Tensor and L · S forces in the triton

Nuclear Physics 17 (1960) 6 7 - - 7 3 ; ~ ) N o r t h - H o l l a n d Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

TENSOR

AND L. S FORCES IN THE TRITON G. I~. D E R R I C K

Institute /or Advanced Study, Princeton, New Jersey and J. M. B L A T T

Applied Mathematics Department, University o] New Sou~h Wales Received 8 F e b r u a r y 1960 A v a r i a t i o n a l e s t i m a t e is m a d e of t h e t r i t o n b i n d i n g e n e r g y for a p o t e n t i a l developed b y G a m m e l , C h r i s t i a n a n d Thaler, modified to include L • S forces. This potential, w h i c h is s i m i l a r in m a n y respects to p o t e n t i a l s b a s e d in p a r t on m e s o n t h e o r y , is f o u n d to be too w e a k to b i n d t h e t r i t o n a t all; t h e s t a t e of lowest e n e r g y is a d e u t e r o n a n d a free n e u t r o n . T h e p r o p o r t i o n of P - s t a t e s coupled in b y t h e L • S forces is less t h a n 0.001 per cent.

Abstract:

1. I n t r o d u c t i o n In two previous papers 1, 9.) a classification b y the methods of group theory was made of the possible total-angular-momentum and isobaric-spin functions of the triton ground state. It was there shown that there are 16 such functions, of which only 10 are independent with respect to permutations of particles. The total wave function for the triton is then given b y the sum of products of these total-angular-momentum and isobaric-spin functions with 16 radial wave functions. The latter depend only on the 3 interparticle distances r23, %1 and r12; again, only 10 of these radial wave functions are independent under permutations of the particles. If we now substitute the above form of the triton wave function in the SchrSdinger equation H~o = E% all angle, spin and isobaric-spin variables can be eliminated from the problem. We are then left with a set oi'coupled differential equations in only 3 variables r23, r31 and r12, for the 16 radial wave functions. The present paper reports the results of some preliminary variational estimates of the triton binding energy made with this formalism. 2. T h e Potential We have used one of the potentials given by Gammel, Christian and Thaler 3) modified to include L • S forces 4, s). The interaction between particles i and j is given b y V,j =

V c (r,1) + V T ( r , ~ ) S i ~ + V L s 67

(r,j)L,~.

( s , + st);

(1)

68

G . H . DERRICK AND J. M. BLATT

S~ is the tensor operator for particles i and f, L~j the angular momentum of these particles relative to their mutual centre of mass, and s, and s~ the two spin yectors. The central, tensor and L . S radial potential functions, Vc(r,j ), VT(r~j) and VLs(r~j) respectively, are each cut-off Yukawa wells of the form

l

- - V 0 e-/*r~J

V(r,j)=i--#r,--~ oo

"

for

r,~>r 0

for

r i j < r o.

The constants Vo, # and r o depend on the various two-particle states; the values adopted are the final " b e s t " values of ref. 5) (p. 1339, 1340). Gammel et al. fitted the central and tensor part of the potential (1) to the low and medium energy two-body data. The spin-orbit terms were then added to improve the fit to the high energy scattering data, without, however, readjusting the central and tensor triplet parameters. The main features of the potential (1) are: (i) The central forces are much weaker than the tensor forces *, e.g. in the triplet even state the non-dimensional central well depth e) is only 0.35 compared with a tensor well depth of 0.96. The intrinsic ranges 6) of these forces are about 2.0× 10-13 cm. (ii) The L . S forces are quite strong, but of very short range. Thus in the even state the well depth is 0.45 with an intrinsic range of only 0.8 x 10 -la cm. it 3. M a t h e m a t i c a l M e t h o d The triton binding energy is the greatest lower bound of (v?, H~p)/(~p, ~) for all quadratically integrable functions ~vfor which the Hamiltonian H is Hermitian. With the aid of the matrix elements given in ref. 2) this expectation value can be written entirely in terms of the radial wave functions, with all angle, Spin and isobaric-spin variables eliminated. 3.1. T H E T R I A L W A V E F U N C T I O N

Since the triton is predominately in the space-symmetric S-state, we need include, for a first survey, only those states which have non-zero matrix elements to the main S-state. In the notation of ref. 2) these are states nos. 1, 3, 6, 7, 8, 9, 10. The predominant S-state no. 1 is connected directly to the mixeds y m m e t r y S-state no. 3 by spin exchange forces, to P-states nos. 6 and 7 by the L . S forces, and to the D-states nos. 8, 9 and 10 by the tensor forces. (State no. 5 is also coupled in to first order by the L • S forces, but its kinetic energy is so high that it m a y be omitted 1, 2). • The p o t e n t i a l we h a v e used s h o u l d be d i s t i n g u i s h e d fro m t h a t of B r u e c k n e r a n d G a m m e l 1,); t h e i r p o t e n t i a l has c e n t r a l a n d t e n s o r c o m p o n e n t s of r o u g h l y e q u a l s t r e n g t h . t* SVe are g r a t e f u l to Mr. T. H a m a d a for c a l c u l a t i n g t h e s e w e l l d e p t h s a n d i n t r i n s i c ranges. T h e n o n - d i m e n s i o n a l w e l l - d e p t h is t h e r a t i o of t h e a c t u a l w e l l - d e p t h to t h a t j u s t n e c e s s a r y to b i n d a 1S s t a t e .

TENSOR AND L" S FORCES ~N THE T~*O~

(~9

L e t us i n t r o d u c e the following n o t a t i o n : e-ydr-ro) u,(r) = - [ 1 - - e -*'(r-r°)]

(i = 1, 3, 6, 7, 8, 9, 10),

v,(r) = e-*"'-'0)u, (r),

S,

= u,(rz3)u,(r31)u,(r12),

S',

= u, (r23)u, (ra)v, (r12) + u, (r23)v ,(ral)u , (r12) + v, (r2z)u , (raft, ` (r12),

P,

1 = 3-½ [{v, (r,3)u , (r3x) + u, (rz3)v , (ra)}u, (r12) -- 2u, (r,a)ui (ra)v , (r 12)J,

Q,

= {v,(r2z)ui(ra)-u,(r23)v,(r-81)}u,(r12 ),

A = area of triangle

= ~[(r2z+ra+rl2)(--r23+ra+r12)(r23.-r31+r12)(r23+ra--r12)]½, 4.3½A R2

=

cos

-

-

-

,

R2 F

=

3 3 - I(r233 2~- ral--2r122),

2" G = ? ~2 - - ¥31 •

Here, r o is the core radius, a n d y~, (~ a n d ** v a r i a t i o n p a r a m e t e r s ; the subscript i refers to the different t r i t o n states. Using the n o t a t i o n of ref. 2) our trial radial w a v e functions were /1

=

O t l S l ' ~ - 0~'1 S t 1 ,

[3,1 ---- 0¢3 P 3 ,

/3.2 =

/e,1 = x6APe,

/6,2 = % A Q 3 ,

/7,1 -- CzTAP7,

/7,2 = o~7A07,

/8,1+ 3-½ sin 2/9,1 ---- % cosZ2Ps,

0¢3 (¢)3,

(2)

/s,2+3-½ sin 2/9,2 = as cos2~Q8,

COS JL/9,2+/10, 2 ~ -

%S9F c o t ~t, %S9Gcot ~,

cos ,t/~.1--/io.1 :

~10(GQ10 - FP10) cot A,

COS )'/9,1+/10.1 =

cos ~/2,~-/10,2 = :qo(G Plo + FQlo) cot ~. T h e a m p l i t u d e s :¢i are a d d i t i o n a l v a r i a t i o n p a r a m e t e r s . T h e radial w a v e functions (2) h a v e the correct a s y m p t o t i c b e h a v i o u r n e a r the t w o triangle c o n f i g u r a t i o n s at which our specification of the E u l e r angles fails, viz. all three particles in a s t r a i g h t line, a n d the particles at the vertices of an equilateral triangle 2).

70 3.2. N U M E R I C A L

G. H. DERRICK AND J. M. TECHNIQUES

BLATT

AND APPROXIMATIONS

The quantities (~2, HW) and (% ~) are given b y triple integrals over all permissible values of r~3, r31 and rx2 of quadratic expressions in the radial wave functions and their first derivatives 3). Those integrals involved in the m a t r i x elements of the main S-state to itself were calculated numerically using Simpson's integration formula ~). All the other integrals were estimated to lower accuracy b y a Monte Carlo m e t h o d 8). The following approximations were made: (i) The hard core radius was taken to be the even-state value of r 0 ---- 0.4 × 10-13 cm in both even and odd two-particle states; in the odd states, where the core is actually slightl)/ larger, the potentials were taken as zero for interparticle distances between r o = 0.4 × 10 -1~ cm and the actual core radius. Since the odd potentials have no direct m a t r i x elements to the space-symmetric S-state, the error introduced t h e r e b y is negligible. (ii) All P-state to D-state m a t r i x elements were set to zero, since the proportion of P and D states is small. (iii) All D-state to D-state matrix elements of the L . S force were taken as zero. t 4. R e s u l t s

The potential (1) was [ound to be too weak to bind the triton at all; the m i n i m u m value of the energy (% Hv2) / (% v2) occurs for parameter values which simulate a deuteron with a neutron infinitely distant. W i t h the wave function (2) there is no m i n i m u m of energy at all when all three particles are within range of one another's forces; as the parameters )h -+ 0 the energy decreases steadily and finally becomes negative only at v e r y small values of these parameters, corresponding to a deuteron and a free neutron. To estimate by how much the potential fails to bind the triton, let us introduce the "force factor" i.e. the factor b y which the potential must be multiplied to lead to the observed binding energy of 8.49 MeV. We have the following variation principle: force factor --< (% (T-4-B)~o) -

(~, - v ~ )

for all functions ~o for which - - V is a positive operator; T denotes the kinetic energy operator, and B the triton binding energy 8.49 MeV. Taking ~0 given b y (2) this variation principle yields the results listed in table 1. The binding energy is a small difference of two large quantities (potential * In a m o r e recent calculation, to be published shortly, we h a v e included these D-D elements of t h e L . S force. T h e i r i n c l u s i o n e f f e c t i v e l y i n c r e a s e s t h e D - s t a t e k i n e t i c e n e r g y , a n d reduces t h e triton binding.

TENSOR

AND

L.

S

FORCES

IN T H E

TRITON

71

TABLE 1 Calculated values of force factor, probabilities of c o n t r i b u t i n g states, Coulomb energy, binding

energy and optimum parameters O p t i m u m p a r a m e t e r s (1013 cm -l)

Force factor Percent. S states,

No. 1 No. 3 Percent. P states, No. 6 No. 7 Percent. D states, Nos. 8, 9, l0 Coulomb energy E n e r g y (~o,Hv2)/(~P, ~)

<1.172 89.5 0.59

4-0.023 4-3.4 4-0.16 O.00008 4- O.00003 0.00072 4- 0.00024 9.9 4-3.5 0.83 4-0.03 MeV 4.0 4-1.4 MeV

State

)p

1 3 6,7 8 9 10

0.18 0.20 1.0 0.22 0.65 0.85

1.1 1.1 ~ 2.0 0.9 0.9 1.15

1

0.4 0.3 1.0 1.0 irrelevant --0.5

L

The errors q u o t e d arise from the Monte Carlo evaluation of the m a t r i x elements. T h e y are not estimates of the errors due to the use of the variation method, which errors could well be larger. The p a r a m e t e r s 7, 6 and e were t a k e n the s a m e for the two P-states, nos. 6 and 7.

energy and kinetic energy), and the force factor applies to the potential energy only. Thus, the excess of the force factor over unity, namely 17 % is a very appreciable amount indeed. Furthermore, much of the potential energy comes from the tensor force, which enters quadratically; hence a significant fraction of the force factor is effectively squared. 5. D i s c u s s i o n

The question arises, whether the lack of binding might be due to a poor choice of trial wave function. To investigate this possibility we have estimated the triton binding energy with the function (2) for one of the potentials used by Pease and Feshbach 9). We obtained a binding energy of 7.0+ 1.1 MeV and a D-state probability of 4.64-1.6 %; Pease and Feshbach's values were 7.63 MeV and 3.5 %. These two sets of results are quite consistent within our large statistical errors. The lack of binding for the Gammel-Thaler force appears to stem from the weakness of the central part of the potential (1), compared with its tensor part. Thus the ratio of the central to tensor force non-dimensional well depths in the triplet even state is only 0.36. This contrasts with potentials used previously in triton energy calculations 9, 10), where the central and tensor forces are roughly equal in strength, e.g. the corresponding central to tensor force ratio for the Pease-Feshbach potential is 0.78. Tensor forces are less effective in binding the triton than the deuteron 11). Hence it follows that of two potentials which both bind the deuteron with the correct energy t, that with the higher * Strictly, it was the potential (1) with L • S forces omitted which was fitted to the deuteron binding energy. However, if we o m i t the L • S forces from our t r i t o n calculation, our results are unaltered to the accuracy of the calculation.

72

c.H. DERmCKA~D j. M. BLATT

proportion of central forces gives the more binding for the triton. This feature of the potential (1) (weak central forces, strong tensor forces) is shared by those potentials which are based in part on meson theory l~). A remarkable feature is the almost complete absence of P-states coupled in by the L • S force. The immediate reason for-this_is the smallness of the S-state to P-state matrix elements of the L • S force *. These elements are only 0.04 times the magnitude of the S-state to D-state elements of the tensor force. In addition the diagonal P-state matrix elements of the L • S force are strongly repulsive. A further inhibitory feature is the very high P-state kinetic energy, caused by the concentration at small distances of the P-state wave functions to take advantage of the very short ranged L • S force. However, the ineffectiveness of the L • S force can not be attributed solely to its very short range. Even if we take a fictitious L • S potential equal in range and strength to the tensor even potential, the proportion of P-states is still less than 0.002 %. We can make a rough estimate of the importance of L • S forces from second order perturbation theory, using closure. For this estimate we include only the central part of the potential in the unperturbed Hamiltonian. We shall assume that the central forces alone suffice to bind the triton, and shall ignore tensor forces altogether. Let 10> denote the unperturbed triton ground state wave function with energy --B, and [~> a three-nucleon scattering state with positive energy E~. We shall assume that the first order perturbation <0[H'[0> vanishes, and shall use closure. Then the increase in triton binding energy produced by a perturbation H ' is given to second order by <0iH'[~> ~B =

~

E~+B

_ _ 1 <01n'I,><~lH'f0> < <

B

~

B

Taking H ' as the L • S force and 10> as a pure space-symmetric S-state with the radial wave function /x(r~3, rax, r12), we obtain

1f

I (1 c9/1 10/1)VLs(rx2)l 2

(3)

As before A denotes the area of the triangle formed by the three particles;/12 has been normalized to give unity when integrated over all permissible r2a, rsl and rl~ with the weight factor r~3r~x rl~. The integrand of the integral in eq. (3) has a second order zero on the surface r23 = r31. In addition it vanishes everywhere on the bounding surface of the domain of integration (A = 0). Thus we would expect 3B to be very small. On the other hand the analogous closure estimate for tensor forces yields an integral whose integrand cannot vanish for a n y finite values of r2a, r31 and rx2 "r The numerical values of t h e normalization, kinetic energy and potential energy m a t r i x elements have been published elsewhere la).

TENSOR AND L ' S FORCES IN THE TRITON

73

outside the hard core (assuming/1 itself has no zeros, which is always true for the lowest S state). Qualitatively, one can_look at the result as follows. The P-state angle-function ~MValis essentially r12 × r23, and is thus anti-symmetric under exchange of particles (although of positive parity). The "natural" P-state is the one with exchange symmetric internal wave-function/p(r23, r31, r12), i.e., state no. 4 in our classification. However, since this function gets multiplied b y r12 × r23, it is exchange-antisymmetric and hence is not coupled in at all, to lowest order. The states which are coupled in, states 5, 6 and 7, have lower exchange symmetry in their internal wave functions, hence have more nodes, and lead to extremely small matrix elements. The fact that the P-states are completely unimportant in the triton ground state is therefore a result of basic symmetries, not merely an accident of our particular calculation. We have much pleasure in thanking Professor H. Messel for the excellent research facilities afforded us in the School of Physics, University of Sydney, where most of this work was carried out. We are also indebted to Dr. J. M. Bennett and the members of the Adolf Basser Computing Laboratories for a very generous allocation of Silliac computing time. One of us (G. H. D.) would like to acknowledge receipt of a research grant from the University of S y d n e y during 1959. He also wishes to thank Professor J. R. Oppenheimer for the hospitality extended to him, and for a grant-in-aid, at the Institute for Advanced Study. References 1) 2 3 4 5 6 7 8 9 10

11) 12)

13) 14)

G. Derrick and J. M. Blatt, Nuclear Physics 8 (1958) 310 G. H. Derrick, Nuclear Physics, 16 (1960) 405 J. L. Gammel, R. S. Christian and R. M. Thaler, Phys. Rev. 105 (1957) 311 J. L. Gammel and R. M. Thaler, Phys. Rev. 107 (1957) 291 J. L. Gammel and R. M. Thaler, Phys. Rev, 107 (1957) 1337 J. M. Blatt and J. D. Jackson, Phys. Rev. 76 (1949) 18 D. R. Hartree, Numerical Analysis (Oxford University Press, 1952) A. S. Householder, Principles of Numerical Analysis (McGraw Hill, New York, 1953) R . L . Pease, Thesis, M.I.T. (1950); R. L. Pease and H. Feshbach, Phys. Rev. 81 (1951) 142L E. Gerjuoy and J. S. Schwinger, Phys. Rev. 61 (1942) 138; H. Feshbach and W'. Rarita, Phys. Rev. 73 (1948) 1272A; 75 (1949) 1384; R. E. Clapp, Phys. Rev. 76 (1949) 873L Thesis, Harvard University (1949); T-M. Hu and K-N. Hsu, Phys. Rev. 78 (1950) 633L; Proc. Roy. Soc. A 204 (1951)476; G. Abraham, L. Cohen and A. S. Roberts, Proc. Roy. Soc. A 68 (1955) 265 R. D. Inglis, Phys. Rev. 55 (1939) 988; J. M. Blatt and V. F. \¥eisskopf, Theoretical Nuclear Physics (John Wiley, New York, 1952) J. Iwadare, S. Otsuki, R. Tamagaki and W. Watari, supplement to Prog. Theor. Phys., no. 3 (1956) S. Gartenhaus, Phys. Rev. 100 (1955) 900; P. S. Signell and R. E. Marshak, Phys. Rev. 109 (1968) 1229 G. H. Derrick, Thesis, University of Sydney (1959) K. A. Brueckner and J. L. Gammel, Phys. Rev. 109 (1958) 1023