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The triton problem with tensor forces
Nuclear Physics 46 (1963) 572--576; (~) North-Holland Publishing Co., Amsterdam N o t to be r e p r o d u c e d
by photoprint
or microfilm without
written
permission from
the
publisher
THE T R I T O N P R O B L E M W I T H T E N S O R F O R C E S B. S. BHAKAR
Department of Physics, University of Delhi, Delhi-6, India Received 9 April 1963 Abstract: The triton wave function in presence of tensor forces is investigated, when full symmetry
of the three-nucleon system in configuration, spin and isobaric spaces is taken into account assuming separable potentials to operate between pairs. The wave function for the threeparticle system involves only single parameter functions, two in the case of central forces and three in the case of tensor forces, resulting in a corresponding number of coupled one-dimensional integral equations.
Recently Mitra ~) considered the three-body bound state problem using a non-local separable potential between two nucleons and showed how the three-particle wave functions could be obtained in an exact form in terms of certain single variable functions which satisfy as many coupled one-dimensional integral equations. This procedure facilitates a calculation of the triton binding energy without any emphasis on variational principles. However, in Mitra's paper the symmetries of the 2 n + p system were only partially taken into account, in so far as the isospin dependence of the wave function was ignored. The purpose of this note is to reproduce the structure of the triton wave function in the presence of tensor forces when the full symmetry of the three-nucleon system in configuration, spin and isospin spaces is taken into account. A corresponding formulation for central forces only has been recently given by Mitra and Bhasin 2) in the context of the n-d scattering at low energies. A full classification of states for the triton wave function is of course available from Derrick and Blatt's 3) work, but it was not possible to adapt their procedure to the present formulation which starts with separable interactions in the S- and Dstates only, It is more convenient to adapt to the present problem the procedure of Gerjuoy and Schwinger 4) (without isospin) and Sachs s) (with inclusion of isospin)). However, the number of possible functions of the correct symmetry is severely restricted by the interaction model, since the overall wave function must satisfy the three-particle Schr6dinger equations, in the spirit of ref. 1). The interaction model considered is of the Yamaguchi type 6), viz.
-M(plglp' ) = 2a,g(p)g(p r )P~-F p~-- +2,3f(p)f( p ! )Po- P~+,
(l)
where, ill the notation of Verde 7), the projection operators P+ ~have been introduced and Aal , ),13 are the strength parameters of the triplet even and singlet even interactions, respectively. Two additional terms in triplet odd (S = 1, T = 1) and singlet 572
THE TRITONPROBLEM
573
odd (S -- 0, T = 0) states have been dropped from eq. (1) with the understanding that H 3 and He 3 have predominantly symmetric wave functions 3, 7). For S-state (central) interaction alone, #(p) and f(p) are functions of pZ only. For tensor forces, we take, like Yamaguchi 6),
9 Iz(P) = c(p) + 8-iT(p)S1 z(q)
(2)
Sl2(q) = 3(0"1" P)(@2" P)--(GI "O'2)
(3)
The three-particle SchrSdinger equation is now
I I~(p2--[-P2--I-P2)-E1~] =- ~ 2M
d
f (pij[Vlp~j)(~3(pk--P'k)d3pkd3p~j@(e~P;ek)
(4)
i,j, kO
where Pi(i = l, 2, 3) are the momenta of the particles such that PI + P2 + P3 = 0 and
2Pu --
P i - P j,
- Pk = P~ + Pj
(5)
AS in ref. 7) taking the basis representation in terms of the particle momenta
P2, P3 the totally antisymmetric ~ has the structure ~O -- 2 - ½ ( A ' ( " - A " ~ ')
(6)
where (', (" are the isospin functions of the [2, 1 ] symmetry 7) in the (P2, Pa) basis, and A', A" are the combined spin and momentum functions of the corresponding symmetry. Substitution of 6 in eq. 4 gives the following coupled integral equations:
D(E)A'=
¼X/3231 Ig(plZ) f
dp'leg(p'12)P+(12)(x/3A'+ A")+g(Pl3 )
X fdplag(P13)P,r(13)(~/3A , , + - , - A ,, )1
+¼)'13 [4f(p23) f L.
,d
dp'23f(P'23)P~(23)A'
(7)
-'~f(~ 12)f dPl2 f'P'l 2)Va (12)(A' -- N/3A"')Jr-f(p 13)5d,',~f(,',3)P: 03)(A'+ ,/~A") 1 ,
[
f,
D@)A" =¼23~ 40(P23) dp23O(P23)P~(23) +g(P~2) X fdp'12g(p'12)P+(12)(x/3A'+A")+g(p13)fdp'lag(p'13)P+~(13)(-x/3A'+A")l -{-¼N/'3"~13[f(Pl2) f
dp'12f(P'12)P:(I2)(-A'Ar N/~m'') +f(p,3) f dp'~3f(P',3)V:(13)(a'+~/SA")] , (g)
where
D(E) = ½(p~+ p~ + p~)+~2,
~2
E= - -
M
(9)
574
B.s. BHAKAR
So far these equations are quite general in so far as #(p) a n d f ( p ) are not yet specified. The structure of A' and A" can now be read off following Verde 7) in terms of 2 x 2 matrices (in spin space) operating on a function Z' antisymmetric in 2 and 3 so that, e.g., Z" -- -3-~0-1 " 0-3Z', 0"2~( t = - 0 " 3 ; ( ' . (10) Thus, for the case of central interactions, we have after routine algebra
D(E)A' = x/319(P~ 2)P~+(I 2)F(p3) + 9(P~3)P~+(13)F(p2)]Z' + x/3[f(pz3)G(pl) +f(plz)P; (12)G(ez) +f(p~ 3)P2 (13)G(p2)]Z',
(11)
D(E)A" = [ - 9(P2 3)°"1 . 0-3F(Pl) + {9(Pl 2)P.+ (12)F(P3) - 9(Pl 3)P2 (13)F(p2)}]Z' - 3[f(p12)Pff (12)a(p3)-f(p13)eff (13)G(p2)]Z', (12) where F(,p) and G(p) satisfy the coupled equations F(P)[23a 1 - h3 I(P)-[ = ½ f d3~ D- 1(¢, p)g(~ + ½p)[g(p + ½¢)F(¢) + 3f(p + ½~)G(¢)], (13)
G(P)I-21-31 - h l zlP)] = ½ f d3¢ D - I(~, p)f(~ q_½p)[3g(p q- ½¢)F(~) +f(p + ½¢)G(¢)3, (14) where
D(~,p)=~Z+p.~+p2+c~2
and
h3~(p),
x [92(¢),f2(¢)].
(15)
The case of tensor forces (2), (3) is slightly more complicated, but the deduction of the structure of A' and A" is greatly facilitated by using relations of the type S12(II))X' = -S13(P)X', S13(~)SI3(q)X t 9[(p. ~)2 @(/). ~){hrl . (/3 X ~) -t-i0-3" (/3 X ~)} =
--(0-1 " P X 4 ) ( ~ 3 " ~ X q ) ] Z ' - - [3 --20-~ " 0-3]X'-- S 1 3 ( P ) Z ' - - S13(0),~', 523(/9)0-1 " 0"3 Z' =
823(p)S13(q)z'
- 2S13(P)Z',
~-- [ - 18(/~- (1)(0-1 " q)(0-3 " P) -}- 2o'1" 0"3 + 2 S 1 3 ( P ) --I-2 S 13(~)]Z', etc. (16)
Thus in tiffs case we have, after some algebra,
D(E)A' = \/319(Pl 2){P~(12)Fl(p3) + 8-~S xz(~z)Fe(p3)} + 9(Pl 3){P+~(13)FdP2) + 8 - ~S~3(I~z)F2(pD}]X'+ x/'3[f(p23)G(p~) +f(p~ 2)Pd (12)G(p3) +f(t, x3)P2(13)G(p2)]Z', (t7) D(E)A' = [29(p23){ - ½0"~ " 0"3F~(pj) + 8- ;S~ 3(P~)Fz(P~)} + 9(P~2){P~+(12)Fdp3) --I ~ ^ ¢ + 8- I~S12(/33)F2(pa)} - 9(Pl 3){Po+(13)Ft(Pz) + 8 -S 13(p2)f2(P2)}])~ -3[f(p12)pff(12)G(p3)-f(p13)P~(13)G(p2)]'Z', (18)
THE TRITON PROBLEM
575
where the functions F and G satisfy the following coupled integral equations: F,(P)[).3) - h3 I(P)] = ½ f d3~D - '(~, p)[K, ~(p, ¢)F1(¢)+ K12(p , ¢)F2(¢)+ K,a(p , ¢)G(¢)], FE(p)[).311 -- h3 I(P)] (19)
= ½ f d3~D - 1(¢, p)[K2,(p, ¢)F1(¢) + KE2(P, ¢)F2(¢) + K23(P, ¢)G(¢)], G(p)[).~31 -- h, 3(P)] = ½ f d3~D - 1(¢, P)[K31(P, ~)FI(~)+ K32(P, ~)F2(¢)+ K33(p, ¢)G(~)], d
with h 3, (P) = f
d
3~(¢2 + ~p2 qt_~ 2 ) - 1 [C2(~) _~_T2(~)] ;
(20)
the quantities KIj(P, ~) are defined in the appendix. The essential simplicity of the separable model is brought about by the structure of A' and A" which, with the inclusion of tensor forces, involves only three single parameter functions F1, F2, G. The model automatically generates all the even parity S, P, D states allowed by the interaction (2), (3). The presence of a P-state can be detected by the terms of the form Sii(3.) Sij(/3) which appear coupled to the Dstates, as expected. The interpretation of the various terms in eqs. (17) and (18) can be made easily on the lines of ref. 1). Numerical calculations in the case of tensor forces have not yet been made. However, for the case of central forces alone, taking g(p) = f ( p ) = (p2+f12)-i with fl = 6.255 c~o (and 3~13 = 23.3 ~3 to fit the two-body data 6), a determination of 231 ---- 30.0 ~3 was made from eqs. (13) and (14), which may be compared with the two-body determination of 231 = 33.36 c~3. It is encouraging to note that the threebody value of ).31 is somewhat smaller than the two-body value, in keeping with the usual belief that an effective central two-body interaction gives somewhat more binding for H 3 than one with a mixture of central and tensor forces. The author would like to express his deep gratitude to Dr. A. N. Mitra for suggesting the problem and continued guidance throughout the course of this work. Finally, he is grateful to Professor R. C. Majumdar for his interest and encouragement.
Appendix K , , ( p , ~ ) = C(¢+~p)C(p+½~)+2r(¢ +TP)T(P+7¢)[T2P i i z7 2¢ sin 2 0(p+½¢) -2
× (¢ +
p)-2
_ 1],
576
S.S. BHAKAR
Ka2(p, ¢) = C(¢ +½p)T(p+½¢)[1 _~.pZ sin 2 0 ( p + ½ ¢ ) - 2 ] + 2T(~ +½p)C(p+½ 0 × lap2 sin 2 0 ( ¢ + ½ p ) - 2 1] +x/Er(¢+½p)r(p+½¢)[1--¼p2 sin 2 0 ( p + ½ ¢ ) - 2 13p E sm 2 0 ( ~ + E1p )
-2
- ~ 2- 7P
2 ¢2
sinE
O(p+½0-2(~+½p)-E],
½p)f(p + ½0, K21 (P, ¢) = 2c(¢ + ½p)T(p + ½¢) [ ~ 2 sin 20(p + ½~)- 2 _ _ ] ] .j_T(¢ + ½p)c(p + ½~) x [1 _ _ 3 ~ 2 sin 2 0 ( ~ + ½ p ) - 2 ] +½x/~T(~+½p)T(p+½~)[2_3~2 sin E 0 ( p + ½ ¢ ) - 2 K x3(P, ~) = 3C(¢ +
2'7 2 1 -2 ~g~ sin E O(~+=p)
KE 2(P, _
27 2 2 --TOP ~ sin20(¢+½p)-E(p+½0-2],
~) = C(¢ + ½p)C(p + ½¢)(1 - 3 cos 2 O)+ x/72C(¢+ ½p)T(p + ½¢)[¼(3 cos 2 0 + 1) ¼(p2 + ¼¢2) sin 20(p + ½~)- 2] + ½x/2T( ¢ + ½p)C(p + ½¢)[(3 cos z 0 - 1)
+ ~ p ~ cos 0 sin E 0(~+½p) - 2 ] -b T(~+½p)T(P+½ )[~3 sin 2 0 - 42t( p 2 - ~1
2
) x sin E O(p + ½0- z _ ¼(~Z_ ~p2) sin 2 0(~ + ½p)- Z+ _~pE~2 sin z O(p + ½0- 2 ×
I,;23(v, ¢) = 3T(
+½V)/(v+½0,
Ka,(p , ¢) --- 3f(¢ +½p)C(p+½~), K3 2(P, ¢) = 3f(¢ + ½P)r(p + ½¢)[1 - 3pZ sin z O(p + ½~)- 2], K3 3(P, ¢) =
f(¢ + ½P)f(P + ½~). References
1) 2) 3) 4) 5) 6) 7)
A. N. Mitra, Nuclear Physics 32 (1962) 529 A. N. Mitra and V. S. Bhasin, preprint G. Derrick and J. Blatt, Nuclear Physics 8 (1958) 310 E. Gerjuoy and J. Schwinger, Phys. Rev. 61 (1942) 138 R. G. Sachs, Nuclear theory (Addison-Wesley Publishing Co.) Y. Yamaguchi, Phys. Rev. 95 (1954) 1628 and 1635 M. Verde, Handbuch der Physik, Vol. 39, 145
+ Iv)-
2],
by photoprint
or microfilm without
written
permission from
the
publisher
THE T R I T O N P R O B L E M W I T H T E N S O R F O R C E S B. S. BHAKAR
Department of Physics, University of Delhi, Delhi-6, India Received 9 April 1963 Abstract: The triton wave function in presence of tensor forces is investigated, when full symmetry
of the three-nucleon system in configuration, spin and isobaric spaces is taken into account assuming separable potentials to operate between pairs. The wave function for the threeparticle system involves only single parameter functions, two in the case of central forces and three in the case of tensor forces, resulting in a corresponding number of coupled one-dimensional integral equations.
Recently Mitra ~) considered the three-body bound state problem using a non-local separable potential between two nucleons and showed how the three-particle wave functions could be obtained in an exact form in terms of certain single variable functions which satisfy as many coupled one-dimensional integral equations. This procedure facilitates a calculation of the triton binding energy without any emphasis on variational principles. However, in Mitra's paper the symmetries of the 2 n + p system were only partially taken into account, in so far as the isospin dependence of the wave function was ignored. The purpose of this note is to reproduce the structure of the triton wave function in the presence of tensor forces when the full symmetry of the three-nucleon system in configuration, spin and isospin spaces is taken into account. A corresponding formulation for central forces only has been recently given by Mitra and Bhasin 2) in the context of the n-d scattering at low energies. A full classification of states for the triton wave function is of course available from Derrick and Blatt's 3) work, but it was not possible to adapt their procedure to the present formulation which starts with separable interactions in the S- and Dstates only, It is more convenient to adapt to the present problem the procedure of Gerjuoy and Schwinger 4) (without isospin) and Sachs s) (with inclusion of isospin)). However, the number of possible functions of the correct symmetry is severely restricted by the interaction model, since the overall wave function must satisfy the three-particle Schr6dinger equations, in the spirit of ref. 1). The interaction model considered is of the Yamaguchi type 6), viz.
-M(plglp' ) = 2a,g(p)g(p r )P~-F p~-- +2,3f(p)f( p ! )Po- P~+,
(l)
where, ill the notation of Verde 7), the projection operators P+ ~have been introduced and Aal , ),13 are the strength parameters of the triplet even and singlet even interactions, respectively. Two additional terms in triplet odd (S = 1, T = 1) and singlet 572
THE TRITONPROBLEM
573
odd (S -- 0, T = 0) states have been dropped from eq. (1) with the understanding that H 3 and He 3 have predominantly symmetric wave functions 3, 7). For S-state (central) interaction alone, #(p) and f(p) are functions of pZ only. For tensor forces, we take, like Yamaguchi 6),
9 Iz(P) = c(p) + 8-iT(p)S1 z(q)
(2)
Sl2(q) = 3(0"1" P)(@2" P)--(GI "O'2)
(3)
The three-particle SchrSdinger equation is now
I I~(p2--[-P2--I-P2)-E1~] =- ~ 2M
d
f (pij[Vlp~j)(~3(pk--P'k)d3pkd3p~j@(e~P;ek)
(4)
i,j, kO
where Pi(i = l, 2, 3) are the momenta of the particles such that PI + P2 + P3 = 0 and
2Pu --
P i - P j,
- Pk = P~ + Pj
(5)
AS in ref. 7) taking the basis representation in terms of the particle momenta
P2, P3 the totally antisymmetric ~ has the structure ~O -- 2 - ½ ( A ' ( " - A " ~ ')
(6)
where (', (" are the isospin functions of the [2, 1 ] symmetry 7) in the (P2, Pa) basis, and A', A" are the combined spin and momentum functions of the corresponding symmetry. Substitution of 6 in eq. 4 gives the following coupled integral equations:
D(E)A'=
¼X/3231 Ig(plZ) f
dp'leg(p'12)P+(12)(x/3A'+ A")+g(Pl3 )
X fdplag(P13)P,r(13)(~/3A , , + - , - A ,, )1
+¼)'13 [4f(p23) f L.
,d
dp'23f(P'23)P~(23)A'
(7)
-'~f(~ 12)f dPl2 f'P'l 2)Va (12)(A' -- N/3A"')Jr-f(p 13)5d,',~f(,',3)P: 03)(A'+ ,/~A") 1 ,
[
f,
D@)A" =¼23~ 40(P23) dp23O(P23)P~(23) +g(P~2) X fdp'12g(p'12)P+(12)(x/3A'+A")+g(p13)fdp'lag(p'13)P+~(13)(-x/3A'+A")l -{-¼N/'3"~13[f(Pl2) f
dp'12f(P'12)P:(I2)(-A'Ar N/~m'') +f(p,3) f dp'~3f(P',3)V:(13)(a'+~/SA")] , (g)
where
D(E) = ½(p~+ p~ + p~)+~2,
~2
E= - -
M
(9)
574
B.s. BHAKAR
So far these equations are quite general in so far as #(p) a n d f ( p ) are not yet specified. The structure of A' and A" can now be read off following Verde 7) in terms of 2 x 2 matrices (in spin space) operating on a function Z' antisymmetric in 2 and 3 so that, e.g., Z" -- -3-~0-1 " 0-3Z', 0"2~( t = - 0 " 3 ; ( ' . (10) Thus, for the case of central interactions, we have after routine algebra
D(E)A' = x/319(P~ 2)P~+(I 2)F(p3) + 9(P~3)P~+(13)F(p2)]Z' + x/3[f(pz3)G(pl) +f(plz)P; (12)G(ez) +f(p~ 3)P2 (13)G(p2)]Z',
(11)
D(E)A" = [ - 9(P2 3)°"1 . 0-3F(Pl) + {9(Pl 2)P.+ (12)F(P3) - 9(Pl 3)P2 (13)F(p2)}]Z' - 3[f(p12)Pff (12)a(p3)-f(p13)eff (13)G(p2)]Z', (12) where F(,p) and G(p) satisfy the coupled equations F(P)[23a 1 - h3 I(P)-[ = ½ f d3~ D- 1(¢, p)g(~ + ½p)[g(p + ½¢)F(¢) + 3f(p + ½~)G(¢)], (13)
G(P)I-21-31 - h l zlP)] = ½ f d3¢ D - I(~, p)f(~ q_½p)[3g(p q- ½¢)F(~) +f(p + ½¢)G(¢)3, (14) where
D(~,p)=~Z+p.~+p2+c~2
and
h3~(p),
x [92(¢),f2(¢)].
(15)
The case of tensor forces (2), (3) is slightly more complicated, but the deduction of the structure of A' and A" is greatly facilitated by using relations of the type S12(II))X' = -S13(P)X', S13(~)SI3(q)X t 9[(p. ~)2 @(/). ~){hrl . (/3 X ~) -t-i0-3" (/3 X ~)} =
--(0-1 " P X 4 ) ( ~ 3 " ~ X q ) ] Z ' - - [3 --20-~ " 0-3]X'-- S 1 3 ( P ) Z ' - - S13(0),~', 523(/9)0-1 " 0"3 Z' =
823(p)S13(q)z'
- 2S13(P)Z',
~-- [ - 18(/~- (1)(0-1 " q)(0-3 " P) -}- 2o'1" 0"3 + 2 S 1 3 ( P ) --I-2 S 13(~)]Z', etc. (16)
Thus in tiffs case we have, after some algebra,
D(E)A' = \/319(Pl 2){P~(12)Fl(p3) + 8-~S xz(~z)Fe(p3)} + 9(Pl 3){P+~(13)FdP2) + 8 - ~S~3(I~z)F2(pD}]X'+ x/'3[f(p23)G(p~) +f(p~ 2)Pd (12)G(p3) +f(t, x3)P2(13)G(p2)]Z', (t7) D(E)A' = [29(p23){ - ½0"~ " 0"3F~(pj) + 8- ;S~ 3(P~)Fz(P~)} + 9(P~2){P~+(12)Fdp3) --I ~ ^ ¢ + 8- I~S12(/33)F2(pa)} - 9(Pl 3){Po+(13)Ft(Pz) + 8 -S 13(p2)f2(P2)}])~ -3[f(p12)pff(12)G(p3)-f(p13)P~(13)G(p2)]'Z', (18)
THE TRITON PROBLEM
575
where the functions F and G satisfy the following coupled integral equations: F,(P)[).3) - h3 I(P)] = ½ f d3~D - '(~, p)[K, ~(p, ¢)F1(¢)+ K12(p , ¢)F2(¢)+ K,a(p , ¢)G(¢)], FE(p)[).311 -- h3 I(P)] (19)
= ½ f d3~D - 1(¢, p)[K2,(p, ¢)F1(¢) + KE2(P, ¢)F2(¢) + K23(P, ¢)G(¢)], G(p)[).~31 -- h, 3(P)] = ½ f d3~D - 1(¢, P)[K31(P, ~)FI(~)+ K32(P, ~)F2(¢)+ K33(p, ¢)G(~)], d
with h 3, (P) = f
d
3~(¢2 + ~p2 qt_~ 2 ) - 1 [C2(~) _~_T2(~)] ;
(20)
the quantities KIj(P, ~) are defined in the appendix. The essential simplicity of the separable model is brought about by the structure of A' and A" which, with the inclusion of tensor forces, involves only three single parameter functions F1, F2, G. The model automatically generates all the even parity S, P, D states allowed by the interaction (2), (3). The presence of a P-state can be detected by the terms of the form Sii(3.) Sij(/3) which appear coupled to the Dstates, as expected. The interpretation of the various terms in eqs. (17) and (18) can be made easily on the lines of ref. 1). Numerical calculations in the case of tensor forces have not yet been made. However, for the case of central forces alone, taking g(p) = f ( p ) = (p2+f12)-i with fl = 6.255 c~o (and 3~13 = 23.3 ~3 to fit the two-body data 6), a determination of 231 ---- 30.0 ~3 was made from eqs. (13) and (14), which may be compared with the two-body determination of 231 = 33.36 c~3. It is encouraging to note that the threebody value of ).31 is somewhat smaller than the two-body value, in keeping with the usual belief that an effective central two-body interaction gives somewhat more binding for H 3 than one with a mixture of central and tensor forces. The author would like to express his deep gratitude to Dr. A. N. Mitra for suggesting the problem and continued guidance throughout the course of this work. Finally, he is grateful to Professor R. C. Majumdar for his interest and encouragement.
Appendix K , , ( p , ~ ) = C(¢+~p)C(p+½~)+2r(¢ +TP)T(P+7¢)[T2P i i z7 2¢ sin 2 0(p+½¢) -2
× (¢ +
p)-2
_ 1],
576
S.S. BHAKAR
Ka2(p, ¢) = C(¢ +½p)T(p+½¢)[1 _~.pZ sin 2 0 ( p + ½ ¢ ) - 2 ] + 2T(~ +½p)C(p+½ 0 × lap2 sin 2 0 ( ¢ + ½ p ) - 2 1] +x/Er(¢+½p)r(p+½¢)[1--¼p2 sin 2 0 ( p + ½ ¢ ) - 2 13p E sm 2 0 ( ~ + E1p )
-2
- ~ 2- 7P
2 ¢2
sinE
O(p+½0-2(~+½p)-E],
½p)f(p + ½0, K21 (P, ¢) = 2c(¢ + ½p)T(p + ½¢) [ ~ 2 sin 20(p + ½~)- 2 _ _ ] ] .j_T(¢ + ½p)c(p + ½~) x [1 _ _ 3 ~ 2 sin 2 0 ( ~ + ½ p ) - 2 ] +½x/~T(~+½p)T(p+½~)[2_3~2 sin E 0 ( p + ½ ¢ ) - 2 K x3(P, ~) = 3C(¢ +
2'7 2 1 -2 ~g~ sin E O(~+=p)
KE 2(P, _
27 2 2 --TOP ~ sin20(¢+½p)-E(p+½0-2],
~) = C(¢ + ½p)C(p + ½¢)(1 - 3 cos 2 O)+ x/72C(¢+ ½p)T(p + ½¢)[¼(3 cos 2 0 + 1) ¼(p2 + ¼¢2) sin 20(p + ½~)- 2] + ½x/2T( ¢ + ½p)C(p + ½¢)[(3 cos z 0 - 1)
+ ~ p ~ cos 0 sin E 0(~+½p) - 2 ] -b T(~+½p)T(P+½ )[~3 sin 2 0 - 42t( p 2 - ~1
2
) x sin E O(p + ½0- z _ ¼(~Z_ ~p2) sin 2 0(~ + ½p)- Z+ _~pE~2 sin z O(p + ½0- 2 ×
I,;23(v, ¢) = 3T(
+½V)/(v+½0,
Ka,(p , ¢) --- 3f(¢ +½p)C(p+½~), K3 2(P, ¢) = 3f(¢ + ½P)r(p + ½¢)[1 - 3pZ sin z O(p + ½~)- 2], K3 3(P, ¢) =
f(¢ + ½P)f(P + ½~). References
1) 2) 3) 4) 5) 6) 7)
A. N. Mitra, Nuclear Physics 32 (1962) 529 A. N. Mitra and V. S. Bhasin, preprint G. Derrick and J. Blatt, Nuclear Physics 8 (1958) 310 E. Gerjuoy and J. Schwinger, Phys. Rev. 61 (1942) 138 R. G. Sachs, Nuclear theory (Addison-Wesley Publishing Co.) Y. Yamaguchi, Phys. Rev. 95 (1954) 1628 and 1635 M. Verde, Handbuch der Physik, Vol. 39, 145
+ Iv)-
2],