Numerical calculations for the neutron-proton system with tensor forces

Nuclear Physics, North-Holland Publishing Co., Amsterdam, 1 (1956)

NUMERICAL NEUTRON-PROTON

CALCULATIONS FOR THE SYSTEM WITH TENSOR FORCES t 1~. H. KALOS tt

Laboratory o/ Nuclear Studies, Cornell University, Ithaca, New York L. C. B I E D E N H A R N

Physics Department, The Rice Institute, Houston, Texas and J. 1VL BLATT

Physics Department, University o/ Sydney, Sydney, Australia Received 27 August 1955 A b s t r a c t : Low-energy properties of the neutron-proton system are tabulated for a

variety of central and tensor potentials.

This report is devoted to a study of some purely phenomenological combinations of central and tensor potentials of varying ranges and strengths. In general, the emphasis is on a survey of the effects of the introduction of tensor forces so that for the most part no a t t e m p t has been made to fit the experimentally known parameters of the system. (Such fits have been interpolated from our results, but will not be included here.) The square well calculations are those of Biedenharn 1), while the rest were obtained with the University of Illinois electronic digital computer. The method of calculation used for the square wells differs from that employed with the automatic computer, but in both cases a ground state wave function is obtained with the correct deuteron binding energy. Let R = ? ~ / ( 2 M B ) ~ = (4.317 4- 0.003)10 -la cm be the 'radius of the deuteron'. Here M = reduced mass of the N - - P system, B ---- binding energy ---- 2.225 ± 0.002 MeV. 2) t Assisted in part b y the joint program of the Office of Naval Research and the U.S. Atomic Energy Commission. tt Present address: Nuclear Development Corporation of America, White Plains, N.Y. 233

2~4

M.H.

KALOS. S. C. m E D E ~ H A R N

A N D J. M. BLATT

Results are c o m p u t e d and t a b u l a t e d with R as the unit of length and B the unit of energy. To change the value of the binding energy, it is necessary simply to change the value of R in relating our n u m bers to the experiments. B o t h the central and tensor potentials are p a r a m e t r i z e d in t e r m s of the intrinsic range, b, and the well depth, s, i n t r o d u c e d b y B l a t t and Jackson 3) ,. The quantities t a b u l a t e d together with the experimental values (in units of R) are as follows: bc = central range; sc = d e p t h p a r a m e t e r of central well; b, = tensor range; s t = d e p t h p a r a m e t e r of tensor potential; PD = percent D state; it is usually estimated t h a t 0.02 ~ P D ----<0.06; Q = quadrupole m o m e n t = (0.01469 ± 0.0001)R 2 4); = Or(-- B, -- B) ---- deuteron effective range 5); a~ = triplet scattering length = (1.245 + 0.006)R s); r o t = O,(0,0) = triplet effective range 5); a~ = singlet scattering length = (--5.487 -¢- 0.006)R 6); ro~ = Q,(0,0) = singlet effective range = (0.58 4- 0.06)R ~); *t P = shape d e p e n d e n t p a r a m e t e r in triplet state 3). In computing singlet state results, it is assumed t h a t the singlet potential is the same as the central potential in the triplet state; t h a t is, the possibility of spin d e p e n d e n t central forces is ignored. The values listed for b,, s,, and b, usually do not contain insignificant zeros. These n u m b e r s can be assumed correct t h r o u g h the fourth decimal place. Otherwise, most of the results h a v e an error no worse t h a n a b o u t two in the last digit q u o t e d (and usually better). One i m p o r t a n t exception occurs for the Y u k a w a potentials of v e r y short range. The c o m p u t e r does not t r e a t the x -1 singularity at the origin of the potential correctly and the resulting error becomes i m p o r t a n t when the range is short. A measure of this error can be deduced from the value of a, -1 for s = 1. This n u m b e r should be zero for all values of b,. We see t h a t for b, = 0.1, it is a b o u t 0.03. The a c c u r a c y of other n u m b e r s derived from the same potential is t l~ote t h a t w h e n the range, b, becomes zero, this p a r a m e t r i z a t i o n is degenerate in the sense t h a t s = 1 identically. F o r this special case, a b e t t e r m e a s u r e of the s t r e n g t h is the scattering length itself. t t "Ihis w o r k gives the best singlet range deduced from low energy I ~ - - P scattering. The analysis of P - - P scattering s) gives a range which depends s o m e w h a t on the potential shape b u t whose average value is (0.62 ± 0.03) R.

b~

,.q¢

L

2.669 5.585 7.6t4 9.096

1.5588 1.7097 1.8617 2,0140

1.4025 1.5924 1,8016 2,0297

0.9307 0.9764

0.2 0.5 0.8 1.1

0.2 0.5 0.8 1.1

0.5 0.8

4.048 5.350

2.965 6.659 9.534 11.786

2.843 7.102 10.00

1.4695 1.7023 1,9537

PD

0.2 0.6 1.0

b~

I

i

! IJ l

0.1453 0,3820 0.5565

0.1361 0.3033 0.4336 0.5353

[

0.1554 0.3544 0.5144 0.6396

0.00839 0.01578

0.2549 0.3608

Morse tensor force

0.00275 0.01340 0.02790 0.04434

Gauss tensor force

0.00323 0.01268 0.02265 0.03194

Y u k a w a tensor force

0.00297 0.01709 0.03386

E x p o n e n t i a l tensor force

Q

No central force

TABLE 1

I. 1464 1.2184

1.0843 1.2172 1.3557 1,4957

1.0751 1.1782 1.2730 1.3522

1.0783 1,2375 1.3911

a,

L

0.2547 0.3570

0.1558 0.3599 0.5359 0.6885

0.1340 0.3015 0.4252 0.5083

0,1451 0.3861 0.5697

%, a8 ~o~

-

-

0.023 0.016

--0.026 --0.031 0.036 --0.039

1.183 0.019 0.024 0.048

0.027 --0.020 --0.020

1 1 1

0 0 0 0 0 0 0 0 0 0 0 0.2939 0.3160 0.3397 0.2666 0.3074 0.3457 0.3923 0.4471 0.4987 0.3733 0.4781 0.5776 0.5433 0.5975 0.6507 0.4645 0.5742 0.6786 0.6427

0.8812 0.8812 0.8812 0.8103 0.8103 0.8103 1.0125 1.0125 1.0125 1.0128 1.0128 1.0128 1.4269 1.4269 1.4269 0.8200 0.8200 0.8200 1.0643

1 1 1 1

1 1 1 1

S¢

......b~

0.6583 0.7455 0.8365 0.6023 0.6583 0.7240 1.0074 0.7063 0.7860 0.8364 0.8970 0.6068 0.6725 0.7473 0.5332 0.6456 0.7606 0.5492 0.6706 0.7979 0.3733 0.5259 0.6931 0.5433 0.7170 0.9110 0.4180 0.5455 0.6786 0.6427

b~" 0.8812 0.9715 1.0662 0.8812 0.8812 0.8812 0.8812 1.0663 1.0663 1.0663 1.0663 0.9388 0.9980 1.0662 0.9532 1.0509 1.1534 0.7249 0.8322 0.9468 0.6044 0.7313 0.8704 0.2980 0.4291 0.5841 0.8514 0.9487 1.0511 0.8249

5f 3.18 3.82 4.51 3.14 3.18 3.19 2.94 4.53 4.53 4.51 4.46 4.09 4.64 5.25 3.97 4.94 5.93 2.91 3.74 4.88 1.73 2.96 4.39 0.69 1.51 2.57 3.00 4.23 5.53 3.94

PD 0.01079 0.01378 0.01722 0.00974 0.01079 0.01194 O.O1530 0.01456 0.01626 0.01722 0.01824 0.01205 0.01459 0.01772 0.01014 0.01444 0.01941 0.00924 O.O1354 0.01882 0.00462 0.00931 0.01586 0.00503 0.00988 0.01642 0.00721 0.01202 0.01807 0.01441

Q

a~ 1.1430 1.1774 1.2162 1'.1369 1.1430 1.1486 1.1549 ].2019 1..2117 1.2162 1.2203 1.2184 1.2452 1.2767 1.1972 1.2440 1.2942 1.2194 1.2644 1.3135 1.1859 1.2458 1.3120 1.218I 1.3149 1.3583 1.2190 1.2772 1.3381 1.3228

Square - Square

TABLE I I

0.2430 0.2914 0.3422 0.2358 0.2430 0.2482 0.2305 0.3304 0.3396 0.3422 0.3421 0.3612 0.3975 0.4384 0.3316 0.3962 0.4610 0.3637 0.4241 0.4860 0.3160 0.3999 0.4851 0.4462 0.4889 0.5411 0.3629 0.4407 0.5168 0.4989

ro~

--6.090 -- 1.770 --1.902 --2.045 --0.925 -- 1.067 --1.200 25.7 29.2 32.6 23.9 30.6 37.0 1.461 1.607 1.750 -- 1.720 --2.126 --2.513 8.612

-- 1.594 1.921 5.875

- - 5.342

4.864

~s 0 0 0 0 0 0 0 0 0 0 0 0.3137 0.3372 0.3625 0.2976 0.3432 0.3859 0.3899 0.4443 0.4956 0.3709 0.4751 0.5740 0.4638 0.5100 0.5555 0.5151 0.6368 0.7525 0.6233

0.263 0.209 0.170 0.207 0.263 0.353 1.555 0.080 0.127 0.169 0.239 --0.026 --0.027 --0.028 --0.029 --0.029 --0.031 --0.037 --0.038 --0.038 --0.036 --0.039 --0.041 --0.040 - - 0.041 --0.042 --0.035 --0.038 -- 0 . 0 4 0 --0.040

P

0.7

0.5

1.O

0.9

0.6

1.0

0.938

0.9

0.6

1.0

0.9

0.5 0.65 0.8 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.67 0.71 0.748 0.79 0.83 0.87 0.5 0.8 1.1 0.5 .65 .8 0.5 0.65 0.8 0.5 0.65 0.8

0.1

0.6

bt

be

1.3086 1.4568 1.5961 0.9228 1.2344 1.5330 0.6460 0.9095 1.1795 1.1476 1.3430 1.5896 0.8875 1.0563 1.2835 0.9341 0.9578 0.9814 1.0087 1.0356 1.0635 0.7898 0.9454 1.1589 1.2135 1.2783 1.3637 0.9963 1.0439 1.1156 0.9171 0.9592 1.0256

St

5.23 6.78 8,18 3.13 5.57 7.80 1.73 3,33 4.94 4.48 6.93 8.98 3.13 5.08 6.78 4.03 4.28 4.50 4.75 4.98 5.21 2,64 4.34 5.86 4,59 5.91 7.08 3.43 4.53 5.53 3.02 4.03 4.95

pD 0.01128 0.01763 0.02471 0.00814 0.01872 0.03163 0.00581 0.01347 0.02280 0.01119 0.02272 0.03590 0.00948 0.01902 0.02974 0.01429 0.01555 0.01677 0.01813 0.01945 0.02079 0.00874 0.01747 0.02719 0.01199 0.01784 0.02404 0.01077 0.01593 0.02131 0.01023 0.01513 0.02019

Q

III

0.3087 0.3966 0.4768 0.2314 0.3886 0.5344 0.1730 0.2909 0.4238 0.3679 0.4755 0.5847 0.3757 0.4506 0.5398 0.4144 0.4243 0.4340 0.4450 0.4559 0.4670 0.3785 0.4412 0.5206 0.4131 0.4601 0.5083 0.4403 0.4713 0.5056 0.4491 0.4752 0.5048

at

1.2350 1.2866 1.3550 1.2604 1.3024 1.3483 1.2826 1.3118 1.3450 1.2902 1.3151 1.3440

1.1830 1.2491 1.3176 1.1301 1.2384 1.3609 1.0940 1.1647 1.2518 1.2269 1.3177 1.4248 1.2328 1.2948 1.3747

G a u s s -- G a u s s

TABLE

Yof

0.3827 0.4502 0.5282 0.4135 0.4690 0.5255 0.4412 0.4795 0.5207 0.4506 0.4832 0.5191

0.3101 0.4016 0.4879 0.2290 0.3818 0.5259 0.1704 0.2751 0.3799 0.3723 0.4894 0.6093 0.3799 0.4607 0.5514 -

-

as

-

-

--0.1202 --0.1202 --0.1202 --0.7067 --0.7067 --0.7067 - - 9 X 10 ~ - - 9 x 102 --9X 102 --0.6007 --0.6007 --0.6007 --3.532 --3.532 3.532 --5.919 --5.919 --5.919 --5.919 5.919 --5.919 --3X 10 a - - 3 X 10 a - - 3 x 10 a --0.8411 --0.8411 --0.8411 --4.946 --4.946 --4.946 - - 5 x 10a - - 5 × 10 a - - 5 X 10 a

~'os

0.1412 0.1412 0.1412 0.1071 0.1071 0.1071 0.1000 0.1000 0.1000 0.7060 0.7060 0.7060 0.5353 0.5353 0.5353 0.5211 0.5211 0.521I 0.5211 0.5211 0.5211 0.5000 0.5000 0.5000 0.9884 0.9884 0.9884 0.7495 0.7495 0.7495 O.7O0O 0.7000 0.7000

--0.020 --0.026 --0.014 --0.002 --0.022 --0.031 --0.003 --0.019 --0.027 --0.004 --0.018 --0.026

--0.013 --0.021 --0.026 0.054 0.029 0.015 0.140 0.186 0.204 --0.023 --0.031 --0.029 --0.020 --0.027 --0.019

,..j

0.7

0.5

0.6

0.1

1.0

0.9

0.6

1.0

0.9

0.6

1.0

0.9

$e

bc

0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1

bt

1.2056 1.4115 1.5998 0.7473 0.9134 1.0713 0.4964 0.6152 0.7301 1.2640 1.3521 1.4675 0.9899 1.0499 1.1411 0.8847 0.9358 1.0175 1.3578 1.4299 1.5312 1.1383 1.1781 1.2555 1.0551 1.0857 1.1551

Sf

3.60 5.37 6.79 1.73 2.77 3.68 0.85 1.41 1.91 3.64 5.17 6.30 2.50 3.64 4.49 2.09 3.06 3.79 3.84 5.48 6,66 2.86 4.19 5.15 2.52 3.72 4.59

0.00915 0.01701 0.02465 0.00584 0.01091 0,01582 0.00397 0.00740 0.01O65 0.01084 0.01833 0.02498 0.00923 0.01529 0.02051 0.00851 0.01402 0.01872 0.01192 0.02017 0.02730 0.01091 0.01808 0.02408 0.01043 0.01718 0.02277

Q

at

1.1274 1.2019 1.2690 1.0845 1.1244 1.1630 1.0663 1.0865 1.1063 1,2073 1.2596 1.3063 1.2212 1.2538 1.2840 1.2258 1.2520 1.2767 1.2420 1.2976 1.3461 1.2719 1.3095 1.3434 1.2815 1.3134 1.3426

Yukawa

0.2279 0.3454 0.4486 0.1573 0.2305 0.3073 0.1251 0,1650 0.2112 0.3433 0.4140 0.4805 0.3615 0.4051 0.4502 0.3674 0.4024 0.4398 0.3924 0.4597 0.5210 0.4290 0.4720 0.5144 0.4402 0.4760 0.5123

Gauss

TABLE I V

0.2237 0.3274 0.4012 0.1538 0.2129 0.2561 0.1233 0.1538 0.1757 0.3432 0.4113 0.4597 0.3630 0.4051 0.4364 0.3694 0.4034 0.4289 0.3868 0.4583 0.5089 0.4258 0.4738 0.5094 0.4382 0.4788 O.5095

¢/'o~

10

a

-

-

--103 --10a -- 0.6007 - - O. 6007 -0.6007 --3.532 --3.532 --3.532 - - 3 x 103 --3xlO 3 - - 3 x 10a --0.8411 --0.8411 --0.8411 --4.946 4.946 --4.946 --5Z lO8 - - 5 X 103 - - 5 x 10a

--

--0.1202 --0.1202 --0.1202 --0.7067 --0.7067 --0.7067

ae

0.1412 0.1412 0.1412 0.1071 0.1071 0.1071 0.1000 0.1000 0.1000 0.7060 0.7060 0.7060 0.5353 0.5353 0.5353 0.5000 0.5000 0.5000 0.9884 0.9884 0.9884 0.7495 0.7495 0.7495 0.7000 0.7000 0.7000

"los

0.101 0.123 0.175 0.281 0.434 0.717 0.275 0.742 1.527 0.003 0.006 0.048 --0.008 -- 0.002 0.036 --0.010 --0.006 0.029 0.025 0.002 0.020 0.011 --0.005 --0.007 0.007 --0.007 0,003

P

t~

0.7

0.5

0.6

0.1

1.O

0.9

0.6

1.O

0.9

0.6

1.0

0.9

s,

b,

0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1

b~

a, 1.1876 1.3221 1.4609 1.1370 1.2489 1.3742 1.0954 1.1677 1.2562 1.2160 1.3175 1.4325 1.2127 1.2849 1.3737 1.2109 1.2702 1.3455 1.2336 1.3324 1.4434 1.2416 1.3138 1.4009 1.2442 1.3059 1.3825

Q 0.3149 0.4814 0.6149 0.2420 0.4011 0.5460 0.1756 0.2953 0.4290 0.3579 0.4780 0.5925 0.3564 0.4443 0.5427 0.3551 0.4284 0.5163 0.3901 0.4990 0.6038 0.4066 0.4848 0.5716 0.4119 0.4784 0.5566

Q 0.01152 0.02506 0.04076 0.00852 0.01946 0.03275 0.00592 0.01368 0.02315 0.01099 0.02302 0.03680 0.00897 0.01886 0.03011 0.00803 0.01691 0.02692 0.01136 0.02357 0.03731 0.00969 0.02013 0.03172 0.00895 0.01861 0.02926

5.39 8.33 10.70 3.37 5.89 8.16 1.79 3.42 5.06 4.48 7.06 9.22 3.00 5.05 6.87 2.40 4.16 5.76 4.49 7.05 9.17 3.16 5.27 7.10 2.67 4.55 6.21

1.3364 1.6160 1.8827 0.9660 1.2770 1.5747 0.6588 0.9229 1.1940 1.1586 1.3859 1.6388 0.8733 1.0823 1.3230 0.7565 0.9500 1.1765 1.1778 1.3808 1.6163 0.9243 1.1077 1.3283 O.8269 0.9988 1.2087

Gauss

PD

-

st

Yukawa

TABLE V

29.4

--

I(P

--0.5303 --0.5303 - - O. 5303 --2.948 --2.948 --2.948 - - 3 X 103 - - 3 X 10a - - 3 × 10 a --0.7424 --0.7424 - - 0.7424 --4.128 --4.128 --4.128 --l(P -10 4

--

--29.4

29.4

--0.018 --0.027 - - 0.033 0.046 0.022 0.006 0.193 0.193 0.188 0.023 --0.020 -- 0.027 0.057 0.001 --0.005 0.073 0.015 0.009 0.077 --0.009 - - 0.025 0.118 0.018 --0.007 0.131 0.032 0.003 0.1845 0.1845 0.1845 0.1154 0.1154 0.1154 0.1005 0.1005 0.1005 0.9204 0.9204 0.9204 0.5748 0.5748 0.5748 0.5002 0.5002 0.5002 1.2886 1.2886 1.2886 0.8046 0.8046 0.8046 0.7003 0.7003 0.7003 --0.1058 --0.1058 --0.1058 --0.5794 --0.5794 --0.5794

0.3171 0.4938 0.6487 0.2397 0.3959 0.5427 0.1722 0.2788 0.3863 0.3533 0.4867 0.6163 0.3460 0.4432 0.5458 0.3425 0.4231 0.5112 0.3708 0.5011 0.6269 0.3766 0.4738 0.5752 0.3784 0.4622 0.5523 --

P

ro,

a.

ro,

1.0

0.9

0.6

1.0

0.9

0.6

1.1

0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.I 0.5 0.8 1.1 0.5 0.8

b~

0.6634 0.6891 0.6783 0.7 0.6 0.5 0.8 1.1 0.9 0.5 ! 0.8 i 1.1 0.5 i 1.0 0.8 1.1

0.5

0.1

bc

1.2482 1.4548 1.6420 0.7823 0.9584 1.1229 0.4981 0.6215 0.7392 1.2013 1.3273 1.4633 0.8871 0.9876 1.0996 0.7629 0.8515 0.9512 1.2218 1.2477 1.3604 1.4866 0.9718 1.0605 1.1650 0.8671 0.9465 1.0416

St

3.77 5.60 7.04 1.85 2.99 3.97 0.85 1.43 1.95 3.44 5.01 6.20 2.16 3.27 4.14 1.69 2.59 3.32 4.12 3.54 5.14 6.34 2.39 3.60 4.54 1.98 3.03 3.85

~D 0.00947 0.01758 0.02542 0.00610 0.01146 0.01666 0.00398 0.00747 0.01077 O.OlOlO 0.01762 0.02447 0.00807 0.01396 0.01920 0.00716 0.01236 0.01695 0.01468 0.01071 0.01861 0.02570 0.00907 0.01563 0.02136 0.00835 0.O1435 0.01957

Q

-

0.2361 0.3559 0.4593 0.1629 0.2418 0.3229 0.1242 0.1656 0.2132 0.3221 0.4000 0.4726 0.3323 0.3799 0.4295 0.3359 0.3727 0.4125 0.4000 0.3600 0.4329 0.5003 0.3871 O.4328 0.4793 0.3959 0.4328 0.4717

Yukawa at

1.1325 1.2098 1.2785 1.0876 1.1313 1.1733 1.0656 1.0866 1.1072 1.1877 1.2444 1.2953 1.1934 1.2280 1.2606 1.1955 1.2222 1.2477 1.2399 1.2076 1.2655 1.3171 1.2238 1.2617 1.2970 1.2294 1.2605 1.2898

Yukawa

TABLE V I

0.2321 0.3388 0.4141 0.1592 0.2236 0.2708 0.1218 0.1535 0.1768 0.3111 0.3869 0.4412 0.3169 0.3640 0.3994 0.3192 0.3557 0.3836 0.3771 0.3319 0.4097 0.4659 0.3497 0.4015 0.4407 0.3561 0.3986 0.4316

Yot

-

-

--0.1058 - - O. 1058 --0.1058 --0.5794 - - O . 5794 - - O . 5794 --29.4 --29.4 --29.4 --0.5303 --0.5303 --0.5303 --2.948 --2.948 --2.948 - - 3 × 103 - - 3 × 103 - - 3 × 103 -- 1.018 --0.7424 --0.7424 --0.7424 --4.128 --4.128 --4.128 - - 104 --104 10 4

as

0.1845 0.1845 0.1845 0.1154 0.1154 0.1154 0.1005 0.1005 0.1005 0.9204 0.9204 0.9204 0.5748 0.5748 0.5748 0.5002 0.5002 0.5002 1.048 1.289 1.289 1.289 0.8046 0.8046 0.8046 0.7003 0.7003 0.7003

0.111

0.074 O. 104 0.152 0.218 0.383 0.620 0.355 0.804 1.528 0.085 0.051 0.086 0.114 0.076 0.110 0.119 0.087 0.119 0.092 0.163 0.072 0.077 0.188 O. 103 0.101 0.189 0.116

P

tO

0.7

0.5

0.1

1.0

0.9

0.6

1.3058 1.5635 1.8005 0.8874 1.1515 1.4025 0.5855 0.7920 O.9996 1.1700

1.3505

1.5501 0.8707 1.0275 1.2065 0.7507 0.8930 1.0574 1.2016 1.3620 1.5459 0.9372 1.0744 1.2381 0.8365 0.9629 1.1157

0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5

0.8

1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1

4.70 7.18 9.17 2,67 4.60 6.34 1.33 2.47 3.58 3.99 6.10 7.81 2.58 4,18 5.55 2.04 3.38 4.55 4.05 6.15 7.84 2.78 4.46 5.86 2.33 3.80 5.05

0.01068 0.02171 0.03355 0.00743 0.01573 0.02501 0.00502 0.01068 0.01695 0.01055 0.02031 0.03043 0.0085I 0.01634 0.02436 0.00757 0.01455 0.02164 0.01104 0.02107 0.03129 0.00936 0.01781 0.02626 0.00863 0.01640 0.02413

Yukawa

VII

0.5243 0.4030 0,4539 0.5121

0.4573

0.2787 0.4279 0.5517 0.2029 0.3296 0.4540 0.1484 0.2303 0.3273 0.3391 0.4390 0.5352 0.3432 0.4104 0.4851 0.3444 0.3984 0.4620 0.3742 0.4656 0.5535 0.3958 1.1616 1.2713 1.3790 1.1120 1.1923 1,2793 1.0794 1.1251 1.1785 1.2011 1.2801 1.3630 1.2021 1.2544 1.3132 1.2023 1.2440 1.2921 1.2199 1.2980 1.3791 1.2318 1.2860 1.3457 1.2360 1,2812 1.3324

- Exponential

TABLE

0.2777 0.4264 0.5488 0.1999 0.3157 0.4196 0.1452 0.2153 0.2797 0.3314 0,4371 0.5312 0.3302 0.4017 0.4701 0,3296 0.3871 0.4435 0.3506 0.4557 0.5483 0.3620 0.4361 0.5061 0.3661 0.4285 0.4890 -

-

-

-

-

-

--0.5794 O.5794 --0.5794 --29.4 --29.4 --29.4 --0.5303 --0.5303 --0.5303 --2.948 --2.948 - - 2.948 - - 3 x 10 a - - 3 × 10 a - - 3 X 10 a --0.7424 --0.7424 -- O.7424 --4.128 --4.128 --4.128 104 --104 104

- - O. 1 0 5 8

--0.1058 --0.1058

0.1845 0.1845 0.1845 0.1154 0.1154 0.1154 0.1005 0.1005 0.1005 0.9204 0.9204 0.9204 0.5748 0.5748 0.5748 0.5002 O.5O02 0,5002 1.289 1.289 1.289 0.8046 0.8046 0.8046 0.7003 0.7003 0.7003

0,013 0.002 0.002 0.091 0,109 0.116 0.298 0.361 0.530 0.047 0.003 0.005 0.084 0.030 0.034 0.097 0.044 0.050 0.114 0.018 0.005 0.151 0.052 0.029 0.160 0.067 0.042

0

O

0

C

t~

0

©

¢5

o

c~

t~

b~

$c

0.5 0.8 1.1 0.5 0.8 1.1 0,5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1 0.5 0.8 1.1

1.2208 1.4274 1.6155 0.7575 0.9274 1.0877 0.4932 0.6133 0.7287 1.2287 1.3343 1.4595 0.9377 1.0170 1.1181 0.8258 0.8953 0.9858 1.3021 1.3912 1.5038 1.0573 1.1190 1.2091 0.9655 1.0185 1.1004

St

3.66 5.46 6.88 1.76 2.84 3.77 0.84 1.40 1.90 3.53 5.07 6.22 2.33 3.45 4.30 1.90 2.84 3.57 3.70 5.30 6.48 2.65 3.90 4.85 2.28 3.39 4.24

0.00926 0.01722 0.02494 0.00591 0.01108 0.01608 0.00395 0.00738 0.01062 0.01045 0.01792 0.02464 0.00865 0.01462 0.01984 0.00786 0.01323 0.01788 0.01133 0.01935 0.02643 0.01003 0.01687 0.02272 0.00944 0.01582 0.02123

0.2309 0.3494 0.4527 0.1589 0.2340 0.3123 0.1242 0.1643 0.2106 0.3332 0.4066 0.4757 0.3483 0.3934 0.4403 0.3534 0.3891 0.4275 0.3776 0.4467 0.5105 0.4103 0.4541 0.4981 0.4206 0.4567 0.4940

(~

Exponential

1,1294 1.2050 1.2726 1.0855 1.1267 1.1663 1.0659 1.0861 1.1061 1.1982 1.2520 1.3004 1.2088 1.2420 1.2732 1.2124 1.2388 1.2639 1.2265 1.2824 1.3319 1.2505 1.2878 1.3221 1.2585 1.2898 1.3190

a.t

- Yukawa

TABLE V I I I

-

-

-

0.1565 0.1565 0.1565 0.1098 0.1098 0.1098 0.1000 0.1000 0.1000 0.7822 0.7822 0.7822 0.5491 0.5491 0.5491 0.5OOO 0.5000 0.5000 1.095 1.095 1.095 0.7688 0.7688 0.7688 0.7000 0.7000 0.7000

--0.1147 --0.1147 --0,1147 --0.6618 --0.6618 --0.6618 +104 +104 +104 --0.5737 --0.5737 --0.5737 --3.309 --3.309 --3.309 - - 9 × 105 - - 9 × 105 - - 9 × 105 --0.8032 --0.8032 --0.8032 --4.632 --4.632 --4.632 lO s _106 lOs O.2264 0.3319 0.4060 0.1556 0.2163 0.2608 0.1223 0.1528 0.1750 0.3284 0.3993 0.45O2 0.3426 0.3866 0.4196 0.3475 0.3825 0.4091 0.3617 0.4353 0.4880 0.3919 0.4410 0.4781 0.4020 0.4431 0.4747 -

Fo$

as

~'ot

0.117 0.113 O. 167 0.262 0.427 0.689 0.351 0.808 1.563 0.034 0.026 0.065 0.035 0.027 0.064 0.035 0.027 0.062 0.081 0.031 0.045 0.073 0.035 0.042 0.068 0.036 0.042

P

2t~

0.7

0.5

0.1

bc

1.0009

0.6369 0.74 0.8425

11.1865

i0 7996 11.0949 1.2378

1.1195 0.8759 0.7427

0.01472 0.01472 0.01471 0.01472

2.50 4.25 3.23 2.65 !

0.01470

3.21

0.8450

0.829

1.0563

0.7119

1 0.00887 0.01860 0.02443

2.69 4.24 5.55

0.8917 1.0257 1.1935

0.5 0.8 1.1

0.01473

0.9

I

0.00492 0.01049 0.01666

4.34

I

i

i

0.01524 0.02428

1.0918

1.29 2.39 3.49

0.5737 0.7782 0.9842

0.5 0.8 1.1

1.0

:

0.01048 0.02143 0.03320

0.679

4.39 6.09

1.1184 1.3686

0.8 1.1

0.9

h

0.7798

4.59 7.07 9.06

1.2819 1.5449 1.7851

PD

0.5 0.8 1.1

bt

0.6

Se

0.4365 0.4641 0.4767

0.4163

0.4097

0.3560 0.4177 0.4876

0.3958 0.3520 0.4168 0.4798

1.0786 1.1230 1.1750

1.2150 1.2634 1.3185

0.1443 0.2114 0.2748

1.1854 1.2701

0.3201 0.4438 0.1467 0.2266 0.3221

0.3057 0.4075

1.1582 1.2675 1.3750

0.2727 0.4215 0.5438

gog

0.2738 0.4235 0.5482

~f

Exponential - Exponential

TABLE I X

0.5491 0.5491 0.5491

0.022 0.001 0.015

0.232 0.403 0.558

0.1000 0.1000 0.1000 1.3 × ]0 ~ 1.3 × 104 1.3 × 104

--3.309 --3.309 --3.309

0.126 0.1 23

0.1098 0.1098

0.013 00.04 0.005

p

--0.6618 --0.6618

I

0.1565 0.1565 0.1565

~'os

--0.1147 -- 0.1147 --0.1147

as

b~

~44

M. H.

K A L O S , L. C. B I E D E N H A R I ~ A N D J .

1~. B L A T T

therefore on the order of a few percent. (Here the actual error will depend upon the importance of the central potential in the triplet state. ) Except for the depth of the exponential well, the parameters for the range and depth of the potentials are taken from Blatt and Jackson 3). Most values of P are computed from the relation

P = ( 1 - - a t -1-½ro, ) ro~3

and m a y suffer from inaccuracies in a~ and r0v For the most part, we estimate an error in P of about 0.005. For the shortest ranges, the error m a y be as large as 0.1. The square well results for P were computed from an integral expression 1) and are accurate to the figures given. In tables I-IX, the notation 'Yukawa-exponential' means that the central potential is of the Yukawa shape and the tensor potential has an exponential shape. Wave functions are not available for most of the cases shown in the tables. For the 'Yukawa-Gauss', 'Yukawa-exponential', 'Yukawa-Yukawa', and 'ExponentialYukawa' combinations, short tables of wave functions which permit a sketch of the shape have been printed. Results on the Yukawa-Yukawa combination have already been published b y Feshbach and Schwinger 9). Because our calculations are usually more accurate and include the scattering state at zero energy, we have included the new results. Interpolation shows that they are in agreement with the earlier work. We wish to acknowledge the excellent work of Miss Patricia Boland in the square well computations and the helpful cooperation of the staff of the ILLIAC digital computer.

References I) L.C. Biedenharn, Thesis submitted to the Massachusetts Institute of Technology, 1949 (unpublished) 2) C . W . Li a al., Phys. Rev. 83 (1951) 512 3) J. M. ]31att and J. D. Jackson, Phys. Rev. 76 (1949) 18 4) G. F. Newell, Phys. Rev. 77 (1950) 141; H. G. Kolsky et al., Phys. Rev. 81

(1951) 1061 5) H. A. Bethe, Phys. Rev. 76 (1949) 38; N. Austern, Phys. Rev. 92 (1953) 670; L. C. Biedenharn and J. M. Blatt, Phys. Rev. 93 (1954) 1387 6) Burgy, 1Ringo and Hughes, Phys. Rev. 84 (1951) 1160; E. Melkonian, Phys. Rev. 76 (1949) 1766 7) Fields, Becker and Adair, Phys. Rev. 94 (1954) 389 8) J. D. Jackson and J. 1VL Blatt, Rev. l~od. Phys. 22 (1950) 77 9) H. Feshbach and J. Schwinger, Phys. Rev. 84 (1951) 194