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On the collective states of C12 and ground-state correlation
Nuclear Not
to
Physirs
44 (1963)
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294-308;
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Publishing
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ON THE COLLECTIVE
STATES
AND GROUND-STATE
CORRELATION
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publisher
OF C”
A. GOSWAMI Saha Institute
of Nuclear
Physics,
Calcutta
and M. K. PAL + Atomic
Energy
Research
Received
Establishment,
21 January
Harwell
1963
The collective electric dipole, quadrupole and octupole states of Cl3 have been calculated, including the effects of ground state correlation, by the Sawada method and using finite range Soper force. A first order calculation of the amount of ground state correlation indicates that this amount is considerable in Cl2 and that most of it is due to quadrupole vibrations. The justification of the Sawada method for Cl2 is discussed in this context. Quantities related to the ground state correlation which can be determined experimentally are also calculated and discussed.
Abstract:
1. Introduction The presence of collective electric dipole, quadrupole and octupole states in the spectrum of C” has been established by many experiments. Inelastic proton scattering experiments ‘) and (r-n) reaction experiments “) show the presence of a dipole state at about 22.2 MeV. The collective nature of the 4.43 MeV 2+ state is also wellknown “) from the enhanced quadrupole y-transition to the ground state and also from electron scattering experiments. Lastly, a careful analysis of experiments 4> has now conclusively shown that the correct spin and parity assignment of the 9.63 MeV state is 3 -. Theoretical calculations have been made (i) by Brown and Vinh-Mau “) for the dipole state and (ii) by the present authors 6, and by Gillet and Vinh-Mau ‘) for rhe quadrupole state. Two methods of calculation are used which differ in the description of the ground state. In the first method the ground state is assumed to be a doubly closed shell state (1~)~ (lp+)*. A y-ray quantum can excite the nucleus by raising a nucleon from any of the closed shells to an empty shell, thereby creating a holeparticle pair (1H-IP). The unperturbed energy of such a lH-1P state is given as the sum of the hole-energy and the particle-energy which can be determined from experimental data. One then diagonalizes the hole-particle interaction with these states as basis to get different states of specified total angular momentum and isobaric spin. This is the usual shell model method and is often referred to as the Tamm-Dancoff method. 7 Present address:
Saha
Institute
of Nuclear
Physics, 294
Calcutta.
C O L L E C T I V E S T A T E S OF C 12
295
The second method differs from the first in that correlations in the ground state consisting of hole-particle pairs are taken into account. The standard method of calculation, taking ground state correlations into account, is the Sawada a) method of linearized equation of motion originally designed for the treatment of plasma oscillations in the electron gas. This method has been applied to the case of nuclear vibrations by many authors 9-13). In the Sawada method, correlations in a doubly closed shell nucleus implies the presence of two-hole, two-particle (2H-2P) pairs of all orders in the ground state. Excited states may then also be obtained by the annihilation of one of the hole-particle pairs present in the ground state. Owing to this new type of excitation amplitude, the ground state correlation may enhance the v-transition probability. Calculations t have been made by both methods for the collective states of C 12 (refs. 5-7)), Or6 and Ca "° (refs. 7, 9)). The results indicate that in order to reproduce the collective behaviour of the T = 0 states, one has to take recourse to the second method, while for the T = 1 states, the effects of ground state correlations is not appreciable. The Sawada method enables one to calculate directly the energy and transition probabilities of the excited states, without having ever to write down the detailed structure of the correlated ground state wave function. However, such a detailed expression in terms of the uncorrelated ground state plus the excited 2H-2P configurations in various orders would be welcome for several reasons: (I) It is necessary, in the first place, for a check on the internal consistency of the Sawada method. A hole-particle pair is assumed, in this method, to be a composite unit behaving as a boson, which assumption is good only if the occupation number of the single nucleon states in the correlated ground-state is not very different from that in the closed shell state. If, for example, the above condition is found to be very badly satisfied for the particular correlated ground state wave function, then the agreement of energy and transition probability of the excited collective states would really be fortuitous. (2) The detailed wave function for the ground state enables one to calculate the occupation probability of an excited single particle state, which can be verified experimentally by performing pick-up reactions on the ground state. The present paper gives the results of our calculations on the basis of a finiterange Yukawa-force with Soper 15) exchange mixture for the electric dipole, quadrupole and octupole states of C 12. For the reasons stated above we have also calculated the detailed ground state-~ave function. Although the method used for this purpose (described later in sect. 3) is applicable, in principle, to the calculation of the amplitudes of excited configurations up to any desired order, we have been content with only the lowest order 2H-2P configurations. In view of the approximations involved in the Sawada method itself, we did not think that a calculation of the higher order configurations, 4H-4P and above, would be meaningful. t After completion of this work we came across the work of Gillet ~,) who has also made calculations for the collective states of C12with a finite range hole-particle force.
296
A. G O S W A M I A N D M. K . P A L
In sect. 2, we briefly review the Sawada method for the treatment of nuclear collective vibrations. Sect. 3 contains the description of the method used for the calculation of the detailed ground state wave function. In sect. 4 the results of numerical calculations are given and sect. 5 presents a summary and discussion of these results. 2. Review of the Formalism
Our unperturbed hole-particle states are of the type [h-lpJM; TMr> where h and p denote the angular momenta of the hole and the particle respectively, and J and T denote the total angular momentum and isobaric spin with projections M and Mr. For brevity, we shall denote this state by Ik> and let At denote the creation operator which, operating on the closed shell ground state, creates this state; A~ is actually composed of a product of the annihilation operator an and the creation operator a~* together with suitable vector-coupling coefficients. The unperturbed energy 2k of this state is given by 2k = eh+ %, where eh and % are the hole and particle energies respectively. Let us further introduce the notations [~> and I•*> for the respective states
]h-apJ-M; T-Mr>,
(--)h+p-S+l-rlp-lhJM ; TMT>.
In the latter state the unusual occurrence of the particle and hole states, playing their reverse roles, has to be understood in the following way: the creation operator for this state is composed of the product a, atn. Let 1~o> denote the actual correlated ground state and [TJ> the excited collective vibrational state. The respective forward and backward going amplitudes are defined, by Xk = <'eolAkl'e >, ~ = s~<~ola~[7'>,
O)
where s~ = ( _ ) J - M + r - M . .
We have already noted that the A-operators are treated as boson operators. Taking the commutator of these operators with the Hamiltonian, and retaining only the terms linear in the A and A t operators in the resultant equations one obtains the following matrix relation satisfied by the amplitudes:
(_;
(2)
where the matrix elements of A and B are given by
A,R, = (k]V[k') + ~k6a,,, Bkk' =
(klV[k"),
(3a) (3b)
297
C O L L E C T I V E S T A T E S OF C 12
V being the two-nucleon interaction potential. An element )~k in the diagonal components of A is a sum of the appropriate unperturbed energy 2k, the hole-shell energy and the particle-shell energy. When the hole and particle energies are determined from the experimental data (as we have done in this work), the numbers already include the contributions of the hole-shell and particle-shell interactions. Eq. (2) determines the energy eigenvalues e of the states 17') relative to the ground state as also the amplitudes x and ~ except for normalization. Let us now introduce two operators Qt and Q by
Qt = E (XkAI-- Sk2kAi) ,
(4)
k
Q being the Hermitian conjugate of Q~. Then in the same approximation as that required to obtain eq. (2) one obtains
[Q,,.~] = eQ,
[Qt,jg~] = _eQ,,
(5)
which shows that Qt is the creation operator of an oscillator quantum, Q being the corresponding destruction operator. Thus, the collective state 17~) is given as Iku) = Qt[kUo),
(6)
where [~o) is the actual ground state determined through the equation Oleo> = 0.
(7)
The normalization requirement for [7'>, viz. < 7~I7J> = 1, now fixes the normalization of x and 2 as 1 = (tPItP) =
= E(x
--2
(8)
k
Eqs. (2), (6) and (8) completely determine the collective state [~P). If we evaluate the commutator [ASrT(hP), ~MMa,'4tJT(h'p')] exactly * we see that it consists of terms containing (1) 6hh, 6pp, giving the exact boson commutation result, (2) 6pp, ah, atn and (3) 6hh' a;, ap. For the operators A to be boson operators, the commutator must be a c-number. Therefore, one replaces the operators ah,a~ and apt, ap, respectively, by their ground state expectation values, i.e., by
(q'olah, a'hi~o) = (1 - Vh)6hW,
(9a)
(Tto[a~,ap[~o) = vpapp,,
(9b)
where vh and v~ are the occupation probabilities of the hole-state h and the particle state p in the ground state. The commutator can then be written as
[agMr(hp), a ,~t'r , , . . t,,. . .p. .)J = ahh.
ASMTrr(h/p)
(10)
t The operator is the same as our previous A~. Only we have n o w written the quantum numbers of the state [k) explicitly, placing these in convenient positions with respect to A.
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A. G O S W A M I A N D M. K. PAL
Thus the A-operators are normalized boson operators only if vh - Vp = 1 or equivalently vh = 1 and Vp -- 0. The c o m m u t a t o r (10) can then be written in the abbreviated notation we used before,.as [-Ak, At,] -- 'Skk..
(11)
Once we assume the result (11), the Q and Qt operators can easily be seen to be exact boson annihilation and creation operators and the states I~u) given by (6) describe normalized vibrational states. It is now apparent that, for a check of the validity of the Sawada approximation, we have to evaluate the quantities vh and Vp. A first order orientation of the magnitudes o f these quantities will be obtained in the next section in terms of the correlation intensities of the ground state. It may be emphasized here that in evaluating vh and Vp we shall be using quantities that are obtained under the assumption of eq. (11) so that our evaluation of vh and vv can only provide a check of the consistency of the method and should not be used to correct quantities like normalization or transition probabilities. 3. Correlation in the Ground State To start with we shall retain only the 2H-2P term in the ground state, neglecting all higher order terms. We shall further assume that the 2H-2P states of J = 0, T = 0 are each formed by angular m o m e n t u m coupling of two hole-particle pairs of equal J and T, with opposite projection quantum numbers. Thus we write [ ~ o ) as ]~Uo) = N I l 0 ) +
Z hph'p"
(--)S-M+T-MT{2[J][T]} -~
Z MTMT
tJT
tJT
t
t
x ChSp ~, h'f A~MT(hp)A-~t-MT( h P )10)] •
(12)
Here 10) denotes the closed-shell state. The factors ( - ) s - M[j] - ~ and ( - ) r - MTIT] - ~ come from the Clebsch-Gordan coefficients. The notation [x] is an abbreviation for (2x + 1) and N is the normalization constant, given by N = [1+ Z
Z(C~r,h'v')2] -~"
(13)
hph'p' J T
Now recall that Q od T
o~ MMT = Z [XhpsrAJmTT(hP)--(--)'-M+r-MT'2~psTA--*~f-MT(hP)],
(14)
hp
where we have introduced the additional quantum number ~ to differentiate between the various perturbed energy states corresponding to the (J, T) type vibration. Then from (7), neglecting the A t A t A t term, we get
X~,sT = Z 2~{[J][T]} -~v CS~h'fX~'p"T •
(15)
h'p'
The ,~ and x can now be regarded as matrices whose rows and columns are labelled by the quantum numbers ct and (hp), respectively. One can then solve eq. (15) for the
299
COLLECTIVE STATES OF C12
coefficients C by the usual method o f matrix inversion. Obviously, the square of C~rh,¢, determines the correlation intensity of the ground state corresponding to the type of correlation indicated by the suffices and indices. With the eq. (12) for ]~o) and from the definitions (9) we can now write down approximate expressions for vh and vp: Vh. . . . = 1 - - N 2 2
~[h]
- lfl,-,JT
t,~h,.h,p,) ~,
(16a)
ph'p' JT
vp, ,,,, =
N 2
JT Z Z [p] - 1 (Ch,, h,,,) 2,
(16b)
hh'p" J T
where m and z denote the projections of angular momentum and isobaric spin, respectively, which we now write explicitly. Eq. (16) clearly shows that for the Sawada approximation to hold good, the correlation intensities must be small. It is interesting to note that the quantities given by eq. (16) are involved in a pick-up process like the (p, d) reaction o n C 12 ground state and can therefore be estimated from experiment as well, thus providing a direct check of the theory. 4. Results of Numerical Calculation
4.1. THE UNPERTURBED ENERGIES In all the cases considered here only hole-particle configurations corresponding to single-particle transitions through nho~ (n = 0, 1, 2, 3) have been taken into account. The hole-energies and particle energies, determined from experimental data, are given in table 1. These energies are denoted respectively by gh and ~p in contradistinction to the unperturbed energies eh and ep, since they now include the holeshell and particle-shell interaction energy, respectively. Note that ~ k ~ ~h-JJ-~p • In table 1, the hole-energy for lp~ level and particle energies up to the lf~_ level are fairly accurately known from the experimental level spectra of C I: and C t3, respectively. The remaining particle energies have been estimated from the data ~6) of single-particle level spacings in Sc4~ except for the 2dl and lgt states, for which TABLE 1 Particle and hole energies States
ls
lp~
lp~
2s~
Unperturbed energy (MeV) n
35
18.72
--4.95
-- 1.85
States
2p~
2pk
I g,~
Unperturbed energy (MeV)
8.92
11.12
12.62
ldg
Idl
lf~,
--1.09
3.39
7.02
2d~
1f~
2d~
1g~
13.02
13.52
20.02
25.22
a By unperturbed energy we mean ~ for hole states and ~ for particle states. For definition of ~h and ~ see text.
300
A . G O S W A M I A N D M. K. P A L
the energies have been estimated very roughly from the already estimated values of gzdl and ~lgI and a rough estimate of the spin-orbit splitting of the 2d and lg levels. The hole-energy ils~ has been taken to be the binding energy of a ls nucleon as determined from (p, 2p) experiments 17). 4.2. DETAILS OF NUMERICAL CALCULATION The matrix elements of V appearing in the matrices A and B are calculated in a straightforward manner by expressing the jj-coupled matrix elements in terms of the LS-coupled ones and then using the Moshinsky-bracket 18) method for evaluating the L - S coupled matrix elements. The potential V is taken to be central, of Yukawa type, with Soper exchange mixture, i.e., V =
-- Vo
e - r/~ r~-[O.3W+O.43M+O.27B],
where W, M and B stand for the strengths of the Wigner, Majorana and Bartlett exchange, respectively. The range ~ is taken to be 1.36 fm from Soper's work 19) and then the depth is adjusted to fit the 4.43 MeV level exactly. The value obtained for V0 in this way is 37 MeV. The matrices were diagonalized with the help of the IBM 7090 machine belonging to the U.K. Atomic Energy Authority. 4.3. THE QUADRUPOLE STATES Table 2 shows the energy levels and their respective percentage quadrupole strengths. We have defined the strength for electric 2S-pole radiation from a particular state J, T, to the ground state as
[~ (X~pj~+ ~.~r)E~]~I~(E~) ~, hp
hp
TABLE t
OVa )
2
Calculated energies and quadrupole strengths of 2+ ( T = 0) states of C1~ Energy (MeV)
Quadrupole strength (~) a)
4.43 18.85 24.09 28.24 30.85 33.10 40.41 Total
226.52 46.11 9.76 1.12 O.52 1.03 0.30 285.36
a) The percentages here and in tables (4) and (5) are with respect to E ~ (Eh~)~for 2S-pole electric radiation. For the definition of E h~ J see subsect. 4.3.
COLLECTIVE STATES OF C 1~
301
where the + ( - ) sign is for J + T ---even (odd), and
[P
° Rp r Jr 2dr, Rh
(17b)
where Rp and Rh denote the radial functions belonging to the particle state p and the hole-state h, respectively. Harmonic oscillator wave functions, with the oscillator well parameter adjusted to 1.64 fm to fit the r.m.s, radius of C x2, were used in the present work. TABLE 3 Experimental
data on the E2 lifetime of the 4.43 MeV 2+ state V a l u e o f m e a n life -r (10 -1~) s
Experimenter 1. 2. 3.
D e v o n s et al. R a s m u s s e n et al. H e l m et al.
2.6~0.9 6.5-~1.2 5.3
Table 3 shows the experimental situation for the y-lifetime measurement for the E2-transition from the 4.43 MeV state to the ground state. It will be found that the values of different measurements differ to a considerable extent. Although the second and the third quoted values seem to be more reliable, we shall continue to compare the theoretical value with the first-quoted experimental value as in ref. 6). This gives
B(E2)exp/B(E2)theor = 0.75, where B(E2) is the reduced E2 transition probability given by B(E2) - (e2/STr)ly, (xhp2o + Xhp20)Ehp] : 2•
(18)
hp
The overestimate of B(E2) in our theory (which may even be by a factor 2 if the second or the third value of v in table 3 is correct) is due to the fact that we have not mixed in the two-hole, two-particle states, which, having no quadrupole strength, are bound to reduce B(E2) if included in the calculation. For the quadrupole states one may easily derive the following sum rule: A
M(4hz ARZ)-1 2 a
'ol 2 (3z,
r~)]~o)] z -- 1,
(19)
i=1
where M is the nucleon mass, A the mass number, R the root mean square radius of the nucleus; the }ku) denote the different quadrupole states, and e, their respective energy relative to the ground state. The value of the sum in the present calculation comes out as 0.86, of which 45.6 % is contributed by the 4.43 MeV state. This assures that even if the theoretical value of B(E2) of the 4.43 MeV state decreases by the inclusion of two-hole, two-particle states, the theoretical value of the sum in eq. (19) will not be much affected.
302
A. GOSWAMIAND M. K. PAL
4.4. THE DIPOLE STATES
Table 4 shows the calculated values of energy and dipole strength. The two dipole states appear at 25 MeV and 37.4 MeV in agreement with the values given by Brown and Vinh-Mau 5). The little over-estimate in our case is due to the use of a different exchange mixture in the two-nucleon interaction and the adjustment of the depth to fit the 4.43 MeV state exactly. TABLE 4 Calculated energies and dipole strengths o f 1- ( T = 1) states of C TM Energy (MeV)
Dipole strength (%)
20.86 25.08 26.30 37.37
1.63 52.42 12.36 19.88 86.29
Total 4.5. THE OCTUPOLE STATES
Table 5 displays the calculated energy and octupole strengths of the various levels. The oetupole state comes a little higher in energy than the experimental value of 9.63 MeV. This is due to (1) the particular choice of the exchange-mixture and (2) the TABLE 5 Calculated energies and octupole strengths o f 3- (T = 0) states o f C 12. Energy (MeV)
Octupole strength (~)
11.58 20.87 28.46 29.89 37.85 39.56 44.82 48.48
63.84 8.12 37.41 7.16 1.19 7.30 2.72 0.77 128.51
Total
uncertainty of the unperturbed energies of the high-lying configurations. It is interesting to note that the level at 28.5 MeV carries about 3 7 ~ octupole strength. This is due to the high value of the radial integral appearing in the octupole strength of the unperturbed 3hco configuration 10p,)- lg > of which it is chiefly composed. 4.6. THE CORRELATION IN THE G R O U N D STATE
The values of x and ~ for each of the collective states as well as the unnormalized correlation amplitudes are given in the appendix. The obvious normalization now is
N
=
(I+C~+C~+C~) -~,
COLLECTIVE STATESOF C12
303
where CD 2, C~ and Co2 denote the total correlation intensity due to the dipole, quadrupole and octupole states, respectively. The value of normalized total correlation intensity due to the different states are then given in table 6. TABLE 6 Calculated total correlation intensity o f the ground state due to the important vibrational states o f C xa Correlation intensity o f ground state
State
(%)
2 +, T = 0 3% T = 0 1-, T = 1
40.5 9.9 3.2 53.6
Total
The 2H-2P correlations in the ground state are assumed to be formed by angular m o m e n t u m coupling o f two 1H-1P pairs o f equal J and T with opposite projection q u a n t u m numbers (see sect. 3),
It is interesting to note that most of the ground state correlation in C ~2 comes from the quadrupole vibration. Again, of the 40.5% correlation from the quadrupole states, about 18 % is due to correlation involving excitation of two (lp+)-1 lp+ pairs. The correlation intensity due to dipole oscillation is small in accordance with the oft-quoted result that the ground-state correlation is not very important for the T = 1 states. The correlation due to the octupole vibration is also small. This fact, coupled with the fact that in O t6 the configuration I(lp+) -1 lp+> is absent, tends to show TABLE 7 Occupation probability o f hole and particle states (vh, m, ~ and l~p,m, ¢) in the correlated ground state Occupation probability
Levels Holes
Particles
Is lp!
,
0.948 0.892
lp~
0.201
if+
0.101 o.o13
ifi 2p! 2P½
0.007 0.005
lch ldt
0.081 0.101
lg t lg t 2d t 2d t
0.007 0.001 0.006 0.007
F o r holes, the occupation probability vh, m, • for a state o f specified m and T is shown. F o r particles, w e h a v e g i v e n the v a l u e s o f [p] t i m e s vp, m, 3, since t h e Vp, m, ~ are small.
304
a.
G O S W A M I A N D M. K. P A L
that the amount of ground state correlation would be smaller in 0 1 6 , SO that the Sawada method is a better approximation there. This is in accordance with general expectation. The quantities Vh. . . . and vp.... given by eq. (16) may now be calculated from tables 11-13 given in the appendix. Table 7 shows these quantities. It may be seen that for any h or p, their values are not very far removed from the closed shell values required by the Sawada approximation. However, the difference is appreciable enough in some of the cases and, therefore, an experimental determination of these quantities might be possible.
5. Summary and Conclusions In this paper, we have calculated the important collective states of C 12 viz. the 2 + (T = 0), 3 - (T = 0) and 1- (T = 1) states, including ground state correlations by the Sawada method of linearized equation of motion. The two-nucleon interaction has been taken to be finite range central Yukawa with Soper ~5) exchange mixture. The range of the force has been taken from Soper's work ~9) as 1.36 fm and the depth adjusted to fit the 4.43 MeV state exactly, the value thus found being 37 MeV. The collective octupole and dipole states, however, appear at 11.63 MeV and 25 MeV, respectively; these calculated values are slightly higher than the experimental values of 9.63 MeV and 22.2 MeV. However, as has been shown by the work of Gillet and Vinh-Mau 20) in the case of 016, it is possible to get an optimum fit of the different levels by an adjustment of the parameters of the force, viz. depth, range and exchange mixture. We have not done so but presume that this could be done in our case by an adjustment of the depth and specially the exchange mixture. Let us now summarize the results obtained. For the 2 + (T = 0) states it is found that almost all the quadrupole strength is concentrated in the lowest state at 4.43 MeV. The probability of E2 transition from this state to the ground state has, however, been overestimated in our calculation. This is due to the fact that we have not mixed in the lh(o 2H-2P states which are energetically on the same footing as the 2hco 1[ t - I P states. Since the 2H-2P states carry no quadrupole strength, their inclusion would naturally decrease the theoretical value of the E2 transition probability. The energy-weighted sum rule, given by eq. (19) is exhausted to 86 % by the calculated 2 ~ states, of which 45.6 % is contributed by the 4.43 MeV state. This assures that even if the theoretical value of the E2 transition probability from the 4.43 MeV state to the ground state decreases considerably by the inclusion of the 2H-2P states in the calculation, the sum rule will still be satisfied to a tolerable extent. Our results for the dipole states are in general agreement with Brown and VinhMau's calculation using zero-range Soper force. In our calculation the dipole state appears at 25 MeV, and it consists almost exclusively of the non-spin-slip state [(lp~) - I ldj). In our calculation the state at 26.3 MeV, which chiefly consists o f the spin-flip state [(lp~)-1 l d~), carries a dipole strength of about 12 ~ . This may
COLLECTIVE STATES OF C 12
305
be significant since there is an experimental indication of a peak besides the 22.2 MeV peak, at about 23.3 MeV, in the (~, n) reaction cross section curve 2). The slightly higher energy values of our theoretical prediction for these two states are once again due to the fact that we did not try to choose an optimum exchangedependence of the two-nucleon force. The other state which is found to carry a significant amount of dipole strength is that at 37.4 MeV in our calculation. This consists almost exclusively of the state I(ls) -1 lp~) with an unperturbed energy of about 30 MeV Obviously the reason that such a dipole state is not found in 016 is that there is no such unperturbed hole-particle state in the case of 016 . The collective octupole state appears at 11.6 MeV in our calculation. We predict another high-lying octupole state carrying 3 7 . 4 ~ octupoie strength at 28.5 MeV. This is not a collective state but is produced by the large octupole strength of the unperturbed 3h~o state [(lp~_)-i lgl) it chiefly consists of" In the calculations of collective states so far carried out by using the Sawada method, no attempt has been made to calculate the correlated ground state explicitly. This is due to the fact that the Sawada method enables one to calculate the energy and transition probabilities directly without having to write down the detailed structure of the correlated ground state wave function. However, such a detailed expression in terms of the uncorrelated ground state plus the excited 2H-2P configurations in various orders is needed, in the first place, to check the internal consistency of the Sawada method. In this paper we have calculated the ground state wave function, assuming it to be the uncorrelated ground state plus excited 2H-2P configurations. The squared amplitudes of these 2H-2P configurations in the correlated ground state serve to calculate the occupation probability of single hole and particle states in the correlated ground state. The calculated values of the quantities justify the treatment of the hole-particle pairs as bosons, thus giving a check of the internal consistency of the Sawada method. However, some of the occupation probabilities might deviate enough from their closed shell values to allow of an experimental determination of them by, for example, a pick-up reaction experiment on the ground state of C 12. The total amount of ground state correlation is also important, since this determines the amount of deviation from the closed-shell picture. The total 2H-2P correlation intensity is found by us to be 53.6 ~ , of which 40.5 ~ is due to the quadrupole vibration. The correlation due to the dipole oscillation is small, as expected for T = 1 oscillations. The correlation due to the 3-, T -- 0, vibration is also small. The large correlation due to the quadrupole vibration is not disturbing, partly because it is slightly over-estimated on account of the non-inclusion of 2H-2P states in the quadrupole state calculation, and partly because 18 ~ of the correlation is produced by the occurrence of two (lp~)-1 lp~ pairs. Even the intermediate-coupling wave-function, calculated earlier with mixing within the lp shell only 19), contains about 25 ~ admixture of the l(lp~) -2 (lP½) 2) configuration. A m a j o r part of the quadrupole-correlation, calculated by us, is from this configuration, and the value of 18 ~ for this configuration in our wave function compares satisfactorily with the intermediate-coupling value.
306
A. G O S W A M I A N D M. K . P A L
W e w o u l d like to t h a n k Dr. A. M. Lane a n d Professor M. K. Banerjee for m a n y helpful discussions. We w o u l d also like to acknowledge the very valuable help received f r o m Dr. Ted Y o r k o f the Harwell C o m p u t i n g Division who provided us with a p r o g r a m m e for finding the eigenvalues a n d vectors of a n o n - s y m m e t r i c matrix. O n e o f the a u t h o r s (A. G ) wishes to t h a n k the D e p a r t m e n t o f A t o m i c Energy, G o v e r n m e n t o f India, for the g r a n t o f a fellowship.
Appendix We give below the tables o f n o r m a l i z e d x a n d ~ for the three collective states of C 12. Tables of u n n o r m a l i z e d correlation amplitudes o f the g r o u n d state due to dipole, q u a d r u p o l e a n d octupole states are also given. TABLE 8 Values of forward-going and backward-going amplitudes x and 3~of the different unperturbed holeparticle states for the 4.43 MeV 2+, T = 0 collective state (Ipt)-~lp½
(ls½)-~ldt
(Is~)-~ld{
0.982 0.496
--0.266 --0.207
--0.383 --0.266
x
(lpi)-a2p4x (lp])-12p] 0.067 0.073
0A18 0.090
(lpt)-~Ift --0.159 --0.116
(lpi)-llf~ --0.532 --0.347
TABLE 9 Values of x and ~ of the different unperturbed hole-particle states for the 25.1 MeV collective 1-, T = 1 state
x
(lpt)-X2s~
(lp|)-Xldj
(lpt)-lld~
(ls½)-~lp4~
0.151 --0.033
--0.114 --0.046
0.984 --0.049
--0.070 --0,051
TABLE 10 Values of x and ~ of the different unperturbed hole-particle states for the 11.58 MeV 3- T = 0 collective state (ls½)-llq x
--0.073 0.041
(lpt)-Hg ~ (ls½)-llf~ (lpt)-~2d! (lpt)-~2dt (lpt)-Xlgt --0.051 0.024
--0,134 0.054
0.099 --0.061
0.114 --0.055
--0.183 0.052
(lpt)-Hd ! 0.553 --0.153
(lp~)-lldt 0,816 --0.127
307
COLLECTIVE STATES OF C 1~
TABLE 11 Unnormalized correlation amplitudes Chp, "iT h"p" for 2 ~, T = 0 states Hole-paxticle states (ls~)-qd~ (ls½)-qd| (ls½)-Xldt (lpa)-~2P½ (lp])-qp! (lpj)-Xlft (lpj)-llf~ (lp])-~lp~
0.056 0.090 --0.033 --0.034 0.049 0.135 --0.197
(ls½)-qd t
(lpt)-X2p~x
(lpj)-X2pt
(lp|)-tlf~
0.090 0.112 --0.031 --0.046 0.055 0.188 --0.241
--0.033 --0.031 0.037 0.040 --0.006 --0.018 0.079
--0.034 --0.046 0.040 0.034 --0.009 --0.029 0.094
0.049 0.055 --0.006 --0.009 0.023 0.073 --0.108
(lP|)-Xlf~ ( l P | ) - q P Jt 0.135 0.188 --0.018 --0.029 0.073 0.233 --0.306
--0.1 °~ --0. 7 0.079 0.094 --0.108 --0.306 0.452
TABLE 12
Unnormalized correlation amplitudes C hp, JT h'p" for 1-, 7"= l states Hole-paxticle states (lpj)-X2s½ (lpt)-Xldt (lp|)-lldt (ls½)-Xlp½
(lPt)-12s½ ( l P t ) - q d l ( l P i ) - q d l (ls½)-XlP½ 0.032 --0.002 0.065 --0.015
--0.002 --0.059 0.091 --0.035
0.065 0.091 0.115 0.107
--0.015 --0.035 0.107 --0.008
r'.~ULE 13 Unnormalized correlation amplitudes C hp, JT h"p" for 3-, T = 0 states Hole-particle (ls½)_t!ft ( l p | ) _ l l g t (is½)_llf] (lp|)_X2d| (lpt)_~2d| (lp|)_Xlgt ( l p | ) _ H d ! (lp|)_Xld| states (ls½)-llft ( 1p | ) - t 1gt (ls½)-tlf½ (lpt)-12dt (lpj)-12dt (lpt)-llg~ (lpg)-lldt (lpt)-qd |
--0.017 -- 0.009 --0.019 0.021 0.016 --0.026 0.054 0.042
--0.009 -- 0.007 --0.012 0.007 0.006 --0.027 0.028 0.025
--0.019 -- 0.012 --0.028 0.020 0.023 --0.035 0.062 0.061
0.021 0.007 0.020 --0.049 --0.036 0.018 --0.082 --0.064
0.016 0.006 0.023 --0.036 --0.033 0.019 --0.068 --0.062
--0.026 -- 0.027 --0.035 0.018 0.019 --0.055 0.078 0.041
0.054 0.028 0.062 --0.082 --0.068 0.078 --0.198 --0.162
References 1) 2) 3) 4)
H. Tyr6n and A. J. Maxis, Nuclear Physics 43 (1957) 52, 4 (1957) 637 J. Miller, G. Schuhl, G. Tamas and C. Tzara, Phys. Lett. 2 (1962) 76 F. Ajzenberg-Selov¢ and T. Lauritsen, Nuclear Physics 11 (1959) 1 F. C. Barker, E. Bradford and L. J. Tassi¢, Nuclear Physics 19 (1960) 101; Richard R. Carlson, Nuclear Physics 28 (1961) 443 5) N. Vinh-Mau and G. E. Brown, Nuclear Physics 29 (1962) 89 6) A. Goswami and M. K. Pal, Nuclear Physics 35 (1962) 544
0.041 0.025 0.061 --0.064 --0.062 0.041 --0.162 --0.141
308
A. G O S W A M I A N D M. K. P A L
7) V. Gillet and N. Vinh-Mau, in Proc. Rutherford Jubilee Int. Conf. (Heywood, London, 1961) p. 315 8) K. Sawada, Phys. Rev. 106 (1957) 372 9) G. E. Brown, J. A. Evans and D. J. Thouless, Nuclear Physics 24 (1961) 1 10) M. Baranger, Phys. Rev. 120 (1960) 957 11) S. Fallieros and R. A. Ferrell, Phys. Rev. 116 (1959) 660 12) M. Kobayasi and T. Marumori, Prog. Theor. Phys. 23 (1960) 387 13) J. Sawicki and T. Soda, Nuclear Physics 28 (1961) 270 14) V. Gillet, Thesis, Paris, 1962 151 J. M. Soper, Ph.D. Thesis, Cambridge, 1958 16) C. M. Class, R. H. David and J. H. Johnson, Phys. Rev. Lett. 3 (1959) 41 17) H. Tyr6n, Peter Hillman and Th. A. J. Maris, Nuclear Physics 7 (1958) 10 18) M. Moshinsky and T. A. Brody, Table of transformation brackets (Monografias del Instituto de Fisica, Mexico, 1960) 19) J. M. Soper, private communication 20) V. Gillet and N. Vinh-Mau, Phys. Lett. 1 (1962) 25
to
Physirs
44 (1963)
be reproduced
by
294-308;
photoprint
or
@
North-Holland
microf;lm
without
Publishing
written
ON THE COLLECTIVE
STATES
AND GROUND-STATE
CORRELATION
permission
Co., Amsterdam from
the
publisher
OF C”
A. GOSWAMI Saha Institute
of Nuclear
Physics,
Calcutta
and M. K. PAL + Atomic
Energy
Research
Received
Establishment,
21 January
Harwell
1963
The collective electric dipole, quadrupole and octupole states of Cl3 have been calculated, including the effects of ground state correlation, by the Sawada method and using finite range Soper force. A first order calculation of the amount of ground state correlation indicates that this amount is considerable in Cl2 and that most of it is due to quadrupole vibrations. The justification of the Sawada method for Cl2 is discussed in this context. Quantities related to the ground state correlation which can be determined experimentally are also calculated and discussed.
Abstract:
1. Introduction The presence of collective electric dipole, quadrupole and octupole states in the spectrum of C” has been established by many experiments. Inelastic proton scattering experiments ‘) and (r-n) reaction experiments “) show the presence of a dipole state at about 22.2 MeV. The collective nature of the 4.43 MeV 2+ state is also wellknown “) from the enhanced quadrupole y-transition to the ground state and also from electron scattering experiments. Lastly, a careful analysis of experiments 4> has now conclusively shown that the correct spin and parity assignment of the 9.63 MeV state is 3 -. Theoretical calculations have been made (i) by Brown and Vinh-Mau “) for the dipole state and (ii) by the present authors 6, and by Gillet and Vinh-Mau ‘) for rhe quadrupole state. Two methods of calculation are used which differ in the description of the ground state. In the first method the ground state is assumed to be a doubly closed shell state (1~)~ (lp+)*. A y-ray quantum can excite the nucleus by raising a nucleon from any of the closed shells to an empty shell, thereby creating a holeparticle pair (1H-IP). The unperturbed energy of such a lH-1P state is given as the sum of the hole-energy and the particle-energy which can be determined from experimental data. One then diagonalizes the hole-particle interaction with these states as basis to get different states of specified total angular momentum and isobaric spin. This is the usual shell model method and is often referred to as the Tamm-Dancoff method. 7 Present address:
Saha
Institute
of Nuclear
Physics, 294
Calcutta.
C O L L E C T I V E S T A T E S OF C 12
295
The second method differs from the first in that correlations in the ground state consisting of hole-particle pairs are taken into account. The standard method of calculation, taking ground state correlations into account, is the Sawada a) method of linearized equation of motion originally designed for the treatment of plasma oscillations in the electron gas. This method has been applied to the case of nuclear vibrations by many authors 9-13). In the Sawada method, correlations in a doubly closed shell nucleus implies the presence of two-hole, two-particle (2H-2P) pairs of all orders in the ground state. Excited states may then also be obtained by the annihilation of one of the hole-particle pairs present in the ground state. Owing to this new type of excitation amplitude, the ground state correlation may enhance the v-transition probability. Calculations t have been made by both methods for the collective states of C 12 (refs. 5-7)), Or6 and Ca "° (refs. 7, 9)). The results indicate that in order to reproduce the collective behaviour of the T = 0 states, one has to take recourse to the second method, while for the T = 1 states, the effects of ground state correlations is not appreciable. The Sawada method enables one to calculate directly the energy and transition probabilities of the excited states, without having ever to write down the detailed structure of the correlated ground state wave function. However, such a detailed expression in terms of the uncorrelated ground state plus the excited 2H-2P configurations in various orders would be welcome for several reasons: (I) It is necessary, in the first place, for a check on the internal consistency of the Sawada method. A hole-particle pair is assumed, in this method, to be a composite unit behaving as a boson, which assumption is good only if the occupation number of the single nucleon states in the correlated ground-state is not very different from that in the closed shell state. If, for example, the above condition is found to be very badly satisfied for the particular correlated ground state wave function, then the agreement of energy and transition probability of the excited collective states would really be fortuitous. (2) The detailed wave function for the ground state enables one to calculate the occupation probability of an excited single particle state, which can be verified experimentally by performing pick-up reactions on the ground state. The present paper gives the results of our calculations on the basis of a finiterange Yukawa-force with Soper 15) exchange mixture for the electric dipole, quadrupole and octupole states of C 12. For the reasons stated above we have also calculated the detailed ground state-~ave function. Although the method used for this purpose (described later in sect. 3) is applicable, in principle, to the calculation of the amplitudes of excited configurations up to any desired order, we have been content with only the lowest order 2H-2P configurations. In view of the approximations involved in the Sawada method itself, we did not think that a calculation of the higher order configurations, 4H-4P and above, would be meaningful. t After completion of this work we came across the work of Gillet ~,) who has also made calculations for the collective states of C12with a finite range hole-particle force.
296
A. G O S W A M I A N D M. K . P A L
In sect. 2, we briefly review the Sawada method for the treatment of nuclear collective vibrations. Sect. 3 contains the description of the method used for the calculation of the detailed ground state wave function. In sect. 4 the results of numerical calculations are given and sect. 5 presents a summary and discussion of these results. 2. Review of the Formalism
Our unperturbed hole-particle states are of the type [h-lpJM; TMr> where h and p denote the angular momenta of the hole and the particle respectively, and J and T denote the total angular momentum and isobaric spin with projections M and Mr. For brevity, we shall denote this state by Ik> and let At denote the creation operator which, operating on the closed shell ground state, creates this state; A~ is actually composed of a product of the annihilation operator an and the creation operator a~* together with suitable vector-coupling coefficients. The unperturbed energy 2k of this state is given by 2k = eh+ %, where eh and % are the hole and particle energies respectively. Let us further introduce the notations [~> and I•*> for the respective states
]h-apJ-M; T-Mr>,
(--)h+p-S+l-rlp-lhJM ; TMT>.
In the latter state the unusual occurrence of the particle and hole states, playing their reverse roles, has to be understood in the following way: the creation operator for this state is composed of the product a, atn. Let 1~o> denote the actual correlated ground state and [TJ> the excited collective vibrational state. The respective forward and backward going amplitudes are defined, by Xk = <'eolAkl'e >, ~ = s~<~ola~[7'>,
O)
where s~ = ( _ ) J - M + r - M . .
We have already noted that the A-operators are treated as boson operators. Taking the commutator of these operators with the Hamiltonian, and retaining only the terms linear in the A and A t operators in the resultant equations one obtains the following matrix relation satisfied by the amplitudes:
(_;
(2)
where the matrix elements of A and B are given by
A,R, = (k]V[k') + ~k6a,,, Bkk' =
(klV[k"),
(3a) (3b)
297
C O L L E C T I V E S T A T E S OF C 12
V being the two-nucleon interaction potential. An element )~k in the diagonal components of A is a sum of the appropriate unperturbed energy 2k, the hole-shell energy and the particle-shell energy. When the hole and particle energies are determined from the experimental data (as we have done in this work), the numbers already include the contributions of the hole-shell and particle-shell interactions. Eq. (2) determines the energy eigenvalues e of the states 17') relative to the ground state as also the amplitudes x and ~ except for normalization. Let us now introduce two operators Qt and Q by
Qt = E (XkAI-- Sk2kAi) ,
(4)
k
Q being the Hermitian conjugate of Q~. Then in the same approximation as that required to obtain eq. (2) one obtains
[Q,,.~] = eQ,
[Qt,jg~] = _eQ,,
(5)
which shows that Qt is the creation operator of an oscillator quantum, Q being the corresponding destruction operator. Thus, the collective state 17~) is given as Iku) = Qt[kUo),
(6)
where [~o) is the actual ground state determined through the equation Oleo> = 0.
(7)
The normalization requirement for [7'>, viz. < 7~I7J> = 1, now fixes the normalization of x and 2 as 1 = (tPItP) =
= E(x
--2
(8)
k
Eqs. (2), (6) and (8) completely determine the collective state [~P). If we evaluate the commutator [ASrT(hP), ~MMa,'4tJT(h'p')] exactly * we see that it consists of terms containing (1) 6hh, 6pp, giving the exact boson commutation result, (2) 6pp, ah, atn and (3) 6hh' a;, ap. For the operators A to be boson operators, the commutator must be a c-number. Therefore, one replaces the operators ah,a~ and apt, ap, respectively, by their ground state expectation values, i.e., by
(q'olah, a'hi~o) = (1 - Vh)6hW,
(9a)
(Tto[a~,ap[~o) = vpapp,,
(9b)
where vh and v~ are the occupation probabilities of the hole-state h and the particle state p in the ground state. The commutator can then be written as
[agMr(hp), a ,~t'r , , . . t,,. . .p. .)J = ahh.
ASMTrr(h/p)
(10)
t The operator is the same as our previous A~. Only we have n o w written the quantum numbers of the state [k) explicitly, placing these in convenient positions with respect to A.
298
A. G O S W A M I A N D M. K. PAL
Thus the A-operators are normalized boson operators only if vh - Vp = 1 or equivalently vh = 1 and Vp -- 0. The c o m m u t a t o r (10) can then be written in the abbreviated notation we used before,.as [-Ak, At,] -- 'Skk..
(11)
Once we assume the result (11), the Q and Qt operators can easily be seen to be exact boson annihilation and creation operators and the states I~u) given by (6) describe normalized vibrational states. It is now apparent that, for a check of the validity of the Sawada approximation, we have to evaluate the quantities vh and Vp. A first order orientation of the magnitudes o f these quantities will be obtained in the next section in terms of the correlation intensities of the ground state. It may be emphasized here that in evaluating vh and Vp we shall be using quantities that are obtained under the assumption of eq. (11) so that our evaluation of vh and vv can only provide a check of the consistency of the method and should not be used to correct quantities like normalization or transition probabilities. 3. Correlation in the Ground State To start with we shall retain only the 2H-2P term in the ground state, neglecting all higher order terms. We shall further assume that the 2H-2P states of J = 0, T = 0 are each formed by angular m o m e n t u m coupling of two hole-particle pairs of equal J and T, with opposite projection quantum numbers. Thus we write [ ~ o ) as ]~Uo) = N I l 0 ) +
Z hph'p"
(--)S-M+T-MT{2[J][T]} -~
Z MTMT
tJT
tJT
t
t
x ChSp ~, h'f A~MT(hp)A-~t-MT( h P )10)] •
(12)
Here 10) denotes the closed-shell state. The factors ( - ) s - M[j] - ~ and ( - ) r - MTIT] - ~ come from the Clebsch-Gordan coefficients. The notation [x] is an abbreviation for (2x + 1) and N is the normalization constant, given by N = [1+ Z
Z(C~r,h'v')2] -~"
(13)
hph'p' J T
Now recall that Q od T
o~ MMT = Z [XhpsrAJmTT(hP)--(--)'-M+r-MT'2~psTA--*~f-MT(hP)],
(14)
hp
where we have introduced the additional quantum number ~ to differentiate between the various perturbed energy states corresponding to the (J, T) type vibration. Then from (7), neglecting the A t A t A t term, we get
X~,sT = Z 2~{[J][T]} -~v CS~h'fX~'p"T •
(15)
h'p'
The ,~ and x can now be regarded as matrices whose rows and columns are labelled by the quantum numbers ct and (hp), respectively. One can then solve eq. (15) for the
299
COLLECTIVE STATES OF C12
coefficients C by the usual method o f matrix inversion. Obviously, the square of C~rh,¢, determines the correlation intensity of the ground state corresponding to the type of correlation indicated by the suffices and indices. With the eq. (12) for ]~o) and from the definitions (9) we can now write down approximate expressions for vh and vp: Vh. . . . = 1 - - N 2 2
~[h]
- lfl,-,JT
t,~h,.h,p,) ~,
(16a)
ph'p' JT
vp, ,,,, =
N 2
JT Z Z [p] - 1 (Ch,, h,,,) 2,
(16b)
hh'p" J T
where m and z denote the projections of angular momentum and isobaric spin, respectively, which we now write explicitly. Eq. (16) clearly shows that for the Sawada approximation to hold good, the correlation intensities must be small. It is interesting to note that the quantities given by eq. (16) are involved in a pick-up process like the (p, d) reaction o n C 12 ground state and can therefore be estimated from experiment as well, thus providing a direct check of the theory. 4. Results of Numerical Calculation
4.1. THE UNPERTURBED ENERGIES In all the cases considered here only hole-particle configurations corresponding to single-particle transitions through nho~ (n = 0, 1, 2, 3) have been taken into account. The hole-energies and particle energies, determined from experimental data, are given in table 1. These energies are denoted respectively by gh and ~p in contradistinction to the unperturbed energies eh and ep, since they now include the holeshell and particle-shell interaction energy, respectively. Note that ~ k ~ ~h-JJ-~p • In table 1, the hole-energy for lp~ level and particle energies up to the lf~_ level are fairly accurately known from the experimental level spectra of C I: and C t3, respectively. The remaining particle energies have been estimated from the data ~6) of single-particle level spacings in Sc4~ except for the 2dl and lgt states, for which TABLE 1 Particle and hole energies States
ls
lp~
lp~
2s~
Unperturbed energy (MeV) n
35
18.72
--4.95
-- 1.85
States
2p~
2pk
I g,~
Unperturbed energy (MeV)
8.92
11.12
12.62
ldg
Idl
lf~,
--1.09
3.39
7.02
2d~
1f~
2d~
1g~
13.02
13.52
20.02
25.22
a By unperturbed energy we mean ~ for hole states and ~ for particle states. For definition of ~h and ~ see text.
300
A . G O S W A M I A N D M. K. P A L
the energies have been estimated very roughly from the already estimated values of gzdl and ~lgI and a rough estimate of the spin-orbit splitting of the 2d and lg levels. The hole-energy ils~ has been taken to be the binding energy of a ls nucleon as determined from (p, 2p) experiments 17). 4.2. DETAILS OF NUMERICAL CALCULATION The matrix elements of V appearing in the matrices A and B are calculated in a straightforward manner by expressing the jj-coupled matrix elements in terms of the LS-coupled ones and then using the Moshinsky-bracket 18) method for evaluating the L - S coupled matrix elements. The potential V is taken to be central, of Yukawa type, with Soper exchange mixture, i.e., V =
-- Vo
e - r/~ r~-[O.3W+O.43M+O.27B],
where W, M and B stand for the strengths of the Wigner, Majorana and Bartlett exchange, respectively. The range ~ is taken to be 1.36 fm from Soper's work 19) and then the depth is adjusted to fit the 4.43 MeV level exactly. The value obtained for V0 in this way is 37 MeV. The matrices were diagonalized with the help of the IBM 7090 machine belonging to the U.K. Atomic Energy Authority. 4.3. THE QUADRUPOLE STATES Table 2 shows the energy levels and their respective percentage quadrupole strengths. We have defined the strength for electric 2S-pole radiation from a particular state J, T, to the ground state as
[~ (X~pj~+ ~.~r)E~]~I~(E~) ~, hp
hp
TABLE t
OVa )
2
Calculated energies and quadrupole strengths of 2+ ( T = 0) states of C1~ Energy (MeV)
Quadrupole strength (~) a)
4.43 18.85 24.09 28.24 30.85 33.10 40.41 Total
226.52 46.11 9.76 1.12 O.52 1.03 0.30 285.36
a) The percentages here and in tables (4) and (5) are with respect to E ~ (Eh~)~for 2S-pole electric radiation. For the definition of E h~ J see subsect. 4.3.
COLLECTIVE STATES OF C 1~
301
where the + ( - ) sign is for J + T ---even (odd), and
[P
° Rp r Jr 2dr, Rh
(17b)
where Rp and Rh denote the radial functions belonging to the particle state p and the hole-state h, respectively. Harmonic oscillator wave functions, with the oscillator well parameter adjusted to 1.64 fm to fit the r.m.s, radius of C x2, were used in the present work. TABLE 3 Experimental
data on the E2 lifetime of the 4.43 MeV 2+ state V a l u e o f m e a n life -r (10 -1~) s
Experimenter 1. 2. 3.
D e v o n s et al. R a s m u s s e n et al. H e l m et al.
2.6~0.9 6.5-~1.2 5.3
Table 3 shows the experimental situation for the y-lifetime measurement for the E2-transition from the 4.43 MeV state to the ground state. It will be found that the values of different measurements differ to a considerable extent. Although the second and the third quoted values seem to be more reliable, we shall continue to compare the theoretical value with the first-quoted experimental value as in ref. 6). This gives
B(E2)exp/B(E2)theor = 0.75, where B(E2) is the reduced E2 transition probability given by B(E2) - (e2/STr)ly, (xhp2o + Xhp20)Ehp] : 2•
(18)
hp
The overestimate of B(E2) in our theory (which may even be by a factor 2 if the second or the third value of v in table 3 is correct) is due to the fact that we have not mixed in the two-hole, two-particle states, which, having no quadrupole strength, are bound to reduce B(E2) if included in the calculation. For the quadrupole states one may easily derive the following sum rule: A
M(4hz ARZ)-1 2 a
'ol 2 (3z,
r~)]~o)] z -- 1,
(19)
i=1
where M is the nucleon mass, A the mass number, R the root mean square radius of the nucleus; the }ku) denote the different quadrupole states, and e, their respective energy relative to the ground state. The value of the sum in the present calculation comes out as 0.86, of which 45.6 % is contributed by the 4.43 MeV state. This assures that even if the theoretical value of B(E2) of the 4.43 MeV state decreases by the inclusion of two-hole, two-particle states, the theoretical value of the sum in eq. (19) will not be much affected.
302
A. GOSWAMIAND M. K. PAL
4.4. THE DIPOLE STATES
Table 4 shows the calculated values of energy and dipole strength. The two dipole states appear at 25 MeV and 37.4 MeV in agreement with the values given by Brown and Vinh-Mau 5). The little over-estimate in our case is due to the use of a different exchange mixture in the two-nucleon interaction and the adjustment of the depth to fit the 4.43 MeV state exactly. TABLE 4 Calculated energies and dipole strengths o f 1- ( T = 1) states of C TM Energy (MeV)
Dipole strength (%)
20.86 25.08 26.30 37.37
1.63 52.42 12.36 19.88 86.29
Total 4.5. THE OCTUPOLE STATES
Table 5 displays the calculated energy and octupole strengths of the various levels. The oetupole state comes a little higher in energy than the experimental value of 9.63 MeV. This is due to (1) the particular choice of the exchange-mixture and (2) the TABLE 5 Calculated energies and octupole strengths o f 3- (T = 0) states o f C 12. Energy (MeV)
Octupole strength (~)
11.58 20.87 28.46 29.89 37.85 39.56 44.82 48.48
63.84 8.12 37.41 7.16 1.19 7.30 2.72 0.77 128.51
Total
uncertainty of the unperturbed energies of the high-lying configurations. It is interesting to note that the level at 28.5 MeV carries about 3 7 ~ octupole strength. This is due to the high value of the radial integral appearing in the octupole strength of the unperturbed 3hco configuration 10p,)- lg > of which it is chiefly composed. 4.6. THE CORRELATION IN THE G R O U N D STATE
The values of x and ~ for each of the collective states as well as the unnormalized correlation amplitudes are given in the appendix. The obvious normalization now is
N
=
(I+C~+C~+C~) -~,
COLLECTIVE STATESOF C12
303
where CD 2, C~ and Co2 denote the total correlation intensity due to the dipole, quadrupole and octupole states, respectively. The value of normalized total correlation intensity due to the different states are then given in table 6. TABLE 6 Calculated total correlation intensity o f the ground state due to the important vibrational states o f C xa Correlation intensity o f ground state
State
(%)
2 +, T = 0 3% T = 0 1-, T = 1
40.5 9.9 3.2 53.6
Total
The 2H-2P correlations in the ground state are assumed to be formed by angular m o m e n t u m coupling o f two 1H-1P pairs o f equal J and T with opposite projection q u a n t u m numbers (see sect. 3),
It is interesting to note that most of the ground state correlation in C ~2 comes from the quadrupole vibration. Again, of the 40.5% correlation from the quadrupole states, about 18 % is due to correlation involving excitation of two (lp+)-1 lp+ pairs. The correlation intensity due to dipole oscillation is small in accordance with the oft-quoted result that the ground-state correlation is not very important for the T = 1 states. The correlation due to the octupole vibration is also small. This fact, coupled with the fact that in O t6 the configuration I(lp+) -1 lp+> is absent, tends to show TABLE 7 Occupation probability o f hole and particle states (vh, m, ~ and l~p,m, ¢) in the correlated ground state Occupation probability
Levels Holes
Particles
Is lp!
,
0.948 0.892
lp~
0.201
if+
0.101 o.o13
ifi 2p! 2P½
0.007 0.005
lch ldt
0.081 0.101
lg t lg t 2d t 2d t
0.007 0.001 0.006 0.007
F o r holes, the occupation probability vh, m, • for a state o f specified m and T is shown. F o r particles, w e h a v e g i v e n the v a l u e s o f [p] t i m e s vp, m, 3, since t h e Vp, m, ~ are small.
304
a.
G O S W A M I A N D M. K. P A L
that the amount of ground state correlation would be smaller in 0 1 6 , SO that the Sawada method is a better approximation there. This is in accordance with general expectation. The quantities Vh. . . . and vp.... given by eq. (16) may now be calculated from tables 11-13 given in the appendix. Table 7 shows these quantities. It may be seen that for any h or p, their values are not very far removed from the closed shell values required by the Sawada approximation. However, the difference is appreciable enough in some of the cases and, therefore, an experimental determination of these quantities might be possible.
5. Summary and Conclusions In this paper, we have calculated the important collective states of C 12 viz. the 2 + (T = 0), 3 - (T = 0) and 1- (T = 1) states, including ground state correlations by the Sawada method of linearized equation of motion. The two-nucleon interaction has been taken to be finite range central Yukawa with Soper ~5) exchange mixture. The range of the force has been taken from Soper's work ~9) as 1.36 fm and the depth adjusted to fit the 4.43 MeV state exactly, the value thus found being 37 MeV. The collective octupole and dipole states, however, appear at 11.63 MeV and 25 MeV, respectively; these calculated values are slightly higher than the experimental values of 9.63 MeV and 22.2 MeV. However, as has been shown by the work of Gillet and Vinh-Mau 20) in the case of 016, it is possible to get an optimum fit of the different levels by an adjustment of the parameters of the force, viz. depth, range and exchange mixture. We have not done so but presume that this could be done in our case by an adjustment of the depth and specially the exchange mixture. Let us now summarize the results obtained. For the 2 + (T = 0) states it is found that almost all the quadrupole strength is concentrated in the lowest state at 4.43 MeV. The probability of E2 transition from this state to the ground state has, however, been overestimated in our calculation. This is due to the fact that we have not mixed in the lh(o 2H-2P states which are energetically on the same footing as the 2hco 1[ t - I P states. Since the 2H-2P states carry no quadrupole strength, their inclusion would naturally decrease the theoretical value of the E2 transition probability. The energy-weighted sum rule, given by eq. (19) is exhausted to 86 % by the calculated 2 ~ states, of which 45.6 % is contributed by the 4.43 MeV state. This assures that even if the theoretical value of the E2 transition probability from the 4.43 MeV state to the ground state decreases considerably by the inclusion of the 2H-2P states in the calculation, the sum rule will still be satisfied to a tolerable extent. Our results for the dipole states are in general agreement with Brown and VinhMau's calculation using zero-range Soper force. In our calculation the dipole state appears at 25 MeV, and it consists almost exclusively of the non-spin-slip state [(lp~) - I ldj). In our calculation the state at 26.3 MeV, which chiefly consists o f the spin-flip state [(lp~)-1 l d~), carries a dipole strength of about 12 ~ . This may
COLLECTIVE STATES OF C 12
305
be significant since there is an experimental indication of a peak besides the 22.2 MeV peak, at about 23.3 MeV, in the (~, n) reaction cross section curve 2). The slightly higher energy values of our theoretical prediction for these two states are once again due to the fact that we did not try to choose an optimum exchangedependence of the two-nucleon force. The other state which is found to carry a significant amount of dipole strength is that at 37.4 MeV in our calculation. This consists almost exclusively of the state I(ls) -1 lp~) with an unperturbed energy of about 30 MeV Obviously the reason that such a dipole state is not found in 016 is that there is no such unperturbed hole-particle state in the case of 016 . The collective octupole state appears at 11.6 MeV in our calculation. We predict another high-lying octupole state carrying 3 7 . 4 ~ octupoie strength at 28.5 MeV. This is not a collective state but is produced by the large octupole strength of the unperturbed 3h~o state [(lp~_)-i lgl) it chiefly consists of" In the calculations of collective states so far carried out by using the Sawada method, no attempt has been made to calculate the correlated ground state explicitly. This is due to the fact that the Sawada method enables one to calculate the energy and transition probabilities directly without having to write down the detailed structure of the correlated ground state wave function. However, such a detailed expression in terms of the uncorrelated ground state plus the excited 2H-2P configurations in various orders is needed, in the first place, to check the internal consistency of the Sawada method. In this paper we have calculated the ground state wave function, assuming it to be the uncorrelated ground state plus excited 2H-2P configurations. The squared amplitudes of these 2H-2P configurations in the correlated ground state serve to calculate the occupation probability of single hole and particle states in the correlated ground state. The calculated values of the quantities justify the treatment of the hole-particle pairs as bosons, thus giving a check of the internal consistency of the Sawada method. However, some of the occupation probabilities might deviate enough from their closed shell values to allow of an experimental determination of them by, for example, a pick-up reaction experiment on the ground state of C 12. The total amount of ground state correlation is also important, since this determines the amount of deviation from the closed-shell picture. The total 2H-2P correlation intensity is found by us to be 53.6 ~ , of which 40.5 ~ is due to the quadrupole vibration. The correlation due to the dipole oscillation is small, as expected for T = 1 oscillations. The correlation due to the 3-, T -- 0, vibration is also small. The large correlation due to the quadrupole vibration is not disturbing, partly because it is slightly over-estimated on account of the non-inclusion of 2H-2P states in the quadrupole state calculation, and partly because 18 ~ of the correlation is produced by the occurrence of two (lp~)-1 lp~ pairs. Even the intermediate-coupling wave-function, calculated earlier with mixing within the lp shell only 19), contains about 25 ~ admixture of the l(lp~) -2 (lP½) 2) configuration. A m a j o r part of the quadrupole-correlation, calculated by us, is from this configuration, and the value of 18 ~ for this configuration in our wave function compares satisfactorily with the intermediate-coupling value.
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A. G O S W A M I A N D M. K . P A L
W e w o u l d like to t h a n k Dr. A. M. Lane a n d Professor M. K. Banerjee for m a n y helpful discussions. We w o u l d also like to acknowledge the very valuable help received f r o m Dr. Ted Y o r k o f the Harwell C o m p u t i n g Division who provided us with a p r o g r a m m e for finding the eigenvalues a n d vectors of a n o n - s y m m e t r i c matrix. O n e o f the a u t h o r s (A. G ) wishes to t h a n k the D e p a r t m e n t o f A t o m i c Energy, G o v e r n m e n t o f India, for the g r a n t o f a fellowship.
Appendix We give below the tables o f n o r m a l i z e d x a n d ~ for the three collective states of C 12. Tables of u n n o r m a l i z e d correlation amplitudes o f the g r o u n d state due to dipole, q u a d r u p o l e a n d octupole states are also given. TABLE 8 Values of forward-going and backward-going amplitudes x and 3~of the different unperturbed holeparticle states for the 4.43 MeV 2+, T = 0 collective state (Ipt)-~lp½
(ls½)-~ldt
(Is~)-~ld{
0.982 0.496
--0.266 --0.207
--0.383 --0.266
x
(lpi)-a2p4x (lp])-12p] 0.067 0.073
0A18 0.090
(lpt)-~Ift --0.159 --0.116
(lpi)-llf~ --0.532 --0.347
TABLE 9 Values of x and ~ of the different unperturbed hole-particle states for the 25.1 MeV collective 1-, T = 1 state
x
(lpt)-X2s~
(lp|)-Xldj
(lpt)-lld~
(ls½)-~lp4~
0.151 --0.033
--0.114 --0.046
0.984 --0.049
--0.070 --0,051
TABLE 10 Values of x and ~ of the different unperturbed hole-particle states for the 11.58 MeV 3- T = 0 collective state (ls½)-llq x
--0.073 0.041
(lpt)-Hg ~ (ls½)-llf~ (lpt)-~2d! (lpt)-~2dt (lpt)-Xlgt --0.051 0.024
--0,134 0.054
0.099 --0.061
0.114 --0.055
--0.183 0.052
(lpt)-Hd ! 0.553 --0.153
(lp~)-lldt 0,816 --0.127
307
COLLECTIVE STATES OF C 1~
TABLE 11 Unnormalized correlation amplitudes Chp, "iT h"p" for 2 ~, T = 0 states Hole-paxticle states (ls~)-qd~ (ls½)-qd| (ls½)-Xldt (lpa)-~2P½ (lp])-qp! (lpj)-Xlft (lpj)-llf~ (lp])-~lp~
0.056 0.090 --0.033 --0.034 0.049 0.135 --0.197
(ls½)-qd t
(lpt)-X2p~x
(lpj)-X2pt
(lp|)-tlf~
0.090 0.112 --0.031 --0.046 0.055 0.188 --0.241
--0.033 --0.031 0.037 0.040 --0.006 --0.018 0.079
--0.034 --0.046 0.040 0.034 --0.009 --0.029 0.094
0.049 0.055 --0.006 --0.009 0.023 0.073 --0.108
(lP|)-Xlf~ ( l P | ) - q P Jt 0.135 0.188 --0.018 --0.029 0.073 0.233 --0.306
--0.1 °~ --0. 7 0.079 0.094 --0.108 --0.306 0.452
TABLE 12
Unnormalized correlation amplitudes C hp, JT h'p" for 1-, 7"= l states Hole-paxticle states (lpj)-X2s½ (lpt)-Xldt (lp|)-lldt (ls½)-Xlp½
(lPt)-12s½ ( l P t ) - q d l ( l P i ) - q d l (ls½)-XlP½ 0.032 --0.002 0.065 --0.015
--0.002 --0.059 0.091 --0.035
0.065 0.091 0.115 0.107
--0.015 --0.035 0.107 --0.008
r'.~ULE 13 Unnormalized correlation amplitudes C hp, JT h"p" for 3-, T = 0 states Hole-particle (ls½)_t!ft ( l p | ) _ l l g t (is½)_llf] (lp|)_X2d| (lpt)_~2d| (lp|)_Xlgt ( l p | ) _ H d ! (lp|)_Xld| states (ls½)-llft ( 1p | ) - t 1gt (ls½)-tlf½ (lpt)-12dt (lpj)-12dt (lpt)-llg~ (lpg)-lldt (lpt)-qd |
--0.017 -- 0.009 --0.019 0.021 0.016 --0.026 0.054 0.042
--0.009 -- 0.007 --0.012 0.007 0.006 --0.027 0.028 0.025
--0.019 -- 0.012 --0.028 0.020 0.023 --0.035 0.062 0.061
0.021 0.007 0.020 --0.049 --0.036 0.018 --0.082 --0.064
0.016 0.006 0.023 --0.036 --0.033 0.019 --0.068 --0.062
--0.026 -- 0.027 --0.035 0.018 0.019 --0.055 0.078 0.041
0.054 0.028 0.062 --0.082 --0.068 0.078 --0.198 --0.162
References 1) 2) 3) 4)
H. Tyr6n and A. J. Maxis, Nuclear Physics 43 (1957) 52, 4 (1957) 637 J. Miller, G. Schuhl, G. Tamas and C. Tzara, Phys. Lett. 2 (1962) 76 F. Ajzenberg-Selov¢ and T. Lauritsen, Nuclear Physics 11 (1959) 1 F. C. Barker, E. Bradford and L. J. Tassi¢, Nuclear Physics 19 (1960) 101; Richard R. Carlson, Nuclear Physics 28 (1961) 443 5) N. Vinh-Mau and G. E. Brown, Nuclear Physics 29 (1962) 89 6) A. Goswami and M. K. Pal, Nuclear Physics 35 (1962) 544
0.041 0.025 0.061 --0.064 --0.062 0.041 --0.162 --0.141
308
A. G O S W A M I A N D M. K. P A L
7) V. Gillet and N. Vinh-Mau, in Proc. Rutherford Jubilee Int. Conf. (Heywood, London, 1961) p. 315 8) K. Sawada, Phys. Rev. 106 (1957) 372 9) G. E. Brown, J. A. Evans and D. J. Thouless, Nuclear Physics 24 (1961) 1 10) M. Baranger, Phys. Rev. 120 (1960) 957 11) S. Fallieros and R. A. Ferrell, Phys. Rev. 116 (1959) 660 12) M. Kobayasi and T. Marumori, Prog. Theor. Phys. 23 (1960) 387 13) J. Sawicki and T. Soda, Nuclear Physics 28 (1961) 270 14) V. Gillet, Thesis, Paris, 1962 151 J. M. Soper, Ph.D. Thesis, Cambridge, 1958 16) C. M. Class, R. H. David and J. H. Johnson, Phys. Rev. Lett. 3 (1959) 41 17) H. Tyr6n, Peter Hillman and Th. A. J. Maris, Nuclear Physics 7 (1958) 10 18) M. Moshinsky and T. A. Brody, Table of transformation brackets (Monografias del Instituto de Fisica, Mexico, 1960) 19) J. M. Soper, private communication 20) V. Gillet and N. Vinh-Mau, Phys. Lett. 1 (1962) 25