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The 39K(p, 2p) reaction at 150 MeV
Nuclear Physics 82 (1966) 513--520; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
THE sgK(p, 2p) REACTION AT 150 MeV D. N E W T O N t
G. L. SALMON and A. B. CLEGG
Nuclear Physics Laboratory, Oxford Received 15 November 1965 Abstract: The low-lying states of 8BAr, which are produced in the 3*K(p, 2p) reaction induced by 150 MeV protons, were found by studying gamma radiation in coincidence with two outgoing protons. The results are compared with the two-hole states expected in an intermediate coupling model. These results suggest that, though there are many more states of 3BAr than would be expected solely from two holes in 4°Ca, the two-hole states are contained in a very few of the actual states. The second excited state of the "rotational band" based on the ground state is tentatively identified at 6.6 MeV, an energy which is 3.0 times the energy of the first excited state. The results are compared with measurements of the 4°Ca(p, 3p) reaction, providing information about the mechanism of this reaction and further information about the structure of 3BAr. E
N U C L E A R REACTIONS SSK(p, 2p), E ~ 150 MeV; measured E~, a(E~). 3*Ar deduced levels, J, zt. Natural target.
1. Introduction In this paper we report observations of the gamma rays from the excited states of ZSAr produced in the 3 9 K ( p , 2p) reaction in which a potassium target was bombarded with 150 MeV protons. Gamma rays were studied when they were in coincidence with two outgoing protons, one on either side of the incident beam and at about 40 ° to it. Similar studies of the 4°Ca(p, 2p) reaction are reported separately 1). In studies 1.2) of the 4°Ca(p, 2p) reaction at 150-185 MeV it is found that the states of 39K produced most strongly are those which are holes in the closed-shell ground state of 4°Ca: the ground state of 39K being a ld~ hole, the first excited state at 2.53 MeV a 2s~ hole, and the ld~ hole state is at about 6.4 MeV. We would therefore expect that in the 39K(p, 2p) reaction we should produce those low-lying states of 3SAr which are formed by adding afurther hole in the 2s or ld shells to the ld~ hole already present in the ground state of 39K. The spectrum expected to be formed by two holes in the ground state of 4°Ca has been calculated under various assumptions (see typically the results of a calculation by Wilmore, which are quoted by Phillips 3). However, it is found 4) that there are many more states present in the experimental spectrum of 3SAr than is expected from this simple model. It is therefore of great interest to attempt to identify such two-hole states in the spectrum of 3SAr t Now at Lawrence Radiation Laboratory, Berkeley, California, U.S.A. 513
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and to see if they are contained in only a few experimental states of this nucleus. In our present experiment we should see only those two-hole states for which at least one of the holes is in the ld~ state, so that we could expect to see the following states in 3SAr: ld~2: J = 0, 2 +, ld~ 12s~ 1 : j = 1, 2 +, ld~lld~l:J
= 1,2, 3,4 + .
In each of these three groups the relative probability of production of a state with angular m o m e n t u m J should be proportional to 23"+ 1. The J = 0 + state would be expected to be largely contained in the ground state of 3SAr, so that its production should not be observed in our experiment. We find that the first excited state of 3SAr, which has J = 2 ÷, is produced strongly, together with production of some other states. We find evidence for production of a state around 6.6 MeV which may have J = 4 +, and of states around 4 MeV.
2. Experimental Arrangement By scattering from a system of rotating targets 5) inside the synchrocyclotron at A E R E Harwell, 150 MeV protons were ~xtracted with a duty ratio of 0.2. They were focussed on to a target of natural potassium metal, containing 93 % of 39K and of thickness 3.93 g • cm -2 so that the mean proton energy in the target was 142 MeV. TABLE 1 Geometry defined by proton counters Side of beam
Size o f counter (cm)
Distance from centre of target (cm)
Angle to beam (degree)
Solid angle dO(sO
d.QxdO z (sr 2)
left right
5.1 ×5.1 6.35 × 6.35
12.7 15.2
40 40
0.16 / 0.174J
0.028
Two outgoing protons were detected by two plastic scintillation counters placed on either side of the incident beam and in a horizontal plane containing the beam. The geometry defined by the proton counters is shown in table 1. The angle between the two protons detected in this way was 80 ° with a POssible spread in this angle of 23 °. G a m m a radiation was detected by a well-shielded sodium iodide crystal, 12.7 cm diam and 15.2 c m long with its front face at 15.2 cm from the centre of the target. This counter was at 120 ° to the incident beam. A triple coincidence between the two proton counters and the gamma-ray counter passed the corresponding pulse from the gamma-ray counter to be analysed by a 100-channel pulse-height analyser.
agK(p, 2p) REACTION
515
3. Results
The coincidence pulse-height spectrum obtained is shown in fig. 1. It is analysed into the gamma rays shown in that figure, giving differential cross sections for production of the various gamma rays which are shown in table 2. The strongest Counts 120
100
r % 2°1-" \
/
|1
~
I
l
I
0
1"5
2.0
2.5
GammQ
-\~
3.0
x
I
I
3.5
&.o
"
t..5
5.0
roy e n e r 0 y (MeV)
Fig. 1. Pulse-height spectrum of the gamma rays from a potassium target, taken in coincidence with two outgoing protons each travelling at 40 ° to the incident beam direction (see table 1). The curves show the experimental shapes of the strongest mono-energetic gamma rays, and their combined spectral shape. TABLE 2
Cross sections for production of gamma rays in the aSK(p, 2p) reaction Gamma-ray energy (MeV)
dtr dO1 d-Q= (mb • sr -~)
1.65±0.07 2.20~0.07 2.9 ±0.I 3.6 ±0.1
0.9±0.3 2.5!0.4 1.1±0.3 1.2
4.4 ±0.2
0.6
g a m m a - r a y is that at 2.20 + 0.07 which we identify with the decay o f the first excited state o f as A at 2.17 M e V : h o w e v e r we shall see that an ap p r eci ab l e p a r t o f the p r o d u c tion o f this state m u s t be by cascades f r o m higher states. Th e 1.65 M e V g a m m a ray
516
D. N~w'roN et
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is presumably the well-known 3,4) cascade decay of the third excited state at 3.82 MeV. The 2.9 MeV gamma ray must also be a cascade decay as there is no excited state of this energy. There are two known cascade gamma rays 6) in 3aAr which are possible candidates: a decay from a state at about 5.0 MeV to the 2.17 MeV state or a decay from a state at about 6.6 MeV to the 3.82 MeV state. We will present arguments in favour of the latter of these two possibilities in the next section. The exact identification of the energies of the two highest energy gamma rays is somewhat uncertain but we believe that the energy quoted as 3.6 MeV cannot be as high as 3.9 MeV and so again must be a cascade decay, probably from a state around 5.8 MeV to the first excited state. We shall assume, for reasons given in the following section, that the gamma radiation near 4.4 MeV is a ground state decay. 4. Discussion
We shall start by describing in more detail which states of 3SAr one would expect to be produced in the 39K(p, 2p) reaction. We have indicated in the introduction those jj-coupling configurations which will be involved. Wilmore has calculated (see Phillips 3)) how these configurations should mix in intermediate coupling. It is possible, however, to estimate in a simpler manner what should be the dominant configurations in the low-lying two-hole states of 3SAr, by the methods used by Redlich 7) to study low-lying states of 1SO and lSF. He obtained excellent estimates of intermediate coupling wave functions from a product of wave functions of a single particle in a spheroidal potential. He showed how, if he projected eigenfunctions of angular momentum out of such a product wave function (which is not itself an eigenfunction of angular momentum) the resulting wave functions were very similar to those obtained from an intermediate coupling calculation. He initially tried as his single-particle wave functions the asymptotic wave functions which correspond to a very large value of Nilsson's ratio r/ of the spheroidal deformation to the spin orbit coupling s), but then found that a better fit could be obtained if he modified his single-particle wave functions in an empirical manner. We note that these resulting modified single-particle wave functions are very similar to those found by Nilsson for mass 18 nuclei with a prolate shape, and r/about 4 to 6. (Note that there is a difference in sign convention between the wave functions of Redlich and those of Nilsson.) This similarity justifies the single-particle wave functions introduced empirically by Redlich. This procedure for obtaining intermediate-coupling wave functions has also been remarkably successful in the lp shell 9). In using this procedure the degree of intermediate coupling is related to the value of 17assumed. The corresponding single-hole wave functions in a deformed potential that should contribute to the lowest states of 3aAr are those corresponding to Nilsson's levels 11 and 8, and are of the forms level 11: ~Ox(K = ___½)= axd~, ± , + b l s ~ , ±~r+ctd~, ±~, level 8: ~k2(K = +__~) = a2d~, ±~+c2d~, ±2"
agK(p, 2p) REACTION
517
F r o m these single-hole wave functions one then forms rotational bands by taking a product of two of them and projecting out eigenfunctions of angular momentum. Each such product wave function has projected out of it the wave functions of the several states which make up the "rotational band". One would expect that, for 3SAr, Ir/[ would not be large so that for ~ i and ~bz the first coefficients, al and a2 respectively, should be distinctly larger than the others. It is then clear that there will usually be one term which is dominant in each intermediate coupling wave function which is projected out of the product. The two lowest such rotational bands in 3SAr, come from these products: ~kl(K = ½)~bl(K = - ½ ) , forming a K = 0 band with states of J~ = 0 ÷, 2 +, 4 +. ~k2(K = ~ ) ~ I ( K = - ½ ) , forming a K = 1 band with states o f J ~ = 1 ÷, 2 +, 3 ÷, 4 ÷. The energies of the states of the K = 0 band should increase with J, but for the K = 1 band the ordering in energy can differ from the ordering in J, as it can for a K --- ½ band ~0). There can also be mixing of states of the same J from different rotational bands. On examining the product wave functions we find that for the following states the dominant contribution includes at least one hole in the d~ state: lower K = 0 b a n d (~klffl): J~ = 0 +, d~ 2 J~ = 2 +, d~ z J~ = 4 + , d~ld~ -1 K = 1 band (~i ~2):
jn = 1 +, d~ls~l,
d~ld~ "1
J~ = 2 + , d ~ l s ~ 1,
d ~ l d ~ 1,
d~ 2
J~ = 3 + , d ~ l d ~ 1 J~ = 4 + , d ~ d ~ 1 Thus we should expect these states to be produced strongly in the 39K(p, 2p) reaction. The J~ = 0 + state is presumably to be identified with the ground state of 3SAr, so here we should only observe production of the other states. It has been found 4) that there are m a n y more low-lying states of 38Ar than there should be for two holes in a #°Ca closed shell. One may therefore ask whether there will be important mixing between the two hole states and the other states. Our results provide some evidence that this mixing is not strong. We will assume that low-lying negative parity states of 3SAr are not produced strongly in the 39K(p, 2p) reaction because they would be due to promotion of a particle into the I f or 2p shell. They could thus be produced in this reaction only if there were appreciable admixture of two-particle-three-hole configurations in the ground state of 39K; we would expect such admixtures to exist, but to be weak. We find no evidence for any strong production of negative parity states of 39K in the 4°Ca(p, 2p) reaction 1).
51 8
D. NEWTON" et aL
We may expect that the J " = 2 + first excited state of aaAr is to be identified with the J " = 2 + state of the K = 0 rotational band, and on this basis we can estimate the expected cross section for its direct production (as opposed to the production by cascade decays from higher states, which we have already remarked on). It was found 2) that the cross sections for ejection of d~ and d~ protons from 4°Ca are much the same, and we found a cross section of approximately 1 m b . sr -2 for ejection of a d~ proton f r o m 4°Ca with a counter geometry similar to that used in the present experiment. One expects the probability of two d~ holes coupling to form a state of angular m o m e n t u m J to be proportional to 2 J + 1, so that approximately ~ of the ejections of a d~ proton from 39K should result in production of this J~ = 2 + state of aSAr. On this basis we deduce an expected cross section of approximately 0.8 m b . sr -2 for its production. Our results are consistent with such a cross section for direct production of the first excited state of 38Ar. As the third excited state of aSAr at 3.82 MeV has 3,11) j = 3 - , we would not expect it to be produced directly in the 39K(p, 2p) reaction. We therefore ascribe its observed production to cascade decay of a state at about 6.6 MeV, as mentioned in the previous section. The agreement between the cross sections for production of the 1.65 and 2.9 MeV g a m m a rays, which we are taking to be successive members of a cascade, supports this assignment. We also remark in further support that if both the 1.65 and 2.9 MeV g a m m a rays were separate cascade decays to the first excited state of 38Ar, these cascades, together with the 3.6 MeV g a m m a ray would be greater than the observed cross section for production of the first excited state. I f we take the 1.65 and 2.9 MeV g a m m a rays to be successive members of a cascade from a 6.6 MeV state, the cross section for direct production of the first excited state is then approximately 0.5__+0.5 mb • sr -2, consistent with the expected cross section. As decay of the 6.6 MeV state to the J = 3 - state at 3.82 MeV is apparently favoured over other conceivable decays with a higher energy release we would think that this state probably has J = 4 +, with the electric dipole decay being faster than higher energy electric quadrupole decays. However d = 2 or 3 ÷ assignments are presumably possible, with the magnetic dipole decays to the first excited state slower than the E1 decay to the second excited state. (We note that electric dipole decays will not here be inhibited by isobaric spin selection rules as aSAr is not a self-conjugate nucleus in contrast to an electric dipole transition we have found 12) to be strongly inhibited in 40Ca). We have seen that there is some evidence for production of states of 3aAr at about 4.4 and 5.8 MeV, though further work is needed to sort out the higher energy region of the gamma-ray spectrum in more detail. We should not ascribe all this g a m m a radiation to cascades through the first excited state as such cascades could exhaust all of the observed production of the first excited state. O f the states that we are observing here, a state that could decay to the ground state would probably have J = 1 or 2 ÷, while one which decays by a cascade through the first excited state would probably have J = 2 or 3 ÷, for which magnetic dipole decay is possible, though d = 4 + is also possible.
3~K(p, 2p) REACTION
519
The calculations of Wilmore with our considerations earlierin this section, would seem to favour a suggestion that the states we have discussed in the last paragraph should be the J~ = 1 ÷ and 2 + states of the K = 1 band. These states would be produced on ejecting a 2s~ proton from 39K. The combined cross section for their production is close to the 1.5 m b • sr -2 we found 1) for ejection o f a 2s½ proton from 4°Ca with the same counter geometry as in the present experiment. This agreement between these cross sections provides support for our assignment. The 6.6 MeV state would then probably be the J = 4 + state of the K --- 0 band: the cross section for its production is not far from the 0.5 mb • sr -2 we deduce for its expected production cross section from the observed cross section 1) for ejection of a ld~ proton f r o m 4°Ca. I f this assignment is correct the ratio of the energy of the J = 4 + state to that of the J = 2 + state of this K = 0 band is 3.0, a ratio which is not far from the value of 3.33 expected for a rotational band in a deformed nucleus. This ratio, for two holes in the 2 s - l d shell, is quite different from the ratio of 2.0 found for the corresponding states of two particles in the 2 s - l d shell, in 180. This may indicate that 4°Ca is more deformable than is 160; further evidence for this has been deduced 13) from the strengths of electric quadrupole excitations in 4°Ca and in 39K. In our study 1) of the reactions induced by b o m b a r d m e n t of 4°Ca with 150 MeV protons we observed production of states of 3SAt in the 4°Ca(p, 3p) reaction. In this reaction we found that the same states of 38Ar were produced as in the 39K(p, 2p) reaction, with much the same relative cross sections, suggesting that the major contribution to the mechanism of the (p, 3p) reaction producing low-lying states of the residual nucleus is a process in which the incident proton makes two separate collisions with different protons in the target nucleus and ejects them both. As the 4°Ca(p, 3p) reaction should produce all states formed by two holes in the 2 s - l d shell, while the 39K(p,2p) reaction should produce only those in which at least one of the holes is in the ld~ state, this r¢sult suggests that states of 38Ar which are largely made up of two holes neither of which is in the ld~ shell must be at high excitation energies for otherwise we should have observed g a m m a radiation from their decay. This conclusion agrees with the picture of 3SAt we have presented. In conclusion we see that this work, even though it is of an exploratory nature, has indicated the possibility of obtaining considerable information about the two-hole states of aaAr. It would be very interesting to study this g a m m a radiation in greater detail to obtain more information about the states between 4 and 6 MeV. It would also be interesting to repeat these measurements for different angles between the two proton counters, as the angular correlation between the two outgoing protons is much sharper for ejection of a proton from an s-state than for ejection from a d-state 2) in this way one could hope to obtain information about the relative contributions of 2s and ld holes to the 38Ar states. We wish to thank Dr. B. Rose and all members of the cyclotron group at A E R E Harwell for their considerable help and the Director, AERE, for permission to use the cyclotron.
520
D. NEWTON et aL
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
D. Newton, G. L. Salmon and A. B. Clegg, Nuclear Physics 82 (1966) 499 G. Tibell, O. Sundberg and U. Miklav~.i6, Phys. Lett. 2 (1962) 100 W. g . Phillips, Nuclear Physics 60 (t964) 544 R. G. Alias, L. Meyer-Schutzmeister and D. yon Ehrenstein, Nuclear Physics 61 (1965) 289 G. L. Salmon, Nucl. Instr. 23 (1963) 338 J. H. Towle, R. Berenbaum and J. H. Matthews, Proc. Phys. Soc. AT0 (1957) 84 M. G. gedlich, Phys. Rev. U 0 (1958) 468 S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) D. Kurath and L. Pi6man, Nuclear Physics 10 (1959) 313 J. P. Elliott and M. Harvey, Proc. Roy. Soc. A272 (1963) 587 P. M. Endt and C. van tier Leun, Nuclear Physics 34 (1962) 1 D. Newton and. A. B. Clegg and G. L. Salmon, Nuclear Physics 67 (1965) 449 D. Newton, G. L. Salmon and A. B. Clegg, Nuclear Physics 73 (1965) 145
THE sgK(p, 2p) REACTION AT 150 MeV D. N E W T O N t
G. L. SALMON and A. B. CLEGG
Nuclear Physics Laboratory, Oxford Received 15 November 1965 Abstract: The low-lying states of 8BAr, which are produced in the 3*K(p, 2p) reaction induced by 150 MeV protons, were found by studying gamma radiation in coincidence with two outgoing protons. The results are compared with the two-hole states expected in an intermediate coupling model. These results suggest that, though there are many more states of 3BAr than would be expected solely from two holes in 4°Ca, the two-hole states are contained in a very few of the actual states. The second excited state of the "rotational band" based on the ground state is tentatively identified at 6.6 MeV, an energy which is 3.0 times the energy of the first excited state. The results are compared with measurements of the 4°Ca(p, 3p) reaction, providing information about the mechanism of this reaction and further information about the structure of 3BAr. E
N U C L E A R REACTIONS SSK(p, 2p), E ~ 150 MeV; measured E~, a(E~). 3*Ar deduced levels, J, zt. Natural target.
1. Introduction In this paper we report observations of the gamma rays from the excited states of ZSAr produced in the 3 9 K ( p , 2p) reaction in which a potassium target was bombarded with 150 MeV protons. Gamma rays were studied when they were in coincidence with two outgoing protons, one on either side of the incident beam and at about 40 ° to it. Similar studies of the 4°Ca(p, 2p) reaction are reported separately 1). In studies 1.2) of the 4°Ca(p, 2p) reaction at 150-185 MeV it is found that the states of 39K produced most strongly are those which are holes in the closed-shell ground state of 4°Ca: the ground state of 39K being a ld~ hole, the first excited state at 2.53 MeV a 2s~ hole, and the ld~ hole state is at about 6.4 MeV. We would therefore expect that in the 39K(p, 2p) reaction we should produce those low-lying states of 3SAr which are formed by adding afurther hole in the 2s or ld shells to the ld~ hole already present in the ground state of 39K. The spectrum expected to be formed by two holes in the ground state of 4°Ca has been calculated under various assumptions (see typically the results of a calculation by Wilmore, which are quoted by Phillips 3). However, it is found 4) that there are many more states present in the experimental spectrum of 3SAr than is expected from this simple model. It is therefore of great interest to attempt to identify such two-hole states in the spectrum of 3SAr t Now at Lawrence Radiation Laboratory, Berkeley, California, U.S.A. 513
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D. NEWTON et aL
and to see if they are contained in only a few experimental states of this nucleus. In our present experiment we should see only those two-hole states for which at least one of the holes is in the ld~ state, so that we could expect to see the following states in 3SAr: ld~2: J = 0, 2 +, ld~ 12s~ 1 : j = 1, 2 +, ld~lld~l:J
= 1,2, 3,4 + .
In each of these three groups the relative probability of production of a state with angular m o m e n t u m J should be proportional to 23"+ 1. The J = 0 + state would be expected to be largely contained in the ground state of 3SAr, so that its production should not be observed in our experiment. We find that the first excited state of 3SAr, which has J = 2 ÷, is produced strongly, together with production of some other states. We find evidence for production of a state around 6.6 MeV which may have J = 4 +, and of states around 4 MeV.
2. Experimental Arrangement By scattering from a system of rotating targets 5) inside the synchrocyclotron at A E R E Harwell, 150 MeV protons were ~xtracted with a duty ratio of 0.2. They were focussed on to a target of natural potassium metal, containing 93 % of 39K and of thickness 3.93 g • cm -2 so that the mean proton energy in the target was 142 MeV. TABLE 1 Geometry defined by proton counters Side of beam
Size o f counter (cm)
Distance from centre of target (cm)
Angle to beam (degree)
Solid angle dO(sO
d.QxdO z (sr 2)
left right
5.1 ×5.1 6.35 × 6.35
12.7 15.2
40 40
0.16 / 0.174J
0.028
Two outgoing protons were detected by two plastic scintillation counters placed on either side of the incident beam and in a horizontal plane containing the beam. The geometry defined by the proton counters is shown in table 1. The angle between the two protons detected in this way was 80 ° with a POssible spread in this angle of 23 °. G a m m a radiation was detected by a well-shielded sodium iodide crystal, 12.7 cm diam and 15.2 c m long with its front face at 15.2 cm from the centre of the target. This counter was at 120 ° to the incident beam. A triple coincidence between the two proton counters and the gamma-ray counter passed the corresponding pulse from the gamma-ray counter to be analysed by a 100-channel pulse-height analyser.
agK(p, 2p) REACTION
515
3. Results
The coincidence pulse-height spectrum obtained is shown in fig. 1. It is analysed into the gamma rays shown in that figure, giving differential cross sections for production of the various gamma rays which are shown in table 2. The strongest Counts 120
100
r % 2°1-" \
/
|1
~
I
l
I
0
1"5
2.0
2.5
GammQ
-\~
3.0
x
I
I
3.5
&.o
"
t..5
5.0
roy e n e r 0 y (MeV)
Fig. 1. Pulse-height spectrum of the gamma rays from a potassium target, taken in coincidence with two outgoing protons each travelling at 40 ° to the incident beam direction (see table 1). The curves show the experimental shapes of the strongest mono-energetic gamma rays, and their combined spectral shape. TABLE 2
Cross sections for production of gamma rays in the aSK(p, 2p) reaction Gamma-ray energy (MeV)
dtr dO1 d-Q= (mb • sr -~)
1.65±0.07 2.20~0.07 2.9 ±0.I 3.6 ±0.1
0.9±0.3 2.5!0.4 1.1±0.3 1.2
4.4 ±0.2
0.6
g a m m a - r a y is that at 2.20 + 0.07 which we identify with the decay o f the first excited state o f as A at 2.17 M e V : h o w e v e r we shall see that an ap p r eci ab l e p a r t o f the p r o d u c tion o f this state m u s t be by cascades f r o m higher states. Th e 1.65 M e V g a m m a ray
516
D. N~w'roN et
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is presumably the well-known 3,4) cascade decay of the third excited state at 3.82 MeV. The 2.9 MeV gamma ray must also be a cascade decay as there is no excited state of this energy. There are two known cascade gamma rays 6) in 3aAr which are possible candidates: a decay from a state at about 5.0 MeV to the 2.17 MeV state or a decay from a state at about 6.6 MeV to the 3.82 MeV state. We will present arguments in favour of the latter of these two possibilities in the next section. The exact identification of the energies of the two highest energy gamma rays is somewhat uncertain but we believe that the energy quoted as 3.6 MeV cannot be as high as 3.9 MeV and so again must be a cascade decay, probably from a state around 5.8 MeV to the first excited state. We shall assume, for reasons given in the following section, that the gamma radiation near 4.4 MeV is a ground state decay. 4. Discussion
We shall start by describing in more detail which states of 3SAr one would expect to be produced in the 39K(p, 2p) reaction. We have indicated in the introduction those jj-coupling configurations which will be involved. Wilmore has calculated (see Phillips 3)) how these configurations should mix in intermediate coupling. It is possible, however, to estimate in a simpler manner what should be the dominant configurations in the low-lying two-hole states of 3SAr, by the methods used by Redlich 7) to study low-lying states of 1SO and lSF. He obtained excellent estimates of intermediate coupling wave functions from a product of wave functions of a single particle in a spheroidal potential. He showed how, if he projected eigenfunctions of angular momentum out of such a product wave function (which is not itself an eigenfunction of angular momentum) the resulting wave functions were very similar to those obtained from an intermediate coupling calculation. He initially tried as his single-particle wave functions the asymptotic wave functions which correspond to a very large value of Nilsson's ratio r/ of the spheroidal deformation to the spin orbit coupling s), but then found that a better fit could be obtained if he modified his single-particle wave functions in an empirical manner. We note that these resulting modified single-particle wave functions are very similar to those found by Nilsson for mass 18 nuclei with a prolate shape, and r/about 4 to 6. (Note that there is a difference in sign convention between the wave functions of Redlich and those of Nilsson.) This similarity justifies the single-particle wave functions introduced empirically by Redlich. This procedure for obtaining intermediate-coupling wave functions has also been remarkably successful in the lp shell 9). In using this procedure the degree of intermediate coupling is related to the value of 17assumed. The corresponding single-hole wave functions in a deformed potential that should contribute to the lowest states of 3aAr are those corresponding to Nilsson's levels 11 and 8, and are of the forms level 11: ~Ox(K = ___½)= axd~, ± , + b l s ~ , ±~r+ctd~, ±~, level 8: ~k2(K = +__~) = a2d~, ±~+c2d~, ±2"
agK(p, 2p) REACTION
517
F r o m these single-hole wave functions one then forms rotational bands by taking a product of two of them and projecting out eigenfunctions of angular momentum. Each such product wave function has projected out of it the wave functions of the several states which make up the "rotational band". One would expect that, for 3SAr, Ir/[ would not be large so that for ~ i and ~bz the first coefficients, al and a2 respectively, should be distinctly larger than the others. It is then clear that there will usually be one term which is dominant in each intermediate coupling wave function which is projected out of the product. The two lowest such rotational bands in 3SAr, come from these products: ~kl(K = ½)~bl(K = - ½ ) , forming a K = 0 band with states of J~ = 0 ÷, 2 +, 4 +. ~k2(K = ~ ) ~ I ( K = - ½ ) , forming a K = 1 band with states o f J ~ = 1 ÷, 2 +, 3 ÷, 4 ÷. The energies of the states of the K = 0 band should increase with J, but for the K = 1 band the ordering in energy can differ from the ordering in J, as it can for a K --- ½ band ~0). There can also be mixing of states of the same J from different rotational bands. On examining the product wave functions we find that for the following states the dominant contribution includes at least one hole in the d~ state: lower K = 0 b a n d (~klffl): J~ = 0 +, d~ 2 J~ = 2 +, d~ z J~ = 4 + , d~ld~ -1 K = 1 band (~i ~2):
jn = 1 +, d~ls~l,
d~ld~ "1
J~ = 2 + , d ~ l s ~ 1,
d ~ l d ~ 1,
d~ 2
J~ = 3 + , d ~ l d ~ 1 J~ = 4 + , d ~ d ~ 1 Thus we should expect these states to be produced strongly in the 39K(p, 2p) reaction. The J~ = 0 + state is presumably to be identified with the ground state of 3SAr, so here we should only observe production of the other states. It has been found 4) that there are m a n y more low-lying states of 38Ar than there should be for two holes in a #°Ca closed shell. One may therefore ask whether there will be important mixing between the two hole states and the other states. Our results provide some evidence that this mixing is not strong. We will assume that low-lying negative parity states of 3SAr are not produced strongly in the 39K(p, 2p) reaction because they would be due to promotion of a particle into the I f or 2p shell. They could thus be produced in this reaction only if there were appreciable admixture of two-particle-three-hole configurations in the ground state of 39K; we would expect such admixtures to exist, but to be weak. We find no evidence for any strong production of negative parity states of 39K in the 4°Ca(p, 2p) reaction 1).
51 8
D. NEWTON" et aL
We may expect that the J " = 2 + first excited state of aaAr is to be identified with the J " = 2 + state of the K = 0 rotational band, and on this basis we can estimate the expected cross section for its direct production (as opposed to the production by cascade decays from higher states, which we have already remarked on). It was found 2) that the cross sections for ejection of d~ and d~ protons from 4°Ca are much the same, and we found a cross section of approximately 1 m b . sr -2 for ejection of a d~ proton f r o m 4°Ca with a counter geometry similar to that used in the present experiment. One expects the probability of two d~ holes coupling to form a state of angular m o m e n t u m J to be proportional to 2 J + 1, so that approximately ~ of the ejections of a d~ proton from 39K should result in production of this J~ = 2 + state of aSAr. On this basis we deduce an expected cross section of approximately 0.8 m b . sr -2 for its production. Our results are consistent with such a cross section for direct production of the first excited state of 38Ar. As the third excited state of aSAr at 3.82 MeV has 3,11) j = 3 - , we would not expect it to be produced directly in the 39K(p, 2p) reaction. We therefore ascribe its observed production to cascade decay of a state at about 6.6 MeV, as mentioned in the previous section. The agreement between the cross sections for production of the 1.65 and 2.9 MeV g a m m a rays, which we are taking to be successive members of a cascade, supports this assignment. We also remark in further support that if both the 1.65 and 2.9 MeV g a m m a rays were separate cascade decays to the first excited state of 38Ar, these cascades, together with the 3.6 MeV g a m m a ray would be greater than the observed cross section for production of the first excited state. I f we take the 1.65 and 2.9 MeV g a m m a rays to be successive members of a cascade from a 6.6 MeV state, the cross section for direct production of the first excited state is then approximately 0.5__+0.5 mb • sr -2, consistent with the expected cross section. As decay of the 6.6 MeV state to the J = 3 - state at 3.82 MeV is apparently favoured over other conceivable decays with a higher energy release we would think that this state probably has J = 4 +, with the electric dipole decay being faster than higher energy electric quadrupole decays. However d = 2 or 3 ÷ assignments are presumably possible, with the magnetic dipole decays to the first excited state slower than the E1 decay to the second excited state. (We note that electric dipole decays will not here be inhibited by isobaric spin selection rules as aSAr is not a self-conjugate nucleus in contrast to an electric dipole transition we have found 12) to be strongly inhibited in 40Ca). We have seen that there is some evidence for production of states of 3aAr at about 4.4 and 5.8 MeV, though further work is needed to sort out the higher energy region of the gamma-ray spectrum in more detail. We should not ascribe all this g a m m a radiation to cascades through the first excited state as such cascades could exhaust all of the observed production of the first excited state. O f the states that we are observing here, a state that could decay to the ground state would probably have J = 1 or 2 ÷, while one which decays by a cascade through the first excited state would probably have J = 2 or 3 ÷, for which magnetic dipole decay is possible, though d = 4 + is also possible.
3~K(p, 2p) REACTION
519
The calculations of Wilmore with our considerations earlierin this section, would seem to favour a suggestion that the states we have discussed in the last paragraph should be the J~ = 1 ÷ and 2 + states of the K = 1 band. These states would be produced on ejecting a 2s~ proton from 39K. The combined cross section for their production is close to the 1.5 m b • sr -2 we found 1) for ejection o f a 2s½ proton from 4°Ca with the same counter geometry as in the present experiment. This agreement between these cross sections provides support for our assignment. The 6.6 MeV state would then probably be the J = 4 + state of the K --- 0 band: the cross section for its production is not far from the 0.5 mb • sr -2 we deduce for its expected production cross section from the observed cross section 1) for ejection of a ld~ proton f r o m 4°Ca. I f this assignment is correct the ratio of the energy of the J = 4 + state to that of the J = 2 + state of this K = 0 band is 3.0, a ratio which is not far from the value of 3.33 expected for a rotational band in a deformed nucleus. This ratio, for two holes in the 2 s - l d shell, is quite different from the ratio of 2.0 found for the corresponding states of two particles in the 2 s - l d shell, in 180. This may indicate that 4°Ca is more deformable than is 160; further evidence for this has been deduced 13) from the strengths of electric quadrupole excitations in 4°Ca and in 39K. In our study 1) of the reactions induced by b o m b a r d m e n t of 4°Ca with 150 MeV protons we observed production of states of 3SAt in the 4°Ca(p, 3p) reaction. In this reaction we found that the same states of 38Ar were produced as in the 39K(p, 2p) reaction, with much the same relative cross sections, suggesting that the major contribution to the mechanism of the (p, 3p) reaction producing low-lying states of the residual nucleus is a process in which the incident proton makes two separate collisions with different protons in the target nucleus and ejects them both. As the 4°Ca(p, 3p) reaction should produce all states formed by two holes in the 2 s - l d shell, while the 39K(p,2p) reaction should produce only those in which at least one of the holes is in the ld~ state, this r¢sult suggests that states of 38Ar which are largely made up of two holes neither of which is in the ld~ shell must be at high excitation energies for otherwise we should have observed g a m m a radiation from their decay. This conclusion agrees with the picture of 3SAt we have presented. In conclusion we see that this work, even though it is of an exploratory nature, has indicated the possibility of obtaining considerable information about the two-hole states of aaAr. It would be very interesting to study this g a m m a radiation in greater detail to obtain more information about the states between 4 and 6 MeV. It would also be interesting to repeat these measurements for different angles between the two proton counters, as the angular correlation between the two outgoing protons is much sharper for ejection of a proton from an s-state than for ejection from a d-state 2) in this way one could hope to obtain information about the relative contributions of 2s and ld holes to the 38Ar states. We wish to thank Dr. B. Rose and all members of the cyclotron group at A E R E Harwell for their considerable help and the Director, AERE, for permission to use the cyclotron.
520
D. NEWTON et aL
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