The T = 1 excited states of C12

Nuclear Physics 29 (1962) 89--99; ~ ) North-Holland Publisldng Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher

THE

T = 1 EXCITED

STATES

O F C 12

N. V I N H - M A U

Laboratoire de Physique Nucl~aire, Facult~ des Sciences de l'Universit~ de Paris, Orsay and G. E. B R O W N Institute for Theoretical "Physics, Copenhagen, D e n m a r k

Received 4 M a y 1961 T h e T ~ 1 s t a t e s of C Is a r e calculated within t h e particle-hole formalism, a n d t h e results are compared with experiment. Calculations are carried o u t both. with a n d w i t h o u t g r o u n d - s t a t e correlations. The a p p r o x i m a t i o n of zero-range forces is used. I n addition to reproducing k n o w n levels, t h e ca2culatlons indicate t h s t t h e strong level a t 19.5 MeV is 2 - - , a n d t h a t a higher dipole level, which h a s n o t y e t been observed, exists a t 34 MeV.

Abstract:

Highly excited states, most of which presumably have an isobaric spin T ---- 1, have been observed in Cx2 in a variety of experiments. These include inelastic proton and electron scattering x, 2), which excite several states with high probability in the 15--25 MeV region, and 7-ray absorption 8) and the inverse (P, 7) process t), which excite the giant dipole resonance at 22.5 MeV. The purpose of this note is to give a simple description of these states. In earlier work 5), the T = 1 dipole states in 0 le were described, beginning from !-7"coupling, as one-particle, one-hole states. The particle-hole interaction mixed the unperturbed configurations, producing final eigenstates with quite an unequal distribution of dipole strength. In that simple description, zero-range forces were seen to give results which were not only qualitatively, but also quantitatively valid in the sense that t h e y reproduced the results both of the finite-range calculation 0) and of experiment. In the same spirit, we view C1~ here as a closed-sheU nucleus in i-i coupling, with the ground-state configuration (ls½)4 (lpt)S. We treat this ground state as the physical vacuum and refer to absent particles as holes. Our unperturbed particle-hole states are, e.g., l p | -1 l d j or ls½-1 lP½. Unperturbed energies are obtained empirically from neighbouring nuclei. For example, in the configuration 1pt-~ ld4, the enelgy of the p~ hole is obtained from the binding energy of Cu and that of the d t particle from the excited state of C13, which can be described as a particle in the d4 orbit. The particle-hole interaction determined for 018 is then diagonalized in this representation. We have, therefore, no adjustable parameters at our disposal. 89

90

14. VINH°MAU AND G. E. BROWN

This approach is novel in several respects. The unperturbed configuration l p f I l p i h a s an energy of ~ 14 MeV. Yet this is the largest componen t of the low T = 0, 2 + state. If we used the procedure here to calculate T =- 0 states, the particle-hole force would have to move this state down to the observed value of 4.4 MeV, which would clearly involve a great deal of configurational mixing. Thus, the method would probably be somewhat more cumbersome than the usual intermediate-coupling approach in this case, although we believe that it m a y be useful in certain respects even in such a calculation. In the case of T = I excitations, the situation turns out to be quite different. Although there is appreciable mixing of configurations of initial energy 1~¢o, where we take t~o as a typical distance between shells, the states axe not really shifted far in energy. We believe this to be the main reason w h y the results are insensitive to the details of the force. This indicates, however, that the approach using zero-range forces m a y be applicable only to light nuclei, since the proportional shift in energy and amount of mixing seem to be greater in heavy nuclei. In section 1 we shall give a short survey of the formalism employed in this work. In sect. g we shall describe the calculations of the electric l - and 2 + and the magnetic 0-, 1+ and 2-levels in C1~. Sect. 3 consists of a discussion to the results.

1. Survey of the F o r m a l i s m The particle-hole interaction is easily obtained from the particle-particle one b y means of the technique of second quantization outlined in ref. J). The matrix elements of the particle-hole interaction V,,,, ,~,, can be written as the difference of two terms D,,, ~, and E,,, t~', represented in fig. 1.

Fig. I. G r a p h s illustrating (a) the m a t r i x element D ~ i , IJ" and (13) t h e n ~ t r i x element E . , , 11"T h e s t a t e s i a n d ~ are t h e initial and final particle s t a t e s a n d i" a n d ~" t h e hole o n e s . E x a m p l e s of s u c h m a t r i x elements axe given in eq. (5) of ref. 6) ?.

We now have to diagonalize the corresponding secular matrix, the off-diagonal elements of which are just the V.., ~. and the diagonal terms e..~.., ~ . + V . . , ~., where e.. is the unperturbed energy, that is, the sum of particle-core and holecore energies. t W e use, however, a slightly different p h a s e c o n v e n t i o n here. I n ref. 6) the p h a s e s (--I)ta_--~ a n d (-- l)Z'1-'m'I converted the particle w a v e functions i n t o hole ones for t h e p u r p o s e s of t r a n s f o r m a t i o n . H e r e we shall use factors so as to c o n v e r t hole w a v e functions into particle ones; these a r e t h e c o n v e n t i o n s of Bell 7))

THE T ~

I EXCITED STATES OF CIs

91

We c a r r y out the calculations in/'-1" coupling a n d calln~,/'~,l~, s~,r~thequanr u m n u m b e r s of the particles a n d m~, ~ , a~, v~, their projections. We use the same b u t with a lower suffix i', for holes. We can t h e n write the particle-hole wave functions as t Iph> =

~ ]j,,L~ L.s,u

• s,, j,, S J

(LSM--MIJO)

x 2 (lJ,,a,a,,ILM)g,~,>
x E. (s,s,,,,,o,,IS--M)ls,,,,>
''+'"

X (~,r,.,,,,,..ITo) I,,,,,,><,,,-~,,.I(-

(1)

x)',"-','.,

•t'tt~ " t ¢ , s

with the definition d = (2aq-1)½. Here the particle and hole are coupled to total angular momentum J. (Since the ground state of C]l is 0-k, J is the angular m o m e n t u m of the level we are considering. Similarly, the p a r i t y of the excited level is t h a t of the particle-hole pair). We performed the calculations using the exchange mixture given in eq. (8.1) of ref. 5) in zero range, i.e., V ---- V0(0.865+0.135 o x • a , ) 6 ( r a - - r , ) .

(2)

F o r this force, the E ~ , ~¢ p a r t of the m a t r i x element is always zero for T = 1 states. We obtain, then, different formulas for electric and magnetic transitions. 1) For a T = 1 electric transition we have

v'~'"'"-

(--1)'"+b'Vel,l,,~il,,(l,l,.J~ ~l,l,,J~ 4=

00

o / ~ o 0 o/

(3)

× [T~TJo~-F T b'¢ T;j] ,,' asC.,, .,, where

fz, ~, i, I T'~;. = ],],,L~ it,, s,, i,q (LS

(3.1)

j)

and as is a coefficient resulting from the presence of the 01 • a s term in V; i t has the form •~t~" TJJ'

T t ~ ' "/'ql'

as = l + 0.27 -'o!- o]----if"

1]

T ~i" T t J ' _ I _ T t~" T t g

~Of ~Of -- ~1 f ~lf

(3.2) "

The G... ~, is the radial integral, which is of the form

G.,a¢ = f R.& R.,a, , R.,z, R.,,b" r~dr t We adopt the conventions of Edmonds

(3.3)

s) f o r t h e v a r i o u s f u n c t i o n s o f a n g u l a r m o m e n t u m .

9~

N. VINH-MAU AI~D G. E. BROWN

for a zero-range force. We take the radial wave functions R to be the eigenfunctions of the simple harmonic oscillator. 2) For T = 1 magnetic transitions, we have V o l , , l , i , I ~ , [ Atit j _ l A jj_f x + ".r+l".r+lJ'-" A , , , nJJ" ~t:_ : ~', V~,ff ~--- 0.73 (--1)',+'f 4~

(4)

where 0

~ 1L"

The factor 0.73 comes from the particular force employed. Until now we have considered the ground state to be a system of noninteracting particles, as is done in most shell-model calculations. It has been pointed out 9-~0), however, that one should include correlations in the ground state consisting of particle-hole pairs, each coupled to ] . In r-excitation, one of these pairs can be annihilated in absorption of the r-ray, as illustrated in fig. 2.

F i g . 2. T h e e f f e c t of g r o u n d - s t a t e c o r r e l a t i o n s o n t h e a b s o r p t i o n of t h e ~ - r a y .

These effects can be included b y diagonalization of a larger secular matrix of the form .-lo)

where A is the matrix considered above without ground-state correlations, i.e., with matrix elements e,, 6,,,~j, + V , , f f and B is a matrix with elements V,,,f~. Thus, the matrix elements of B can be obtained from those of A b y interchange of the particle and hole in either the final or the initial state. Within the schematic model formulated earlier xl), the ground-state correlations have a great effect on both the energies and the transition probabilities of those states which are shifted far b y the perturbation. In this model, V,¢, ~,~ was set equal to V u , f t . In actual fact, in j-j coupling the matrix elements of B tend to be smaller than those of A, so that the schematic model grossly overestimates their effect. The reason for this is that, whereas the contribution to

THE

T ~

I EXCITED

STATES

OF

CIt

9B

(3) coming from the part of the particle-hole state coupled to S = 0 behaves as in the schematic model, the part coupled to S = 1 takes on the opposite sign. This comes about because TJ0~ and TJl~ behave in opposite fashions on interchange of i and i'. If TJ0~does not change sign, then T ~ does, and vice versa. The opposing effect of S = 0 and S = 1 contributions tends to make the net effect small. In the calculations we performed the effect from inclusion of ground-state correlations was so small t h a t we shall not bother to list the results, except in the case of the 2- states, where we shall give them by way of illustration.

2.

Calculations

The strength Vo of the particle-hole interaction was determined in the case o f O le to be6)

vo 4rib3

--

8.50 MeW- fm -3,

where b is the range parameter in the harmonic oscillator well. We take Vo to be the same for C12, but change b so that (r~) varies as AJ. This gives V0 = 10.2 MeV • fin -3 4~b 3

- -

for 0 2. We take the unperturbed energies e,, from the experimental data 11) on the levels of C11 and C18. If ~,, = ~+~i', then one has e, = B ( C 1 3 ) - - B ( C l S ) + E o

e,, = B ( C l l ) - - B ( C l S ) + E v ,

where B denotes the binding energy of the relevant nucleus, E~ is the excitation energy of the relevant single-particle state in C18 and E~, that of the hole state in Cu. Experimentally we know very accurately the energy ,~, of a hole in the l p t shell, corresponding to the ground state of C11, but it is more difficult to find the energy of a hole in the ls½ shell. The Uppsala and Oxford (p, 2p) experiments 1~ 13) give a binding energy of about 35 MeV for the ls nucleon. (This hole level is very broad, and we settle on 35 MeV as its centre). In the case of the particle-core energies e~, the energy of a particle in the lP½ shell can be found from the ground state of C13" The !+ state at 3.09 MeV 2 and the ~-+ state at 3.85 MeV can be interpreted as the Clz ground state plus a 2s½ and ld t neutron, respectively 1~). The ld t excitation lies somewhat higher, and probably does not consist of a single compound state. The (d, p) reaction, which is especially suitable for picking out single-particle states, shows 15) a maximum in the region of 8.33 MeV, which is interpreted as coming from the

N. VINH-MAU AND G. E. BROWN

94

excitation of a {+ state or states. We choose this as the energy of the l d t state. This completes the particle energies needed for the calculation of odd-parity states. It is, however, interesting to consider the even-parity states in Cis as well, since the energy of the lp~ -1 lP½ configuration lies in the region of the negative parity states. This configuration can be coupled to J = 1 and J = 2 so as to give T ---- 1, 1+ and 2 + states, both of which have been observed. In order to have an idea of how much other configurations mixed into these states, we introduced the 21ko states * into the calculation. This involved the use of particle-core energies for lfg, l f{, 2pt, 2P½ states, and these cannot be obtained empirically. In order to have some values, we fitted the paramete~ of a harmonic oscillator well so as to reproduce as well as possible the known particle-core and hole-core energies, and then extrapolated to find the unknown ones. Since we only use these higher levels to give qualitative indications of the amount of configurational mixing and distribution of multipole strength, this procedure should be adequate. '~Ve indicate all values of particle-core and hole-core energies that we used in table 1. We now list the various results. TABLE 1 Particle-core and hole- coreiuteraction energies

States

ls

i lp~

et t'i,

35

18.7

2.1. T H E

lp~ I 2s.~ ! ldt --4.9

--1.9

--1.1

ldt 3.4

14 7.0

tI I

~Pt

2Pt

1%

8.7

14.7 t 15.5

I

1- L E V E L S

If we consider only excitations of I//to, we have the following four possible configurations for the particle-hole system: lpt-1 2s½,

lp} -1 ld},

lpt-1 l d t,

ls] -1 l p t .

The results of the diagonalization are given in table 2. We have listed here the perturbed eigenstates in the column corresponding to their largest component in the unperturbed configuration. The non-spin-flip dipole state comes at 22.2 MeV, and this is in reasonable agreement with the experiments s, 4), i n w h i c h t h e maximum comes at 22.3 MeV. On the other hand, our level at 18.7 MeV comes too high, in t h a t it presumably corresponds to the 1-, T = 1 level le) observed at 17.2 MeV. Here the particle is almost completely in the 2s½ state, and just this level m a y be especially sensitive to our assumption of zero range, since the 2st-wave function has a node. ? W h e n we s p e a k of nTtco states, we m e a n s t a t e s in w h i c h the particle in s m a j o r shells above the hole. I n a h a r m o n i c oscillator well, such states w o u l d h a v e the u n p e r t u r b e d energy mrtoJ, b u t here t h e y can h a v e energies differing b y several MeV because of spin-orbit coupling etc.

THE T =

1 EXCITED

STATES OF C 11,

95

A reasonable proportion of the dipole strength is concentrated at the much higher energy of ~ 34 MeV, where no state has yet been observed. This state should be observable in inelastic electron or proton experiments, when they are extended to such high energies. TABLE 2 Calculated energies and s t r e n g h t s of the T = l, 1- levels in C I~ U n p e r t u r b e d configuration U n p e r t u r b e d e n e r g y (MeV) U n p e r t u r b e d s t r e n g t h (%) s) P e r t u r b e d e n e r g y (MeV) P e r t u r b e d s t r e n g t h (~o)

l p | -1 2st

l p | -1 l d |

l p | -1 l d t

16.9

22.1 7 23.9 0.5

17.6 67 22.2 75

15 18.7 6.5

Is½-l l p t 30 11 34.3

18

i) B y s t r e n g t h we m e a n the s q u a r e of t h e m a t r i x element of the dipole operator, w i t h o u t a n y e n e r g y factors.

2.2. T H E 2- LEVELS Here we include the configurations Ipi-1 2s½, Ipi-1 Idl, Ip~ -I Id i. The energies of the levels and their strenghts are given in table 3. TABLE 3 Calculated energies a n d s t r e n g t h s of the T = 1, 2 - levels ~,Vithout g r o u n d state correlations E(MeV)

22.9

19.2

P e r t u r b e d s t r e n g t h (~/o)

22

72

18.1

6

~:ith ground state correlations 22.9

19.2

24

86

18.1

7

XVe obtain a level at 19.2 MeV which carries 72 % of the M2 strength. This is interesting, because we can identify this with the peak in inelastic proton scattering 1) observed at 19.5 MeV. The angular distribution of protons exciting this peak is similar to that coming from the excitation of the 22.3 MeV 1- level, and many people have interpreted it as 1-. Our interpretation as 2would explain w h y the lower peak has not been observed in y-ray reactions, where the probability of emission of an M2 7-ray is small. The interpretation of the peak at 19.5 MeV as a 2- excitation is supported b y work of E. Sanderson 17), who has calculated the angular distributions for excitation of 1- and 2- levels in inelastic proton scattering. The theoretically predicted angular distributions are similar in the small-angle region, in agreement with experiment. The angular distribution from the 2- excitation would, however, be expected to predominate over that from the 1- at larger angles,

96

N. V I N H - M A U AND G. E . BRO W N

since it can be excited through transfer of both angular momentum l = 1 and l = 3 to the nucleus, coupled with spin flip, whereas the 1- level can be excited only through l = 1. This predominance is also observed experimentally is). Using the compositions of the 1- and 2- states calculated here, Sanderson's calculations reproduce the experimental data reasonably. 2.3. T H E 0 - L E V E L S

We include here the two configurations ls½-1 lp½ and lpt-1 l d t. After diagonalization we obtain the results of table 4, showing levels at 24.7 and 33.9 MeV. No experimental data exist to compare with these results. TABLE 4 Calculated energies and s t r e n g t h s of the T = I, 0 - levels E (MeV)

P e r t u r b e d s t r e n g t h (%)

24.7

75

33.9

25

2.4. T H E 1+ L E V E L S

Here we take into account both 0 and 27~o excitations, which gave the six configurations lpt-1 lP½, ls½-1 l d t , ls~ -1 2s½, lpt-1 2pt, lpt-1 2P½, l p t - I l f | . The lpt-1 l p t configuration lies at a lower energy than the others, and carries all of the M1 strength. Our calculations show that the other states are mixed TABLE

5

Calculated energies of t h e T -~ 1, I + levels E (MeV) 16.1 28.7 34.2 35.7 38.0 41.0

P e r t u r b e d s t r e n g t h (%) 100

only weakly into it, so that it remains almost pure. We do not believe that uncertainties as to the particle-core and hole- core energies affect this conclusion appreciably. In our calculations, the 1+ level comes at 16.1 MeV, whereas experimentally it is observed le) at 15.1 MeV. Whereas the theoretical level is a bit too high, it is close enough to indicate that our use of the surprisingly high value of ~ 14 MeV (see table 1) for the "spin-orbit splitting" between the l p | and lp½ levels is not unreasonable.

THE T---~ 1 EXCITED STATES OF Cli

97

2.5. T H E 2 + L E V E L S

Here we include excitations of 0 and 27/~ which gives seven different particlehole configurations: lpt-1 l p t , lpt-1 lft, lpt-1 If t, lpt-1 2pt, lpt-1 2p[, ls -1 l d t , Is -1 ld~. The unperturbed energy of only the first of these is known with any accuracy, so that our results will only give qualitative trends for the distribution of quadrupole strength among the upper levels. Furthermore, we have neglected here two-particle, two-hole excitations, which can have the same unperturbed energy of 27/~o, and these m a y mix in appreciably. We give our results in table 6. TABLE 6 Calculated energies and s t r e n g t h s of t h e T ~ I, 2+ levels E (MeV)

P e r t u r b e d s t r e n g t h (%) 9 41 1 19 1 13 16

16.5 27,9 28.1 35 35.2 36.6 41.5

The only level well determined experimentally lies at 16.1 MeV (ref. 16)); we find a level at 16.5 MeV which carries 9% of the total quadrupole strength. Most of the T---- 1 quadrupole strength is found in the region 28-42 MeV. There can, therefore, be appreciable interference between quadrupole and dipole processes. An example of this has been calculated for the (p, P'7) process 19). 3. D i s c u s s i o n

In the preceding development, we have given a simple explanation of most of the highly-excited states observed in Clz. In m a n y cases, the energy of the state was determined mainly b y the particle-core plus the hole-core energy, but in some important cases, e. g. the dipole state, the particle-hole interaction shifted the energy appreciably. The 19.5 MeV state observed in inelastic proton scattering is identified as 2-. We find a T ~ 1, 2- state at 19.2 MeV, but there m a y also be a T ~ 0, 2- level at about the same energy. The T = 0 magnetic levels appear, however, to be much more sensitive to the 1-ange of the force t, and we have not calculated This is so because t h e direct a n d exchange p a r t s of t h e particle-hole interaction a l m o s t cancel each o t h e r for these states, a t least for t h e t y p e of e x c h a n g e m i x t u r e we employ. T h e r a t i o of direct t o exchange t e r m depends, however,~/airly sensitively on range, so t h a t t h e small difference b e t w e e n t h e m is v e r y sensitive to t h e range. Indications are t h a t t h e particle-hole force is w e a k in t h e T ~ 0 m a g n e t i c s t a t e s (0% 1 + and 2-), a n d t h a t these do n o t m o v e far f r o m their u n p e r t u r b e d

values.

98

N. VINH~MAU AND G. E. BROWN

them here. Furthermore we predict a 1- level at 34 MeV, which has not yet been observed. There are, of course m a n y more levels in the energy region considered t h a n we obtain in our calculations. However, we obtain the single-particle excitations, and these are most strongly excited in experiments such as inelastic proton scattering. Each excitation m a y well be made up of a number of compound states ~ Sl), but this should not affect the results of experiments with an energy resolution broad compared with the distance between compounds levels. The inclusion of ground-state correlations in the calculations did not change the results appreciably; thus the calculation was essentially the same as in the usual shell-model formalism. The schematic model of refs. s, 11) did indicate the general trend t h a t the results would take, although in the calculations here the off-diagonal matrix elements were relatively small compared with those in the schematic model, so t h a t the configurational mixing was not so great. The main reason why the ground-state correlations were not so important here as in the schematic model is t h a t the S = 1 part of the particle-hole matrix element behaves differently to t h a t of the S = 0 part, and to t h a t required in the schematic model, in going from matrix A to matrix B in eq. (5). This indicates t h a t in applying the schematic model to T = 1 states it is better to leave out the ground-state correlations. We are pleased to acknowledge m a n y helpful discussions with J. A. Evans, and wish to t h a n k E. Sanderson for giving us the results of his calculations on the inelastic scattering of protons b y Cit. This work was begun when both authors were at the University of Birmingham, and t h e y want to t h a n k Professor R. E. Peierls for encouragement. One of us (N. V. M.) wishes to t h a n k the Institute of Theoretical Physics, University of Copenhagen, for hospitality. References 1) H. Tyr6n a n d Th. A. J. Marls, Nuclear Physics 3 (1957) 52 2) J. P. Garrou, J. Phys. et Rad. 21 1 (960) 317 3) See for references M. E. Toms, Bibliography of photonuclear reactions, Naval Research Laboratory Bibliography, No 18, October 1960 4) Cove, Litherland and Batchelor, Phys. Rev. Lett. 3 (1959) 177 5) Brown, Castillejo and Evans, Nuclear Physics 2 2 (1961) 1 6) J. P. Elliott a n d B. H. Flowers, Proc. Roy Soc. 241 (1957) 57 7) J. S. Bell, lquclear Physics 12 (1959) 117 8) A. R. Edmonds, Angular m o m e n t u m in q u a n t u m mechanics (Princeton University Press, Princeton 1957) 9) Brown, E v a n s and Thouless, Nuclear Physics 24 (1961) 1 10) D. J. Thouless, Nuclear Physics 22 (1961) 78 11) G. E. Brown and M. Bolsterli, Phys. Rev. Lett. 3 (1959) 572 12) Tyr6n, Hillman and MarLs, Nuclear Physics 7 (1958) 10 13) T. J. Gooding a n d H. G. Pugh, Nuclear Physics 18 (1960) 46 14) A. M. Lane, Phys. Rev. 9 2 (1953) 599

THE

T ~

1 EXCITED

STATICS OF

C 12

99

15) Mc Gruer, W a r b u r t o n and Bender, Phys. Rev. 1O0 (1955) 2S5

16) F. A|zenberg-Selove and T. Lauritsen, Nuclear Physics l l (1955) 1 17) E. Sanderson, Nuclear Physics, t o be published. 18) Riou, Garron, J a c m a r t and /luhla, Proc. of the Int. Conf. on Nuclear Structure, Kingston, Canada (North-Holland Publi_qhlug Co., Amsterdam, 1960) p. 965; see also rcport b y I Katz. ibid, p. 710 19) N. Gilbert, Proc. Phys. Soc. 77 (1961) 362 20) Lane, Thomas a n d Wigner, Phys. 1Rev. 98 (1955) 693 21) G. E. Brown, Revs. Mod. Phys. 31 (1959) 893