Nacdear PA~ysicr A274 (1976) 1 Sl -167 ; © North-Eopand Publlsh4rp Co., Amsterdant
Not to be nepeodooed h7 D~topzlat or m1aa~81m without wrletm pamiedon ßro~ the p~oblieher
MïCROSCOPIC INEB1YAL FUNCTIONS
FOR NUCLEI iIN THE BARIUM REGION
TERESA KANIOWSI{A and ADAM 30HICZEWSIü
Institute for Nuclear Renmrh, Soso 69, 00-681 War:sawa, Polcord
RRZYSZ1bF POMORSKI
Insütute ojPhyaics, The Maria Sklodowaka-Carle Unftwrüy, LubUn, Poland
and
3TANL4LAW (3. ROHOZI~TSKI
InsHtuts of Thearstfcol Physici, Warsaw Unioersity, Eoza 69, 00-681 Wamzawo, Poland
Received 3 May 1976
(Revised 1 July 1976)
A6atraua : Dynamical cbnracteristia of doubly area nuclei is the neutron-deâci~t barium region,
30 < Z, N < 82, are investistuted microscopically . The three vibtational (BIl, BR B~) and
three rotational (~ B B.) iuertisl functions are studied as fnctions of both axial ~ and nonuuaial y-deformations. Important effexts flrom the shins structure are found.
1. Introduction
There is an ever increasing amount of experimental data [cf. e.g. references given
in refs. i, ~) and also the later refs. s-a)] concerning the collective properties of the
neutron-deficient nuclei in the region 50 < Z, N < 82 . The data give evidence of the
existence of all three kinds of nuclei : spherical, transitional and deformed in this
region, similarly as in the traditional rare~arth and actinide regions.
Extensive studies of the potential energy 9 _ i 4) and also a study of the moments of
inertia 1) ofnuclei with SO < Z, N < 82 have already been performed. No systematic
study, however, of all inertial functions, in particular the vibrational ones, has been
done yet. In the dynamical studies of the collective states performed up to now ~" t s),
only a simplified form was assumed for them. However, as, specifically for this region,
the potential energy is rather flat, especially in the y-degree of frcedom, knowledge
of the detailed deformation dependence of the inertial functions seems to be important. It may influence the results for the collective properties ofthe nuclei considerably .
A systematic microscopic study of all the inertial functions for nuclei in the considered region is just the goal of the present paper. Ellipsoidal shapes of a nucleus,
described by the axial ß- and non-axial y-deformation parameters, are assumed. All
the six inertial functions, three vibrational BRA, B~,, Br, and three rotational B,~, B,,
Bs are investigated as functions of both ß and y.
lsl
T. xnxiowsxw a ~
In sect. 2, the parametrization of the shapes is specißed and the inertial functions
are deßned. In sect . 3, the deformation dependence of these functions, as obtained in
the liquid-dropmodel, is given. Description ofthemicroscopic calculations is presented
in sect. 4 and the results of these calculations are given in sect. s. Sect. 6 presents
the conclusions drawn from the research and the appendix provides a demonstration
that the microscopic inertial functions' studied satisfy the proper symmetry conditions. The appendix gives also a discussion of the role of the pairing-vibration terms
in the inertial functions.
Z . lDeßnitioos
As mentioned in the introduction, we assume ellipsoidal shapes of a nucleus. The
somi~axes of the ellipsoid are
whore
~, Y) ~ Ro(ß, Yxl+k coa y~),
k
=
s~~ß,
YK = Y-~~"
(2.1a)
with i~ = 1, 2, 3 corresponding to x ~ x, y, z, respectively. Thus, Qx, Oy, Oz are the
principal axes of the ellipsoid and ß, y are the Hohr deformation parameters 1 s).
The volume conservation condition gives
where ~o is the radius of a sphere which has the same volume as the ellipsoid.
It is sufficient to take Y and ß from the iatervals
0°sYs~ °,
,
osß~~
1
s coe(60-y)
(2.~)
to describe all shapes of the ellipsoid.
To get a general form of the kinetic energy of a nucleus corresponding to its vibrations and rotation, we introduce the collective variables a, describing the deformation and the orientation of the micleus in the lab coordinate system
where B~ _ ~, S, ~ are the Euler angles between the intrinsic and the lab systems,
~i,(Bt) are the standard rotation functions and u ~ 0, f l, f 2. Thus, the collective
variables constitute a spherical tensor of rank 2.
The classical kinetic energy connected with the motion a~, = a(t) is of the form
where B. is the inertial tensor. Expressing Tby the deformationparameters ß and y,
MICROSCOPIC INER'ITAL FUNCTIONS
133
Eulen angles B i and their time derivatives, we get i~" ie)
T = ~fBpi, Y)~=+ZßBp,(ß, Y)~3+ß~B»~, Y) ~~+E ~K(rs, Y~K},
K
(2.
where mK are components of the angular velocity of the intrinsic system with respect
to the lab system . The moments of inertia of the nucleus ~K may be written in the
form
(2.6a)
~K~, Y) = 4ß~BK(ß, Y) ~~ YK~
Eqs. (2.6) and (2.6a) define the six inertial fondions (or equivalently : inertial or
mass parameters) of a nucleus : threevibrational Bpp, Bp, and B~, and three rotational
B=, B~ and Bs. Sometimes, slightly different vibrational functions Dpp, Dp, and D~
aro introduced i 9). They are defined. as
~
~ ß=Brn .
Dpt
Drr
DIp Bpp,
~ ßBp, ,
(2.6b)
The components of the inertial tensor BH,. of eq . (2.5) are assumed regular and
isotropic functions of the variables a~,. This leads to some symmetry conditions for
them, discussed extensively in ref. ie) . The following relations between the inertial
functions are an immediate consequence'of these conditions :
Bpr ° 0,
B m B
B= a Br ,
fur y = 0°,
(2.7a)
Bp, a 0,
B, = B,,
B= = Bs,
for y = 60°,
(2.7b)
Bp, = 0,
Bpp = Bn ° B= ° B, °= Bs,
for ß = 0.
(2.7c)
Relations (2.7) guarantee the fulfilment ofthe proper boundary conditions by the oolledive wave functions.
3. Yneräal fm ctlooe in the liquid-drop model
It is well known that the values of the nuclear inertial functions obtained in the
liquid-drop (hydrodynamic) model are too small i 9) . Still, it is instructive to look at
the deformation dependence of these functions.
We assume, as usual, an irrotational and incompressible liquid. Solving the Neumann problem for the velocity potential for a vibrating and rotating drop of the
ellipsoidal shape, eq . (2.1), and finding the corresponding kinetic energy, eq. (2.6),
of the liquid, we get the following expressions for the inertial functions :
B~ =
Ba~(1+âk4a6bs sing 3y),
BK = Ba~
(1-~k COS yK)~
~
bz -k oos yK =~k~ cos 2y=
(3 .1)
T. KANIOWSKA et a~
1S4
where
3
B = AMltâ,
8n
bl = 1-~}k cos 3y, bs ® 1+}k2,
(3.1a)
and a = Ro/~o [see eq. (2.2)]. Here A is the mass number, Mis the mass of a nucleon
and k is given bY eq . (2.1a) .
In the zero-order approximation in ß, we get
(3 .2)
B~~=B = Bx=B~=Bs=B,
BR,=O.
This approximation was used in the early paper t s) by Hohr.
It is easy to check that the hydrodynamic functions (3.1) fulfil the symmetry conditions (2.7), the condition (2.7c) being just the Bohr approximation (3.2).
Keeping two lowest-order terms in ß in each of the inertial functions, we get
ßs)
B~~ = B (1 + 15
8n
ß3.
5
Bp, = B ~ ain 3y / 1- 1
8n 4~x
4
B ~ B (1 +
~ ß cos 3y) ,
Mc
5 ßsi
8~
It is seen from eqs. (3 .3) that BRA, B and BR have the lowest order zero, in ß, and
have a minimum at ß = 0, while B~, has the order three and has a deflection point at
ßm0.
r
Fib. 1. Dependence of the i~ydrodynamic vibrational inertiel Fonctions oa the y-deformation. Bach
line cornsponds to the 8aed value of ß indicated on the lino.
MICR03COPIC INERTIAL FUNCTIONS
135
Fib. 2. 3am~e as in ~ 1 for tho rotational functions.
The deformation dependence of the liquid-drop functions (3.1) is illustrated in
figs. 1 and 2 and also in figs . 4=9. For a few flied values of ß, indicated at each line
of figs. 1 and 2, the q-deformation dependence is shown explicitly. It is seen that all
the functions, both vibratioaal (fig . 1) and rotational (flg . 2), except for only B,~, are
increasing functions of ß . The ß-dependence of rotational functions is weaker than
that of vibrational ones . The strongest dependence on ß is obtained for BRA.
For the region of deformations usually considered, say ß ~ 0.5, most important
for collective states analysis, the dependence of all inertial parameters on both ß and
y is rather weak and smooth. The Bp parameter changes by less than about 30 ~,
B by less than about 6 ~, and B~, is smaller than 0.02B. Each of the rotational
parameters changes by less than about f4 ~. All parameters are practically
independent of y within the deformation region discussed (i.e. ß ~ 0.5). Thus,
inside the liquid-drop model, the zero-order approximation (3.2) is rather good.
However, a strong deformation dependence of the inertial parameters may come from
the shell effects, as pointed already out is the early paper ~ °) and as will be seen explicitly in sect. 5.
4. Deecripüon of the microscopic calcalatbns
4.1 . METHOD OF THE CALCULATION3
We assume the cranking approximation of Inglis . The cornsponding formula for
the inertial function connected with collective variables q, and q~ is aras)
B~r
ki<kI ala4~l0i ~
= 2liZ ~ ~O~a~agil
tfo
~t- ~o
(4.1)
where ~0~ and ~k) are the ground and excited states ofa system, respectively, and d°
T. xAxiowsxw ~ ~r
iss
and dt are the energies of these states. The vibrational functions Bop, BRA and B~,,
are directly obtained from eq. (4:1) when the deformation parameters ß and y are
taken' as the collective variables q. To get the moments of inertia J,~, the rotation
angles around the intrinsic axes should be taken (sce the appendix) .
After inclusion of the pairing interaction betwcen constituents of the system, by
the BCS formalism, one gets for the moments of inertia si),
(4.2)
where jR is the single-particle angular momentum operator and x = x, y, z. For the
vibrational parameters, one obtains ~=- :a, is)
(4.3)
where His the single-particle I3amiltoaian, a, and acf stand for ß or y and, according
to eq. (2.6b), DRR = BRR, DRr ' ßBR, and D, ~ ß1Bn. In both eqs. (4.2) and (4.3),
u, and n, are the HCS variational parameters corresponding to the singlo-particb
state w~ with the energy e and E, ~ "/(e,-~1)s+d= is the quasiparticle energy
corresponding to w). The quantities ~, and 2d are the Fermi energy and the pairing
~8Y BaP, ~P~~Y"
Theterm Pu ofeq . (4.3) comes from the dependeaoe ofA and d on the deformation
a. It is given by the formula a4, ss, is)
where
d
a2 + e,-2 ad a ,
a~,
2d a~,
a~
~ (act+d~bd,)/D,
a~,
a=~ e,-~,
,
E,s
c,~~
ad~
a~,
1
b=~ s,
. E,
E,a
~~ _
m
<y~
aH ~,,~,
a~,
2e~ad,-bc,)ID,
D = as+d~b~,
dt=~~s .
. ,
The modified harmonic oscillator potential =° s~) is used for description of the
singlo-particle motion. The corresponding I3amiltonian is
MICROSCOPIC INERTIAL FUNCTION3
157
where the oscillator part V~ of the potential is
and the part Ya,K modifying it is
Tt was pointed out in sect. 2 that an important question for inertial functions is to
satisfy the symmetry conditions (2.7). We mentioned in sect. 3 that the liquid-drop
inertial functions satisfy them. Also the functions calculated by Kumar and Baranger
in their pairing-plus-quadrupole model fulfil these conditions 1$). We show in the
appendix that our inertial functions obtained by the, formulae (4.2) and (4.3) with
the use ofthe modified harmonic oscillator potential satisfytho relations (2.7) as well.
Now, let us specify a few practical details connected with the treatment of the
Hamiltonian (4.4). The transformation to the stretched coordinates as) ~, q, Z
(~ = x./Mm=li~, etc.), which ensures as automatic inclusion of the couplings between
different oscillator shells for the pure oscillator part of the Iiamiltonian, leads to the
following form of the oscillator part Y~ of the potential
where p~ _ ~~ +ri2 +Z~, the functions Yzf are the spherical harmonics in the stretched
coordinate system and the coefficients coo, c1o, csz are
As the only part of the Hamiltonian to which we introduce the deformation a is
the oscillator part V, of the potential, the derivatives ôH/âa appearing in eq. (4.3)
for the vibrational paramaters are ôH/âa = aV~/a«. According to eq. (4.5), in the
stretched coordinates in which wo work, they are
ôH - bpa dcoo + ôc2o
ôczs.~
Y2s+Yz_z)J ~
(4.6)
~YZO+
ôa
Côa
ôa
ôa
Transformation of the single-particle angular momentum operators j~, appearing
in formula (4.2) for the moments of inertia, to the stretched coordinates gives
~ aRi~-bRfR+'a ;.
(4.7)
where l,` are the orbital angular momentum . operators in the stretched coordinates
and s,~ are the internal spin operators. The formulae for l,`, jR and thé coefficients
ax and bR are
jR
(4.7a)
158
T. KAHIOWSKA u aL
and cyclically for y and z.
The operatorsfx have non-vanishing matrix elements only between states of different major shells. For the spherical harmonic oscillator states, which we use as a basis
in all the calculations, the matrix elements are
<N'I'~'Q'I1;INI~In>
s
~N-N')~N'I'~'Q'IP~Y2i+Y2-i)INI~) .
<N'I'~t'o'If;INl~te~ = i~I-i~(N-N')<N'I'~'Q'IP~Yzi-Yz-i)INIAD)~
<N'I'd'A'L1~INI~1.0) _ -~N-N~(N'I'd'O'Ip2(Y22 -Yz_ 2)INIAO~.
(4.8)
As already mentioned above, only the oscillator part of the potential is deformed .
The deformation is parametrized in the following way as. 1s)
(4.9)
where iR = 1, 2,3 corresponds to x = x, y, z. The volume~onservation condition for
an equipotential surface gives
(4.10)
Formulae (4.9) lead to the following expressions for the coefficients coo, c2o, c22
of eq. (4.Sa)
coo = ~0(8+ Y~~
cso = -~ cos Y.~o(8 . Y~.
C22 ° ~~ sin Y.~o(~ Ya"
The requirement of an equipotential surface to have the same shape as the nuclear
surface, eq. (2.1), gives
coo(e, Ya = ~(ß. Y)
(4.12)
~~(~ Y~ Ro(ß. Y)
which leads to the relation between the potential e, y, and the nuclear surface ß, y
deformation parameters
k sin 2y+2 sin y
Y. ar~an
.
k oos 2y-2 cos y
e=
3k cos y
= 3k (4+k2 -4k oos 3y)},
(4.13)
where k = "/SJ4n ß, as given in eq . (2.1a).
~.2. NUMERICAL CALCULATIONS
Numerical calculations are performed for transitional and deformed nucl~ from
the neutron-deficient region around barium : SO < .Z, N < 82. The nuclei are
iis-isoXe~ 121-134Ha and 116-136~~ being altogether
, nucleides .
The results are obtained for the 36 grid points, ß s 0(O.1x1.S and y = 0°(10°)60°.
MICR03COPIC INERTIAL FUNCTIONS
139
The "A = 165" parameters z9) of the Nilsson potential are taken. They are:
xP = 0.0637, ~ = 0.60 for protons and xo = 0.0637, ~o = 0.42 for neutrons. The
oscillator frequency
6a3° = 41A-} MeV
(4.14)
is used. T'he pairing interaction strengths
GpA = 28.5 MeV
GeA z 25.0 MoV
for protons,
for neutrons,
(4.15)
with 24 energy levels nearest to the Fermi level taken into account when solving the
pairing equations, are employed t °) . The strengths G are kept constant as functions
of deformation.
S. Rewlts aod dfecaeeion
We illustrate here the results for the example of t~ 613a. The results obtained for
the other nuclei investigated, its-i3o~~ 122-134 $IId 126-136~~ ~ $enetally
Smular t0 ihCm.
Fig. 3 shows the potential energy, figs . 4~ the vibrational and figs. 7-9 the rotational inertial functions. Each of the figures gives the microscopic results on the left
andthose ofthe liquid drop onthe right. All the microscopic values and also the liquiddrop potential energy are calculated for tsel3a. The liquid-drop inertial functions,
divided by B, eq . (3.1), are general, independent of the mass number A.
Let us concentrate first on the microscopic results.
The potential energy is calculated as a sum of the smooth part, given by the liquid
drop, and the shell correction obtained by the Strutinsky method. For the liquid
Fig. 3. Contour maps of the microscopic (left) and the liquid-drop (right) potential energies V
calculated as functions of the deformations ß and y. Minimal value of the energy, specißed at each
map, is obtained at the point denoted by a cross and servos as the referencevalue. The values corresponding to the contours 1,2,3, . . . are higher by therespective numbers 1,2,3, . . . of the units specißed as the scale. The dashed line in the left 5gurocorresponds to 0.3 and dotted line to 0.23 of the unit .
160
T. RADTIOWBKA et at:
Fin. 4. game as ila. 3 for the vibrational inertial function B~~. As the right Baure wes the hydrodynamic values normalized to B, eq . (3 .1eß the minimal value 1 is obtained at the point ß ~ 0.
MICR03COPIC INERTIAL FUNCTIONS
161
Fig. 7. Same as ~ 4 for the rotational inertial fimcdoa B,. Negative number at the contour liana
is the right flgote mean that the lines correspond to values lower than 1.
162
T. KANIOWSIU st al.
drop, the Myers-Swiatecki parameters 3°) are taken. Fig. 3 implies that the nucleus
12sBa is well deformed in ß. The deformation energy is around 2.5 MeV. However,
as with most nuclei in the neutron-deßcient region 50 < Z, N < 82, it is soft to the
non-axial y-deformation. The bottom of the potential energy valley going from oblate
(Y ~ ~°) ~ 1?rolate (Y = 0°) ~i?~ is flat and the prolate-oblate energy difference
is small. In other words, the potential energy in itself localizes the wave function of
the collective motion to some region around ß ib 0 (ß s;s 0.25). It does not localize,
however, the wave function in the y-degree of freedom. The behaviour of tlu wave
function with y may be strongly influencedthen by the y-dependence ofthe dynamical
parameters of the nucleus.
Conoeming the dependence of the inertial functions on deformation, considerable
fluctuations in them with deformation are seen in figs. 4-9. The fluctuations come
from the shell structure of the nucleus and are largtr for the vibratioaal than for the
rotational parameters. Their amplitude amounts to about 40 ~ of the average value
for B and even more for Bar in the investigated interval of the deformation. For the
function B, some barrier between the prolate and the oblate shapes is obtained. The
height of the barrier comes to around 30 ~ of the value of Br, for axial shapes . The
barrier may have some effect of favouring the axial shapes .
On what concerns the average behaviour of the microscopic mass parameters, i.e.
the behaviour obtained after averaging the shell effects, we can say the following. The
rotational parameters decrease quite strongly with increasing ß. They decrease
roughly .(there is some prolate-oblate asymmetry) to only about half value when ß
increases form 0 to 0.5. This is quite unlike what we get in the liquid-drop model and
means that the moments of inertia increase with increasing ß slower than ß2. The
vibrational parameters are or about constant (B,) or increase slowly (B~,) with ß.
The parameter BpR increases with ß quite strongly for oblate (y = 60°) but is about
constant for prolate (y = 0°) shapes.
A general statement for the microscopic values is that B~, is alwayas smaller than
both Bpp and B, similarly as it is in the liquid-drop model.
The liquid-drop inertial functions change with deformation much less (note the
scale in the plotsl) than the microscopic functions. The largest change, obtained for
BRp, is around 25 ~ of the minimal value, while it is of the order of 400 ~ in the
microscopic ~ for the interval of deformation considered. The other liquid-drop
functions change by less than 6 ~.
Resides the different scales of the changes, it is also seen that the character of the
deformation dependence of the microscopic inertial functions, even averaged over
the shell effects, is different from that of the liquid-drop functions.
Only systematic research on the collective properties of nuclei, such as energies and
transition probabilities, can tell us how good are the microscopic inertial functions
calculated by us and how important is their deformation dependence. Preliminary
results of such calculations, performed by us, indicate that the microscopic values of
these functions are about two to three times too small to reproduce experiment.
MICROSCOPIC IIIERTIAL FUNCTIONS
163
Also a simple analysis of the experimental data on the energy of the first 2+ state and
the respective transition probability B(E2) leads to similar results i 9). Namely, the
inertial functions, treated in the zero-order approximation eq. (3.2), deduced from
this data in the harmonic approximation are about two to three times larger than our
microscopic values. One should stress however that the "experimental" values
deducedinthedynamicalcalculations arerather sensitive tothe potential energy used. A
change in the energy due to an inclusion of as additional degree offreedom, forexample
the hexadecapole deformation may change them significantly. Detailed results of the
dynamical calculations will be published elsewhere si).
It is possible that our microscopic inertial functions are really too small and that
to describe the collective motion of the vibrational and rotational type requires the
inclusion of some additional residual interaction a 2- '4). or some other approximation in the method of the calculations. It is interesting, however, that the cranking
inertial functions calculated inthe same way as in the presentpaper, seemto be about
right for the de8Cr1pt10II 96-38) of the fission process (half-lives).
It is worth mentioning here that the possibility of our inertial parameters being too
small to reproduce the experimental collective energies is not contradictory with the
results of ref. 1). In this reference, the rotational parameters, calculated in the same
way as in the present paper, allowed tho reproduction with a 15 ~ accuracy of the experimental energy E2 ~ of the lowest 2+ state of deformed nuclei in the barium region .
The states were interpreted as the states of pure axially symmetric rotors. Ifwe assume,
however, that instead of good stabilization of an axial shape there is a complete
instability in the y-degree offreedom [Whets-Jean model a 9)] we wouldneed the mass
parameters twice as large to reproduce E2 *. Our preliminary dynamical calcuations
of the wave functions of the 2+ states for a few nuclei in the barium region indicate
that we are close to the second (i.e. y-instability) rather than to the first case.
6. Cànclaeiooe
The following conclusions may be drawn from our research :
(i) The microscopic values of the inertial functions are three to eight times larger
than the hydrodynamic ones for the interval of deformation investigated . This is
close to what was obtained in the similar microscopic calculations of the function
BRA for the actinide region s s). The last indicates only that the dependence of the
microscopic inertial functions on the mass number A is close to that of the hydrodynamic functions, i.e.close to the Af dependence.
(ü) Shell effects areimportant for the inertial functions, especially for the vibrational
ones . The corresponding fluctuations are up to about f40 ~ of the average value,
in the deformation interval investigated.
(iü) The dependence of the microscopic inertial functions, averaged over the shell
fluctuations, on deformation seems to be different from that obtained in the hydrodynamic model. In particular, all rotational functions (B,~, Br, B= decreaso, on the
T. KANIOWSRA et d.
164
average, with increasing .ß. The latter means that the microscopic moments of inertia
J~, J,, Js increase with increasing ß more slowly than ß2.
(iv) Preliminary results of the calculations of the collective energies and transition
probabilitiesfor the investigated nuclei indicato that the microscopic inertial functions
are still two to three times too small to reproduce wcperiment . However, these values
are rather sensitive to the potential energy used in the calculations . A change of the
potential energy, e.g. due to the inclusion of the hexadecapole deformation not
considered here,.may chan>3e them considerably. .
The authors wouldlike tothankProf. R. ICumar and Drs. Ch. Droste, T. Ragnarason
and J. Srebrny for very useful discussions. Very helpful comments by Profs. A. Bohr
and B. R. Mottelson are gratefiilly acknowledged.
ai. sn~x~rRY coxnrrioxs
To show that our microscopic inertial functions ïulßl the symmetry conditions
(2.7) it is enough, according to acct. 2, to demonstrate that they form a tensor in the
lab systemIn that system, the cranking formula for the inertial functions gives
8k-do
k*o
in analogy with the formula (4.1). Using the relation
eq. (A. l) may be written as
F
f
k" 0
where ~° is the Hamiltonian of thé system :
(8k-8o)3
(A.3)
A
Here, His the single-particle and Ep the pairing Hamiltonian.
To demonstrate that BF. of eq. (A.3) are components of a tensor, it is enough to
show that ~° is scalar, and even only that its single-particle part H is scalar as the
pairing part Hp is éxplicitly rotationally invariant. In our modißed oscillator Hamiltonian (4.4), the kinetic energy and the correction part V~ of the potential are obviously scalars. To see the scalar character of the oscillator part Yo, we write it in the
form
(A.s)
MICROSCOPIC INERTIAL FUNCTIONS
165
d00 ° 7l~s+CO,+.COs),
d2f - ~lio3l~i-~s_~~)+Y3l~ß+~~,_Z7V3l~s-~7)f
the angles 9, ~ in the argument of YzF are in the lab system and .~~ is a shortened
notation for the rotation functions ~r,(Be) of eq . (24). In this form, V, has explicit
solar charactar. It is easy to check that it talaes the form (4.4a) in the intrinsic system
what ends our demonstration .
It may be mentioned that the tensor dz can be easily expressed in forma of the
dynamical variables af. The expression is
where a is given byeq . (3.1a) and
6r = -J~[«~7~,v
where the brackats in the last equation denote tha tensor coupling to a given multipo~Y"
The way of geüing our microscopic inertial functions from BF., eq. (A .3), is the
transformation of the derivatives a~aa F to the intrinsic system
sin yx
x
sin y,
'
sin gs
'~
A.7
( )
where I~, I,, Is are the components of the angular momentum operators in the intrinsic
system .
For the vibrational functions we get
`e r -
t*o
(dk- ~o)3
which uader treatment of the pairing interaction Hp by the BCS formalism gives
formula (4.3).
For the moments of inertia, we obtain
I~~ILIcf ~°]Ik>h f
~s a Z~~ ~
t*o (d,~ - do) a
which gives
~x
= 2~1=
E I<ol[!~, ~°71k>I~
= 2~2
~ I<oliRlk>l'
(A.9)
,
166
T. KANIOWSKA et aL
because [IR +j~, a!°] = 0, due to the scalar nature of the Hamiltonian .~!°. Formula
(A.10) leads to eq. (4 .2) when the pairing interaction is treated in the BCS way.
A.2. ROLE OF TH8 PAIRING-VIBRATION TERMS
Role of the pairingvibration terms ~h eq . (4.3), was discussed in ref. s s). However, only for the axially symmetric prolate shapes (y = 0°) of a nucleus sad only for
the fuaction Bpp. Now, we can extend the discussion to the non-axial shapes and to
all three vibrational functions B.
Let us denote the contributions of the term P~ to the functions Bpp, Bp i and B,
by aBpp, 3Bp, and bB,~, respectively . According to the notation of eq. (4 .3), aBpp
= Ptl, ôBp ~ = P`~Iß and bB ~ P ty/ßs for (at, a!) _ ~, ß). ~, y) and (y, y), respectively.
The investigation, performed by us for the twenty nuclei specified in sect. 5, shows
that the contributions bB may be positive as well as nogative. They fluctuate around
zero as functions of both the deformation and the nucleon (proton or neutron)
numbers. Let us discuss then the absolute values of SB. The ratio SBppfB~p as well as
ôB~,lB°~, wham B° is the value of B when no pairing-vibration contribution is included, rarely exceed 15 ~. Their average values are around 10 ~.
Concerningthe contribmion bBpr we can state that on average aBpr rs aBpp ~ SB.
However, as B°p, is a few times smaller than ~p and B°,v, the ratio bBp,/B%, is a few
times larger than each of the ratios bBpplBwp and SB,JB°. . ,Thus, for Bpi only
the behaviour of B depends remarkably on whether the pairing-vibration term is
included or not.
For the axial shapes (y ~ 0° or 60°), both contributions ôBp, and bB vanish
(Bh, itself, also vanishes but ~~ does not).
To complete the discussion, let us give a more particular illustration . For the deformation ß = 0.40, y a 30°, we get the following numbers for the twenty nuclei considered. The maximum value of aBpplB%p is 20 ~ and the average one is 9 ~. The
corresponding values for aB~lB;r are 15 ~ and 5 ~ and for SBp~lB°p~ they are 70
and 25 ~.
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