, 1, 2,...; 2, < 0, a! > 0). With Z, = 8, - 0 and 01 = -$@,I

(3.53)

- ie, E + +O, Eq. (3.46) becomes

(3.54)

with the variable u. defined by Eq. (3.26). The series in Eq. (3.54) is an asymptotic expansion, valid only in the shadow region 0 > Bn . In the “illuminated region” 8 < en we rewrite fLi@) in the form

&i(e) = f,(e) + + (2n ,i’, e)l,2 1” dA X1’zeid+Q*s).

(3.55)

112

With the expansion (3.44) in the integrand of the second term we can write

&i(e) = fo(e) + + ( 2n2n e )I” eid+(“~e)L(,<)(e, - e),

(3.56)

where p(zs)

=

J)+‘2

(+

A+(-p)

eiLz.e+h/2b/l’P~)~

(3.57)

If now the main contribution to the integral comes from the vicinity of /J, w 0, we may expand ,4+(-p) in a Taylor series similar to Eq. (3.49). Since the stationary point of the integrand in (3.57) is now at ps = -z,/l 0,’ 1 = (0 - eA)/l On’ 1, this is permissible in the region 6’ < 8, . Because the integrand in (3.57) is supposed to contain a convergence factor exp(-+) (see footnote5), we can take the upper integration limit A - 4 > 1 to infinity without changing the asymptotic value of L:<‘(z8). Then we obtain I$‘(zJ

= 2 (-)”

Wl=O

A$qO) I&(-ZJ,

(3.58)

536

FRAHN

AND

GROSS

and with Eq. (3.53)

fk@) = fo(Q + $ ( 2rr;n J . i.

e-i(l14hm

ei*t(‘Q) ei;;;A;)y;y’

4%) (- @A’)(1/2h(-)“D-,-l(-2”‘u,).

(3.59)

The series in (3.59) is a valid asymptotic expansion in the illuminated region O
w4

+ (A - 4 0, + x2@ - A),

(3.60)

where

x2@- A) = 5 0 yQp @‘P-l’(A).

(3.61)

p=2

Then we have f:;;(e)

- - i

A

1’2 id-(A 8) ’ L-VA + 01, e

k ( 2rr sin 8 1

(3.62)

where L&J

= lrn dh (A/A)1/2 exp{i[(h - A) z, + x2(X - A)]}

A

=s

m dp A-(p) 0

with

eizs”

A-(p) = (1 + (p/A>)“” eixzb)

(3.63) (3.64)

and z, = ell + 8. By expanding A&L) in a Taylor series about p = 0 we obtain (3.65)

where the integrations are carried out by means of a convergence factor exp(-ep) (see footnote5). This yields

eim-m) f:;;(e) = -L A k ( 2rsin0 1 112

(3.66)

Because of the approximation inherent in expressions (3.43) for j$~(@, which are already asymptotic in that terms of relative order X-l have been neglected (see Section 2. and Ref. [ll]), the asymptotic expansions (3.54), (3.59), and (3.66)

HEAW

537

ION SCATTERING

are meaningful only for the first few terms of the m-series containing the leading contributions of order (1-1/z or n-1/2. These are evaluated in Appendix V, with the result &o(e)

= f,(e)

i

+ ( 21i;n

-I-w-9,

e )“2[eid+(Ase)r<(eA

-

e> + eim-(n,e) &]

eat

(3.67a)

fsco(8)= + ( 2n$n e)“2[e’d+(~-V,(eA - e) + eid-(nse) +]

+ w-‘1,

8 3 en,

(3.67b)

where r2(eA - e) = rr(2d.43ti

+ de,

(3.68)

- e)i - ao,

with the function y defined by Eq. (V. 15) and a, , a, by Eq. (V.13). The expressions (3.67) are differentiably continuous over the point 0 = tin , where they have the same value &,(Bn) as given by Eq. (3.40) (see Eq. (V.16)). In Section 4 we shall compare other properties of Eqs. (3.67) and (3.30), (3.31) and show, in particular, that they have the same asymptotic behaviour for 1 8, - e 1 > (-2~~‘)1/2. 3.5 The Total Scattering Amplitude To summarize our results, the elastic scattering amplitude ,F112 can be written as f(e) = .m(i

- F+) T(e, - 0) + f%(e) F+ + fk$e)

to leading order F- ,

(3.69)

where F* = F[d(O, F @I, and fJ,$(e) are given by Eqs. (3.30) and (3.31) or by (3.34) and (3.35), or by the alternative form (3.67). With the use of the latter, Eq. (3.69) is identical with the result derived in Ref. [9, Eq. (7.54)]. 4. PHYSICAL INTERPRETATION 4.1. Limit Conditions We now discuss the physical content of the formulas derived in Section 3 in the limits of strong and weak absorption. These limits may be roughly distinguished by the conditions strong absorption, A
538

FRAHN

AND

GROSS

absorption situation is characterized by a rather sharply defined edge (of width d) of the reflection function q(X) in X-space, irrespective of how this is produced by both a depletion of particle flux in the elastic channel due to nonelastic processes and by reflections from the surface of the real and imaginary potentials. How the strong absorption form of v(X) is generated by optical potentials has been clarified by Austern [ 161. A basic condition for our methods is a large value of the critical (or “grazing”) angular momentum fl. This corresponds to short wavelengths characteristic of heavy-ion collisions near and above the Coulomb barrier, but also applies to the scattering of lighter projectiles at high enough energies. It is thecondition underlying the basic approximations (i) and (ii) of Section 2 that lead to the integral representation (2.6) of the scattering amplitude. Precisely how this form is obtained from the exact partial-wave expansion (2.1) as the leading term in an asymptotic expansion with respect to the large parameter fl, is discussed in detail in Ref. [I 11. Heavy-ion collisions in general are characterized by large values of the Sommerfeld parameter n, and it is useful to define two further limits distinguished by the relative magnitude of the two large parameters (1 and n. Using a terminology introduced in Ref. [7] we define 0) UJ)

the “V-limit”: the “M-limit”:

n > 1 and d > 1, such that n/d = const, (4.2) 1 < IZ < A, such that n = const., n//l --+ 0, (4.3)

Looking at the various possible scattering situations, the “%‘-limit” may be realized by using increasingly heavier projectile and target nuclei at a fixed energy, while the “M-limit” is approached by increasing the energy for a given pair of colliding nuclei. In the presence of strong absorption, the V-limit represents conditions in which the Coulomb interaction is dominant, whereas in the N-limit it is negligible compared with the real part of the nuclear interaction. As has been shown in Ref. [7], these limits correspond, respectively, to the regime of Fresnel diffraction scattering (including, for n ---f co, the classical “geometrical optics” limit) and the opposite, purely quanta1 regime of Fraunhofer diffraction scattering. A general classification of nuclear scattering situations from this point of view has been given in Ref. [8]. To identify the leading terms of our asymptotic expressions for f(e) in each of these limits, we estimate the order of contributions containing @(h,) or @(A) and their derivatives @tm)(X,) or @(m)(A) by using the values for Rutherford scattering, obtained from the asymptotic form (2.11) of the Coulomb phase shifts, 0,(X,) = 0 = 2 arctan(n/h,), @,‘(A,) = -(2/n)(sin(1/2)8)2, @;;(A,) = (2/n”) sin B(sin(l/2)0>2,

0,(/l) O/(A)

= 8, = 2 arctan(n//l), = -(l/A)

sin 13,)

O;;(A) = (2/D) sin eR(cos(1/2)eR)2,

(4.4)

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539

SCATTERING

etc. Thus we have @y/l)

= 0(1/L@) = o(l/n’n)

in the W-limit,

O’“‘(A)

= O(n/Ll”“)

in the M-limit.

4.2. Weak Absorption,

(4.5)

Semiclassical Regime

Under these conditions the scattering amplitude has the form (3.2). It is essentially given by the potential scattering amplitude (3.3), obtained to leading order in h by the stationary phase method. If there is only one stationary value he (as for our assumption of a monotonic deflection function), the scattering cross section is given by the classical expression u,,(e) multiplied by the probability for reflection P(0), 49 = ucl(e) zw, (4.6) where acl(~) = ihm2, w) = hhd12, (4.7) The attenuation factor I 5 1 is a slowly varying function of 8, which tends to unity as 6’ ---f 0, so that always u(e) -+ u,,(0) for small angles. In leading order of the stationary phase approximation only the positive branch ft;“(@ contributes; fb-j(e), having no real stationary point, is negligible in the SPA. In the g-limit, fO(0) is essentially given by the Rutherford scattering amplitude 1 h(e)

-h(e)

=

-

-$

(sin(lj2)e)2

i~26~~O~-21aln~sin~1/2~e~1 e

(4.8)

,

which in the presence of weak absorption is multiplied by y(he) w ~(n * tan 40). In the N-limit,&(e) may have a complicated angular dependence for nonmonotonic nuclear deflection functions with several rainbows, orbiting, etc. Thus the weak absorption limit covers the semiclassical regime characterized by the absence of those quanta1 diffractive effects that are associated with a rapid variation of q(h). 4.3.

Strong Absorption

4.3.1. g-limit: Fresnel diffractive scattering. leading term off&?) for 8 < en is given by

In the sharp-cutoff

limit,

the

(4.9) The potential scattering amplitude is now modulated by the function & erfc(u,,) which describes Fresnel diffractive scattering [7]. This results from quanta1 dilfraction by the sharp edge of r](X) at X = /1, in the presence of a strong repulsive

540

FRAHN

AND

GROSS

Coulomb field which diverges the diffracted wave so as to appear to originate from a virtual source point at a finite distance [S]. The angular distribution is thereby divided into the “illuminated region” 8 ,< 19~) where the cross section ratio ~(@/a,,(@ shows characteristic Fresnel oscillations described by the function 1erfc(u,)12 for U, < 0, and the “shadow region” 193 8, where ~/cr,, decreases smoothly to zero. In the extreme U-limit, IZ -+ co, 8, = const, Eq. (4.9) tends to .fkLce)

-h(e)

(4.10)

de, - e),

except for a Gibbs phenomenon at 0 = eA. Thus the ratio u/u,, tends (nonuniformly) to the unit step function as expected in the classical (geometrical optics) limit. The other contributions to j&,(0), namely, f~~~,2(0) = jjli(@ - fade,, and f:;!(e), are of relative order A--l12 compared tojo( They represent further quantaldiffractive effects essentially of Fraunhofer type, which may be interpreted as arising from diffraction “at opposite sides of the nucleus”. The Fraunhofer contributions in the (+) and (-) branches offSc0(8) are, however, of different magnitude due to the distorting effects of the Coulomb field. The Fresnel term (4.9) has the exact “one-quarter-point” property, first noted empirically by Blair [5] in the sharp-cutoff model for Rutherford scattering: 4eR)bR(eR) M a. Because of the contributions from ff,~~,2(0) and f;;:(e) this property holds only up to terms of order A-1/2 or n-lj2. From Eqs. (3.40) or (3.67) we obtain ~fscouu hut4

= z1 1 + i

where a,(O) = -c,(A). (2/n)1/2 sin(l/2) eR and

@/4)n

For

+ igq

(+)1’2[2a,(0)

pure

Coulomb

ao(~)= &&-

+ o(n-l)~,(4

phases, with

[--@,‘(A)]‘~”

11) =

+ i cot i eR, R

we have

+ i 2 sin(l/2) OR

8R

exp

(iheR

cot i

eR)]

+

o(+)l

.

(4.13)

The quarter-point property has often been used as a recipe for determining nuclear radii. From experimental data one determines the angle 8,,, where u/OR = 4. Supposing that this angle is given by OR = 2 arctan(n/Ll), with the VdUe

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SCATTERING

of (1 so obtained one then calculates the “strong absorption radius” R from the semiclassical relation kR

=

n +

(n2

+

(4.14)

A2)lj2.

Eqs. (4.11) and (4.13) show that not only is this procedure subject to corrections of order /1-1j2 or ICI/~, but also that the true quarter-point 8,,, is approximately given by Bn rather than by eR . Amusingly, since 0, is often somewhat smaller than 8, , these effects tend to oppose each other so that Blair’s recipe usually works quite well in practice. For 0 # 8,) Eqs. (3.30) and (3.31) can be written as = fo(e) i erfc(u,) - $ ( ,-?i’,

&i(e)

)l” c+(e, e,) Z.ZY 8, - e (4.15) (4.16)

where we have used Eq. (3.28), and (3.27) in the form

’ c+(e, - 8, e,) = -Jen - e [ (4any2

- c+(e, kd].

(4.17)

we can replace Well away from 8, , for j u0 1> 1 or / e - en 1 > (-20,‘)1/2, erfc(u,) by the leading term of its asymptotic expression for arg u,, = &T (see Eq. (5.46)). Then the component f~$,(B) becomes

f%,.,(e) = h(@ - .h(e)2 erfc(--u,) g h(e) - fo(e) 2n$~~uo, en - e > (-20Ay,

(4.18a)

f%de) g fo(e)& -@*’

A

= 7Z

(

257 sin

0

p+bl.e) I/2 e 1 c+ut 64 0, _ 8 T

e - en > (-20ny2,

(4.18b)

542

FRAHN

AND

GROSS

where we have used Eq. (3.32) forfO(6’). Thus fkA,<@ r-L)(e) + ;

( 27i;n

e)1’2 g

)

en - e > (-20n’y2,

(4.19a) l/2 eiQ+bl,o)

e - en > (-20ny2.

Tc-&---

(4.19b)

It is shown in Appendix V that the same asymptotic expressions follow from Eqs. (3.67a, b). The effects of smooth cutoff are described by the functions F+ = F[Ll(B, F O)]. Because F(dz) is defined as the Fourier transform of the absorptive shape function, for strong absorption it is a broad but strongly (e.g., exponentially) decreasing function of angle. While F+ has its maximum at 8 = 0, , F- peaks at 0 = -0, so that only its exponential tail extends into the range of physical angles. This causes a strong damping of the negative branch of j&,(O) and thus of the Fraunhofer-type oscillations superimposed on the Fresnel pattern in the sharpcutoff cross section. The Fresnel oscillations themselves are damped towards smaller angles by the function F+ , so that often only the first Fresnel maximum is seen in experimental data. Further discussion of these features can be found in Refs. [6-91. 4.3.2. M-limit: Fraunhofer dlfiaction scattering. Because in this limit the critical angle On tends to zero, the amplitude jade reduces to the expression (4.19b) for the asymptotic shadow region 6’ > On . Therefore we have f%Xek

112 ei6+bl.s) ___ e-e43

-+(,,4,,)

e>

en,

(4.20)

112 ,i&(A,d

f:;:(e)

By introducing

=

.L

A

k ( 2rrsine

the asymptotic

-zF-Jy*

(4.21)

form of the Fraunhofer diffraction amplitude

1 z-ke

1

n

im+(n.e) _ e-id-Li.e) 1 ( 2nsin8 1 [e

= fkm + f&Lm

l/2

[9]

(4.22)

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ION

543

SCATTERING

we can write

and the total scattering amplitude f(e) = M4(@

(3.69) becomes

JJf+v> + fkim>

M-(Q

e>

en

(4.24)

where

Thus as long as 8, is small but still finite, the positive and negative branches of the Fraunhofer scattering amplitude are modulated by different functions M+(e) and M-(B), respectively, where M+(B) > M-(B). This results in a damping of the Fraunhofer diffraction oscillations described by the ratio M-(0)/M+(0), whose magnitude depends strongly on the Sommerfeld parameter and has therefore been termed “Coulomb damping” [6-91. In the limit en ---f 0 we obtain

This means that for unsymmetrical absorptive shape functions D(h) the positive and negative branches of fFRA(8) are still modulated differently; only for symmetrical D(h) we have (4.27) Fraunhofer diffractive scattering is a purely quanta1 phenomenon, of relative order A-@ compared to the “semiclassical” potential scattering amplitude &(0). It represents as it were the opposite extreme to potential scattering, in that the two branches fp&,(e) are of equal magnitude, giving rise to strong diffraction minima (zeros), whereas to leading order of the stationary-phase approximation for fO(0) only the positive branch contributes. The derivation of Eq. (4.23) from Eqs. (4.15), (4.16) shows clearly how the amplitude in the N-limit originates from that in the V-limit: the negative branchf&jA(0) arises directly fromf$A(@ given by Eq. (4.16), while the positive branch f”’FRA(0 ) r esults from a cancellation of part of the order-A-1/2 contribution f Lzi .2( 0) by the asymptotic form of the Fresnel term fi$A,&?) =&(e) &erfc(u,,) in the shadow region, where the latter by itself is of order A-112. For heavy-ion collisions, in which Coulomb effects are always strong, Fraunhofer-type diffraction is of lesser importance than Fresnel scattering; it does however appear at high energies when 8, is sufficiently small.

544

FRAHN AND GROSS

5. EVALUATION

FOR A GAUSSIAN ABSORPTIVE

SHAPE FUNCTION

5.1. Motivation

In the preceding sections we have derived explicit expressions for the elastic scattering amplitude without specifying the detailed form of the functions q(h) and 6(h). In fact, these functions enter our expressions only through the Fourier transform F(Az) of the absorptive shape function D(A), and through the values and derivatives of 6(h) at A, and A. These expressions are valid in the limits of strong absorption (A Q A) and weak absorption (A 2 A). We now choose a simple specific form of D(X) which enables us to derive asymptotic expressions that are valid over the whole range of the width parameter d. The main features of typical absorptive shape functions can be represented by a Gaussian form D(h) = &

corresponding

exp [-

&$j2],

(5.1)

to the reflection function q(h) = f erfc (q).

(5.2)

The Fourier transform of D(X) is given by F(h)

= exp[-(*

Az)~].

(5.3)

The methods we shall use to evaluate f(e) with the form (5.1) provide a more rigorous justification for the approximations of Section 3 by which the amplitude in the strong absorption limit was expressed in terms of the sharp-cutoff amplitude and the Fourier transform F(Az). Moreover, it will be shown that the transition between the limits of weak and strong absorption is controlled by a characteristic parameter TV which enables us to give quantitative criteria for the validity of the “semiclassical” and “quantal” regimes of heavy ion scattering. 5.2. Asymptotic Expansion for

f c*)(8)

Starting once again from the partial wave expansion (2.1) we now proceed in a more general way by not immediately using the asymptotic form (2.3) of the Legendre polynomials. First the summation over I is converted into a series of integrals by means of the Poisson sum formula. As discussed in detail in Ref. [l 11, for our present purpose of obtaining asymptotic expressions for f (0) to leading orders in a large parameter /3 defined below, we may restrict our considerations to

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ION

545

SCATTERING

the zeroth-order term of the Poisson series. The two branches of the scattering amplitude are then given by [l l]

f’*‘(e) = - + (2r ,; e)l,2j- dhA@,se>q(h)ei6+(A’e),

(5.4)

112

with (5.5)

where F(cz, b; c; z) is the hypergeometric function. We want to explicitly display the dependence of all quantities on a large parameter p, which may be the Sommerfeld parameter n (in the V-limit), or the critical angular momentum A (in the M-limit), or any other large parameter. Our aim is to derive asymptotic expansions of f(0) with respect to /3. Denoting “reduced” quantities by a tilde, we write6

h = /3x,

f!l+A,

etc.

(5.6)

A(@( e) = @X)1/2+ O(/w2),

(5.7)

For instance, for large /3 and fixed x we have

where the leading term yields the asymptotic form (2.6) of f(+)(e). By introducing the integral representation (3.4) in the form (5.8)

we can write

- Irn

dX A(& &i3) exp{i/3[2@)

-

(Z

+ 0)x]},

1128

(5.9)

where’ 6(A) = @(A) and A@, 0) = p1/2 @, 6) (see Eq. (5.7)). 6 The convergence of the integrals (5.4) at h ---t m may be ensured by taking p to be slightly complex, fl = p,,(l F k), and letting c -+ +0 after integration; this will be understood in what follows. ’ For instance, Eq. (2.11) shows that 6,(A) is of order n or A.

546

FRAHN

AND

GROSS

The integral over x can be evaluated by the stationary phase method with the result [l l]

- {expb%WWd- X&l + O(PW>,

(5.10)

where A, is the (reduced) stationary values pertaining to the angle 9 = z f 8, B < 8 < P, and where we have defined (see Eqs. (4.4) and (4.5)) G(X) = ; [2&i)]

= $ [26(A)] = O(A),

we)

= $7 (5.11)

@(X) = $ o”(X) = /30’(A),

&yJ)

= p@~‘@).

With the definition (5.12) Eq. (5.9) may be written as

Now we adopt our choice (5.3) for F@z) and define the phase function &i,

) 8) = 28(X,) - (X, - A) 8 + i@(L!l F ey + o(p),

(5.14)

where LI = /31120”. Then we have (5.15) with (5.16) 8 For simplicity of notation we denote the different stationary values x8 and other quantities pertaining to the angles 6 = z + B and 8 = z - 0, respectively, by the same symbols except where confusion may arise.

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547

SCATTERING

The integrand of J(*)(8) has a saddle point where d$?/dG = 0. For the first and second derivatives of + we obtain

pLi,+ip(g.q?), 1 a’&)

d3=-da2

+

(5.17) iiJ2

=

2

‘(‘) -@($)

’

(5.18)

where we have defined .$(a) = 1 + From Eq. (5.17) the condition Xg,

i@2[-@(3;,)]

=

1+

b(8).

(5.19)

for the saddle point 9, is -

A

=

iJQi2(8,

;C

tl),

(5.20)

which is independent of /3. If 9 = en, we obtain the solution $I;“’ = en = 8, otherwise we have ai+’ # Bn for 0” f 0. After multiplication by fi this condition becomes x6, - (1 = o-1(8,) - o-ye,) = igyo, F e). (5.21) We now expand +@, ,6) about the saddle point 6, explicitly up to second order, so

3$1 = c&

3$8) + (Es/(-mw9

where ES = &as), &i,, = @‘(&J

- $,I2 + $@ - ~3,

(5.22)

and (5.23)

By defining the functions

~+(a,8,) = qx, , ke) eioJ(-),

(5.24)

the integrals (5.16) can be written as

(5.25)

where & = +(&, ,6,). The main contribution to the integral (5.25) comes from the inside of the integration interval (the pole and the stationarity point). The contributions from the boundaries are of the order /3-l or even exponentially smaller because

548

FRAHN

AND

GROSS

the total phase has an imaginary part @o”“(ti - @2. Moreover, we have already neglected the contributions coming from the boundaries in the SPA result (5.10). Therefore we should only evaluate contributions coming from the inside of the integration interval in Eq. (5.25) up to order /!-’ exclusively. These will be the contributions of leading order coming from the pole in J(+)(B) and from the stationarity point 9, in both J(+)(B) and J(-)(0). The contribution of the pole in J(+)(B) can be separated from that of the stationarity point in the following way. We define the auxiliary function (5.26) which because of

iii C+(S- 0,8,) = [(a/28)C+@,zq&%e is finite and analytic including two parts

(5.27)

9 = 0. Then the integral J’+)(8) can be split into

Y+)(e)= Y+)(e) + Y+)(e) 1 2 9

(5.28)

where

exPb~(~sl(-2~8’N(~- 6J21 8-W-k

(5129) &‘(e)

= ei(8@8,,-t(1’4)n) &. 1” d8 C+(iI+ - 0,19,) exp [i/l 0

and where we have extended the integration in J:+‘(B) to & co in order to get rid of the boundary contributions and to isolate the contribution of the pole. Asymptotic evaluation of the integrals in (5.29) and (5.30), carried out in Appendices I and II, respectively, yields J?)(e) = -c+(e,

8,)

ei(fl@8+(1/4)n)e~P23

erf+‘l”s),

(5.31)

where - as),

(5.32)

c+(e, a,) = G& , e) PJ(e-aJ

(5.34)

12 = (i&/--26,;)1/2(0

From

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549

SCATTERING

it is seen that the exponential in Eq. (5.31) is ides + PC2 + if+@ - as) = i/3+, + (5s/(-26,‘>)(0 = i~$&

- 8J2 + $(e - 8,)]

, 0)

(5.35)

= i/3[2&ji,) - (X0 - ff) e] + O(p),

where we have used successively Eqs. (5.32), (5.22), and (5.14). Thus

f(+)(e) = & e-i(aAe-r1’4)n) J(+)(e) 1 k (sin ey l ia[2d(n‘B)-ie,sl z1erfc(/31’2zl).

. P We , 4

- --I k (sin e)v e

(5.36)

Now we note that the potential scattering amplitude

f,(e) = - $ (2n ,it, e)l,2 j- dx A(A, e) 2b+(Ase)

(5.37)

l/2

can be evaluated by the stationary phase method. Using the result (5.10) and the definition (5.12) we obtain fo(e)

=

-i

&

G(Xe

94

p[2S&+nss1

k (sin @l/2

+ 0 ($j,

(5.38)

and Eq. (5.36) becomes f(+)(e) 1

= f 0 (e)g erfc(/3%).

The rest of the amplitude f(+)(e) = f:“(0)

(5.39)

+ fi+)(e) is given by

f(+)(e) = & e-i(B’e-(1’4)n)J(+)(e) 2 k (sin @l/2 2 z---e

Bl” k

ib(+

(5.40) -&

6-Ze)+(l/4hl (

5,27~sin

e

For the (-) branch integral Y-)(0) we define the auxiliary function c-p

+ e,9,) = c-p, s,)/p

+ e).

(5.41)

Then the integral P(e)

= P@~+(l’~)n)

&

=

J’ & C-(8 0

+ e, 9,) exp [ifi

(5.42)

550

FRAHN

AND

GROSS

is of the same type as Eq. (5.30) for Ji”(@ and its asymptotic evaluation can be carried out by the method of Appendix II. Thus the negative branch of the scattering amplitude becomes (5.43) _

-&

P 1/Z ,irB(~~+&0-(1/4h1

k

(

f,2~ sin

)liZ f

e

c!?‘(a,

+ e,s,> -$ (+g)‘.

s

?=O

5.3. Explicit Expressions for f(B) to Leading Orders in /3

We can now discuss the order in the large parameter /3 of the various terms in f(e) = f,“)(e) + f?(e)

+ f(-)(e) 2

(5.44)

given by Eqs. (5.39), (5.40) and (5.43). First we note that all phase factors in these expressions are of order /?. Since the wave number k scales proportional to /3, so that we may write k = @, Eq. (5.38) shows that the potential scattering amplitudef,(B) is of order /3” (with corrections of order ,8-l to the stationary phase approximation). Because of 4 erfc(-z)

(5.45)

= 1 - 4 erfc(z),

the contribution f:+‘(e) in the far “illuminated region” B < 8, is of the same order (PO) asf@), while in the far “shadow region” 0 > 9,) because of

[1 + OW2)1,

erfc(z) = -$$

1 arg z = - r,

IZl>l1,

4

(5.46)

j:+)(e) is of the same order (/3-1/2) as the other contributions fi+“(e) and f(-)(e). In all our subsequent discussions we have to keep in mind that there is an inherent inaccuracy in our method of 0(/3-l) due to the use of the SPA in Eq. (5.10). Therefore we shall only collect terms of O(p) with q > - 1. In the expansion (5.40) forf:+‘(@ the first term with r = 0 is given by f~)(qrxo

=

L$!T

ei[s(+~@)+(l/4)771

( ,,2;T;i

e)1'2

c+(~,

_

0, 8,).

(5 47)

From Eqs. (5.26), (5.24) (5.12), and (5.7) we have

c+p, - 8, ss) = =-

c+(h,a,)- c+v,8.4 G& , e) 8,--e

8, -

=

I/=

X8,

1

8

-a’&,)

I

1, -

[ -By&J

G(J, , e) eisJt(s-ea) 9, - 8

(5.48)

1'2 .we-eq

3

+

O(fi-1).

HEAVY

ION

551

SCATTERING

Thus the term (5.47) is of order /I- 1/2. The remainder of the sum over r in Eq. (5.40) would be of order /P/2, were it not for the dependence of c+ on the function 6. In forming the derivatives Cy)(aS - 0,6,) this function gives rise to contributions of the form (f@yo))[2'/31 CF-t2r/31)(& _ 0, $3, (5.49) (among others of lower order in p), where [x] denotes the largest integer < x. This has the consequence that at the point 0 = en = 6, there is an additional contribution of order p-l12 from the term with r = 1 in the expansion (5.40) (Gibbs phenomenon). From the calculation carried out in Appendix IV we find f$+'@

=

8,)

_

fy2

co(x8J

eiM+%)+(l/4hl

=

-

(

5 [ R.&.)

h8

($2~ sin i+, iI 1/Z co + ww, 6:

+

( 1 --

(-&)2

1 3‘$$ 11’

(5.50) (5.51)

while for 8 # On , 0 # 9, all contributions of order /3--l/2 are given by Eq. (5.47), with C($, - 6,6,) given by the last expression in Eq. (5.48). In the expansion (5.43) for f(-)(e) the term with r = 0 is f(-)(e),=,

_

8-1'2

$tB(Gs+iie)-(114)nl

-6:'

(

)I"

f,2~ sin

K'

c-(8,

+

0, $3,

(5.52)

8

where from Eqs. (5.41), (5.24), (5.12), and (5.7) we have

e-(8, + 8, as) =

f.Xi+I , -0) = -& h-to 8

($-J”

+ 0(/3-l).

(5.53)

The remainder of the r-series is of order jgPi2, so that the term (5.52) is the sole contribution to order j3-l12. As a result, the explicit asymptotic expression for f(0) for 0 # 8, to order /3-l/2 is f(e)

=

fo(e)

erfc [ ( ~~$+t~l

i

s

j v2 K

(e - a?))]

eits(~y'-Aee)+(l/4M

-@+) (

I;

) P2

)1’2

1/Z c@+'

5;+)27~ sin

e)

X'-' (

5’,--)2~ sin

-

8, a',"))

112

8

eiIBb$$+'+m-(l/aM

s

e)

&-,l+ e + O(P), J

(5.54)

552

FRAHN

AND - GROSS

where

fo(8)i - $ [ -@(;I) sine]ljr ei4r2~Qe)-x@l + 0(/3-l),

(5.55)

where we have now distinguished by superscripts (A) between the different stationary values A, for f(+)(e) and f(-)(e). 5.4. Strong Absorption

Limit

5.4.1. Saddle point condition. In order to see how our expressions (5.39), (5.40), and (5.43) reduce in the limit of strong absorption, we return to the saddle point conditions (5.20) for f(+)(0) and f(-)(8), respectively,

Xk) - f$ = i*J”($F’

- e),

(5.56)

Xji’ _ J = ilJ”?($(-) 2 s + 0)

(5.57)

In the strong absorption limit d” is small, so that we have xe) w 2 and 8F) NN e-J . We therefore expand in powers of 82) - 8,)

a(*) - 8, * @(fT) ’

- (3 ml!+ h

(5.58)

and obtain from Eqs. (5.56), (5.57) a(*) s - en M ii P(8F) @(ii)

T e)

(5.59)

e),

(5.60)

or a?) -

8, M -iTA

T

where we have defined the parameter TA = Alternatively,

T(e,) = -@(2T)

22.

(5.61)

,

(5.62)

Eq. (5.60) can be written as s?) F e - (e, T w5

where fn =

ge,)

= 1 + iTn .

(5.63)

The functions c(8) and ~(8) were introduced in Eq. (5.19). We shall see that the parameter 7n or ,$A controls the transition between the “quantal” and “semi.

HEAVY

ION

553

KATIERING

classical” regimes of scattering and also defines the limits absorption by T/l - 0, T/l-+

a(+) s ---f 8A, a(+) s + e,

en-+19

00, L+iQ

of strong and weak

strong absorption limit,

(5.64)

weak absorption limit.

(5.65)

The physical significance of these parameters will be discussed in more detail in Section 7. Note that because o”l = PO’ and n”z = /1-‘02, the functions ~(8) and ((8) are independently of p also determined by 10’ 1A2. From Eq. (5.62) we obtain further g(i) 9 68 en - i(~A/EAWA -f e>, so that in the strong absorption

(5.66)

limit

Im 7YF’ M

-T,(e,F

e)+o.

(5.67)

Eqs. (5.66) and (5.67) justify the expansion (5.58) in the strong absorption limit. By using Eqs. (5.62) in the conditions (5.56) and (5.57) we obtain XE) - ii- = i(J2/2LWA and from the definition

T e),

(5.68)

6:” = .&(8~*‘) = .$A[1 + O(P)].

(5.69)

(5.19)

5.4.2. Sharp-cutofamplitude. In the limit (5.64), n” = 0, 74 = 0, the scattering amplitude f(0) becomes the sharp-cutoff amplitude. For the component f!+)(e) we obtain from Eqs. (5.39) and (5.32) with 8:‘) = tin and e:” = 1,

(5.70) where

ii, = ei’l/4’” In Eq. (5.47) forfi+‘(@ Fs ZE &’

e - 8, [4@(4]‘/”

’

(5.71)

we have

z.zz+(X2’ ,8?‘)

= 2S(e’)

- (Q) - A) 82) + jjJ”($f’

- 6)”

= 2$(A) + (32’ - d)(eA - 192’) + i)d2(616f’ - 0)” = 2&ll”) + i&i2(eA - 8)’ + o(4”).

(5.72)

554

FRAHN

Thus, +‘,” = 2&l) j$i,,(@

=

AND

GROSS

in the limit d = 0, and Eq. (5.47) reduces to

F

,is[zs(d)-/Tsl+i(l/4,a

( 2-;;;)1’2 ?r

C+(s,

_

0, 0,)

+

e# with the abbreviation

ocp-Ij,

en,

(5.73)

@A1= @‘(~). At 8 = Bn we obtain from Eq. (5.50)

f$i,2(e,) = .$!Y

eiB[2S(;I)-~~~l+a(1/4)n

(2 7l

where

st e )l” A

co(Jj + ~(p--I),

(5.74)

6; &Q = - 3[/&‘) + 3 (-GJ2I *

(5.75)

In Eq. (5.52) for f(-)(e) we have qs = $j-’ = q(xJy’, p>

= 2&A&‘)

= 28(A) + (Fq) - A)(0‘, - sj-‘) = 28(A) +

Thus, &’

i@(eA

-
+

+ 0)”

e)”

(5.76)

+ 0)2 + O(B).

= 26”(z) in the limit 0” = 0, and Eq. (5.52) with (5.53) reduces to

fj,(@ _- - v2

k

erEt28(K)+~el-i(1/4)n

( 2n;n

e)1’2 &

+ qp-1).

(5.77)

5.4.3. Smooth-cutoff corrections. Now we want to determine, in the context of the Gaussian model for the shape function D(X), the nature of the procedure in Section 3.2. which led to the extraction of the “form factors” F[d(8, - @] and F[d(dn + ($1, where I;(dz) was dehned as the Fourier transform (3.5) of D(h). It is only through these functions that the form of the smooth cutoff in v(X) near X = (1 enters our expression (3.69) forf(r3). This is clearly an approximation which, however, should be good in the strong absorption limit when F(dz) varies slowly in the neighborhood of the stationary points 8j*) = zj+) + 0 = tin in the integrals (3.7) and (3.19). On the other hand, Eqs. (5.39), (5.40), and (5.43) give an exact expression of f(e), in the Gaussian model, valid for all values of d. To compare this with the approximation procedure of Section 3.2. we identify the Gaussian form factors

F[Ll(B, F 0)] = F[~1/2iT(0A F 0)] = exp[-@2(0n

F @al

in approaching the strong absorption limit of Eqs. (5.39), (5.40), and (5.43).

(5.78)

HEAVY

ION

555

SCATTERING

This is straightforward for the componentsfi++‘(B) andf(-j(O), where the approximation consists in retaining.linear terms in Jz in thephusefunctions only, +*)

M 2&A) + i@ye,

+ e>z,

(5.79)

so that

f(-)(e) w FgWi(eA + e)]fi&e).

A+)(e) w m1’24eA - e)i&2(e),

(5.80)

In the componentf:+‘(O)

the argument u’ becomes, to first order in TV, (5.81)

so that w

f(+)(e) 1

f‘(e)+ 0

We now expand the error function given in Appendix VI, &erfc[/31/2z&,(1- i+TJ]

erfc[f11’2i&(l - i&rJ].

(5.82)

by means of the addition

= exp[i/3fi02(Tn -

i$~h)]

theorem (VI.7)

&erfc@1/2z20)+ W(zZ,>, (5.83)

where

* j.

[i~A(#l)1/2 ZZOlmilD-m-2[(219)1/2 co].

(5.84)

In order that the series (5.84) converges even for large /3 we may use Eq. (5.83) only in the shadow region 0 > 8, (arg CO= $T), where the D-functions have the asymptotic form Ln-2Km1'2

GOI E

[(q3)1,2

6 p+2 0

(5.85)

*

In the region 6’ -C en we write 9 erfc[/31/2zZo(1- i&rJ]

= 1 - 4 erfc~1/2(-iZo)(l

= 1 - exp[i@02(T11 - if~~~)] 4 erfc[/N”(-Coo)]

- i&-J] - 9(-Co).

(5.86)

556

FRAHN

AND

GROSS

Then the first terms in Eqs. (5.83) and (5.86) are dominant Since ~/?z&~T~= --@A2(0 - 0J2, we have fl+‘(@ m f;(O)& erfc(/Y222,) F[/?i’2Li(~A - e)], f!+‘(e)

~.m>U

e3eA,

FpQi(e,

- [l - ; erfc@““&J]

for small TV and large p.

-

e)]},

(5.87)

eden.

This, together with Eqs. (5.80), agrees with the expression (3.69). 5.5. Weak Absorption

Limit

5.5.1, Saddle point condition. From the saddle point condition (5.56) for f (+)(O) it is seen that for large 2 we have a’,” = l?. Then we expand in powers of SI;” - 8,

xy dXe+

a(+)- e s BI(X,) ’

(5.88)

and obtain from Eq. (5.56) (5.89)

Using

- en- e A = &I + @(Q ’

(5.90)

it is seen that Im 9:) M s

(e - e,),

(5.91)

which is small if 1 S(e)/ > 1, even if i;, and e are quite different from ii and en, respectively. Therefore the expansion (5.88) is justified. From Eqs. (5.88)-(5.90) we have in the weak absorption limit

8, a.(+) .s - e m ___

e=

0(7;1),

(5.92)

m - g+‘-J,- - -A--h, = U(T,l), 8 -34 ((+I = ((8”‘) = f(e)[i + s s 5.5.2. Scattering amplitude.

ii=

if@(‘)) [

-2@;x;;l,

(5.93) (5.94)

o(T;l)].

Using Eqs. (5.32) and (5.89) the variable zi becomes 1”’ (0 - J$+)) M [ -iEjy

1”’ (J - ,Q,),

(5.95)

557

HEAVY ION SCATTERING

and, since t(6) + L-(0) = -i@@(&), 22+ (ii - X,)/J

for

74 + co.

(5.96)

Thus the component f(+)(e), given by Eq. (5.39), tends to fj+‘(O) -fo(e)

i erfc (/31i2 ’ j A, ).

(5.97)

Furthermore, Eqs. (5.40) and (5.43) show that the components f:+‘“(0) and f(-)(e) are both or order [-112, therefore A+‘@> - 0, and the total amplitude

f’-‘(e)

3 0,

for

TV -+ co,

(5.98)

in the weak absorption limit reduces to (5.99)

for the special form (5.2), in agreement with Eq. (3.2).

6. NONMONOTONIC

DEFLECTION

FUNCTIONS

As discussed in Section 2, our methods of evaluating the scattering amplitude so far are based on the assumption that the quanta1 deflection function O(X) is a smooth, monotonically decreasing function of the angular momentum variable h. In particular, the stationary phase approximation is valid only if O’(X) # 0 for all finite X. This is at variance with semiclassical models of the deflection function, especially with the forms of &i(b) calculated in WKB approximation for strong real nuclear potentials and weak absorption. In these functions there are one or more points b, at which Oi,(b,) = 0 (“rainbow scattering”), and there may be a singularity at a point b,, where O(b,,) = - co (“orbiting”). However, if quanta1 effects due to reflections at the nuclear surface are taken into account, by means of complex turning points in the WKB approximation for complex potentials [3,4], the orbiting singularity is removed and the deflection function is smoothed out. This is confirmed by optical model calculations of O(X) for realistic complex potentials [15]. Nevertheless, for some of these potentials small deviations from the average monotonic decrease of O(h) may occur in the vicinity of the critical angular momentum. Such deviations do contain points X, at which @‘(XV) = 0, where the SPA breaks down and must be replaced by a uniform approximation. One might, therefore, expect that the occurrence of such deviations would give rise to complicated Airy-type patterns in the angular distribution near the critical

558

FRAHN

AND

GROSS

angle On and thus invalidate our results. The following considerations will show that this is not the case under conditions of strong absorption when the deviations are relatively small. Considerable changes will occur only for weak absorption when the angular distribution is dominated by potential scattering. Since these have been discussed extensively in the literature (e.g., in [4; 9, Chaps. 4-6; 171) we confine ourselves to the strong absorption case. We assume that the quanta1 deflection function may be written as O(h) = @(A) + @(A),

(6.1)

where 8(h) is a smooth monotonically decreasing average (dominated by the Rutherford contribution O,(h)), while o”(h) represents the nonmonotonic deviations confined to a relatively small region around h = d. If we write the corresponding phase shifts as 6(h) = S(h) + 8(A) (6.2) and define S(x) = v(h) @*(A) I S(h) ei28(A), (6.3) then, under conditions of strong absorption, the function S(A) = q(A) eizsfA) represents the rapidIy varying part of S(h). Now we introduce an integral representation

(6.4)

of this function as in Eq. (3.4),

where F(z)

= jrn dAxed&v -cc

-i(A-AM =

Codh [D(h) + &(A) q(h)] ei28(A)e-“cA-AJz. s --m (6.6)

Because s’(h) is rapidly varying, its Fourier transform s(z) will be a smooth (complex-valued) function of z. Assuming that g(A) and &(A) are small compared to unity so that we may expand exp[i2@)] in the integrand of s(z) to first order, we can write S(z)

= F(dz)

+ G+(z),

(6.7)

where F(dz) is defined by Eq. (3.5) and F(z) = Irn dh [&(A) $A) + 2&‘(h) D(h)] e-t(A-A)s. --m

w9

559

HEAVY ION SCATTERING

Now we can evaluate the scattering amplitude Section 3.2, with the result

by the same method

as in

f(e) = ~o(e)(l - F+ - iF+) .((%I - 0) + (F+ + @+>J$(c+d(~) + V" + ip-2 f:&e),

(6.9)

where &o(0) and j’sCO(B) are the potential scattering and sharp-cutoff amplitudes for the phase shift function 8(A), and where we have used the abbreviations F+ = F[@, F e)],P+ = P(e, i e) with 8, = S(A). To indicate the main effect of the nonmonotonic deviations on the angular distribution, we approximate j’&0) by the Fresnel term

(6.10) Then the differential cross section becomes

40) m~~48, - w + c.t;;cfL - em I~k(w.

(6.11)

Thus we find, perhaps somewhat unexpectedly, that under conditions of strong absorption, small deviations from a monotonic deflection function lead mainly to a small addition p+” to the “smoothing function” F+2 and therefore do not appreciably affect the main features of the Fresnel diffraction pattern.

7. CONCLUSION.SEMICLASSICALANDQUANTALREGIMESOFHEAVY-IONSCATTERING In this paper we have derived explicit formulas for the elastic heavy-ion scattering amplitude. These have been obtained as the leading terms of asymptotic expansions with respect to the large parameters A and n that characterise heavy-ion collisions. Our expressions are valid for general realistic forms of the elastic scattering function S(h) in the limits of weak and strong absorption. In Section 4 we have discussed the physical content of these results by considering two further limiting conditions, the V-limit and the M-limit defined by Eqs. (4.2) and (4.3), respectively. In order to investigate the transition between the strong and weak absorption limits we have adopted a Gaussian model of the absorptive shape function D(A). In this case the scattering amplitude can be asymptotically evaluated for arbitrary values of the Gaussian width parameter A. The results of Section 5 show that the transition between weak and strong absorption is controlled by the parameter 7n = &I2 [ o,(A)l.

(7.1)

560 This quantity widths”,

FRAHN

AND

GROSS

has a simple physical interpretation w, = (d/a)

as a ratio of two “angular

w, = d/A.

I O’(A)l,

(7.2)

We may regard w, as the range of angles (around 0 = 0,) that is associated with the width A/S? in angular momentum space (around h = A) via the quanta1 deflection function @(A). If the latter were equal to the classical deflection function, W, would be the range of deflection angles of trajectories associated with the range A/d/z of orbital angular momenta via the classical equations of motion. By contrast, w, measures the spreading in &space associated with quantum diffraction by the h-space “window” of width A/d/z and profile D(h). Thus w, and w, may be termed the “classical-dynamical width” and “quanta1 width”, respectively, and were first defined by Strutinsky [lS] for heavy-ion transfer reactions. Their significance for elastic scattering was pointed out in Ref. [9]. Since 7‘4 = w,lw, 3

(7.3)

the conditions (5.64) and (5.65) for strong and weak absorption also characterise the “quanta1 regime” (w, > w, , 74 < 1) and the “semiclassical regime” (wo > w, 9 74 > 1) of heavy-ion scattering. Similar conditions apply to transfer reactions as was pointed out in Ref. [19]. Together with the conditions (4.2) and (4.3) for the q- and &“-limits, we may classify the physical processes in nuclear scattering by the following scheme, giving criteria for the main regimes in which they dominate: %-limit A>l, e,iu

1

N-limit Ll>l, e,<1

semiclassical regime (weak absorption) Tn> 1

(I)

nondiffractive (potential) scattering (IV) with strong with negligible Coulomb interaction Coulomb interaction

quanta1 regime (strong absorption) 74 < 1

(11)

Fresnel Fraunhofer diffractive scattering

(III)

For heavy-ion scattering the main regimes are I (below the Coulomb barrier) and II (near and above the Coulomb barrier), which gradually changes into region III at higher energies, whereas region IV never applies. For A = 0 and pure Coulomb phases this classification agrees with the systematics given in Ref. [S]. Using 1 @,‘(A)[ = sin e,lA we can write TA = M442,

(7.4)

561

HEAVY ION SCATTERING

with the diffraction pattern parameter p = A2 1O’(A)] = d sin OR defined in Ref. [8]. Equation (7.4) shows how the quantity TV controlling the transition between the semiclassical and quanta1 regimes is composed of the parameters defining the type of diffractive scattering (p) and the strength of absorption (O/A). Similar methods to those described in this paper have been developed for explicit evaluation of the amplitude for inelastic heavy-ion scattering [20-221.

APPENDIX

I: ASYMPTOTIC

EVALUATION

OF Jj"@),

EQ. (5.29)

With the substitutions

he-d,

5, = 0 - 6,)

(1.1)

Eq. (5.29) becomes

(I.3

We further substitute iB

5s

(-26,‘)

52 =

--2 1 ’

2

Or

s

=

2

I e& -4,

)1’2

ei"1/2)8+(1/4)n)

,

(1.3)

where 8 is defined by (1.4) The pole in Eq. (1.2) is then at z. = i.5,

or

(1.5)

Its location and the path of integration in the s-plane is shown in Fig. 3. We may deform the integration path to run along the imaginary s-axis, because in the shaded sectors we have Re s2 < 0, since

562

FRAHN AND GROSS

FIG. 3. Location of pole and integration path for the integral in Eq. (1.2).

for the positive sector, and similarly becomes

for the negative sector. Eq. (1.2) therefore

1 Jam ds i exp [k ~2 - (2/?)1/261, - 2rri --irn

(1.7)

where

ii = ($-)‘”

2, = (+y

(e - 8,).

W

Using [23, Formulas (19.5.4) and (7.2.13)] we obtain J("(@ 1

=

(8 +

=

APPENDIX

-c

3

6 ) &3@,+(1/4M s

-C+((j$,)

II: ASYMPTOTIC

e

eIb3b,+(1/4h)

(la3~a

&

~-1Km1’2

e8i-P 4 erfc(/?‘222).

EVALUATION

OF .T~"(e>,

J?)(8) = ei(6Qs+(1’4)n) & [

(1.9)

EQ. (5.30), AND OF J(-)(8),

EQ. (5.42) In order to extract the contributions

4

of

d8 C+($ - 0, 8,) exp [Z-P

HEAVY

ION

563

SCATTERING

coming from the saddlepoint alone we proceed as follows. We expand C+($ - 8, ir,) in a Taylor series about 9 = 6,) assuming C, to be analytic between the real axis and 9,) (11.2) Then we can write ,

Jp'(@

=

eib3%+(l/4h)

(11.3)

where

having extended the integration spurious boundary contributions The substitution

ifi

limits to + 03 and - co, in order to get rid of which are anyway of order /3-r.

5,

(6 - 6,)2 = $2

(4447

(11.5)

yields

where we have used [23, (19.5.4) and (19.3.1)], and have deformed the path of integration in the same way as in Eq. (1.7). With D,(O)

= (-)’ = 0,

$

, .

m = 2r, m = 2rf

1,

we obtain (11.8) The integral J(-)(O), given by Eq. (5.42),

564

FRAHN AND GROSS

where c-(8 + O,S,) is defined by Eq. (5.41), has similar properties to J:+‘(O) and can be evaluated by the same method. The result is

APPENDIX

III:

THE CORRECTIONS R(8 f k, 8,) IN EQS. (3.12)

AND

(3.14)

We consider only the first term of an expansion in Eq. (3.15) in powers of d,

8- en nd7Y [ ht9 T2 79- (ef ic)

* lim

fo

l +o

-

ei[2s(Asbha-n)sl

O’(h,)

(III. 1)

+ 0 [Lw”(o)p].

8, = (8 - 0) + (e- e,),and using Eqs. (3.13) and (3.9), we can write (111.2) R(e - ie, e,)= m(o)k(e) + (e - e,)fkh3)i + a~2~wp-v, R(e+ ie, e,)= qe- k 0,) - (0 - 0,) ~wwm

With 9 -

= myo)bm- (e- eAwxe) - fs’:bm+ okw~op~, u11.3 where

g(B) =-L &Al3 s [ A9 2rrk

* dt9

(sin 8)112 o

112esr2s(A&(Aa-n)el

- Wd

I

(I11.4)

The integral in (111.4) receives its dominant contribution from the vicinity of the point of stationary phase, 6 = ed . By expanding about this point we can write g(e) = 1 k

(zy--i(lie-w4)n)

(sin 0)112 J&9,

(111.5)

where

J,(e)= eit2S(A)+(l/4hl

& f”o dac+@,e,)exp[i ‘;“-1$~]

(111.6)

and (III.7)

565

HEAVY ION SCATTERING

is the asymptotic form of the function defined by Eq. (5.34). The integral J,(8) is analogous to .74+‘(o), with C+(a, 0,) replacing C+(a - 8, $3, and can be evaluated by the same method as described in Appendix II. The result is (cf. Eq. (11.8))

J,(8) = ,@*(A) The leading contribution, yields

(111.8)

of order js-lf2, is given by the term with r = 0, which

g(e) = P+(A,@+ ( 27izn e )

112 = o(p2).

(111.9)

Going back to Eqs. (111.2) and (111.3) we note that jade is of the same order jg--lj2 in the shadow region 8 > tin but of order /3Oin the illuminated region 0 < 0, . The opposite holds for fo(t9) -f::;(e). Thus R(0 - ic, 0,) is small of order &“(O) fi--lj2 for 8 > en , while R(B + ic, 0,) is small of the same order for 8 < 8,) R(e

- ie, e,) =

o[dyo)

R(e + k, e,) = o[.~F~o)

p-1/2],

e>e,,

p-q,

e
(111.10)

Similar results hold for the higher terms of order LI~F”(O)/~-~, etc. in the expansion 111.1. Indeed, for symmetrical F((dz) the first-order terms (111.10) vanish and the corrections R are of order n2F”(0) p-l. APPENDIX

IV.

CALCULATION

OFT;+)@)

FOR 0 = 6,

The expansion (5.40) for fi+,+‘(8),

would progress in descending powers of the parameter /3, were it not for the dependence on j3 of the coefficients C,(2r) through the phase function fi$. From the definitions (5.26) and (5.24) with (5.12) and (5.7) we have

c+(8 - 8, as) =

--&

[ (&)1’2

pm-%)

_ (AL)“’

ewe-%q

+ 0(/s-l),

(IV.2)

566

FRAHN

AND

GROSS

with the abbreviations Go’ = @‘&) and Q8’ = @‘&). If we want to retain only terms of order /3-lla in j:“‘(0), it follows from the structure of expression (IV.2) that for 0 + 6, the only contribution of this order comes from the term with C+(CJs- 13,a,), while for 0 = 8,Ythere is an additional contribution from the term with cy)(O, 8,). We calculate these quantities from the Taylor expansion (11.2), (IV.3) Setting 6 - 9, = x, we expand to order x2,

x, = &l(8) = x, + alx+ a2x2 -/-0(x3),

Thus

&’

(IV.4) (IV.5)

= o”,’ + &x + lQ2x2+ 0(x8).

t&j" =&-J2 [1+clx+c2x2 +o(x3)1,(IV.6) where cl=*-*;

Bl

(IV.7)

the expression for c2 will not be needed. With 8 - 8, = 2, we have

c+(a - 8,as) = &

I(&-~”

[I + clx -Jrc2x2+

0(x3)] eiBJ(“)

(IV.8) Since q(x) is of order x8, we obtain to order x2,

112(1 + This yields C+(@, -

P(8 +*

e,7!l,)= $ [(-j&-)“”

-e 7s 6) = -$ [ (&)l’”

.@(‘~) - (&) eiM(aa) -

(&)l”

“‘1,

(IV. 10) (1 + c,Z, + c,fF)].

(IV. 11)

HEAVY

ION

567

SCATTERING

To calculate the values at 0 = 8, , i.e., for Z, = 0, we expand the square brackets in (IV. 10) and (IV. 11) to orders Z, and ZS3,respectively. With

(A&“’ = (22-r;’

[I

+

Cl?,

+

c2zs2

+

+

0(/3g,4),

c,z,3

s

+ O(.zs4)] (IV.12)

and p@s)

=

1 +

iy,g5,3

(IV.13)

where (lV.14) we obtain (IV.15)

Xa, ~:'(o,k>

=

112

(rd;)

2(&'3

+

(IV.16)

Cd.

s

To leading order in /3 we neglect c3 compared to i/3r3 in (IV.16). Thus, (IV. 17) where (IV.18) From Eq. (IV.7) with (IV.19) we obtain Cl =

c&i,,)

=

1

1

Q/f -

%ps’

(IV.20)

+

2(0,‘)2 ’

and with Eq. (IV.14)

co = co&,, = 4,)

@If 1 =y-y----$..e et, @;)2 a*%

+ -2-s

(9 2(@SY2 (

1-l

-

. (IV.21) 3L )

The difference co&,) - c,(&. ) represents a “Gibbs phenomenon” in the asymptotic expansion (IV.1) at the pbint 6 = 6, (i.e., the saddle point al,“). Since this

568

FRAHN AND GROSS

effect is proportional to [;‘, it vanishes in the weak absorption and is maximal in the strong absorption limit 5, + I, where

limit

t, -+icc

(IV.23)

APPENDIX

V: EVALUATION

OF EXPANSIONS (3.54), (3.59), AND (3.66) TO LEADING ORDERS

The coefficients A:m’(0)/(-@On’)u/2)m in the expansions (3.54), (3.59) would be of order p-(1/2)m, were it not for the dependence of A+&) on the function X&L) = 0(j3p3). In forming the derivatives &J”‘(O) this function gives rise to contributions of the form (/3~;(o))[“‘31

W.1)

(among others of lower order in /3), where [x] denotes the largest integer
W.2)

with 5 = 21/2uo , and where the upper and lower signs refer to the m-sum in (3.54) and (3.59), respectively. We now use the recursion formula for the parabolic cylinder functions

D,+1(5) - u&(5) + a-l(i)

= 0

(V.3)

to express De2 and D-, in terms of D-, and Do, where Del(<) = D-1(21/2uo)= (2z-)li2 e(112)uo”& erfc(u,),

Do({) = Do(21&4,) = e--(1/2)%2.

(V.4) (V.5)

HEAVY

ION

569

SCATTERING

This yields D-2(5) = &l(L) - 5~-1(5),

W.6)

D-,(i)

(V.7)

= g1 + g?) &(5> - X(1 + @I”>~-l(C),

and we obtain 2

(ItI

= ~-,kt21’2ucJ[~

- de - e/l)1

tkn.i.3

* D,( *21/2uo) e--i(1/4)n(- OA’)1/2 a, ,

CJ.8)

where A:‘(O) “1=(-o,,,

1 At’(O) - i j (-0 A

AS_“(O) 1 A?‘(O) - i - ___ Qo=(-On’T3 (-@A’)2

b = -iA

+ We - eJ2,

(V.9)

en)2,

+ ib(8 -

(V.10)

A:‘(O)

(V. 11)

,j(--0,‘)3’

With At’(O)

1

= 2n,

A?‘(0

(V.12)

= --!8A3 + i@i = iO> + 0(j3-3)

we have U, = -co(A) + ib(e - eA)2,

U, = -C,(A)

+ ib(8 -

en)2,

b=I

@=i

6(-0,‘)2’ (v.13)

where c,(A), c,(d) are the quantities defined by Eqs. (IV.22) and (IV.23). Using Eqs. (V.8) with (V.4) and (V.5), we obtain from Eqs. (3.59) and (3.54), f%(e)

= fo(e) + + ( 27iin

e )1’2 ei6+(n*eYy(--uo)[l - de

- e,)l - 4, ea,

&&e) = $ ( ,,,$,

e )1’2 e”m+‘“Se’(-r(-uO)[l

- al(e - e,)] - aOj,

(V.14a)

8 2 8, , (V.14b)

where ~(24~)= 3 erfc(u,) (+)1’2

ei(1/4)n+u~2.

(V. 15)

570

FRAHN

AND

Equations (V. 14a, b) are differentiably where they have the value

GROSS

continuous

fkA(On)= $fo(O,)- +- ( 2rrtn e )l”

over the point

eid+(n*sA)czo(0).

In the asymptotic regions I U, [ > 1 we find from the asymptotic the parabolic cylinder functions, for arg 5 = &r, D-,(Q

e-(1/4K2

= 5”

[I - “(Y$

8 = en,

l) + Nrn + :“r”,t;j”

(V.16)

expansion of

+ 3, + o(c-$ (V.17)

with m = 1 and using Eqs. (V.4) and (V.15), for arg(Ffu,,) = &T,

=-____

(V.18)

1 e-

en

Hence we obtain

and for Eqs. (V.14),

Me) = fo(e) + $ ( 2nin e )‘:, G

[I + 0(p)], en - e g (-2~3~y2,

(V.20a)

e - en > (-20ny2. (V.20b) Finally, in Eq. (3.66) the coefficients A’m)(O) descend in powers -m + [m/3] of /3, SO the only contribution to f:;;(e) of order ,!W2 comes from the term with m = 0. Thus we have 112eid-(A.O) &A(e) = J-k ( 2rsin’ tI 1 Jyg- [l + W-Y.

(V.21)

HEAVY

ION

APPENDIX

571

SCATTERING

VI: ADDITION

THEOREMS

In this Appendix we derive some useful special cases of the addition for the parabolic cylinder functions (see [25, p. 327]), D,(x

+

v)

=

e-~l/zkcY-(1/4w

=

e-w2)zY-(l/4)s2

theorems

Ym~"-97dx)

f ( ,1(, ) pD"+(y) m=0

(VI. 1)

(VI.2)

(VI.3)

(VI.4) For Y = - 1 we obtain, because of the relation D-,(z) = (n/2)lj2 e(1/4)22erfc(z/2’/“), the following addition theorems for the complementary error function as far as we know have not been given in the literature). Noting that (2) = (- l)m we obtain from Eqs. (VI.1) and (VI.2)

(VI.5) (which

4 erfc(x + y) = (2J)1,1 e-(1~2)22-2sU-v2 mzo (--21j2y)” D-m-1(21/2x) (VI.6) = e-2xu-Y2.&&c(x) ____ + (2& + erfc(x + y) = &

e-(l/2)~2-2~V-y2 f (-2l/zy)m+l Demd2(21/2x), (‘1.7) ?72=0

e-z2-22v-u/2)~2 z. (-2112x)nl O-,-,(21i2y) (VI.8)

= e-2xy-r2& erfc( v)

With the relation D,(~)

=

2-(112)ne--(l/4)22~,(z/2112),

(VI.10)

572

FXAHN AND GROSS

where H, are the Hermite polynomials,

we obtain from Eqs. (VI.3) and (VI.4),

cc (--y)m+l 1 -kerfc(x + JJ) = 3 erfc(x) + ,ljz 67”’ 2 m=. (m + l)! HWL(x) * 1 = 3 erfc(v) + yr1/2 e-‘* 1

(-x)“+l

m=O (m + 1) ! H&)*

(VI.1 1)

(V1.12)

ACKNOWLEDGMENTS A first version of this paper was written while W.E.F. was a guest of the Hahn-Meitner Institute for Nuclear Research during March 1975. He thanks Professors J. Eichler and K. H. Lindenberger for their kind invitation and hospitality.

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Y.) 72 (1972), 524. 9. W. E. FRAHN, in “Heavy-Ion, High-Spin States and Nuclear Structure,” Vol. I, p. 157. International Atomic Energy Agency, Vienna, 1975. 10. W. E. FRAHN AND T. F. HILL, to appear. 11. D. H. E. GROIN, Nucl. Phys. A206 (1976), 333. 12. R. A. BROOLIA, S. LANDOWNE, AND A. WINTHER, Phys. Lett. 40B (1972), 293. 13. B. C. ROBERTSON, J. T. SAMPLE, D. R. GOOSMAN, K. NAGATANI, AND K. W. JONES,Phys. Rev. C4 (1971), 2176. 14. K. W. FORD AND J. A. WHEELER, Ann. Phys. (N. Y.) 7 (1959), 259. 15. C. K. GELBKE, Z. Phys. 271 (1974), 399; Phys. Lett. 55B (1975), 134. 16. N. AUSTERN, Ann. Phys. (N. Y.) 15 (1961), 299. 17. D. H. E. GROIN, in “Heavy-Ion, High-Spin States and Nuclear Structure,” Vol. I, p. 27. International Atomic Energy Agency, Vienna, 1975. 18. V. M. STRUTINSKI, Phys. Lett. 44B (1973), 245. 19. P. J. SIEMENS AND F. D. BECCXIEITI, Phys. Lett. 42B (1972), 389.

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ION

SCATTERING

573

20. W. E. FRAHN, in “Classical and Quantum Mechanical Aspects of Heavy Ion Collisions,” Lecture Notes in Physics, Vol. 33, p. 102. Springer-Verlag, Berlin, 1975; W. E. FRAHIN AND K. E. REHM, to appear. 21. W. E. FRAHN, Nucl. Phys. A, in press. 22. W. E. FRAHN AND K. E. REHM, Phys. Rep., to appear. of Mathematical Functions,” Dover 23. M. AEXAMOWITZ AND I. A. STEGUN, “Handbook Publications, New York, 1965. 24. I. S. GRADSHTEYN AND I. M. RYZHIK, “Table of Integrals, Series and Products,” Academic Press, New York, 1965. 25. W. MAGNUS, F. OBERHETTINGER, AND R. P. SONI, “Formulas and Theorems for the Special Functions of Mathematical Physics,” 3rd ed. Springer-Verlag, Berlin, 1966.