Geometrical analysis of the interacting boson model

)

~ = ~ ( A--x -+

i

(1.3)

'

Ap

with Ax = / ~--~-- and Ap = f ~ 2mm By taking hamiltonian

and

O

the expectation value of is replaced by a c-number:

H

with

respect

to

the

states

la> the

2

,

<~IHl~> = ~x~ = =

2m

+ ½ mm22

'

(1.4)

the classical limit of (i.I). The coherent states (1.2) have a number of useful properties [17] which I give without proof: (i) they are eigenstates of the destruction operator al~> = =I=>; (ii) they minimize the uncertainty principle, (Ax)2"(Ap)2 = ~ 4

; (ill) they are quantum states that obey c l a s s i c a l

motion

B

<~lx(t)l~>

= A sin(~t+~)

(the latter

p r o p e r t y makes c o h e r e n t s t a t e s a u s e f u l

tool in discussing the classical quantum c o r r e s p o n d e n c e ) ; overcomplete set, resolving the identity

i = ~

1

f ~

finally

y d~* i~><~[ .

they

form an

(1.5)

For use in connection with compact Lie algebras the coherent states (1.2) must be generalized [18]. In the following I restrict myself to the group G = SU(6), the symmetry group of the IBA-model. In this model [19] one considers a system of N bosons, that can occupy six levels, namely an s (L=0) level and a flve-fold degenerate d (L=2) level, interacting through a hamiltonlan that can be expressed in terms of the 36 generators

s~s, d's, stdg and d'd~~' of U(6).

Coherent states can be introduced as follows. In the Hilbert space of the IBA model, spanned by the totally symmetric representations [N] of G one considers an extremal (or maximum weight) state, IN,ext>, which can conveniently be taken as the state with all bosons in the s-level:

IsN>. Consider the subgroup H c G that

Geometrical Analysis of the Interacting Boson Model

123

consists of all elements h & G with the property hlN,ext> = IN,ext> * e i~(h), where ~(h) is a phase factor. H is referred to as stability group; in the present case H contains the generators sTs and dTd ,, i.e. H = U(5) ~) U(1). Clearly every group element g • G can be decomposed ~n~an element h and a representative Q of the coset G/H: g=Q h. An SU(6) coherent state is than defined as IN,Q> = QIN,ext>. The parametrization of the states IN,Q> is not unique [15]. A general one, sometimes referred to as the algebraic realization, is:

IN,~> = exp{ Z D d~S - ~ d sT}IsN> ,

(1.6)

where the ~ represent five complex parameters. However, have also b~en used, for example the group realization: iN,a> =

i

( l_~-~aj2

st +

Z

other

parametrizatlons

(1.7)

T~N

and the projective realization IN,~> = (N!(I+IzI2)N) -~ (s t + Z z d~)N[0>

,

(1.8)

which are related via the transformations:

a

ffi

~/l~lsinl~l

and z =

~/t~ltanl~l.

A most useful property of the coherent (analogous to eq. (1.5)) da da 1 = D([N]) 51/g~l ~ 2 ~ i

IN'=>
(1.9) states is the resolution

'

of the identity:

(I.I0)

where D([N]) denotes the dimension of the representation

[N].

Note that eq. (1.7) has a simple interpretation: for Nffil it represents a singleparticle wave function (expressed in terms of the basis states Is>,Id >) ; for N > 1 it represents a many-boson wave function in product fo~m, a boson condensate.

1.3 Classical Limit of IBA-I The classical limit [4,20] of any normal ordered number conserving k-body operator 0 can be defined as the coherent state expectation value of the operator (up to a normalization)

Ocl = /N . With the parametrization

(1.11) (I.7) this amounts to the substitutions

dt ÷ a ~ t , d ~ ÷ a- , and st,s ÷ / l - a * ~

If one is only interested

in the leading

order (in N) result, one can ignore commutators and evaluate the classical imit of any function f(Eij ) of the group generators Eij by replacing every Eli by its classical limit. In order to obtain transformations [6]

a more physical interpretation it is useful to perform the complex variables ~ : first it is convenient to

on

two

124

A.E.L. Dieperink

introduce conjugate momenta and coordinates 1 ~ = 7 ~ (q + i p~)

(1.12)

with q~ = (-)~q ~ and p~ = (-)~p_~. Secondly,

the

rotational

lnvariance

of

the

problem can be exploited by transforming to the intrinsic frame:

qp = E a

V

D(2-(@) ) ~LV

(1.13a)

D(2-*(O) )

(1.13b)

and

p~ = E u

where ~ denotes the three Euler angles (~,8,¢). The intrinsic be parametrized as

coordinates a

i a 0 = ~cosy, a±2= 7 ~ ~siny , a±l= 0. The five parameters u u0= p~cosy

v

U~l u~2

~ siny , L2

2~/2sin(y- 2~)

= -~

(1.14)

can be expressed in terms of the momenta:

-i L 1 =

can

±

2~/2sin(y - ~ )

i L3 1 2~#2siny + ~-2 cosy

~+ ~

,

(1.15)

i

~-~ siny p~ ,

where L k (k=1,2,3) are the angular momentum components in the intrinsic frame: L 1 = (pc- p~cosO) cos____~ sln8 - Pe sine , sln¢ L 2 = (pc- pc cosS) s~n0 + P8 cos¢,

(1.16)

L 3 = pC. Of course for the study of ground state properties (static limit) one can set all momenta equal to zero; then the classical hamlltonlan becomes a function of the intrinsic coordinates ~ and y only.

1.4 Application to IBA-I The hamlltonlan. The semi-classlcal analysls can be applled to the most general IBA-I hamlltonlan [19]; I flnd it convenient to scale the parameters of the twobody terms by (N-I) and express H in the form:

"

H=en d + ~

6

^2

nd+

¼cO et 2K2 Q(2) Q(2) K5 K1 ~ "e + ~-~ • + ~ C5 +

L(1).L(1), (1,17)

Geometrical Analysis of the Interacting Boson Model where p = pt.p = (dt.dt_sts~)(~.~_ss) ~2)

= d's + s'd

, the 0(6) pairing operator

~ ½~7(d'~)~2),

125

(1.18)

,

(1.19)

the SU(3) quadrupole operator

I L(1).L(1)}, C5 = ~1 {(dt~)(3) .(d%~) ( 3)+ ~-~

(1.20)

the 0(5) Casimir invariant

and e (I)

= / 1 0 ( d % ~ ) (I)

.

(i .21)

Below I first discuss the classical limit of each of the symmetries which occur for special choices of the parameters.

The U(5) limit. The classical hamiltonian corresponding (1.17) is given by *

e

2

w h e r e T2

dynamical

to the first term in eq.

T2

2

Hcl = = c ~ .~ = ~ (p8+ 8

three

+~

) ,

(1.22)

2 + ¼ Z 2~k) " = PY k=l sin2(y - -

(1.23)

The r.h.s, of eq. (1.22) corresponds to the usual expression for a fivedimensional hamiltonian describing harmonic quadrupole vibrations around a spherical equilibrit~n [21]. I will discuss the quantization of this hamiltonian in section III. Here I restrict myself to a few remarks. First it is clear that •

A 2

if I had taken into account anharmonic terms of the type 6 n d the classical limit (1.22) would have been much more complicated, including higher order terms in the 4 (p~, etc.), which usually do not appear in the geometrical ~scription. 2 2 T Secondly, in the present case the phase space is bounded: ~(~ + p~ +--~ ) < i; no

momenta

such a condition appears in the geometrical approach. Thirdly another difference with the conventional description becomes apparent if one considers the classical limit of the IBA quadrupole operator (1.19), Q(2) 2 T2 cl,~ = q ( 2 - p ~ - ~

_~2)~ + X Q~I,~

(1.24)

Whereas the first term at the r.h.s of eq. (1.24), apart from the square root associated with the finiteness of the phase space, has the usual coordinate-like form of the geometrical model, the second term has a complicated dependence on coordinates and momenta, and has no obvious geometric counterpart.

0(6) limit. The classical (I.17) is given by

limit

of

the 0(6),

K 0 He I =--4 (~2p~+(i_~2)2)2 + . <5 ~ T 2 + KI L2 ,

0(5)

and 0(3)

invarlants

in eq.

(1.25)

where T 2 is defined in eq. (1.23), and mO > 0 . It is seen that the 0(5) and 0(3) invariants contribute only a kinetic energy; the resulting potential has a (deformed) minimum at ~0=I and is independent of y, suggesting the y-unstable deformed rotor character of the dynamics. Small vibrations in the ~-dlrections can be described by writing ~=~0+A~ and expanding around 60:

126

A.E.L. Dieperink 2 2B P~

Hcl/K0

+ ~ C(~8)2 + (anharmonic terms) ,

(1.26)

1 show that (AS)2 ~ one finds that the 1 anharmonic terms are smaller by a factor 7N' which suggests that the 0(6) pairing with

C = B-I

= <0/2.

Since

one

can

operator describes a nearly harmonic vibration in A~.

SU(3) limit. In this limit I consider a combination of the SU(3) and 0(3) Casimir operator invariants in eq. (1.17): !

K2 ( 2Q(2)'Q(2)+ %3 L(1).L(1)) + ~ H =N---~

IL(1) .L(1) •

(1.27)

3 where ~2 < 0, and K~ = E l- ~ <2 The classical hamiltonian is more complicated than in the previous cases: _T4 _ - -T2 (582-p~) - 462+ 28 2 (B 2 +Ps) 2 2 2 %1 (82+ PS) Hcl = -w2{- 4~4 + 262 T2 2 2 2 P 2 2 - Z L2± 2[i . . . . B +P~ )~((__~- 6(6 +Ps)) cos 3y + 262 2

8 pyp~Sin 3y + Zk 82sin2(y -~

--23k) c o s ( y - ~ ) ) }

2 + K~Z L k

•

(1.28)

I consider the limit of large N and expand Hcl around the equilibrium values Y0

=

0 ° (60°), ~0= 7~2

up to order I/N. By using p~ ~ (A~) 2

~

P Y2

Y

2 ~' i

2 1 and L k ~ 7 ' one finds 2 I 2 9 )2 9 2 + 9 L3/4 Hcl/4K2 = ~ P8 + ~ (A~ + ~ py - ~ - - 7 - + 2y 2. (1.29) Y In this limit the ~ and y degrees of freedom decouple and become harmonic, in agreement with the geometrical picture in which the ~ and y degrees of freedom are considered as small amplitude harmonic surface vibrations of dimension one and two, respectively [21]. Standard quantization of the harmonic oscillator hamiltonian (see Section III) yields the energy spectrum E(ns,ny) = 12~2(n 8 + ny),

(1.30)

where n8 = 0,1,2, .... , and ny = 2m + ~K, with m = 0,1,2, and K ~ 0,2,4,... One can easily verify that in leading order in N the result 1.30) agrees with the spectrum of the SU(3) Caslmlr operator. Transitional re$ions. Ground state properties of the general hamiltonian can conveniently be studied by considering the classical hamiltonian with all momenta put equal to zero [2,3]. The resulting energy surface E(8,y) depends on the two real parameters 8 and y, and has the property that in the limit of N ÷ ~ it approaches the exact binding energy per particle [2,3,20] EO/ N = ~

{E(8,y) } + 0(~)

.

(1.31)

Geometrical Analysis of the Interacting Boson Model

127

Therefore the ground state properties of the general IBA hamiltonian simply follow from an examination of min {E(~,y)} as a function of the interaction parameters.

(1.17)

For the dynamical symmetries these energy surfaces are given by [2,3]:

I)

U(5) limit: El(~,y) = e~2/(I+~ 2) ,

II)

SU(3) limit: Ell(~,y) = 2K2(½~±2/2~3COS3y + 4~2)/(i+~212 ,

1111 0(61 limit: Elll(~,y) = KO/4 (I-~2)2/(I+~2) 2 ,

(1.32)

where I have used eqs. (1.22,1.25,1.281 and the projective parametrization: --~J2//1+~ 2 . Transitional

regions

can

be

studied

(for N + =)

by considering

the energy surface for an hamiltonian that is an admixture of two limiting cases [2,3]. For example, in the transitional region I -> III one finds that K0 El-lll(~,y)/e = ~2/(I+~2) + ~ (i-~2)2/(I+~2) 2, as a function of the strength K0 parameter ~ = ~ has a minimum that shifts from the spherical value ~0 = 0 for ~ ¼ to a deformed value T0 = /(I-4D)/(I+4~]) for ~ > ¼.

The nature of this shape

phase transition for N ÷ =0 is determined by the behaviour of El-lll(~,y) at the 2 critical point Dc; for the I-III case ~-~]~ I,rc~ is discontinuous, which corresponds to a second-order phase transition. characterized that

the

by

II-III

a first-order transition

Similarly one finds that the I-II region is

phase

transition

is completely

smooth.

(discontinuity

in 5~E,l~c) , and

More details

can be found in

ref. [3]. From this analysis one can conclude that all ground state shapes in IBA-I are axially symmetric, i.e. Y0 = 0" or 60". It can be shown that in order to have triaxial shape cubic interaction terms of the form E c~ (dTdTdT1(~).(~'d~)(~) , % whose classical limit is proportional to ~ cos23y, must be included.

1.5 Derivation of Differential E~uation Several other approaches have been proposed [7-11] with the aim to construct the geometrical analog of the IBA hamiltonian. Here I briefly discuss a method suggested by Ginocchio and Kirson [7,8], which uses the concept of coherent states. The idea is to transform the original IBA eigenvalue problem (H-EN~LIINvL> = O,

(1.33)

where E. . are the eigenvalues and v a label that distinguishes different NL eigensta~es with the same L, into a differential form by introducing a complete set of states I~> in a continuous variable. Defining an overlap function %NvL(~) = ~ I N v L >

,

(1.34)

128

A.E.L. Dieperink

the eigenvalue problem (1.33) can be written as

<~]HINvL>

=

(1.35)

ENv L XNvL(~ ) .

By taking for I=> the coherent states in the projective (1.8)] one can construct the following correspondence: S~slNz> = A-I(N - Z z

d~s]Nz> = A -I

s~

d

parametrization

[eq.

~ zd ) AINz >,

AINz >

[Nz> = A -I z g ( N - Z z

dT~ ~ v iNz> = A-I "~ ~ d

d ~z

) A[Nz>

AiNz > '

(1.36)

where A = (N!(I+IzI2)N) ½. The eigenvalue equation (I.33) becomes H(Z,Tz) XN~L(Z) = ENvLXNvL(Z) J

(1.37)

which corresponds to an unusual SchrBdinger equation in the complex variables z. Only in special cases (as e.g. the 0(6) and U(5) limits this equation can be solved in closed form. In their application Ginocchio and Kirson do not use a general coherent state for INz>; instead they use a real parametrization and introduce an intrinsic wave function INSy> i~J z 18Jyei~J z [N~yQ> = e

e

[NRy>

,

(1.38)

with =

~(

t ~ sin ~ t s q~3cosyd0+~(d2+d_2)

N ) I0>

.

(1.39)

In this way eq. (1.37) reduces to a differential equation in the real coordinates ~d~nd~ y, in which (for two-body hamiltonians) no more than second derivatives d=2~ appear; the resulting hamiltonian

[8] has many resemblances with the Bohr

hamiltonian [21] especially in the limit of large N. However, it has been pointed out [i0] that there is also a serious problem connected with this approach, namely the lack of a scalar product in the real Hilbert space, and therefore it seems that no orthogonal set of eigenfunctions can be defined. The reader can see ^

this

problem

already

replaced by E ~

case:

the U(5)

hamiltonian,

H = E n d , is

, which is not a self-adjoint operator.

1.6 Transformation Hamiltonians In practice,

for the simple

to

Coordinate

many phenomenological

Space:

Relation

calculations

between

IBA

and

of nuclear collective

Frankfurt

properties

Geometrical Analysis of the Interacting Boson Model

129

have been carried out by the Frankfurt group [24] using a geometric hamiltonian, HFR (q ,p ), which is a polynomial in the collective quadrupole coordinates q and ~ome~ta p . The relation of this hamiltonian with the IBA model ~ can conveniently b~ discussed using a method developed by Moshinsky and coworkers for the calculation of the matrix elements of the IBA-I hamiltonian in coordinate space [9,10]. A complete set of states for the U(6) D U(5) ~ 0(5) ~ 0(3) chain of groups is given by INndA nAe M > = IN-nd>IndAnAe M >.

(1.40)

The ket IN-n.> stands for a one-dimensional oscillator and the second ket represents t~e U(5) basis states which can be expressed in terms of first quantized variables as [22,23] AnAL L* (Qi) , IndA nA e M> = ~n(6) Z ~K (Y) DMK K where ~-(6) is related to a Laguerre polynomial (n= (nd-A)/2) , and the functions ~ are derived in ref. [23].

(1.41)

We note that the states (1.41) span the Hilbert space of the Frankfurt hamiltonian [24]. As shown in ref. [9] this hamiltonian can be expressed in terms of the coordinates 62 , 63cos3y,

L 2 and the operator nd (the latter arises from a 5 transformation of the kinetic energy: ½p2 = ~d_~62 + ~): HFR

=

H(~2,63cos3y

L 2 , n^ d )

,

.

(1.42)

It turns out that the IBA model can also be discussed in the Hilbert space spanned b~ the functions (1.41). To see this one notes that the boson nLunber operator N can be replaced by its eigenvalue N~ and furthermore that the s-boson can be eliminated by using s~[N-nd > = (N-nd+l)~IN-nd+l> ,

(1.43)

and similarly for s. By going to first quantized variables d ~ = /~

~

=

/~ ( ~

+~)

one can easily

,

show

(1.44)

that the IBA hamiltonian can be expressed in terms

same variables 6 2 , ~ 3 c o s 3 y ,

L 2 and nd a s t h e F r a n k f u r t

of the

hamiltonian.

In conclusion one has the result that (for given N) both models can be described in the Hilbert space of the states (i~41). However, in practice the hamiltonians differ in several aspects: and

has

a

HIB A is linear in 63cos3y,

complex n d dependence;

on

the

other

hand

at most quadratic in 62 , HFR

has

a

more

general

dependence on 62 and ~3cos3y, but is in general linear in nd" In addition HIB A is diagonalized in a finite space which has a great numerical advantage over the diagonalization of HFR in an infinite space.

130

A.E.L. Dieperink II. GEOMETRY OF THE NEUTRON-PROTON IBA (IBA-2)

2.1 Introduction In an attempt to relate the collective boson degrees of freedom to a microscopic theory a generalized version (IBA-2) of the IBA model has been proposed [25], in which two distinguishable types of bosons were introduced, one representing correlated neutrons pairs and another representing correlated proton pairs. The hamiltonian [19] for the coupled system can be expressed as H = H

p

+ H

n

+ H

pn

.

(2.1)

Th 9 @igensta~e~ span the irreducible representations [Np] ® [Nn] of the group su~P)(6)~su~n](6). The most general rotationally inva~iant two-body hamiltonian contains a large number of terms; in most phenomenological studies a schematic hamiltonian has been used of the type H = ~(ndp+ndn)+^ ^ ~ 0~2).Q~2)+ Vnn + Vpp+ ~M ,

(2.2)

where -nd (nd-) is the number operator for the proton (neutron) d-bosons, 0(~)^ is p n the general quadrupole operator (i=p,n) Q!2)= d~ s + s ~ d + xi(d I 7 ) (2) i,~ i,~ i i i,~ i-~ '

(2.3)

and M denotes the Majorana interaction , t~*

.~ T

= ~Span-apSn

)(2.(Sp~n_~pSn)(2)_2

-t-t-(X).(~p'an)(~) E (apanJ ~=1,3

(2.4)

This last term is added for the following reason. Since the neutron and proton bosons are distinguishable the decomposition of the tensor product [Np] ~ [Nn] of the su(P)(6) ~ su(n)(6) representations contains in addition to the totally symmetric representation [N] (with N=ND+Nn) , which correspond to those occurring in IBA-I, also representations of mixe~ symmetry [N-I,1] etc. The Majorana force splits these different representations and can therefore be used to remove states of mixed proton-neutron symmetry about which, at present, there is no experimental information.

2.2 Dynamical Symmetries The mutual interaction between the like bosons (Vnn and VDD in eq. (2.2)) is usually approximated by a weak d-boson number conserving inte'faction. However, in order to be able to discuss analytic solutions of the eigenvalue problem in terms of dynamical symmetries [26] I will replace it by a quadrupole-quadrupole force: (2) ^(2) ^(2) ^(2) Vnn+ Vpp= K' (O n "~ n + ~p .~p ),

(2.5)

with K' = ~K . (In case of X_ = Xn this choice does not mix the various [N-k,k] multiplets and therefore contains the IBA-I eigenstates as a subset). Therefore I consider the schematic hamiltonian, H = E (ndp+ ndn) + K Q(2).Q(2) , ^ with Q(2)

^(2) + ^(2) Wp

Wn

'

where I have also dropped the Majorana term.

(2.6)

Geometrical Analysis of the Interacting Boson Model

131

By generalizing the results of IBA-I one easily sees that dynamical symmetries of (2.6) occur in the following cases:

Case

parameters

subgroup chain

I

< =0

u(P)(5) ~ u ( n ) ( 5 )

II A

e = 0,Xp=Xn =

su(P)(3) ~ su(n)(3) D SU(3) m 0(3)

II B

/7 e = O,Xp= -Xn = e~--

su(P)(3)® S--~Un)(3) D SU*(3) D 0(3)

III

e = O,Xp=Xn=0

o(P)(6) ~ o(n)(6) D 0(6) D 0(5) D 0(3)

D U(5) m 0(5) D 0(3)

Here I restrict myself to a discussion of the deformed situations II+III (~=0). To examine case II (IXl=½/7) in more detail, it is convenient to rewrite (2.6) in terms of Casimir invariants: H = ~ K C2(SU 3) - ~3 < L(1).L(1) ,

(2.7)

where C2(SU3) = 2 Q(2).Q(2) + 3/4 L(1).L (I),

(2.8)

and L (I) the angular momentum operator: L (1) = L(1)+ L (I) p n The corresponding energy spectrum can be expressed as E(k,~,e) = ~(k2+~2+k~+3(k+~)) where k,~ denote the quantum distinguish two possibilities:

_

~3 K e(e+l) ,

labels

of

the

(2.9) irreps

of

SU(3).

Now

one

can

Case II A: Xp = Xn = - ~ /7. The irreps of su(i)(3) are given by (k4,~i) = (2Ni,0) ~ ( 2 N j - 4 , 2 ) ~ ..... (i=p,n). The allowed (k,~) values are given by The decomposition of ~he tensor product ( k p , ~ p ) ~ (kn,~n). In addition to the irreps (k,~) = (2N,0) • (2N-4,2) O .... , which correspond to the ones occurring in IBA-I, there also occur other representations which are not totally symmetric in the neutron-proton degree of freedom, such as (k,~) = (2N-2,1). Its geometrical interpretation will be discussed below. (It is clear that in this case the Majorana term in eq. (2.2) can be added without breaking the dynamical symmetry). Case II B: Xp = - Xn = - ~/7. Here the irreps of SU*(3) are found* by combining the representation (kp,~p) su(P)(3) with the conjugate of the representation (kn,~n)~Of su(n)(3): ( k , g )=(2Np,2Nn)~ (2Np-4,2Nn+2) • (2Np+2,2Nn-4) O (2Np-l,2Nn+l)~

....

*To distinguish this case from the SU(3) chain in II A I have put an asterisk over SU*(3); the bar over ~ ( n ) ( 3 ) conjugate representations.

indicates that this group is built from the

of

132

A.E.L. Dieperink

The K quantum number that can be introduced to distinguish states with the same L within the irreps (X,~)takes on the values K= min(l,~), min(l,~)-2,...l,0, and the allowed L values are given by K ~ L < K + max(l,p) with the restriction that if K=0: L--max(%,g) , max(l,~)-2, ... I or 0.. In fig. 2.1 the spectrum of H is shown for the case Np=4, Nn=3. It is seen that the angular momentum structure of the lowest representation

(X=2Np,~=2Nn)

is similar to that of the triaxial rotor

with y=30* of the geometrical model [27]. It is clear that the angular momentum degeneracy can be split by adding any symmetry breaking term to the hamiltonian (2.7); in this case also the Majorana interaction breaks the symmetry. Note that in the present approach higher excited multiplets including 0+ states occur in a natural way; in the geometrical model these states must be introduced ad hoc [28].

E (MeV)

!

Nn=4

3.01

NP=3 (X,F)= (8,6)

/z 2.C

I 6 "

LO

3-

4

O.,/.L~00,2)

\,

~

6

6

5

5

4

4

2

2i

6 0

3 2 0.0

2

0

Fig. 2.1 Energy spectrum for the hamiltonian (2.7) with Xp = -Xn = -½¢7 for the case N n = 4, Np = 3. Case III: Xp = Xn = O, E = 0. In this case the spectrum can be labeled in terms numbers (oi, 02) and (~I' x2 )' and L, respectively:

of 0(6) 9 0(5) D 0(3)

E(o1,G2,~I,~2,L)=A(oI(OI+4)+d2(d2+2))+B(~1(~I+3)+~2(~2+I))

+ CL(L+I)

,

quantum

(2.10)

where the constants A, B and C can be related to ~ in eq. (2.6): A = ~, B = -~, C = O. The lowest representation has o I = Np+Nn, o2=~2=0, ~I=Oi , oi-I,...0 , and can be identified with the o=N multiplet of the 0(6) limit in IBA-I.

2.3 Geometrical

Interpretation

of the Dynamical S~rmmetries

It has been shown in I that the relation between the algebraic formulation of collective properties and the geometric picture can (for large boson n,nnber N) can be established by considering the classical limit of the boson hamiltonian. This can be achieved by taking the expectation of H in the coherent state representation. In the present case the generalized su(P)(6) ~ su(n)(6) coherent states can be expressed as

Geometrical Analysis of the Interacting Boson Model i

~

INp~pNn~n >= ~

a

( p

s~+

P

.d')NP(/~-~-~s'+a P . n. n

133

.d~)Nnlo> n

'

n

where the a_ and a n represent five in general complex quadrupole variables. Since I consider ~ere only ground state (static) properties I can take without loss of generality the a's real. It is then convenient to transform to the intrinsic frame of the neutron and proton bosons separately (i=p,n) (compare eq. (l.13a)) = E D(2)(Oi ) ai,v v ~v where ai,0= ~icosYi

(2.11)

'

,

(2.12a)

1 and ai, 2 = ai,_2 = 7 ~ ~isinYi"

(2.12b)

Subsequently one can introduce a transformation from the (QD,Qn) system to the Euler angles ~ for the orientation of the mass distribution a~d three angles {X ,X^,X^} describing the relative orientation of the neutron intrinsic system Z J wi~h respect to the proton intrinsic frame. In this way using the rotational invariance of H results in an energy surface E(ap,an) =

,

(2.13)

which depends in general on seven intrinsic variables: XI,X2,X3.

Minimization

with

respect

to

these

shape. Here I will not present a complete summarize the most important results. The

SU(3)

case.

One

finds

for

the

~n,~p,Yn,y p and

variables analysis

equilibrium

defines

the

equilibrium

of E(~i,Yi,Xk) , but

value

only

80,n =80,p =2//3

'

and Y0,p = Y0,n = O° " Therefore both neutron and proton intrinsic states possess axial symmetry and only one relative angle between the two symmetry axis, X, suffices to specify the relative orientation, E(X) ~ K Pp(cosx), resulting in an equilibrium value X-O = 0 • This analysis suggests that the ground state shape can be characterized by an axially symmetric deformed rotor, symmetrlc in proton and neutron bosons. I note that in comparison with the IBA-I picture where there occur two fundamental modes of excitation (namely ~ and y vibrations of the matter distribution), in the present approach there are several new possible eigenmodes, which represent vibrations of the neutrons against the protons. A complete survey of the latter requires a more detailed study of the classical limit of H, using the complex variables a p, a n. Here I mention only one interesting possibility, namely an oscillation in terms of the angle X between the two symmetry axis. For small X the classical hamiltonian takes on the form 2 2 L3 + ~ X 2 • Hcl ~ ~ PX + 2X2

(2.14)

One can interpret this expression as a two-dimensional isotropic harmonic oscillator with X as radial coordinate and L 3 as the a~gula~ m o m e n t ~ of the polar angle ,; the corresponding quantum eigenvalues are L~ = K z - ¼ and E(nx) = mx(nx+I), where n X = 2m + K. On top of each excitation there is a rotational band starting with L=K. The lowest eigenmode has m~0 and K=I with angular momenta L= = 1+, 2+ , 3+ , . . . . This particular neutron-proton mode has been investigated in terms of the geometrical model by Palumbo and Loludice [29], and has also been suggested by Suzuki and Rowe [30]. In section 2.4 1 briefly discuss a way to locate this mode experimentally.

134

A.E.L. Dieperink

T_he S_U*_(3) limit.

A similar analysis

for the SU*(3)

case leads

to the following

results: 80,n = ~0,p= 2//3, YO,p = 0", Y0,n = 60 ° and X0 = 90". Its geometrical interpretation is that of a prolate proton and an oblate neutron axial rotor coupled so as to maximize their overlay. The resulting mass distribution can be parametrized by an asymmetry parameter y in the following way [21] ^(2),^(2) tan [ - /2 Vm,2/Um,0

,

(2.15)

where

2) ,0 = < 2z2-x2-y2>, and Um, 2^(2) =



,

(2.16)

characterize the mass quadrupole distribution (sum of neutron and proton contributions). By taking the prolate proton distribution with respect to the zaxis and the oblate neutron distribution with respect to the x-axis one finds 2) ,0

For

=

Np

~(2) Wp,O

Nn

=

borating

-

^(2) ~n,0

one

^(2) and Um, 2

,

has ^(2)~ Wp, 0

the interpretation

=

-

^(2) and

-Wn, O

~

^(2) Vn, 0

therefore

(2.17)

•

from

eq.

(2.15) ~ = 30 °

corro-

of a triaxial rotor.

0(6) limit. This case corresponds to a generalization of the classical limit of the IBA-I 0(6) hamiltonian, discussed in I. The ground state shape can be interpreted as a y-unstable rotor symmetric in proton and neutron bosons. In addition excited bands occur that can be interpreted in terms of oscillations of neutron versus proton bosons.

2.4 M1 Excitation of the K ~ = i+~ L ~ = 1+ State in the SU(3) limit It is an interesting question whether it is possible to locate collective states which are not totally symmetric in neutron-proton degrees of freedom. An obvious procedure would be to try to locate the L~ = 1+ state of the K~ = 1+ band in SU(3) discussed above via an inelastic M1 excitation in electron scattering. In the long wave length limit the collective operator can be expressed as T(1)_ .(I)+ .(I)_ gp+gn L(1) + g p - g n (L(1)_ .(I)~ - gpLp,~ gnLn,~ 2 ~ --7"-p,~ Un,~)

(2.18)

"

An estimate for the M1 strength in the SU(3) limit between the ground state and the L~ = 1+ state of the K ~ = i+ band for large Np, N n can be obtained by constructing the intrinsic state for the latter:

IK~1>= 7~ where

Ig.s.>=

1

p

b kO = By e v a l u a t i n g

(2.19)

p,K~l p,0- -~

(bLo) P(bLo)nlo> N

n

/2 the matrix

o)

N

(k-p n)

elements

of the operator

(2.18)

one f i n d s

Geometrical Analysis of the Interacting NN B(M1; 0+ + i+) ~ 21I 2 = 4 - - ~

(gp-gn)2

Boson Model

(nm) 2

•

135

(2.20)

An estimate for the value of gp and gn can be obtained by analyzing magnetic dipole properties of known low-lying collective states, such as magnetic moments + + + + of 2 , 2 states and the B(MI; 22 + 21). A recent analysis [31] of magnetic 1 2 moments around Z ~ 50 nuclei seems ~o indicate gp ~ I and gn ~ 0. I n an inelastic electron scattering experiment one would also like to distinguish between M1 excitation of this particular collective neutron-proton mode and possible spin-flip excitation of two-quasiparticle states. However, in the latter case the form factor would arise mainly from the magnetization distribution (peaked at the surface) whereas in the former case the form factor is expected to have a purely convection current type behaviour. Of course a difficulty is that it is not easy to give a reliable estimate for the excitation energy. In terms of the hamiltonian (2.2) the excitation energy depends on the strength k of the Majorana interaction which cannot be determined accurately in phenomenological studies.

2.5 Intermediate situations and application

to 104Ru

It has been shown [19] that for all three dynamical symmetries that occur within the IBA-I model experimental examples can be found, although even in the best examples the 0(6) symmetry appears only approximate. While apparentl~ no nuclei seem to exist that can be interpreted in terms of the extreme SU (3) limit discussed above a closer inspectation [32,33] suggests that several nuclei can be described in terms of an interplay of 0(6) (y-soft) and SU*(3) (y-rigid) characteristics, and therefore it seems of interest to investigate this transitional region in IBA-2 in more detail. The shape phase diagram corresponding to the IBA-2 hamiltonian (2.2) can be represented by a tetrahedron as shown in fig. 2.2 with each corner denoting a dynamical symmetry. For K=0 (and ~ 0 ) one has the U(5) or vibrational limit; for e=0 one encounters the three limiting cases of deformed rotors discussed above.

vibrator

U (5)

IBA-I

~

SU~i3') rotor

SU~13)rotor

Fig. 2.2 Shape phase diagram of IBA-2. The corners represent symmetries, the legs transitional regions.

dynamical

A particular transitional region can be described by considering a hamiltonian which is an admixture of the extreme limits; its eigenvalues can be solved by numerical diagonalization. The transition between the 0(6) and SU*(3) limits can conveniently be parametrized in terms of a variation of the parameters Xn = -Z between 0 and - ~ 7 . Qualitatively the energy spectrum varies in the following p

136

A.E.L.

Dieperink

way: for I~I = + ~ 7 the level scheme is given by fig. 2.1; by decreasing I~I the degeneracy of the various L states is broken without affecting (in first approximation) the ratio R = E31/IE2" i + E221. which has the value 1 in the SU*(3) limit; for still smaller values of IX| an 0(6) type spectrum developes:

the

ratio R tends to the value 9/7, and a 0+ state comes down in energy to join the 0(6) ground state multiplet. As an illustrative example I d~scuss the case of l ~ R u 6 0 . (It seems worth noting that nuclei with 0(6) and/or SU-(3) characteristics are ~ound in particular in situations where the neutrons are hole-like and the protons are partlcle-llke 104132~ with respect to closed shells and vice versa; examples are 44Ku60, 56ma76 . This seems to give some support to the geometrical

interpretation

of the SU*(3) case

as a competition between opposite intrinsic quadrupole deformations of the neutrons and protons.) This nucleus has been described [34] in terms of the triaxial rotor model [27], the generalized collective model (GCM) [24], IBA-I [35] and IBA-2 [36]. From fig. 2.3 it is clear that whereas in the triaxial rotor model there is a strong clustering of states L = (2,3), (4,5), (6,7) .... in the y-band, the GCM (and also the IBA-2 calculation) produce too much (3,4), (5,6),... clustering, typical for a y-soft potential.

104Ru 8-9--

MeV // 3

,

,

m

9

I0--" "

,

7 m

8---~-

~ - -

6--

m -

--"

4

_

--6 _

4

---

2-AR

"--8 m 7

- - 5

--3 __

--2

EXP

GCM

4

2~ 0

AR

EXP

GCM

Fig. 2.3 Ground state and gamma band energies in 104Ru calculated with triaxial rotor model (AR) and generalized collective model (GCM) (from ref. [34]) Recent Coulomb excitation measurements of electromagne,tic properties suggest an interplay between y-unstable (0(6)) and y-rigid (SU (3)) features [34]. For example, besides the fact that the experimental ratio R = 1.00 precisely agrees with the triaxial rotor value, the measured values of the quadrupole moments QL in the ground state band seem to follow the y-rigid prediction. On the other hand the energies of higher spin states in the y-band do not follow the strong clustering predicted by the triaxlal rotor model and the observed B(E2) values and ratios for y- to ground state band transitions are in general closer to the predictions of the y-unstable approach. We have repeated [26] the calculation of ref. [36] for 104Ru in the framework of IBA-2 with the schematic hamiltonian (2.6). The only essential difference with ref. [36] is the inclusion of a quadrupole-quadrupole interaction between like bosons (eq. (2.5)). Since the protons are assumed to be hole-like (Np= 7) and the

Geometrical Analysis of the Interacting

137

Boson Model

neutrons particle-like (N n = 5) (with respect to the N=Z=50 core) I have assumed opposite and almost equal values of Xp and Xn: + 0.80, X_ = -0.90. The values of E = .70 MeV and K = -.i0 MeV have ~een f i t t e d X ~ the overa~l properties of the ground state and y-bands of the spectrum. From fig. 2.4 where the experimental and calculated spectrum are compared, two features emerge: in the revised calculation th~ odd-even level spacings in the yband are reproduced correctly, and the known 01 state at 1.0 MeV is not described at all. However, there are several indicatlon~ [34,37] that this state does not belong to the IBA model space consisting of pairs of valence nucleons, but instead arises by promoting a pair of protons across the Z=50 shell.

Mev! 3~

104Ru

s.... __ 7---.,

8----

s--

6 . . . .

5

2

--

--,, ---" "'~

(4) __

~--

0 3 --(0)--

2 .......... O ..........

A

Exp.

2 3 --

!

A

B

Fig. 2.4 Low-lying spectrum of 104Ru calculated in the framework of IBA-2: A: taken from ref. [36], B: present calculation [26].

The electric quadrupole matrix simplest possible E2-operator

~

elements

have

been

calculated

by

taking

the

^(2)+ ~(2)~ 2) = eB(~p,g ~n,~ J'

and the boson effective charge, eB, has been fixed by fitting the observed + + B(E2;21÷OI) value. Some results are shown in figs. 2.5-7. The calculated values of 0 L in the ground state band fall in between the results for the trlaxlal rotor and those of ref. [36], In fair agreement wlth experiment. Also the rapid falloff of the B ( E 2 , L ÷ Lg-2) with increasing ~ , which is typical for the yunstable model,

is well reproduced

We note, however, present analysis.

in the present calculation

(fig. 2.6).

that it is difficult to draw definite conclusions from the In particular the observed strong enhancement of the

B(E2;6~ ÷ 4~) over the rigid rotor value

[34] suggests that yet another degree

of

freedom p ~ s ~0 role. Also it would be interesting to investigate whether the assumed QX~j.Q~Z) interaction between the like bosons can be justified from a microscopic picture.

2.6 Discussion It is worth while to mention that there has been proposed [38] another generalization of the IBA-I model that gives rlse to K~=I + rotational bands. If a g-boson (L=4) degree of freedom is added to the s and d bosons the group

138

A.E.L. Dieperink

O0

i

0008

i

i

B(E2;I),~Ig-2) \ 0,16 ~

-05

0.006

%

...... ....

IP present AR

8(E2;17~Ig) ~

\y~

O12

.....

\~

....

_

IP

present

AR

%

-IO

o.oo~

-15 --IP ..... ....

~0.08I

\'\ /

i

present AR

I

I

Fig. 2.5 0uadrupole matrix elements (LIIM(E2)I[L> (in eb) in the ground state band of 104Ru; IP=ref.[36]; AR: asymmetric rotor; data are from ref.[34]

Fig. 2.6 The B ( ~ ; L ÷ L - 2 ) values in 104R~; g see caption fig. 2.5

\\

I ".~

\

I

Fig. 2.7 The B(E~;L ÷L ) values in IU4R{I;'"g caption fig. 2.5

see

structure becomes SU(15). It "s c ~ T ~ in this structure there are basis states with L~ =i +, e.g. [(d½dt)" ~ j ' ~ 1 0 > . In the special case of a quadrupole-quadrupole interaction the states can be classified according to the group chain SU(15) D SU(3) D 0(3) and the lowest SU(3) representations are given by

[38] (k,p) = (4N,0) e

(4N-4,2) @

(4N-6,3) @ (4N-8,4) 2 •

A new feature is the occurrence of the (4N-6,3) is not present in the SU(6) model.

.....

representation

with K=I,3 which

III. FUNCTIONAL INTEGRAL METHODS 3.1. Introduction In recent years functional integral methods have been applied in several areas of physics. The work of Levit et al. [39,40], Alhassid and Koonin [41,42] and Blaizot and Orland [43] suggests that these methods ~ould also be useful in describing the nuclear many-body problem. Up to now most studies have dealt with the general formalism and applications to simple schematic, mostly onedimensional, models. By comparison with the exact solutions it is possible to study the validity of these methods, which in lowest order essentially reduce to the mean field or Kartree-Fock approximation. Although the ultimate aim of the theory would be to derive the collective parameters for the nuclear many-body problem from fermion degrees of freedom there are several reasons why application of functional integral methods to the

Geometrical Analysis of the Interacting

139

Boson Model

algebraic boson models is of interest. First for n > 2 the SU(n) models constitute important extensions of the two-level Lipkin model [44] (n=2), that has frequently been used as a soluable model to test many-body approximation schemes; the multilevel extensions are useful in studying new complications that arise, such as the quantization in a multi-dimensional phase space. Secondly, although the general IBA eigenvalue problem can be solved exactly by numerical diagonalization, with increasing N the dimensionalities become exceedingly large and therefore the availability of approximate methods could be of interest (such as e.g. a 1/N expansion). Thirdly the work of ref. [41,42] on the calculation of the S-matrix suggests that the path integral method is a useful approach for the description of nuclear reactions that involve collective degrees of freedom (e.g. Coulomb excitation).

3.2. Fe~nman Path Intesral for Coherent States. Coherent states were introduced in I as an overcomplete set of states and used to convert the algebraic hamiltonian into a SchrSdinger equation. Here I discuss more general applications of coherent states for the calculation of the propagator e -iHT. The matrix elements of the time evolution can be written as a functional integral [43,45]



=

a (T)~" * iS[a ,= ] / D[a(t),~ (t)] e ,

(3.1)

~(0)~' where the action S is given by T S[~,~ * ] = I dt < N , a l l" ~~ - HIN,a> - i %n 0


with

IN'a>

= iN

*

- ~ (~ "~ - = a * )

.

(3.2)

(3.3)

The semiclassical approximation to eq. (3.1) is obtained by applying the stationary phase approximation (SPA), i.e. one considers only those paths =(t) for which the action S is stationary with respect to small variations of a(t): 6S=0. It then follows [46] that the classical paths ~(t) satisfy Hamilton's equations of motion: ,

i ~

~Hcl

Da

, and

i ~=

~Hcl ,

,

where Hcl = /N.

(3.4)

(3.5)

One can show that equations (3.4) are equivalent with the time-dependent HartreeFock (TDHF) equations, which describe the evolution of single-particle wave functions in the one-body field of a linearized hamiltonian. To see this I note that the llnearized version of the Hamiltonian H = ~ gjGj + E VjkGjG k ,

(3.6)

(where G i are the generators of the SU(n) group) takes on the form H o = Z EjGj + Z Vkj (GkOj+ OkGj) where oj are the mean fields

,

(3.7)

140

<+IGjI+>,,

o

A.E.L. Dieperink

=

(3.8)

J

<+[+>

' --

and ff denotes the s.p. wave function: ff ~ (= s + It is i~portant to keep in mind however, that in and ~ are independent, whereas in the TDHF condition that ff (T) is the complex conjugate of

Z = dt)10>' . the~p~esent formulation approach one has as a boundary if(T).

3.2.1. Bound states. To calculate energies and other properties of bound states in the semi-classical approximation it is convenient to consider the propagator [45] / dT e lET / D[~,~ * ] eiS[~,~ ] , (3.9) 0 ~(0)~(r) where I have expressed the trace of an operator in terms of an integral over coherent state space. By evaluating the path integral in the SPA one finds that T with period T: the solutions of eq. (3.4) must be periodic orbitals ~cl G(E) = Tr E_--IH= d i

•

- T

*T-

= O~i

fdT e iET Z eiS[~cl'~cl | . cl The integral over T can be done also stationary points Ti: G(E)

E

=

-

8S

~--~ J T

(3.10) in SPA,

yielding

the condition for

"

the

(3.11)

1 Since the classical action has the property ~S _ _ H ( T *T ~T cl' ~cl"

(3.12) Ti one finds that only those periodic classical trajectories =cl contribute that satisfy E = H(~

'

,~

i), i.e. all periods T i which are a multiple of the

fundamental period TF, and therefore TF ~ TF G(E) ~ ~ exp[im ~F dt = 2 ~m. (m fFdt = ~N Z(-)~(p ~ p - q _ ~

> ]

(3.13)

if integer)

(3.14)

momenta (see eq. (1.12)) we have

) ,

(3.15)

and eq. (3.14) becomes N Z (-)~

p_~ dq~ = 2 ~m,

(3.16)

which is a Bohr-Sommerfeld type quantization rule.

3.2.2 Application to the U(n-l)-O(n) transitional re$ion in IBA-I (n=6). Consider the hamiltonian [6] H = (l-~)dt.d + with 0 < ~ < I. has the form

~ (dt.dt-stsT)(d.~-ss) 4(N-I)

(3.7)

Using the results of I the corresponding classical hamiltonian

Geometrical Analysis of the Interacting Boson Model 2 T2 ) + ~ 2 2 2) . (6 P~+ (I-6 2 ) Hcl = (I-~)(½~ 2+ ½P6+ -262

14

(3.18)

Note that similar e ~ r e s s l o n s are obtained for n=4 [12] (with T 2 replaced by the angular momentum L ) and n=2 (where the pairing operator has the doubly degenerate ~Inlmum at ~= ~I). In applying the quantization condition (3.16) I note that T ~ i n eq. (3.18) is a constant of the motion 2 T 2 = ~---~ = 0,1,2, .... (3.19) N2

'

The radial condition N~p6d ~ = 2=nB ,

(3.20)

must in general be solved numerically by using the relation p~ = p(H,~) . In fig. 3.1 some classical trajectories are shown in the [~,p~) plane for the cases n-2 and n=6, ~=0.7. For the two dynamical symmetries the integration can be performed analytically. In the U(5) limit (~=0) one finds E(~,~)-2~+

~ , with ~ = 0,I ..... N - 2 ~

,

(3.21)

and in the 0(6) limit (~=i) E(n6) = n~(N-n~) N

(3.22)

L°~s

Fig. 3.1 Some classical trajectories (N=I4) in the(~, pe) plane in the U(n-l)-O(n) transitional region for the case n=2 (left) a~nd n=6 (IBA) (right). The value of the strength parameter ~ = 0.7; in the SU(6) case the orbltals are labeled b y (n,~ = I).

Note that, whereas in the U(5) limit the exact energy the 0(6) limit eq. (3.22) agrees with the exact result

spectrum is obtalned,

in

42

A.E.L. n~ (N-n~+2) N-I

E(n 8) =

Dieperink

'

(3.23)

only in leading order in I/N. In fig. 3.2 the excitation energies E(ns,z) in the transitional region are compared with the exact ones as a functlon of the strength parameter ~. It is seen that in the semiclassical approximation the overall trend of the energy levels is reproduced quite well; as expected the largest deviations occur near the phase transition (~=0.5) •

4.o 5

EXA~T

3°

~

~

~=~

t

4,o

MEAN FIELD

'

\

2.0

0.0

Fig.

2,4

\\

OI

02

a3

04

Q5

06

0.7

08

0.9

ID

QI

Q2

0.3

a4

0.5

'

1,6

0.6

0.7

-

0.8

0.9

m

3.2 Some excitation energies, E(n,z) in the U(5)-0(6) transitional region as a function of the strength parameter ~ for N=I4. The exact results (left) are obtained with the computer program PHINT; the mean field results (right) are shown for T = ~/N (broken llne) and T = (~+ 3/2)/N (full line).

3.2.3. Corrections to the mean field approximation. The mean field approximation provides the energy in leading order in I/N. In general higher order corrections to the SPA are difficult to calculate. However, for the ground state the quadratic correction to the energy can easily be obtained [43]: AE (2) = ~ E (~(i) _ e(i)) , i where ~(i) denote the roots of the RPA matrix (~* ~*)

(3.24)

with

(3.25)

A v = ~ --------~~2H and B v

= ~

~2H v

(evaluated at the minimum of H(~,~*)), and E (i) are the TDA energies. In table 3.1 a c o n ~ r i s o n is made [12] between the mean field energies E (0), the RPA corrections AE'-', and the exact energies, Eex , in the U(n-1)-O(n) transitional region, for n=2,4. It is seen that the quadratic corrections are much more important in the higher dimensional case in particular in the strong coupling region (~ > 0.5).

Geometrical Analysis of the Interacting Boson Model Table 3.1. Ground state energies in the U(n-l)-O(n)

transitional

su(4) E(0)+AE (2)

E (o)

143

region of SU(n)

SU(2)

E(O)+AE (2)

Eex

Eex

0.2

1.5000

1.4892

1.4890

1.4964

1.4963

0.5

3.7500

3.3750

3.5440

3.6250

3.6726

0.8

2.6250

2.5248

2.5252

2.6248

2.6248

E(0): meanfield approximation,

AE(2): quadratic correction,

N=30.

3.3. Dynamical problems. It seems promising to apply the seml-classical methods to situations where a reaction can be described in terms of a few collective coordinates. Examples are Coulomb excitation of collective states in heavy nuclei, or laser excitation of rotatlonal-vlbratlonal states in molecules. The basic idea is to calculate the Smatrix as the matrix elements of the seml-classlcal propagator in the limit of T ÷ ~. Some recent work of Alhassld and Koonin [41] who applied these ideas to simple schematic hamiltonlans (forced oscillator and forced Lipkin model) indicates that in this way a satisfactory approximation to the exact solution can be obtained. I consider a general tlme-dependent algebraic hamiltonlan H 0 of the type (3.6) to which I add a time-dependent one-body potential H = H 0 + Hl(t )

,

(3.26)

where Hi(t ) = Z fj(t) Gj . (3.27) 3 It is convenient to define [41] the S-matrlx as the long-tlme limit of the interaction picture evolution operator: Sif = t÷lim + ~ <$flUo(0,t)U(t,-t)U0(-t,O)l+i> where U(t2,tl) describes Hamiltonian, and U0(t2,tl) i ~

U(t,t') = H(t)U(t,t')

,

the evolution from time t I the evolution under H 0 only: .

(3.28) to

t2

under

the

full

(3.29)

In the coherent state representation one can write down a functional integral expression for the matrix elements of U similar to eq. (3.1). In practical calculations of matrix elements <~fIul~i > between arbitrary initial and final states it appears more convenient to use instead of the coherent state path integral (3.1) a representation in which the mean fields are introduced explicitly. The idea is that one can express the evolution of a system under a general m a ~ - ~ o d y hamiltonian in terms of an integral over one-body evolutions, U_(T) - e- no~ , corresponding to one-body hamiltonlans H [39-41]. In the semic~asslcal approximation the mean fields for a hamiltonian°of the type (3.26) are determined by the condition [41]

144

A.E.L. Dieperink

oj(t) =

<+f(t)IGjl+i(t)> <~f(t)l+i(t) >

,

(3.30)

where I+ l (t)> = U~(t,0)u [+-> i and <+f(t). l=<+flU~ (0,t) • It can be shown [47] that in case l+i > and <+fl are represented by coherent states this approach, which has been referred fo [41] as tlme-dependent mean field approximation (TDMF), is equivalent with the coherent state path integral method, provided the fields o i ( t ) a r e allowed to be complex. (However, in the applications given in ref. [41] o was defined in terms of the real part of the r.h.s, of eq. (3.30)). It should be pointed out that the TDHF approximation to the S-matrix corresponds to a different condition for o (t), namely 3 ~j(t) = Re [

<$i(t) [G~ [¢i(t)> <$i(t)I$i(t) > ]

,

(3.31)

where <4~i(t)I = l$~(t)> T. It is clear TDMF equations is-much more involved equations.

that the self-conslstent solution of the than the solution of the tlme-local TDHF i.o II

O5~ i

l

i

,

I

I

f

i

I

i

v

3

4

5

0.9 0.8

°-

Z ,, ,Soo-I

0.7

z~ o

v

0.6 03 0.5

02--

~

~

0,4

"

03

,, 02

0.1

1.

2.

3.

4..

5.

6

l

8.

9.

I

~

Fig. 3.3 The elastic S-matrlx elements .[S^oI and Im SON for the hamiltonian (~.32) with A=O.~ and B=0.2 as a function of the adiabaticity parameter to; the full lines represent the exact, ( O ) the TDHF, and (A) the TDMF result for N=30.

--L Fig. 3.4 The inelastic matrix elements SOL for the hamiltonlan-(3.32) with t0ffi4.0; (V) represents the exact results, ( a ) TDHF and and (A) TDMF.

146

A.E.L. Dieperink

20. R. Gilmore, J.Math.Phys. 20___(1979)891. 21. A. Bohr and B. Mottelson, Nuclear Structure, Vol. II (W.A. Benjamin, Reading, 19755. A. Bohr, Mat.Fys.Medd. Dan.Vid.Selsk., Vol. 26 no. 14. 22. E. Chac6n, M. Moshinsky and R.T. Sharp, J.Math.Phys. 17___(1976)668. 23. E. Chac6n and M. Moshlnsky, J.Math.Phys. 18(1977)870. 24. G. Gneuss and W. Greiner, Nucl.Phys. A171(1971)449. 25. T. Otsuka et al., Phys.Lett. 76B(1978) 139. 26. A.E.L. Dieperink and R. Bijker, Phys.Lett.B, to be published. 27. A.S. Davydov and G.F. Filippov, Nucl.Phys. 8(1958)237. 28. A.S. Davydov and A.A. Chaban, Nucl.Phys. 20_.__(19605499. 29. N. Loludlce and F. Palumbo, Nucl.Phys. A326(1979) 193. 30. T. Suzuki and D.J. Rowe, Nucl.Phys. A289(1977)461. 31. M. Sambataro and A.E.L. Dieperink, Phys.Lett. I07B(1981)249. 32. L. Hasselgren and D. Cline, in ref. [2], p. 59. 33. D. Cline, in ref. [2], p. 241. 34. S. Stachel et al., preprint GSI 82-11. 35. J. Stachel, P. van Isacker, and K. Heyde, Phys.Rev. C25(1982)650. 36. P. van Isacker and G. Puddu, Nucl.Phys. A348(19805125. 37. P. van Isacker, in ref. [2], p. 115. 38. W. Hua-Chuan, Phys.Lett. 110B(1982)I. 39. S. Levit, Phys.Rev. C21(1980)1594. 40. S. Levit, J.W. Negele and Z. Paltiel, Phys.Rev. C21(1980)1603. 41. Y. Alhassid and S.E. Koonin, Phys.Rev. C23(198151590. 42. Y. Alhassid, S.E. Koonin and B. Miiller, Phys.Rev. C23(19815487. 43. J.P. Blalzot and H. Orland, Phys.Rev. C24(1981)1740. 44. H.J. Lipkin, N. Meshkov van A.J. Glick, Nucl.Phys. 62(1965)188. 45. S.E. Koonin, Lectures presented at Nucl. Theory Summer Workshop at Santa Barbara (1981). 46. D.S. Feng and R. Gilmore, Phys.Lett. 90B(1980)327. 47. O.S. van Roosmalen, private communication.

Geometrical Analysis of the Interacting

Boson Model

145

The SU(n) models discussed above can conveniently be used to test the accuracy of various approximations. In Groningen we have extended the study of [41] to the more realistic case of SU(4). As an illustration I show some results for the Smatrix obtained by van Roosmalen [47] for the group chain SU(4) ~ 0(4) ~ 0(3) (generators:

(pt~)~l) and (pts+stp)~l)-)

with

the

time-dependent

hamiltonian

of

the form: -16(t/t0)2 H = A(ptp)(1).(pT~) (1) + B e to

(pt s+stp) (~)

(3.32)

In terms of such a hamiltonian one could attempt to describe multiple excitation of a rotational band of a diatomic molecule in a time-dependent dipole field; t O has the function of an adiabaticity parameter. In fig. 3.3 the elastic S-matrix elements for the TDHF and TDMF approximation are compared with the exact results, and in fig. 3.4 the inelastic matrix elements SOL. In general it is found that the mean field approximations work better near the sudden limit (t0 small) than in the adiabatic limit.

ACKNOWLE DGEMENT It is a pleasure to numerous discussions.

thank

O.

van

Roosmalen,

R.

Bijker

and

F.

lachello

for

This work has been performed as part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM) with financial support of the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO).

REFERENCES i. A.E.L. Dieperink, O. Scholten and F. lachello, Phys.Rev.Lett. 44(1980) 1747. 2. A.E.L. Dieperink and O. Scholten, in Interacting Bose-Fermi Systems in Nuclei, F. lachello, ed., Plenum Press, New York (1981), p. 167. 3. D.H. Feng, R. Gilmore and S.R. Deans, Phys.Rev. C23(1981)1254. 4. D.H. Feng and R. Gilmore, in ref. [2], p. 149. 5. R. Hatch and S. Levit, Phys.Rev. C25(1982)614. 6. O.S. van Roosmalen and A.E.L. Dieperink, Phys.Lett. I00B(1981)299. 7. J.N. Ginocchio and M.W. Kirson, Phys.Rev.Lett. 44__._(1980)1744. 8. J.N. Ginocch~o and M.W. Kirson, Nucl.Phys. A350(1980)31. 9. M. Moshi~sky, Nucl.Phys. A338(1980)156. I0 O. Castanos, A. Frank, P. Hess and M. Moshinsky, Phys.Rev. C24(1981)1367. ii. A. Klein and M. Valli~res, Phys.Rev.Lett. 46(1981)586. A. Klein, C.T. Li and M. Valli~res, Phys.Rev. C25_(1982)2733. 12. O.S. van Roosmalen and A.E.L. Dieperink, Ann.Phys.(N.Y.) 139(1982) 198. 13. S. Levit and U. Smilansky, preprint WIS 81/61. 14. F. lachello, Chem.Phys.Lett. 78(1981)581. 15. R. Gilmore, C.M. Bowden and L.---M. Narducci, Phys.Rev. A12(1975)1019. 16. R. Gilmore, Rev.Mex.Fis. 23(1974) 143. 17. M.M. Nieto and L.M. Simmons, Phys.Rev. A19(1979)438; in Foundations of Rad. Th. and 0u. Eleetrod., A.O. Barut, ed., Plenum Press, New York (1980), p. 203. 18. A.M. Perelomow, Comm. Math.Phys. 26(1972)222. 19. For a recent review of this model, see A. Arima and F. lachello, Ann.Rev.Nuel.Part.Sci 31(1981)75.