E-Book Overview
This classic volume, reprinted twenty years after it was first published, takes a close look at the theory of direct nuclear reactions. It emphasizes the microscopic aspects of these reactions and their description in terms of the changes induced in the motion of individual nucleons, except where collective motion in nuclei gives a more succinct description. Assuming only a modest knowledge of quantum mechanics and some acquaintance with angular momentum algebra, the book begins essentially at the beginning. Its goal is to provide the novice with the means of becoming competent to do research on direct reactions, and the experienced researcher with a detailed discussion of advanced topics. For completeness, appendices on angular momentum algebra and special functions are included.
E-Book Content
257
D. Form Factor
This is a fully antisymmetric two-nucleon shell-model wave function in spin, isospin, and space. The Cp on the right side of (15.17) is (15.19)
C~nie,j,(l)~nze2j*(')IJM,
is a shell model spin-orbit wave function [see (7.89)]. where To prepare for the integration over spin coordinates in use can be made of an angular momentum transformation (see Appendix)
c
I(~I~I)~I,
LZ
l(ele2)L,
(15.20)
where the square bracket, denoting a transformation coefficient, is related to the 9 j symbol. The bracket notation of (15.20) is a commonly used one when many angular-momenta couplings are involved. The wave funcfunction, as contrasted tion on the right is sometimes referred to as an with that on the left, called a function. In more detail, (~1~2)s;
CCCpnlelCpn2e,lLCXs1Xs2ISIJM.
At this point the Talmi-Moshinsky transformation from individual to relative and c.m. coordinates can be performed (Talmi, 1952; Moshinsky, 1959; Brody and Moshinsky, 1960). This can be done exactly if the singleparticle states dnej are harmonic-oscillator wave functions in their radial part (see Appendix). Otherwise it can be done approximately by expanding the wave function in a finite number of oscillator wave functions (Glendenning, 1965). The transformation reads
rl
rl
r2,
2R
rl
r2.
This is a very useful and important transformation in nuclear physics because nuclear wave functions are typically expressed in terms of the coordinates of the nucleons, whereas potential interactions are a function of relative coordinates. The summation is restricted by a condition that follows from the for a particle fact that the energy associated with an oscillator state of mass is [2(n
1.
1)
Therefore energy conservation implies that Ll
2(n
A,
(15.224
258
15. Two-Nucleon Transfer Reactions
and parity (see A8) implies
In terms of these transformations, the two-particle wave function can now be written as
x
L J rJ &,
3
j , (n/ZNA;L[n,L,n,tf2;L )
[4nI(mr)&NA(fiR)I
L 7 xS(
9
2)]y.
(15.23)
Throughout, for spin or isospin two-particle wave functions we use the notation
X Y ( L 2)
=~
~ 1 / 2 ~ ~ 1 ~ ~ =1 XY(01, / 2 ~ ~0 22) . ~ 1 Y
(15.24)
The integration over r, as well as over spin and isospin in the expression for F [Eq. (15.15)], can now be performed and the result expressed by
where the radial part of the form factor is
fjL”;(~)= 1 ~
jN iL j~ 2J
uNJJ~VR),
(15.26)
N
and the coefficients G are referred to as structure amplitudes for two-particle transfer reactions. They have been tabulated for the entire range of quantum numbers that occur in nuclei (Glendenning 1975a) and from the preceding development are defined by