eichler, yamamura

1.D.l: I I Nuclear Physics AI82 (1972) 33-53; @ North-Holland Publishing Co., Amsterdam

I.D.2 Not to be reproduced by photoprint or microfilm without written permission from the publisher

QUADRUPLING AND PAIRING IN THE SHELL MODEL

J. EICHLER t

Kahn-~ejt~er-I~stitut fiir Kernforschang Berlin, Sektor Kernphysik

and

Freie Universitiit Berlin, Fachbereich Physik, Berlin- West, i&many

and

M. YAMAMURA

Department of Physics, Faculty of Science, Kyoto Uniuersity, Kyoto, Japan

Received 5 April 1971

(Revised 8 October 1971)

Abstract: A system of Z protons and N neutrons (Z+N = even) moving in non-degenerate j-levels is treated for an isoscalar T = 1 pairing interaction and an effective four-body force. A classification scheme is introduced in which the basis states are explicitly classified with respect to the number of quadruples (systems of two pairs with J = 0, T = 1 coupled to 2 = 0) and the number of T = 1 pairs in each of the levels. For the particular case of two j- levels the system has been solved exactly. The two-particle transfer, the four-particle transfer and the four-particle scattering exhibit a phase transition between a normal and a superfluid phase. The superfluid phase is shown to consist mainly of quadruple;, not pairs, even for a pure isoscalar pairing force. A quadrupling seniority scheme is discussed in close analogy to the pairing seniority scheme.

1. Introduction

In light nuclei, protons and neutrons move in the same shell-model orbits. As a consequence, it becomes necessary to consider proton-neutron pairing in addition to proton-proton and .neutron-neutron pairing. Furthermore, a new building stone of nuclei is excepted to become important, namely a subunit in which two protons and two neutrons are coupled to J = 0 and T = 0. The a-like couping of four nucleons is by no means unique. In competition with the conventional pairing it is suggestive, however, to consider associations of two pairs with J = 0, 7 = 1 to composite structures with J = 0, T = 0. These specific a-like systems are usually called quadruples I).

Of course, a description of light nuclei in terms of T = 1 pairing and quadrupling cannot be quite realistic. As is known from N = Z = odd nuclei the coupling to J = J,,,, T = 0 is about as important as the J = 0, T = 1 pairing. Correspondingly, it is also favourable to construct four-nucleon systems from J = J,,,,,, T = 0 pairs to form quartets ).

t An essential part of this work was performed while the author was visitor at the Research Institute for Fundamental Physics, Kyoto University, Kyoto, Japan, in the autumrr 1970.

33

34 J. EICHLER AND M. YAMAMURA

Various authors 3- ) have discussed the connection of quartets with the observed deformations in light nuclei. The relation to deformations is not surprising since it is easily seen that within a j-shell the quartet coupling scheme is equivalent to mini- mizing the expectation value of a simple attractive Q,,Qe force (where Q, is the zero component of the quadrupole operator) which is known to lead to deformations. The coupling scheme, then, will not be self-consistent unless it is used in conjunction with a quadrupole-quadrupole or a similar interaction.

While there is no doubt that four-particle correlations are important ) it is not immediately clear which type of coupling will be dominant. We believe that both, quartetting and quadrupling should be important for light nuclei since the underlying pairs (J = J,,, T = 0 and J = 0, T = 1, respectively) are both favoured by realistic nuclear interactions.

So far, both types of correlations have only been considered separately. Quartet systems are treated ) y 3, 4 b using physical arguments to justify a drastic truncation (at the expense of the completeness) of the configuration space in which the Hamil- tonian is diagonalized. In a second step, the Pauli principle is incorporated and re- dundancies are eliminated ). In practice, the method is limited to a very small number of quartets and pairs.

Our aim lies in a different direction. We want to use a Hamiltonian which is simple enough to be amenable to an exact diagonalization in a basis which is complete with respect to this Hamiltonian. The Pauli principle is taken into account rigorously, a point which is essential for the description of typical many-body effects.

A predecessor ) to the present calculations was motivated by the desire to explain the observed anomaly in elastic a-scattering ** ) in the Ca region by studying c1- type correlations and the effect of blocking by excess neutrons on such correlations. Using isopairs lo) with T = 0, J = odd (predominantly ) J,_) as basic building stones and introducing an effective four-body interaction on physical grounds a sharp phase transition to cl-superfluidity for a critical strength of the four-body interaction was found and furthermore a strong sensitivity to blocking by excess neutrons.

With the final aim of treating both T = 0 and T = 1 pairing along with the corresponding four-body structures we presently confine ourselves to T = 1 pairing and quadrupling. Starting from the physical task of finding orthogonal basis states which are composed of a definite number of quadruples and a definite number of pairs we succeeded to construct a complete set which we later found + to be equivalent to the R(5) classification scheme 11-14). In our case, however, with the seniority and the reduced isospin both equal to zero, we can avoid the complications of group theory.

As an illustration, we applied the classification scheme to a two-level model in sect. 3. Clearly, an extension to more than two levels is just a matter of angular momentum coupling in isospace. Numerical results are discussed in sect. 4 and conclusions drawn in sect. 5.

t The authors are indebted to D. H. E. Gross for pointing out this connection.

QUADRUPLING AND PAIRING 35

2. Formulation of the theory

For a detailed treatment of quadrupling and pairing it is useful to construct all relevant states explicitly. Thus we may gain a more direct physical insight into the structure of the states than by a mere classification in terms of a set of quantum numbers.

2.1. BASIC OPERATORS

The elementary building blocks of all states which we are considering in the present paper are particle pairs coupled to J = 0 and T = 1. They are defined by

PZ, = $x(-)j-+&, m

B,+ = 3C(-)-m(ni,pit,+pi+mnif-m), m

PI, = Lx(-)i-mp;mp;_m, J2 In

(2.1)

where pi, and njn are fermion operators which create protons and neutrons with angular momentumj and projection nz. Furthermore, we introduce the isospin operators

pi-1 = - $; njf,Pj,,

rT, = 3 C (n_Lnjm-P_LPjm), m

(2.2)

and finally the number operator

(2.3)

These operators together with the hermitian conjugates of the pair operators (2.1) form a closed algebra, which can be shown to be identical with the group R(5) of rotations in a five-dimensional space 11-14). The commutation relations can be easily worked out and are given in appendix 1.

In the framework of a T = 1 pairing theory there is only one way to construct a four-nucleon system with T = 0 and J = 0. This object is called a quadruple 192P1 ) and is defined by the creation operator

(2.4)

36 J. ElCHLER AND M. YAMAMURA

Trivially, 0, obeys the simple commutation relations

L-G, +1, Qo+] = 0, (2.5)

which largely simplify further calculations. Other commutation relations are given in appendix 1.

It is our aim to construct exphcitly the basis states in terms of the operators fi, p and &z. In the following subsection we show that this problem may be reduced to the familiar problem of constructing the spherical harmonics.

2.2. CORRESPONDENCE TO THE SOLID HARMONICS

If, in a single&level, we want to classify arbitrary states built from pairs with J = 0 and T = 1 an arbitrary state will have the general structure.

with iz+ +n, +n_ = N and n, --PI_ = Te. Of course, the states (2.6) will not, in general, form an orthogonal basis.

However, it is easy to construct an orthogonal basis characterized by the number N of pairs, by the isospin T and its projection To if we observe that the vector operators r, with

7-O = 2,

r2 =kil(M)krkr-kF

and the angular momentum operators &., with

(2.7)

L, = L,,

obey exactly the same commutation relations as the operators p:, & and $. There- fore, we have a one-to-one correspondence between the following operators

rk 2 fsk+,

2, 2 $, (2.9)

r-2 2 $o*.


Now, it is well known that the orthogonal set of eigenstates of L2 and Lo constructed from the operators rk is given by the solid harmonics

Here, the first factor is due to the specific normalization of the spherical harmonics and the second factor arises entirely from the successive application of the operators z*. It is now a simple step to construct the basis explicitly.

2.3. CONSTRUCTION OF THE BASIS STATES

Using the operator correspondence (2.9) of the last subsection and eq. (2.10) we can immediately write down special states charac~rized by N = T, Tand To. Further- more, by multiplying (2.10) with some power of rz and using the last eq. (2.9) we obtain t the general orthonormal set of basis states

IN, T, +lT,I) = $- 1/ 2r=+(T+ lTOl)! NT (2T)!(T-IT& (+-~~l)-(~~,)=(Q^,f)ff~--~~O>. (2.11) Evidently, N-T must be an even number. The factor (C&)-l arises from the normalization of the quadruple operators and is given by

c NT = T!(SZ+l)!(N-T)!!(N+T+l)!!

(2T+l)!!(G+l+T-N)!!(&T-N)!! (2.12)

where Sz = 2j-t- 1 is the degeneracy of the level. The derivation of eq. (2.12) can be found in appendix 2.

It is important to observe that the structure of the orthonormal set IN, TTo) clearly exhibits the content of pairing and quadrupling. The limiting cases of pure pairing and pure quadrupling are given by T = N and T = 0, respectively. Thus, the representation (2.11) has the advantage that quadrupling appears in a very natural and necessary way, similarly as the pairing in the conventional seniority scheme.

The basis states (2.11) are not only eigenstates of the operators fi0, p2 and .?e but also of the number operator for quadruples 15), Q,&. We have

8,$,lK TT,> = QmINt T&h (2.13) The eigenvalue

QNr = (~+T+l)(~-T)(~-~+T+3)(~-~-T+2), (2.14)

may be obtained from the observation that

Q,*tK TT,> = Q~~W+2, TT,>, &IN TT,) = QiG2N-2, TT,),

t A formula related to ours has been given by Parikh 13).

(2.15)

38 3. EICHLER AND M. YAMAMURA

with $I,, = C,,,, rJC,.,r and Q&1 = C&./C,_,, r. Explicit values are given in appendix 3.

2.4. EXPANSION OF GENERAL STATES IN THE BASIS

If we want to calculate matrix elements of operators belonging to our algebra or to expand arbitrary products of pair operators in terms of the basis states (2.11) we may proceed in either one of two ways: (i) We may expand the states (2.6) in the basis (2.1 I> by using the well-known expansion of arbitrary products of r+, r,, and r_ in terms of solid harmonics and observing the correspondence (2.9) and (2.10). (ii> More conveniently, we may use the Clebsch-Gordan series for solid harmonics

together with the identifications (2.9) and (2.10). As a particularly simple example for the application of (2.16) we obtain

p(i) G+,,, NTL = ~

(2L--l)!!T! T 1 1;

C NT f ii (2T-l)!!L! 0 0 0 (2.18)

Similarly, we have

(-I-*~-&% TT,> = AZ* P;L ' '(; :1,"+~)

IN-i,L,T,+k>, (2.19)

with

c NT p&i = - ___ (2T-I)!&! 2T+l

G-,,L (I&-Q!!T! 2X+-t-1 (2.20)

The explicit form of the Goe~~ients P$$$, is given in appendix 3.

3. A two-level model

The formulation presented in sect. 2 may be used to study the competition of qua- dr~pl~ng and pairing in the framework of the shell model. The i~~e~a~tion appropriate to our coupling scheme consists of an isoscalar pairing force acting between nucleon pairs coupled to .I = 0, T = 1 and an additional effective force which partly simulates the influence of other terms in the interaction, in particular T = 0 terms. For convenience, we consider a simple system of two levels with the degeneracies Sz, = 2j, -I- 1, Q2 = 2jz f 1 and the level distance D. The extension to more than two levels is straightforward and only involves more cumbersome angular momentum coupling in isospace.


3.1. EFFECTIVE FOUR-BODY INTERACTION

If we would restrict ourselves to a T = 1 pairing force we would miss an important effect which manifests itself in the systematics of binding energies in light nuclei. Let us start from a nucleus with 2 = N = even and successively add T = 1 pairs in such a way that after every second step we again have an cc-nucleus with Z = N = even. Then, the resulting gain in binding energy clearly shows an odd-even effect in pairs.

ENERGY SPECTRUM OF THE QUADRUPLING FORCE

S-l=12

N 0 1 2 3 4 5 6 I 0 9 10 11 12

T=O 1 2 3 L 5 6 shell model value ------ - ________ _____-_-__-____ _-__

5

T:O 0

4 L

1 1

2 - 3 - 3 2

3

0 0

2 I -

2 - 1

1 0 - 0

Fig. 1. Energy spectrum (in arbitrary units) of an attractive quadrupling force as a function of the number N of T = 1, J = 0 pairs in a single level with the degeneracy P = 2j+ 1 = 12. Eachlevel is labelled by the isospin quantum number T. In all cases, the level with the lowest possible isospin

T = 0 (T = 1) for even (odd) N has the lowest energy.

While the first pair (we take the average over the three isospin projections) contrib- utes about 12 to 13.5 MeV to the total binding energy in the Ca region (between 32S and 44Ti) the second pair completing a quadruple or cc-particle gives about 21.5 MeV. Such an odd-even effect may be accounted for by a pairing force between pairs or effective four-body interaction in just the same way as the ordinary pairing force accounts for the odd-even effect with respect to nucleons.

Of course, this odd-even effect is due to the T = 0 interaction which gives additional contributions to the binding energy each time when two T = 1 pairs complete a quadruple and thus an isospin recoupling results into T = 0 pairs. The inclusion of T = 0 pairing (isopairing), however, would break our scheme or the R(5) scheme


and necessitates the introduction of the group R(6). We are forced, therefore, to simulate this force by an effective quadrupling interaction.

This force is mediated by the operator 0, Q,, which according to eq. (2.13) is diagonal within one level and has the eigenvalue QlvT given by eq. (2.14). The odd-even effect is simply a result of the selection rule that N-T has to be an even number. As a consequence, the lowest levels for nuclei with an even number of pairs and an odd number of pairs have T = 0 and T = 1, respectively. Thus there is a complete analogy between our scheme, the &iQe force, and the isospin quantum number T (counting quasi-pairs) on the one hand and the seniority scheme, the ordinary pairing force, and the seniority quantum number s (counting quasi-particles) on the other hand.

Fig. 1 shows the energy spectrum of an attractive pure $gf& force for a single j-level with the degeneracy D = 12 corresponding to the sd shell. The picture has a close resemblance to the spectrum of a pure pairing force in the seniority scheme. The odd-even effect may be seen quite clearly. Of course, it is known that also a pure isoscalar pairing force shows an odd-even effect in pairs i6). However, this effect is much too small (if the number of pairs is larger than two) to be comparable with the observed fluctuations.

It seems, therefore, reasonable to introduce a quadrupling interaction 11r7) into the Hamiltonian in order to simulate the effect of interactions which, in real nuclei, produce the quadrupling effect. The strength of this force is considered as a parameter in the present paper. For more realistic calculations ) it may be taken from the systematics of experimental binding energies.

3.2. MATRIX ELEMENTS OF THE HAMILTONIAN

If we adopt a T = 1 pairing force with the strength GP and a quadrupling force with the strength Go we are able to write down the Hamiltonian of our system entirely in terms of operators belonging to the algebra (2.4). Measuring all energies in terms of the energy difference D between the upper level 2 and the lower level 1 and putting the Fermi energy halfway between the shells we have

H = #e(2)-i?,(l))-+, i i P,(.@&)-tGoj J$C1a:cj,e&). (3.1) j,f=i k=-1

This Hamiltonian is diagonalized in the complete and orthonormal basis of states which are obtained by vector-coupling states of the structure (2.11) for each of the levels

INI C N2 T,; NW) =,& (z. ;. j ;) IN, T1 T,o)IN, r, Go>,

N,+N2 = N. (3.2)

The explicit form of the matrix elements is readily obtained from the formulae of


subsects. 2.3 and 2.4. The diagonal matrix element is


The cross section may be readily calculated by using the expansion (3.5) of the eigenvectors and using the formulae (2.17) and (2.15), respectively.

It is fur~ermore of interest to calculate a matrix element which leads from the initial channel back to the initial channel by first adding and then removing a quadruple. Neglecting shape effects which are not defined in the present model and the energy dependence of intermediate states the square of this matrix element is proportional to the cross section for elastic quadruple scattering. Just to have a name we denote it this way and take

t$) = I(ilY~Y~li)l, (3.11)

as a measure for the quadrupling correlations of the target nucleus. The correlations contained in the eigenvectors may also be expressed more directly

in terms of the expansion coefficients themselves. For a given eigenvector we may ask, for example, how likely it is that N2 fermion pairs are in the upper level. This proba- bility is given by

(3.12)

Furthermore, we may wish to know the distribution of the isospin in the upper level given by

xaivr(T2) =&IG&% Tl Nz TX (3.13) 2

The comparison of both quantities will be of particular interest. If, with increasing interaction strength, the maximum of K(N2), shifts to higher Nz values and simultaneously the maximum of K(T2) shifts to higher T2 values, so that approximately T, M IV,, we will say that the upper (and consequently also the lower) level is paired. If, conversely, T2 has its dominant weight close to the smallest possible value even for large N,, we will say that the levels are quadrupled off.

4. Numerical rest&s

Using the results of sect. 3 it is easy to diagonal&z the Hamiltonian on a computer and to calculate energy eigenvalues and eigenvectors. Aside from the degeneracies aI and Sz, of the levels, the number N of pairs, and the total isospin T, we have two parameters, Gr and Go characterizing the strength of the pairing and quadrupling interaction, respectively. For convenience, we introduce

x = QG,, (4.1) Y = Q2G, (4.2)

where 52 = (Q, 52,)*. With this convention, the random phase approximation (RPA) for the pure pairing case and S2, = 8, would give a phase transition ** 1 ) for x = 1. Using the estimates for the force parameters from subsect. 3.2, the level distance D m 7.6 MeV between the d, and the f+ shell and the degeneracy D = 8 we obtain the values x M 0.63 and y m 0.76.

Our task is now to study the correlations contained in the eigenvectors as a function of x and y.


4.1. FOUR-NUCLEON TRANSFER AND BLOCKING BY EXCESSS NEUTRONS

In the case of pairing correlations it is well known 18* ) that the cross section for two-nucleon transfer is a suitable measure for the degree of pairing. The cross section rrc2) shows a strong increase around the phase transition point x = 1. Corresponding- ly, we take the cross section oc4) for quadruple transfer defined in eq. (3.9) as a measure for four-nucleon correlations.

,lLl

I50

100

SO

L

I , I I I I

QUADRUPLE TRANSFER

AND THE INFLUENCE OF EXCESS NEUTRONS

R, =R*=20 N.ZO,T=i

Y = 0.2

I I I I I I

0.1 0.2 0.3 0.L 0.5 0.6 y

Fig. 2. The cross section ~(0 for quadruple transfer as a function of the strength y of the quadrupling interaction for a fixed pairing force x = 0.2. The influence of blocking due to excess neutrons is given in broken lines. T = 1 and 7 = 2 correspond to 2 and 4 excess neutrons, respectively.

The degeneracy is D, = Qz = 20.

Fig. 2 shows af4) as a function of y for a symmetric model with 8, = Q, = 20, N = 20, T = 0 and a pairing force with the strength x = 0.2 well below the pairing phase transition. There is clearly a phase transition to quadruple superfluidity at a critical strength of y w 0.375. The increase in the cross section is more pronounced and sharper than in the pairing case I). This is to be expected since a quadrupling force, by definition, lifts four particles at a time from the lower to the upper level. It turns out that at the same force strength y also (r c2) shows a strong increase. Further- more, the phase transition is reflected in a minimum for the excitation energy of the first excited state. The cross section for quadruple transfer leading to the first excited state of the final nucleus is much smaller than to the ground state for all sets of parameters considered.

It is now an interesting question how the system behaves if the upper level is blocked in some way. In the case of pair transfer in a pure pairing model the blocking of the upper level by a single nucleon which is not affected by the pairing force leads ) to a slight decrease of the cross section due to the decrease of the effective degen-


oiel)

SCATTERING

15000

6.2 * ii6 y

Fig. 3. The cross section for elastic quadruple scattering de&ted by eq. (3.11) as a function of the strength y of the quadrupling force is plotted for various strength parameters x of the pairing force.

ELASTIC QUADRUPLE SCATTERING

3000

Fig. 4. The cross section for elastic quadruple scattering defined by eq. (3.11) as a function of the strength y of the quadrupling force. The degeneracies sd, = 12 and Sa, = 8 are chosen to correspond to the {sd) shell and the fs shell, respectively. For two values of x the solid lines represent a closed- shell system f40Ca) while the broken lines show the influence of two excess neutrons (42Ca). [It should be noted that for the calculation of these curves the transfer operator f14+ has been taken to

be (ja,Sa,)-~(a,+(l)+~,+(2)) instead of the definition (3.10).]

QUADRUPL~G AND PAIRING 45

olel! 1 ELASTIC QUADRUPLE SCATTERING

--I _-

1 1 1 I 1 1 2 3 4 5 x

Fig. 5. The cross section for quadruple scattering defined by eq. (3.11) is plotted as a function of the strength x of the pairing force in the absence of a quadrupling force. The solid line represents the closed-shell system N =t G, = 4 = 20, T = 0 while the broken line shows the influence of four

excess neutrons (T = 2).

eracy and thus of the collectivity which the system is able to develope. On the other hand, in the model of ref. ) the four-nucleon transfer showed an extreme sensitivity to blocking by two inert neutrons which are neither affected by the T = 0 pairing force nor by the four-body force.

In the present model, blocking by two (four) neutrons is equivalent to the choice N= Q,+landT= l(N= 8,+2andT= 2) for the target nucleus. For these cases, the cross section c#~) is also plotted in fig. 2 as a function of y. We notice that the CIOSS section is not smaller but slightly larger than the unblocked one below the phase transition point.

This result is clear from the energy spectrum of the quadrupling force given in fig. 1. The blocked transition from Nz = 1 and N, = Q2, to Nz = 1 -t-.&V, and N1 = 51, = St1 -AN, is energetically more favourable than the unblocked transition from Nz = 0 and IV1 = 52, to Nz = AN, and Nr = 52, -AN,. Thus the pair already present in the upper level helps to pull up quadruples, This enhancement effect is even increased somewhat if already four neutrons are present in the upper level. We have, in fact, a complete analogy to the seniority scheme which gives an energy gap for seniority s = 0 but no gap for s > 0.

As to the influence of blocking by excess neutrons on alpha transfer and elastic a-scattering *, ) we would conclude from the present calculations (see figs. 2,4 and 5) that it is not possible to make the blocking effect responsible for a destruction of CI-


1 I I I I I I III I I I I

ISOSPIN DISTRIEUTION -

PAIR DlSTRlBUTtON ----

IN THE UPPER LEVEL

IO 15

Fig. 6. The distribution K(T2) of the isospin (solid lines) and K(N2) of the number of pairs (broken lines) in the upper level of a closed-shell system (T = 0, N = Q1 = Qz = 20) in its ground state are given for various strength parameters x of the pairing force and y = 0. While the isospiu T2 tends to be small even for large values of x the number Nz of excited pairs increases strongly with increasing X. This discrepancy shows that the pairs arrange themselves predominantly in quadruples.

iilti:l , , , , , , , , 1 , ( , , , / & to.80 i

I ISOSPIN DiSTRlBUTlON -

AND PAiR DlSTRIBUTION---

IN THE UPPER LEVEL

D,=R2=N=20

110

x.0.t

2

Fig. 7. The distribution K(L) of the isospin (solid lines) and K(NZ) of the number of pairs (broken lines) in the upper level of a closed-shell system (T = 0, N = ai = Q, = 20) in its ground state are given for pairing force x = 0.4 and the two values y = 0.3 and y = 0.5 for the strength of the

quadrupling force.

QUADRUPLING AND PAIRING

W2) 0.6 ISOSPIN DISTRIBUTION -

AND PAIR DISTRIBUTION

IN THE UPPER LEVEL

T-4

R,R220

N-24 y.0

0 E .I

Fig. 8. The distribution K(T2) of the isospin (solid lines) and K(N2) of thenumber of pairs (broken lines) in the upper level of a system with 8 excess neutrons (T = 4). For an increasing strength parameter x of the pairing force the average isospin in the upper level decreases while the number of excited pairs increases. Apparently, the formation of quadruples is maximized by dist~buting the

excess neutrons equally between the levels.

correlations and thus for the disappearance in 44Ca of the anomaly in cl-scattering observed in 40Ca and other N = 2 = even nuclei ).

4.2. QUADRUPLE SCATTERING

The phase transition becomes more pronounced if we consider quadruple scattering instead of quadruple transfer. Scattering has the further advantage that only the target nucleus itself is involved. Fig. 3 shows the cross section g() for elastic quadruple scattering as a function of y for various values of x. It is seen that for an increasing strength x of the pairing force the strength y of the quadrupling force needed to in- duce the phase transition becomes smaller and smaller. The dependence of x and y at the phase transition point may be used to construct a phase diagram in the xy plane.

Fig. 4 shows #) for 8, = 12 and f2, = 8 ~rresponding to the degeneracies of the (sd) and f+ shell, respectively. We have the same qualitative behaviour as before except that the phase transition is less pronounced due to the decreased degeneracy. Again we notice that an excess neutron pair in the upper level (N = 52, + 1, T = 1) enhances the quadruple scattering similarly as it enhances the transfer.

Even in the absence of a quadrupling force (fig. 5) we observe a phase transition in quadruple scattering. The critical value x = 1.4 is similar to the value at which the two-nucleon transfer exhibits an increase la* )*


4.3. CORRELATION FUNCTIONS

Having considered pair transfer, quadruple transfer and quadruple scattering it may be desirable to find a more direct measure of the pairing and quadrupling correlations. Such a measure is given by the isospin distribution K(T,) and the pair distribution K(N,) in the upper level as defined by eqs. (3.12) and (3.13).

Fig. 6 simultaneously displays the isospin distribution and pair distribution of the upper level - reflecting also the distribution in the lower level - for y = 0 and P, = L?, = N = 20, T = 0. Just below the phase transition, for x = 1.0, we have predominantly the excitation of no pair or one pair and, consequently, the distributions of isospin and pairs are the same. For x = 2.0 the maximum of K(T,) is at T2 = 1 with T, = 0, T, = 2 being about equally important. The number of excited pairs centres around NZ = 2. If we increase the strength of the pairing force to x = 4 this is usually considered as a clear pairing situation lg) in which the BCS theory is valid. From fig. 6 we see, however, that the isospin distribution does not change any more while now about 7 pairs are excited on the average.

If the four-body interaction is different from zero (see fig. 7) we obtain the same general pattern as in fig. 6, except for an odd-even staggering of the pair distribution which favours even numbers for NZ.

Considering our unique representation (2.11) and the subsequent discussion the result shown in fig. 6 clearly means that even for a pure pairing force all pairs except for one or two arrange themselves in quadruples. Thus we do not have a phase transition to ordinary pairing superfluidity but rather to quadruple superfluidity. This property, of course, cannot be adequately tested by just considering two-nucleon transfer.

The discrepancy between isospin distribution and pair distribution becomes even more apparent (see fig. 8) for the extreme case of eight excess neutrons (T = 4). For an increasing strength x of the pairing force it becomes energetically favourable to distribute the excess neutrons equally between the levels. In this way, the quadrupling is maximized such that for x = 5.0 in each of the levels all pairs are arranged in quadruples except for an average of two excess neutron pairs.

It is worth mentioning that the discrepancy between isospin distribution and pair distribution continues to exist for the first excited state of the system.

Summarizing we may state that we have not been able to find any set of parameters {a,, Q,, T, x, y} which would have produced a typical pairing solution with NZ z T2 > 1.

5. Discussion

The results of exact calculations lend themselves to the comparison with various approximation methods. For the case of identical nucleons, the solutions of a solv- able two-level model have been compared by Broglia et al. 18) with the results of the RPA and BCS theory. Dussel et al. ) h ave used a model like ours with an isoscalar


pairing force between nucleon pairs with T = 1. Their discussion concentrates on isospin properties, in particular on permanent deformations and rotations in isospace ). They show that in the case of a strong pairing force, x = 4.0, a Nilsson like treatment ) gives an excellent approximation for the energy and satisfactory agreement for the two-particle transfer.

On the other hand, it is known ) that for an isoscalar pairing force the BCS treatment, or more generally, the Hartree-Bogoliubov treatment leads to certain ambigui- ties which show up most clearly in the case of N = 2 nuclei. Here, the state vector of the ground state may be expressed in various ways, the limiting cases being (i) a product of two identical special Bogoliubov transformations performed separately on neutrons and protons ). (ii) A BCS state constructed from proton-neutron pairs [ref. 23)]. Both solutions have the same coefficients u and u in the Bogoliubov trans- formation 21) and h ave identical energies. However, the structure of the state vectors (i) and (ii) is entirely different. The degeneracy of the solutions is, of course, a consequence of the charge independence of the Hamiltonian.

Our results presented in sect. 4 give a more explicit insight into the problem. We have found, that even in the absence of a quadrupling force the ground state vector of a system with N = Z = even consists predominantly of quadruples for a strong isoscalar pairing interaction. In practice, there is never more than one quadruple in one level broken up into pairs. The formation of quadruples in a system of protons and neutrons subject to an isoscalar pairing force thus seems to play the same role as pairing for a system of identical nucleons. In analogy, we may introduce the notion of a quadrupling seniority w as the number of quasi-pairs which are not condensed into quadruples. Energetically, states with lowest quadrupling seniority which can be reconciled with the given isospin are preferred.

The condensation into quadruples is certainly a consequence of the isospin being a good quantum number. Suppose we wanted to construct a T = 0 state which is paired. Pairing would mean that each of the levels (e.g. in a two-level model) had a large isospin Ti z Ni >> 1. The coupling to a total T = 0 is achieved by constructing a complicated linear combination whose extreme components would contain all protons in the upper level and all neutrons in the lower level and vice versa. It must be conceded that such a coupling appears to be rather artificial and cannot be expected to have a low energy. Conversely, a state which is essentially built from quadruples, in each of the levels, minimizes the isospin coupling and hence appears to be more natural. It is seen that the requirement of good isospin is indeed a very strong con- straint.

Coming back to our discussion of the pairing approach we arrive at the conclusion that for a system composed of protons and neutrons the pairing theory is not com- pletely adequate. (a) The ambiguity 22* 23) discussed above just arises from the various ways in which pairs may be projected out from quadruples. (b) In a generalized pairing theory 24) for N = Z doubly even nuclei in which T = 0 and T = 1 pairing is treated simultaneously, the mean feature of the solution is the mutual exclusion of


T = 0 and T = 1 pairing. The T = 1 pairing solution gives an isospin intrinsic state from which one has to project the various T-states. Since it is unlikely that states sep- arated by NN 9 MeV could be contained in the same intrinsic state ) the T = 1 pairing solution is disregarded and the T = 0 solutions are accepted as the physical solutions ). Evidently, such a reasoning ) is based on the shortcomings of the theory (namely isospin violation). It does not mean that T = 1 pairing is unimportant altogether but only shows that BCS type pairing theories have to be considered with some caution.

Since the pairing scheme is not entirely satisfactory it is the next task to take into account the quadrupling structure from the outset. Attempts in this direction have been made by various authors Zs 26n ). Th e p resent investigation will give a more clear-cut starting point and a testing ground for approximate methods.

An interesting application of our method to treat quadrupling in light nuclei will be a discussion of the nuclear Josephson effect 27) for quadruples instead of pairs. In a sub-Coulomb collision between light nuclei one might hope to observe an enhancement of the a-transfer cross section due to quadruple correlations ).

One of the authors (J.E.) wishes to express his gratitude to Prof. H. Yukawa, Prof. Z. Maki and Prof. R. Tamagaki for the warm hospitality extended to him during his stay at the Research Institute for Fundamental Physics, Kyoto University, Kyoto. He also would like to thank Dr. D. H. E. Gross for fruitful discussions and helpful comments. Thanks are due to Mrs. A. Valentien for her help in computer programming. The financial support of the German Academic Exchange Service, DAAD, is grate- fully acknowledged.

Appendix 1

COMMUTATION RELATIONS

For convenience, we give a list of the commutation relations of the operators defined in subsect. 2.1. Using the definition Sz = 2j+ 1 we have

For the isospin operators we have the usual relations

(Al.l)

(A1.2)


The mixed commutation relations are

CT*, , Gl = 0,

[cd%! = *i%,

CL &I = HT,

m, Cl1 = 2% 9

@*I, Kl] = WI?

[Co, JTI = 0,

[&, P,] = 2P5,+. (A1.3)

Finally the following relations involving the quadruple operator 00 are useful:

[pfl, @] = -21i:@+l-&)+4& &P,T4~,+~~,,

[b;,, &] = 2&$2+1--&J-4& f+f,,+4p: T-I. (A1.4)

Appendix 2

CALCULATION OF THE NORMALIZATION COEFFICIENTS C,,

Using the product rule for commutators and the relations (A1.4) we obtain

P,($,)+(N-T)IO) = (N- T)(B+3+ T-N)~,+(&,+)"N-T-2)~0), (A2.1)

E@,)+-~o) = (N-T)(S2+3+T-N)(S2+4+2T-2N)(&,+)*N-T-210)

+(N-T)(N-T-2)(8+3+T-N)(S2+5+T-N)(~,+)Z(&,+)~N-T-4)lO). (A2.2)

Similar expressions hold, if we replace PO by P+ 1. Combining both and using the definition (2.4) of &l we get

Qlo(Qo)*N- Tlo)

= (N-T)(N+1-T)(i2+2-T-N)(B+3+T-N)(~,+)+N-T-2)10). (A2.3)

From this we get the result of Q0 acting on the general state

&,(~=,)T($,>f(N-T)IO)

= (N-T)(N+T+1)(S2+3+T-N)(S2+2-T-N)(B:,)T(~,+)3N-T-2)l0). (A2.4)

From eq. (A2.4) we may directly calculate the normalization coefficient

C;, = (Ol(&,)fN-T)(~f l)T(p; l)T(~o+)(N-T)jO>

= (0~($,)*-~-~~(f3*~)~Q~(~f~)~(~O+)~~~-~~~O>

= (N-T)(N+T+1)(52+3+T-N&2+2-T-N)C;_,,.. (A2.5)

52 J. EICHLER AND M. YAMAM~RA

Working on with this recursion formula down to N = T we get

C& = (N-T)!! (Nf T+l)!!(Q+ l)!!(B-2T)!! _ C2 (2T+1)!!(8+1+T-N)!!(&T-N)!! ==

The remaining expression has no Ionger quadrupie operators. Then

CL = Gw* l)V:: I)%>>

is readity cabdated via another recursion farmuh to give

c;, = T!i2!!

(&2T)!! *

Inserting C& into the expression for C& yields the result (2.12)

EXPLICIT FORMWLAE FOR EXPANSION COEFFICIENTS

The coefficients QN1^ () defined in eq. (2.15) are given by

QrT) = Q- n_N,T = ((N+T+3)(N-T+2)(52--N+T+l)(&-N-T))*,

(A2.6)

(A2.7)

(A2.8)

Q&-j = QcN T I

= ((N+T+l)(N-~)~~-Nf-T+3)(9-N-T+2))f~. (A34

The coe%cients PNTL f&a) d efined in eqs. (2.18) and (2.20) have the explicit form t

pf+) NT,T+l = -Pg-& -f Tfl =

t

(T+l)(N+T+3)(~-N-T) - - I l

2T+3 I

P$& 1 = -P I - sY?vTT -,.- I= - i

T(N-T+2)(56-N4Tfl) +

2T--l I f

P& 1 = I - - PpN, T, T_ 1 = T(m-Tfl)@-N-Tf2) *

I .

2T-1 (A3.2)

t The coefficients are proportional to the R(5) Wigner coefficients given in table 3 of the second ref. 12).

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