Remarkable properties of the eikonal approximation

Physzca 66 (1973) 33-42 0 North-Holland Publzshzng Co

REMARKABLE

PROPERTIES

OF THE EIKONAL

APPROXIMATION

*

F W BYRON, Jr Department of Physzcs and Astronomy, Unzverszty of Massachusetts, Amherst, Massachusetts, USA and C J JOACHAIN Physzque Thkorzque et Mathkmatzque, Faculte’des Sczetzces, Unzverszte Lzbre de Bruxelles, BruxelleA, Belgzque

Received 4 December

1972

Synopsis We discuss the relatIonshIp between the elkonal approxlmatlon and the Born senes and show that a reassessment of the condltlons of vahdlty of the elkonal approxlmatlon IS necessary for a large class of mteractlon potentials

1 Introductzon The purpose of this article 1s to point out some remarkable properties of the elkonal method1*2) as apphed to quantum colhslon problems Although some of our results are also vahd m more general sltuatlons3), we shall confine our attention to potential scattering, where “exact” solutions are readily computed to check the accuracy of our statements We begin m section 2 by recalling a few basic formulae relative to the elkonal approxlmatlon Section 3 IS devoted to a comparison of the Born and elkonal multiple-scattermg series We suggest that for an arbitrary superposltlon of Yukawa potentials, and for all momentum transfers, each term of the elkonal senes gives the asymptotic value (for large incident wave numbers) of the correspondmg term of the Born series This property, together with the requirements of umtanty imply that m the weak-couphng limit, I e when the Born series 1s rapldly convergent, the elkonal amphtude gives a consistently poorer approxlmatlon to the exact amplitude than does the second Born approxlmatlon As the couplmg mcreases the elkonal method improves steadily. In fact, we find m section 4 that for mtermedlate couplings and high wave numbers the elkonal method 1s excellent at all angles for potentials of the Yukawa type Finally, we also study * Supported

by the NATO Sclentlfic Affaalrs DIVISION,Research Grant no 586 33

34

F W BYRON,

Jr AND C J JOACHAIN

m section 4 the strong-couplmg case where we find that the elkonal amphtude remams qmte good at small momentum transfers These results strongly suggest that a reassessment necessary

of the condltlons

A systematic

tlon, based on the present action

of vahdlty

of the elkonal

study of the hmlts of vahdlty work, 1s made m another

approxlmatlon

of the elkonal

IS

approxlma-

paper4) for a variety of Inter-

potentials

2 The elkonal scattermg amplrtude Let us consider the nonrelatlvlstlc scatterof mass m by a local, real potential V(r) of range a We

mg of a spmless particle

denote by k, and kr the lmtlal and final wave vectors of the particle, while I? 1s the scattermg angle between ki and /if It will also prove convenient to mtroduce the “reduced” potential U(r) = 2mV(r)/h2 The energy of the particle 1s E = h2k2/2m, where k = lkij = lkrl IS its wave number We shall also call V, a typical potential strength and U, = 2mV,,/h’ the corresponding strength of the reduced potential We first recall that the elkonal approxlmatlon 1s usually presented’T2) as a high wave number (ka P l), high energy (IVol/E 4 1) and small angle (0 < (ka)-‘) approxlmatlon In addltlon, the elkonal scattering amphtude fk (k, A) = (k/2x1) 1 e” ’ (e’x(k,*) - 1) d2b, where d = k, - kf IS the wave-vector

transfer

(2 1) and

x (k, 6) = - (1/2k) +r” U(b, z) dz, --m exhlblts the important theorem2) In writing

(2 2)

property of umtarlty m the sense that It satisfies the optical eqs (2 1) and (2 2) we have chosen a cyhndrlcal coordinate

system such that r=b+zii

(2 3)

and the unit vector A 1s perpendicular to d Hence the elkonal phase-shift functlon x (k, b) IS evaluated along a dlrectlon parallel to the blssector of the scattering angle 0 If the potential possesses azlmuthal symmetry, then the elkonal amplitude of eq (2 1) reduces to the Fourier-Bessel transform

fE (k, 0) = (k/l) i JO (db) (eiX(k*b)- 1) b db,

(2 4)

0

where the elkonal phase-shift function, as seen from eq (2 2), now depends only on k and b = Ibl We recall that for high-energy, small-angle scattermg the phase

PROPERTIES

OF THE EIKONAL

APPROXIMATION

35

x (k, b) may be related to the phase shifts & appearmg m the partial-wave series for the scattermg amphtude. The result lsz) (2 5)

x (k, b) = 26, (k), where b and 1 are related by I z kb

3 The elkanal and Born multiple-scattermg series Let us expand the elkonal amplitude (2 1) m powers of the phase shift-function (z e , of the mteractlon potential) as

fE=

“@En,

(3 1)

where

fEn =

-5 5

s

e"

’

[I]”

d2b

(3.2)

In particular, for potentials which possess azimuthal symmetry, eq (3 2) reduces to _& = -Ik $1

JO (db) [X(b)]”b db

(3 3)

0

worth notmg that the terms (3 3) are alternatively real and Imaginary also define the quantity

It IS

fE,= il Jim

We

(3 4)

so thatf,, = f~, .fh = _/k + 3~2,etc The exact amphtude f has an expansion similar to (3 1) m powers of the potential, namely the Born series _f-= fIxi”

(3 5)

We shall also write m this case (3 6) Let us analyze the relatlonshlps between the terms fEn and JIB”of the elkonal and Born series when ka % 1 Firstly, we recall that’)

fEl

=fBl,

(3 7)

36

F W BYRON, Jr AND C J JOACHAIN

for all momentum transfers We emphasize that this result 1s true for all scattering angles only when the z axis of the cyhndrlcal coordinate system [see eq (2 3)] 1s chosen along a direction perpendicular to the wave-vector transfer A In what follows we shall always adopt this choice We now consider the second-order termsf,, andfBf,, and concentrate our attention on central potentials We first note that RefE,, = 0 while m general RefB,, # 0, so that there 1s no analogue of the result (3 7) for the quantities Re_& and RefEZ However, for an arbitrary superposition of Yukawa potentials, namely U(r) = U,

i Q(U)(e-“/r) oro>o

dcr,

(3 8)

one has

ka+oo

Imf,,

&,A)

(3 9)

’

for all values of the momentum transfer A To prove eq (3 9), we first write fBZ in momentum space as (3 10) where we have defined (KI U IK’) = (2~)~~ J eICKPK’)’ U(r) dr

(3 11)

Hence, for the interaction (3 8), we find that _L =

-$ ldrri(4 JDdiMP, “0

X

s dq

a0 1

(3 12)

(q2 - kZ - 1~) [(q - ki)2 + u’] [(q - kd2 + B’l

Using the Feynman parametrlzatlon straightforward calculation *,

techmque5)

one then obtams,

after a

m

hm Imf,, ILs-a,

(k, A) = -$-

da@(a) s

X

s

4MB)

’

CC/I (u2 - l)+

where u = (LX”+ b’ + A2)/2@ * More details about this calculation are gwen m ref 4

log[u+(u2-

I)+], (313)

PROPERTIES

OF THE EIKONAL

37

APPROXIMATION

Let us now evaluate the quantity IrnfEz (k, A). It 1s given by

(3 14) where the erkonal phase-shift function (2 2) becomes, for the potential (3 8), (3 15)

x (k, b) = (uo/M .I?+) Ko (a@ da a0

Here K, IS a modified Bessel function of order zero The integral on the b variable m eq (3 14) may then be performed, with the result

and the relation (3 9) follows by comparing eqs (3 13) and (3 16) In order to mvestlgate the relationship between Ref‘, andf“,,, we proceed as follows We first obtalnjE’,,, which 1s purely real, by a numerical integration of eq. (3 3), m which we have set n = 3 Then we evaluate the exact amplitude f by TABLE I Comparison

of Re f&

U(r) = - exp (-r)/r

and fE3 for a simple Yukawa potential The wave number 1s k = 5 The notation

-384(-3)means e

-384~10~~

Re fs3

f E3

0”

-3

84 (-3)

-3

91 (-3)

30”

-3

01 (-3)

-3

02 (-3)

60”

-205

90”

-1

52 (-3)

(-3)

-204

(-3)

-1

51 (-3) 23 (-3)

120”

-1

24 (-3)

-1

150

-1

lO(-3)

-109

(-3)

180”

-1

06 (-3)

-105

(-3)

integrating numerically the partial-wave Schrodmger equations and obtalnfBf”,, by subtracting_&, fromf. This, of course, IS an approximate procedure but it 1s quite accurate d the coupling 1s weak Table I shows the comparison of RefsJ wlthJIE3 for a simple Yukawa potential U(r) = U, e-@/r,

(3 17)

38

F W BYRON, Jr AND C J JOACHAIN

with U,, = - 1, a = 1 and a wave number k = 5 The agreement 1s seen to be excellent for all values of the momentum transfer A slmllar procedure was used to compare Imj”4 with Im fEf,, As m the case of RefB3 and fE3, the agreement between Imf,, and Im fE4 IS excellent at all momentum transfers We have also found analogous results for superposltlons of Yukawa potentlals4) This strongly suggests that the relatlonshlps llm -RefBn = 1 @odd), J’En

llm -Im& = 1 ImfEn

ka-+m

@even),

ko+m

(3 18)

hold for all n and all momentum transfers for mteractlons of the form (3 8) Two remarks should be made at this point Firstly, the quantity ka does not have to be much greater than one for the asymptotic behavlour to set m An exammatlon of the expression for the second Born approxlmatlon m the case of a simple Yukawa potentlaP) shows that this 1s not unreasonable Secondly, the relations (3 18) [and m particular the second-order result (3 9)] are not true for an arbitrary interaction potentla14) This fact 1s m contradlctlon with some suggestlons about the relatlonshlps between the elkonal and Born series recently made by Moore7) A close exammatlon of Moore’s arguments shows that his derlvatlon is Incorrect Let us now analyze some of the consequences of the relations (3 18) We first consider the weak-couplmg case such that I I’,( a/h

= I CJ,l a/2k 4 1,

(3 19)

where v = iik/m IS the velocity of the particle and I VA/E = lU,l/k’

6 1

(3 20)

In this case we may rewrite the Born series (3 5) for ka $ 1 as f(k, A) =fBl(A)

+ T

+ Iy

+ F

+ O(k-3)

(3 21)

Here A and B come fromfB’,, and hence are proportional to U,’ while C, arlsmg from fB83,1s proportional to U,” The k dependence of the various terms 1s easdy checked by requiring the scattering amphtude to satisfy the optical theorem order by order m powers of U, and k- ’ Since, as we have shown above, the corresponding elkonal amplitude 1s given by .fE (k, A) = _fBl(A) + 1 y

C(A) + - kZ

+ @%k-3),

(3 22)

neltherfs2 nor fE2 will be correct through order ke2 Thus, if IU,,l IS large, then C(A) will dominate A(A) and the elkonal amphtudef, ~111be a better approxlmatlon to f than fBZ On the other hand, If IUOlIS small, thenf,, will be more accu-

PROPERTIES

OF THE EIKONAL

39

APPROXIMATION

rate thanf, This comment applies obviously also to the calculation of the dlfferentlal cross sectlon Indeed, the terms A(d), B(d) and C(0) contrlbute equally m correcting the first Born dlfferentlal cross sectlon to order kw2 Only B(A) and C(0) are given by the elkonal approxlmatlon In order to Illustrate this point we consider a superposltlon of two Yukawa potentials of different ranges, namely U(r) = U, (e-l - 1 125 ee2’)/r

(3 23)

Such a potential gives a nontrIvIa structure m the differential cross sectlon, even In first Born approxlmatlon and can be used to reproduce some of the features of strong interaction forcesa) Table II illustrates the sltuatlon when U, = -3, k = 5 In this case the first Born term gives a rather good approxlmatlon to the TABLE II Four different approxlmatlons

to the real part of the exact scattermg amphtude for a potential

given by eq (3 23) with U, = -3

The wave number IS k = 5

Re fE

Re (fE + .&A

0

RefBl

0

2 156 (0)

2 162 (0)

2 149 (0)

2 154 (0)

2 154 (0)

30”

7 422 (-2)

7 236 (-2)

7 172 (-2)

6 986 (-2)

6 986 (-2)

RefBz

Ref

60

-9

947 (-4)

-1

589 (-3)

-1

169 (-3)

-1

763 (-3)

-1

90

-3

676 (-3)

-3

761 (-3)

-3

565 (-3)

-3

649 (-3)

-3604(-3)

120

-3

248 (-3)

-3

210 (-3)

-3

123 (-3)

-3

085 (-3)

-3062

(-3)

150”

-2

873 (-3)

-2

805 (-3)

-2

758 (-3)

-2690(-3)

-2677

(-3)

180

-2

749 (-3)

-2

675 (-3)

-2

639 (-3)

-2

-2

565 (-3)

689 (-3)

553 (-3)

real part of the scattermg amphtude, however, neither the second Born approxlmatlon [which lacks C(A)] nor the elkonal amplitude [which lacks A(d)] offers a slgmficant Improvement over the first Born result Nevertheless, if we add RefB2 to Ref,, table II shows that a major improvement m the real part of the amphtude 1s obtained, as we expect from the above dlscusslon 4 The mtermedrate and strong-coupbng cases We now turn to the case of “mtermedlate couplmg”. We stdl require ka 9 1 and 1V,I/E < 1, but we now have

lU,l a/2k zz 1

(4 1)

In fig 1 we show the real part of the elkonal scattermg amphtude as a function of the scattering angle for a superposrtlon of Yukawa potentials of the form given in eq (3 23), for the case U, = -20, k = 5 For companson, we have also plotted the first Born, the second Born and the exact results We see that the elkonal amplitude reproduces correctly the exact amplitude, with only a shght shift of the second zero Even at B = 180” the agreement between the elkonal and exact results is strlkmg We also note that the first and second Born approxlmatlons give

.-.

e 30'

60*

90°

1200

1500

1600

Fig 1 The real part of the scattermg amphtude for a superposltlon of Yukawa potentials of the form given m eq (3 23) with U, = -20, k = 5 The sohd curve shows the exact result, the dashed curve gives the elkonal result, the dotted curve represents the first Born approxlmatlon and the dash-dotted curve IS the second Born approxlmatlon Smce a logarlthmtc scale IS used we also mdlcate the sign of the various quantltles

Fig 2 Same as fig 1 except that the lmagmary part of the amplitude IS shown

PROPERTIES

OF THE EIKONAL

APPROXIMATION

41

Ftg 3 The real part of the scattermg amphtude for a superposmon of Yukawa potenttals of the form gtven m eq (3 23) wtth U, = -20, k = 2 The soltd curve shows the exact result, the dashed curve gtves the etkonal result and the dash-dotted curve represents the second Born approximatton

7 6

n i

\ \ \

‘\

5

\

\

:

4

\

'1

3

: \

\ \ \ \ \ \ \ \ \ \ \

Ftg 4 Same as fig 3 except that the tmagmary part of the amphtude IS shown

42

F W BYRON, Jr AND C J JOACHAIN

results which are consistently

worse than those obtained

Fig 2 gives the correspondmg amplitude,

with essentially

at large angles asymptotic rapidly term

properties persist

discussed

couplmg

m section

(even though m the total

from the elkonal

for the Imaginary

the same conclusions

m the mtermedlate

m this example properties

results

part

The reasons

method

of the scattering for this agreement

case IS undoubtedly

related

to the

3 The Born series 1s converging

fairly

IUOi,l/k = 4), and the large-angle, amphtude

Also, the fact that

means that the Born-series terms mlssmg from the elkonal compared to the terms Included m the elkonal series

term-by-

lU,l IS large

series are unimportant

Finally, we come to the strong-couplmg case for which I V,,l/E > 1 We compare m fig 3 the real parts of the elkonal and exact amplitudes for the potential of eq (3 23) with U, = -20 and k = 2 We also show the second Born results which, as we expect, are disastrous m this case On the other hand, the agreement between the elkonal and exact results is seen to be fairly good at small angles Slmllar conclusions may be drawn from the exammatlon of the Imaginary part of the scattering amplitude, shown m fig 4 In view of the conventlonal crlterla for the apphcablhty of the elkonal approxlmatlon, the agreement between the elkonal and exact results m figs 3 and 4 1s qmte unexpected A slmllar situation occurs m the problem of scattering by a potential of the form V(r) = Vo/rs (s > 2), where a straightforward evaluation of the elkonal scattering amplitude by the method of stationary phases leads to a result for low-energy scattermg which agrees with the semlclasslcal formula for small angles, even though I Vol/E 9 1 The reason for this agreement IS that, although IV,l/E $ 1 for certain Impact parameters (small ones), at Impact parameters important for small-angle scattermg, the strength of the potential IS considerably less than the crude estimate given by I V,l Along with the points discussed previously, this strongly suggests that the traditional crlterla for the validity of the elkonal approximation are only sufficient condltlons which are often unnecessarily strrngent It IS a pleasure to thank Dr E H Mund for his assistance Acknowledgment m helping us to solve the numerlcal problems connected with this work REFERENCES 1) Mohbre, G , Z Naturforsch 2A (1947) 133 2) Glauber, R J , m Lectures m TheoretIcal Physics, Vol I, W E Brlttm, ed , Intersclence Pub1 , Inc (New York, 1959), p 315 3) Byron, F W , Jr and Joacham, C J , Phys Rev, to be published 4) Byron, F W , Jr , Joacham, C J and Mund, E H , Phys Rev , to be pubhshed 5) Feynman, R P , Phys Rev 76 (1949) 769 (appendix) 6) Dahtz, R H , Proc Roy Sot A206 (1951) 509 7) Moore, R J , Phys Rev D2 (1970) 313 8) Ball, N F , Shu-Yuan Chu, Haymaker, R W and Chung-I Tan, Phys Rev 161 (1967) 1450