A TEST FOR THE COHEN-KURATH WAVE FUNCTIONS IN (p, d)-REACTIONS S. Abdel-Kariem Department of Physics, Faculty of Science, Ain Shams University, Abbasia, Cairo, Egypt Received 25 October 1990
The experimental data for (p, d)-reaetions o n the Ip-shell nuclei at Ep ~ 33 MeV are used to test the C o h e n and Kurath wave functions and the intermediate coupling. The experimental excitation energies and their order for all law transitions of the lp-shell nuclei are compared with the theoretical predictions of C o h e n and Kurath. The integrated cross-sections ~r(0--90 ~ for these transitions are also compared with their theoretical predictions for one-neutron pick-up. The good agreement between these data indicates a success for the C o h e n - K u r a t h wave functions and for their model of calculations.
1. Introduction
The direct transfer reactions are scattering processes. They can be described with the help of the Distorted Wave Born Approximation Theory (DWBA). The transition amplitudes are dependent on the quantum numbers and on the wave functions of the target and residual nucleus. The excitation energies, the form of the angular distributions and the comparison between the strengths of the transitions and the predictions of the nuclear models (especially those of the shell-model) serve as a test for the accuracy of the wave functions used and also for the model itself. This has been established for two- [1, 4, 7] and three-particles transfer reactions on l p - a n d lsld-sbell nuclei [1 - 3, 5 - 7 , 20]. The one particle transfer nuclear reactions are the simplest ones, they lead to the excitation of one-particle - or one-hole states in the nucleus. The examination of such reactions gives information on the structure of nucleons in these states. The strength of the transition in a direct nuclear reaction is proportional to its spectroscopic factor, which leads also to correlation between the total cross-sections atot ( 0 - 9 0 ~ and the spectroscopic factors for all excited states in a transfer reaction. This means that the transition strengths can be described by the bare SU(3) spectroscopic factors indicating the minor importance of the dynamics of the processes for the strengths. Hauser et al. [7] tested the shell-model predictions with regard to two- and threenucleons transfer reactions on lp- and ls- ld-shell nuclei by comparing their experimental results and the spectroscopic factors for a number of transitions. This was a test for the shell-model wave functions for lp-shell nuclei [8] and for ls- ld-shell nuclei [9]. This work is devoted to a test of Cohen and Kurath wave functions [8] using the experimental results of one neutron pick-up reactions (p, d). Czechoslovak Journal of Physics,Vol. 41 (1991), No. 6
545
S. Abdel-Kariem
7
'~ \.
axX 9,s.
9
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-
~'~200
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60
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Fig. 1. Experimental angular distributions for the transitions of the lp-shell nuclei obtained from (p, d)-reactions [10--15]. The uncertainty of the absolute crosssection is taken for all states to be
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Czech. J. Phys. 41 (1991>
A test f o r Cohen-Kurath wave f u n c t i o n s . . .
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Czech.J. Phys.41 (1991)
547
S. Abdel-Kariem
2. Experimental data for the (p, d)-reactions The experimental data of the reaction (p, d) on the nuclei 7Li, 9Be, l~ and 11B are taken at incident proton energy Ep = 33.6 MeV [10, 11], on a2C and 14N nuclei at Ep = 30.3 MeV [12, 14], cn 13C at Ep = 65 MeV [13] and on I5N at Ep = 39"84 MeV [15]. Figure 1 shows the experimental angular distributions for ground and low-excited states for the lp shell nuclei9 They have diffraction pattern indicating pure direct mechanisms (see 4.2).
3. Spectroscopic factors According to Cohen and Kuratb [16], the wave functions [8] could be obtained by deducing the effective interactions for the 1p-shell by fitting the energy levels. The coefficients of fractional parentage (CFP) are equal to where
fftr'(1..- N) =
Z .[~'~176176
q~J(N)]tr
(1)
loToaoJ
This equation relates the wave function for N active lp nucleons to the wave functions for (N - l) lp nucleons. In equation (1), 1 is the angular momentum, T is the isospin and ~ is the energy eigenvalue. The subscript 0 always refers to the parent states of ( N - 1) nucleons9 This means that the (CFP) is a measurement for the overlap between the target ground state wave function and the wave functions of the residual nucleus for different excited states ~I~176176... N - 1) plus a lp-shell neutron q~. The spectroscopic factor [17] for the transfer of a nucleon which has angular momentum j in the target nucleus is given by
S(ITo~; Io To%, j) = N 2 .
(z)
The spectroscopic factor is a quantity which has the same meaning as the probability given above for the (CFP). Table 1 contains the results of Cohen and Kurath for one neutron-pick-up-reactions on the lp-shell nuclei9 The first two columns of this table identify the state of the residual nucleus and the transferred neutron, Exca I is the calculated excitation energy of this identified state, Exe~p is the corresponding experimental value and S are the calculated spectroscopic factors using equation (2), which are summed in the seventh column and normalized to the isospin for the final state of the residual nucleus in the eighth column. Cohen and Kurath have used two approximations in calculating the effective interactions for the lp-shell nuclei [8], but there is no effective difference between the two methods on the resultant spectroscopic factors. 548
Czech. J. Phys. 41 (1991)
A test for Cohen-Kurath wave functions... Table 1. The excitation energies, coefficients of fractional parentage (CFP) and spectroscopic factors for all transitions in the Ip-shell nuclei.
(Irt, TR)
Exca I (MeV)
Exexp [18] (MeV)
CFP
3/2 1[2 3/2 3/2 3/2 1/2
0"0 0'0 2.144 2"508 6"037 6.037
g.s. g.s. 2"186 3-563 5"366 5.366
0.3979 0.3105 0.4296 --0r5442 --0-3296 0.3016
0"4313 0"2893 0-5538 0"8884 0"3259 0"2728
0"7206
0'7206
0"5538 0"8884 0'5987
0.5538 0.2961 0"1996
3/2 3/2 1/2 3/2 1/2 3/2 1/2 3/2 1/2 3/2 1/2 3/2
0"0 3.412 3"412 15'800 15"800 14'431 14.431 16.881 16.881 14'979 14'979 17"493
g.s. 3"04 3"04 16"626 16"626 16"922 16.922 17"640 17"640 18.150 18.150 19.070
0"3406 --0.3652 0"1084 0.5189 0-1748 0"2632 0"1289 --0"2486 0-2435 0'0558 --0"0920 0"3243
0.58130 0"6668 0"0588 1"3461 0"1528 0-3465 0"0830 0"3091 0'2965 0.0156 0"0423 0"5260
0"5800 0"7256
0'5800 0-7256
1"4989
0"4996
0"4295
0"4295
0-6056
0-2019
0"0579
0"0579
0"5260
0"1753
3/2 3/2 1/2 3/2 1/2 3/2 1/2 3/2 1/2
0"0 2"942 2"942 6"201 6"201 9"855 9.855 12"517 12"517
g.s. 2"361 2.361 6"97 6"97 11"70 11"70 14"7 14"7
--0-4473 --0"3760 0"0975 --0"5067 --0"0407 0"3892 0"2367 0-2537 --0.1666
1-2004 0"8484 0"0570 1'5406 0'0099 0"9087 0"3361 0-3862 0.1665
1"2004 0"9054
0"6002 0"4527
1"5505
0"77525
1"2448
0"6224
0-5527
0"27635
3/2 3/2 1/2 3/2 3/2 1/2
0"0 0.902 0.902 1.418 2.384 2"384
g.s. 0.7184 0.7184 1.74015 2'1543 2.1543
--0-3954 --0-1011 --0.1652 --0.3035 --0.2569 0.0994
1.0943 0.0715 0.1910 0"6449 0.4621 0.0692
1.0943 0"2625
1.0943 0.2625
0.6449 0"5313
0.2150 0.5313
nly
S
~-~S
S (2TR + 1)
The target 7Li (1,0) (3,0) (0, 1) (2, I)
lp lp lp lp lp lp
The target 9Be (0, 0) (2, 0) (2, 1) (2, 0) (1, 1) (1, 0) (3, 1)
lp lp lp lp lp lp lp lp lp Ip lp lp
The target 1~ (3/2, 1/2) (5/2, 1/2) (7/2, 1/2) (7/2, 1/2) (5/2, 1/2)
lp lp lp lp lp lp lp lp lp
The target 11B (3, 0) (1, 0) (0, 1) (1,0)
Ip lp lp lp lp lp
(continued) Czech. J. Phys. 41 (1991)
549
S. Abdel-Kariem Table 1 (continued)
EXcal (MeV)
Exexp [18] (MeV)
CFP
3/2 1/2 3/2 3/2 1/2 3/2 1/2 3/2 1/2
3"339 3"339 4"718 5"578 5"578 6-188 6-188 5"530 5-530
3"5871 3.5871 4"774 5"1639 5"1639 5.180 5"180 5'9195 5"9195
--0"0814 --0"1730 0"1404 0"4827 0"1056 0"0627 --0"0657 --0"0595 0"0698
0"0464 0.2096 0"1381 1"6312 0"0781 0"0275 0"0303 0.0248 0"0341
0"256
0"256
0"1381 1"7093
0"1381 0"5698
0"0578
0"0578
0"0589
0"0589
(3/2, 1/2)
lp 3/2
(i/2, 1/2)
lp 1/2
(3/2, I/2)
lp 3/2
0.0 1.708 5.390
g.s. 2.000 4.8042
--0.8440 0.4339 0.3072
5.6989 1.5060 0.7549
5.6989 1.5060 0.7549
2.84945 0.7530 0-37745
(,rR, rR) (2, 0) (3, 0) (2, 1) (1, 0) (2, 0)
nl/
lp Ip lp lp lp Ip lp lp lp
S
Es
(2TR + I)
The target 12C
The target 13C
(o,o)
lp 1/2
(1, 0)
lp lp lp lp lp
(I, 1) (2,1)
o-o
3/2 1/2 3/2 1/2 3/2
12.449 12.449 15.084 15.084 16.691
g.s. 12"710 12"710 15"11 15"ll 16"1
0-2610 0"2710 0"0106 --0"4470 0"0276 --0"5807
0.6132 0"6611 0"0010 1'7983 0"0069 3'0348
0"6132 0"6621
0-6132 0"6621
1.8052
0"6017
3'0348
1'0116
3/2 I/2 3/2 1/2 3/2 3/2 1/2 3/2 I/2
0"0 0'0 3-587 3"587 7"404 8-781 8-781 10"433 10.433
g.s. g.s. 3.511 3"511 7"376 8"918 8"918 11-74 11.74
--0"0282 --0"3769 --0"1664 --0"0710 --0"6096 --0"3506 0'1053 --0'4799 --0"0325
0.0080 1.3756 0-2768 0'0504 3"7166 1'2290 0.1109 2-3033 0.0105
1.3836
0-6918
0.3272
.0"1636
3/2 1/2 1/2 3[2 1/2
0'0 0"0 2"690 3"616 3"616
g.s. g.s. 2"313 3"9481 3"9481
0'0542 --0"3601 0"3376 0"2434 0"0642
0'0323 1"4267 1-2534 0'6515 0"0453
The target 14N (1/2, 1/2) (3/2, 1/2) (5[2, 1/2) (112, 1/2) (3/2, 1/2)
lp lp lp lp lp lp lp Ip lp
3"7166 1'3399
1-8583 0"66995
2"3138
1"1569
1"4590
1"4590
1"2534
0"4178
0'6968
0"6968
The target tSN (1,0) (0, 1) (1, 0)
lp Ip lp lp lp
(continued) 550
Czech. J. Phys. 40 (1990~
A test for Cohen-Kurath wave functions... Table 1 (continued)
(IR, TR)
Exca I
nly
(MeV)
(2, O) (2, 1)
lp 3/2 lp 3f2
6"991 9"524
(I, 1)
lp 3/2 lp 1/2
11.783 11.783
Exex p [18]
(MeV)
7"02912 9"17225 + 10"432 13"74 13"74
CFP
S
~2S
S (2TR -at- l)
--0"3371 --0"5704
1"2500 3-5792
1"2500 3"5792
1"2500 1"1931
--0"4523 0"0000
2"2500 0"0000
2"2500
0-7500
4. Discussion
As mentioned earlier, we try to test the Cohen-Kurath wave functions for the 1p-shell nuclei and their intermediate coupling model for calculating the (CFP) using these wave functions. A comparison between the excitation energies given by this model and the experimental values is undertaken. The mechanisms of the (p, d)-reactions on the lp-shell nuclei are also discussed. 4.l The e x c i t a t i o n e n e r g i e s Assuming that the (p,d)-reactions on lp-shell nuclei at Ep ~ 33 MeV have direct mechanisms and the transferred neutron is mostly picked-up from the lp-shell, thus it has 1 = 1, there is parity change between the target- and the residual nucleus wave function in a (p, d)-reaction. Cohen and Kurath's calculations are devoted only to the natural parity states for the residual nucleus (positive parity states for the nuclei 6Li, 8Be, I~ 12C and a4N and negative parity states for the nuclei 9B, 11C and 13N). In Table 1 and Fig. 2 the values of excitation energies obtained as eigenvalues of the Hamiltonian operators for the Schr6dinger equation for the lp-shell nuclei are compared with their corresponding experimental values. The experimental values for the excitation energies for all excited states given in Table 1 and Fig. 2 are taken from literature [18]. It is clear from Table 1 and Fig. 2 that there is a good correspondence between theoretical and experimental energy levels for all nuclei except some levels in 8Be and J~ Also the positions for most levels are exactly predicted by Cohen and Kurath. The mean difference between calculated and experimental values is of the order of 700 keV. A difference of about 2 MeV is found for the states t6"922MeV (2+; 0 + 1); 17.64MeV (1+; 1 ) i n 8Be, 14.7MeV (5/2-; 1 / 2 ) i n 9B and 13.70 MeV (1 +; 1)in ~4N. These states are probably excited through a neutron pick-up from the ls-ld-shell. As a final result, Cohen and Kurath's calcuatlions predict the position of excited states for the 1p-nuclei in good agreement with experimental values. Czech. J. Phys. 41 (1991)
551
S. Abdel-Kariem
EX
"
6Li
MeV 20
~.
8B e
Exp. HeV 20
Th.
Tt,.
Exp. MeV 20
//
2~
15 -
__s~,15 st~ - - /
10
106 Th. Exp.
9B
IZmp. ~ V 20
-
10
-
.~ ?/2-
?rE-/
10
" 10
71i__/-?t/ I* - -
S -
5 -
2*
.
0
--
Ex MeV 20
1"
0
------I*
-
11C Th.
Exp,
-
HeY 20
-
2"
S
o'------o"
~
2*
3
0
12C Th. Exp. l~eV 20
13N Th. Exp. ME~V 20
3* 14N Th. Exp.
211--~__~.,I 1S
15
--
1,1-
l*dlS
IS
r __I---1' 1",1- - /
10
10
10
10
-
-
/--2";1
vf__~--ln" sn-------st~
~z---~ __ )/T S
2" - - - - - - 2 " S
I* _ _ i - -
o
3tf------~f
0
r ------o"
o -
~i'~----~
o
I9
r ~----~*
Fig. 2. Energy levels for the lp-shell nuclei calculated by Cohen and Kurath in comparison with the experimental values.
552
Czech. J. Phys 41 (1991)
A test for Cohen-Kurath wave functions...
4.2. A n g u l a r d i s t r i b u t i o n s and r e a c t i o n m e c h a n i s m s Referring to the experimental angular distributions for (p, d)-reactions on lpnuclei at E v ~- 33 MeV as shown in Fig. 1, they have shapes indicating that these reactions have direct mechanisms. This is estab!ished by fitting all given angular distributions with the predictions of the DWBA-theory [10-15]. In most cases the experimental angular distributions have diffraction patterns characterized by a transfer angular momentum of I --- 1; this means that the transferred neutron is picked-up from the lp-shell. Furthermore the non-excitation or a weak excitation of the unnatural parity states in some cases (~3C-,14N- and 1SN-residual nucleus)indicates that (p, d)-reactions on lp-shell nuclei have direct mechanisms. In 14N (p, d) ~3N-reaction, the excitation of the level 2-37 MeV (1/2 +) is due to the existence of the configuration admixtures of 2s- ld-shell in the wave function of the g.s. of X4N. Also the excitation of the two states 6.38 MeV (5/2 +) and 6.91 MeV (3/2 +) in the nucleus 13N have been interpreted as belonging principally to a configuration of the a2C (2 +) core plus a 2sl/z nucleon [19] with some ds/2 admixture in the g.s. wave function of a4N in the Case of the level 6"38 MeV. The excitation of the level 5-106 MeV (2-) in ~4N in XSN (p, d) a4N-reaction represents a ld3/2 admixture in aSN g.s. wave function [15]. Figure 3 replesents Cohen and Kurath's calculations for the level strengths compared with the total cross-section O'tot (0--90 ~ for lp-shell nuclei which indicate that, the (p, d)-reactions on the lp-shell nuclei at Ep "-" 33 MeV have direct mechanisms. 4.3. S p e c t r o s c o p i c r e s u l t s The spectroscopic factors for a picked-up neutron from the lp-shell-nuclei, which have been calculated by Cohen and Kurath in the SU(3) formalism, can be tested by comparing with the observed experimental strengths for excited states in direct nuclear reaction. This is already made for all transitions listed in Table 1 and presented in Fig. 3. In this representation the comparison process between the two values is made for all transitions with the same normalization constant of N = 7"577 rob. There is a good agreement between both values for all transitions including high excited states [for example (16-922; 17.64; 18.15 and 19.069 MeV) in 8Be, (12.71; 15.11 and 16"1 MeV)in 12C, (11.74 MeV) in 13N and (13.74 MeV)in ~4N]. A big difference between both values is observed for the state [7-376 MeV (5/2-; 1/2)] in a3N. This good agreement between experimental and theoretical strengths indicates minor importance of the reaction-dynamics in the (p, d)-reaction on 1p-nuclei. This was also found previously for two-nucleons transfer reactions on lp-shell nuclei [1, 4] and for (p, e)-reaction on lp- [1, 2, 20] and on 2s-ld-shell nuclei [ 5 - 7 ] . The data in Fig. 3 are normalized to the g.s. transition of 6Li. The spectroscopic factors for the two states at Exc.l = 15.238 MeV (1+; 0) and 1.7.879 MeV (2+; 1) in 14N are very small. Czech. J. Phys. 41 (1991)
553
2
o
?
JI
1
16 2
1(~1
10 ~
10
mb
10 2
10 ~
Li
o
[
'B,"
.o
/
%
o~.
/,-
:-7
~B
12C
1]C
lt'H
lsN
Target
Fig. 3. Comparison between angle integrated cross-sections for (p, d)-reaction of lp-shell auclei and spectroscopic factors, which calculated with the C o h e n - K u r a t h wave functions. l'he sum of the spectroscopic factors is normalized to the sum of the experimental data. iThe axis to the back represents the different excited states for the residual nuclei.)
N
~z
k~
A test f o r Cohen-Kurath wave f u n c t i o n s . . .
Accordingly one can conclude that the calculated spectroscopic factors for oneneutron-picked-up-reactions can show the general trend o f the experimental strengths for all transitions for the whole lp-shell nuclei. This process has been f o u n d for (p, ~)-reactions on l p and 2s-ld-shell nuclei [5, 7].
5. Conclusion By means o f the wave functions for the 1p-shell nuclei [8] and the intermediate coupling model, Cohen and K u r a t h have calculated the excitation energies and the C F P values for all transitions related to the lp-shell nuclei. The obtained predictions for the excitation energies are in g o o d agreement with the experimental values (4.1). The (p, d)-reactions on the lp-shell nuclei at Ep ~_ 33 MeV have pure direct mechan-isms (4.2). The C F P values calculated by C o h e n and K u r a t h could predict the transition strengths for all transitions in the lp-shell nuclei exactly (4.3). This indicates that the intermediate coupling model and the C o h e n and K u r a t h wave functions are successful in describing and interpreting the lp-shell nuclei. References
[1] Abdel-Kariem S.: Doctor thesis, Tiibingen University, 1984. Abdel-Kariem S., Rohwer T., Staudt G., Leitner W.: Nucl. Phys. A 487 (1988) 550. [2] Weng F.: Doctor thesis, Tiibingen University, 1979. Weng F., Rohwer T., Staudt G.: Annual Report 1979, Inst. Ffir Kernphysik der KFA-Jfilich, Jfil-Spez- 72, p. 12. [3] Hauser H.-J.: Diploma thesis, Tfibingen University, 1981. [4] Rapp V.: Doctor thesis, Ti.ibingen University, 1982. [5] Rohwer T.: Doctor thesis, Tfibingen University, 1980. [6] Hoyler F.: Doctor thesis, Tfibingen University, 1982. Buck W., Hoyler F., Stfibler A., Staudt G., Klapdor H. V., Oesehler H.: Nucl. Phys A" 398 (1983) 189. [7] Hauser H.-J., Rohwer T., Hoyler F., Staudt G., Abdel-Kariem S., Grasshoff P., Klapdor H. V., Korber A., Leitner W., Rapp V., Walz M., Weinmann D.: in AIF Conf. (USA), No. 125 r p. 701--704. [8] Cohen S., Kurath D.: Nucl. Phys. 73 (1965) 1. [9] Chung W., Wildenthal B., private communication. [10l Kull L. A.: Phys. Rev. 163 (1967) 1066. [11] Kull L. A., Kashy E.: Phys. Rev. 167 (1968) 963. [12] Chant N. S., Fisher P. S., Seatt D. K.: Nucl. Phys. A 99 (1967) 669. [13] Hosono K., Kondo M., Saito T., Matsuoka N., Nagamachi S., Noro T., Shimizu H., Kato S., Okada K., Ogino K., Kadota Y.: Nuel. Phys. A 343 (1980) 234. [14] Kozub R. L., Kull L. A., Kashy E.: Nucl. Phys. A 99 (1967) 540. [15] Snelgrone J. L., Kashy E.: Phys. Rev. 187 (1969) 1259. [16] Cohen S., Kurath D.: Nucl. Phys. A 101 (1967) 1. [17] Macfarlane M. H., French J. B.: Rev. Mod. Phys. 32 (1960) 567. [18] Ajzenberg-Selove F.: Nucl. Phys. A 490 (1988) 1; A 433 (1985) 1; A 449 (1986) 1. [19] Reich C. W., Phillips G. C., Russell J. L., Jr.: Phys. Rev. 104 (1956) 143. [20] Kurath D.: in Proe. 2nd Int. Conf. on Clustering Phenomena in Nuclei College Park, Maryland 1975, p. 439. Czech. J. Phys. 41
(1991)
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