- Email: [email protected]
A study of collective motion
I 1.D.2
]
Nuclear Physics A l l 4 (1968) 289--308; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
A STUDY OF COLLECTIVE M O T I O N (I). Rigid, liquid and related rotations R. Y. CUSSON
lnstitut fiir Theoretische Physik, University of Heidelberg, Germany Received 4 September 1967 Abstract: Models of nuclear collective rotations are defined by giving the form of the velocity field versus position. Only fields which are linear in the particle coordinates are considered. The canonical formalism is developed in the presence of the constraints represented by the field specification. The usual quantization rules are then applied to form the quantal collective Hamiltonian. The rigid-body problem is first treated. The liquid rotator Hamiltonian is obtained, starting from a vorticity free field, for the quadrupole mode only but at arbitrary deformation. The quantal problem corresponding to a field with arbitrary amounts of rigid and liquid components is shown to correspond to liquid motion with quantized vorticity as in nonlinear quantum superfluids and is treated using the components of the complete moment of momentum tensor which generate rotations and shear deformations. Connections with the algebra of SL(3, R) are pointed out. Fields where the ratio of liquid to rigid components are given, before quantizing, with the help of an ansatz connecting this ratio with the pairing strength, yield values of the moment of inertia which are intermediate between the rigid and liquid values. Qualitative agreement with the experimental situation in heavy nuclei is obtained for this last case.
1. Introduction Considerable progress both in atomic and nuclear physics has been made over the last decade in obtaining realistic solutions of the many-body Schr~Sdinger equation in the presence of mutual two-body interactions between the particles. As a consequence, one shows, in the context of the unified nuclear model for example, that collective effects can be obtained from the microscopic description 1). Since, however, the coupling between the collective modes and the particle modes is rather strong in the nuclear physics applications, these modes are often mixed in a given set of states. This has made it difficult to describe precisely just which collective modes do occur. This paper will be concerned with a somewhat new approach to the description of collective modes, in which the classical flow pattern is considered as primary data from which the quantum dynamics is to be constructed in the usual manner. But before we elaborate further on the possibilities of such a method and on its connection with the microscopic description, it is instructive to review briefly the development of the relevant ideas concerning collective motions. Fission phenomena and the existence of a fairly well-defined nuclear surface led early 2) to a model where the nucleus has the dynamics of a liquid drop. That the predictions of such a model are in accord with the exclusion principle requirements 289
290
R. Y. CUSSON
discussed by Teller and Wheeler 3), was pointed out by Bohr and Mottelson 4), inasmuch as a properly symmetrised or antisymmetrised system cannot physically be rotated about one of its symmetry axis. This property is characteristic of the liquid drop whose effective moment of inertia about any one of its symmetry axis vanishes. Experimentally it was found that the moment of inertia was several times higher than the liquid-drop value 5) and about one half of the rigid value. This could be explained using the semiclassical cranking model of Inglis 6) in the presence of residual attractive short-range two-body forces. Belyaev 7) showed that good fits to the experimental data could be obtained in this way. It was also shown that axial symmetry of the system was not always realised 8). Finally various improved treatments 9-~1) in the context of the Hartree-Fock theory have appeared recently and further explore the mechanism which generates collective motion from the microscopic dynamics. One outcome of these developments seems to be that the collective rotational motion is intermediate between pure rigid motion and pure liquid motion as exemplified by the moments of inertia. Since this particular type of collective dynamics is the end result of a quantal calculation, it is not clear however that it is obtainable as the quantized version of a classical model, for it is neither that of the rigid top nor that of a rotating drop. What we shall do is to consider models where the total velocity field is the sum of two components, a rigid component whose vorticity is proportional to an angular velocity and a vortex-free (liquid) component. By fixing the ratios of these two components before the quantization of the system is performed, a new class of rotators is obtained, whose moments of inertia can take any values intermediate between the rigid and liquid case depending on these given ratios. A specific dependence of these ratios on the pair correlation length, and the nuclear radius is proposed in order to obtain a connection with the microscopic properties and to remove the arbitrariness of these new adjustable parameters. An alternative scheme of quantization is also developed in which no attempt is made to fix the ratios of the two velocity fields at the beginning. The amount of vorticity then effectively becomes a parameter subject to quantization. The result of this scheme is that a generalisation of the liquid drop model is obtained where the vorticity instead of being strictly vanishing is quantized in discrete units which contribute to an effective "vortex angular momentum" as in the theory of quantum fluids. It is proposed that the absence of such vortices in finite nuclei is due to the fact they are too large, on account of the weakness of the pairing force, to fit inside one nuclear radius. In the second part of this study it will be shown that quadrupole vibrations can be included in the present formalism. The techniques used to calry out this program are those of the classical canonical formalism supplemented by the traditional rules of quantization. Yet there is a rather complex group theoretical context associated with our approach. Because of its complexity, the group aspect will not be discussed in detail, but since we first discovered the algebraic manipulations presented here through its use, we list some of the main ideas that are involved. To begin with there is the apparently isolated fact
COLLECTIVE MOTION (I)
291
that the algebra U(3) of the harmonic oscillator is useful in discussing the rotational structure of nuclei 12). However, it was shown by Lipkin and Goshen 13) that the Elliot algebra is part of a larger one, that of the linear canonical transformations SP(6) (Bogolyubov transformations for a spin 1 boson field, the coordinate), and that this larger, non-compact algebra, did predict rotations and vibrations at least in its two-dimensional form. This appears natural since SP(6) also contains the algebra GL(3, R) of the rotations and deformations. The non-compact algebra GL(3, R) has unitary representations consisting of infinite rotational bands 14). The operators which generate this algebra rotate and distort the coordinate system. The collective models where the particles are at rest in such rotating and distorting coordinate system are the ones which will be considered in this study. Thus it appears that the Elliot model, the rigid top and the linear liquid drop can all be treated as phenomena related to linear canonical transformations of the position and momentum variables. We begin by discussing the rigid body (sect. 2) and the liquid drop (sect. 3) in some detail, mostly in order to introduce the notation and the techniques. Although these sections contain few new results, they serve as convenient reviews of these two wellknown models, and their presence greatly facilitates the understanding of the following ones. Sect. 4 develops the alternative schemes of quantization mentioned above and outlines the method of solving the resulting Schr~dinger equations. The discussion (sect. 5) brings out the physical meaning of the equations obtained in sect. 4 and proposes a connection with the microscopic description. A later publication will include the possibility of vibrations since this involves an appreciable amount of new algebraic manipulations and does not greatly affect our conclusions.
2. The rigid body Our starting point is the classical Lagrangian kinetic energy of a collection of N (large) point particles with mass m N
T = Z ½mlr(")(t)l 2,
(1)
n=l
where J:(")(t) is the laboratory velocity of the nth particle. We assume throughout that the centre of mass is at rest in the laboratory. The condition that the distance between any two points in the body be independent of time leads to the condition on the velocity field ~(")(t) F(")(t) = to × r(")(t) = •(t)" r(")(t),
n = 1. . . . . N,
(2)
where to(t) is an arbitrary vector, the angular velocity and is independent of the particle label n and 12(t) the antisymmetric matrix equivalent to to x. Eq. (2) thus defines the flow pattern for rigid motion and is l i n e a r in the coordinates of the particles. The two equations above define the dynamics of rigid motion as a many-body problem with constraints. The first step is to deal with the constraints.
292
R . Y. C U S S O N
To do this we rewrite eq. (2) as constraint equations expressed as vanishing differentials 3
dC~")(r ("), t) = 0 = drl " ) - Z g2ij(t)r~ n)dt, j=l
i = 1, 2, 3,
n = 1. . . . . N,
(3)
The 3N constraints represented by (3) will be holonomic if we can find some integration factors and/or changes of variables such that they become exact differentials in the 3 N + 1 variables. This we do as follows: let R~k(t ) be a real 3 x 3 orthogonal matrix depending on the time only and consider equivalent constraint equations dCt,(") = Z l~k,dr~")+ Z r}")QJiRik dt = 0, i
i,j
k = 1, 2, 3, n -- 1,...,
N.
It is easily verified that the differentials will be exact if we put f~ = R . R -1 '
(4)
and that a first integral is then simply
C'k(") = Z R~ X(t)r~n)(t) = rfk"O )"
(5)
i
Condition (4) is possible when R is orthogonal as can be seen by taking the time derivative of R. R. Condition (5) states that the position is independent of time in the rotating coordinate system. This system is defined by giving the relation between the components V(s)i of an arbitrary vector V in the old, that is, lab or space system and its components V(b)j in the new, namely, rotating or body system as V(s), = Z Rij(t)V(b)J" J
(6)
,.(n) By comparing (5) and (6) we see that the constants "k0 are the body components of the position vectors. The integrated constraint condition (6) may be substituted in the Lagrangian (1) to yield in an obvious matrix notation
T = ½ Xr JR" G(b) • ~] = ½ Vr J R - ' . / ~ - G(b) • ( R - I -
~)],
(7)
where the mass quadrupole moment in the body coordinate system is N
= •
H~jO
~kO "
(8)
n=l
Since G(b ) is constant in time, it can be diagonalised by performing a time-independent rotation in the body system. We let its diagonal components in this new body system be Goi, i = 1, 2, 3. We see that the constraints have effectively eliminated all the particle variables from eq. (7) and that the Lagrangian is now expressed in terms of
COLLECTIVE MOTION (I)
293
new (collective) variables R-a (t)./~(t). Before we proceed with the solution of this system, it will be useful to develop some notation regarding the parametrisation of the mass quad~upole moment. We follow a procedure similar to that used in discussing the ellipsoid of inertia ~5), namely, we define the mass quadrupole ellipsoid in a three-dimensional space with coordinates p~, P2, P3, using the equation p2 + p~2 2 + p 32 = 1. Go1
Go2
(9)
Go3
We introduce new parameters G 0, t 1, t2, t3, such that t l + t 2 + t 3 = O.
Got = Go e2t',
(10)
All ellipsoids which are characterised by the same value of G o enclose the same volume and thus look like a sphere in some coordinate system which is linearly distorted at constant volume. This is seen by writing Pi
=
et'p~,
so that (9) becomes 3
Z
= Go,
i=1
which is the equation of a sphere in the system p'. From the definition of Go and eq. (10), we have M
Go = ~ e -2ti~[-~(n)']2 ' ~ ' L ' i O _l
i = 1, 2, 3.
n=l
Corresponding to the p' system, we can define a new distorted body coordinate sysp tem r o so that rtn) t. ,(n) iO = e 'rio ,
i = 1, 2, 3,
N
GO = Z
m krr l ,(n)-I o d 2,
i = 1, 2, 3.
n=l
In short we see that the r~ system is such that, in it, the mass quadrupole tensor is proportional to the identity matrix. The transformation to this particular system will be seen to enter explicitly in the formalism of the liquid drop to be discussed in sect. 3. N o w we deal explicitly with the constant volume condition ~itl = 0 by introducing two new parameters s~, s2, forming a vector s in a two-dimensional plane. We also define three constant vectors z~, z2, z3, in this plane, as in fig. 1, and we let t k = Z k ' S = Z k ~ S l + Z k 2 S 2 , k = 1, 2, 3. Because ~ k Z k = 0, the constant volume condition is fulfilled. This particular way of realising the condition is chosen for reasons of group theory related to the definition of the matrices II 4 and fls of table 1. Corresponding to eq. (10), we may express the moments of inertia of the system as 3
Iok = [ E j=l
Goj]-Gok
= 2Goe-'k'Scosh(flk'S),
k = 1,2,3,
(11)
294
R.Y. CUSSON
where the vectors flk are also defined in fig. 1. If we now represent the vector s in polar coordinates by giving its length a and the angle y, it forms with the positive s 2 axis as shown in fig. 1, we find easily that
,7 ~-o" c o s ( ~ - ~ k r c ) , //k" s = 2a sin ( ~ - aZkn),
• k" s =
k = 1, 2, 3.
(12)
1-~=(k- ~-3) S2
-TI ~
V
~
-1-2
-f13 •
b
P SI
1-2
l~l = 1-2- 1-5 ,8 2 = -t-5 - l- I
fil
-
2
fis =rl-1-2
Fig. 1. T h e vectors [3i a n d ~ . T h e Euclidian c o o r d i n a t e s o f this d e f o r m a t i o n s plane are st, s2 a n d its p o l a r c o o r d i n a t e s a a n d ~. F o r small a, this p l a n e is the usual/3, ~, plane o f the r o t a t i o n a l v i b r a t i o n a l m o d e l w h e r e fl = ~/(16~/15)a. TABLE 1
A basis for the eight traceless 3 × 3 real matrices
111=
[io!] 0 --
,
112=
1
116 =
E! o 0 1
E i oil
,
E o o !1
,
0
--
1
,
11~ =
113 =
0
0 1
0 0
1l 8 =
[O_,ooO] 1
0
0
0
E° 1 0
'
0 0
0 ,
0
114=
E,o!] E-i o i] 0 --1 0
,
115 =
0
--1 0
Finally if we define 16 fl = x/~-rco" ~ 1.83a,
(13)
the quantities fl and ~ will coincide for small fl with those defined by Bohr 16). We shall come back to this in sect. 3. There are several ways of obtaining the quantal Hamiltonian from the Lagrangian (7). For example we can take the Lagrangian to be a function of the Euler angles 0 z which define thc orthogonal matrix R and their time derivatives Oz. The general form of (7) is then 3
T = Z 2., ~t= 1
9a~,(o¢)OzO~,"
(14)
COLLECTIVE MOTION (I)
295
In order to define the Hamiltonian and quantize this system, we should use the prescription of Pauli 17) because eq. (14) corresponds to a curvilinear coordinate system. However, we can avoid this complication and gain additional physical insight by considering the body angular velocities as being the 0-values of the system. Using the defining eq. (A. 16) of the appendix for the body angular velocity, we can rewrite (7) as 3
T = +Z
(15)
I=l
where ~J
~(O~b) - - J(b)k :
k=l
_
k
(16)
lOk CO(b),
k=l Iok
1 -~ J~b)ke'~'~ 4G o k= 1 c o s h fig" $
(17)
Eq. (17) is the familiar expression of classical mechanics for the rigid-body Hamiltonian H
Since liquid motion is much more complex than rigid motion, we shall resort to a simple model instead of trying to solve the complete problem. This model we call a vortex-free linear rotator or liquid rotator for reasons which will soon become apparent. The Hamiltonian for this model will be shown to yield for small deformations, the Bohr Hamiltonian for liquid rotations. The model constitutes in fact a special case of general liquid motion. We replace eq. (2) by the following specification of the velocity field: g")(t) = ~ ( t ) .
r<")(t),
qgij = ~pj,,
n = 1.....
N,
(18)
where q~ij(t) is now an arbitrary s y m m e t r i c t r a c e l e s s tensor and plays a role similar to the angular velocity of the rigid case, which was represented by an antisymmetric
296
R . Y . CUSSON
tensor. In the limit where the number of mass elements N goes to infinity, eq. (18) becomes equivalent to 3
~i(r, t) = E
tPiJ(t)rj(t),
(19)
j=l
from which it is easily verified that
Or-~O#j(r, t) = ,,JZe k,j ~cr, i'j(r, t) = 0,
k = 1, 2, 3.
(20)
These last equations state that the flow represented by (19) is both incompressible and irrotational (vanishing vorticity). The reader is referred to the discussion of Bohr 19) for some more details concerning the derivation and validity of eqs. (18) and (19) in nuclear physics. Here we consider eqs. (20) to be sufficient justification for our use of this type of flow pattern. The flow (18) is also of special interest because (i) it is linear in the particle coordinates, (ii) it represents the lowest multipole moment (quadrupole) in the expansion of the complete velocity potential, and (iii) it is complementary to rigid motion within the linear group of rotations and distortions SL(3, R), in the sense that a body which is being linearly rotated and deformed at constant volume has a velocity field (internal mass current) which is the sum of two terms, one of the form (2) and the other of the form (I 8) (iv) the resulting dynamical problem is completely soluble. We shall consider arbitrary values of the deformation parameter ft. But when the deformation parameter is large, octupole and higher-order terms should be added to eq. (19), for their contribution is no longer negligible. Gustafson 2 o) has considered this effect in connection with fission, whereas we shall neglect it here because of the non-linearity involved. Thus our treatment is in fact more suited to the case of small deformations. We now repeat the steps of the second paragraph of sect. 2 replacing ~2 by ~ and R by a real 3 x 3 matrix with unit determinant Nl(t). An integral of the constraint differentials
M-~(t) • r(")(t) = r(o"),
n = 1,..., N,
(21)
is obtained if we impose the condition X = M" NI-I = ~ .
(22)
Eq. (22) can again be viewed as a differential constraint, but on NI this time. This new constraint on the collective variables contained in M would be holonomic if we could integrate it to obtain the explicit form of M. In fact we shall show presently that this is not possible and that the constraint (22) is non-holonomic. For the matrix NI, we use the convenient general parametrization 21) NI = RI(0~,). So(S) • k2(0a2),
(23)
where Ra and R2 are two orthogonal matrices defined by two independent sets of
COLLECTIVE MOTION (I)
297
Euler angles 0~1, 042; 21, 2 2 = 1, 2, 3, and So a diagonal matrix with elements
Sou = 3kt e"k" 5,
k, I = 1, 2, 3.
(24)
We have used the notation Vk" S introduced in sect. 2 in order to make the determinant ofS o equal to unity. Eqs. (21) and (23) imply that the particles are at rest in a coordinate system which is reached by three successive transformations. The components of some vector V can now be given in any one of four systems. I f we let these four sets of components be labelled V(s)i, Vi', V i'', V(b)i, we have
V~s)i = Z Ri,jVj,
V/ = Z Soi~Vj',
J
J
V i " = Z~2,jV~b),, J
V~s)i= ZM,2V~b)j. J
(25)
The space to body transformation now involves eight parameters which however, cannot all be arbitrary since t# in eq. (22) has at the most five independent parameters. To interpret the three transformations defined above, we look at the mass quadrupole tensor components in the various systems. We have
Cat =
G(s) = R 1 • G'" R1,
SO • G t t , SO '
G t t ~--- h E . ~ ( b ) "
R2-
(26)
The tensor G~b) is independent of time because the particles are at rest in the body system. By performing a time-independent rotation and deformation in the body system, we can insure that G(b) is proportional to the identity matrix in this body system, or equivalently we suppose that the body system has been cleverly chosen so that G(b) already has that property. This is clearly always possible. Thus G" is also equal to G o • I, by rotational invariance, for any R2. Again from (26) we find
G'kt = 6kt Goe2"k's,
k, l = 1, 2, 3,
which is of course precisely eq. (10). Since we are not interested in vibrations we choose s to be a constant, independent of time. Finally R1 transforms G' to the lab system. Thus So and R~ work as in sect. l, and they represent the constant shape and variable orientation of the mass quadrupole ellipsoid. Note, however, that the deformation S o now enters explicitly in the passage from space to body. Fortunately enough R2 still remains at our disposal, so we shall use it to fulfill the condition (22). Using (23) we compute X~) and X' X(~) = M ' M - 1
= RI" [/~1 " R I + S O " ~2" R2 "S01] " R1 = R1 " X " R1 •
(27)
We can expand X' using the basis of table 1 for the eight traceless 3 x 3 matrices and the definition of the appendix for the angular velocity which reads, in the notation of this section 3
3
k=l
k=l
0.) 2 ~'Lk .
This simple algebra yields 3
X
' =
Z k=l
{f~k[CO~ ,k --cosh (fig . $)(Dt2tk]-~-~'~k+5 sinh (ilk" $)('Ot2tk) •
(28)
298
1,,. Y. CUSSON
Thus ×' will be symmetric if we put to~,k =
1 eo~k cosh/~k" s
(liquid),
(29)
and as a matter of interest it will be antisymmetric if a~zt t k = 0
(rigid).
(30)
These last two equations are of considerable interest in that they show how one obtains either liquid or rigid motion from the same general situation, by imposing suitable constraints on the collective variables. In either case, M has only three remaining degrees of freedom, but the two types of dynamics are quite different. The rigid constraint can be satisfied with Rz = constant, but the liquid constraint is non-holonomic because the angular velocities involved in (29) are not integrable, as pointed out in sect. 2. The need for a time-dependent Rz in the case of liquid rotations is now easily understood. The first rotation R~ takes us to a rotating frame where the mass quadrupole moment is at rest (no vibrations!). If the particles were at rest in this system as in the rigid case, the vorticity would be infinite. To cancel out the vorticity implied by the rotation of the mass quadrupole tensor, one must first distort the coordinate system until the mass quadrupole ellipsoid becomes spherical and then, in this deformed system, perform a further rotation in the opposite direction from the original t t¢ one. For a spherical drop s vanishes and ¢91 = to z so that NI = R1-R1 = l, and the drop is at rest in the lab system. This is the well-known statement that a spherical liquid drop cannot have rotational excitations. The successive transformations involved in (23) are illustrated qualitatively in fig. 2 of Bohr's paper 29). We m a y now proceed to develop the canonical formalism in the presence of the non-holonomic constraints (29). To do this we first ignore the constraint completely in order to define the canonical m o m e n t a and the Hamiltonian H(g) of the complete system with its 6 degrees of freedom. This Hamiltonian will be a function of the six canonical m o m e n t a only. Then we shall express the angular velocity constraints (29) as constraints on the momenta. The non-independent momenta will then be eliminated from H(g) to give the Hamiltonian of the constrained system. This procedure works equally well for the liquid or rigid case (29) or (30), but we shall carry it out only for the liquid case, having already obtained the rigid-body Hamiltonian in sect. 2. This method has the advantage of supplying us with H(g) which we shall need in sect. 4. We substitute X(s) for ~ in eq. (18) and use the result in eq. (1) to obtain the Lagrangian as a function of X(s). Using (27) to relate X~) and X', we obtain 3
T = ½Go Z [X~k]2ea'~'s = ½Tr [ X ' ' G ' - ~']. i.k=l
(31)
COLLECTIVE MOTION (I)
299
The diagonal elements of X' vanish according to (28). We consider the six remaining independent elements to be the canonical velocities of the system. The three antisymmetric components of X' are the angular velocities of the pure rotation part and the three symmetric ones are the rates of shear deformation. This follows from the definition of X(s) as ~ • M-1, eq. (27); G' is a constant of the motion. The canonical momenta are as usual
,
aT
NiJ
i ~ j, i,j = 1, 2, 3,
~Zji
(32)
except perhaps for the transposition of the matrix indices. This is required here because the group SL(3, R), which the quantum operators corresponding to N' will generate, is non-compact. Therefore the metric in the algebra of its generators is not the identity. It can be proved that this implies (32). From (31) and (32) we obtain N
Nij = Z Gi'kZjk = Z ri("~P)("), k
(33)
n=l
showing that the canonical momenta are the components of the moments of momentum of the system in the primed system. The Hamiltonian is H(g)
=
Tr [tO'. N ' ] - T = -}2Tr [/~/' • G' - 1 " N'].
(34)
As expected H(g) is a bilinear function of the six independent moments of momenta N;j. We may now incorporate the constraints (29) by substituting (29) in (28) and the result in (33) to obtain the constrained expression for N;j, N[L)ij 3
N(L)ij '
=
2 k=l
Go e2"''s
tgh
(ilk • s) 2
jkcoi.,k
(35)
This tensor has but three independent components, which we take to be its antisymmetric components labelled J~k" We find 3
J'lk = Z ekijN(L)ij = lOk tghE (ilk" $)(Olk. i,j=l
(36)
Next we use (36) to eliminate ~ k from (35). This gives 3
e(,~i - Tj) • s
' = ±z y ' N(L)ij
k'~1 sinh//k" S
2
I
~ijkJlk •
(37)
Note that the symmetric components of N~b ) a r e non-vanishing so that when we substitute (37) in (34) to obtain H(L) they will contribute to the Hamiltonian. Doing this yields 3 = ½Z
k = 1 I0k
1
tgh 2 l~k" s
,2 Jlk"
(38)
300
R.Y.
CUSSON
The reader may repeat steps (35) through (38) using (30) instead of (29) and check that eq. (17) for H(r ) is indeed obtained with the notation J~k instead of J(b)k for the total angular momentum in the primed system. The liquid rotator Hamiltonian is seen to differ from the rigid one only by the presence of the additional factor tgh2flk • s in the denominator. This factor vanishes at sphericity and when the system is axial along the k'th axis. The quantity Iok tgh2flk • S may be loosely referred to as the liquid moment of inertia. It is always smaller than the rigid moment of inertia. It now remains to show how to obtain the Bohr expression s) for the liquid rotator from eq. (38). We suppose that we have a drop with uniform density up to its surface which is assumed ellipsoidal. It can be shown 19) that this is consistent with the constraint (18). Because of the constant density and ellipsoidal surface, the parameter s which characterises the mass quadrupole ellipsoid also describes the semi-axis R~, of the surface as R;, = Ro Ck's. For small deformations of the surface, we expand the exponential and use (12) and (13) to get
R~ ~- Ro [I+flV-~ c o s ( , - ~ k ) l ,
k=1,2,3,
(39)
which is the usual expression for the semi-axis. Letting the drop density be p, we 1 5 compute G O to be 8B/15rq where B is ~pR o. Upon expansion of eq. (38) to lowest order in fl and substitution of the above value for G o, one obtains
1 H(L) ~
Jg
8Bfl 2 k= 1 sin 2 (y--a~xk) "
(40)
This last expression has been used by Davydov 8) to compute the energy levels of a non-axial liquid drop. 4. The intermediate models
We have seen in the previous section how a system possessing both pure rotational and shear rotational degrees of freedom can yield either pure rigid or liquid motion when suitable constraints are imposed on the collective angular velocities. In this section we shall generalize these results in two ways. We shall derive the Hamiltonian for the case where the velocity field consists of a fixed mixture of rigid and liquid components and in the more general case where both components of this field are independent. It will be more convenient to study the second case first. Eq. (18) holds in the space system for the liquid case. We replace ~ by ×(s) and transform to the primed system. This gives the following equations for the primed -(n) ~'¢") of the general velocity field .(g) components -(g)i 3
f,(,,) (g)i
~--- ~,(,,) " ( r ) i ±,v(.) ~ ~(L)/
=
Y' Z.a j=l
Z i 'j
f'(") j
~
n
=
1~ . . .~ N .
(41)
COLLECTIVE MOTION (I)
301
Using eq. (28) for X' we write 3
~'(") ' " r'("), [r) = Z nk(oYlk - c ° s h (//k" s)c°~'~) " r'(") = X(r) k=l 3 i'(") (e) = Z
sinh
( i l k " S J, ( D 2t tkt'~ aLk+5
• r ' ( n ) = X(L) • r'(n),
(42)
k=l
where o'l and o[' are assumed independent. As we have seen the field (41) leads to the general Hamiltonian H(g) given by (34). We must now express H(g) in a form which is easy to quantize. The simplest way to do this is to express N' as a function of angular momentum variables. To do this we use the fact that the moment of momentum tensor N can be expressed in terms of its component Ni~ in the primed system and its components N~ '~ in the double-primed system. Since the metric in the doubleprimed system is not unity due to the stretch involved in 30, covariant and contravariant indices are to be distinguished in that case. Actually, N is a mixed tensor because it must transform a contravariant vector into another such vector. Some simple algebra yields the rule N" = 5o 1" N ' - 5 . (43) The antisymmetric components of N' and N " are angular momenta in their respective systems. Using (28), (33), and (43) we find
Jig
~ ekdNij
Iok 0)~
ii
Jzk = ~ ek~pNa = Iok .
.
.
0~2 cosh (ilk" s
.
.
(44)
cosh{
These two angular momenta are independent as long as o~'1 and o~[' are independent. Eq. (44) may be inverted to give the e) as functions of the J. This result is used in (28), (33) and (34) to give
H(g) =
1
LOlk~OZk
½ ~ k= 1 lokB k
]
cosh (fig" s)d
,
(45)
where B 2 = tgh2(flk • s). This expresses H(g) in terms of the primed total angular momentum of the system and a new angular momentum J~. It takes into account the fact that the vorticity is finite. For example when we substitute the liquid constraint (29) in (44), we find that J;' vanishes identically. We shall call J~' the "vortex angular m o m e n t u m " of the system, since a comparison of (42) and (44) shows that it is proportional to the finite vorticity component of the matrix X' which specifies the flow pattern. Inserting (29) or (30) in (44) the yield relations between the two J in those cases J~, = 0 (liquid),
J'2'k =
1
cosh (ilk" S)
J'xk
(rigid).
(46)
302
R. Y. CUSSON
For a rigid system with a spherical mass quadrupole, the vortex angular momentum coincides with the total angular momentum. When the rigid body is axial, only the components along the symmetry axis will coincide. In order to quantize H ( g ) w e must give the operator form of J'l and J~'. From their definition (44) we see that these operators are the generators of infinitesimal rotations in the primed and doubleprimed systems. These may be obtained by repeating the steps of the appendix using /d instead of R to find J(s)l and J(b)2 which are related to the present ones by J(s)l = R~ "J'l,
J(b)2 =
R2" J;'.
(47)
The result is that J;k(O~,, ((3/00~)) has the form of J(b)k(O~,((?/(?Oal) ) given in eq. (A.17), and that J;'k(O;.2((?/t?Ox2)) is minus J(b)k(Oa2, @/00~2)). In particular J i is given in terms of the first set of Euler angles and J'2' in terms of the second set only. We may diagonalize Jlz, J2z, IJll 2, IJ212 since they commute with H(g). The eigenstates of H(g) have the form T I , ~2 to~t,, K~(J1 , JE)Dj,M,, r, (0a,)Djg2, -J,, J2~t2, ~,, ~ra t"a,, 042) = ~ aK~, 2 r2 (0k2),
(48)
KI, K2
with aK~,K~(J~, ~,~2 J2) acting as eigenvectors. The matrix elements of H(g) o n the evensymmetric s t a t e s 2 2 ) form a tri-diagonal matrix which must be diagonalized numerically, even when the system is axial, to give the a in (48) and the energy eigenvalues as a function of J~, J 2 , ~'1 and z2. The subset of eigenvalues with J 2 = 0, "C2 = 0 reproduce the level spectrum of the liquid rotator which was computed by Davydov 8), while the J 2 = l , 2 . . . . eigenvalues correspond to liquid motion with 1, 2 , . . . quantas of vorticity. These levels are characteristic of a quantum liquid only. The energy spectrum of H(g) bears no resemblance to the rigid-body spectrum when the distortion is large. This is because the rigid constraint (46) cannot be realised when both J'~ and J~' are quantal operators, except if s = 0. We now see that if we want a level sequence which can approximate either the rigid or liquid sequence we must specify the ratio of the two velocity field in (41) before we apply the quantization rules. To do this we shall write an intermediate constraint condition
CO,2,k _
Ak
e),k,
0 < A k < 1,
k = 1, 2, 3,
(49)
cosh/I k • s where the A are numerical constants to be determined a priori as some fundamental property of the body under study. We insert (49) in (42) to get 3
3
k=l
k=l
from which we define the ratios of rigid to liquid field ,ok as the ratios of the components of Ilg in Xlr) to I/k+ 5 in X(L)
Pk-
1--Ak
A~lB~l
(51)
COLLECTIVE MOTION (l)
303
This shows that the intermediate constraint (49) is equivalent to a specification of Pk. Eq. (49) is now substituted in (44) to yield the new relations between the J as ,,
J2k
=
(1--Ak) ch fig" s , Jlk" ch2 (Pk " s ) - A k
(52)
This momentum constraint can be substituted in (45) to give the intermediate Hamiltonian H(i )
H(i)
3
=
½2
1
,2
(53)
- Jlk, k = i lo~f~
where ,_
[P~ + IBkl] z
J~' -
1 +2pklBkl
+P#
"
The intermediate Hamiltonian differs from H(L ) or H(r ) only in the magnitude of its effective moment of inertia Iokfk. The fraction fk(Pk, [Bkl) varies smoothly from B z to 1 when Pk goes from zero to infinity. Thus by selecting some ratio of rigid to liquid flow, we can achieve any value of the moment of inertia between the liquid and rigid value. For example if Bk is small and Pk is 1, fk is about ½.
5. D i s c u s s i o n
Here we shall present some physical considerations in connecting with the applications of the various models of collective rotations studied in this paper. Since the physical properties of systems undergoing pure rigid or liquid motion are well known, we shall concentrate our discussion on the properties of the models presented in the last section and refer to the other ones as the need arises. It is easiest to study the system represented by the general Hamiltonian (45) first, because it involves no new adjustable parameters. Its energy eigenvalues are labelled by J1, J2, zl, Zz, where Jl is the total angular momentum quantum number, as implied by eq. (47), and -/2 is the vortex angular momentum quantum number. We can express J2 in a way which exhibits clearly its connection with the vorticity by writing it in the following form: "
Iok
J 2k - -
'
fD( v)k ,
(54)
cosh ilk" S with ! =
i, j, l
' 21 ~ k i j ~ t Z j'l Fl" 01" i
(55)
These equations give (44) when eqs. (42) for X' are used. Thus J~' is proportional to the curl of the velocity field which is itself the vortex strength and the quantization of J~' is equivalent to the quantization of the vorticity. This may be compared with the treatment of quantized vortices e 3) in the superfluid 4He. Using Stokes' theorem,
304
R . Y . CUSSON
one replaces the vanishing vorticity condition V x ~: = 0 by the contour integral m ~ i . dl = 0.
(56)
Since this has the units of an angular momentum one allows the values 27c/, l = 0, 1. . . . for the contour integral, as a quantal generalization of liquid motion. Vortices with l = 1 are supposed to be responsible for the inertial properties of rotating liquid helium. If we consider a drop whose diameter is of the order of the diameter of a vortex tube, the linearized expression (42) for the velocity field might hold so that H(g) could represent the collective motion of this drop. This suggests that H(g) should be applied to a system which has undergone Bose condensation. This could happen in a nucleus if the pairing force were sufficiently strong. A measure of the pairing strength is the pair correlation length 24) L = 2el .
(57)
kiA If Lp were much smaller than the nuclear diameter one would expect a true quantum fluid behaviour and the level spectrum of H(g). Actually Lp is somewhat larger than the nuclear diameter in most cases, indicating again that H(g) is not a suitable Hamiltonian to describe nuclear collective rotations. Insofar as the general Hamiltonian H(g) is nothing more than the complete Hamiltonian for a linear quantum liquid including vorticity quantization, it might not come as a great surprise that is it inadequate for the description of the collective rotations of finite deformed nuclei since it is well known that nuclear moments of inertia are much larger than the liquid values. Yet physical intuition strongly suggests liquid motion, with its vanishing moment of inertia at sphericity. This dilemma must find its resolution upon detailed consideration of the microscopic dynamics of nuclear motion. We may suppose that the nucleons interact (i) with the average distorted onebody potential which is often taken to be a distorted harmonic oscillator well and (ii) with each other via a weaker, short-range attractive two-body force, which is usually represented by the pairing interaction. Neglecting the pairing force, it can be shown that if the distorted well is forced to rotate slowly, the mass quadrupole ellipsoid will follow 6). Thus the well angular velocity is to be identified with the angular velocity to'1. One then shows that the velocity field in the primed system must vanish on account of the weakness of the Coriolis force 5), so that we have to;' = 0. This means that the motion is rigid. In the presence of pairing, the primed-system velocity field has not been estimated but it has been shown that the moment of inertia is decreased and that in the limit of strong pairing the moment of inertia attains the liquid value : 5). This effect is consistent with our previous considerations involving the pair correlation length, and may be reproduced by using the following ansatz: Lp Pk oc - -
Ro
(58)
COLLECTIVE MOTION (I)
305
where R 0 is the average nuclear radius, and Pk and Lp have been defined above. It states that the ratio of rigid to liquid components of the velocity fields should be proportional to the parameter ~:/fl used by Migdal 25). When (58) is inserted in the expressions (53) for H(1), the various limits of the moment of inertia computed by Migdal 25) are obtained. Assuming equality instead of mere proportionality in (58) and taking Lp = R o as a reasonable estimate, yields about one half of the rigid value for the moment of inertia. These approximations are consistent with the experimental situation in heavy nuclei. These considerations allow us to form the following picture of collective nuclear rotations. Starting with weak residual forces and the average distorted field, the motions tends to be rigid. As the attractive pairing force strength is increased the boson pair density becomes better defined and the velocity fields goes over from rigid to fluid. In the limit of strong pairing H(i ) goes over to H(L) and we have a superfluid of bosons as in liquid 4He. Then H(g) may be substituted for H(L) as a quantization of the vortex motion of this superfluid. One can only speculate that the new collective degrees of freedom involved in vortex quantization are to be understood as rotational motion of the pair correlation field if not rotational motion of the particles themselves. We have not been able to derive a criterion for the onset of vortex quantization as the pairing force is increased. Such a "phase transition" probably does not take place in finite nuclei due to the rather large values ofpk. The transition might occur in infinite nuclear matter where R o ~ oo but there our linear approximation (422) would not hold. We are grateful to Professor J. H. D. Jensen for his kind hospitality at the Institute for Theoretical Physics where much of this work was conducted. We are also happy to acknowledge the support of an O.N.R. grant at the Kellogg Radiation Laboratory of the California Institute of Technology during the early stages of this work. We have benefited greatly from conversations with Professor M. Gell-Mann, Professor H. A. Weidenmtiller, Dr. K. Hara and J. Hiifner.
Appendix T H E A N G U L A R M O M E N T U M O P E R A T O R S F O R A R I G I D BODY"
Corresponding to a rotation of the coordinate system there is a unitary transformation which acts on the wave function and which is generated by the angular momentum operators. We shall first define the usual angular momentum operators for a point particle and generalize the definition to the case of a rigid body. We may consider the coordinate system as a rigid object defined by the location of its three unit vectors e ("), n = 1, 2, 3, each having components e~"), i ---- 1, 2, 3, in the lab system. We suppose that the vectors e (") depend on some (time) parameter such that 3
e~")(t) = ~, Ro(t)e~")(O), j=l
(A.1)
306
R . Y . CUSSON
where Re(O ) = t5e and R(t) is an orthogonal matrix having an analytic dependence on t. We say that e~")(0) are the lab coordinates of the old coordinate system unit vectors and e~n)(t) those of the new unit vectors. Let ri be the components of some vector in the old system, e.g. before R is applied, and r/' those in the new system, namely 3
3
r = Erie(°(O) = Z r~e(°(t) • i=l
(A.2)
/=1
Eqs. (A.1) and (A.2) imply the relation
r, = ~ Ri~r;. J
(A.3)
Now let $(ri) be a scalar function of the old coordinates r i of a point particle. We have
$(r~) = ~b(~, g,jr)) = $'(r~,),
(A.4)
J
which is the definition of a new function of r~, $'(r~). We assume that this new function can be obtained by a unitary transformation S acting on the old function of the new variables ~b(r~) that is $'(r~,) = S$(r~,).
(A.5)
Thus we have the equivalent properties
d/( E R,j r)) = S~(r'k), J ~b( ~ g ~ lrj) = S - l $ ( r k ) .
(g.6)
J
For an infinitesimal R and S, we write 3
S = 1 +i~
3
e,klk,
R,j = ~5i~+ ~ ek[i'~di~,
k=l
(g.7)
k=l
where the e k a r e infinitesimal, the matrix f~k are given in table 1 and lk are the generators of infinitesimal rotations. We insert (A.7) in (A.6), expand and compare terms of order e k to get
Ik = Tr [flk" t],
tu = rtpj' 3
PJ -- i r~rj' (A.8)
Ik = ~ e~ljrtPj • i,j=l
A derivation of the generators of the unimodular group SL(3, R) would proceed along similar lines. In the case of a rigid body, the coordinate manifold is R~j itself instead of r~. Since the coordinate is now a matrix we can transform each index separately. It will be easier to keep track of these transformations by recalling that the two indices of R
COLLECTIVE MOTION (I)
307
are space and body indices. We rewrite the result (A.3) as r(s)i = ~
Rijr(b)j,
(A.9)
J
where Rij depends on three Euler angles which we choose according to the prescription of Rose 2 6 )
R = R~(a). R2(fl). R3(y),
(g.10)
where R1 and R3 rotate about the z-axis and R2 about the y-axis. To find the transformation properties of R, we must take into account the fact that independent rotations may be performed in both the space and body system. We write r(s)i---- ~
j
g(s)ijr~s)j,
r(b)i =
~
j
R(b)ijr~b)j,
r~s)j = ~, g~k~b)k. k
(A.11)
This yields the transformation law for R !
--1
(A.12)
Rij = ~ R(s)i k Rkt R(b)l J . k,l
The transformation law (A.12) is the equivalent of (A.3). We can now define two unitary transformations S(~) and S(b) in the following manner: S~)I~b(R) =
~b(R~)I'R)'
(a.13)
The definition of Sty) follows the conventions of (A.3) while S(b) is defined so that its generators will be simply related to those of St,). We consider the infinitesimal transformations 3
3
S(s) = 1 + i ~ e(s)J(s)k, k
R(s)U =
k=l 3
t~ij- ]- ~,, ~,~s)[~'~k]ij, k=1 3
S(b) = 1 + i 2 e~b)J(b)k, k=l
g(b)U
=
(A.14)
t~ijq" ~ e(kb)[ak]ij. k=l
The independent variables of the scalar function ~b(R) are the three Euler angles 0~, k = 1, 2, 3, called a, fl, 7 in (A.10). We shall need the following 3 x 3 matrices ¢o(s)g, k . to(b)g"
= k=l
to(s)¢[fll]ij, I=1
3 -
i,j, ~ = 1, 2, 3.
(A.15)
=
k
I=1
They have inverses to(~)¢k-~and CO(~)~¢k. The angular velocity pseudo-vectors ta(s) and tO(b) are related to these matrices by 3
tO~)
=
~ 4=I
3
k
k
=
g=l
(A.16)
308
R.Y. CUSSON
We insert (A.14) in (A.13) a n d use (A.15) to o b t a i n - , 1 =~lCO(s)k - (~ J(s)k = i t~0¢ _
-
-
,
=
J(b)k
I =~103 I g
_
a d0¢
(A.17)
The definition (A. 15) of the co supplies the relation k co(s)~ -~
3 /=1
l
R k t co(b)~,
which, when inserted in (A. 17), yields 3 J(s)i = ~ R u J ( b ) j . j=l
(A.18)
Eqs. (A.9) a n d (A.18) show that J(s) a n d 'J(b) are the c o m p o n e n t s of a pseudo-vector in the space a n d b o d y system. As in the case of a p o i n t particle, the derivation given here is easily generalized to include d e f o r m a t i o n s of the body.
References 1) G. E. Brown, Unified theory of nuclear models (North-Holland, Pub1. Co., Amsterdam, 1964); A. M. Lane, Nuclear theory (W. A. Benjamin, New York, 1964) 2) N. Bohr, Nature 137 (1936) 344 3) E. Teller and J. A. Wheeler, Phys. Rev. 53 (1938) 778 4) A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953) 5) A. Bohr and R. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 30, No. 1 (1955) 6) D. R. Inglis, Phys. Rev. 103 (1956) 1786 7) S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No. 11 (1959) 8) A. S. Davydov and F. G. Filippov, Nuclear Physics 8 (1958) 237 9) D. J. Thouless and J. G. Valatin, Nuclear Physics 31 (1962) 211 10) S. T. Belyaev, Nuclear Physics 64 (1965) 17 11) A. Klein, L. Celenza and A. K. Kerman, Phys. Rev. 140 (1965) B245 12) J. P. Elliott and M. Harvey, Proc. Roy. Soc. A272 (557) 1963 13) S. Goshen and H. J. Lipkin, Ann. of Phys. 6 (1969) 301 14) M. Gell-Mann, unpublished 15) E. T. Whittaker, Analytical dynamics (Cambridge University Press, 1959) p. 153 16) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 14 (1952) 17) W. Pauli, in Handbuch der Physik, V/l, (Springer-Verlag, Berlin, 1958) p. 39 18) E. T. Whittaker, op. cit., p. 41 19) A. Bohr, Rotational states of atomic nuclei (Ejnar Munksgaards Forlag, Copenhagen 1954) 20) T. Gustafson, Mat. Fys. Medd. Dan. Vid. Selsk. 30, No. 5 (1955) 21) F. R. Gantmacher, The theory of matrices (Chelsea, New York, 1960) 22) C. van Winter, Physica 20 (1954) 274 23) R. P. Feynman, in Progress in low temperature physics, Vol. I, ed. by C. J. Gorter (NorthHolland Publ. Co., Amsterdam, 1955) chapt. 2 24) J. M. Blatt, Theory of superconductivity (Academic Press, New York, 1964) p. 157 25) A. B. Migdal, ZhETF (USSR) 37 (1960) 176 26) M. E. Rose, Elementary theory of angular momentum (John Wiley, New York, 1957)
]
Nuclear Physics A l l 4 (1968) 289--308; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
A STUDY OF COLLECTIVE M O T I O N (I). Rigid, liquid and related rotations R. Y. CUSSON
lnstitut fiir Theoretische Physik, University of Heidelberg, Germany Received 4 September 1967 Abstract: Models of nuclear collective rotations are defined by giving the form of the velocity field versus position. Only fields which are linear in the particle coordinates are considered. The canonical formalism is developed in the presence of the constraints represented by the field specification. The usual quantization rules are then applied to form the quantal collective Hamiltonian. The rigid-body problem is first treated. The liquid rotator Hamiltonian is obtained, starting from a vorticity free field, for the quadrupole mode only but at arbitrary deformation. The quantal problem corresponding to a field with arbitrary amounts of rigid and liquid components is shown to correspond to liquid motion with quantized vorticity as in nonlinear quantum superfluids and is treated using the components of the complete moment of momentum tensor which generate rotations and shear deformations. Connections with the algebra of SL(3, R) are pointed out. Fields where the ratio of liquid to rigid components are given, before quantizing, with the help of an ansatz connecting this ratio with the pairing strength, yield values of the moment of inertia which are intermediate between the rigid and liquid values. Qualitative agreement with the experimental situation in heavy nuclei is obtained for this last case.
1. Introduction Considerable progress both in atomic and nuclear physics has been made over the last decade in obtaining realistic solutions of the many-body Schr~Sdinger equation in the presence of mutual two-body interactions between the particles. As a consequence, one shows, in the context of the unified nuclear model for example, that collective effects can be obtained from the microscopic description 1). Since, however, the coupling between the collective modes and the particle modes is rather strong in the nuclear physics applications, these modes are often mixed in a given set of states. This has made it difficult to describe precisely just which collective modes do occur. This paper will be concerned with a somewhat new approach to the description of collective modes, in which the classical flow pattern is considered as primary data from which the quantum dynamics is to be constructed in the usual manner. But before we elaborate further on the possibilities of such a method and on its connection with the microscopic description, it is instructive to review briefly the development of the relevant ideas concerning collective motions. Fission phenomena and the existence of a fairly well-defined nuclear surface led early 2) to a model where the nucleus has the dynamics of a liquid drop. That the predictions of such a model are in accord with the exclusion principle requirements 289
290
R. Y. CUSSON
discussed by Teller and Wheeler 3), was pointed out by Bohr and Mottelson 4), inasmuch as a properly symmetrised or antisymmetrised system cannot physically be rotated about one of its symmetry axis. This property is characteristic of the liquid drop whose effective moment of inertia about any one of its symmetry axis vanishes. Experimentally it was found that the moment of inertia was several times higher than the liquid-drop value 5) and about one half of the rigid value. This could be explained using the semiclassical cranking model of Inglis 6) in the presence of residual attractive short-range two-body forces. Belyaev 7) showed that good fits to the experimental data could be obtained in this way. It was also shown that axial symmetry of the system was not always realised 8). Finally various improved treatments 9-~1) in the context of the Hartree-Fock theory have appeared recently and further explore the mechanism which generates collective motion from the microscopic dynamics. One outcome of these developments seems to be that the collective rotational motion is intermediate between pure rigid motion and pure liquid motion as exemplified by the moments of inertia. Since this particular type of collective dynamics is the end result of a quantal calculation, it is not clear however that it is obtainable as the quantized version of a classical model, for it is neither that of the rigid top nor that of a rotating drop. What we shall do is to consider models where the total velocity field is the sum of two components, a rigid component whose vorticity is proportional to an angular velocity and a vortex-free (liquid) component. By fixing the ratios of these two components before the quantization of the system is performed, a new class of rotators is obtained, whose moments of inertia can take any values intermediate between the rigid and liquid case depending on these given ratios. A specific dependence of these ratios on the pair correlation length, and the nuclear radius is proposed in order to obtain a connection with the microscopic properties and to remove the arbitrariness of these new adjustable parameters. An alternative scheme of quantization is also developed in which no attempt is made to fix the ratios of the two velocity fields at the beginning. The amount of vorticity then effectively becomes a parameter subject to quantization. The result of this scheme is that a generalisation of the liquid drop model is obtained where the vorticity instead of being strictly vanishing is quantized in discrete units which contribute to an effective "vortex angular momentum" as in the theory of quantum fluids. It is proposed that the absence of such vortices in finite nuclei is due to the fact they are too large, on account of the weakness of the pairing force, to fit inside one nuclear radius. In the second part of this study it will be shown that quadrupole vibrations can be included in the present formalism. The techniques used to calry out this program are those of the classical canonical formalism supplemented by the traditional rules of quantization. Yet there is a rather complex group theoretical context associated with our approach. Because of its complexity, the group aspect will not be discussed in detail, but since we first discovered the algebraic manipulations presented here through its use, we list some of the main ideas that are involved. To begin with there is the apparently isolated fact
COLLECTIVE MOTION (I)
291
that the algebra U(3) of the harmonic oscillator is useful in discussing the rotational structure of nuclei 12). However, it was shown by Lipkin and Goshen 13) that the Elliot algebra is part of a larger one, that of the linear canonical transformations SP(6) (Bogolyubov transformations for a spin 1 boson field, the coordinate), and that this larger, non-compact algebra, did predict rotations and vibrations at least in its two-dimensional form. This appears natural since SP(6) also contains the algebra GL(3, R) of the rotations and deformations. The non-compact algebra GL(3, R) has unitary representations consisting of infinite rotational bands 14). The operators which generate this algebra rotate and distort the coordinate system. The collective models where the particles are at rest in such rotating and distorting coordinate system are the ones which will be considered in this study. Thus it appears that the Elliot model, the rigid top and the linear liquid drop can all be treated as phenomena related to linear canonical transformations of the position and momentum variables. We begin by discussing the rigid body (sect. 2) and the liquid drop (sect. 3) in some detail, mostly in order to introduce the notation and the techniques. Although these sections contain few new results, they serve as convenient reviews of these two wellknown models, and their presence greatly facilitates the understanding of the following ones. Sect. 4 develops the alternative schemes of quantization mentioned above and outlines the method of solving the resulting Schr~dinger equations. The discussion (sect. 5) brings out the physical meaning of the equations obtained in sect. 4 and proposes a connection with the microscopic description. A later publication will include the possibility of vibrations since this involves an appreciable amount of new algebraic manipulations and does not greatly affect our conclusions.
2. The rigid body Our starting point is the classical Lagrangian kinetic energy of a collection of N (large) point particles with mass m N
T = Z ½mlr(")(t)l 2,
(1)
n=l
where J:(")(t) is the laboratory velocity of the nth particle. We assume throughout that the centre of mass is at rest in the laboratory. The condition that the distance between any two points in the body be independent of time leads to the condition on the velocity field ~(")(t) F(")(t) = to × r(")(t) = •(t)" r(")(t),
n = 1. . . . . N,
(2)
where to(t) is an arbitrary vector, the angular velocity and is independent of the particle label n and 12(t) the antisymmetric matrix equivalent to to x. Eq. (2) thus defines the flow pattern for rigid motion and is l i n e a r in the coordinates of the particles. The two equations above define the dynamics of rigid motion as a many-body problem with constraints. The first step is to deal with the constraints.
292
R . Y. C U S S O N
To do this we rewrite eq. (2) as constraint equations expressed as vanishing differentials 3
dC~")(r ("), t) = 0 = drl " ) - Z g2ij(t)r~ n)dt, j=l
i = 1, 2, 3,
n = 1. . . . . N,
(3)
The 3N constraints represented by (3) will be holonomic if we can find some integration factors and/or changes of variables such that they become exact differentials in the 3 N + 1 variables. This we do as follows: let R~k(t ) be a real 3 x 3 orthogonal matrix depending on the time only and consider equivalent constraint equations dCt,(") = Z l~k,dr~")+ Z r}")QJiRik dt = 0, i
i,j
k = 1, 2, 3, n -- 1,...,
N.
It is easily verified that the differentials will be exact if we put f~ = R . R -1 '
(4)
and that a first integral is then simply
C'k(") = Z R~ X(t)r~n)(t) = rfk"O )"
(5)
i
Condition (4) is possible when R is orthogonal as can be seen by taking the time derivative of R. R. Condition (5) states that the position is independent of time in the rotating coordinate system. This system is defined by giving the relation between the components V(s)i of an arbitrary vector V in the old, that is, lab or space system and its components V(b)j in the new, namely, rotating or body system as V(s), = Z Rij(t)V(b)J" J
(6)
,.(n) By comparing (5) and (6) we see that the constants "k0 are the body components of the position vectors. The integrated constraint condition (6) may be substituted in the Lagrangian (1) to yield in an obvious matrix notation
T = ½ Xr JR" G(b) • ~] = ½ Vr J R - ' . / ~ - G(b) • ( R - I -
~)],
(7)
where the mass quadrupole moment in the body coordinate system is N
= •
H~jO
~kO "
(8)
n=l
Since G(b ) is constant in time, it can be diagonalised by performing a time-independent rotation in the body system. We let its diagonal components in this new body system be Goi, i = 1, 2, 3. We see that the constraints have effectively eliminated all the particle variables from eq. (7) and that the Lagrangian is now expressed in terms of
COLLECTIVE MOTION (I)
293
new (collective) variables R-a (t)./~(t). Before we proceed with the solution of this system, it will be useful to develop some notation regarding the parametrisation of the mass quad~upole moment. We follow a procedure similar to that used in discussing the ellipsoid of inertia ~5), namely, we define the mass quadrupole ellipsoid in a three-dimensional space with coordinates p~, P2, P3, using the equation p2 + p~2 2 + p 32 = 1. Go1
Go2
(9)
Go3
We introduce new parameters G 0, t 1, t2, t3, such that t l + t 2 + t 3 = O.
Got = Go e2t',
(10)
All ellipsoids which are characterised by the same value of G o enclose the same volume and thus look like a sphere in some coordinate system which is linearly distorted at constant volume. This is seen by writing Pi
=
et'p~,
so that (9) becomes 3
Z
= Go,
i=1
which is the equation of a sphere in the system p'. From the definition of Go and eq. (10), we have M
Go = ~ e -2ti~[-~(n)']2 ' ~ ' L ' i O _l
i = 1, 2, 3.
n=l
Corresponding to the p' system, we can define a new distorted body coordinate sysp tem r o so that rtn) t. ,(n) iO = e 'rio ,
i = 1, 2, 3,
N
GO = Z
m krr l ,(n)-I o d 2,
i = 1, 2, 3.
n=l
In short we see that the r~ system is such that, in it, the mass quadrupole tensor is proportional to the identity matrix. The transformation to this particular system will be seen to enter explicitly in the formalism of the liquid drop to be discussed in sect. 3. N o w we deal explicitly with the constant volume condition ~itl = 0 by introducing two new parameters s~, s2, forming a vector s in a two-dimensional plane. We also define three constant vectors z~, z2, z3, in this plane, as in fig. 1, and we let t k = Z k ' S = Z k ~ S l + Z k 2 S 2 , k = 1, 2, 3. Because ~ k Z k = 0, the constant volume condition is fulfilled. This particular way of realising the condition is chosen for reasons of group theory related to the definition of the matrices II 4 and fls of table 1. Corresponding to eq. (10), we may express the moments of inertia of the system as 3
Iok = [ E j=l
Goj]-Gok
= 2Goe-'k'Scosh(flk'S),
k = 1,2,3,
(11)
294
R.Y. CUSSON
where the vectors flk are also defined in fig. 1. If we now represent the vector s in polar coordinates by giving its length a and the angle y, it forms with the positive s 2 axis as shown in fig. 1, we find easily that
,7 ~-o" c o s ( ~ - ~ k r c ) , //k" s = 2a sin ( ~ - aZkn),
• k" s =
k = 1, 2, 3.
(12)
1-~=(k- ~-3) S2
-TI ~
V
~
-1-2
-f13 •
b
P SI
1-2
l~l = 1-2- 1-5 ,8 2 = -t-5 - l- I
fil
-
2
fis =rl-1-2
Fig. 1. T h e vectors [3i a n d ~ . T h e Euclidian c o o r d i n a t e s o f this d e f o r m a t i o n s plane are st, s2 a n d its p o l a r c o o r d i n a t e s a a n d ~. F o r small a, this p l a n e is the usual/3, ~, plane o f the r o t a t i o n a l v i b r a t i o n a l m o d e l w h e r e fl = ~/(16~/15)a. TABLE 1
A basis for the eight traceless 3 × 3 real matrices
111=
[io!] 0 --
,
112=
1
116 =
E! o 0 1
E i oil
,
E o o !1
,
0
--
1
,
11~ =
113 =
0
0 1
0 0
1l 8 =
[O_,ooO] 1
0
0
0
E° 1 0
'
0 0
0 ,
0
114=
E,o!] E-i o i] 0 --1 0
,
115 =
0
--1 0
Finally if we define 16 fl = x/~-rco" ~ 1.83a,
(13)
the quantities fl and ~ will coincide for small fl with those defined by Bohr 16). We shall come back to this in sect. 3. There are several ways of obtaining the quantal Hamiltonian from the Lagrangian (7). For example we can take the Lagrangian to be a function of the Euler angles 0 z which define thc orthogonal matrix R and their time derivatives Oz. The general form of (7) is then 3
T = Z 2., ~t= 1
9a~,(o¢)OzO~,"
(14)
COLLECTIVE MOTION (I)
295
In order to define the Hamiltonian and quantize this system, we should use the prescription of Pauli 17) because eq. (14) corresponds to a curvilinear coordinate system. However, we can avoid this complication and gain additional physical insight by considering the body angular velocities as being the 0-values of the system. Using the defining eq. (A. 16) of the appendix for the body angular velocity, we can rewrite (7) as 3
T = +Z
(15)
I=l
where ~J
~(O~b) - - J(b)k :
k=l
_
k
(16)
lOk CO(b),
k=l Iok
1 -~ J~b)ke'~'~ 4G o k= 1 c o s h fig" $
(17)
Eq. (17) is the familiar expression of classical mechanics for the rigid-body Hamiltonian H
Since liquid motion is much more complex than rigid motion, we shall resort to a simple model instead of trying to solve the complete problem. This model we call a vortex-free linear rotator or liquid rotator for reasons which will soon become apparent. The Hamiltonian for this model will be shown to yield for small deformations, the Bohr Hamiltonian for liquid rotations. The model constitutes in fact a special case of general liquid motion. We replace eq. (2) by the following specification of the velocity field: g")(t) = ~ ( t ) .
r<")(t),
qgij = ~pj,,
n = 1.....
N,
(18)
where q~ij(t) is now an arbitrary s y m m e t r i c t r a c e l e s s tensor and plays a role similar to the angular velocity of the rigid case, which was represented by an antisymmetric
296
R . Y . CUSSON
tensor. In the limit where the number of mass elements N goes to infinity, eq. (18) becomes equivalent to 3
~i(r, t) = E
tPiJ(t)rj(t),
(19)
j=l
from which it is easily verified that
Or-~O#j(r, t) = ,,JZe k,j ~cr, i'j(r, t) = 0,
k = 1, 2, 3.
(20)
These last equations state that the flow represented by (19) is both incompressible and irrotational (vanishing vorticity). The reader is referred to the discussion of Bohr 19) for some more details concerning the derivation and validity of eqs. (18) and (19) in nuclear physics. Here we consider eqs. (20) to be sufficient justification for our use of this type of flow pattern. The flow (18) is also of special interest because (i) it is linear in the particle coordinates, (ii) it represents the lowest multipole moment (quadrupole) in the expansion of the complete velocity potential, and (iii) it is complementary to rigid motion within the linear group of rotations and distortions SL(3, R), in the sense that a body which is being linearly rotated and deformed at constant volume has a velocity field (internal mass current) which is the sum of two terms, one of the form (2) and the other of the form (I 8) (iv) the resulting dynamical problem is completely soluble. We shall consider arbitrary values of the deformation parameter ft. But when the deformation parameter is large, octupole and higher-order terms should be added to eq. (19), for their contribution is no longer negligible. Gustafson 2 o) has considered this effect in connection with fission, whereas we shall neglect it here because of the non-linearity involved. Thus our treatment is in fact more suited to the case of small deformations. We now repeat the steps of the second paragraph of sect. 2 replacing ~2 by ~ and R by a real 3 x 3 matrix with unit determinant Nl(t). An integral of the constraint differentials
M-~(t) • r(")(t) = r(o"),
n = 1,..., N,
(21)
is obtained if we impose the condition X = M" NI-I = ~ .
(22)
Eq. (22) can again be viewed as a differential constraint, but on NI this time. This new constraint on the collective variables contained in M would be holonomic if we could integrate it to obtain the explicit form of M. In fact we shall show presently that this is not possible and that the constraint (22) is non-holonomic. For the matrix NI, we use the convenient general parametrization 21) NI = RI(0~,). So(S) • k2(0a2),
(23)
where Ra and R2 are two orthogonal matrices defined by two independent sets of
COLLECTIVE MOTION (I)
297
Euler angles 0~1, 042; 21, 2 2 = 1, 2, 3, and So a diagonal matrix with elements
Sou = 3kt e"k" 5,
k, I = 1, 2, 3.
(24)
We have used the notation Vk" S introduced in sect. 2 in order to make the determinant ofS o equal to unity. Eqs. (21) and (23) imply that the particles are at rest in a coordinate system which is reached by three successive transformations. The components of some vector V can now be given in any one of four systems. I f we let these four sets of components be labelled V(s)i, Vi', V i'', V(b)i, we have
V~s)i = Z Ri,jVj,
V/ = Z Soi~Vj',
J
J
V i " = Z~2,jV~b),, J
V~s)i= ZM,2V~b)j. J
(25)
The space to body transformation now involves eight parameters which however, cannot all be arbitrary since t# in eq. (22) has at the most five independent parameters. To interpret the three transformations defined above, we look at the mass quadrupole tensor components in the various systems. We have
Cat =
G(s) = R 1 • G'" R1,
SO • G t t , SO '
G t t ~--- h E . ~ ( b ) "
R2-
(26)
The tensor G~b) is independent of time because the particles are at rest in the body system. By performing a time-independent rotation and deformation in the body system, we can insure that G(b) is proportional to the identity matrix in this body system, or equivalently we suppose that the body system has been cleverly chosen so that G(b) already has that property. This is clearly always possible. Thus G" is also equal to G o • I, by rotational invariance, for any R2. Again from (26) we find
G'kt = 6kt Goe2"k's,
k, l = 1, 2, 3,
which is of course precisely eq. (10). Since we are not interested in vibrations we choose s to be a constant, independent of time. Finally R1 transforms G' to the lab system. Thus So and R~ work as in sect. l, and they represent the constant shape and variable orientation of the mass quadrupole ellipsoid. Note, however, that the deformation S o now enters explicitly in the passage from space to body. Fortunately enough R2 still remains at our disposal, so we shall use it to fulfill the condition (22). Using (23) we compute X~) and X' X(~) = M ' M - 1
= RI" [/~1 " R I + S O " ~2" R2 "S01] " R1 = R1 " X " R1 •
(27)
We can expand X' using the basis of table 1 for the eight traceless 3 x 3 matrices and the definition of the appendix for the angular velocity which reads, in the notation of this section 3
3
k=l
k=l
0.) 2 ~'Lk .
This simple algebra yields 3
X
' =
Z k=l
{f~k[CO~ ,k --cosh (fig . $)(Dt2tk]-~-~'~k+5 sinh (ilk" $)('Ot2tk) •
(28)
298
1,,. Y. CUSSON
Thus ×' will be symmetric if we put to~,k =
1 eo~k cosh/~k" s
(liquid),
(29)
and as a matter of interest it will be antisymmetric if a~zt t k = 0
(rigid).
(30)
These last two equations are of considerable interest in that they show how one obtains either liquid or rigid motion from the same general situation, by imposing suitable constraints on the collective variables. In either case, M has only three remaining degrees of freedom, but the two types of dynamics are quite different. The rigid constraint can be satisfied with Rz = constant, but the liquid constraint is non-holonomic because the angular velocities involved in (29) are not integrable, as pointed out in sect. 2. The need for a time-dependent Rz in the case of liquid rotations is now easily understood. The first rotation R~ takes us to a rotating frame where the mass quadrupole moment is at rest (no vibrations!). If the particles were at rest in this system as in the rigid case, the vorticity would be infinite. To cancel out the vorticity implied by the rotation of the mass quadrupole tensor, one must first distort the coordinate system until the mass quadrupole ellipsoid becomes spherical and then, in this deformed system, perform a further rotation in the opposite direction from the original t t¢ one. For a spherical drop s vanishes and ¢91 = to z so that NI = R1-R1 = l, and the drop is at rest in the lab system. This is the well-known statement that a spherical liquid drop cannot have rotational excitations. The successive transformations involved in (23) are illustrated qualitatively in fig. 2 of Bohr's paper 29). We m a y now proceed to develop the canonical formalism in the presence of the non-holonomic constraints (29). To do this we first ignore the constraint completely in order to define the canonical m o m e n t a and the Hamiltonian H(g) of the complete system with its 6 degrees of freedom. This Hamiltonian will be a function of the six canonical m o m e n t a only. Then we shall express the angular velocity constraints (29) as constraints on the momenta. The non-independent momenta will then be eliminated from H(g) to give the Hamiltonian of the constrained system. This procedure works equally well for the liquid or rigid case (29) or (30), but we shall carry it out only for the liquid case, having already obtained the rigid-body Hamiltonian in sect. 2. This method has the advantage of supplying us with H(g) which we shall need in sect. 4. We substitute X(s) for ~ in eq. (18) and use the result in eq. (1) to obtain the Lagrangian as a function of X(s). Using (27) to relate X~) and X', we obtain 3
T = ½Go Z [X~k]2ea'~'s = ½Tr [ X ' ' G ' - ~']. i.k=l
(31)
COLLECTIVE MOTION (I)
299
The diagonal elements of X' vanish according to (28). We consider the six remaining independent elements to be the canonical velocities of the system. The three antisymmetric components of X' are the angular velocities of the pure rotation part and the three symmetric ones are the rates of shear deformation. This follows from the definition of X(s) as ~ • M-1, eq. (27); G' is a constant of the motion. The canonical momenta are as usual
,
aT
NiJ
i ~ j, i,j = 1, 2, 3,
~Zji
(32)
except perhaps for the transposition of the matrix indices. This is required here because the group SL(3, R), which the quantum operators corresponding to N' will generate, is non-compact. Therefore the metric in the algebra of its generators is not the identity. It can be proved that this implies (32). From (31) and (32) we obtain N
Nij = Z Gi'kZjk = Z ri("~P)("), k
(33)
n=l
showing that the canonical momenta are the components of the moments of momentum of the system in the primed system. The Hamiltonian is H(g)
=
Tr [tO'. N ' ] - T = -}2Tr [/~/' • G' - 1 " N'].
(34)
As expected H(g) is a bilinear function of the six independent moments of momenta N;j. We may now incorporate the constraints (29) by substituting (29) in (28) and the result in (33) to obtain the constrained expression for N;j, N[L)ij 3
N(L)ij '
=
2 k=l
Go e2"''s
tgh
(ilk • s) 2
jkcoi.,k
(35)
This tensor has but three independent components, which we take to be its antisymmetric components labelled J~k" We find 3
J'lk = Z ekijN(L)ij = lOk tghE (ilk" $)(Olk. i,j=l
(36)
Next we use (36) to eliminate ~ k from (35). This gives 3
e(,~i - Tj) • s
' = ±z y ' N(L)ij
k'~1 sinh//k" S
2
I
~ijkJlk •
(37)
Note that the symmetric components of N~b ) a r e non-vanishing so that when we substitute (37) in (34) to obtain H(L) they will contribute to the Hamiltonian. Doing this yields 3 = ½Z
k = 1 I0k
1
tgh 2 l~k" s
,2 Jlk"
(38)
300
R.Y.
CUSSON
The reader may repeat steps (35) through (38) using (30) instead of (29) and check that eq. (17) for H(r ) is indeed obtained with the notation J~k instead of J(b)k for the total angular momentum in the primed system. The liquid rotator Hamiltonian is seen to differ from the rigid one only by the presence of the additional factor tgh2flk • s in the denominator. This factor vanishes at sphericity and when the system is axial along the k'th axis. The quantity Iok tgh2flk • S may be loosely referred to as the liquid moment of inertia. It is always smaller than the rigid moment of inertia. It now remains to show how to obtain the Bohr expression s) for the liquid rotator from eq. (38). We suppose that we have a drop with uniform density up to its surface which is assumed ellipsoidal. It can be shown 19) that this is consistent with the constraint (18). Because of the constant density and ellipsoidal surface, the parameter s which characterises the mass quadrupole ellipsoid also describes the semi-axis R~, of the surface as R;, = Ro Ck's. For small deformations of the surface, we expand the exponential and use (12) and (13) to get
R~ ~- Ro [I+flV-~ c o s ( , - ~ k ) l ,
k=1,2,3,
(39)
which is the usual expression for the semi-axis. Letting the drop density be p, we 1 5 compute G O to be 8B/15rq where B is ~pR o. Upon expansion of eq. (38) to lowest order in fl and substitution of the above value for G o, one obtains
1 H(L) ~
Jg
8Bfl 2 k= 1 sin 2 (y--a~xk) "
(40)
This last expression has been used by Davydov 8) to compute the energy levels of a non-axial liquid drop. 4. The intermediate models
We have seen in the previous section how a system possessing both pure rotational and shear rotational degrees of freedom can yield either pure rigid or liquid motion when suitable constraints are imposed on the collective angular velocities. In this section we shall generalize these results in two ways. We shall derive the Hamiltonian for the case where the velocity field consists of a fixed mixture of rigid and liquid components and in the more general case where both components of this field are independent. It will be more convenient to study the second case first. Eq. (18) holds in the space system for the liquid case. We replace ~ by ×(s) and transform to the primed system. This gives the following equations for the primed -(n) ~'¢") of the general velocity field .(g) components -(g)i 3
f,(,,) (g)i
~--- ~,(,,) " ( r ) i ±,v(.) ~ ~(L)/
=
Y' Z.a j=l
Z i 'j
f'(") j
~
n
=
1~ . . .~ N .
(41)
COLLECTIVE MOTION (I)
301
Using eq. (28) for X' we write 3
~'(") ' " r'("), [r) = Z nk(oYlk - c ° s h (//k" s)c°~'~) " r'(") = X(r) k=l 3 i'(") (e) = Z
sinh
( i l k " S J, ( D 2t tkt'~ aLk+5
• r ' ( n ) = X(L) • r'(n),
(42)
k=l
where o'l and o[' are assumed independent. As we have seen the field (41) leads to the general Hamiltonian H(g) given by (34). We must now express H(g) in a form which is easy to quantize. The simplest way to do this is to express N' as a function of angular momentum variables. To do this we use the fact that the moment of momentum tensor N can be expressed in terms of its component Ni~ in the primed system and its components N~ '~ in the double-primed system. Since the metric in the doubleprimed system is not unity due to the stretch involved in 30, covariant and contravariant indices are to be distinguished in that case. Actually, N is a mixed tensor because it must transform a contravariant vector into another such vector. Some simple algebra yields the rule N" = 5o 1" N ' - 5 . (43) The antisymmetric components of N' and N " are angular momenta in their respective systems. Using (28), (33), and (43) we find
Jig
~ ekdNij
Iok 0)~
ii
Jzk = ~ ek~pNa = Iok .
.
.
0~2 cosh (ilk" s
.
.
(44)
cosh{
These two angular momenta are independent as long as o~'1 and o~[' are independent. Eq. (44) may be inverted to give the e) as functions of the J. This result is used in (28), (33) and (34) to give
H(g) =
1
LOlk~OZk
½ ~ k= 1 lokB k
]
cosh (fig" s)d
,
(45)
where B 2 = tgh2(flk • s). This expresses H(g) in terms of the primed total angular momentum of the system and a new angular momentum J~. It takes into account the fact that the vorticity is finite. For example when we substitute the liquid constraint (29) in (44), we find that J;' vanishes identically. We shall call J~' the "vortex angular m o m e n t u m " of the system, since a comparison of (42) and (44) shows that it is proportional to the finite vorticity component of the matrix X' which specifies the flow pattern. Inserting (29) or (30) in (44) the yield relations between the two J in those cases J~, = 0 (liquid),
J'2'k =
1
cosh (ilk" S)
J'xk
(rigid).
(46)
302
R. Y. CUSSON
For a rigid system with a spherical mass quadrupole, the vortex angular momentum coincides with the total angular momentum. When the rigid body is axial, only the components along the symmetry axis will coincide. In order to quantize H ( g ) w e must give the operator form of J'l and J~'. From their definition (44) we see that these operators are the generators of infinitesimal rotations in the primed and doubleprimed systems. These may be obtained by repeating the steps of the appendix using /d instead of R to find J(s)l and J(b)2 which are related to the present ones by J(s)l = R~ "J'l,
J(b)2 =
R2" J;'.
(47)
The result is that J;k(O~,, ((3/00~)) has the form of J(b)k(O~,((?/(?Oal) ) given in eq. (A.17), and that J;'k(O;.2((?/t?Ox2)) is minus J(b)k(Oa2, @/00~2)). In particular J i is given in terms of the first set of Euler angles and J'2' in terms of the second set only. We may diagonalize Jlz, J2z, IJll 2, IJ212 since they commute with H(g). The eigenstates of H(g) have the form T I , ~2 to~t,, K~(J1 , JE)Dj,M,, r, (0a,)Djg2, -J,, J2~t2, ~,, ~ra t"a,, 042) = ~ aK~, 2 r2 (0k2),
(48)
KI, K2
with aK~,K~(J~, ~,~2 J2) acting as eigenvectors. The matrix elements of H(g) o n the evensymmetric s t a t e s 2 2 ) form a tri-diagonal matrix which must be diagonalized numerically, even when the system is axial, to give the a in (48) and the energy eigenvalues as a function of J~, J 2 , ~'1 and z2. The subset of eigenvalues with J 2 = 0, "C2 = 0 reproduce the level spectrum of the liquid rotator which was computed by Davydov 8), while the J 2 = l , 2 . . . . eigenvalues correspond to liquid motion with 1, 2 , . . . quantas of vorticity. These levels are characteristic of a quantum liquid only. The energy spectrum of H(g) bears no resemblance to the rigid-body spectrum when the distortion is large. This is because the rigid constraint (46) cannot be realised when both J'~ and J~' are quantal operators, except if s = 0. We now see that if we want a level sequence which can approximate either the rigid or liquid sequence we must specify the ratio of the two velocity field in (41) before we apply the quantization rules. To do this we shall write an intermediate constraint condition
CO,2,k _
Ak
e),k,
0 < A k < 1,
k = 1, 2, 3,
(49)
cosh/I k • s where the A are numerical constants to be determined a priori as some fundamental property of the body under study. We insert (49) in (42) to get 3
3
k=l
k=l
from which we define the ratios of rigid to liquid field ,ok as the ratios of the components of Ilg in Xlr) to I/k+ 5 in X(L)
Pk-
1--Ak
A~lB~l
(51)
COLLECTIVE MOTION (l)
303
This shows that the intermediate constraint (49) is equivalent to a specification of Pk. Eq. (49) is now substituted in (44) to yield the new relations between the J as ,,
J2k
=
(1--Ak) ch fig" s , Jlk" ch2 (Pk " s ) - A k
(52)
This momentum constraint can be substituted in (45) to give the intermediate Hamiltonian H(i )
H(i)
3
=
½2
1
,2
(53)
- Jlk, k = i lo~f~
where ,_
[P~ + IBkl] z
J~' -
1 +2pklBkl
+P#
"
The intermediate Hamiltonian differs from H(L ) or H(r ) only in the magnitude of its effective moment of inertia Iokfk. The fraction fk(Pk, [Bkl) varies smoothly from B z to 1 when Pk goes from zero to infinity. Thus by selecting some ratio of rigid to liquid flow, we can achieve any value of the moment of inertia between the liquid and rigid value. For example if Bk is small and Pk is 1, fk is about ½.
5. D i s c u s s i o n
Here we shall present some physical considerations in connecting with the applications of the various models of collective rotations studied in this paper. Since the physical properties of systems undergoing pure rigid or liquid motion are well known, we shall concentrate our discussion on the properties of the models presented in the last section and refer to the other ones as the need arises. It is easiest to study the system represented by the general Hamiltonian (45) first, because it involves no new adjustable parameters. Its energy eigenvalues are labelled by J1, J2, zl, Zz, where Jl is the total angular momentum quantum number, as implied by eq. (47), and -/2 is the vortex angular momentum quantum number. We can express J2 in a way which exhibits clearly its connection with the vorticity by writing it in the following form: "
Iok
J 2k - -
'
fD( v)k ,
(54)
cosh ilk" S with ! =
i, j, l
' 21 ~ k i j ~ t Z j'l Fl" 01" i
(55)
These equations give (44) when eqs. (42) for X' are used. Thus J~' is proportional to the curl of the velocity field which is itself the vortex strength and the quantization of J~' is equivalent to the quantization of the vorticity. This may be compared with the treatment of quantized vortices e 3) in the superfluid 4He. Using Stokes' theorem,
304
R . Y . CUSSON
one replaces the vanishing vorticity condition V x ~: = 0 by the contour integral m ~ i . dl = 0.
(56)
Since this has the units of an angular momentum one allows the values 27c/, l = 0, 1. . . . for the contour integral, as a quantal generalization of liquid motion. Vortices with l = 1 are supposed to be responsible for the inertial properties of rotating liquid helium. If we consider a drop whose diameter is of the order of the diameter of a vortex tube, the linearized expression (42) for the velocity field might hold so that H(g) could represent the collective motion of this drop. This suggests that H(g) should be applied to a system which has undergone Bose condensation. This could happen in a nucleus if the pairing force were sufficiently strong. A measure of the pairing strength is the pair correlation length 24) L = 2el .
(57)
kiA If Lp were much smaller than the nuclear diameter one would expect a true quantum fluid behaviour and the level spectrum of H(g). Actually Lp is somewhat larger than the nuclear diameter in most cases, indicating again that H(g) is not a suitable Hamiltonian to describe nuclear collective rotations. Insofar as the general Hamiltonian H(g) is nothing more than the complete Hamiltonian for a linear quantum liquid including vorticity quantization, it might not come as a great surprise that is it inadequate for the description of the collective rotations of finite deformed nuclei since it is well known that nuclear moments of inertia are much larger than the liquid values. Yet physical intuition strongly suggests liquid motion, with its vanishing moment of inertia at sphericity. This dilemma must find its resolution upon detailed consideration of the microscopic dynamics of nuclear motion. We may suppose that the nucleons interact (i) with the average distorted onebody potential which is often taken to be a distorted harmonic oscillator well and (ii) with each other via a weaker, short-range attractive two-body force, which is usually represented by the pairing interaction. Neglecting the pairing force, it can be shown that if the distorted well is forced to rotate slowly, the mass quadrupole ellipsoid will follow 6). Thus the well angular velocity is to be identified with the angular velocity to'1. One then shows that the velocity field in the primed system must vanish on account of the weakness of the Coriolis force 5), so that we have to;' = 0. This means that the motion is rigid. In the presence of pairing, the primed-system velocity field has not been estimated but it has been shown that the moment of inertia is decreased and that in the limit of strong pairing the moment of inertia attains the liquid value : 5). This effect is consistent with our previous considerations involving the pair correlation length, and may be reproduced by using the following ansatz: Lp Pk oc - -
Ro
(58)
COLLECTIVE MOTION (I)
305
where R 0 is the average nuclear radius, and Pk and Lp have been defined above. It states that the ratio of rigid to liquid components of the velocity fields should be proportional to the parameter ~:/fl used by Migdal 25). When (58) is inserted in the expressions (53) for H(1), the various limits of the moment of inertia computed by Migdal 25) are obtained. Assuming equality instead of mere proportionality in (58) and taking Lp = R o as a reasonable estimate, yields about one half of the rigid value for the moment of inertia. These approximations are consistent with the experimental situation in heavy nuclei. These considerations allow us to form the following picture of collective nuclear rotations. Starting with weak residual forces and the average distorted field, the motions tends to be rigid. As the attractive pairing force strength is increased the boson pair density becomes better defined and the velocity fields goes over from rigid to fluid. In the limit of strong pairing H(i ) goes over to H(L) and we have a superfluid of bosons as in liquid 4He. Then H(g) may be substituted for H(L) as a quantization of the vortex motion of this superfluid. One can only speculate that the new collective degrees of freedom involved in vortex quantization are to be understood as rotational motion of the pair correlation field if not rotational motion of the particles themselves. We have not been able to derive a criterion for the onset of vortex quantization as the pairing force is increased. Such a "phase transition" probably does not take place in finite nuclei due to the rather large values ofpk. The transition might occur in infinite nuclear matter where R o ~ oo but there our linear approximation (422) would not hold. We are grateful to Professor J. H. D. Jensen for his kind hospitality at the Institute for Theoretical Physics where much of this work was conducted. We are also happy to acknowledge the support of an O.N.R. grant at the Kellogg Radiation Laboratory of the California Institute of Technology during the early stages of this work. We have benefited greatly from conversations with Professor M. Gell-Mann, Professor H. A. Weidenmtiller, Dr. K. Hara and J. Hiifner.
Appendix T H E A N G U L A R M O M E N T U M O P E R A T O R S F O R A R I G I D BODY"
Corresponding to a rotation of the coordinate system there is a unitary transformation which acts on the wave function and which is generated by the angular momentum operators. We shall first define the usual angular momentum operators for a point particle and generalize the definition to the case of a rigid body. We may consider the coordinate system as a rigid object defined by the location of its three unit vectors e ("), n = 1, 2, 3, each having components e~"), i ---- 1, 2, 3, in the lab system. We suppose that the vectors e (") depend on some (time) parameter such that 3
e~")(t) = ~, Ro(t)e~")(O), j=l
(A.1)
306
R . Y . CUSSON
where Re(O ) = t5e and R(t) is an orthogonal matrix having an analytic dependence on t. We say that e~")(0) are the lab coordinates of the old coordinate system unit vectors and e~n)(t) those of the new unit vectors. Let ri be the components of some vector in the old system, e.g. before R is applied, and r/' those in the new system, namely 3
3
r = Erie(°(O) = Z r~e(°(t) • i=l
(A.2)
/=1
Eqs. (A.1) and (A.2) imply the relation
r, = ~ Ri~r;. J
(A.3)
Now let $(ri) be a scalar function of the old coordinates r i of a point particle. We have
$(r~) = ~b(~, g,jr)) = $'(r~,),
(A.4)
J
which is the definition of a new function of r~, $'(r~). We assume that this new function can be obtained by a unitary transformation S acting on the old function of the new variables ~b(r~) that is $'(r~,) = S$(r~,).
(A.5)
Thus we have the equivalent properties
d/( E R,j r)) = S~(r'k), J ~b( ~ g ~ lrj) = S - l $ ( r k ) .
(g.6)
J
For an infinitesimal R and S, we write 3
S = 1 +i~
3
e,klk,
R,j = ~5i~+ ~ ek[i'~di~,
k=l
(g.7)
k=l
where the e k a r e infinitesimal, the matrix f~k are given in table 1 and lk are the generators of infinitesimal rotations. We insert (A.7) in (A.6), expand and compare terms of order e k to get
Ik = Tr [flk" t],
tu = rtpj' 3
PJ -- i r~rj' (A.8)
Ik = ~ e~ljrtPj • i,j=l
A derivation of the generators of the unimodular group SL(3, R) would proceed along similar lines. In the case of a rigid body, the coordinate manifold is R~j itself instead of r~. Since the coordinate is now a matrix we can transform each index separately. It will be easier to keep track of these transformations by recalling that the two indices of R
COLLECTIVE MOTION (I)
307
are space and body indices. We rewrite the result (A.3) as r(s)i = ~
Rijr(b)j,
(A.9)
J
where Rij depends on three Euler angles which we choose according to the prescription of Rose 2 6 )
R = R~(a). R2(fl). R3(y),
(g.10)
where R1 and R3 rotate about the z-axis and R2 about the y-axis. To find the transformation properties of R, we must take into account the fact that independent rotations may be performed in both the space and body system. We write r(s)i---- ~
j
g(s)ijr~s)j,
r(b)i =
~
j
R(b)ijr~b)j,
r~s)j = ~, g~k~b)k. k
(A.11)
This yields the transformation law for R !
--1
(A.12)
Rij = ~ R(s)i k Rkt R(b)l J . k,l
The transformation law (A.12) is the equivalent of (A.3). We can now define two unitary transformations S(~) and S(b) in the following manner: S~)I~b(R) =
~b(R~)I'R)'
(a.13)
The definition of Sty) follows the conventions of (A.3) while S(b) is defined so that its generators will be simply related to those of St,). We consider the infinitesimal transformations 3
3
S(s) = 1 + i ~ e(s)J(s)k, k
R(s)U =
k=l 3
t~ij- ]- ~,, ~,~s)[~'~k]ij, k=1 3
S(b) = 1 + i 2 e~b)J(b)k, k=l
g(b)U
=
(A.14)
t~ijq" ~ e(kb)[ak]ij. k=l
The independent variables of the scalar function ~b(R) are the three Euler angles 0~, k = 1, 2, 3, called a, fl, 7 in (A.10). We shall need the following 3 x 3 matrices ¢o(s)g, k . to(b)g"
= k=l
to(s)¢[fll]ij, I=1
3 -
i,j, ~ = 1, 2, 3.
(A.15)
=
k
I=1
They have inverses to(~)¢k-~and CO(~)~¢k. The angular velocity pseudo-vectors ta(s) and tO(b) are related to these matrices by 3
tO~)
=
~ 4=I
3
k
k
=
g=l
(A.16)
308
R.Y. CUSSON
We insert (A.14) in (A.13) a n d use (A.15) to o b t a i n - , 1 =~lCO(s)k - (~ J(s)k = i t~0¢ _
-
-
,
=
J(b)k
I =~103 I g
_
a d0¢
(A.17)
The definition (A. 15) of the co supplies the relation k co(s)~ -~
3 /=1
l
R k t co(b)~,
which, when inserted in (A. 17), yields 3 J(s)i = ~ R u J ( b ) j . j=l
(A.18)
Eqs. (A.9) a n d (A.18) show that J(s) a n d 'J(b) are the c o m p o n e n t s of a pseudo-vector in the space a n d b o d y system. As in the case of a p o i n t particle, the derivation given here is easily generalized to include d e f o r m a t i o n s of the body.
References 1) G. E. Brown, Unified theory of nuclear models (North-Holland, Pub1. Co., Amsterdam, 1964); A. M. Lane, Nuclear theory (W. A. Benjamin, New York, 1964) 2) N. Bohr, Nature 137 (1936) 344 3) E. Teller and J. A. Wheeler, Phys. Rev. 53 (1938) 778 4) A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 16 (1953) 5) A. Bohr and R. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 30, No. 1 (1955) 6) D. R. Inglis, Phys. Rev. 103 (1956) 1786 7) S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No. 11 (1959) 8) A. S. Davydov and F. G. Filippov, Nuclear Physics 8 (1958) 237 9) D. J. Thouless and J. G. Valatin, Nuclear Physics 31 (1962) 211 10) S. T. Belyaev, Nuclear Physics 64 (1965) 17 11) A. Klein, L. Celenza and A. K. Kerman, Phys. Rev. 140 (1965) B245 12) J. P. Elliott and M. Harvey, Proc. Roy. Soc. A272 (557) 1963 13) S. Goshen and H. J. Lipkin, Ann. of Phys. 6 (1969) 301 14) M. Gell-Mann, unpublished 15) E. T. Whittaker, Analytical dynamics (Cambridge University Press, 1959) p. 153 16) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 14 (1952) 17) W. Pauli, in Handbuch der Physik, V/l, (Springer-Verlag, Berlin, 1958) p. 39 18) E. T. Whittaker, op. cit., p. 41 19) A. Bohr, Rotational states of atomic nuclei (Ejnar Munksgaards Forlag, Copenhagen 1954) 20) T. Gustafson, Mat. Fys. Medd. Dan. Vid. Selsk. 30, No. 5 (1955) 21) F. R. Gantmacher, The theory of matrices (Chelsea, New York, 1960) 22) C. van Winter, Physica 20 (1954) 274 23) R. P. Feynman, in Progress in low temperature physics, Vol. I, ed. by C. J. Gorter (NorthHolland Publ. Co., Amsterdam, 1955) chapt. 2 24) J. M. Blatt, Theory of superconductivity (Academic Press, New York, 1964) p. 157 25) A. B. Migdal, ZhETF (USSR) 37 (1960) 176 26) M. E. Rose, Elementary theory of angular momentum (John Wiley, New York, 1957)