Relation of scattering phases to interaction parameters

Nuclear Physics 7 (1958) 389--396; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permission from the publisher

R E L A T I O N OF S C A T T E R I N G P H A S E S TO I N T E R A C T I O N PARAMETERS E. VAN D E R SPUY" and H. J. P I E N A A R

Department o] Physics, University, Stellenbosch Received 27 March 1958 An analysis is introduced which relates the parameters of suitable functions of experimental scattering phases, in the presence of Coulomb forces, to parameters which define the nuclear interaction. A feature of the analysis is to circumvent the need for the troublesome continuous bivariate interpolation of Coulomb-functions, and requires instead the knowledge of a number of coefficients which are functions of one nuclear parameter which assumes only certain discrete values, corresponding to nuclear quantities (such as charge, reduced mass, and outer nuclear radius) which need in practice be tried only at discrete values. A further advantage of the analysis is the direct relation which it establishes between experimental phase parameters and the nuclear interaction parameters.

Abstract:

1. Introduction A previous analysis 1) established a direct connection between scattering phase parameters and the parameters defining interaction of the two-body velocity-independent type, without Coulomb-forces, for any angular momentum quantum number l. It is now proposed to extend this analysis to the case where Coulomb-forces have to be included, and to show how this can be done in a way avoiding the troublesome use of a number of Coulombfunctions. Instead the analysis introduces certain coefficients which are functions of r N, the outer radius of the nuclear interaction concerned, and the charges and reduced masses of the particles concerned. The latter nuclear quantities have discrete values: even in a phenomenological analysis r~ need only be tried at certain discrete values which m a y be generally selected for all cases over quite a limited range. With the use of Coulomb-functions one requires them to be known for the whole range of the discrete set of nuclear quantities and continuously over the energy range. One is then generally involved in bivariate continuous interpolation. 2. The General A n a l y s i s We consider a two-body scattering problem, assuming velocity-independent forces. After the usual elimination of the angular dependence from the 389

390

E. VAN DER SPUY

A N D H. J .

PIENAAR

Schr/Sdinger equation, the radial part of the wave-function, R~(r)/r, is governed b y the equation d2

{

dr---~R,(r)+ k S -

/(l+l)

r2

}

--Ue(r)--UN(r ) Rz(r ) = 0,

(1)

where k 2 = 2/~E/?/2; here # is the reduced mass of the two particles having centre-of-mass energy E, and U e (r) ---- 2FZ z Zz e2/h2r = 0

for r > rN, for r < rN,

Zle , Z,e being the charges of the two interacting particles; for r < r N the Coulomb-potential is included in the nuclear potential; ON(r ) =

2FVN(r)/~ 2,

with VN(r ) = 0

for r > r N.

Although velocity-dependence will not be discussed, simple types of the latter, as for example spin-orbit coupling, could be treated b y a simple extension of the present analysis. The boundary conditions of R, (r) are defined b y R, (0) = 0 and R, (rN)= 1. Then, the logarithmic derivative at the interaction boundary, at r = r N, namely

1'=1' Rz~-) ',=,,R '=/r ( r ), l ,dar '-

,,=,."

(2)

Now select a comparison function R',(r) with the same defining equation and boundary conditions as Rl(r), except that now VN(r) ~ 0 for all r. Let/'z be the logarithmic derivative of R', (r). Clearly R', (r) = j~ (kr)/j, (krN) for r < r N, ]l(kr) being the spherical Bessel-function. Also

{rd]~(kr)]

/'1=

i,(k,)

(3)

As usual the Wronskian

d R'z(r) dr Rz(r)--R,(r) d R',(r) ---- constant for r > r N, and d

co

f UN(r)R'z(r)R~(r)dr.

---Therefore, 1

(/~--1'3" rN

1

i~(krN) f ~ U N (r)R, (r)], (kr)dr.

(4)

R E L A T I O N OF S C A T T E R I N G PHASES

391

One can proceed with the fight-hand side of (4) as in the previous analysis a) without Coulomb-forces: the function G ( r ) ~ UN(r)Rz(r) is fairly generally = 0 for r = 0, r N. This means one can expand G (r) in terms of the orthogonal functions i z ( k . r ) where k~ is given b y ~z(k.rN) = 0; n indicates the number of loops in j,(knr ) for 0 ~ r ~ r N. Thus

G(r) = ~ Ani~(k.r ).

(5)

n=l

Therefore (4) now changes to (--A+/'z)

1

--

.~Z=li, (kr~) Jo

~'N

iz(k.r)i,(kr)dr = ~

A.kn

. = 1 (k.-~k--k2) " i ~ - l ( k n r N ) .

(6)

Eqs. (5) and (1) with the R~(r) boundary conditions lead to

R,(r) --

i,(kr) ~ A. + .=x ~ (k2--k. 2) • i,(knr), i~(krN)

(7)

~_ An " i , ( k . r ) } = .,.. A . / , ( k . r ) , n=l (k2--k. 2) n=l

(8)

and U.(r)

[iz(krN)( i~ (kr) +

for r --< r~. Eq. (8) clearly shows how the An can be regarded as nuclear interaction parameters: the A . determine UN (r) and vice versa. As in the analysis without Coulomb-forces 1), the fight-hand side of eq. (6) can be expanded in powers of k S (or of the energy E). To do this, and put the result in its simplest form, one expands A.

kl 2 - -

Cno+C.l(k/kl)2+C.~(k/~)4+

....

(9)

and uses the relations between the C.~ which can be deduced from the equations one obtains b y expanding (8) in powers of k S and equating coefficients of the same powers. In the process the coefficient of each power of k S in the expansion of (8) gives rise to a separate energy-independent equation. The latter equations relate the C.,. to UN(r ). Though these steps can be effected generally, they will be exemplified here in the case of l = 0 aione. Here jo(knr) = sin knr, klr N = :~, and k. = nk 1. For l = 0,

(1'0-/o)" ~ q- ~

n=l,

(--1) n 2C"o

.=1

n

Cnl

-~+(--a)"~+~,

and the expansion of (8) leads to

.=1.

~-

~ (Cno

+~

n4j

C.,oC,a~t +

~4 /J(k/kl)~+ . . . .

0o)

392

E. VAN DER SPUY AND H. J. PIENAAR

kl 2

sio

n=l

sin

n=l

or

UN(r )

{

=2

sin (knr)

r 3 n=l

n2

(Cnl+ -

-

--

,,

)/

Cn0

n2

~ C~o

C~o sin (k~r),

C~1--

~.~1

(11)

6-

etc.. Eq. (11) shows the C~m as interaction parameters; the lowest order equation suffices to determine U N(r) in terms of Cno, and vice versa, and the higher order equations fix successively C,~1, etc.. Eq. (10) hence establishes that (/'o--/o) m a y be given explicitly as a k ~ (or E) power series whose coefficients are functions of the interaction parameters. For the first two terms in the expansion of (10) one needs only the C~o; but for higher order expansions, a knowledge of C~, etc. is successively needed. Note the zero-energy solution R°(r)*=°

r

-

-

rN

~ Cno

n=l ~ y sin (nk 1 r).

Now (11) gives an infinite number of energy-independent equations. Another set m a y be obtained from eq. (10), by expanding the left-hand side into a k S (or E) power series, and equating coefficients of like powers of k 2. Consider now/0: it m a y be obtained from the outside solution of eq. (1): R o(r) = {cos doNF o(r) + sin doNG O(r)}/N o, where F o(r) and GO(r) are the regular and irregular Coulomb-wavefunctions as normalized by Breit et al. 2); doN is the nuclear phase-shift with Coulomb forces present, and N o is selected to make Ro(rN) = 1. With these F o(r), Go(r), d d GO(r) drr F° (r) -- F o(r) drr GO(r) = k. Therefore,

/O/rN----

\

~ ~

= (~o(r)

+ sin 6oNGo(r)

/,=~

Fo(r)Go(r ) + cot doN Foe(r ) ,=rN

Therefore,

/o = /"o--krN{Fo(rN)Go(rN)+Fo2(rN) cot doN}-1 = /"o--/'"o, where

(12)

RELATION OF SCATTERING PHASES

/"o.:

\.

393

-

is the logarithmic derivative of the regular Coulomb-wavefunction at the nuclear boundary. I n t r o d u c e the symbols PN :

krN,

aN = P N ~ :

~ : ~ Z 1 Z 2 e~/ti ~ k, ~AZI Z ~ e 2 r N / 1 ~ .

The dimensionless q u a n t i t y a N is generally quite small; for example, a N = 1.38 for the a - - ~ interaction at r N ---- 5 × 10 -13 cm. Using the auxiliary functions of Breit et al. ~) one immediately gets from (19.) 1

/'H 0

-- { F o(r N) G O(r.) + Fo 2(r.) cot 6o"}/PN

---- ~o(rN)~(r~)+2aNq~o2(rN)

( e 2 ~ - l ) cot d O N + Q + 2 ~ ' - - I + log 2a N , (13)

where Q = Re

r'(i~)

r(i~)

log ~

and ~' is Euler's constant. The q u a n t i t y --

(e~--1)

cot ~oN+Q

is an experimental phase function of k 2 (or E) which m a y be written as a k~-power series 3). This will also appear explicitly from the following analysis. One assumes t h a t the experimental phase ~oN has already been analyzed in this form: -= o ~ + / s k 2 + y k * + .

. . =

~+ (/5/rN2)p.2+ (v/r~)p.~+

....

(14)

or

@ + 2 y ' - - l + l o g 2a N = ~ : + / 5 : p N 2 + V : p N 4 + . . . . where ~1 = ~ + 2 ~ , ' - - l + l o g 2an,

/51 = /5/rN 2,

Yi = 7/rN 4.

Now one introduces the convergent series for q~0(rN), T0(rN) , and q~0*(rN), given b y Breit et al. ~), noting t h a t

/"o =

these are

~o* (r.) .

394

E. VAN DEE SPUY" AND H. J. PIENAAR

=

O~2

0~2-~-fl2 p N 2 - ~ - 7 2 p 4 N @ - . . . . 1

2

1

3

1

4

1

5

1

4

1

11

.i

1

6

l + a N + g a N + i g a N +vggaN +2--g-ff~aN +56--gf~aN + . . . . X

1

1

2

1

3

5

~+gaN+-fgaN +~-~aN +3~TeaN +~6--f~aN + . . . , 1

~

23

~

L

72 --~ ] - ~ 6 n - ~ - - 0 5 ~ N n - ~ N

40* (rs) =

.

2A_

- - ~ ~ N

1 ?

2

23

--~4

~'4

•

....

2

3

49

~

1

4

1

5

1

6

~

~

2£

11

a

3_1_

•

a

28

~''',

(16)

0~4 "AFf14 p N 2-~- 7 4 p N 4 - ' f - . . . . '~a

~4

~

1-~-2aN-~-aN - ] - 9 a N -{-3-6aN -{-7~5-ffaN - ~ - ~ g a N - ~ - ' " "' __1 __4 5 2 1 3 7 4 8 5 2~-9aN~-gga N -~-~a N ~ - ~ a N - ] - ~ a N -]- . . . .

7 3 = ~-~n-9---ff~N~-~l--6~N ~ ~ N ~Po(rN) =

3A_

~X3"31-/~3 tON2 ~ - ~3 P N 4-~- . . . . ,

--f13=

7

(15)

2

14a

3

35

- - ~ N -- 9 "N - - ~ N

a

11/, __ 43 a 2 77 a 2 9 ~*N 1-~N ~ N 1 2 a __ 172 a 2 24 2--25 -~N ~ * N

4

101

5

a

6A_

--yV66-N --101--6yy-g~N-- . . . .

(17)

3__

373 a 4 247 a 5A_ ~ N 11-34-00 ~N t " " ", 2687 a 3 56~5~0~N - - " • ""

Note t h a t / ' 0 = PN c o t tON ~---

1--2BlPN2--§B2p4N

-

. ..,

where Br is the rth Bernoulli number. The formulae have been given explicitly in powers of pN2 up to the third order (this is sufficient for most practical purposes); they show the ~ , fl~, 7n to the lower orders in aN, but they could be calculated to any order of accuracy by using the appropriate recursion formulae 2). The fact is that %, fin, 7 . . . . . for n = 2, 3, 4, m a y be calculated and tabulated for any discrete set of values which the dimensionless q u a n t i t y a N would normally take. In practice it is useful to take for r s a few of a discrete set of values extending over quite a small range which will cover all problems. One will finally interpolate from the final results for such trial values of r N (4 values for a given problem, say) to the precise r N demanded by the experiments. Since one is inevitably limited to the class of 2-body problems that m a y effectively be analysed, /~Z1 Z 2 and a N = t t Z 1 Z 2 e 2rN/h ~ need have only a limited number of values. For the required values of aN, the formulae (15), (16), (17) give the an fin, 7~ for n = 2, 3, 4, which is normally all that is required, since the experiments usuallyonly permit to derive from @ 2 or 3 of the parameters a, r, 7. Thus for any particular rN, or a N, one has energy-independent values of %, fl~, 7n for n = 1, 2, 3, 4. Using these expansions of #o(rN), #o*(rN), ~o(rN), o~, PN cot pN one m a y expand the left-hand side of (10) in a power series in ON2, or (k/kl) 2. (kl rN) 2 ~2(k/kl) 2, and hence by equating coefficients of like powers of (k/kl) 2 obtain a number of equations relating interaction parameters to the phase parameters ~1, ill, 71 through the intermediary of certain energy-independent coefficients ~ , fl~, 7n (for n = 2, 3, 4) which have to be known only for a discrete set of values of a N. Since the latter equations relating interaction and phase para-

R E L A T I O N OF S C A T T E R I N G PHASES

395

meters may be more simply arrived at with numerical values for the e~, fl~, 7~ the general relations will only be quoted for the first two coefficients of the (k/kl)*-power expansion of (10): 1- ~ +(~,,+2a~1~) o~

~

8.

~2

~2

=

_I ~r~ n = l

1

-1 - - ~ ~ ( - 1 ) ~ ,*=1

C~o n

~$4+fl~,+ 2 a N ( ~ h ~ + ~ l )

(18)

( ~ aa-]- 2aN czl cz22)2

( - 1)- - g ~ +

-2 " n~J "

Being energy-independent, the equations of the type (18) presumably have a wider validity than indicated b y the radius of convergence of the expansions involved. Using these relations has the advantage of establishing the connection between the phase parameters and interaction parameters only with the help of energy-independent coefficients e~, fl~, 7~ (n = 2, 3, 4), which have to be known in terms of the discrete variable a N. In any case, a N also determines the values of the normal Coulomb-functions, but in using the above parameters ~ , fix, 7 . . . . . . one avoids the general need for continuous interpolation of Coulomb-wave functions over a practically continuous energy range. This is even more important if one realizes that in the normal use of Coulomb-wavefunctions one has bivariate continuous interpolation in terms of p and ~. Furthermore the advantage of the direct connection of phase and interaction parameters in the above form is that it gives a direct method of selecting in a general way useful phenomenological parameters corresponding to the amount of information contained in the experimental data. In such a general phenomenological analysis, equations such as (11) relate U~(r) and C~m while eq. (18) relates C~,~ and the experimental el, ill, 71, etc. The selection of useful interaction parameters proceeds on the same lines as in the previous analysis without Coulomb-forces 1), and one may refer to this paper for a further discussion of this aspect; in fact, for phenomenological potentials which are smooth, reasonably slowly varying and non-oscillating with distance, one can limit the analysis to the lower harmonics, i.e. the lower n in C ~ ; b y (11) higher harmonics correspond to rapidly changing or oscillatory potentials, and (18) shows the reduced effect of higher harmonics). In the opposite sense, any velocity-independent potential VN(r ) of finite range can be analysed by relations such as (11) to as many harmonics as are required, and relations (18), (14) will immediately give the phases. The analysis for higher l can follow in the same way, the relations becoming, however, progressively more complicated.

396

E. VAN DER SPUY AND H. J. PIENAAR

One of the authors (H. J. P.) wishes to acknowledge the benefit of a bursary of the South African Council for Scientific and Industrial Research during the present investigations. References l) E. v a n der S p u y , N u c l e a r P h y s i c s 1 (1956) 381 2) Breit, W h e e l e r a n d Yost, P h y s . Rev. 4 9 (1936) 174 3) G. F. Chew a n d M. L. Goldberger, P h y s . Rev. 75 (1949) 1637