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Shell model with σ · x potential
Nuclear Physics 78 (1966) 625--640; ~ ) North-Holland Publishiny Co., Amsterdam Not to
be reproduced by photoprint or mlerofilm without written permission from the publisher
SHELL MODEL
WITH ~. x POTENTIAL
GENTARO ARAKI
Department of Nuclear Engineering, Kyoto University, Yosida, Kyoto Received 20 September 1965 The one-nucleon state of a definite j-value is constructed as a linear combination of two states which belong to the azimuthal q u a n t u m numbers j--½ and j+~-, respectively. The invariance of the state for space inversion is guaranteed by introducing a unit pseudoscalar. The Hartree-Fock equation satisfied by such one-nucleon states and the isospin and spin-angle dependence of the Hartree and exchange fields are discussed on the assumption that the twonucleon interaction consists of three parts, i.e. the central, the ~ • ~' and the tensor-force type. It is shown that the Hartree-Fock equation reduces to a form of eigenvalue equation of the Hamiltonian with central plus o • x type potentials. Addition theorems for spin-orbital harmonics are obtained in the lira scheme.
Abstract:
1. Introduction
In a previous note 1) it was shown that the spectrum of the one-nucleon Hamiltonian with a a • x type potential is an inverted doublet, and it was suggested that the nuclear shell structure may be interpreted by this spectrum. The eigenstate of this Hamiltonian consists of two parts belonging to different azimuthal quantum numbers and opposite parities. In order to maintain invariance of the total nuclear Hamiltonian under space inversion, the total nuclear state was assumed to be represented by a projection of the Slater determinant, composed of such one-nucleon states of mixed parity, onto a state of a definite parity 1, 2). This assumption deprives the Hartree-Fock scheme of its simplicity. Nevertheless, one may wish to look for its theoretical foundation in this scheme. The undesirable feature arises from the fact that the a • x type potential is non-invariant under space inversion. In the present paper the formalism is modified to remove this undesirable complication by introducing an invariant o- • x type potential. The formal consistency of the basic assumption is discussed in the Hartree-Fock scheme. It is shown that the Hartree-Fock equation reduces to a form of eigenvalue equation of the Hamiltonian with a central type plus a a • x type potential. 2. One-Nucleon States and Levels We consider a free-particle model of nuclei in which one-nucleon states are eigenstates of the Hamiltonian H
=
-
hz -A+
U(r)+V(r)2
2M 625
a" x, r
(2.1)
626
G. ARAKI
where M is the nucleon mass, x and r are the position vector and its magnitude, respectively, ½ha is the nucleon spin angular momentum, U(r) and V(r) are scalar radial functions and ~, is a unit pseudoscalar. The last quantity guarantees the invariance of the Hamiltonian with respect to space inversion. If ~ j , , denotes the normalized simultaneous one-nucleon eigenstate of the squares of the orbital and total angular momenta and of a component of the latter, the spin-angle dependence of the normalized eigenstate of H is given by 1) O = { F ( r ) + G(r);.,, • x/r)~,,~,
(2.2)
where F(r) and G(r) are radial scalar functions. We denote the norms of the two orthogonal parts of this state by ci and c2, respectively, and their normalized parts by ~l and ~2: clO1 = F(r)~bU~,
C2O 2 = G(r)(a" x/r)~tim.
(2.3)
We have then
'k
.
. cl¢q+~c2'kz,
. 1101112.
ii~zll 2
11@112
c,+c 2z = 1. z
(2.4)
As was shown in the previous note x), ~2 belongs to the azimuthal quantum number' 2 j - I and the parity - ( - 1 ) k Nevertheless ~ belongs to the definite parity ( - 1 ) k because ,~a • x is invariant with respect to space inversion. The eigenvalue of H and the corresponding eigenstate are respectively given by E (+) and ~(+) or E (-) and ~b(-) according to E 1 > E 2 or E, < E2, where these quantities are defined as follows: E (+) = E l + 5 ,
~(+) = ( r O l + ~ , 2 ) / N
(El > E2),
(2.5)
E (-) = E i - 6 ,
0 (-) = (rO~-X~0d/N
(E~ < E2),
(2.6)
Ei = (~l, H ~ l ) ,
E2 = (~t2, H~t2),
~ = (Ol, HO2),
(2.7)
= ½IE1-E2I ,
= (g2--[-~2)~--/3,
N = (72+62)~.
(2.8)
The scalar radial parts F(r) and G(r) can be taken to be real, consequently y is real. If the effect of V(r) is not too large, E 1 is higher than E2 when l is greater than 2 j - l (that is, when l = j + ½, 2 j - l = l - 1) and El is lower than E2 when l is smaller than 2 j - l (that is, when l = j - ½, 2 j - l = l + i). Hence the spectrum of one-nucleon levels is an inverted doublet as was explained in the previous note 1). In the present case the total nuclear state, which is represented by the Slater determinant composed of onenucleon states of the above mentioned type; belongs to a definite parity. If the oddnucleon state of an odd-mass nucleus is given by c1~/'1+ 2e2~/'2, the magnetic moment 2 +c2#2, 2 # of this nucleus is given by # = c~#~ where #1 and #2 are two values of the magnetic moment in the Schmidt model corresponding to l = j_+½. 3. Hartree Field
In order to examine whether the one-nucleon Hamiltonian (2.1) is compatible with
SHELL MODEL
627
two-nucleon interactions or not, we first consider a closed-shell nucleus containing A nucleons in q~, q52. . . . . q5A states in the Hartree-Fock scheme. The Hartree-Fock equations for the normalized one-nucleon states ~bv (v = 1, 2 . . . . . A) are given by
{ - hZA/(2M) + Uri(q) - Uex(q))(~,,(q) = E,, O,,(q), A
(3.1)
A
Un(q) = 2 V#.~(q),
Uex(q)~,,(q) = 2 Vu~(q)~u(q),
g=l
(3.2)
/Z=I
Vu,,(q) = f Ou(q '
q')O,(q')dq',
(3.3)
where q stands for one-nucleon coordinates including space, spin and isospin variables, B(q, q') denotes the two-nucleon interaction and E~ is the diagonal element of the Lagrange multipliers (the Hermitian matrix). We assume that the isospin dependence of the two-nucleon interaction is given by
B(q, q') = B(x, x', ~, ~')+'r,. ,~'B,~(x, x', ~, a')+zpz've2/lx-x'[,
(3.4)
where ~ denotes the isospin operator whose three components satisfy the same algebraic relations as those satisfied by the components of o'. The last term is the Coulomb interaction between protons where Zp denotes the projection operator to the proton state. We further assume that B and B~ each contain central, a • ~' and tensorforce terms. We denote the normalized isospin part of ~b~ by Z, and the normalized spin-orbital part by 0~,~j,~, where t is the eigenvalue of %. In accordance with eq. (2.2), the latter part is given by
~.~nt/m = {F~ntj(r) + Gtnlj(r)2a" x/r}~tjm"
(3.5)
We now examine the spin-angle dependence of the Hartree field Un. For this purpose we assume the a dependence of the two-nucleon interactions as follows:
B(x, x', ,~, ,¢) = w(Ix-x'l)+,7" ,¢u(lx-x'l)+(,~" v)(~' • V ' ) K i x - x ' l ) .
(3.6)
As is well-known, the last term is a sum of a • a' and tensor-force type interactions. We first carry out a sum of Vuu with respect to m. Corresponding to the three parts of eq. (3.6) we have to calculate the three sums
V~ =
x' O,,Zjm(X')>w(lx- x'l)d(x'),
(3.7)
m
ga =
U(IX-- X I)d(x ),
~ " m
'
VT = (o' " V )
(3.8)
j
'
'
.V'v([x-x'l)d(x'),
(3.9)
m
where the isospin quantum numbers are omitted for simplicity, < > means the Hermitian inner product of spin parts and d(x') the volume element in the x' space.
628
G. ARAKI
To carry out the sums we first look for addition theorems for ~tim. A set of 2j+ 1 spin-orbitals ~tj,, with definite l and j is orthonormal in any system of space coordinates. Therefore the transformation of these 2j+ 1 spin-orbitals for the space rotation is unitary and J
E
<~tj,~, ~tjm> = C(l, 1),
(3.10a)
in= - - j
m= - j
<'#,in, ~2j--Hm> = ~
m= - j
eljm
r
~,jm
= C(1, 2j--l)
(3.10b)
are consequently invariant for the space rotation, that is, C(l, l') is independent of space coordinates. In the latter equation we have made use of the relation
¢2j-u .
=
(3.11)
The constant C(l, l') can be determined by integrating both sides of eq. (3.10) with respect to angle variables. We have then J
2 <~b,j~, #rjm> = 3u. 2j+____l,
(l' = 1, 2 j - l ) .
(3.12)
4re
-j
m=
Next we consider the sum ~,,(#u,,, ¢r#rJ,*>" This is a homogeneous polynomial in
x/r, y/r and z/r because ~0Umis a spherical surface harmonic. On the other hand, this is a vector for the space rotation. Therefore this sum is equal to Ct(l, l')x/r, where Ct(l, l') is independent of space coordinates, and the scalar product of the sum by x/r is equal to Cl(l, l'). This gives just the same equation as eq. (3.10) [note the relation (3.11)]. The constant Ct(/, l') can be determined by comparing this equation with eq. (3.12). We have thus
J <~)ljm, ff~Dl'jm> = (1_ flu,) 2j+ 1 X, 4~ r
Z
(l' = l, 2j-- O"
(3.13)
m= --j
By making use of eqs. (3.12) and (3.13) we can carry out the summation of integrands in eqs. (3.7)-(3.9) as follows: m= --j
2j+l
{F.tj(r) + G,u(r)},
J
2j+l
Z
--
m= -j
4re
J
2j+l
,.= - j
(3.14)
47z
<~n,jm, a~n,jm> " V = - -
4re
x
Fnlj(r)Gnlj(r))c - ,
(3.15)
r
3
F,tj(r)G.tj(r)2 - - . ar
(3.16)
We denote x • x'/rr' by ~ and expand the radial parts of the two-nucleon interactions into the Legendre series in {. For example, we have
u(lx-x'[) = ~ (2k+ 1)Uk(r, r')Pk((), k=O
(3.17)
SHELL MODEL
629
where Pk(~) is the Legendre polynomial of degree k in ~. The angular integrations of the integrals (3.7)-(3.9) can be easily carried out by making use of the orthogonality of spherical surface harmonics. The results are given by 2 t )+ G,a.i(r 2 , )}wo(r, r')r'Zdr ', vo = (2j+1) I ; {F,,lj(r
(3.18)
V~ = ). a ' x 2(2j+ 1) I®Fntj(r')G, tj(r')ux(r, r')r'2dr ',
(3.19)
r
./o
VT = 2 ~r " x 2(2/+1)
r
F.tj(r ,)G.lj(r , )r ,2 - - vo(r, r')dr', OrOr' "
fo °
(3.20)
where wo(r, r') and vo(r, r') are the zero degree term in the Legendre series of w ( l x - x ' l ) and v(Ix-x'l), respectively, [see eq. (3.17)]. By summing the last three expressions we have the Hartree potential due to B(x, x', ~r, ~r') in the form O H ~--- UI(F)--~- V / ( F ) ~ • x/r. (3.21) The radial functions are given by
Ui(r ) = Vi(r) =
fo
{pp(r')+ pn(r')}wo(r, r')dr',
{qp(r')+~/n(r')} ul(r, r')+
02 vo(r, r') dr', c~r~r'
(3.22a) (3.22b)
pp(r) = ~ (2j + 1){Fzlj(r) + Gz,zj(r)}r 2,
(3.23 a)
%(r) = ~ 2(2j+ 1)F, tj(r)G,,ti(r)r z,
(3.23b)
where the summations extend over all occupied proton shells and pn(r) and t/,(r) are the similar sums for occupied neutron shells t If we take into account the v-dependent terms of eq. (3.4), the Hartree potential is modified as follows:
UH = U,(r)+ zz U~(r)+zp U¢(r)+ {Vi(r)+ z 3 V~(r))2a • x/r,
(3.24)
where U~(r) and V~(r) are given by similar integrals, including radial parts of B~, as eqs. (3.22a) and (b), respectively, except that the density functions in their integrands are replaced by p p ( r ) - pn(r) and % ( 0 - t/n(r). (We assume that the Proton state belongs to the plus eigenvalue of v3. ) The contribution from the Coulomb term is given by
Uo(r) =
50
pp(r')(eZ/rg)dr',
(3.25)
where rg denotes the greater one of r and r'. ¢ E q . (3.22b) as w e l l as t h e r e s u l t o b t a i n e d i n t h e n e x t s e c t i o n s h o w t h a t the a r g u m e n t g i v e a b y
Amiet and Huguenin 2) contains a mistake.
630
G. ARAKI
4. Exchange Field In this section we examine the spin-angle dependence of the exchange field Uex. For this purpose we have to look for addition theorems for ~b~j,,with different arguments. This spin-orbital function is referred to as a spin-orbital harmonic. Let YZm" be the simultaneous normalized eigenstate of the square of the orbital momentum and of its component. The spin-orbital harmonics ~Ijm and ~ u - l m (J = l+½) are linear combinations of aY, ,,_~ and flY, ,,+~ with the given azimuthal quantum number l, where e and fi are spin eigenstates corresponding to spin magnetic quantum numbers +1 and - 1 , respectively. The systems {eY*m', flY~m'} and {~*Sm,~U-lm} are both orthonormal, and the transformation between them is consequently unitary. Therefore we have j
j--1
m=-j
m=--(j--
1)
I
= Z {c~*(1)Y~*(1)c~(2)):},,(2)+fl*(1)Yz*(l)fi(2)Y~m(2)}'
(4.1)
m = --l
wherej = l+½. The sum on the right can be carried out by making use of the wellknown addition theorem for the zonal harmonic, and the right-hand side reduces to {(2/+ 1)/(4rc)}Pl((,2)6~(1, 2), where 6~ and ~,2 are defined by fi~(1, 2) = e*(1)e(2)+fi*(1)fi(2),
(12
=
Xt"
x2/rlr2.
(4.2)
The first and the second sums in the left-hand side of eq. (4.1) can be separated by projecting the left-hand side o n j = l+½ a n d j ' = l - ½ states, respectively. The corresponding projection operators are given by (l+ 1 +0-2 "L2)/(21+ 1) and ( l - - o " 2 " L2)/(21+ 1), respectively, where hL is the one-particle orbital angular momentum. Since / + 1 = j + ½ , 2 ( j - l ) = +1, l = j ' + ½ , and 2 ( j ' - l ) = - 1 , we have J ~jm(1)~tj.~(2) = {j+½+2(j-I)0-2"L2}PI(~12)6~,(1 , 2)/4rc.
(4.3)
m = -j
if we multiply both sides by 0-2 "x2/r2 from the left and make use of eq. (3.11), we have m = --j
~;.m(1)~2j_Um(2) __ 0-2 "X2 {j+½+2(j--l)0-2" L2}P,((,2) 6~'(1' 2) r2 4re
(4.4)
The last two identities together with eq. (3.12) are addition theorems for spin-orbital harmonics. We first calculate the contribution to Uex from the z independent terms of the twonucleon interaction. In this case the exchange field is given by
U~xZ,(2)~.,tjm(2 ) = Z,(2)U~x~,,,tj,,,(2), Uext]lmUm(2)
=
Z ' t'(@m,rj,m,(1), B(I, 2)0,,um(1)>~b,,,rj,,,,(2)d(xl ), d
(4.5a) (4.5b)
SHELL MODEL
631
where ~ ' denotes the sum extending over all occupied n', l', j', m' states. All spinorbitals in these equations belong to the same isospin quantum number t, on account of the orthonormality of isospin states. Therefore the interaction between protons and neutrons are not involved in this exchange field. In order to facilitate the calculation we write the two-nucleon interaction in the form
B(xl, xz,
al, az) = w(Ix~21)+,rl
" ,r2u(Ix121)+
(,rl " x l ~ ) ( , r 2 •
xlz)v(Ixlzl),
(4.6)
where Xla stands for xl - x 2. The third term is a sum of a 1 • a 2 and tensor-force type interactions. Therefore the functions u and v are not the same as those in eq. (3.6). Corresponding to three parts of this interaction we have to calculate the three integrals Rc = f ~ ' @',Ti,m,(1), ~.ljm(1))W(lXl 2[)~'nTj'm'(2)d(xl),
(4.7)
R~ = f z ' (tPn'l'J'm'(1)' ~1 ~lnljm(1)) "aEO,'Vj'm'(Z)u(lxlzl)d(xl),
(4.8)
RT = f Z' (~tn't'j'm'(l)' al " Xl2tPnljm(1))~r2 " Xlz~lln't'J'm'(Z)v(Ix121)d(Xl),
(4.9)
where the isospin quantum numbers are omitted for simplicity. By making use of the addition theorems (4.3) and (4.4), the summation with respect to m' in the integrand of the first integral can be carried out as follows: j,
£ m' = -- j r
, ~n,rj,m,(1)~tn,r j,m,(2) = [D I { j' + ½+ 2(j'-- I')a 2 " L2}Pv((a2)
+D2{J' + ½ - 2 ( j ' - l ' ) a z • L2}Pzj,_r(~12)]6,(1, 2)/4rc, (4.10) where
D1 = F, T2,(r~){Fn,rj,(r2)+ G,,rj,(r2)2a2 " x2/r2},
(4.11a)
Dz = G,,t,j,(ri){G,,rj,(r2)+
(4.11b)
r,,ra,(rz)2a2
" ~;r~}.
If we carry out the differentiation, we have
0"2" L2Pt((,2 ) = i(x2 x a2). x~ P;((~2), /'2
(4.12)
rl
whereP;(() denotes the derivative. The function of Ix 1 -x21 is the function of r l, r 2 and ~lz. Therefore we can expand the function of Ixl-x21 multiplied by P l ( ~ l z ) o r P'l(~12)in the Legendre series of (tz. For example, we have W(IXl
--X2])Pl(~12)
=
~ (2k+
1)Pk(~z)w(k, l),
(4.13)
k=0
w(lxl- x2l)P;((~z) = ~ (2k+ 1)Pk(~2)w(~)(k, I), k=O
(4.14)
632
G. ARAKI
where the coefficients are given by
w(k, l, rl, r2) = ½
Pk((12)w(lxl--x21)P,((iz)d(12,
(4.15)
-1
win(< t, rl, "2) -- ½
-xd)P glz)d l .
(4.16)
In the series (4.13) and (4.14) the arguments q and r 2 are omitted from their coefficients in order to avoid unnecessary complications. If we substitute appropriate parts in the integrand of eq. (4.7) with expressions (4.10), (4.12), (4.13) and (4.14), the integrand turns out to be the sum of two kinds of terms; one is the product of the Legendre series and 0,~jm(1), and the other is the product of the Legendre series and (xl/r~)Ontjm(1). The angular integration of the former can easily be carried out by making use of the delta function property of Pk(~12)6~(1, 2) as well as the orthogonality of spherical surface harmonics: co
+ 1 Pk(~,2)(6~(1, 2), ~,,jm(1)>dco, ~ ow( k' l , ).J( 2 k4=
= {w(l, l')F,,zj(1) + w ( 2 j - I, l')G, tj(1)2a2" xz/r2}~tjm(2),
(4.17)
where do), denotes the solid-angle part of the volume element d(xl). The angular integration of the second part containing (Xl/q)~k, tjm(1) can be carried out by a projection technique. The angular part of (Xl/q)~tjm(1) consists of spherical surface harmonics of degrees l - 1 and l+ 1. They can be separated by the projection operators defined by E}_)(1) = (l+ 1)(1+2)-L 2 2(21+1) '
(4.18a)
E~+)(1) =/,21 - ( l - 1)/ 2(2/+1)
(4.18b)
The sum of these operators is equal to unity and we have
(xl/rl)~ljm(1) = {E}-)(1)+ E} +)(1)}(xl/rl)~jm(1).
(4.19)
By using this identity and also the above-mentioned delta function property, we have }~ w(a)(k, l ' ) ~ 2 k + 1 Pk(~12)(J~(1, 2), O,tj,,(1))(xl/rl)de), k=o
d
4~
"" (1)( I -- 1 , l')E}-l(2)+wO)(l+l, l')E}+)(2)}(x2/r2)~tjm(2) = Fntj(1){w (+) + G,tj(1){wO)(2j- l - 1,/')E(z7 )_,(2)+ w(1)(2j- l+ 1, 1 )Eej-z(2)}(xz/re) x (2trz " x2/r2)~btjm(2 ). (4.20) ,
Then the integrand of eq. (4.7) contains the operators which are given by the left side of
SHELL MODEL
633
the identities
i(x2 x a2) E}+)(2). x2 = T az'__L2 r2 r2 21 + 1
(4.21)
The left side can easily be changed into the right side. Since ~bu.m(2) is an eigenstate of the operators on the right, these operators reduce to mere numbers. Thus R~ takes the form
Ro = { U~(r) + V~(r)2a " x/r}F,,tj(r)~lj m
+ (v~(r) + v~(r)2a, x/r)G,.,j(r)(~a • x/~)~,j~,
(4.22)
where
U}(r) =
Q~(j, l, r, r')F~.u(r')dr'/F~.u(r ),
(4.23a)
V~(r) =
Q~(j, l, r, r')F~tj(r')dr'/F,.lj(r),
(4.23b)
U~(r) =
v;(r) =
f o°Q~(
fo
j, 2 j - l, r, r')G,.u(r')dr'/G,.u(r ),
e~(J, 2 j - l , ~, r')a,.,j(r')dr'/G,.,j(r),
(4.23c) (4.23d)
Q~j(j, l, r, r') = Z' WC(J, l, j', l', r, r')F~.,rj,(r')F,.,vj,(r )
+ Z' W°(/, I, £, 2 ; ' - r , r, r')C,...Ar')~,.,.Ar), Q.~(j, t, r, r') = Z' w°(J, I, j', r, r, r')F,...Ar')a,.,.AO + Z' We(J, l, j', 2j'-l', r, r')G,.,rj,(r')F~..rj,(r),
(4.24a)
(4.24b)
w°(:, I, j', r, r, r') = (j'+ ½)w(t, r, r, r')r '2 +2(j'-I')K(j,
~ t2 l){w°)(1-1, l,t r, r ' ) - w ( 1 ) ( l + l , l t, r, r t ')~r ,
K(j, l) = { j ( j + 1 ) - / ( / + 1)-4s-}/(2/+ 1).
(4.25) (4.26)
The last quantity is the eigenvalue of the operator (4.21) of plus sign and ~ ' in eqs. (4.24) the sum extending over all occupied n', l', j ' shells. The second integral R~ can be obtained from Re by replacing ~,u=(1) in its integrand with a l e , urn(1) and multiplying the result by a2 from the left. Then the integrand contains a 2 " Dia 2 and a2 • D i ( a 2 " L 2 ) a 2 (i = 1, 2). We can eliminate two a2 from these expressions by using the identities
a" (a" a)a = --a" a, a" (b + ca" x)(a" L)a = - (b+ ca" x ) ( a . L),
(4.27) (4.28)
634
G. ARAKI
where a, b and e commute with a. Thus R~ is reduced to R~ = { U~-(r) + V [ ( r ) 2 a • x/r}F,,,j(r)cPtj
m+ { U~(r) +
V~(r)2~ • x/r}
G,,lj(r)(2a. xlr)e&j~, (4.29)
a t") and Vg(r) denote the radial integrals which are obtained where U~(r), V~(r), U~( from integrals (4.23a)-(4.23d) by replacing Q~(j, l, r, r') and Q~(j, l, r, r') in their integrands with the following Q(~(j, l, r, r') and Q~(j, l, r, r'), respectively:
Q[j(j, l, r, r') = Z ' W(j(j, l, j', l', r, r')F,,,rj,(r')F,,,rj,(r) + Z ' W(j(j, l, j', 2 j ' - i ' , r, r')G,,,rj,(r')G,,,vj,(r),
(4.30a)
Q~(j, l, r, r') = ~ ' W~(j, l, j', l', r, r')F,,,,rj,(r')G,,,rj,(r) (4.30b)
+ ~,' We(j, l, j', 2 j ' - l ' , r, r')Gm,rj,(r')F,n,rj,(r), W(j(j, l,j', l', r, r') = 3(j' +½)u(l, l', r, r')r 'z -2(j'-l')K(j, l){u(1)(l-1, l', r, r')-u(1)(l+ i, l', r, r')}r '2,
(4.31a)
W(:(j, l,j', l', r, r') = W~:(j, l,j', l', r, r')-4(j'+½)u(l,
(4.31b)
l', r, r')r '2,
where u(l, l', r, r') and u°)(l, l', r, r') are the coefficients of the Legendre series pertaining to u(Ix-x'l) [see eqs. (4.15) and (4.16)]. We examine the spin-angle dependence of R T by dividing the corresponding interaction into four parts { - ( ~ 1 " xl)(~2 • x 2 ) + ( , , ~ • x ~ ) ( ~ , x~) "t- (O" 1 " X l ) ( O " 2 " X l ) - - ( l ~ "
1 " X2)(a
2 "
xl)}v([xlzl).
(4.32)
The contributions to RT from the respective parts are denoted by R (~), R (2), R (3) and R (4). If the functions ~,Um(1), Cn'Vj',,'(2) and w(lx12I) in eq. (4.7) are substituted by (,tal"xl/rx)~k.u,,(1), (2az'xz/r2)~,,vj,m,(2), and rlrzv(Ix12]), respectively, it turns out to be R u). The former two substitutions are equivalent to interchanges F,,j(rl) ~--- G,u(rl) and F,,vj,(rz) ~- G,,w,(r2) in ~,Um(1) and ~.,Vy,m,(2) of eq. (4.7), respectively. Therefore the spin-angle dependence of R (1) is the same as eq. (4.22). We have only to substitute Uxg, V~, U~ and V~ by the following U(F~), V(F~), "~orr
U(v~)(r) =
L oQ(ul)(j, 2 j - l ,
V(vl)(r) = U(1)z G t r)~ =
r, r')F,,zj(r')dr'/F,,zj(r),
(4.33a)
Q(vl)(j, 2 j - l , r, r')F~,lj(r')dr'/F,,lj(r),
(4.33b)
L oQ(ul)(j,. l, r, r ,)G,~,j(r ,)dr/G,~u(r , ),
(4.33c)
SHELL MODEL
V(~l)(r) = I, r,
r')
f;
= Z'
635
Q(vi)(j, I, r, r')G~n,j(r')dr'/G,nzi(r), 1,j', r, r,
(4.33d)
)
+ 2 ' W(1)(J, l, j', 2 j ' - l', r, r')G,n,vj,(r')G,~,vj,(r),
(4.34a)
Q(vl)(j, l, r, r') = 2 ' W°)(J, l, j', l', r, r')F,n,rj,(r')G,~,ri,(r) + ~ ' W(1)(j, l, j', 2 j ' - l ' , r, r')G,,,vi,(r')F,,,vi,(r),
(4.34b)
WO)(j, l,j', l', r, r') = (j'+½)v(l, I', r, r')rr '3 +2(j'-I')K(j,
l){vO)(l-1, l', r, r ' ) - v ° ) ( l + l ,
l', r, r')}rr '3,
(4.35)
where v(l, l', r, r') and v(1)(l, l', r, r') are the coefficients of the Legendre series pertaining to v(Ix-x'l) [see eqs. (4.15) and (4.16)]. The integral R (2) can be obtained by substituting 0,1i,~(1), ~k~,Vj,m,(2) and w([xl21) in eq. (4.7) with (el • x2/r2)O, lim(1), (a2" x2/r2)O,,Vi,m,(2) and r22v([xi21), respectively. The first substitution is equivalent to multiplying eqs. (4.17)and (4.20)by a 2 " x e / r 2 from the left. The second substitution is equivalent to multiplying eq. (4.7) by a 2 • x2/r 2 from the left. These two multipliers reduce to unity if the multiplicant does not contain x2 x a 2 in its factor. If the multiplicant contains x 2 x a 2, one of them appears on the left of x z x a z and the other between x 2 x a2 and E}-+)(2) of the products such as the left side of eq. (4.21). We can easily verify the anticommutation relation
(x x a)(a " x) = - (a " x)(x x a).
(4.36)
Consequently the effect of (a t • x2/rz)(a 2 • xz/r2) in the interaction is merely to change the signs of K(j, l) and K(j, 2 j - l ) in the integrands of eq. (4.23). (Note that (2l+ 1) K(j, l) is an eigenvalue of a z • L2) • Therefore R (2) is of the same form as eq. (4.22) in which we replace U~, V~, U~ and V~ with U(r2), V(F2), ~rf(2)and rG~(2),respectively. The last four are defined by eq. (4.23) (a), (b), (c) and (d) by substituting Q~(j, l, r, r') and Q~(j, l, r, r') in their integrands with Q(~)( J, l, r, r') and Q~)( J, l, r, r') respectively; these two functions are given by eq. ( 4 . 2 4 ) ( a ) a n d (b), respectively, by substituting W¢(j, l,j', l', r, r') with W(2)(j, I,j', l', r, r') = (j' +½)v(l, I', r, r')(rr') 2
-2(j'-l')K(j,
l){v(l)(l-1, l', r, r')-v(a)(l+ 1, I', r, r')}(rr') 2.
(4.37)
The integrand of R (3) contains (a i • xl)(a 2 • xl). The effect of the first factor is to interchange F,u(rl) and 2G, u(rl) in 0,zjm(1). The effect of the second is to multiply eq. (4.10) by a 2 .x~ from the left. This multiplier can be moved to the right of D1 (or D2) by making use of their anticommutator (a 2 " xa/rl)D1-t-DI(o"
2 " xl/rl)
=
2L,t.y(rl)Gn.rj.(r2)2(12 + 2F,.vj.(r~)F,.rj.(r2)a 2 • x l / r l .
(4.38)
636
G. ARAKI
Then the integrand of R (3) contains the left side of the equation ( ~ . x,/r,)(¢~ "Lz)Pr(~2) = (az" xzlr2)P;,((~2)
--(tr 2 . x,/rl){l'Pv(¢,2)+ n;,_,(~,2)}.
(4.39}
This can be changed into the right-hand side by carrying out the differentiation and making use of the recurrence formula for the Legendre polynomial. The recurrence formula can also be made use of to absorb ~12 of the first term in the right side of eq. (4.38) into the Legendre series. The projection operators reduce to 1 or 0 according to the equations
E(+-) 2 j - I @Um= { 4- ( l - j ) + ½} ~ljm E} +)(a" x/r)~um = { +_(j-- l) + ½}(a- x/r){#tj,,.
(4.40a)
(4.40b)
Now the angular integration of R (3) is easily carried out as was done in the case of Re. The result is represented in the form of eq. (4.22). Its radial parts are denoted by U(F3), V(F3), '-'GrT(3)and --G1/(3).They are given by the integrals of the same form as in eqs. (4.33) (a), (b), (c) and (d), respectively, in which the functions Q~I) and Q(v1) are substituted by the following Q~) and Q(v3), respectively: Qb3)(j, l, r, r') = y ' W~a)(j, l, j', I', r, r')F,,,t,~,(r')F,,Ti,(r )
+ Y.' W(ua)(j, I, j', 2j'-l', r, r')G,.,rj,(r')G,.,vj,(r),
(4.41a),
Q(va)(j, l, r, r') = ~ ' Wv(a)(j, l,j', l', r, r')F,.,vj,(r')G,.,vj.(r)
+ ~,' W(va)(j, l, j', 2j'-l', r, r')G,..vr(r')F,.,rj,(r),
(4.41b),
Wu(a)(j, l, j', l', r, r') = { j' + ½- 2 ( j ' - l')I'}{ ( l - j + k)v( t - 1 , l', r, r')
+0-1+½)v(t+ 1, r, r, r')}r'* + 2(j'-l'){v°)(l, l', r, r ' ) - ( 1 - j +½)v(1)(1-1, l ' - l, r, r') - ( j - l +½)vO)(l + 1, l'-- l, r, r')}r '4, / ' + 1+-i v ( l , l ' + l , r , r ' ) + - - v2r+1 (l,l'-l,r,r') w43 (j, l,j,"' t,' r , / ) = ( 2 ; + 0 tt2/
(4,42a} r '4
+ 4 ( j ' - l ' ) K ( j , l){l'v(l-1, l', r, r')-l'v(l+ l, l', r, r') +vO)(I-1, I'-1, r, r ' ) - v ° ) ( l + l , l ' - l , r, r')}r'4--wda)(j, l,j', l', r, r'). (4.42b} The angular integration of R (4) is carried out in the same way, and its spin-angle dependence is the same as in eq. (4.22). If its radial parts are denoted by U(F4), V(F4), U(4) they are given by the integrals of the same form as eqs. (4.23) (a), (b), G and rrA4) G (e) and (d), respectively. The functions Q~ and O~v in the latter integrands are substituted by Q(u4) and Q(v4) as follows: Q(u4)(j, l, r, r') = Z ' Wd4)(J, l, j', l', r, r')Fi.,rj,(r')V,.,vj.(r )
+ Y,' W(u4)(j, I, j', 2j'-l', r, r')G,.,r~,(r')G,.,vr(r),
(4.43a}
SHELL MODEL
637
Q~>(j, I,~,/) = y: w~*)(:, t,;, v, ~,/)F,..,.A/)a,,,..A,.) + •' W(v*)(j, l, j', 2:'--l', r, r')G,,,vs,(r')F,,,t,s,(r ),
(4.43b)
where Wu(4)(j, l, j', l', r, r')
= {j'+½-2(j'-l')l'}
tl+l . 121~+1 v ( 1 - 1 ' I ' ' r ' r ' ) +
I
21+1-- v ( l + l , l ' , r , r ' ) t
t
rr'a
+ 2 ( j ' - - / ' ) {v(~)(/, l', r, r')-- l + l v(l)(/_l ' l'--l, r, r') 2l+1 l v ( ~ ( / + l ' 1'--1, r, r')} rr '3, 2l+1 Wv(4)(j, l, 3,' ' I, r, r') = (2j'+ 1) I~ 2~'+l /'~
~(l,r+l,~,/)+
+ 4 ( j ' - l ' ) { 2 ( 2 j - 1 ) + l} K(j, 2 j - l ) { I ' v ( I - 1 , 2/+1
(4.44a)
"
21' + 1 ~(/,v-l,~,/)
} ,/~
l', r, r')-I'v(l+ l, I', r, r')
+v(1)(l-1, 1'-1, r, r')-v(*)(/+ 1, l ' - l , r, r')}rr'3-W(u4)(j, l,j', l', r, r').(4.44b) The exchange field due to the "r independent interaction is thus given by
UcxO,,,j,, = { U~(r) + V:(r)2a • x/r}F,n,s(r)~,: m + { U~(r) + l/~(r)2a • x/r}G,,~s(r)(2a" x/r)(~lsm,
(4.45)
where the scalar radial parts U~ and V~ are given by U~(r)
~ + Uv(r), t = U ~ ( r ) + Uv(r)
(4.46a)
V~(r) = V~(r) + V~(r) + V~(r),
(4.46b)
and U~ and V~ are given by the equations of the same type if suffices F are replaced with G. The radial parts with a superscript t are given by
Urn(r) = - U(vl)(r) + U(vZ)(r)+ U(vS)(r)--
U(v4)(r),
(4.47)
and so on. These are the contributions from the third term of eq. (4.6). If the two-nucleon interaction is given by the second term in eq. (3.4), eq. (4.5) is modified as follows:
U~xx~(2)~,,tjm(2 ) = Z,(2)Uex~/,nU,,(2),
(4.48a)
Uex~,o,J,,(2) = Z ( 2 - 6a,) Y',' (O,,,,z,j,,,,(1), B(1, 2)~),,~Sm(1))O,,,,,,S,,,,(Z)d(x, ), (4.488) z"
d
where ~ ' denotes the sum extending over all occupied n', l', j', m' states, and B(1, 2) stands for B,(x,, x2, a,, a2). Therefore the spin-angle dependence of this exchange
638
G. A R A K I
field is the same as eq. (4.45). If we denote its scalar radial functions by U~(r), V~(r), i i U~(r) and VG(r) they are given by the integrals of the same form as UF(r), VF(r), ui~(r) and V~(r), i respectively. Besides the appropriate substitutions of w, u and v, we have merely to modify the Q functions, that is, the isospin quantum number t of F~,,l,j,(r') and G,,,vy(r' ) in them should be changed into t' and the summation symbol ~ ' should be replaced with Z , ( 2 - 6 , ' ) Z ' , where Z ' means the sum extending over all occupied n', l', j ' shells as before. The contribution from the Coulomb interaction is given by
U~ Z, ~.tj~ = zp Z,(U~(r) + V;(r)2~ • x / r } F , . t j ( r ) ~ t j ~ + zv Z,{ U~(r) + V~(r)2~ • x/r} G,,u(r)(2~r • x/r)~u, . .
(4.49)
The radial functions U~(r) etc. are given by eqs. (4.23)-(4.26) where w(l, l', r, r') and w°)(l, l', r, r') are substituted by q~(l, l', r, r') and @1)(1, l', r, r'), respectively. The latter two are the coefficients of the Legendre series pertaining to e2/Ix-x'l [see eqs. (4.15) and (4.16)]. 5. Discussion
From the calculations in the preceding sections the Hartree and exchange fields in the closed-shell nucleus are given by
UHZ, ~ltnljm = Zt{ O(P)2Fmlj(F)q- g(F)Gtnlj(F)}~ljm + z.{ v(~)~'..,(,) + v ( ~ ) G , ( ~ ) } ( ~
• x/r)%~,
(5.a)
UexZ, l[ImUm = Z,{ UF(V)FmU(r) q- I/G(r)G,nlj(r) }fl#Urn q- Z,{ VF(F)F, nlj(F) Jy UG(F)Gmlj(F)}(~ " X/F)~ljm,
(5.2)
v(r) = vi(r)+, G(~)+ ~+~, vo(~),
(5.3a)
v(~) = v # ) + ~v~(r),
(5.3b)
UF(r) = U~(r) + U~(r) + 6 +t, U~(r),
(5.4a)
VF(r) = l/~(r) + V~(r) + 6 +1, V~(r),
(5.4b)
where
= Ue(r) + Ue(r) + 6 +t, UG(r),
Vc(r) = G ( r ) + G ( r ) + 5 +1, G(r).
(5.4c) (5.4d)
Since spin-orbital harmonics ~tjm and (2a'x/r)qhjm are linearly independent, the Hartree-Fock equation (3.1) resolves itself into simultaneous equations for the radial functions as follows:
(~+ v - u~-G,)F,°,+(v- V~)G,,j = 0,
(5.5a)
(T2j_t+ U - Ue-E,,tj)G,,tj+(V-l/'v)F,,q = 0,
(5.58 /
SHELL MODEL
639
where
h2 (~ T~
2M
62
/(/+1)}
dr 2
r2
(5.6) "
Therefore we see that the spin-angle dependence of the one-nucleon states assumed in eq. (3.5) is consistent with the two-nucleon interaction which has been generally believed to exist between nucleons. The fact that the Hartree and exchange fields are isospin-dependent shows that the proton radial function of the nlj shell is different from the neutron radial function of the same shell in accordance with our initial assumption. If these functions are the same, the contributions to U, and V, from protons and neutrons in the same shell cancel out. In the case of the non-closed shell nucleus the Hartree-Fock equation should be modified. In this case the Lagrange multipliers Ev, cannot be completely diagonal, and the Hartree-Fock equation takes the form
I
I
-- - - A + U n ( q ) - Uex(q) qSv(q) = Z 2M
E,~u(q)"
(5.7)
The Lagrange multipliers are the substitutes for the orthonormality condition imposed on unknown one-nucleon states. There is no need to impose the orthogonality condition upon one-nucleon states of different zjm because isospin and spin-orbital harmonics X~bu,~ are orthogonal. Consequently the right side of eq. (5.7) contains no non-diagonal element connecting such states. If an tj shell is completely filled the non-diagonal elements of E ~ connecting different states in this shell can be removed by a unitary transformation of one-nucleon states in this shell, as is well-known, without changing the total nuclear state. If the shell is non-closed this is impossible. Therefore q~v and all q5u in eq. (5.7) belong to the same tim and their isospin and spinangle dependences are the same. The non-diagonal Lagrange multipliers can be represented by integrals containing radial parts of wave functions and interactions 3). Hence the non-diagonal part on the right of eq. (5.7) can be considered as a part of the exchange field, and the existence of the non-diagonal Lagrange multipliers in the case of the non-closed-shell nucleus makes no essential difference as compared with the case of the closed-shell nucleus. An important difference between the closed-shell and non-closed shell nuclei is in the spin-angle dependence of the Hartree and exchange fields. On account of the fact that the addition theorems for the spin-orbital harmonics cannot be applied to the summation over non-closed shells, the spin-angle dependence of the fields coming from the non-closed shells is generally not the same as those from the closed shells. This is the situation which is similar to the one encountered in the case of the central interaction between atomic electrons. Hartree's procedure to manage the situation was to substitute the exact fields by their angular averages. We cannot make use of this procedure here. An appropriate averaging process in the present case is to modify the summation over m of the non-closed shell. If the shell is specified by
~40
G. ARAKI
n lj and contains Nj nucleons, we replace ~mwith {Nfl(2j+ 1)}y'~= _j. Then we have the fields of the same spin-angle dependence as those in the case of the closed-shell nucleus. The effect of the deviation of the exact fields from such averages can be considered as the angular-correlation effect, and it can be treated by the weU-known method of correlation functions or of configuration interactions 4). In order to understand the situation we consider the simplest configuration which consists of a closed-shell core and one nucleon outside it, the latter being referred to as an odd nucleon. There is no deviation of the Hartree and exchange fields from their averages for the odd nucleon, because the contributions to both fields f r o m the odd nucleon cancel out and there remain only the contributions from the core. On the contrary, for nucleons in the core, the contributions from the odd nucleon make the fields deviate from their averages. This deviation gives rise to the deformation of the core and this deformation produces the deviation of the fields for the odd nucleon as a secondary effect. Such a phenomenon has been observed in the electron problem of helium 4, s). In conclusion the author wishes to express his hearty thanks to members of the theoretical seminar in our department for their valuable discussions. The present p a p e r is dedicated to Professor Masao Kotani on the occasion of his sixtieth birthday.
References 1) 2) 3) 4) 5)
G. Araki, J. Phys. Soc. Japan 19 (1964) 1514 J. P. Amiet and P. Huguenin, Nuclear Physics 46 (1963) 171 J. C. Slater, Quantum theory of atomic structure, Vol. 2 (McGraw-Hill, New York, 1960) p. 23 G. Araki, Festskrift til Egil hyl!eraas (I Kommisjon hos F. Bruns Bokhandel, Trondheim, 1958) G. Araki, Prog. Theor. Plays. 16 (1956) 197
be reproduced by photoprint or mlerofilm without written permission from the publisher
SHELL MODEL
WITH ~. x POTENTIAL
GENTARO ARAKI
Department of Nuclear Engineering, Kyoto University, Yosida, Kyoto Received 20 September 1965 The one-nucleon state of a definite j-value is constructed as a linear combination of two states which belong to the azimuthal q u a n t u m numbers j--½ and j+~-, respectively. The invariance of the state for space inversion is guaranteed by introducing a unit pseudoscalar. The Hartree-Fock equation satisfied by such one-nucleon states and the isospin and spin-angle dependence of the Hartree and exchange fields are discussed on the assumption that the twonucleon interaction consists of three parts, i.e. the central, the ~ • ~' and the tensor-force type. It is shown that the Hartree-Fock equation reduces to a form of eigenvalue equation of the Hamiltonian with central plus o • x type potentials. Addition theorems for spin-orbital harmonics are obtained in the lira scheme.
Abstract:
1. Introduction
In a previous note 1) it was shown that the spectrum of the one-nucleon Hamiltonian with a a • x type potential is an inverted doublet, and it was suggested that the nuclear shell structure may be interpreted by this spectrum. The eigenstate of this Hamiltonian consists of two parts belonging to different azimuthal quantum numbers and opposite parities. In order to maintain invariance of the total nuclear Hamiltonian under space inversion, the total nuclear state was assumed to be represented by a projection of the Slater determinant, composed of such one-nucleon states of mixed parity, onto a state of a definite parity 1, 2). This assumption deprives the Hartree-Fock scheme of its simplicity. Nevertheless, one may wish to look for its theoretical foundation in this scheme. The undesirable feature arises from the fact that the a • x type potential is non-invariant under space inversion. In the present paper the formalism is modified to remove this undesirable complication by introducing an invariant o- • x type potential. The formal consistency of the basic assumption is discussed in the Hartree-Fock scheme. It is shown that the Hartree-Fock equation reduces to a form of eigenvalue equation of the Hamiltonian with a central type plus a a • x type potential. 2. One-Nucleon States and Levels We consider a free-particle model of nuclei in which one-nucleon states are eigenstates of the Hamiltonian H
=
-
hz -A+
U(r)+V(r)2
2M 625
a" x, r
(2.1)
626
G. ARAKI
where M is the nucleon mass, x and r are the position vector and its magnitude, respectively, ½ha is the nucleon spin angular momentum, U(r) and V(r) are scalar radial functions and ~, is a unit pseudoscalar. The last quantity guarantees the invariance of the Hamiltonian with respect to space inversion. If ~ j , , denotes the normalized simultaneous one-nucleon eigenstate of the squares of the orbital and total angular momenta and of a component of the latter, the spin-angle dependence of the normalized eigenstate of H is given by 1) O = { F ( r ) + G(r);.,, • x/r)~,,~,
(2.2)
where F(r) and G(r) are radial scalar functions. We denote the norms of the two orthogonal parts of this state by ci and c2, respectively, and their normalized parts by ~l and ~2: clO1 = F(r)~bU~,
C2O 2 = G(r)(a" x/r)~tim.
(2.3)
We have then
'k
.
. cl¢q+~c2'kz,
. 1101112.
ii~zll 2
11@112
c,+c 2z = 1. z
(2.4)
As was shown in the previous note x), ~2 belongs to the azimuthal quantum number' 2 j - I and the parity - ( - 1 ) k Nevertheless ~ belongs to the definite parity ( - 1 ) k because ,~a • x is invariant with respect to space inversion. The eigenvalue of H and the corresponding eigenstate are respectively given by E (+) and ~(+) or E (-) and ~b(-) according to E 1 > E 2 or E, < E2, where these quantities are defined as follows: E (+) = E l + 5 ,
~(+) = ( r O l + ~ , 2 ) / N
(El > E2),
(2.5)
E (-) = E i - 6 ,
0 (-) = (rO~-X~0d/N
(E~ < E2),
(2.6)
Ei = (~l, H ~ l ) ,
E2 = (~t2, H~t2),
~ = (Ol, HO2),
(2.7)
= ½IE1-E2I ,
= (g2--[-~2)~--/3,
N = (72+62)~.
(2.8)
The scalar radial parts F(r) and G(r) can be taken to be real, consequently y is real. If the effect of V(r) is not too large, E 1 is higher than E2 when l is greater than 2 j - l (that is, when l = j + ½, 2 j - l = l - 1) and El is lower than E2 when l is smaller than 2 j - l (that is, when l = j - ½, 2 j - l = l + i). Hence the spectrum of one-nucleon levels is an inverted doublet as was explained in the previous note 1). In the present case the total nuclear state, which is represented by the Slater determinant composed of onenucleon states of the above mentioned type; belongs to a definite parity. If the oddnucleon state of an odd-mass nucleus is given by c1~/'1+ 2e2~/'2, the magnetic moment 2 +c2#2, 2 # of this nucleus is given by # = c~#~ where #1 and #2 are two values of the magnetic moment in the Schmidt model corresponding to l = j_+½. 3. Hartree Field
In order to examine whether the one-nucleon Hamiltonian (2.1) is compatible with
SHELL MODEL
627
two-nucleon interactions or not, we first consider a closed-shell nucleus containing A nucleons in q~, q52. . . . . q5A states in the Hartree-Fock scheme. The Hartree-Fock equations for the normalized one-nucleon states ~bv (v = 1, 2 . . . . . A) are given by
{ - hZA/(2M) + Uri(q) - Uex(q))(~,,(q) = E,, O,,(q), A
(3.1)
A
Un(q) = 2 V#.~(q),
Uex(q)~,,(q) = 2 Vu~(q)~u(q),
g=l
(3.2)
/Z=I
Vu,,(q) = f Ou(q '
q')O,(q')dq',
(3.3)
where q stands for one-nucleon coordinates including space, spin and isospin variables, B(q, q') denotes the two-nucleon interaction and E~ is the diagonal element of the Lagrange multipliers (the Hermitian matrix). We assume that the isospin dependence of the two-nucleon interaction is given by
B(q, q') = B(x, x', ~, ~')+'r,. ,~'B,~(x, x', ~, a')+zpz've2/lx-x'[,
(3.4)
where ~ denotes the isospin operator whose three components satisfy the same algebraic relations as those satisfied by the components of o'. The last term is the Coulomb interaction between protons where Zp denotes the projection operator to the proton state. We further assume that B and B~ each contain central, a • ~' and tensorforce terms. We denote the normalized isospin part of ~b~ by Z, and the normalized spin-orbital part by 0~,~j,~, where t is the eigenvalue of %. In accordance with eq. (2.2), the latter part is given by
~.~nt/m = {F~ntj(r) + Gtnlj(r)2a" x/r}~tjm"
(3.5)
We now examine the spin-angle dependence of the Hartree field Un. For this purpose we assume the a dependence of the two-nucleon interactions as follows:
B(x, x', ,~, ,¢) = w(Ix-x'l)+,7" ,¢u(lx-x'l)+(,~" v)(~' • V ' ) K i x - x ' l ) .
(3.6)
As is well-known, the last term is a sum of a • a' and tensor-force type interactions. We first carry out a sum of Vuu with respect to m. Corresponding to the three parts of eq. (3.6) we have to calculate the three sums
V~ =
x' O,,Zjm(X')>w(lx- x'l)d(x'),
(3.7)
m
ga =
~ " m
'
VT = (o' " V )
(3.8)
j
'
'
.V'v([x-x'l)d(x'),
(3.9)
m
where the isospin quantum numbers are omitted for simplicity, < > means the Hermitian inner product of spin parts and d(x') the volume element in the x' space.
628
G. ARAKI
To carry out the sums we first look for addition theorems for ~tim. A set of 2j+ 1 spin-orbitals ~tj,, with definite l and j is orthonormal in any system of space coordinates. Therefore the transformation of these 2j+ 1 spin-orbitals for the space rotation is unitary and J
E
<~tj,~, ~tjm> = C(l, 1),
(3.10a)
in= - - j
m= - j
<'#,in, ~2j--Hm> = ~
m= - j
eljm
r
~,jm
= C(1, 2j--l)
(3.10b)
are consequently invariant for the space rotation, that is, C(l, l') is independent of space coordinates. In the latter equation we have made use of the relation
¢2j-u .
=
(3.11)
The constant C(l, l') can be determined by integrating both sides of eq. (3.10) with respect to angle variables. We have then J
2 <~b,j~, #rjm> = 3u. 2j+____l,
(l' = 1, 2 j - l ) .
(3.12)
4re
-j
m=
Next we consider the sum ~,,(#u,,, ¢r#rJ,*>" This is a homogeneous polynomial in
x/r, y/r and z/r because ~0Umis a spherical surface harmonic. On the other hand, this is a vector for the space rotation. Therefore this sum is equal to Ct(l, l')x/r, where Ct(l, l') is independent of space coordinates, and the scalar product of the sum by x/r is equal to Cl(l, l'). This gives just the same equation as eq. (3.10) [note the relation (3.11)]. The constant Ct(/, l') can be determined by comparing this equation with eq. (3.12). We have thus
J <~)ljm, ff~Dl'jm> = (1_ flu,) 2j+ 1 X, 4~ r
Z
(l' = l, 2j-- O"
(3.13)
m= --j
By making use of eqs. (3.12) and (3.13) we can carry out the summation of integrands in eqs. (3.7)-(3.9) as follows: m= --j
2j+l
{F.tj(r) + G,u(r)},
J
2j+l
Z
m= -j
4re
J
2j+l
,.= - j
(3.14)
47z
<~n,jm, a~n,jm> " V = - -
4re
x
Fnlj(r)Gnlj(r))c - ,
(3.15)
r
3
F,tj(r)G.tj(r)2 - - . ar
(3.16)
We denote x • x'/rr' by ~ and expand the radial parts of the two-nucleon interactions into the Legendre series in {. For example, we have
u(lx-x'[) = ~ (2k+ 1)Uk(r, r')Pk((), k=O
(3.17)
SHELL MODEL
629
where Pk(~) is the Legendre polynomial of degree k in ~. The angular integrations of the integrals (3.7)-(3.9) can be easily carried out by making use of the orthogonality of spherical surface harmonics. The results are given by 2 t )+ G,a.i(r 2 , )}wo(r, r')r'Zdr ', vo = (2j+1) I ; {F,,lj(r
(3.18)
V~ = ). a ' x 2(2j+ 1) I®Fntj(r')G, tj(r')ux(r, r')r'2dr ',
(3.19)
r
./o
VT = 2 ~r " x 2(2/+1)
r
F.tj(r ,)G.lj(r , )r ,2 - - vo(r, r')dr', OrOr' "
fo °
(3.20)
where wo(r, r') and vo(r, r') are the zero degree term in the Legendre series of w ( l x - x ' l ) and v(Ix-x'l), respectively, [see eq. (3.17)]. By summing the last three expressions we have the Hartree potential due to B(x, x', ~r, ~r') in the form O H ~--- UI(F)--~- V / ( F ) ~ • x/r. (3.21) The radial functions are given by
Ui(r ) = Vi(r) =
fo
{pp(r')+ pn(r')}wo(r, r')dr',
{qp(r')+~/n(r')} ul(r, r')+
02 vo(r, r') dr', c~r~r'
(3.22a) (3.22b)
pp(r) = ~ (2j + 1){Fzlj(r) + Gz,zj(r)}r 2,
(3.23 a)
%(r) = ~ 2(2j+ 1)F, tj(r)G,,ti(r)r z,
(3.23b)
where the summations extend over all occupied proton shells and pn(r) and t/,(r) are the similar sums for occupied neutron shells t If we take into account the v-dependent terms of eq. (3.4), the Hartree potential is modified as follows:
UH = U,(r)+ zz U~(r)+zp U¢(r)+ {Vi(r)+ z 3 V~(r))2a • x/r,
(3.24)
where U~(r) and V~(r) are given by similar integrals, including radial parts of B~, as eqs. (3.22a) and (b), respectively, except that the density functions in their integrands are replaced by p p ( r ) - pn(r) and % ( 0 - t/n(r). (We assume that the Proton state belongs to the plus eigenvalue of v3. ) The contribution from the Coulomb term is given by
Uo(r) =
50
pp(r')(eZ/rg)dr',
(3.25)
where rg denotes the greater one of r and r'. ¢ E q . (3.22b) as w e l l as t h e r e s u l t o b t a i n e d i n t h e n e x t s e c t i o n s h o w t h a t the a r g u m e n t g i v e a b y
Amiet and Huguenin 2) contains a mistake.
630
G. ARAKI
4. Exchange Field In this section we examine the spin-angle dependence of the exchange field Uex. For this purpose we have to look for addition theorems for ~b~j,,with different arguments. This spin-orbital function is referred to as a spin-orbital harmonic. Let YZm" be the simultaneous normalized eigenstate of the square of the orbital momentum and of its component. The spin-orbital harmonics ~Ijm and ~ u - l m (J = l+½) are linear combinations of aY, ,,_~ and flY, ,,+~ with the given azimuthal quantum number l, where e and fi are spin eigenstates corresponding to spin magnetic quantum numbers +1 and - 1 , respectively. The systems {eY*m', flY~m'} and {~*Sm,~U-lm} are both orthonormal, and the transformation between them is consequently unitary. Therefore we have j
j--1
m=-j
m=--(j--
1)
I
= Z {c~*(1)Y~*(1)c~(2)):},,(2)+fl*(1)Yz*(l)fi(2)Y~m(2)}'
(4.1)
m = --l
wherej = l+½. The sum on the right can be carried out by making use of the wellknown addition theorem for the zonal harmonic, and the right-hand side reduces to {(2/+ 1)/(4rc)}Pl((,2)6~(1, 2), where 6~ and ~,2 are defined by fi~(1, 2) = e*(1)e(2)+fi*(1)fi(2),
(12
=
Xt"
x2/rlr2.
(4.2)
The first and the second sums in the left-hand side of eq. (4.1) can be separated by projecting the left-hand side o n j = l+½ a n d j ' = l - ½ states, respectively. The corresponding projection operators are given by (l+ 1 +0-2 "L2)/(21+ 1) and ( l - - o " 2 " L2)/(21+ 1), respectively, where hL is the one-particle orbital angular momentum. Since / + 1 = j + ½ , 2 ( j - l ) = +1, l = j ' + ½ , and 2 ( j ' - l ) = - 1 , we have J ~jm(1)~tj.~(2) = {j+½+2(j-I)0-2"L2}PI(~12)6~,(1 , 2)/4rc.
(4.3)
m = -j
if we multiply both sides by 0-2 "x2/r2 from the left and make use of eq. (3.11), we have m = --j
~;.m(1)~2j_Um(2) __ 0-2 "X2 {j+½+2(j--l)0-2" L2}P,((,2) 6~'(1' 2) r2 4re
(4.4)
The last two identities together with eq. (3.12) are addition theorems for spin-orbital harmonics. We first calculate the contribution to Uex from the z independent terms of the twonucleon interaction. In this case the exchange field is given by
U~xZ,(2)~.,tjm(2 ) = Z,(2)U~x~,,,tj,,,(2), Uext]lmUm(2)
=
Z ' t'(@m,rj,m,(1), B(I, 2)0,,um(1)>~b,,,rj,,,,(2)d(xl ), d
(4.5a) (4.5b)
SHELL MODEL
631
where ~ ' denotes the sum extending over all occupied n', l', j', m' states. All spinorbitals in these equations belong to the same isospin quantum number t, on account of the orthonormality of isospin states. Therefore the interaction between protons and neutrons are not involved in this exchange field. In order to facilitate the calculation we write the two-nucleon interaction in the form
B(xl, xz,
al, az) = w(Ix~21)+,rl
" ,r2u(Ix121)+
(,rl " x l ~ ) ( , r 2 •
xlz)v(Ixlzl),
(4.6)
where Xla stands for xl - x 2. The third term is a sum of a 1 • a 2 and tensor-force type interactions. Therefore the functions u and v are not the same as those in eq. (3.6). Corresponding to three parts of this interaction we have to calculate the three integrals Rc = f ~ ' @',Ti,m,(1), ~.ljm(1))W(lXl 2[)~'nTj'm'(2)d(xl),
(4.7)
R~ = f z ' (tPn'l'J'm'(1)' ~1 ~lnljm(1)) "aEO,'Vj'm'(Z)u(lxlzl)d(xl),
(4.8)
RT = f Z' (~tn't'j'm'(l)' al " Xl2tPnljm(1))~r2 " Xlz~lln't'J'm'(Z)v(Ix121)d(Xl),
(4.9)
where the isospin quantum numbers are omitted for simplicity. By making use of the addition theorems (4.3) and (4.4), the summation with respect to m' in the integrand of the first integral can be carried out as follows: j,
£ m' = -- j r
, ~n,rj,m,(1)~tn,r j,m,(2) = [D I { j' + ½+ 2(j'-- I')a 2 " L2}Pv((a2)
+D2{J' + ½ - 2 ( j ' - l ' ) a z • L2}Pzj,_r(~12)]6,(1, 2)/4rc, (4.10) where
D1 = F, T2,(r~){Fn,rj,(r2)+ G,,rj,(r2)2a2 " x2/r2},
(4.11a)
Dz = G,,t,j,(ri){G,,rj,(r2)+
(4.11b)
r,,ra,(rz)2a2
" ~;r~}.
If we carry out the differentiation, we have
0"2" L2Pt((,2 ) = i(x2 x a2). x~ P;((~2), /'2
(4.12)
rl
whereP;(() denotes the derivative. The function of Ix 1 -x21 is the function of r l, r 2 and ~lz. Therefore we can expand the function of Ixl-x21 multiplied by P l ( ~ l z ) o r P'l(~12)in the Legendre series of (tz. For example, we have W(IXl
--X2])Pl(~12)
=
~ (2k+
1)Pk(~z)w(k, l),
(4.13)
k=0
w(lxl- x2l)P;((~z) = ~ (2k+ 1)Pk(~2)w(~)(k, I), k=O
(4.14)
632
G. ARAKI
where the coefficients are given by
w(k, l, rl, r2) = ½
Pk((12)w(lxl--x21)P,((iz)d(12,
(4.15)
-1
win(< t, rl, "2) -- ½
-xd)P glz)d l .
(4.16)
In the series (4.13) and (4.14) the arguments q and r 2 are omitted from their coefficients in order to avoid unnecessary complications. If we substitute appropriate parts in the integrand of eq. (4.7) with expressions (4.10), (4.12), (4.13) and (4.14), the integrand turns out to be the sum of two kinds of terms; one is the product of the Legendre series and 0,~jm(1), and the other is the product of the Legendre series and (xl/r~)Ontjm(1). The angular integration of the former can easily be carried out by making use of the delta function property of Pk(~12)6~(1, 2) as well as the orthogonality of spherical surface harmonics: co
+ 1 Pk(~,2)(6~(1, 2), ~,,jm(1)>dco, ~ ow( k' l , ).J( 2 k4=
= {w(l, l')F,,zj(1) + w ( 2 j - I, l')G, tj(1)2a2" xz/r2}~tjm(2),
(4.17)
where do), denotes the solid-angle part of the volume element d(xl). The angular integration of the second part containing (Xl/q)~k, tjm(1) can be carried out by a projection technique. The angular part of (Xl/q)~tjm(1) consists of spherical surface harmonics of degrees l - 1 and l+ 1. They can be separated by the projection operators defined by E}_)(1) = (l+ 1)(1+2)-L 2 2(21+1) '
(4.18a)
E~+)(1) =/,21 - ( l - 1)/ 2(2/+1)
(4.18b)
The sum of these operators is equal to unity and we have
(xl/rl)~ljm(1) = {E}-)(1)+ E} +)(1)}(xl/rl)~jm(1).
(4.19)
By using this identity and also the above-mentioned delta function property, we have }~ w(a)(k, l ' ) ~ 2 k + 1 Pk(~12)(J~(1, 2), O,tj,,(1))(xl/rl)de), k=o
d
4~
"" (1)( I -- 1 , l')E}-l(2)+wO)(l+l, l')E}+)(2)}(x2/r2)~tjm(2) = Fntj(1){w (+) + G,tj(1){wO)(2j- l - 1,/')E(z7 )_,(2)+ w(1)(2j- l+ 1, 1 )Eej-z(2)}(xz/re) x (2trz " x2/r2)~btjm(2 ). (4.20) ,
Then the integrand of eq. (4.7) contains the operators which are given by the left side of
SHELL MODEL
633
the identities
i(x2 x a2) E}+)(2). x2 = T az'__L2 r2 r2 21 + 1
(4.21)
The left side can easily be changed into the right side. Since ~bu.m(2) is an eigenstate of the operators on the right, these operators reduce to mere numbers. Thus R~ takes the form
Ro = { U~(r) + V~(r)2a " x/r}F,,tj(r)~lj m
+ (v~(r) + v~(r)2a, x/r)G,.,j(r)(~a • x/~)~,j~,
(4.22)
where
U}(r) =
Q~(j, l, r, r')F~.u(r')dr'/F~.u(r ),
(4.23a)
V~(r) =
Q~(j, l, r, r')F~tj(r')dr'/F,.lj(r),
(4.23b)
U~(r) =
v;(r) =
f o°Q~(
fo
j, 2 j - l, r, r')G,.u(r')dr'/G,.u(r ),
e~(J, 2 j - l , ~, r')a,.,j(r')dr'/G,.,j(r),
(4.23c) (4.23d)
Q~j(j, l, r, r') = Z' WC(J, l, j', l', r, r')F~.,rj,(r')F,.,vj,(r )
+ Z' W°(/, I, £, 2 ; ' - r , r, r')C,...Ar')~,.,.Ar), Q.~(j, t, r, r') = Z' w°(J, I, j', r, r, r')F,...Ar')a,.,.AO + Z' We(J, l, j', 2j'-l', r, r')G,.,rj,(r')F~..rj,(r),
(4.24a)
(4.24b)
w°(:, I, j', r, r, r') = (j'+ ½)w(t, r, r, r')r '2 +2(j'-I')K(j,
~ t2 l){w°)(1-1, l,t r, r ' ) - w ( 1 ) ( l + l , l t, r, r t ')~r ,
K(j, l) = { j ( j + 1 ) - / ( / + 1)-4s-}/(2/+ 1).
(4.25) (4.26)
The last quantity is the eigenvalue of the operator (4.21) of plus sign and ~ ' in eqs. (4.24) the sum extending over all occupied n', l', j ' shells. The second integral R~ can be obtained from Re by replacing ~,u=(1) in its integrand with a l e , urn(1) and multiplying the result by a2 from the left. Then the integrand contains a 2 " Dia 2 and a2 • D i ( a 2 " L 2 ) a 2 (i = 1, 2). We can eliminate two a2 from these expressions by using the identities
a" (a" a)a = --a" a, a" (b + ca" x)(a" L)a = - (b+ ca" x ) ( a . L),
(4.27) (4.28)
634
G. ARAKI
where a, b and e commute with a. Thus R~ is reduced to R~ = { U~-(r) + V [ ( r ) 2 a • x/r}F,,,j(r)cPtj
m+ { U~(r) +
V~(r)2~ • x/r}
G,,lj(r)(2a. xlr)e&j~, (4.29)
a t") and Vg(r) denote the radial integrals which are obtained where U~(r), V~(r), U~( from integrals (4.23a)-(4.23d) by replacing Q~(j, l, r, r') and Q~(j, l, r, r') in their integrands with the following Q(~(j, l, r, r') and Q~(j, l, r, r'), respectively:
Q[j(j, l, r, r') = Z ' W(j(j, l, j', l', r, r')F,,,rj,(r')F,,,rj,(r) + Z ' W(j(j, l, j', 2 j ' - i ' , r, r')G,,,rj,(r')G,,,vj,(r),
(4.30a)
Q~(j, l, r, r') = ~ ' W~(j, l, j', l', r, r')F,,,,rj,(r')G,,,rj,(r) (4.30b)
+ ~,' We(j, l, j', 2 j ' - l ' , r, r')Gm,rj,(r')F,n,rj,(r), W(j(j, l,j', l', r, r') = 3(j' +½)u(l, l', r, r')r 'z -2(j'-l')K(j, l){u(1)(l-1, l', r, r')-u(1)(l+ i, l', r, r')}r '2,
(4.31a)
W(:(j, l,j', l', r, r') = W~:(j, l,j', l', r, r')-4(j'+½)u(l,
(4.31b)
l', r, r')r '2,
where u(l, l', r, r') and u°)(l, l', r, r') are the coefficients of the Legendre series pertaining to u(Ix-x'l) [see eqs. (4.15) and (4.16)]. We examine the spin-angle dependence of R T by dividing the corresponding interaction into four parts { - ( ~ 1 " xl)(~2 • x 2 ) + ( , , ~ • x ~ ) ( ~ , x~) "t- (O" 1 " X l ) ( O " 2 " X l ) - - ( l ~ "
1 " X2)(a
2 "
xl)}v([xlzl).
(4.32)
The contributions to RT from the respective parts are denoted by R (~), R (2), R (3) and R (4). If the functions ~,Um(1), Cn'Vj',,'(2) and w(lx12I) in eq. (4.7) are substituted by (,tal"xl/rx)~k.u,,(1), (2az'xz/r2)~,,vj,m,(2), and rlrzv(Ix12]), respectively, it turns out to be R u). The former two substitutions are equivalent to interchanges F,,j(rl) ~--- G,u(rl) and F,,vj,(rz) ~- G,,w,(r2) in ~,Um(1) and ~.,Vy,m,(2) of eq. (4.7), respectively. Therefore the spin-angle dependence of R (1) is the same as eq. (4.22). We have only to substitute Uxg, V~, U~ and V~ by the following U(F~), V(F~), "~orr
U(v~)(r) =
L oQ(ul)(j, 2 j - l ,
V(vl)(r) = U(1)z G t r)~ =
r, r')F,,zj(r')dr'/F,,zj(r),
(4.33a)
Q(vl)(j, 2 j - l , r, r')F~,lj(r')dr'/F,,lj(r),
(4.33b)
L oQ(ul)(j,. l, r, r ,)G,~,j(r ,)dr/G,~u(r , ),
(4.33c)
SHELL MODEL
V(~l)(r) = I, r,
r')
f;
= Z'
635
Q(vi)(j, I, r, r')G~n,j(r')dr'/G,nzi(r), 1,j', r, r,
(4.33d)
)
+ 2 ' W(1)(J, l, j', 2 j ' - l', r, r')G,n,vj,(r')G,~,vj,(r),
(4.34a)
Q(vl)(j, l, r, r') = 2 ' W°)(J, l, j', l', r, r')F,n,rj,(r')G,~,ri,(r) + ~ ' W(1)(j, l, j', 2 j ' - l ' , r, r')G,,,vi,(r')F,,,vi,(r),
(4.34b)
WO)(j, l,j', l', r, r') = (j'+½)v(l, I', r, r')rr '3 +2(j'-I')K(j,
l){vO)(l-1, l', r, r ' ) - v ° ) ( l + l ,
l', r, r')}rr '3,
(4.35)
where v(l, l', r, r') and v(1)(l, l', r, r') are the coefficients of the Legendre series pertaining to v(Ix-x'l) [see eqs. (4.15) and (4.16)]. The integral R (2) can be obtained by substituting 0,1i,~(1), ~k~,Vj,m,(2) and w([xl21) in eq. (4.7) with (el • x2/r2)O, lim(1), (a2" x2/r2)O,,Vi,m,(2) and r22v([xi21), respectively. The first substitution is equivalent to multiplying eqs. (4.17)and (4.20)by a 2 " x e / r 2 from the left. The second substitution is equivalent to multiplying eq. (4.7) by a 2 • x2/r 2 from the left. These two multipliers reduce to unity if the multiplicant does not contain x2 x a 2 in its factor. If the multiplicant contains x 2 x a 2, one of them appears on the left of x z x a z and the other between x 2 x a2 and E}-+)(2) of the products such as the left side of eq. (4.21). We can easily verify the anticommutation relation
(x x a)(a " x) = - (a " x)(x x a).
(4.36)
Consequently the effect of (a t • x2/rz)(a 2 • xz/r2) in the interaction is merely to change the signs of K(j, l) and K(j, 2 j - l ) in the integrands of eq. (4.23). (Note that (2l+ 1) K(j, l) is an eigenvalue of a z • L2) • Therefore R (2) is of the same form as eq. (4.22) in which we replace U~, V~, U~ and V~ with U(r2), V(F2), ~rf(2)and rG~(2),respectively. The last four are defined by eq. (4.23) (a), (b), (c) and (d) by substituting Q~(j, l, r, r') and Q~(j, l, r, r') in their integrands with Q(~)( J, l, r, r') and Q~)( J, l, r, r') respectively; these two functions are given by eq. ( 4 . 2 4 ) ( a ) a n d (b), respectively, by substituting W¢(j, l,j', l', r, r') with W(2)(j, I,j', l', r, r') = (j' +½)v(l, I', r, r')(rr') 2
-2(j'-l')K(j,
l){v(l)(l-1, l', r, r')-v(a)(l+ 1, I', r, r')}(rr') 2.
(4.37)
The integrand of R (3) contains (a i • xl)(a 2 • xl). The effect of the first factor is to interchange F,u(rl) and 2G, u(rl) in 0,zjm(1). The effect of the second is to multiply eq. (4.10) by a 2 .x~ from the left. This multiplier can be moved to the right of D1 (or D2) by making use of their anticommutator (a 2 " xa/rl)D1-t-DI(o"
2 " xl/rl)
=
2L,t.y(rl)Gn.rj.(r2)2(12 + 2F,.vj.(r~)F,.rj.(r2)a 2 • x l / r l .
(4.38)
636
G. ARAKI
Then the integrand of R (3) contains the left side of the equation ( ~ . x,/r,)(¢~ "Lz)Pr(~2) = (az" xzlr2)P;,((~2)
--(tr 2 . x,/rl){l'Pv(¢,2)+ n;,_,(~,2)}.
(4.39}
This can be changed into the right-hand side by carrying out the differentiation and making use of the recurrence formula for the Legendre polynomial. The recurrence formula can also be made use of to absorb ~12 of the first term in the right side of eq. (4.38) into the Legendre series. The projection operators reduce to 1 or 0 according to the equations
E(+-) 2 j - I @Um= { 4- ( l - j ) + ½} ~ljm E} +)(a" x/r)~um = { +_(j-- l) + ½}(a- x/r){#tj,,.
(4.40a)
(4.40b)
Now the angular integration of R (3) is easily carried out as was done in the case of Re. The result is represented in the form of eq. (4.22). Its radial parts are denoted by U(F3), V(F3), '-'GrT(3)and --G1/(3).They are given by the integrals of the same form as in eqs. (4.33) (a), (b), (c) and (d), respectively, in which the functions Q~I) and Q(v1) are substituted by the following Q~) and Q(v3), respectively: Qb3)(j, l, r, r') = y ' W~a)(j, l, j', I', r, r')F,,,t,~,(r')F,,Ti,(r )
+ Y.' W(ua)(j, I, j', 2j'-l', r, r')G,.,rj,(r')G,.,vj,(r),
(4.41a),
Q(va)(j, l, r, r') = ~ ' Wv(a)(j, l,j', l', r, r')F,.,vj,(r')G,.,vj.(r)
+ ~,' W(va)(j, l, j', 2j'-l', r, r')G,..vr(r')F,.,rj,(r),
(4.41b),
Wu(a)(j, l, j', l', r, r') = { j' + ½- 2 ( j ' - l')I'}{ ( l - j + k)v( t - 1 , l', r, r')
+0-1+½)v(t+ 1, r, r, r')}r'* + 2(j'-l'){v°)(l, l', r, r ' ) - ( 1 - j +½)v(1)(1-1, l ' - l, r, r') - ( j - l +½)vO)(l + 1, l'-- l, r, r')}r '4, / ' + 1+-i v ( l , l ' + l , r , r ' ) + - - v2r+1 (l,l'-l,r,r') w43 (j, l,j,"' t,' r , / ) = ( 2 ; + 0 tt2/
(4,42a} r '4
+ 4 ( j ' - l ' ) K ( j , l){l'v(l-1, l', r, r')-l'v(l+ l, l', r, r') +vO)(I-1, I'-1, r, r ' ) - v ° ) ( l + l , l ' - l , r, r')}r'4--wda)(j, l,j', l', r, r'). (4.42b} The angular integration of R (4) is carried out in the same way, and its spin-angle dependence is the same as in eq. (4.22). If its radial parts are denoted by U(F4), V(F4), U(4) they are given by the integrals of the same form as eqs. (4.23) (a), (b), G and rrA4) G (e) and (d), respectively. The functions Q~ and O~v in the latter integrands are substituted by Q(u4) and Q(v4) as follows: Q(u4)(j, l, r, r') = Z ' Wd4)(J, l, j', l', r, r')Fi.,rj,(r')V,.,vj.(r )
+ Y,' W(u4)(j, I, j', 2j'-l', r, r')G,.,r~,(r')G,.,vr(r),
(4.43a}
SHELL MODEL
637
Q~>(j, I,~,/) = y: w~*)(:, t,;, v, ~,/)F,..,.A/)a,,,..A,.) + •' W(v*)(j, l, j', 2:'--l', r, r')G,,,vs,(r')F,,,t,s,(r ),
(4.43b)
where Wu(4)(j, l, j', l', r, r')
= {j'+½-2(j'-l')l'}
tl+l . 121~+1 v ( 1 - 1 ' I ' ' r ' r ' ) +
I
21+1-- v ( l + l , l ' , r , r ' ) t
t
rr'a
+ 2 ( j ' - - / ' ) {v(~)(/, l', r, r')-- l + l v(l)(/_l ' l'--l, r, r') 2l+1 l v ( ~ ( / + l ' 1'--1, r, r')} rr '3, 2l+1 Wv(4)(j, l, 3,' ' I, r, r') = (2j'+ 1) I~ 2~'+l /'~
~(l,r+l,~,/)+
+ 4 ( j ' - l ' ) { 2 ( 2 j - 1 ) + l} K(j, 2 j - l ) { I ' v ( I - 1 , 2/+1
(4.44a)
"
21' + 1 ~(/,v-l,~,/)
} ,/~
l', r, r')-I'v(l+ l, I', r, r')
+v(1)(l-1, 1'-1, r, r')-v(*)(/+ 1, l ' - l , r, r')}rr'3-W(u4)(j, l,j', l', r, r').(4.44b) The exchange field due to the "r independent interaction is thus given by
UcxO,,,j,, = { U~(r) + V:(r)2a • x/r}F,n,s(r)~,: m + { U~(r) + l/~(r)2a • x/r}G,,~s(r)(2a" x/r)(~lsm,
(4.45)
where the scalar radial parts U~ and V~ are given by U~(r)
~ + Uv(r), t = U ~ ( r ) + Uv(r)
(4.46a)
V~(r) = V~(r) + V~(r) + V~(r),
(4.46b)
and U~ and V~ are given by the equations of the same type if suffices F are replaced with G. The radial parts with a superscript t are given by
Urn(r) = - U(vl)(r) + U(vZ)(r)+ U(vS)(r)--
U(v4)(r),
(4.47)
and so on. These are the contributions from the third term of eq. (4.6). If the two-nucleon interaction is given by the second term in eq. (3.4), eq. (4.5) is modified as follows:
U~xx~(2)~,,tjm(2 ) = Z,(2)Uex~/,nU,,(2),
(4.48a)
Uex~,o,J,,(2) = Z ( 2 - 6a,) Y',' (O,,,,z,j,,,,(1), B(1, 2)~),,~Sm(1))O,,,,,,S,,,,(Z)d(x, ), (4.488) z"
d
where ~ ' denotes the sum extending over all occupied n', l', j', m' states, and B(1, 2) stands for B,(x,, x2, a,, a2). Therefore the spin-angle dependence of this exchange
638
G. A R A K I
field is the same as eq. (4.45). If we denote its scalar radial functions by U~(r), V~(r), i i U~(r) and VG(r) they are given by the integrals of the same form as UF(r), VF(r), ui~(r) and V~(r), i respectively. Besides the appropriate substitutions of w, u and v, we have merely to modify the Q functions, that is, the isospin quantum number t of F~,,l,j,(r') and G,,,vy(r' ) in them should be changed into t' and the summation symbol ~ ' should be replaced with Z , ( 2 - 6 , ' ) Z ' , where Z ' means the sum extending over all occupied n', l', j ' shells as before. The contribution from the Coulomb interaction is given by
U~ Z, ~.tj~ = zp Z,(U~(r) + V;(r)2~ • x / r } F , . t j ( r ) ~ t j ~ + zv Z,{ U~(r) + V~(r)2~ • x/r} G,,u(r)(2~r • x/r)~u, . .
(4.49)
The radial functions U~(r) etc. are given by eqs. (4.23)-(4.26) where w(l, l', r, r') and w°)(l, l', r, r') are substituted by q~(l, l', r, r') and @1)(1, l', r, r'), respectively. The latter two are the coefficients of the Legendre series pertaining to e2/Ix-x'l [see eqs. (4.15) and (4.16)]. 5. Discussion
From the calculations in the preceding sections the Hartree and exchange fields in the closed-shell nucleus are given by
UHZ, ~ltnljm = Zt{ O(P)2Fmlj(F)q- g(F)Gtnlj(F)}~ljm + z.{ v(~)~'..,(,) + v ( ~ ) G , ( ~ ) } ( ~
• x/r)%~,
(5.a)
UexZ, l[ImUm = Z,{ UF(V)FmU(r) q- I/G(r)G,nlj(r) }fl#Urn q- Z,{ VF(F)F, nlj(F) Jy UG(F)Gmlj(F)}(~ " X/F)~ljm,
(5.2)
v(r) = vi(r)+, G(~)+ ~+~, vo(~),
(5.3a)
v(~) = v # ) + ~v~(r),
(5.3b)
UF(r) = U~(r) + U~(r) + 6 +t, U~(r),
(5.4a)
VF(r) = l/~(r) + V~(r) + 6 +1, V~(r),
(5.4b)
where
= Ue(r) + Ue(r) + 6 +t, UG(r),
Vc(r) = G ( r ) + G ( r ) + 5 +1, G(r).
(5.4c) (5.4d)
Since spin-orbital harmonics ~tjm and (2a'x/r)qhjm are linearly independent, the Hartree-Fock equation (3.1) resolves itself into simultaneous equations for the radial functions as follows:
(~+ v - u~-G,)F,°,+(v- V~)G,,j = 0,
(5.5a)
(T2j_t+ U - Ue-E,,tj)G,,tj+(V-l/'v)F,,q = 0,
(5.58 /
SHELL MODEL
639
where
h2 (~ T~
2M
62
/(/+1)}
dr 2
r2
(5.6) "
Therefore we see that the spin-angle dependence of the one-nucleon states assumed in eq. (3.5) is consistent with the two-nucleon interaction which has been generally believed to exist between nucleons. The fact that the Hartree and exchange fields are isospin-dependent shows that the proton radial function of the nlj shell is different from the neutron radial function of the same shell in accordance with our initial assumption. If these functions are the same, the contributions to U, and V, from protons and neutrons in the same shell cancel out. In the case of the non-closed shell nucleus the Hartree-Fock equation should be modified. In this case the Lagrange multipliers Ev, cannot be completely diagonal, and the Hartree-Fock equation takes the form
I
I
-- - - A + U n ( q ) - Uex(q) qSv(q) = Z 2M
E,~u(q)"
(5.7)
The Lagrange multipliers are the substitutes for the orthonormality condition imposed on unknown one-nucleon states. There is no need to impose the orthogonality condition upon one-nucleon states of different zjm because isospin and spin-orbital harmonics X~bu,~ are orthogonal. Consequently the right side of eq. (5.7) contains no non-diagonal element connecting such states. If an tj shell is completely filled the non-diagonal elements of E ~ connecting different states in this shell can be removed by a unitary transformation of one-nucleon states in this shell, as is well-known, without changing the total nuclear state. If the shell is non-closed this is impossible. Therefore q~v and all q5u in eq. (5.7) belong to the same tim and their isospin and spinangle dependences are the same. The non-diagonal Lagrange multipliers can be represented by integrals containing radial parts of wave functions and interactions 3). Hence the non-diagonal part on the right of eq. (5.7) can be considered as a part of the exchange field, and the existence of the non-diagonal Lagrange multipliers in the case of the non-closed-shell nucleus makes no essential difference as compared with the case of the closed-shell nucleus. An important difference between the closed-shell and non-closed shell nuclei is in the spin-angle dependence of the Hartree and exchange fields. On account of the fact that the addition theorems for the spin-orbital harmonics cannot be applied to the summation over non-closed shells, the spin-angle dependence of the fields coming from the non-closed shells is generally not the same as those from the closed shells. This is the situation which is similar to the one encountered in the case of the central interaction between atomic electrons. Hartree's procedure to manage the situation was to substitute the exact fields by their angular averages. We cannot make use of this procedure here. An appropriate averaging process in the present case is to modify the summation over m of the non-closed shell. If the shell is specified by
~40
G. ARAKI
n lj and contains Nj nucleons, we replace ~mwith {Nfl(2j+ 1)}y'~= _j. Then we have the fields of the same spin-angle dependence as those in the case of the closed-shell nucleus. The effect of the deviation of the exact fields from such averages can be considered as the angular-correlation effect, and it can be treated by the weU-known method of correlation functions or of configuration interactions 4). In order to understand the situation we consider the simplest configuration which consists of a closed-shell core and one nucleon outside it, the latter being referred to as an odd nucleon. There is no deviation of the Hartree and exchange fields from their averages for the odd nucleon, because the contributions to both fields f r o m the odd nucleon cancel out and there remain only the contributions from the core. On the contrary, for nucleons in the core, the contributions from the odd nucleon make the fields deviate from their averages. This deviation gives rise to the deformation of the core and this deformation produces the deviation of the fields for the odd nucleon as a secondary effect. Such a phenomenon has been observed in the electron problem of helium 4, s). In conclusion the author wishes to express his hearty thanks to members of the theoretical seminar in our department for their valuable discussions. The present p a p e r is dedicated to Professor Masao Kotani on the occasion of his sixtieth birthday.
References 1) 2) 3) 4) 5)
G. Araki, J. Phys. Soc. Japan 19 (1964) 1514 J. P. Amiet and P. Huguenin, Nuclear Physics 46 (1963) 171 J. C. Slater, Quantum theory of atomic structure, Vol. 2 (McGraw-Hill, New York, 1960) p. 23 G. Araki, Festskrift til Egil hyl!eraas (I Kommisjon hos F. Bruns Bokhandel, Trondheim, 1958) G. Araki, Prog. Theor. Plays. 16 (1956) 197