References
Abstract

The model suggested by Love and Franey for description of the nucleon-nucleon interaction was used as the base. The new fitting of the model parameters was done in the energy range from 100 MeV up to 1100 MeV. It is based on the modern partial-wave-analysis solution for NN-amplitudes. The three observables: differential cross section, vector analyzing power, and spin correlation coefficient – were obtained at every energy. The results are compared with existing the experimental data.

New parameterization of effective nucleon-nucleon -matrix interaction for scattering at intermediate energies.

N.B.Ladygina

E-mail:nladygina@jinr.ru

In order to solve the problems of the nucleon-nucleus scattering, it is very important to know the coupling between two nucleons. A quantitative knowledge of the nucleon-nucleon coupling from first principles over a wide energy range must await a more complete theory of strong interaction.

In this connection it is useful to have a model which describe the effective nucleon-nucleon interaction. Such phenomenological model was suggested by W.G.Love and M.A.Franey in ref.[1]. According to this paper -matrix can be expressed in the following form

(1)

where is the center-of-mass energy; is the scattering angle in the c.m. system. The unit vectors form a right-handed coordinate system with

(2)

where and are the initial and final momenta in the c.m. system.

The usual tensor operator , the singlet spin-projection operator , and the triplet spin-projection operator were used in Eq.(1). It should be noted, that t-matrix, here, is the antisymmetrized one, i.e. , where is the permutation operator of the particles of ”1” and ”2”.

This -matrix is connected with -amplitude by the relationship

(3)

The amplitudes are parameterized according to the symmetric properties. In case, when isospin NN-pare equals to zero, , these amplitudes have the following form

(4)

For these amplitudes are

(5)

In this paper the new set of the coefficients and were obtained by fitting SP07 [2] amplitudes. The energy region from 100 MeV up to 1100 MeV was considered. In order to get the amplitudes subset for isospin equal to zero, the ”NP” and ”NP1” subsets were used. The new sets of parameters are presented in the tables below. The three observables: differential cross section, vector analyzing power and spin correlation coefficient – were obtained for every energy and were compared with the experimental data [2]. On the figures below the fitting results are presented by the red lines. The black curves correspond to the results obtained with the amplitudes from SAID [2].

The NN -matrix can be also expressed in the standard form

(6)

The relationships between amplitudes and can be got with comparison of the two definitions of -matrix, Eq.(1) and Eq.(6).

(7)

All notations used in this paper are the same as in ref.[1].

The work has been supported in part by the Russian Foundation for Basic Research under grant 07-02-00102a.

References

  • [1] W.G.Love, M.A.Franey, Phys. Rev.C 24, 1073 (1981); W.G.Love, M.A.Franey, Phys.Rev.C 31, 488 (1985).
  • [2] http://gwdac.phys.gwu.edu

, Re part

R VSE VTO VLSO VTNO
0.11 8352.18 4669.53 0 -600000
0.15 -680.002 -2523.94 0 23340.9
0.25 383.456 142.706 0 -200
0.4 -753.348 -895.424 0 288.248
0.55 220.39 360.761 0 -47.7974
0.7 -50.1382 -91.3391 0 5.97377
1.4 -0.966063 0.55162 0 -0.00201629

, Im part

R VSE VTO VLSO VTNO
0.11 -6519.78 -6037.04 17149.7 -1.17288e+06
0.15 7112.78 2053.56 -39.1544 857602
0.25 -7831.77 -5679.67 228.483 -55084.2
0.4 1901.84 962.641 1076.17 2887.14
0.55 -1.10827 21.4285 -79.9082 -310.513
0.7 62.9981 33.4869 2.20799 10.0006
1.4 9.77397 -3.48397 -0.173829 -0.0710843

, Re part

R VSO VTE VLSE VTNE
0.11 29001.4 -3280.68 0 1.27875e+06
0.15 -78611.4 43315.1 0 -20000
0.25 2329.43 -4559.44 0 920
0.4 -9198.27 704.897 0 -800
0.55 4694.37 -1299.19 0 145.923
0.7 -936.359 64.2434 0 -38.7549
1.4 14.9161 -6.81892 0 -0.026915

, Im part

R VSO VTE VLSE VTNE
0.11 -2103.44 56000 -40003.8 -400000
0.15 220004 -2500 4999.68 400
0.25 -15781.1 450 -50.032 -30000
0.4 800.06 -3476.29 1370.94 4251.08
0.55 -2027.65 1400 -80.0005 -514.627
0.7 680 -220 11.1557 92.3779
1.4 -40.7887 2.67444 -0.65136 0.140121

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 21605.3 5564.15 0 -1e+06
0.15 -1247.82 -5599.74 0 124038
0.25 333.265 149.951 0 -800
0.4 -1299.38 -908.962 0 153.371
0.55 434.175 359.347 0 -35.7073
0.7 -91.1403 -82.007 0 5.46177
1.4 0.00100003 0.498954 0 -0.00872548

, Im part

R VSE VTO VLSO VTNO
0.11 -8624.53 -8271.96 18202.5 -816552
0.15 7230.76 5172.77 -87.9493 300032
0.25 -8081.44 -6332.04 230.637 -20160.7
0.4 2166.57 1008.42 1149.09 1677.25
0.55 -164.434 24.2187 -151.771 -290.396
0.7 131.628 20.3244 7.41174 19.102
1.4 7.79064 -3.39373 0.0362834 -0.111059

, Re part

R VSO VTE VLSE VTNE
0.11 25999.9 -5359.08 0 850000
0.15 -41962.5 27454.7 0 -40000
0.25 2185.95 -3999.81 0 972.163
0.4 -5768.09 700.001 0 -800
0.55 3625.01 -799.977 0 117.309
0.7 -950 37.101 0 -26.6337
1.4 23.5844 -7.1173 0 -0.0774762

, Im part

R VSO VTE VLSE VTNE
0.11 -4000.02 35929.1 -60000.4 -200000
0.15 249994 -4938.93 2999.99 76436.7
0.25 -20027.1 280.001 -100.002 -16000
0.4 1226.74 -2999.9 1099.95 2810.16
0.55 -2389.81 1370.36 -159.609 -351.581
0.7 704.295 -150 22.2239 68
1.4 -33.4615 2.34164 -0.450722 0.320879

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 26672.7 6019.47 0 -1.18252e+06
0.15 -440.936 -7940.31 0 209184
0.25 338.321 462.706 0 -4417.48
0.4 -1955.99 -910.192 0 330.769
0.55 662.155 357.713 0 -64.9937
0.7 -124.504 -81.137 0 10.3199
1.4 -0.211034 0.654291 0 -0.0221523

, Im part

R VSE VTO VLSO VTNO
0.11 -8700.55 -8520.21 30105 -315784
0.15 8994.24 6830.08 -559.126 95000
0.25 -8691.5 -6492.94 259.858 -7914.95
0.4 2398.75 1018.76 963.971 972.465
0.55 -248.855 49.6393 -151.998 -229.982
0.7 132.005 -2.2 12.2475 17.9979
1.4 7.21693 -3.03145 0.0281493 -0.13225

, Re part

R VSO VTE VLSE VTNE
0.11 15872.8 -9959.96 0 700000
0.15 -41011.9 16785.5 0 -74674.8
0.25 1519.37 -2500.03 0 1000
0.4 -3005.64 699.998 0 -581.599
0.55 2834.9 -498.014 0 95.6785
0.7 -976.34 20.0001 0 -25.4931
1.4 27.6585 -9.56093 0 -0.0611397

, Im part

R VSO VTE VLSE VTNE
0.11 -6059.03 25992.6 -83997.3 -100000
0.15 199774 -5155.48 2001.06 115271
0.25 -21601.1 400.025 -200.298 -14500
0.4 1505.84 -2299.71 1177.83 2146.74
0.55 -2451.54 1098.88 -248.234 -220
0.7 829.524 -155.576 39.9428 53.6106
1.4 -36.9757 2.83388 -0.474437 0.276692

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 35128.7 8969.36 0 -1.4e+06
0.15 -573.388 -7987.51 0 305229
0.25 343.641 823.923 0 -9317.37
0.4 -2767.96 -915.473 0 529.21
0.55 929.159 300.001 0 -65.1609
0.7 -164.348 -65.5461 0 7.57547
1.4 -0.657018 0.697277 0 0.00274612

, Im part

R VSE VTO VLSO VTNO
0.11 -14196.8 -9755.82 34347 -174487
0.15 30704.2 7612.04 -1027.46 30005.5
0.25 -16339.6 -6599.66 259.974 -3000.55
0.4 3549.85 1022.54 839.925 599.838
0.55 -428.424 49.9533 -152 -178.924
0.7 144.327 -4.00007 16.5247 14.6032
1.4 6.439 -3.02676 -0.00171169 -0.138766

, Re part

R VSO VTE VLSE VTNE
0.11 10003.5 -6500.01 0 700000
0.15 -37996.9 19984.3 0 -105666
0.25 3011.6 -4313.26 0 2500
0.4 -1899.16 509.9 0 -605.942
0.55 1997.57 -400.106 0 101.344
0.7 -786.388 66.1521 0 -26.8545
1.4 27.8255 -8.85684 0 -0.0519602

, Im part

R VSO VTE VLSE VTNE
0.11 -9000.74 26096.1 -86161.3 -140000
0.15 150000 -6880.2 3602.31 147643
0.25 -20695.4 800.026 -500.093 -14459.2
0.4 2202.03 -2394.61 1199.98 1878.77
0.55 -2500.01 1161.22 -267.554 -180
0.7 850.16 -199.975 40.3628 52.5685
1.4 -38.5451 2.80001 -0.392178 0.178394

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 37862.8 14504.7 0 -1.74716e+06
0.15 -4954.58 -837.824 0 455147
0.25 3767.37 1500.01 0 -19986.4
0.4 -4346.01 -1839.63 0 1541.32
0.55 1293.77 637.077 0 -251.049
0.7 -208.814 -134.014 0 31.7401
1.4 -0.995248 1.26681 0 -0.0407424

, Im part

R VSE VTO VLSO VTNO
0.11 -14492.5 -20517.6 60377 -194388
0.15 89149.1 9483.94 -8021.8 16807.9
0.25 -43841.8 -6649.3 260.464 -800
0.4 9572.99 1074.55 758.391 424.257
0.55 -1771.73 49.9999 -152.473 -150.147
0.7 294.519 -6.0001 19.8721 12.6668
1.4 6.39985 -3.09034 -0.0364889 -0.142317

, Re part

R VSO VTE VLSE VTNE
0.11 33722.8 -12212.7 0 750000
0.15 -43521.8 23397.1 0 -161042
0.25 4500.25 -5499.99 0 7500
0.4 -1350.43 1000.49 0 -1068.63
0.55 1098.49 -664.344 0 179.948
0.7 -449.229 150 0 -36.1737
1.4 23.9514 -6.09722 0 -0.0409734

, Im part

R VSO VTE VLSE VTNE
0.11 -15019.4 29775.1 -93983.5 -220000
0.15 99907.1 -10333 8001.18 152897
0.25 -16195.6 1528.95 -728.691 -14042.5
0.4 3005.14 -2544.99 1246.66 1927.52
0.55 -2900.47 1099.96 -321.017 -200
0.7 946.45 -170.001 52.1427 51.7188
1.4 -40.3178 2.75396 -0.413609 0.191684

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 39223.1 39999.2 0 -1.9818e+06
0.15 -16823.7 -859.535 0 594371
0.25 8156.47 3249.9 0 -32689.4
0.4 -4906.45 -3297.6 0 2833.79
0.55 1302.73 1137.8 0 -501.582
0.7 -214.687 -233.693 0 65.2895
1.4 -1.00004 2.0221 0 -0.108178

, Im part

R VSE VTO VLSO VTNO
0.11 -22964.1 -26608.4 71451 -489266
0.15 116414 25000.2 -15177.1 85000
0.25 -53112.9 -9611.59 861.466 -1897.11
0.4 10672.9 1353.9 609.355 318.405
0.55 -1961.67 50.3023 -127.863 -97.8803
0.7 300.803 -17.8269 17.5571 4.03453
1.4 6.39 -2.90834 -0.0387547 -0.113632

, Re part

R VSO VTE VLSE VTNE
0.11 34999.8 -23999.8 0 762619
0.15 -46778.3 25015 0 -197296
0.25 8064.22 -5701.42 0 11698
0.4 -2764.45 1834.76 0 -1422.61
0.55 1191.14 -965.507 0 220
0.7 -256.618 180.001 0 -37.7607
1.4 10.4361 -3.30028 0 -0.0672715

, Im part

R VSO VTE VLSE VTNE
0.11 -30002.5 35000.3 -95999.2 -350000
0.15 84998.8 -15973.9 11000 191095
0.25 -15228.7 3114.64 -971.086 -16483.9
0.4 3300.84 -3000 1295.6 2200
0.55 -2866.78 1170.99 -379.887 -260.55
0.7 907.596 -170.001 67.0043 57.0783
1.4 -39.6339 2.5781 -0.479642 0.180795

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 45263.7 34976.6 0 -1.80674e+06
0.15 -61485 -1082.61 0 616534
0.25 26601 8006.56 0 -40869.1
0.4 -7902.72 -5585.28 0 3907.26
0.55 1662.15 1885.61 0 -737.9
0.7 -216.65 -379.801 0 99.4035
1.4 -1.94428 3.12775 0 -0.183929

, Im part

R VSE VTO VLSO VTNO
0.11 -34920 -26764.2 46699.4 -656304
0.15 120864 75446.9 -18638.6 160970
0.25 -53431.5 -19034.7 2628.57 -6500
0.4 11362.7 1408.82 304.796 458.73
0.55 -2532.68 348.91 -66.2242 -89
0.7 445.215 -111.874 9.37959 1.91424
1.4 4.84994 -1.92047 -0.0220682 -0.112063

, Re part

R VSO VTE VLSE VTNE
0.11 38678.4 -33538.1 0 819999
0.15 -60050 26277.9 0 -241958
0.25 17152.3 -5845.67 0 17400.3
0.4 -5998.57 3101.48 0 -2300.32
0.55 1622.22 -1813.63 0 432.924
0.7 -84.9992 399.808 0 -70.4326
1.4 1.64853 -2.06173 0 0.0163029

, Im part

R VSO VTE VLSE VTNE
0.11 -53124.9 49726.5 -98084.6 -680000
0.15 64437.3 -31519.1 28523.1 270000
0.25 -11324.3 6008.46 -2698.22 -18737.3
0.4 3287 -3045.78 1332.24 2252.89
0.55 -2808.41 1306.58 -387.504 -286.462
0.7 857.454 -319.919 69.3883 61.7145
1.4 -37.3773 9.29109 -0.485126 0.131033

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 45300.9 34887.9 0 -800000
0.15 -79869.3 -917.097 0 320054
0.25 36206.3 6224.42 0 -25343.3
0.4 -10176.9 -4412.2 0 2511.69
0.55 2127.49 1327.29 0 -497.576
0.7 -261.752 -258.907 0 70.3886
1.4 -2.43463 2.48021 0 -0.128731

, Im part

R VSE VTO VLSO VTNO
0.11 -56467.2 -13305.2 25165.4 -920000
0.15 99981 121165 -18003.1 291900
0.25 -36248.1 -31106.7 3546.83 -17421.9
0.4 6145.54 2806.01 119.925 1236.76
0.55 -1034.03 146.599 -25.1756 -156.989
0.7 204.387 -112.002 3.38124 2.05897
1.4 4.72574 -1.29686 -0.00598186 -0.0544541

, Re part

R VSO VTE VLSE VTNE
0.11 39000 -34171.7 0 808495
0.15 -99999.9 29618 0 -259064
0.25 27341.1 -7497.5 0 19949.5
0.4 -7133.84 3443.67 0 -2646.22
0.55 1700.12 -1865.03 0 516.977
0.7 -103.451 400 0 -82.5902
1.4 4.47402 -0.481941 0 0.0493507

, Im part

R VSO VTE VLSE VTNE
0.11 -82029.8 85585.8 -97896.2 -380000
0.15 54576.5 -46437.8 40754.2 175805
0.25 -8000.48 4000.48 -4054.97 -14975.3
0.4 3146.73 -2561.78 1336.19 2303.39
0.55 -2717.57 1632.37 -380 -398.557
0.7 799.87 -498.918 67.7489 86.4692
1.4 -35.4606 11.2218 -0.464504 0.0050183

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 45401.6 14223.1 0 -280001
0.15 -74533.3 -909.33 0 139758
0.25 33478.5 4815.19 0 -13235
0.4 -9375.58 -2655 0 1360.05
0.55 1809.97 429.457 0 -305.773
0.7 -189.67 -47.8691 0 48.2382
1.4 -2.9259 0.8333 0 -0.0968045

, Im part

R VSE VTO VLSO VTNO
0.11 -84664.2 -9301.41 53744.1 -600000
0.15 99433 116767 -27889.2 225933
0.25 -28765 -30906 3551.32 -17130.4
0.4 4099.44 2829.68 159.308 1316.68
0.55 -600 114.117 -41.1986 -165.379
0.7 163.771 -116.036 6.12693 2.03006
1.4 3.70942 -0.897898 -0.0157595 -0.042234

, Re part

R VSO VTE VLSE VTNE
0.11 40555.1 -28315.4 0 683621
0.15 -108846 28844.1 0 -244909
0.25 28152.1 -6934.63 0 21046.9
0.4 -6500.3 1971.89 0 -2813.95
0.55 1500.04 -1021.26 0 561.755
0.7 -135.43 215.321 0 -88.4074
1.4 8.8632 -0.214113 0 0.0688726

, Im part

R VSO VTE VLSE VTNE
0.11 -93840.9 65153.1 -96000 -259059
0.15 49515.6 -36529.1 39991.4 134984
0.25 -5111.04 3922.52 -4000.1 -11979
0.4 2153.8 -2439.75 1272.95 1782.36
0.55 -2070.87 1212.83 -374.312 -273.571
0.7 583.589 -270.009 67.6282 66.716
1.4 -31.7335 2.15297 -0.456572 0.0514867

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 60666.9 2888.67 0 -230463
0.15 -72709.5 -808.002 0 98816.7
0.25 28017.3 4547.87 0 -7827.76
0.4 -7447.1 -2165.55 0 644.909
0.55 1117.53 230.777 0 -164.754
0.7 -50.0001 -15.2833 0 29.8561
1.4 -3.56802 0.979312 0 -0.0626672

, Im part

R VSE VTO VLSO VTNO
0.11 -98711.2 -9057.76 129026 -300001
0.15 97688.4 96145.3 -54998.6 143127
0.25 -24155 -26263.4 4350.96 -14093.3
0.4 2897.23 2389.94 167.686 1192.58
0.55 -390.274 113.728 -60.9247 -156.562
0.7 163.013 -116.033 10.5639 2.00019
1.4 2.2639 -0.415819 -0.0387636 -0.0420831

, Re part

R VSO VTE VLSE VTNE
0.11 44901.6 -15179 0 605715
0.15 -70514.2 23029.8 0 -240000
0.25 14013.4 -6293.02 0 22538.7
0.4 -3749.13 1127 0 -2967.2
0.55 1270.4 -600.001 0 592.673
0.7 -199.734 137.29 0 -91.3122
1.4 4.82993 -0.142917 0 0.0768991

, Im part

R VSO VTE VLSE VTNE
0.11 -119999 64686.9 -90000 -120000
0.15 59750.4 -34026.4 23438.1 93524
0.25 -6000 2150.45 -1500 -10164.8
0.4 2797.14 -1557.28 906.58 1576.55
0.55 -2312.91 769.397 -309.155 -235.409
0.7 600 -150.109 60.0015 60.937
1.4 -30.1323 0.190335 -0.439257 0.0573121

, pp

, np

, Re part

R VSE VTO VLSO VTNO
0.11 69410.4 550 0 -185846
0.15 -71738.4 -808 0 72199.7
0.25 25344 4289.49 0 -4635.81
0.4 -6844.69 -1986.87 0 250
0.55 993.501 175 0 -98.0508
0.7 -36.194 -9.61922 0 22.7208
1.4 -3.27555 1.13202 0 -0.0626293

, Im part

R VSE VTO VLSO VTNO
0.11 -102796 -9017.06 181874 -120000
0.15 95875.2 81222.3 -74712.7 85623.9
0.25 -22050.9 -23137.9 4967.06 -11257.2
0.4 2184.63 2204.78 168 1058.8
0.55 -100.491 87.9625 -76.9202 -149.126
0.7 94.7553 -117.096 14.3827 2.58782
1.4 2.25999 -0.100002 -0.0596146 -0.0456317

, Re part

R VSO VTE VLSE VTNE
0.11 48439.2 -13708.4 0 551305
0.15 -59038.5 22873.1 0 -235000
0.25 7698.42 -6077.29 0 23953.4
0.4 -1908.28 650.012 0 -3143.67
0.55 1121.09 -398.936 0 631.914
0.7 -319.695 109.256 0 -95.9452
1.4 4.6774 -0.110743 0 0.0900328

, Im part

R VSO VTE VLSE VTNE
0.11 -218411 41234.9 -86000 -60000
0.15 86984 -21169.5 15583.9 69533.6
0.25 -6075.13 1340.98 -600 -8687.87
0.4 5346.6 -1196.53 818.464 1358.4
0.55 -4490.38 500.001 -300 -186.977
0.7 1269.38 -85.6839 58.3339 53.0555
1.4 -38.4122 0.104405 -0.402291 0.0701362

, pp

, np

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