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Collective rotations in nuclei
ASNALS
OF
PHYSICS:
6,
224242
Collective
(19%)
Rotations
in Nuclei
FELIX VILLARS Department
of Physics
and Laboratory of Technology,
of Nuclear Cambridge,
Science, filassachusetts Massachusetts
Institute
A systematic approach is made to the problem of determining the parameters of collective motion in nuclei. Formulas are given for the moment of inertia of the rotational energy and for the collective g-factor of the magnetic moment. These expressions are not based on the use of a model of the nucleus. Such a model will be needed, however, for the numerical evaluation of the espressions derived. The essential elements of a usable model are briefly discussed. INTRODUCTIOX
Many properties of rotational states in nuclei, and of transition amplitudes between such states, may be expressed in terms of simple, essentially kinematic relations, Examples of this are the ratios between EZ-transition amplitudes within a rotational band, and their connection with the static quadrupole moments, and the corresponding relations between the Ml-transition amplitudes, and the magnetic moment (1-S). These relations hold irrespective of the dynamics, but they contain parameters whose theoretical determination requires a detailed understanding of the interplay between collective rotation and intrinsic motion of the nucleons. Such parameters are the moment of inertia 9, determining the spacing of rotational levels, and the collective magnetic g-factor g,: ~ resulting from the collective current flow. The purpose of this paper is to present a systematic approach to the problem of determining 9, gC and other parameters pertaining t’o states of collective rotational excitation. The method used is that of coruact transformations, with the help of which the variables describing collective dynamics are stepwise decoupled from the “intrinsic” motion. It is shown that the desired result can be achieved in two steps. In the first step, a body fixed coordinate system is introduced, defined by the principal axes of the mass distribution in the nucleus. The Euler angles of that system, and the project,ions of the t)otal angular momentum on the body fixed axes are introduced as new dynamical variables. This leads to t,he “canonical form” of the nuclear Hamiltonian (4). Tn the canonical Hamiltonian, the total angular momentum is still coupled to t&he intrinsic motion; t,he decoupling can however, now, be achieved by perturbation theory. 224
COLLECTIVE
ROTATIONS
IN
225
NUCLEI
In Section I, the canonical form of H is presented, together with a few relevant additional relations. In Section II, the rotational moment of inertia is calculated by a simple, classical method. In Section III, a more rigorous quantum-mechanical version of the same calculation is given, and the validity of the approximation is discussed. In Section IV a few remarks are made on nuclear models simulating the structure of the canonical Hamiltonian. Such models are needed for any numerical work. Finally, in Section V an expression for the magnetic moments of rotational st.ates is derived. I. SEPARATION
OF TOTAL ANGULAR MOMENTUM OF INTERACTING PARTICLES
IN A SYSTEM
It is well known that the Hamiltonian of a rigid body may be written in the form’
(1.0 Jp being the components of the total ahgular momentum, and gP the principal moments of inertia. As a model of a quantum-mechanical system, the rigid body represents generally a gross idealization of actually occurring situations. Nevertheless, spectra characteristic of the eigenvalue system of (1.1) are found in many instances, in particular in molecules and nuclei. For the former, the Hamiltonian can be written in the form (5, 6):
H = &
+ c (Jc 29 Jc’)2 + H,,(T, E), c C
(1.2)
Jc - Jc’ = Rc being the angular momentum associated with the rotation of the body-fixed frame of reference ABC, and l, ?r being additional position and momentum variables in the rotating frame. Jc is the total angular momentum, projected on the body fixed axes, whereas J C’ is the sum of intrinsic (electronic and
nuclear) angular momenta, calculated in the body fixed frame. It is not generally . true that a system such as (1.2) should have an eigenvalue spectrum-as far as
1Kotation: P, Q, R, and A, B, C will be used as subscripts to indicate vector components in a “body fixed” coordinate system. Vector components in a space fixed system will be indicated by subscripts (Y,8, 7. For axial vectors we use the tensor notation, if convenient. In this case we use a square bracket to indicate antisymmetry: JIPQI
In connection with this notation, rotation-matrices RPA :
=
RPA
RQB
-
RQ* RPB =
RCPQI.[ABI
=
JR.
it will often be useful to introduce
R lPQ1.A P&R being an even permutation
- JIQP]
-R[QPI,A
=
=
RRA
double indices into
,
of 123. Use may then be made of the relations =
RRC
)
ZP
RIPQI~A
RPB
=
RQ,IABI
j etc.
226
FELIX
VILLARS
dependence of energy levels on angular momentum is concerned-similar to sy: tern (1 .l). That this is so for molecules is due to the near constancy of the functiojl s,(t) on the one hand, and on the other hand to the extreme smallnessof the ratio (fi’/24)/AEint,. , AEi,t,. being an intrinsic (vibrational or electronic) excitation energy. As a result of this, the coupling term
-c
JcJc’lgc
(1.3)
is an extremely small perturbation, and the rotational spectrum remains to a high accuracy determined by the quasi-rigid body Hamiltonian
F Jc%&
0.4)
alone. The observation of rotational bands (and even of systems of rotational bands) in nuclei leads to the question whether of a term of type (1.4) can be separated out in the nuclear Hamiltonian in certain cases.Clearly this cannot be generally possible. On the other hand, it can indeed be shown that the Hamiltonian of any closedsystemof particles can actually be written in the form (1.2). This form, then, doesnot in general lead to rotational spectra as given by (1.4); but it should provide a convenient starting point in case of their actual occurrence, and also for an analysis of the conditions under which this may happen. Characteristic of both (1.1) and (1.2) is the introduction of 3 “body-fixed” axes. In a non rigid system such as a nucleus, such axes may be introduced in many ways, in terms of the principal axes of any tensor T&z; , pi) symmetric in the nucleons. Actually the close connection between occurrence of nuclear rotational spectra and nonspherical shape of nuclei suggests the choice of the mass tensor2 (15) to define those principal axes: Iaa = c &mRA8te,)IA A
.
(1.6)
In (1.6) I, are the principal moments, and the Ra,(f?,) rotation matrices connecting the spacefixed coordinate axes c& with the principal axes ABC. In the case of nuclei, the structure (1.2) of the Hamiltonian is obtained through a coordinate transformation, which isolates the total angular momentum components JA and the associated Euler angles 0, as independent variables. The residual intrinsic variables .$ , 7rS, u = 1, 2, 3 . . * (3A - 6) appear only through the functions ziA’(&) and piA’(?r# , t.), which we call “intrinsic” positions and * Nucleons will always be numbered
by an index i: i = l...A;
X, is the c.m. position.
COLLECTIVE
n ,menta. It was shown be written as3
grn
H=
ROTATIONS
IN
NUCLEI
227
in (4) that in terms of the latter, the Hamiltonian
+ & c pi/
+ V(GA’) + f A$ QACJA - JA’)~,
can
(1.7)
%,A
m being the nucleon mass, A the mass number, and V the potential energy. intrinsic angular momentum JA’ Barring spin-whose inclusion is trivial -the is the function
J,’
= J&
= c
(xa’pic
- xic'pi~')
(1.8)
and the zero order moments of inertia &A are expressible in terms of the principal moments IA (1.6) as
QAdl = m(IB - Id2/(IB
+ Ic).
(1.9)
It must be realized that the variables xin’, piA’ do not form canonical pairs; they are not even independent variables, but satisfy the identities: F
%A’
= 0,
F
PiA’
=
F XiA’xia) = F (zi~‘pis’ +
0,
(l.lOa)
IA6.48,
xiB’piA’>
(A # I?).
= 0.
(Mob)
As a result, the properties of Ja’ (1.8) are not exactly those of an angular momentum; instead, one has the Poisson-bracket relations: (JA’>
{JwI’,
JB’)
=
xic’l
= ~In + IB kif8BC
{ J~.oI’, pit’)
-(IA
+
Ic)(IB
+
21,
= I,
21,
Ic>
- m
- &
P~A’~Bc
(1.11)
Jc’~
21, XiB’aAC
pi~‘Lc
,
(1.12a)
+ 0(1/A).
(1.12b)
B
[These A-’ terms may be important, occasionally. The derivation of these results and of Eq. (1.14) will be sketched in the Appendix.] As a consequenceof these relations, the angular momenta Jn’ do not commute with the intrinsic energy
Ho = &
c pin12 + V(zd).
(1.13)
GA
Their commutator,4 however, is proportional to the deviation from spherical 3 Similar expressions have been derived by various authors (7-10). * We shall use, in the text, synonymously the word commutator and Poisson-bracket; but writing the expressions as Poisson bracket relations, to be replaced by i/C times commutator in quantum mechanics.
228 symmetry
FELIX
VILLARS
:
For nuclei, the structure (1.7) of the Ha.miltonian represents a much lesser degree of coupling of the intrinsic motion (governed by Ho) from the collective rotation than the corresponding expression (1.2) does for molecules. In fact!, the ratio
is not very small in the nuclear case. [As an example, we may take the band system of Wls3 (11, 12), where this ratio is about fh]. Of course, the occurrence of rotational spectra does not follow from (1.7j, since this holds for an arbitrary system.5 If such spectra occur, we must assume the existence of a set of axes P&R, which allows a term of type (1.4) to be separated from the intrinsic motion, wit,h the addition of small correction term (rotation vibration interaction). The experimental evidence about the kinematics of collective rotation indicates that these axes P&R must be closely related to the principal axes ABC of the mass tensor as defined by (1.6). This leads to t,he physical picture of the axes PQR fluctuating about ABC, these latter defining their average position. The analysis of next chapter will show to which extent, this picture is verified. II.
DETERMINATION
OF
THE
PRINCIPAL
MOMENTS
The existence of rotational spectra implies the possibility of writing the nuclear Hamiltonian, in appropriate variables, as the sum of 3 terms: Center of mass energy, rotational energy, and intrinsic energy. The first two of these terms have the structure given by Ey. (1.1); the third is an operator which, to a good approximation, commutes with all the variables describing center of muss- and rot.nGonal motion. Bohr and Mottelson (IS), as well as Lipkin, de Shalit, and Talmi (1 /t), have pointed out that if this structure of the nuclear Hamiltonian is admitted, the principal moments may be found in principle from the set, of relations:
JP = c Rd&)J,
,
{JP, R,,j = R,,, c {H, %7)&a = JR/~, t 7 5 Observe, however, that the transformat.ion leading to (1.7) becomes singular cal principal moments IA(E). Only if two of the Qa fluctuate about an essentially value does this coordinate transformation assume meaning.
(2.1) (2.2) (2.3) for identinonzero
COLLECTIVE
ROTATIONS
IN
229
NUCLEI
the RPa being rotation matrices relating the body fixed frame (P&R) to the space fixed axes c@y. These equations may be thought of as written in terms of the original variables zirr , pi, with Euler-angles 0, = I!?*(xi.!, piol) and associated rotations matrices Rprr(Os), since in these variables the nuclear Hamiltonian has a known structure. We shall, in this chapter, carry out this program, in the classical limit; but we shall take advantage of the form (1.7) of the nuclear Hamiltonian. In this case, the matrix Rp, may be broken up into two parts: RpaW
= 5 RpC@“)Rca(e’>.
(2.4)
The system of “intermediate” principal axes A, B, C was introduced and used in (1.7). With the help of the relations
{Ho, Rca) = 0,
{J,,
Rcaj =
We find from Eq. (2.3), using the Hamiltonian
R Pa F
&a,
in Eq. (1.6)
a,
(1.7) and expression
(2.4) for
:
{H,, , RPc)RQc + F --- (Ja - JA’)(RA~ - F {JA’, Rpc)RQc) = JR/sR.
In the interest of simplicity, we have assumed here that QA commutes with the matrices RpC , and may be replaced by a constant 1/9A”. This point will be reconsidered in Section IV. To satisfy (2.5), we must put the terms not containing J equal to zero, and equate the coefficients of JR with the help of the projection
JA = c
R
JRRBA .
In this way we find from (2.5) the two basic equations: . 5
{Ho,
Rpc)Roc
- T 2
J*‘(RnA
- F
(JA’,
RPC}RQC)
=
0
(2.s)
and
F $0 RwA(RRA - F (JAI,
R~c}RQc)
Z 8Rw/gR .
(2.7)
Equation (2.6) defines the rotation matrix RpA(eM). It is clear that 0” will be a function of the variables &, , ?T#; the RpA do therefore commute with the Ja but not with the JAI, of course. Equation (2.7) is then the equation defining g, by means of RIIA . It is easily seen that Eq. (2.7) cannot generally be consistent (such consistency would be equivalent to the unconditional existence of rotational states). Indeed, from (2.7) we may derive:
230
FELIX
R RA - 5
VILLARS
(Ja’, RPC) Rw Y $
(2.7a)
Rm ,
a relation which certainly imposes restrictive conditions on the structure of the system to be even approximately true. But assuming (2.7a) to hold, we may feed this relation back into (2.6), to get
F {Ho , RPC] Rot = ;
F JA’R,u .
Equations (2.7a) and (2.8) are the basic results. To explore their implications, let us assume that the angles 8” are small, so that
R,u = 8zz.4+ %A, To lowest
Q~RA=
;
--SlAR
/&A/
((1.
cw
order in 3, (2.7a) reduces to
I Jzi’, fb2pql =
6RA(1
-
(2.10)
gA”/&)
or
{JR’, QppJ = 1 -
&$ti,
In Eq. (2.8) a similar simplification
A #
(JA , QPQ} = 0,
;
[P&l.
occurs, to lowest order in 8: (2.12)
{Ho, QPQ) = JR’Igx. At this place, let us introduce
the notation
s= and the inversion
(2.11)
{Ho,F}
(2.13a)
E;
of this relation
F = s. . With this notation,
Eq. (2.12) may be formally f2PQ = JR’/Sg .
For a first orientation
(2.13b) solved as
- ((&)’
J,‘).
(2.14)
.
let us assume that
(Ho, 4E) r 0,
{J,‘,
9R) S 0;
These assumptions simply imply that gR is approximately follows from Eqs. (2.11) and (2.14) that
a constant.
$R = 9; + (JR’, JR’}. .
This is indeed a familiar we have
result. Translated
into quantum-mechanical
It then (2.15) language,
COLLECTIVE
ROTATIONS
IN NUCLEI
an expression reminiscent of the “Inglis-formula”
231
(X), and (2.15) becomes
In this present equation, the term 4 llo refers to the zero-order moment as appearing in (1.7)) and whose value is given by the expectation value of (1.9)) which is the rigid body moment times the square of the deformation. The measured values of gR are well known to be much larger than this; the main contribution must therefore arise from the second term of (2.16). Equation (2.16) however diiers significantly from the Inglis-formula
in two respects: (a) In the Inglis formula (2.17) the states I n) and 1k) are independent particle states in a deformed well, and JI1 is the actual angular momentum of the system, which in this model is not a constant of the motion, due to the artificially introduced anisotropy of the potential energy. In contrast to this, our Equation (2.16) contains the matrix-elements of the “intrinsic angular momentum JR’, which does not commute with Ho ; but both J’ and HO commute with the total angular momentum J, which has been separated off by a coordinate transformation, as outlined in Section 1. (b) The energy denominators in (2.16) refer to the eigenstates of HO and thus correspond (apart from small level shifts) to the actual intrinsic excitation energies, and may be taken from experimental data. In the Inglis model they refer to single particle excitations.6 III.
CRITICAL
APPROACH
TO THE
PROBLEM
The developments of Chapter II were deliberately sketchy, in order to present clearly the main line of argument. Here we shall take up the problem anew, but apply the method of canonical transformations to separate the collective rotation from the intrinsic motion. This more powerful approach will also enable us to establish conditions for the occurrence of rotational spectra. In order to carry out this program, we must begin with a few preliminary statements concerning the eigenstates of the Hamiltonian (1.7). As will be seen, a few basic approximations have to be made from the outset; unless they a.re 6 The Inglis model can be made more “realistic” by introducing inter-particle couplings, which strongly affect the energy-denominators in (2.7). If this is done, the results obtained from (2.17) can be brought into reasonable agreement with experiment (16).
232
FELIX
VILLARS
made, the situation seemsto become entirely opaque. But we bear in mind that they may somewhat restrict the general validity of our results. (A) THE EIGENSTATES
(1.7)
OF THE HAMILTONIAN
The coordinate transformation discussed in Chapter I, brings the nuclear Hamiltonian into the form H = Ho + Hrot ,
(3.1)
Ho = C piAr2/2m -I V(Si.d’), iA
(3.2)
H rot = ; F &“(JA
- JA’)“,
with QA being given by (1.9). A characteristic property of the zero order moments is that, whenever two of them are equal, the third has to be zero. This axially symmetric situation seemsto be prevalent, according to our experimental information, and the present discussion shall be restricted to cover this case: We assumethat for all states of Ho of interest, we have (n 1&I 1n) = (n I Qz1n) = l/k’!
(3.4)
It follows then that (n j Q31n) >> l/‘.%?.
(3.5)
((Q3) is not infinite, since Q3does not commute with Ho.) In order to exploit, the advantage of the cylindric symmetrical case, another approximation has to be made. As seen from Eq. (1.14), the commutator’
[Ho, Js’l becomes very small for cylindrical symmetry; and we shall approximate HO by an operator no defined as follows: Let a be the eigenvalues of J,‘, and introduce the projection operator pn = n (J3’ - L”)/(fl - 8’). n*+n Then Ho may be decomposed
Ho = cn PslHepsl+ F (1 - PdHoPn
(3.6)
= i?o + 6Ho. This enables us to define a complete set of simultaneous eigenstales of go and JZ’ 7 In t,his
and the
following
chapters,
we put
ii = I.
COLLECTIVE
ROTATIONS
flo I vf2) = Eva I vi?}, There is a degeneracy
of E with
IN
233
NUCLEI
Js’ 1 vn) = n 1 vf2}.
(3.7)
respect to the sign of 3: Evn = -t-%,-n
but apart from this we may assume the levels of f10 to be nondegenerate. 6Ho has no diagonal elements then, and its effects will not be considered here. From Eq. (1.11) we derive the commutation relation:
[J3’, JI’ + iJ2’l = JI’ + iJ2’ - r&)reJ;
- ieJ:>
and a similar relation for [J3’, J1’ - i Jz’]. This justifies an additional mation, in which the only nonzero matrix-elements of J1’ f i Jz’ are
(3.8) approxi-
(vQ 1 J1’ f i Jz’ 1 v’, D T 1). From Eq. (1.12) it follows in addition that the zero order moments QA cornmu& with a.11the J,‘; their matrix elements are therefore diagonal in 0. With these approximations in mind, the problem is determined by the zero order Hamiltonian RO (thereafter written as HO), and the perturbation
$4 QI(JI There is an additional
- JI’)’
+ 34 QdJ2 - Jz’)‘.
(3.9a)
term (3.9b)
45 QdJs - JA2
which is dropped in the cylindric symmetrical case considered here: In a representation in which J3 is diagonal, Ja = K, this term has matrix elements
$5 (vn ( Qs 1 ~‘0) (K - fQ2. The diagonal elements of Q3 were found to be extremely large; hence only states to low energy excitation. States with K # fi can be coupled in only through the part
K = D correspond
34 (&I - Q2) ((J,
- J+‘12 + (J- - J-‘j2)
of (3.9a), and there only through the off -diagonal elements of Q1 since (Q1) = (Q2). Thus, provided the cylindrical symmetry is stable:
((&I - Q2)“) << (&I>~, states K # D are not coupled in, and the term (3.9b) may be dropped altogether. This is done in the following sections. (B)
CANONICAL
TRANSFORMATION
It is obvious enough that the rotation canonical transformation
RPA(O”)
[Eq. (2.4)] is equivalent
to a
234
FELIX j=-j
with
S being the rotation
VILLARS =
pJJe-is
(3.10)
operator: S = (a.J).
(3.11)
In II, the problem was “solved”, assuming (Y to be a small angle. We shall first carry out the transformation (3.10) under this assumption, and have a look at the higher approximation later. Let us introduce the notations
wo , {I = !.I
wo >!21= &, for any operator
q. Additional
useful abbreviations
(3.12)
will be
(A = 1, 2 only)
DA = QAJ~‘,
(3.13)
and
K AB = i[DA , D,& . It then follows
(3.14)
that
c J, (ck + DA - ; F (J&A,
B = H,+;TQAJA”-
+ [aA , DB]JB’)
+ ; 3
[JA , Jsl+
- bA,bB,~
(
-ita~,
’ cc Q-‘D;]l
&I
Del - %x4,
+ &A&o
>
Dd
(3.15)
+ . ** .
The problem is now to cancel the terms linear in JA . This may be done approximately by means of the ansatz . --DA, (3.16) &A = K being a numerical then be written as
(1 -
constant
K)DA
yet to be determined.
-
i
F
(KBAQBwlDB
The coefficient
of - JA may
+ DBQB’K,,).
(3.17)
Now we notice that
K
- KBA = i[DA . , DB]. . part (KeA + KAB)/2 has a nonzero expectation
Hence only the symmetric this latter has the additional
AB
value;
property (n
1 KAB
( n>
=
ARAB
.
(3.18)
COLLECTIVE
This follows
+
K2,
IN
235
NUCLEI
that A, B = 1, 2 only, and
from the observation Kl2
which
ROTATIONS
=
pi
(D,
D+ .
-
B-
D-1, .
of Jsl. For
has no diagonal elements with respect to eigenstates (n I K4n I 4,
the independence on A follows from the assumed cylindrical (3.17) may then be written as (1
6Xn = ‘2 (C The numerical in J
B
constant
K
LB
-
K +
KS’A)&
(I-L),
-
+
Jil,
is now determined
symmetry.
Equation (3.19a)
6x,
+ 2A(l - tiO&adJ:
>
.
so as to minimize the linear coupling
1
(3.20)
x=1+ Looking (K”
-
at the quadratic
~K)&IA
+ ; c
c
terms, the coefficient
f&i&c-%
(3.19b)
of x [JA , JB]+ becomes
+ Kc~a&c-~Kc~)
- d&, [DA, . Qc-‘Gil+ + iI+,
J,lr+
, Qcll) + &As
This again will be split into a main term and a residue ((K2 6x~e
=
(K2
-
+ f 7
~K)(KBI
~K)A
+
K2goA2
+
(g”)-‘)8,,
+
6XAB
(3.21a)
- (Ksn)) + (&A - 1/gob,
&A&C-%
+
Km&c-‘Kc,
The main term, using the value (3.20) of &i&go
K,
- 24’A2 - i[ ]+ + i[ 1).
(3.21b)
may be written as
+ go2A).
Collecting our results, the transformed Hamiltonian has the structure
il = Ho + f A.2 Q,J,12 + ; i2ig$
+ *+dJeei -9
(3.22)
+ ‘4 z 6-TdJm.a, JBI+ + . . . , Provided 6X, and 6XnB are truly small, we have isolated a rotational energy;
236
FELIX
VILLARS
the moment of inertia is 9 = g”(l + 4’A) = g”(l + i4’ (n j [D, , ?,I / n)) or, slightly rewritten:
This is equivalent, within the approximation used, to the result’ obtained in Chapter II. (C) DISCUSSION
OF THE APPROXIMATION
First a word of explanation is in order concerning Eq. (3.15). One might wonder why the term >aCA QAJA’” was not incorporated into Ho , rather than splitting H as in (3.2), (3.3). This procedure would have led to a considerably simpler equation. We have avoided this in order to exhibit the essential fact that 9 is a positive constant. The above mentioned alternative procedure unfortunately does not let this fact become manifest. Indeed, instead of 9 = g”(l + $'A), we would obtain g = go/(1
-
goA')
(3.24)
with A' constructed as A, but using eigenstat,esand energy-levels of
instead of those of Ho . Equation (3.24) is clearly an undesirable expression, inasmuch as no obvious limits can be set on the value of A'. Also, as will be seen in Chapter IV, any use of a model simulating the structure of the Hamiltonian (1.7), makes our procedure appear the most natural one. The present approximation is based on the smallness of CL From (3.16) and (3.20) it follows that -i(n’ 1(Y1n} = ~(12’ 1Q.iJA’ 1n)/(E,,’ - EJ). This is of the order of (n’ I J’ I n)/'SAEi,,,. . The mat)rix-elements of J’ will be of order unity (but this ran only roughly be ascertained through the use of models); CYis then essentially the ratio LY-
g-l/AEi,t,,
.
(3.251
The interest, ‘df ihis expression is that both quantities are experimentally accessible (rather than theoretically). It is indeed a characteristic feature of nuclei exhibiting rotational states that the ratio (3.25) is small compared to unity. This of course does not tell us for which nuclei this will happen; but it indicates that, provided it happens, the present approach should give t’he right answer.8A simi8 Notice parameter
that the alternative cy’ m (g')-l/AEi"t,
method leading which, experimentally.
to Eq. (3.24) makes use of an expansion is 3-5 times larger than (3.25).
COLLECTIVE
ROTATIONS
IN
237
NUCLEI
lar situation prevails with respect to the second condition (besides the smallness of a) for the occurrence of rotational states: The smallness of the rotation-vibration interaction, x c 6XAB[JA , JB]+ . Smallness of this term implies smallness of the off-diagonal elements of QA and K,, :
Sufficient-but the inequalities
perhaps not necessary-conditions
may thus be expressed,through
(f n I Q-4” 1n>/(~“)” - 1) c 1, ‘h 1 F KAB&IA
1n)/A”
(3.26a)
- 1) << 1.
Equation (3.26a) has a direct kinematic interpretation, as the condition of the smallness of the amplitude of shape-fluctuations compared to the amplitude of deviation from spherical shape, A minimum deformation is clearly necessary to satisfy this condition. The analysis of (3.26) can hardly be pushed further without actually introducing a model simulating the structure H = HO + H,,, . We shall indicate in the next chapter what we consider to be the essence of such a model. Here we just observe one basic point: Assuming (3.26) to hold, it also follows that the operator FjXA (3.19b) has negligible dynamical effect. Were this not so, 6X, would affect the value of the effective moment of inertia. Thus, provided the rotation-vibration coupling is small, (3.23) is actually the correct value of the effective moment of inertia. These considerations do not answer the question whether the condition 1 (n’ ] (Y 1n) 1 << 1 is in fact necessary for the occurrence of rotational spectra. All that has been shown is that, together with some additional conditions [roughly expressed by Eqs. (3.26)], the smallness of a is sufficient. It has to be recognized that the methods presented here do not allow to establish necessary conditions on CY.This difficulty hinges on the fact that in the generating function S, Eq. (3.11), both the components of (Y and of J do not commute among themselves, so that p becomes an infinite series in powers of both a! and J. IV.
SOME
REMARKS
ON
THE
USE
OF
NONSPHERICAL
MODELS
The results developed in the previous two chapters are “basically correct” in the sense that we have derived them by dealing with the actual Hamiltonian of the nucleus. But one realizes that, in proceeding in this way, the analysis cannot be pushed beyond a certain stage, and results like Eq. (3.23) cannot be exploited numerically. At this point, a model of the actual system may be helpful. The essential step in constructing a model will consist of ignoring the identities (1.10) satisfied by the variables xi*’ and piA’ and considering them as free pairs
238
FELIX
of canonical
variables.
These variables
bh’, am’1 = 0,
VILLARS
satisfy
the relations
[PiA’,pm’1 = 0(1/A),
(4.1)
b.h’, JIRB’I = i&RBAB + 0(1/A). We thus drop the l/A-terms, except where they add up to a significant correction, as in the commutators (l.lZa, 12b) and (1.14). But there they can be taken care of in the following way: First we also replace the principal moments IA by constants :
II = I2 = lo(l - NP), keeping p-the define
deformation
parameter-as
Jtl = 7 h’pi;
- zi;pil’>,
13 = Jo(l + 4;iP) an adjustable
constant.
(4.2) If then we (4.3a)
(4.3b)
and assume normal commutation relations between the zia’ and piA’, Eqs. (1.12) and (1.14) will be satisfied. Finally, the total angular momentum J may, for the purpose of constructing a model, replaced by a constant vector. This is justified by the fact that in the approximation used to derive the basic result (3.23), no use was made of the commutation relations of the JA . We also keep in mind that for the axially symmetric case JB = J3’, so that the model-Hamiltonian will only contain J1 and J, as “external” constants. It may then be written as: H
model
=
&
c
pi/l“
+
V&A’)
+
&
(.I,
rx3
-
er,‘)“.
(4.4)
In (4.4), JI is a constant vector perpendicular t#o the &axis; JL’ is the vector (4.3b); grip. = 210 , Eq. (4.2), is the rigid-body moment around the 3-axis;g,i,.@’ is the value of Q1-’ and Q2? according to (1.9) and (4.2). The parameter p is to be determined from (4.2), using la = (xi Xia”). The most natural way to use Eq. (4.4) is to derive from it the single particle equations of a Hartree-Fock approximation; the effective moment of inertia follows then from the dependence of (E) on 5’. The single particle states will be those of an axially symmetric potential; in the end, /3 is to be determined by a self-consistency condition. It is evident that this model is closely related to the Inglis “cranking model”, (16), but not exactly identical, mainly due to the different structure of the operator J’, which here is given by (4.3).
COLLECTIVE
V. THE
ROTATIONS
COLLECTIVE
IN
239
NUCLEI
MAGNETIC
MOMENT
The magnetic moment of rotational states contains the two parameters and ga , the collective and intrinsic g-factors, respectively (9, 17): P = g&
($37- Se> + Igc.
gC
(5.1)
It is well known (9, 18, 19), that gC deviates considerably from the value Z/A, expected on the basis of both rigid and irrotational collective motion. The methods developed here make it possible to determine the deviation of gCfrom Z/A. The magnetic moment operator is a tensor with space-fixed components
(gli and gSi being orbital the relations :
and spin g-factors
of the nucleon i). With the help of
xia = x-a+ c RA,GA’ ,
(5.3a)
A
and of
I;: eixiA’xid it follows
iZ
e z
(5.4b)
116~~)
that:
MLAB1’ being defined as the operator MrnBl’
= F
(g&iA’&
-
%$%A’)
+
gs&JABl).
(5.5)
Hence
(M-J)
=
a I&
= & @This tively.
expression
is derived
MIABIJIABI
{(M’.J) in (4).
+ f (J” - J’.J)j.
X, and P, are c.m
position
and
momentum,
respec-
240
FELIX
VILLARS
that J3 = J3’ = K, and using the notation
Remembering we have
/et = { (M,‘)K
(a.b)
= al.bl
+ (M,‘. J,) + ; @(I + 1) - K” - (J,‘. J1))j
&.
+ a&
(5.6)
The expectation value (Af,l) is equal, by definition, t.o g& = gaK. The two other expectation values, (MI’. Jl) and ( JA’ . J I ) must be calculated in the same approximation that leads to (3.23) (5.7a) (5.7b) The k-sum in (5.7b) is nothing we get,
but doA
Both (5.7a) and (5.7b) contribut’e
goA.
=
Trsing the value (3.20) of K: K
go/g,
1 - go/g.
to gC only. It follows:
.)))
(5.8)
(5.9)
go = (M,‘)IK.
The approximations that have been used to derive t,hese two last equations are those made in Section III, plus Eq. (5.4b). This latter st’ates that the charge distribution in a deformed nucleus is essentially proportional to the mattIer distribution. Equation (5.9) is a familiar result (1, 20). Equation (5.8) is new, but again a model such as outlined in Section IV will be needed for its evaluation. This, and the evaluation of (3.23) will be the subject of :I subsequent, publication. APPENDIX
We shall sketch here the derivation of the basic commutation relations (1.1 I), (1.12), and (1.14). Starting point are the two Eqs. (5.3)) which have been derived in Ref. 4. The body fixed frame from ilBC is defined by the propert,y of the variables zia’, and given in Eqs. (1.6), or, equivalently, the second Eq. (l.lOa). Using this definition, and (5.3), it follows that
Let us also define the function
COLLECTIVE
ROTATIONS
IN
241
NUCLEI
(A.21 Inserting
the expressions
(5.3) for xia and pi, , one finds, for (A Z B) :
NAB E z
RAoRs$Vap = ‘e
(Jr-m - JWI ‘I.
With (A.2) and (A-3), it is possible to evaluate all commutators between J[As) , N AB , and RAor , using their representation in terms of the original variables zirr , pi, . For instance:
[J~ABI, JCACII = -iJ[scl t [J [ABI
, Ncol
‘3.4)
~{{B~JIA~I~B~
=
+
{EZCJ[ACI~BD
{ADJ[&AC
-
(A.51
-~AcJ~B~I~AD), IN AB,
NcDl=
+
~((CAD
+
+
TBc)(NAc~LsD
+
((AC
NH&AC)
[BD)(NA~Bc
+
- (~AcJ~BD~ + ~BDJLACI + In (A.4)-(A.6),
(A-6)
NBCSAD) ~ADJLBCI
+
~B~J~AD~)J.
the abbreviation l
AS
=
(IA
+
IB)/(IA
-
IB>
has been used. Using these relations, we can now calculate the commutator the intrinsic angular momentum J’. From (A.3) it follows indeed that [J~ABI’,
JIACI’I
=
[JIABI
,
JIA~JI
-
WAS,
-
of
J~ACIIICAB
[J[ABI
,
NAC]/{AC
+
[NAB
,NAc]/b~dAc.
Inserting (A.4)-(A.6), Eq. (1.11) is gained. A similar, but rather lengthy procedure gives (1.12). Finally, Eq. (1.14) follows directly from (A.3), using the fact that V(zJ commutes with the total angular momentum, J[AB] . RECEIVED:
August
11, 1958 REFERENCES
1. A. BOHR AND B. R. MOTTELSON, Kgl. Danske Videnskab. No. 16 (1953). Ejnar d. A. BOHR, “Rotational States of Atomic Nuclei.” hagen,.1954. 3. K. ALDER, A. BOHR, T. Huus, B. MOTTELSOPIT, AND A. 28, 432 (1956) (Chapter V). $. F. VILLARS, Nuclear Physics 3, 240 (1957). 6. H. H. NIELSEN, Revs. Modern Phys. 23, 90 (1951).
Selskab, Munksgaards
Mat.-fys.
Medd.
Forlag,
WINTHER, Revs. Modern
27,
CopenPhys.
242
FELIX
VILLARS
6. R. DE L. KRONIQ, “Band Spectra and Molecular Structure.” MacMillan, New York, 1930. 7. H. J. LIPKIN, A. DE &ALIT, AND I. TALMI, Nuovo cimento Ser. 10, 2, 773 (1955). 8. T. TAMURA, Nuovo cimento Ser. 10, 4, 713 (1956). 9. R. NATAF, Nuclear Physics 2, 497 (1957). 10. S. HAYAKAWA AND T. MARUMORI, Progr. Theoret. Phys. Japan 18, 396 (1957). 11. J. J. MURRAY, F. BOEHM, P. MARMIER, AND J. W. M. DUMOND, Phys. Rev. 97, 1007 (1955). 12. A. KERMAN, Kgl. Danske Videnskub. Selskab, Mat.-jys. Medd. 30, No. 15 (1956). IS. A. BOHR AND B. R. MOTTELSON (in press). 14. A. J. LIPKIN, A. DE &ALIT, AND I. TALMI, Phys. Rev. 103, 1773 (1956). 15. D. R. INGLIS, Phys. Rev. 96, 1059 (1954). 16. A. BOHR AND B. R. MOTTELSON, Kgl. Danske Videnskab. Selskab, Afat.-jys. Medd. 30, No. 1 (1955). 17. R. J. BLIN-STOYLE, Revs. Modern Phys. 28, 75 (1956). 18. S. OFER, Nuclear Physics 3, 479 (1957). 19. G. GOLDRING AND R. P. SCHARENBERG, Phys. Rev. 110, 701 (1958). 80. R. G. NILSSON, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 29, No. 16 (1955).
OF
PHYSICS:
6,
224242
Collective
(19%)
Rotations
in Nuclei
FELIX VILLARS Department
of Physics
and Laboratory of Technology,
of Nuclear Cambridge,
Science, filassachusetts Massachusetts
Institute
A systematic approach is made to the problem of determining the parameters of collective motion in nuclei. Formulas are given for the moment of inertia of the rotational energy and for the collective g-factor of the magnetic moment. These expressions are not based on the use of a model of the nucleus. Such a model will be needed, however, for the numerical evaluation of the espressions derived. The essential elements of a usable model are briefly discussed. INTRODUCTIOX
Many properties of rotational states in nuclei, and of transition amplitudes between such states, may be expressed in terms of simple, essentially kinematic relations, Examples of this are the ratios between EZ-transition amplitudes within a rotational band, and their connection with the static quadrupole moments, and the corresponding relations between the Ml-transition amplitudes, and the magnetic moment (1-S). These relations hold irrespective of the dynamics, but they contain parameters whose theoretical determination requires a detailed understanding of the interplay between collective rotation and intrinsic motion of the nucleons. Such parameters are the moment of inertia 9, determining the spacing of rotational levels, and the collective magnetic g-factor g,: ~ resulting from the collective current flow. The purpose of this paper is to present a systematic approach to the problem of determining 9, gC and other parameters pertaining t’o states of collective rotational excitation. The method used is that of coruact transformations, with the help of which the variables describing collective dynamics are stepwise decoupled from the “intrinsic” motion. It is shown that the desired result can be achieved in two steps. In the first step, a body fixed coordinate system is introduced, defined by the principal axes of the mass distribution in the nucleus. The Euler angles of that system, and the project,ions of the t)otal angular momentum on the body fixed axes are introduced as new dynamical variables. This leads to t,he “canonical form” of the nuclear Hamiltonian (4). Tn the canonical Hamiltonian, the total angular momentum is still coupled to t&he intrinsic motion; t,he decoupling can however, now, be achieved by perturbation theory. 224
COLLECTIVE
ROTATIONS
IN
225
NUCLEI
In Section I, the canonical form of H is presented, together with a few relevant additional relations. In Section II, the rotational moment of inertia is calculated by a simple, classical method. In Section III, a more rigorous quantum-mechanical version of the same calculation is given, and the validity of the approximation is discussed. In Section IV a few remarks are made on nuclear models simulating the structure of the canonical Hamiltonian. Such models are needed for any numerical work. Finally, in Section V an expression for the magnetic moments of rotational st.ates is derived. I. SEPARATION
OF TOTAL ANGULAR MOMENTUM OF INTERACTING PARTICLES
IN A SYSTEM
It is well known that the Hamiltonian of a rigid body may be written in the form’
(1.0 Jp being the components of the total ahgular momentum, and gP the principal moments of inertia. As a model of a quantum-mechanical system, the rigid body represents generally a gross idealization of actually occurring situations. Nevertheless, spectra characteristic of the eigenvalue system of (1.1) are found in many instances, in particular in molecules and nuclei. For the former, the Hamiltonian can be written in the form (5, 6):
H = &
+ c (Jc 29 Jc’)2 + H,,(T, E), c C
(1.2)
Jc - Jc’ = Rc being the angular momentum associated with the rotation of the body-fixed frame of reference ABC, and l, ?r being additional position and momentum variables in the rotating frame. Jc is the total angular momentum, projected on the body fixed axes, whereas J C’ is the sum of intrinsic (electronic and
nuclear) angular momenta, calculated in the body fixed frame. It is not generally . true that a system such as (1.2) should have an eigenvalue spectrum-as far as
1Kotation: P, Q, R, and A, B, C will be used as subscripts to indicate vector components in a “body fixed” coordinate system. Vector components in a space fixed system will be indicated by subscripts (Y,8, 7. For axial vectors we use the tensor notation, if convenient. In this case we use a square bracket to indicate antisymmetry: JIPQI
In connection with this notation, rotation-matrices RPA :
=
RPA
RQB
-
RQ* RPB =
RCPQI.[ABI
=
JR.
it will often be useful to introduce
R lPQ1.A P&R being an even permutation
- JIQP]
-R[QPI,A
=
=
RRA
double indices into
,
of 123. Use may then be made of the relations =
RRC
)
ZP
RIPQI~A
RPB
=
RQ,IABI
j etc.
226
FELIX
VILLARS
dependence of energy levels on angular momentum is concerned-similar to sy: tern (1 .l). That this is so for molecules is due to the near constancy of the functiojl s,(t) on the one hand, and on the other hand to the extreme smallnessof the ratio (fi’/24)/AEint,. , AEi,t,. being an intrinsic (vibrational or electronic) excitation energy. As a result of this, the coupling term
-c
JcJc’lgc
(1.3)
is an extremely small perturbation, and the rotational spectrum remains to a high accuracy determined by the quasi-rigid body Hamiltonian
F Jc%&
0.4)
alone. The observation of rotational bands (and even of systems of rotational bands) in nuclei leads to the question whether of a term of type (1.4) can be separated out in the nuclear Hamiltonian in certain cases.Clearly this cannot be generally possible. On the other hand, it can indeed be shown that the Hamiltonian of any closedsystemof particles can actually be written in the form (1.2). This form, then, doesnot in general lead to rotational spectra as given by (1.4); but it should provide a convenient starting point in case of their actual occurrence, and also for an analysis of the conditions under which this may happen. Characteristic of both (1.1) and (1.2) is the introduction of 3 “body-fixed” axes. In a non rigid system such as a nucleus, such axes may be introduced in many ways, in terms of the principal axes of any tensor T&z; , pi) symmetric in the nucleons. Actually the close connection between occurrence of nuclear rotational spectra and nonspherical shape of nuclei suggests the choice of the mass tensor2 (15) to define those principal axes: Iaa = c &mRA8te,)IA A
.
(1.6)
In (1.6) I, are the principal moments, and the Ra,(f?,) rotation matrices connecting the spacefixed coordinate axes c& with the principal axes ABC. In the case of nuclei, the structure (1.2) of the Hamiltonian is obtained through a coordinate transformation, which isolates the total angular momentum components JA and the associated Euler angles 0, as independent variables. The residual intrinsic variables .$ , 7rS, u = 1, 2, 3 . . * (3A - 6) appear only through the functions ziA’(&) and piA’(?r# , t.), which we call “intrinsic” positions and * Nucleons will always be numbered
by an index i: i = l...A;
X, is the c.m. position.
COLLECTIVE
n ,menta. It was shown be written as3
grn
H=
ROTATIONS
IN
NUCLEI
227
in (4) that in terms of the latter, the Hamiltonian
+ & c pi/
+ V(GA’) + f A$ QACJA - JA’)~,
can
(1.7)
%,A
m being the nucleon mass, A the mass number, and V the potential energy. intrinsic angular momentum JA’ Barring spin-whose inclusion is trivial -the is the function
J,’
= J&
= c
(xa’pic
- xic'pi~')
(1.8)
and the zero order moments of inertia &A are expressible in terms of the principal moments IA (1.6) as
QAdl = m(IB - Id2/(IB
+ Ic).
(1.9)
It must be realized that the variables xin’, piA’ do not form canonical pairs; they are not even independent variables, but satisfy the identities: F
%A’
= 0,
F
PiA’
=
F XiA’xia) = F (zi~‘pis’ +
0,
(l.lOa)
IA6.48,
xiB’piA’>
(A # I?).
= 0.
(Mob)
As a result, the properties of Ja’ (1.8) are not exactly those of an angular momentum; instead, one has the Poisson-bracket relations: (JA’>
{JwI’,
JB’)
=
xic’l
= ~In + IB kif8BC
{ J~.oI’, pit’)
-(IA
+
Ic)(IB
+
21,
= I,
21,
Ic>
- m
- &
P~A’~Bc
(1.11)
Jc’~
21, XiB’aAC
pi~‘Lc
,
(1.12a)
+ 0(1/A).
(1.12b)
B
[These A-’ terms may be important, occasionally. The derivation of these results and of Eq. (1.14) will be sketched in the Appendix.] As a consequenceof these relations, the angular momenta Jn’ do not commute with the intrinsic energy
Ho = &
c pin12 + V(zd).
(1.13)
GA
Their commutator,4 however, is proportional to the deviation from spherical 3 Similar expressions have been derived by various authors (7-10). * We shall use, in the text, synonymously the word commutator and Poisson-bracket; but writing the expressions as Poisson bracket relations, to be replaced by i/C times commutator in quantum mechanics.
228 symmetry
FELIX
VILLARS
:
For nuclei, the structure (1.7) of the Ha.miltonian represents a much lesser degree of coupling of the intrinsic motion (governed by Ho) from the collective rotation than the corresponding expression (1.2) does for molecules. In fact!, the ratio
is not very small in the nuclear case. [As an example, we may take the band system of Wls3 (11, 12), where this ratio is about fh]. Of course, the occurrence of rotational spectra does not follow from (1.7j, since this holds for an arbitrary system.5 If such spectra occur, we must assume the existence of a set of axes P&R, which allows a term of type (1.4) to be separated from the intrinsic motion, wit,h the addition of small correction term (rotation vibration interaction). The experimental evidence about the kinematics of collective rotation indicates that these axes P&R must be closely related to the principal axes ABC of the mass tensor as defined by (1.6). This leads to t,he physical picture of the axes PQR fluctuating about ABC, these latter defining their average position. The analysis of next chapter will show to which extent, this picture is verified. II.
DETERMINATION
OF
THE
PRINCIPAL
MOMENTS
The existence of rotational spectra implies the possibility of writing the nuclear Hamiltonian, in appropriate variables, as the sum of 3 terms: Center of mass energy, rotational energy, and intrinsic energy. The first two of these terms have the structure given by Ey. (1.1); the third is an operator which, to a good approximation, commutes with all the variables describing center of muss- and rot.nGonal motion. Bohr and Mottelson (IS), as well as Lipkin, de Shalit, and Talmi (1 /t), have pointed out that if this structure of the nuclear Hamiltonian is admitted, the principal moments may be found in principle from the set, of relations:
JP = c Rd&)J,
,
{JP, R,,j = R,,, c {H, %7)&a = JR/~, t 7 5 Observe, however, that the transformat.ion leading to (1.7) becomes singular cal principal moments IA(E). Only if two of the Qa fluctuate about an essentially value does this coordinate transformation assume meaning.
(2.1) (2.2) (2.3) for identinonzero
COLLECTIVE
ROTATIONS
IN
229
NUCLEI
the RPa being rotation matrices relating the body fixed frame (P&R) to the space fixed axes c@y. These equations may be thought of as written in terms of the original variables zirr , pi, with Euler-angles 0, = I!?*(xi.!, piol) and associated rotations matrices Rprr(Os), since in these variables the nuclear Hamiltonian has a known structure. We shall, in this chapter, carry out this program, in the classical limit; but we shall take advantage of the form (1.7) of the nuclear Hamiltonian. In this case, the matrix Rp, may be broken up into two parts: RpaW
= 5 RpC@“)Rca(e’>.
(2.4)
The system of “intermediate” principal axes A, B, C was introduced and used in (1.7). With the help of the relations
{Ho, Rca) = 0,
{J,,
Rcaj =
We find from Eq. (2.3), using the Hamiltonian
R Pa F
&a,
in Eq. (1.6)
a,
(1.7) and expression
(2.4) for
:
{H,, , RPc)RQc + F --- (Ja - JA’)(RA~ - F {JA’, Rpc)RQc) = JR/sR.
In the interest of simplicity, we have assumed here that QA commutes with the matrices RpC , and may be replaced by a constant 1/9A”. This point will be reconsidered in Section IV. To satisfy (2.5), we must put the terms not containing J equal to zero, and equate the coefficients of JR with the help of the projection
JA = c
R
JRRBA .
In this way we find from (2.5) the two basic equations: . 5
{Ho,
Rpc)Roc
- T 2
J*‘(RnA
- F
(JA’,
RPC}RQC)
=
0
(2.s)
and
F $0 RwA(RRA - F (JAI,
R~c}RQc)
Z 8Rw/gR .
(2.7)
Equation (2.6) defines the rotation matrix RpA(eM). It is clear that 0” will be a function of the variables &, , ?T#; the RpA do therefore commute with the Ja but not with the JAI, of course. Equation (2.7) is then the equation defining g, by means of RIIA . It is easily seen that Eq. (2.7) cannot generally be consistent (such consistency would be equivalent to the unconditional existence of rotational states). Indeed, from (2.7) we may derive:
230
FELIX
R RA - 5
VILLARS
(Ja’, RPC) Rw Y $
(2.7a)
Rm ,
a relation which certainly imposes restrictive conditions on the structure of the system to be even approximately true. But assuming (2.7a) to hold, we may feed this relation back into (2.6), to get
F {Ho , RPC] Rot = ;
F JA’R,u .
Equations (2.7a) and (2.8) are the basic results. To explore their implications, let us assume that the angles 8” are small, so that
R,u = 8zz.4+ %A, To lowest
Q~RA=
;
--SlAR
/&A/
((1.
cw
order in 3, (2.7a) reduces to
I Jzi’, fb2pql =
6RA(1
-
(2.10)
gA”/&)
or
{JR’, QppJ = 1 -
&$ti,
In Eq. (2.8) a similar simplification
A #
(JA , QPQ} = 0,
;
[P&l.
occurs, to lowest order in 8: (2.12)
{Ho, QPQ) = JR’Igx. At this place, let us introduce
the notation
s= and the inversion
(2.11)
{Ho,F}
(2.13a)
E;
of this relation
F = s. . With this notation,
Eq. (2.12) may be formally f2PQ = JR’/Sg .
For a first orientation
(2.13b) solved as
- ((&)’
J,‘).
(2.14)
.
let us assume that
(Ho, 4E) r 0,
{J,‘,
9R) S 0;
These assumptions simply imply that gR is approximately follows from Eqs. (2.11) and (2.14) that
a constant.
$R = 9; + (JR’, JR’}. .
This is indeed a familiar we have
result. Translated
into quantum-mechanical
It then (2.15) language,
COLLECTIVE
ROTATIONS
IN NUCLEI
an expression reminiscent of the “Inglis-formula”
231
(X), and (2.15) becomes
In this present equation, the term 4 llo refers to the zero-order moment as appearing in (1.7)) and whose value is given by the expectation value of (1.9)) which is the rigid body moment times the square of the deformation. The measured values of gR are well known to be much larger than this; the main contribution must therefore arise from the second term of (2.16). Equation (2.16) however diiers significantly from the Inglis-formula
in two respects: (a) In the Inglis formula (2.17) the states I n) and 1k) are independent particle states in a deformed well, and JI1 is the actual angular momentum of the system, which in this model is not a constant of the motion, due to the artificially introduced anisotropy of the potential energy. In contrast to this, our Equation (2.16) contains the matrix-elements of the “intrinsic angular momentum JR’, which does not commute with Ho ; but both J’ and HO commute with the total angular momentum J, which has been separated off by a coordinate transformation, as outlined in Section 1. (b) The energy denominators in (2.16) refer to the eigenstates of HO and thus correspond (apart from small level shifts) to the actual intrinsic excitation energies, and may be taken from experimental data. In the Inglis model they refer to single particle excitations.6 III.
CRITICAL
APPROACH
TO THE
PROBLEM
The developments of Chapter II were deliberately sketchy, in order to present clearly the main line of argument. Here we shall take up the problem anew, but apply the method of canonical transformations to separate the collective rotation from the intrinsic motion. This more powerful approach will also enable us to establish conditions for the occurrence of rotational spectra. In order to carry out this program, we must begin with a few preliminary statements concerning the eigenstates of the Hamiltonian (1.7). As will be seen, a few basic approximations have to be made from the outset; unless they a.re 6 The Inglis model can be made more “realistic” by introducing inter-particle couplings, which strongly affect the energy-denominators in (2.7). If this is done, the results obtained from (2.17) can be brought into reasonable agreement with experiment (16).
232
FELIX
VILLARS
made, the situation seemsto become entirely opaque. But we bear in mind that they may somewhat restrict the general validity of our results. (A) THE EIGENSTATES
(1.7)
OF THE HAMILTONIAN
The coordinate transformation discussed in Chapter I, brings the nuclear Hamiltonian into the form H = Ho + Hrot ,
(3.1)
Ho = C piAr2/2m -I V(Si.d’), iA
(3.2)
H rot = ; F &“(JA
- JA’)“,
with QA being given by (1.9). A characteristic property of the zero order moments is that, whenever two of them are equal, the third has to be zero. This axially symmetric situation seemsto be prevalent, according to our experimental information, and the present discussion shall be restricted to cover this case: We assumethat for all states of Ho of interest, we have (n 1&I 1n) = (n I Qz1n) = l/k’!
(3.4)
It follows then that (n j Q31n) >> l/‘.%?.
(3.5)
((Q3) is not infinite, since Q3does not commute with Ho.) In order to exploit, the advantage of the cylindric symmetrical case, another approximation has to be made. As seen from Eq. (1.14), the commutator’
[Ho, Js’l becomes very small for cylindrical symmetry; and we shall approximate HO by an operator no defined as follows: Let a be the eigenvalues of J,‘, and introduce the projection operator pn = n (J3’ - L”)/(fl - 8’). n*+n Then Ho may be decomposed
Ho = cn PslHepsl+ F (1 - PdHoPn
(3.6)
= i?o + 6Ho. This enables us to define a complete set of simultaneous eigenstales of go and JZ’ 7 In t,his
and the
following
chapters,
we put
ii = I.
COLLECTIVE
ROTATIONS
flo I vf2) = Eva I vi?}, There is a degeneracy
of E with
IN
233
NUCLEI
Js’ 1 vn) = n 1 vf2}.
(3.7)
respect to the sign of 3: Evn = -t-%,-n
but apart from this we may assume the levels of f10 to be nondegenerate. 6Ho has no diagonal elements then, and its effects will not be considered here. From Eq. (1.11) we derive the commutation relation:
[J3’, JI’ + iJ2’l = JI’ + iJ2’ - r&)reJ;
- ieJ:>
and a similar relation for [J3’, J1’ - i Jz’]. This justifies an additional mation, in which the only nonzero matrix-elements of J1’ f i Jz’ are
(3.8) approxi-
(vQ 1 J1’ f i Jz’ 1 v’, D T 1). From Eq. (1.12) it follows in addition that the zero order moments QA cornmu& with a.11the J,‘; their matrix elements are therefore diagonal in 0. With these approximations in mind, the problem is determined by the zero order Hamiltonian RO (thereafter written as HO), and the perturbation
$4 QI(JI There is an additional
- JI’)’
+ 34 QdJ2 - Jz’)‘.
(3.9a)
term (3.9b)
45 QdJs - JA2
which is dropped in the cylindric symmetrical case considered here: In a representation in which J3 is diagonal, Ja = K, this term has matrix elements
$5 (vn ( Qs 1 ~‘0) (K - fQ2. The diagonal elements of Q3 were found to be extremely large; hence only states to low energy excitation. States with K # fi can be coupled in only through the part
K = D correspond
34 (&I - Q2) ((J,
- J+‘12 + (J- - J-‘j2)
of (3.9a), and there only through the off -diagonal elements of Q1 since (Q1) = (Q2). Thus, provided the cylindrical symmetry is stable:
((&I - Q2)“) << (&I>~, states K # D are not coupled in, and the term (3.9b) may be dropped altogether. This is done in the following sections. (B)
CANONICAL
TRANSFORMATION
It is obvious enough that the rotation canonical transformation
RPA(O”)
[Eq. (2.4)] is equivalent
to a
234
FELIX j=-j
with
S being the rotation
VILLARS =
pJJe-is
(3.10)
operator: S = (a.J).
(3.11)
In II, the problem was “solved”, assuming (Y to be a small angle. We shall first carry out the transformation (3.10) under this assumption, and have a look at the higher approximation later. Let us introduce the notations
wo , {I = !.I
wo >!21= &, for any operator
q. Additional
useful abbreviations
(3.12)
will be
(A = 1, 2 only)
DA = QAJ~‘,
(3.13)
and
K AB = i[DA , D,& . It then follows
(3.14)
that
c J, (ck + DA - ; F (J&A,
B = H,+;TQAJA”-
+ [aA , DB]JB’)
+ ; 3
[JA , Jsl+
- bA,bB,~
(
-ita~,
’ cc Q-‘D;]l
&I
Del - %x4,
+ &A&o
>
Dd
(3.15)
+ . ** .
The problem is now to cancel the terms linear in JA . This may be done approximately by means of the ansatz . --DA, (3.16) &A = K being a numerical then be written as
(1 -
constant
K)DA
yet to be determined.
-
i
F
(KBAQBwlDB
The coefficient
of - JA may
+ DBQB’K,,).
(3.17)
Now we notice that
K
- KBA = i[DA . , DB]. . part (KeA + KAB)/2 has a nonzero expectation
Hence only the symmetric this latter has the additional
AB
value;
property (n
1 KAB
( n>
=
ARAB
.
(3.18)
COLLECTIVE
This follows
+
K2,
IN
235
NUCLEI
that A, B = 1, 2 only, and
from the observation Kl2
which
ROTATIONS
=
pi
(D,
D+ .
-
B-
D-1, .
of Jsl. For
has no diagonal elements with respect to eigenstates (n I K4n I 4,
the independence on A follows from the assumed cylindrical (3.17) may then be written as (1
6Xn = ‘2 (C The numerical in J
B
constant
K
LB
-
K +
KS’A)&
(I-L),
-
+
Jil,
is now determined
symmetry.
Equation (3.19a)
6x,
+ 2A(l - tiO&adJ:
>
.
so as to minimize the linear coupling
1
(3.20)
x=1+ Looking (K”
-
at the quadratic
~K)&IA
+ ; c
c
terms, the coefficient
f&i&c-%
(3.19b)
of x [JA , JB]+ becomes
+ Kc~a&c-~Kc~)
- d&, [DA, . Qc-‘Gil+ + iI+,
J,lr+
, Qcll) + &As
This again will be split into a main term and a residue ((K2 6x~e
=
(K2
-
+ f 7
~K)(KBI
~K)A
+
K2goA2
+
(g”)-‘)8,,
+
6XAB
(3.21a)
- (Ksn)) + (&A - 1/gob,
&A&C-%
+
Km&c-‘Kc,
The main term, using the value (3.20) of &i&go
K,
- 24’A2 - i[ ]+ + i[ 1).
(3.21b)
may be written as
+ go2A).
Collecting our results, the transformed Hamiltonian has the structure
il = Ho + f A.2 Q,J,12 + ; i2ig$
+ *+dJeei -9
(3.22)
+ ‘4 z 6-TdJm.a, JBI+ + . . . , Provided 6X, and 6XnB are truly small, we have isolated a rotational energy;
236
FELIX
VILLARS
the moment of inertia is 9 = g”(l + 4’A) = g”(l + i4’ (n j [D, , ?,I / n)) or, slightly rewritten:
This is equivalent, within the approximation used, to the result’ obtained in Chapter II. (C) DISCUSSION
OF THE APPROXIMATION
First a word of explanation is in order concerning Eq. (3.15). One might wonder why the term >aCA QAJA’” was not incorporated into Ho , rather than splitting H as in (3.2), (3.3). This procedure would have led to a considerably simpler equation. We have avoided this in order to exhibit the essential fact that 9 is a positive constant. The above mentioned alternative procedure unfortunately does not let this fact become manifest. Indeed, instead of 9 = g”(l + $'A), we would obtain g = go/(1
-
goA')
(3.24)
with A' constructed as A, but using eigenstat,esand energy-levels of
instead of those of Ho . Equation (3.24) is clearly an undesirable expression, inasmuch as no obvious limits can be set on the value of A'. Also, as will be seen in Chapter IV, any use of a model simulating the structure of the Hamiltonian (1.7), makes our procedure appear the most natural one. The present approximation is based on the smallness of CL From (3.16) and (3.20) it follows that -i(n’ 1(Y1n} = ~(12’ 1Q.iJA’ 1n)/(E,,’ - EJ). This is of the order of (n’ I J’ I n)/'SAEi,,,. . The mat)rix-elements of J’ will be of order unity (but this ran only roughly be ascertained through the use of models); CYis then essentially the ratio LY-
g-l/AEi,t,,
.
(3.251
The interest, ‘df ihis expression is that both quantities are experimentally accessible (rather than theoretically). It is indeed a characteristic feature of nuclei exhibiting rotational states that the ratio (3.25) is small compared to unity. This of course does not tell us for which nuclei this will happen; but it indicates that, provided it happens, the present approach should give t’he right answer.8A simi8 Notice parameter
that the alternative cy’ m (g')-l/AEi"t,
method leading which, experimentally.
to Eq. (3.24) makes use of an expansion is 3-5 times larger than (3.25).
COLLECTIVE
ROTATIONS
IN
237
NUCLEI
lar situation prevails with respect to the second condition (besides the smallness of a) for the occurrence of rotational states: The smallness of the rotation-vibration interaction, x c 6XAB[JA , JB]+ . Smallness of this term implies smallness of the off-diagonal elements of QA and K,, :
Sufficient-but the inequalities
perhaps not necessary-conditions
may thus be expressed,through
(f n I Q-4” 1n>/(~“)” - 1) c 1, ‘h 1 F KAB&IA
1n)/A”
(3.26a)
- 1) << 1.
Equation (3.26a) has a direct kinematic interpretation, as the condition of the smallness of the amplitude of shape-fluctuations compared to the amplitude of deviation from spherical shape, A minimum deformation is clearly necessary to satisfy this condition. The analysis of (3.26) can hardly be pushed further without actually introducing a model simulating the structure H = HO + H,,, . We shall indicate in the next chapter what we consider to be the essence of such a model. Here we just observe one basic point: Assuming (3.26) to hold, it also follows that the operator FjXA (3.19b) has negligible dynamical effect. Were this not so, 6X, would affect the value of the effective moment of inertia. Thus, provided the rotation-vibration coupling is small, (3.23) is actually the correct value of the effective moment of inertia. These considerations do not answer the question whether the condition 1 (n’ ] (Y 1n) 1 << 1 is in fact necessary for the occurrence of rotational spectra. All that has been shown is that, together with some additional conditions [roughly expressed by Eqs. (3.26)], the smallness of a is sufficient. It has to be recognized that the methods presented here do not allow to establish necessary conditions on CY.This difficulty hinges on the fact that in the generating function S, Eq. (3.11), both the components of (Y and of J do not commute among themselves, so that p becomes an infinite series in powers of both a! and J. IV.
SOME
REMARKS
ON
THE
USE
OF
NONSPHERICAL
MODELS
The results developed in the previous two chapters are “basically correct” in the sense that we have derived them by dealing with the actual Hamiltonian of the nucleus. But one realizes that, in proceeding in this way, the analysis cannot be pushed beyond a certain stage, and results like Eq. (3.23) cannot be exploited numerically. At this point, a model of the actual system may be helpful. The essential step in constructing a model will consist of ignoring the identities (1.10) satisfied by the variables xi*’ and piA’ and considering them as free pairs
238
FELIX
of canonical
variables.
These variables
bh’, am’1 = 0,
VILLARS
satisfy
the relations
[PiA’,pm’1 = 0(1/A),
(4.1)
b.h’, JIRB’I = i&RBAB + 0(1/A). We thus drop the l/A-terms, except where they add up to a significant correction, as in the commutators (l.lZa, 12b) and (1.14). But there they can be taken care of in the following way: First we also replace the principal moments IA by constants :
II = I2 = lo(l - NP), keeping p-the define
deformation
parameter-as
Jtl = 7 h’pi;
- zi;pil’>,
13 = Jo(l + 4;iP) an adjustable
constant.
(4.2) If then we (4.3a)
(4.3b)
and assume normal commutation relations between the zia’ and piA’, Eqs. (1.12) and (1.14) will be satisfied. Finally, the total angular momentum J may, for the purpose of constructing a model, replaced by a constant vector. This is justified by the fact that in the approximation used to derive the basic result (3.23), no use was made of the commutation relations of the JA . We also keep in mind that for the axially symmetric case JB = J3’, so that the model-Hamiltonian will only contain J1 and J, as “external” constants. It may then be written as: H
model
=
&
c
pi/l“
+
V&A’)
+
&
(.I,
rx3
-
er,‘)“.
(4.4)
In (4.4), JI is a constant vector perpendicular t#o the &axis; JL’ is the vector (4.3b); grip. = 210 , Eq. (4.2), is the rigid-body moment around the 3-axis;g,i,.@’ is the value of Q1-’ and Q2? according to (1.9) and (4.2). The parameter p is to be determined from (4.2), using la = (xi Xia”). The most natural way to use Eq. (4.4) is to derive from it the single particle equations of a Hartree-Fock approximation; the effective moment of inertia follows then from the dependence of (E) on 5’. The single particle states will be those of an axially symmetric potential; in the end, /3 is to be determined by a self-consistency condition. It is evident that this model is closely related to the Inglis “cranking model”, (16), but not exactly identical, mainly due to the different structure of the operator J’, which here is given by (4.3).
COLLECTIVE
V. THE
ROTATIONS
COLLECTIVE
IN
239
NUCLEI
MAGNETIC
MOMENT
The magnetic moment of rotational states contains the two parameters and ga , the collective and intrinsic g-factors, respectively (9, 17): P = g&
($37- Se> + Igc.
gC
(5.1)
It is well known (9, 18, 19), that gC deviates considerably from the value Z/A, expected on the basis of both rigid and irrotational collective motion. The methods developed here make it possible to determine the deviation of gCfrom Z/A. The magnetic moment operator is a tensor with space-fixed components
(gli and gSi being orbital the relations :
and spin g-factors
of the nucleon i). With the help of
xia = x-a+ c RA,GA’ ,
(5.3a)
A
and of
I;: eixiA’xid it follows
iZ
e z
(5.4b)
116~~)
that:
MLAB1’ being defined as the operator MrnBl’
= F
(g&iA’&
-
%$%A’)
+
gs&JABl).
(5.5)
Hence
(M-J)
=
a I&
= & @This tively.
expression
is derived
MIABIJIABI
{(M’.J) in (4).
+ f (J” - J’.J)j.
X, and P, are c.m
position
and
momentum,
respec-
240
FELIX
VILLARS
that J3 = J3’ = K, and using the notation
Remembering we have
/et = { (M,‘)K
(a.b)
= al.bl
+ (M,‘. J,) + ; @(I + 1) - K” - (J,‘. J1))j
&.
+ a&
(5.6)
The expectation value (Af,l) is equal, by definition, t.o g& = gaK. The two other expectation values, (MI’. Jl) and ( JA’ . J I ) must be calculated in the same approximation that leads to (3.23) (5.7a) (5.7b) The k-sum in (5.7b) is nothing we get,
but doA
Both (5.7a) and (5.7b) contribut’e
goA.
=
Trsing the value (3.20) of K: K
go/g,
1 - go/g.
to gC only. It follows:
.)))
(5.8)
(5.9)
go = (M,‘)IK.
The approximations that have been used to derive t,hese two last equations are those made in Section III, plus Eq. (5.4b). This latter st’ates that the charge distribution in a deformed nucleus is essentially proportional to the mattIer distribution. Equation (5.9) is a familiar result (1, 20). Equation (5.8) is new, but again a model such as outlined in Section IV will be needed for its evaluation. This, and the evaluation of (3.23) will be the subject of :I subsequent, publication. APPENDIX
We shall sketch here the derivation of the basic commutation relations (1.1 I), (1.12), and (1.14). Starting point are the two Eqs. (5.3)) which have been derived in Ref. 4. The body fixed frame from ilBC is defined by the propert,y of the variables zia’, and given in Eqs. (1.6), or, equivalently, the second Eq. (l.lOa). Using this definition, and (5.3), it follows that
Let us also define the function
COLLECTIVE
ROTATIONS
IN
241
NUCLEI
(A.21 Inserting
the expressions
(5.3) for xia and pi, , one finds, for (A Z B) :
NAB E z
RAoRs$Vap = ‘e
(Jr-m - JWI ‘I.
With (A.2) and (A-3), it is possible to evaluate all commutators between J[As) , N AB , and RAor , using their representation in terms of the original variables zirr , pi, . For instance:
[J~ABI, JCACII = -iJ[scl t [J [ABI
, Ncol
‘3.4)
~{{B~JIA~I~B~
=
+
{EZCJ[ACI~BD
{ADJ[&AC
-
(A.51
-~AcJ~B~I~AD), IN AB,
NcDl=
+
~((CAD
+
+
TBc)(NAc~LsD
+
((AC
NH&AC)
[BD)(NA~Bc
+
- (~AcJ~BD~ + ~BDJLACI + In (A.4)-(A.6),
(A-6)
NBCSAD) ~ADJLBCI
+
~B~J~AD~)J.
the abbreviation l
AS
=
(IA
+
IB)/(IA
-
IB>
has been used. Using these relations, we can now calculate the commutator the intrinsic angular momentum J’. From (A.3) it follows indeed that [J~ABI’,
JIACI’I
=
[JIABI
,
JIA~JI
-
WAS,
-
of
J~ACIIICAB
[J[ABI
,
NAC]/{AC
+
[NAB
,NAc]/b~dAc.
Inserting (A.4)-(A.6), Eq. (1.11) is gained. A similar, but rather lengthy procedure gives (1.12). Finally, Eq. (1.14) follows directly from (A.3), using the fact that V(zJ commutes with the total angular momentum, J[AB] . RECEIVED:
August
11, 1958 REFERENCES
1. A. BOHR AND B. R. MOTTELSON, Kgl. Danske Videnskab. No. 16 (1953). Ejnar d. A. BOHR, “Rotational States of Atomic Nuclei.” hagen,.1954. 3. K. ALDER, A. BOHR, T. Huus, B. MOTTELSOPIT, AND A. 28, 432 (1956) (Chapter V). $. F. VILLARS, Nuclear Physics 3, 240 (1957). 6. H. H. NIELSEN, Revs. Modern Phys. 23, 90 (1951).
Selskab, Munksgaards
Mat.-fys.
Medd.
Forlag,
WINTHER, Revs. Modern
27,
CopenPhys.
242
FELIX
VILLARS
6. R. DE L. KRONIQ, “Band Spectra and Molecular Structure.” MacMillan, New York, 1930. 7. H. J. LIPKIN, A. DE &ALIT, AND I. TALMI, Nuovo cimento Ser. 10, 2, 773 (1955). 8. T. TAMURA, Nuovo cimento Ser. 10, 4, 713 (1956). 9. R. NATAF, Nuclear Physics 2, 497 (1957). 10. S. HAYAKAWA AND T. MARUMORI, Progr. Theoret. Phys. Japan 18, 396 (1957). 11. J. J. MURRAY, F. BOEHM, P. MARMIER, AND J. W. M. DUMOND, Phys. Rev. 97, 1007 (1955). 12. A. KERMAN, Kgl. Danske Videnskub. Selskab, Mat.-jys. Medd. 30, No. 15 (1956). IS. A. BOHR AND B. R. MOTTELSON (in press). 14. A. J. LIPKIN, A. DE &ALIT, AND I. TALMI, Phys. Rev. 103, 1773 (1956). 15. D. R. INGLIS, Phys. Rev. 96, 1059 (1954). 16. A. BOHR AND B. R. MOTTELSON, Kgl. Danske Videnskab. Selskab, Afat.-jys. Medd. 30, No. 1 (1955). 17. R. J. BLIN-STOYLE, Revs. Modern Phys. 28, 75 (1956). 18. S. OFER, Nuclear Physics 3, 479 (1957). 19. G. GOLDRING AND R. P. SCHARENBERG, Phys. Rev. 110, 701 (1958). 80. R. G. NILSSON, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 29, No. 16 (1955).