Coupling constants of commonly used nuclear probes

Nuclear Physics A365 (1981) 8-12 @ North-Holland Publishing Company

COUPLING

CONSTANTS

OF COMMONLY

USED

NUCLEAR

PROBES* G.R. Institut fiir Physik,

PLATTNER

and

Universitiit Received

R.D.

VIOLLIER

Basel, CH-4056

16 December

Basel, Switzerland

1980

Abstract: We present a compilation of couplingconstants of the light nuclear probes d, T, 3He, (I, 6Li, i3C and “0. These constants are related to quantities such as pole residue, effective range, zero-range constant and reduced width, which play an important role in the quantitative description of nuclear reactions.

1. Introduction Many reactions used to study nuclear structure and spectroscopy involve the transfer of a cluster b from a light projectile A = (b + X) to the target nucleus D under investigation (or, similarly, of a cluster b from the target nucleus to the projectile X) i.e. they are of the type D+A=D+(b+X)+(D+b)+X, or (D+b)+X+D+(b+X)=D+A, and involve the vertex AH b + X. In quantitative studies of this kind one often to know the coupling constant c AbX which describes the virtual decay A + b the bound-state momentum, i.e. the “asymptotic strength” or the magnitude asymptotic bound-state wave function of the configuration b + X in A. The study of some of these coupling constants has a long history.

needs + X at of the As a

consequence, the definitions, the associated notation and of course the canonical values have changed many times, so that now a certain confusion exists in the literature. Also, in recent years several new coupling constants have been determined reliably for the first time, and the precision of some of the older values has been increased considerably. In our short note we present a consistent compilation of the current best values as well as of the relations between the different definitions, notations and methods of determination which are found in the literature. * Condensed body problems

version of an invited talk presented by the first author to the European in nuclear and particle physics, Sesimbra, Portugal, June 1980. 8

Symposium

on few

G.R. Plattner, RD.

9

Viollier / Coupling constants

2. Definition of the coupling constant c The asymptotic radial bound-state wave function of A in the configuration b +X is given for P+ a3 by u(r) - C’JZ

W-ll,,+~,2 (2w)/r,

where we consider the bound state A, virtuahy decaying into b + X, i.e. the vertex A+b+X with r=r& I=&,& ~=~~~=~~~~/(~~+~~); K=K&&x== J2pB~&h; B the binding energy; q = q&,x = .&&e2~/fi2~; e = cAbX; W_ 7),1+1,2 (2Kr) the Whittaker function I); and W0,1i2(2w) = emKr. The wave function u(r) is not normalized to unity, but such that: \u(r)1’r2 dr sprobability = probability

to find particle b between r and r +dr to find particle X between r and r + dr ,

i.e. C, JOmIts(r)12r2dr 3 total number of partitions of A(iV, 2) into b(P&,, Z,) and X(.P&, 2,) without regard to spin, isospin etc.

M is the set of all possible states of b and of X. 3. Relation to other notations The relations of c to the other commonly used notations are as follows: (&cc)* (27d22K)-1

G2

Lecher and Mizutani 2>,

=

ww2

-2 c

Lecher and Mizutani 2, ,

=

(dFvP I2

C2

Lecher and Mizutani 2),

=

( dFv)2

c2

Kim and Tubis

= (-1)’

~2(2~~4K)-~(d~v)z

D2

Goldfarb et al. “) .

t?‘= (-1)’

‘) ,

Here, for the state under consideration, p2 is the integral “cluster probability” 2, for A being formed from b and X, and (dFv)* is a structure factor 2, (calculated on the basis of the independent-particle shell model) which takes into account antisymmetrization with regard to the constituent nucleons. For the vertices reviewed in this paper we consider the dominant configurations and find the values shown below.

G.R. Plattner, R.D. VioNier / Coupling constants

10

Vertex

Configuration

dPn dpn

S-state D-state

Tdn 3He dp 3He 8p cu3He n

61,2)3

~TP 6Liold 13C’2Cn 170160n

ch,2)4

(dFv)’

P2 1-PJJ -0.95 1-Ps - 0.05 l-P,-0.92 l-P,-0.92 l-P, -0.92 l-P,--0.90 l-P,-0.90 0.5ztO.2 0.81ztO.04 -0.9

1 1

(s1,2)3 (S1,213 61,2)4

a + (P3,2Y -c+ I60

Ref.

(P1,2T + (&,d

Note that of these quantities only our c2 and the commonly used vertex constant are free from ambiguities introduced by considering model wave functions.

4. Methods for the determination We list various methods and their relation to spinless, structureless clusters b and X forming effects. (a) Forward dispersion relations (FDR). (b) Analytic continuation in cos 8 (AC). The the pole corresponding to the vertex A+ b +X

G2

of c*

the coupling constant for the case of the nucleus A, neglecting Coulomb

empirical [ref. “)I:

c2 = (-l)‘&fi#R

quantity

is the residue R of

.

(4)

(c) Effective range theory (ER). For I = 0, the empirical range lo) p(-E, -_E):

quantity

is the effective

C2 = (1 - Kp)-* .

(3

(d) Distorted-wave Born approximation (DWBA). For I= 0, the empirical tity is the zero-range constant “) D(iK) at the bound-state momentum iK : c2 = (-1);*.2(2rrh4K)-1D2(iK)

.

quan-

(6)

(For 1> 0, the bound-state wave function has to be considered directly.) D(0) is the constant used in codes like, e.g., DWUCK. (e) R-matrix theory. The empirical quantity is the reduced width y2, evaluated with an interaction radius a [ref. “)I

C2= pa WIc,r+Ij2

(2Ka>(h2K)-1(iVJ1y2

where IV, = 1 + c iEM’

7: (dSi/dE),zE,

.

)

(7)

G.R. ~lattner, R.D.

11

Viallier / Coupling c~n~~a~~s

M’

is the set of all channels. NC, a correction factor, is -1 except for bound states immediately below an I= 0 neutron threshold I’). Of the methods mentioned in this section we regard the application of forward dispersion relations as particularly reliable, and the analytic continuation technique as beset with the danger of large systematic errors [for a thorough discussion of these points see Lecher and Mizutani “) and references therein]. We have therefore chosen the FDR results where available, except for the deuteron coupling constant ci which is known most precisely from n +p scattering via effective range theory (ER). For the heavier nuclear probes 13C and “0 as well as for T the interpretation of transfer reactions by the distorted wave Born approximation (DWBA) has yielded reliable results, which have been included in our list. 5. Numerical values of l? dpn

K = 0.232 fm-’

c; = 1.69j~O.02 +D’(k) = 1.69~kO.02 + D’(O) = 1.58 + 0.05 - -3 c,c, = 0.0740* 0.004 -+c& =(1.13iO.12)x1O-3 +q =0.0259*0.0015

ER

10 )

Noyes 1972 (H&h&n wave function)

AC

Conzett et al. 1979 Griiebler et al. 1980

12 1 13

1

11=&,Cs

Bornand ei al. 1978 Franey et al. 1979 Durell el al. 1980 Goldfarb et al 1973 (D2(k)/D2(0) = 1.4

14

s,” = 3.940.6 D2(irc) = 3.2* 0.3 c-c; = 4.4*0.7) +e; =4.2*0.5 -+0’(O) = 2.210.5

FDR DWBA

c; = 5.1*0.4 +D2(iK)=3.5*o.3 --)D2(0) = 2.5 f 0.6

FDR

s; = 2.9*0.3 -, D’(k) = 7.6~tO.8 +D2(0)=5.4*1.7

FDR

c?$ = 26rk4 +D2(k)=21*3 +D”(O)= 10*3

FDR

Plattner at al. 1973

aTp K = 0.846 fm-’

c; =26.6*1.4 +D2(irc)=21.6~l.2 +D2(0)= 10*3

FDR

4.19 1 18 Plattner er al. 1973 f Lecher and Mizutani 1978 ‘) 4.19 (D2(iK)/D2(0) - 2 )

6Licud K = 0.308 fm-’

c; =4.6*0.3 *D’(k) =0.87+0.06

FDR

Bornand et al. 1978

14 )

c& =(l+t)~lO~~

FDR

Bornand

34 )

13c’2c n K = 0.469 fm-’

e;

DWBA

Gubler et al. 1977 Franey et al. 1979

‘70f60 n K = 0.434 Em-’

cg

Tdn I( = 0.448 fm-”

3He dp K = 0.420 fm-’ 3He bp effective K = 0.54 fm-’ CZ3Hcn K = 0.863 fm-’

=2.55*0.10 =0.77rtO.O8

Plattner et al. 1977 (D2(iK)/D2(0)

Plattner et al. 1977 (D2(iK)/D2(0)

=

(~z(~~)/~z(o)

DWBA

= 1.4

1.4

IS : 16 4)) 4, 17 ) 4, 17 1

4, 18 1

-2

et al. 1978

Franey et al. 1979

?

1s )

15 1

G.R. Plattner, R.D. Viollier / Coupling constants

12 c2

is

a dimensionless

quantity.

The value of 1 is indicated

All values O2 are in units of 104MeV2 quote an average

* fm3. Where

with error bar matched

by the subscripts

several reliable

S, P, D.

values exist, we

to their dispersion.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

H. Buchholz, The confluent hypergeometric function (Springer, Berlin) M.P. Lecher and T. Mizutani, Phys. Reports PLC 46 (1978) 43 Y.E. Kim and A. Tubis, Ann. Rev. Nucl. Sci. 24 (1974) 69 L.J.B. Goldfarb, J.A. Gonzalez and A.C. Phillips, Nucl. Phys. A209 (1973) 77 D.W.L. Sprung, Proc. Int. Conf. few body problems, Laval, 1974, ed. R.J. Slobodrian et al. (Lava1 U. Press, 1975) p. 475 Y. Akaishi, J. Hiura, M. Sakai and H. Tanaka, Proc. Int. Conf. few body problems, Los Angeles, 1972, ed. I. Slaus et al. (North-Holland, Amsterdam, 1972) p. 353 G.R. Plattner, M. Bornand and K. Alder, Phys. Lett. 61B (1976) 21 H.P. Gubler, G.R. Plattner, I. Sick, A. Traber and W. Weiss, Nucl. Phys. A284 (1977) 114 F. Ajzenberg, Nucl. Phys. A281 (1977) 93 H.P. Noyes, Ann. Rev. Nucl. Sci. 22 (1972) 465 A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257 H.E. Conzett, F. Hinterberger, P. von Rossen, F. Seiler and E.J. Stephenson, Phys. Rev. Lett. 43 (1979) 572 W. Griiebler, V. K&-rig, P.A. Schmelzbach, B. Jenny and F. Sperisen, Phys. Lett. B92 (1980) 279 M.P. Bornand, G.R. Plattner, R.D. Viollier and K. Alder, Nucl. Phys. A294 (1978) 492 M.A. Franey, J.S. Lilley and W.R. Phillips, Nucl. Phys. A324 (1979) 193 J.L. Durell, C.A. Harter, J.N. MO and W.R. Phillips, Nucl. Phys. A334 (1980) 144 G.R. Plattner, M. Bornand and R.D. Viollier, Phys. Rev. Lett. 39 (1977) 127 G.R. Plattner, R.D. Viollier, D. Trautmann and K. Alder, Nucl. Phys. A206 (1973) 513 A. Moalem and Z. Vardi, Nucl. Phys. A332 (1979) 205