Interacting Bosons in Nuclei: Proceedings of the Fourth Topical School Held in Granada, Spain September 28 – October 3, 1981

211 Pages·1982·2.614 MB·English

by  F. Iachello (auth.), J. S. Dehesa, J. M. G. Gomez, J. Ros (eds.)

ECAFERP
ehT present emulov contains the proceedings of the Fourth Topical School held in
,adanarG Spain, rebmetpeS 82 to October 3, 1981. ehT subject of the School saw inter-
acting snosob in nuclei. Since its original formulation yb Professors Arima dna
lachello in 4791 the interacting nosob approximation (IBA) sah proved itself a very
fruitful dna powerful approach for understanding ynam collective properties of nuclei+
ehT School intended to present a evisneherpmoc view of the main achievements,
microscopic foundation, dna open smelborp of the interacting nosob ledom (IBM) dna
its relationship with other collective models. ehT invited lecturers of the loohcS
erew .A Arima, .F Iachello, dna ,M .yksnihsoM
Professors Arima dna Iachello discussed the microscopic foundation of the .ledom
In particular, Professor Arima dezisahpme the role played yb the seniority emehcs
described yb using the quasi-spin formalism. Professor lachello presented the wide
exploitation of the algebraic dna geometric techniques which the IBA-I is founded on,
which allow a simple, but yetdetailed description of ynam xelpmoc spectroscopic
properties of nuclei. Professor yksnihsoM presented the present status of the attempts
to unify macroscopic dna microscopic collective sledom in a historical dna group
theoretical yaw in his lectures, clarifying the relationship of those sledom with
the IBA.
In addition to the main series of lectures, seminars were given yb H.J. Daley,
J. Dobaczewski, J.M.G. ,zemOG .P Ring, dna .A .ynugieW ehT last three are included
in this volume. J.M.G. zem~G used interacting snosob to describe the eNO2 nucleus.
ehT theoretical aspects involved in the derivation of the lacigolonemonehp Hamiltonian
in the ABI were analyzed yb Professor Ring. Finally na attempt to relate the IBrl of
amirA dna Iachello with the collective ledom of Bohr-Mottelson yb using the generator
coordinate dohtem saw described in the seminar of .A .ynugieW
ehT School saw organized yb the otnematrapeD ed Fisica Nuclear dna the otnematrapeD
ed Fisica Te6rica of adanarG University. ynaM institutions dna individuals helped
ekam the School possible. eW thank particularly the financial support received from
the opurG Interuniversitario ed Flsica Te6rica (GIFT) within its margorp of sosruC
para Postgraduados, from the Instituto ed Ciencias ed la Educaci6n (I.C.E.) ed
adanarG within its margorp nOicamroF del Profesorado, the aimedacA ed Ciencias
,:sacitametaM Fisico-Quimicas y Naturales ed adanarG dna Caja General ed Ahorros ed
.adanarG roF the hospitality offered to the participants ew thank the otneimatnuyA
dna Diputacii6n Provincial ed .adanarG It is finally a pleasure to egdelwonkca Professor
.R Guardiola dna the naeD of the Faculty of Sciences for their cooperation. eW are
also greatly indebted to Professor .W Beiglb~ck dna Professor H.A. rellUmnedieW for
giving su the possibility to publish these proceedings in "Lecture Notes in Physics."
,adanarG hcraM 2891 J.S. aseheD
.G.I~.J zemoG
J. soR
ELBAT FO STNETNOC
Algebraic dna Geometric Properties of the Interacting nosoB Model-1
.F OLLEHCAI .................................................................... I
Interacting Boson Fbdel dna Its Microscopic Foundation
.A AMIRA ....................................................................... 64
Unified Approach to Nuclear Collective sledoM
.M YKSNIHSOM ................................................................... 79
Microscopic Structure of Interacting snosoB in eNO2
J.M.G. ZEMOG .... ............................................................... 361
Microscopic Theory of Interacting snosoB
.P RING, Y.K. ,RIHBMAG .S ,IKASAWI .P KCUHCS .......... . ........................ 471
nO the Relation neewteB the Interacting nosoB Model of Arima dna lachello
dna the Collective ledo1~ of Bohr dna Mottelson
H.J. ,MUABNESSA .A YNUGIEW ..................................................... 291
List of Participants ........................................................... 802
ALGEBRAIC AND GEOMETRIC PROPERTIES OF THE INTERACTING BOSON MODEL-I
F. lachello
Kernfysisch Versneller Instituut, Rijksuniversiteit Groningen, Nederland
and
Physics Department, Yale University, New Haven, Connecticut 06520, USA
.I Introduction
In the last few years, a model of collective states in nuclei has been
developed, known as interacting boson model. In this model, one assumes that the
observed properties of low-lying collective states in nuclei arise from the
interplay of two effects: )I( the strong pairing interaction between identical
particles (proton-proton and neutron-neutron) and )2( the strong quadrupole-
quadrupole interaction between non-identical particles (proton-neutron). The strong
pairing interaction suggests that it may be appropriate to consider as building
blocks of collective excitations in nuclei correlated pairs of nucleons, similar to
the Cooper pairs of the electron gas, and to treat them as bosons [I]. In first
approximation, only pairs with angular moment~n L=O and L=2 are kept (s- and d-
bosons). One can improve on this approximation by including other pairs (g-
bosons,...). Since in the nucleus there are both protons )~( and neutrons (v), there
appear in this model proton and neutron bosons denoted by s~(s v) and d~(dv).
Although it may not be necessarily ,os it is usually assumed that only valence
particles, i.e. particles outside the major closed shells at ,2 ,8 20, ,82 50, 82
and 126 contribute to the excitation. This fixes the number of proton, N~ and
neutron, Nv, bosons. In order to take into account the particle-hole conjugation in
particle Space, the number of proton, N~, and neutron, v N bosons is counted from the
nearest closed shell, i.e. if more than half of the shell is full, N~(v) is taken as
the number of hole pairs. Thus, for example, for 118. 54xe64, Fig. ,i N = (54-50)/2=2,
.821
v N = (64-50)/2=7, while for 54xe74 , N~ = (54-50)/2=2 and v = N (82-74)/2=~. A bar is
sometimes placed over the number N (v) in order to denote the fact that these are
hole states. This description is often referred to as interacting boson model-2 or
IBA-2. A simpler version of the model can be obtained if no distinction is made
between proton and neutron bosons. In this description, often referred to as
interacting boson model-I or IBA-I, an even-even nucleus is treated as a system of
N=N= v + N bosons. Although a detailed description of the properties of nuclei
requires the use of the interacting boson model-2, it is still of interest to study
the properties of the interacting boson model-l, especially in view of its relation
with the description of collective states in nuclei in terms of shape variables [2].
My lectures will be primarily devoted to the study of the algebraic and geometric
properties of the interacting boson model-l. My intention is not that of discussing
the latest developments fo the model but rather of providing to the interested
reader the mathematical background needed to understand the structure of the
resulting spectra. This mathematical background relies on the use of group theory.
Although I will try to make the lectures self-contalned, introducing some basic
concepts fo group theory as they appear, there will not be obviously enough time to
provide a detailed account of the theory of Lie groups. The interested student may
find a slightly more detailed account in my lecture notes from the Gull Lake Summer
School 3 or in textbooks on group theory 4,5.
(o) (b)
811 X- 118v~
54 e64 54 ^ ~ 64
"17 V
®
2dw2
dlr L=2 ,zd RR~. L'2
~'~ Ihll,
ST' XX L=0 %v x~x~ L=O
. . . . . . 2dsp z
2=w.N 7=,%1
notorP Bosons NeJtron snosoB
nlr=4 nv=14
118
Fig. .al A schematic representation of the shell-model problem for ..Xe.,
(n and n are the number of protons and neutrons outside ~e ~m~jor closed
ll~hs at~O).
.81I
.bl The boson problem which replaces the shell-model problem for 54xe64 .
.2 The interacting boson model-i
In order to calculate observable quantities in any model one begins by writing
the appropriate operators within the model space. For energies, the appropriate
operator is the Hamiltonian, .H In the interacting boson model-I it is assumed that,
in first approximatlon~ H contains only one-body and two-body terms. Thus,
N N
H = Z E i + Z vii . (2.1)
i=l i<j
Higher order (three-body, four-body,..) terms can be added, if needed, without much
complication. For calculations, it has been found to be convenient to make use of a
second quantized formalism, thus introducing creation (st,d t) and annihilation
(s,~) operators, where the index ~ = ,0 el, ±2. These operators satisfy Bose
commutation relations
s,s t = ,1 s,sl=0, st,st=0,
d ,d t 6=1 ,, dl~,dl~ , =0, d~ ,d~, 1=0, (2.2)
s,d~=0 , st,d'=0,
s,d =0 , st,d =0 .
The six operators s t , d t ~ will be denoted altogether by bin t , ~=0,2. tI is well known
that, while the creation operators h~ transform as spherical tensors under
rotations, the annihilation operators do not. However, one can easily construct
spherical tensors by introducing the operators b~g= (-1) ~'~. In particular, this
gives d = (-)~d and s=s. Although there is no need to introduce ,s since s=s, I
will still do so in these lectures in order to keep formulas symmetric.
With tensor operators one can form tensor products. The tensor product of two
operators T(kl ),'" T(k2 ) is defined as 6
1~ 2K
T(k3 )= Z <k.~.k~.Ik3K3> T(kl ) T(k2 ) (2.3)
K 3 ~I,K2 ~ i ~ z K 1 K 2
and denoted by
T(k3 ) = T(kl ) x T(k2 ) (k3). (2.4)
The scalar product of two operators T (k) and U (k) is defined as
Kz ~)_( ¢(k) )k(u
(T (k). U (k)) = (-)kc~T T (k) x u(k)(~ ) iff
K --K
(2.5)
Thus, for example,
(dt.d) = ~ dtxd(~ )= /~ Z <2~12~2100> d t d =
~I'~2 ~I ~2
= Z d t d iff (2.6)
~1 ~1 ~1 nd'
the number operator for d-bosons.
Using the notation (2.4) and (2.5) one can wrlte the second quantized form of
the most general Hamiltonian (2.1) as
H iff ~s(St.s)+Ed(dt.d)+L=0Z2,4 ½~L+'~ cLtdtxdt(L)xdxdl(L)(0)
+/½~2td*xd*(2)xdx~(2)+tdtxst(2)xdxd(2)(0)
(2.7)
+ ½ v0dtxdtl(0)x~x~l(O)+stxstl(0)xdxd(0) )0(
.)°(~x~x)°(tsxts0u½
+ u2dtxst(2)xdx~(2)(0)+
This Pmmiltonian is specified by 9 parameters, 2 appearing in the one-body
terms, es,ed, and 7 in the two-body terms, c L (L=0,2,4), v L (L=0,2) and u L (Ip0,2).
However, since the total number of bosons (pairs) is conserved, N=nd+ns, Eq. (2.7)
can be rewritten as
H iff +Nse )d.td('e+)l+N(N0u½ 4,2~0=L+ ½ ~ L)0()L(ldxdx)L(tdxtd'c
)2(~x~x)2(tsxTd+)2(~dx)2(tdxtd2v½/+ )0( )8.2(
)0+ (ldxdx)0(l*sx*st+)0½ (l~x~x)0(l*dx*d0~ ,)°(
where
1
e'=(ed-e s) +7~u2(N-1) - ½ u0(2N-1) ,
(2.9)
c~ iff c L + u 0 - 2u 2.
The first two terms in Eq. (2.8) contribute only to binding energies. For a
phenomenological analysis of excitation energies within the framework of the
interacting boson model-1 one thus needs at most 6 parameters, e',c~(L-0,2,4) and
.)2,OiffL(LV
There are several other equivalent ways of writing the Hamiltonlan, .H Another
form, often used in phenomenological analyses, is obtained by introducing the
operators
n d = (dt.d)
P = - ½(sb
L iff /~ dt.d )I(
(2.10)
= dt~+stx~ )2( - ½/~dtx~ (2)
~3 dtxd )3( ,
ffi
T4 iff dtx~ )4( •
In terms of these operators, the most general Hamiltonian, omitting terms which
contribute only to binding energies, can be written as
+dn"ciff'H a0(~t.~) + al(~.~) + a2(Q. ~) + a3(!3.~3) + a4(!4.~4) . (2.II)
The reason why it may be convenient to write H in this form is that it has been
found empirically that often only few terms in (2.11) are sufficient to describe
accurately the spectrum. Note that Eq. (2.11) contains 6 independent parameters~
E", a i )4#..,1,0iffi( as the corresponding part of .qE (2.8), E', 'c L (L=0,2,4) and
VL(L=0,2).Uslng some simple algebra, it is possible to convert the two forms into
each other.
Having written down the Hamiltonlan, ,H energy levels can now be found by
diagonalizlng H in an appropriate basis. The construction of an appropriate basis for
this problem is not at all trivial and it is best done by making use of group theory.
We therefore turn now to the study of the group structure of the interacting boson
model. In addition to providing an orthonormal basis, this study will allow us to
find all situations for which the eigenvalue problem for H can be solved analytical-
ly. These situations, although rarely met in the actual spectra fo nuclei, shed
considerable light into the structure of the low-lylng collective states in nuclei.
.3 Group structure of the boson Hamiltonlan
I will keep the discussion here as short as possible, referring the reader to
the more detailed description given in Refs. 3 and 7 and in the original
articles. I begin by introducing the operators
G(k)(%%') = blx~oi,(k) %,~.' = 0,2 = s,d , (3.1)
"K "K
with
b£= ,b I ,., = 6j~%, 6=c t , • (3.2)
The operators (3.1) satisfy the commntation relations
G(k)(%£ '),G =)'''£''%(),'k( r ¢'(2k~-l)C2k'+l) <l~k'K'k"K"> (_)k-k'
K K k" ,K"
)3.3(
x t• -J .k+k ' +k "t~.,,, • k £'k £k" ,} G~,~,,v o(k")c K" ,xx .... , )-{~,, k kl', .~k"l~ j~jU~,,v c(k")t.,~,, .,). ....
Operators, ,X satisfying relations of the kind
Xa,Xb = ~ C c ab Xc' (3.41
are said to form a Lie algebra with structure constants C abc " One can verify that the
commutation relations (3.31 define the Lie algebra of the group U(61 of unitary
transformations in six dimensions. Since the Hamiltonian R in (2.7) is built out fo
operators G(k)(%%'), one says that it has the group structure of U(6). The operators
K
G(k1(£% )' are called generators. There are a total of 36 = 62 , which written down
K
explicitly read
o(G °) (ss)-. tst(cid:127)l(°)
G )0( )dd( = dtxd 0)0(
G(l~dd) = dtxd )I( 3
K K
G (3)(dd) = dtxd(3) 7
K K
(3.s)
G(4)(dd) = dtx~l (4) 9
K
G(2)(ds) = d*x~ )2( 5
K K
G(Z)(sd) = s*xd )z( 5
K
36 = 62
Once the full algebraic structure of the problem has been identified, the next step
is that of identifying all possible subalgebras of the full algebra. A subalgebra i8
a subset of generators which is closed under commutation. tI turns out that in the
present case there are three possible chains of 8ubalgebras.
Subalgebrasl
Delete from the 36 operators the II operators G~O)(ss), G~2)(ds), G~2)(sd)°
The remaining 25 operators close under the algebra U(5), the group of unitary
transformations in five dimensions.
)A U(5)
C(1)(dd) = d*x~l(1) 3
K
G(2)(dd) = 2xt'd (2) 5
K K
(3.6)
G(3)(dd) = ~'x'id (3) 7
K K
G(4)(dd) = d*x~ (4)
9
K K
25 = 52
Delete from the 25 operators the 15 operators G ()(dd), G(2)(dd) and G(4)(dd).
K K
The remaining I0 operators close under the algebra of 0(5), the orthogonal group in
five dimensions.
)B 0(5)
G (1)(dd) = dtx~ (1)
K K
~(3)(dd) = d*x~(3)
7 (3.7)
K
,J
2/4x5=,0I
Delete from the I0 generators the 7 operators G ~ ¢3"(dd) ~ . The remaining 3 operators
K
close under the algebra of 0(3), the ordinary rotation group.
c) 0(3)
G (1)(dd) = d%x~ (1) 3 (3.8)
K K
Finally, delete from the 3 operators the 2 operators "l+)dd()1(G - and --.)dd()~!G
The remaining operator is the generator of 0(2), the group of rotations around the
z-axis.
)D 0(2)
)1(~xtd
G(~)(dd) iff 1 . (3.9)
0
Thus, one possible chain of subalgebras is
U(6) D U(5) D 0(5) D 0(3) D 0(2) . (1) (3.1o)
Subalgebras II
A) U(3)
Consider the operators
)ssc)O(G + o(O)(dd), ts' l(0) + i
G )dir()1( = dtx~ (1) 3
K K
dt~+stx~(2)-
G(2)(ds) + G(2)(sd) - ~f~G(2)(dd) iff ~v~dtx~(: ) 5
'K K K
9 iff 3 2
(3.11)
These operators close under commutation into the algebra of U(3)° Obvious subalgebras
are now
)B 0(3)
G(1)(dd) = dtx~(~) 3 )21.3(
K