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Moment of inertia and rotational frequency in hot rotating 158Yb nuclei
Nuclear Physics AS04 (1989) 413-435 Noah-Holland, Amsterdam
MOMENT OF INERTIA AND ROTATIONAL FREQUENCY IN HOT ROTATING ‘*Yb NUCLEI Alan L. GOODMAN’
Received 4 July 1989 Abstracti . The nucleus rs%% has a transition from quasivibrat~onal structure at low spins and temperatures to quasirotational structure at high spins and moderate temperatures. Thermal shape fluctuations produce fluctuations in 9 and w. However, these fluctuations do not eliminate the structural transition. The standard deviations in 9 and w decrease at moderate temperatures. Consequentfy, thermal shape fluctuations produce minimal dispersion in E2 gamma energies, for moderate temperatures and high spins.
1. Transition from ~uasiyibrationa~ to quasirotational structures The observable attributes of a nuclear state include its energy E and angular momentum I, However, to study the collective properties of a band of states, it is often more usefut to consider the moment of inertia 9 and the rotational frequency w. In the cranking model, these are defined by
(1) .%==J/w,
(2)
where J = [r(I + I)]“‘.
(3)
Although $ and o are not observable properties of nuclear states, they have been invaluable for understanding the structure of the yrast line and low-lying colfective bands. It is common to plot nuclear spectra in the form .9(w’), In hot nuclei, the level density is so high that individual bands cannot be resolved experimentally. We have relatively little experience in analysing the behavior of the moment of inertia and rotational frequency in hot nuclei. Nevertheless, a study of 9 and o can reveal structural transitions in hot nuclei, even though individual bands are not resolved. For an E2 transition, the value of Ey determines o through eq. (1)
itI=[t(r+
Et - Et-z l)]‘/’ -l(r-2)(1-I)1’“1StE,.
’ Supported in part by the National Science Foundation. ~3~5-9474/S9~$03.5~ @ Eisevier Science Publishers B.V. ~No~h-Holland Physics Pub~jshing Division)
A.L. Goodman / Hoi rotating “‘Yb
414
We first consider independent.
cold
nuclei.
Assume
From eq. (4), it follows
eq. (2), 3 increases Next consider
linearly
nuclei
a vibrational
spectrum.
that o is also independent
Then
E, is spin
of the spin. From
with J.
a rotational
spectrum,
where
E = J2/2$, and $ is constant.
From
(5)
either eq. (1) or eq. (2), it follows
to J. Then eq. (4) shows that E, is proportional As a variation, suppose I > I,,, the energy is
that o is proportional
to J.
that the state at I, is a rotational
band
head.
Then
E =R2/24,,,+E(I,)=(J-J,)2/24,,,+E(I,), where 9,,, is constant.
From eq. (l), it follows
(6)
that
J=.Yjrotw+JO.
(7)
The function J(w), or J($E,), has a slope equal Jo. The E2 transition energy is
to 4,,, and a vertical
E, = 2w = 2( J - Jo)/,a,,, which
is proportional
to J-J”.
Eq. (2) defines
moment
of inertia
is defined 9W_=$
From eqs. (7), (9) and (lo),
it follows
intercept
of
(8) moment
of inertia (9)
by
dJ aw
,
the kinematical
@“=4=J/w.
The dynamical
for
rot ’
(10)
that
s’2’/.9 ‘“=l-Jo/J.
(11)
Consider an ideal vibrator and an ideal rotor. The function ,a(,‘) is shown in fig. la. If Jo> 0, then the rotational 9 decreases with spin, in accordance with eqs. (10) and (11). Fig. lb gives the function J(w) or J(fE,). Now consider hot rotating nuclei. Suppose that for some particular path in the energy-spin plane, the functions .Y(w’) and J(w) resemble fig. 1. Then we could say that this path undergoes a transition from quasivibrational to quasirotational structure. For the quasivibrational states, E, is independent of J. For the quasirotational states, E, is proportional to J, or J -Jo.
A.L. Goodman / Hot rotating “‘Yb
nuclei
415
Rotational 3
J. -0
5 5 E
E -z e .e >
iti
w2 Fig. 1. Schematic
A recent Argonne tinuum
spectrum
National
provides
transition
from vibrational
Laboratory
evidence
(ANL)
CAJ
(Ey/2)
to rotational
spectrum.
measurement
for this structural
transition
However, the authors of this experiment raise some interesting peak which exhibits collective rotational behaviour is produced
of the E2 quasiconin hot ‘54Dy nuclei
‘).
questions. The E2 by a cascade which
flows through noncollective oblate states. What causes the rotational behaviour, where E, is proportional to the spin? The nucleus ‘s8Yb, which is an N = 88 isotone of ‘54Dy, has been studied at Oak Ridge National
Laboratory
(ORNL).
It was determined
that the positive-parity
yrust
states with spins 22-36 are quasivibrational ‘,‘). Although the yrast line is noncollective for spins above 40, increasing the temperature promotes collective structures 4,5). In this article we study ‘5xYb. We calculate 4 and w, as well as other properties, for various
paths in the energy-spin
Fock-Bogoliubov paths.
cranking
For each path,
plane.
(FTHFBC)
we find a transition
In sect. 2 the finite-temperature equation
hmX)is solved
from quasivibrational
for these structure
Hartreedifferent at low
spins and temperatures to quasirotational structure at high spins and moderate temperatures. In sect. 3 we show how thermal shape fluctuations produce fluctuations in the moment of inertia and the rotational frequency. Even though the FTHFBC theory predicts a structural transition, it is well known that thermal fluctuations can eliminate these transitions ‘-” ). So it is interesting to see whether fluctuations in 4 and w wash out the transition from quasivibrational to quasirotational structure. Normally one expects that thermal fluctuations will increase as the temperature rises. For example, if Aw is the standard deviation in rotational frequency for given spin, one anticipates that Aw would increase with temperature. However we find that at moderate temperatures, Aw and A4 exhibit a sudden decrease, rather than an increase. This may have implications for the description of motional narrowing in the damping of giant resonances in hot nuclei. A preliminary account of this work is given in ref. “).
A.t.
416
Goodman / HOI rotating rssYb rrdei
2. Finite-temperature The FTHFBC
equation
HFB cranking calculations
is
__l*)( z)=Ei( T)(__T* The Hartree-Fock
(HF)
hamilton~a~
(12)
is
X=e-&&u,N,-@,N*-&IX+~S where the sphericat and p,.
singe-parti~ie
energy
The HF and pair potentials rij
(131
is e, and the chemical
potentials
x
$
( 2%
/ ?JIjbfk
(141
9
Ai,-$C(ij[uIkZ)tk,. kl The HF and pair densities
V’(l-f)V],j,
t~j=(CjCi>={rifV*+ o~~npation
where T is the temperature and free energy are
(15)
are
pij={c;ci)=[fifu*+
The quasiparticie
probability
V’(l-f)tr],. is
and Ej is the quasiparticle
energy.
E =(H)=Tr[(e+fl‘)p+~At’], S=
are (-tp
are
The energy,
entropy,
(19)
-C[~ln~+(l-.J;)ln(l-J;)l,
F=E-TS,
(20) (21)
where H is the pa~r~ng_~~us~quadruQo~e (PPQ) ham~~tonian of Kumar and frequency w is adjusted to satisfy the spin constraint Baranger ‘47is). The rotational .T=(.JJ=[r(l+l>]‘/‘_
(221
The FTHFBC phase diagram for “‘Yb is given by the solid line in fig. 2. For low spins and temperatures, there is prolate (or slightly triaxial) collective rotation. Otherwise there is oblate noncollective rotation. Fig. 3 transforms the phase diagram to the energy-spin plane. Fig. 2 also contains several straight-line trajectories. We first study path b, which is defined by T = (0.02 MeV) (I - 10)
{Path b) .
(231
The FTHFBC equation is solved for every even spin from 10 to 60 and for the spin-dependent temperature of eq. (23). The resulting energy trajectory is path b in fig. 3. This path is very similar to the path observed in the ANL experiment on t54Dy with Ebeam = 15.5 MeV.
A.L. Goodman / Hot rotating ‘saYb nuclei
2.0
,
‘.
I ‘,
”
I “,
I “.
8.
”
417
* I
“, J
9
9
1.6-
OBIATE SPHEROID NON-COLLECTIVE ROTATION
ls8
Yb
_ /
//’ 1.2
-
0.8
-PROLATE
AND
COLLECTIVE
0
TRIAXIAL
ROTATION,,
10
20
’
30
40
50
60
SPIN for ‘58Yb. The critical
Fig. 2. Phase diagram
The function
9(w’)
temperature
r, versus the spin I.
for path b is given by the solid curve (filled circles)
This curve has two distinct
in fig. 4.
sections.
For spins 18-36, 9 increases very rapidly, and there is relatively little variation in w2. However for spins greater than 40,9 decreases somewhat, and becomes nearly constant for spins above 50. This curve bears some resemblance to fig. la. The function Z(w) for path b is given by the solid curve in fig. 5. This curve also has two distinct components. The slope of Z(w) is much larger for spins 18-36 than it is for spins above 40. This should be compared to the schematic transition in fig. lb. Schematic rotational structures were characterized in eq. (7), where J(w) is a straight
line. For spins 48-60 the HFB function
In accordance
with eqs. (7) and (lo),
40 t -
0
J(o)
is very close to a straight
the slope is $(*I = L+,,~-61
OBLATE ROTATION
10
20
30
40
50
SPIN Fig. 3. The energy
E versus the spin I for “sYb.
60
MeV-‘,
line.
and the
418
A.L. Goodman 1
Hot mating 15X Yb nuclei
vertical intercept is & = 10, For this spin-interval, eq. (8) shows that E, F=(0.033 MeVf (J - IO). Our anaIysis
of ..P(w’)and i(w) enables us to charac~~r~~~path b as a~~r~x~rnat~~y ~uasirot,a~~o~a~ for spins above 40, and ~p~roxirnat~~y quas~vibra~~o~~~ for spins below 36, This agrees with the ANL experiment on ““Dy and the QRNL experiment on rsxyb_ Why do the moment of inertia and rotational frequency experience a change in structure on path b at spins -38-40:” We first consider the reasons for the rapid rise in $(w’) for spins up to 36. Fig. 6 shows how the pair gaps d,, and A, vary along path b. The proton pair gap exhibits a gradual decline for spins 18-36. How much of the change in the moment of inertia is caused by this decline in A,?
/ Hot rotating
A. L. Goodman
“‘Yb
a
1.6
nuclei
419
u “‘Yb
1.2
-
0.6
-
SJ 2
a 0.4
h
n T=O.O2MeV
i
0.0
1 * ’
.
10
0 20
(I-
10) ’ 30
.
’
”
.
1 40
SPIN Fig. 6. The proton
For example,
pair gap A, and the neutron pair gap A, versus the spin I for path b.
suppose
that
the pairing
is omitted
from the calculations.
The
FTHFBC equation is solved again for path b with A = 0. The function 9(w’) is given by the A = 0 curve in fig. 7. This should be compared with the HFB curve in fig. 7. For the spin interval 20-36, the increase in 4 is 260% larger for the HFB curve than for the A = 0 curve. For this spin interval, the increase in w2 is 30% smaller for the HFB curve than for the A = 0 curve. Both of these factors make the HFB ,a(~‘) function much steeper (and 9(w’) function calculated with A = 0. Next we keep A fixed at the ground
hence
more
state value,
quasivibrational)
instead
of letting
than
the
A be spin and
temperature dependent. The 9(w’) function is given by the A,,, curve in fig. 7. Once again, this should be compared with the HFB curve in fig. 7. Keeping A fixed at
70 60 50
If?20
36
20
10
40
/
.-. .-. 0-o
HFB A 9.s.
A=0
T=O.O2MeV
(l-l
0)
_
10
30 0.0
0.2
0.4
O2 Fig. 7. The moment
0.6
0.6
1.0
(UeV2)
of inertia 4 versus the square of the rotational frequency w for path b. Calculations displayed use the FTHFBC A, the ground state A, and A =O.
A. L. Goodman
420
the ground-state
value
I = 36. Furthermore, rotation
to oblate
spin interval
causes
the moment
the transition noncollective
20-36,
/ Hoi rotating
of inertia
from
rotation
the increase
tssYb nuclei
prolate occurs
slightly
triaxial)
at I = 30, instead
in 4 is 230%
the A,.,. curve. The increase in 6.1~is 58% A,.,, curve. From these comparisons using fig. 7, reasons for the quasivibrational structure decline in the proton pair gap in this spin
to peak at I = 30, instead
(and larger
smaller
of
collective
of I = 36. For the
for the HFB curve than for
for the HFB curve than for the
we conclude that one of the significant of path b for spins 18-36 is the gradual interval. Either setting A = 0 or keeping
A fixed at the ground-state value significantly of path b for spins 18-36.
weakens
the quasivibrational
structure
The rapid increase in 4 for spins 18-36 is also caused, in part, by the shape transition and associated rotation alignments. The quadrupole deformation parameters /3 and y for path b are given by the solid curves in figs. 8 and 9. At spins 36-40 there is a rapid transition from nearly prolate collective rotation to oblate noncollective rotation (y = -6O”), and there is a decrease in /3. For I = 30 on path b, there are four orbitals which are rotation be the spin projection along the x rotation JcQ, are essentially iIjI2 neutrons
aligned.
axis. Then two of these orbitals,
=?)=0.97,
(&,2mX
(~uz~i,z,2m.X= y} = 0.93 , and two of the orbitals,
IcY~)and IQ>, are primarily
hI,,,z
protons
(axlh , ,/zmX = $) = 0.99 , (a,lh ,,/,m,=~)=0.96. 0.24
. . .
I
,
I
.
.
.
I
.
.
.
,
.
.
'5aYb
-
,,.._...... .
.
.
.
.
.
.
.
.
.
.
.
.
.._
,_,.._...~~~
T=O.O2MeV
0.00 -<‘. 10
’ . ’ . . ’ . * ’ * ’ . 20
30
40
(l-l
0)
.
’ ’ ’ ’ ’ ’ 50
60
SPIN Fig. 8. The quadrupole
deformation
parameter
p versus the spin I for path b.
Let m, Ia,) and
A.L. Goodman
40 I
/ Hot rotating
rs8Yb nuclei
421
Yb
15’
10
20
40
30
50
60
SPIN Fig. 9. The quadrupole
deformation
Since path b has finite temperature,
parameter
these orbitals
y versus the spin I for path b.
are not fully occupied.
The particle
occupation probabilities are p, = 0.94, p2 = 0.61, p3 =0.97 and p4= 0.96. As the nucleus makes the transition from collective to noncollective rotation, more orbitals become rotation aligned. At Z = 40 all nucleon spins are rotation aligned. Next we study the high spin (I > 40) section of path b. The functions ,a(~‘) and Z(w) indicate a structure which is approximately collective rotational. This is in spite of the fact that the phase diagram (fig. 2) shows noncollective rotation for all temperatures when the spin exceeds 40. Some insight into this apparent contradiction may be obtained by first considering the yrast line produced by noncollective rotation. (Assume that the spin Z is sufficiently large that the pairing gap has vanished.) In noncollective rotation the x rotation axis coincides with the symmetry axis of the oblate shape. Then each nucleon orbital has rn, as a good quantum number. Therefore the HF hamiltonian X and J, commute, so that the eigenvalues of 2, i.e., the single-nucleon
energies
in the rotating
frame,
are
E&(W) = E,(W) -Wn,, where m denotes spin is simply
m,, and E,(W) is the eigenvalue
(24) of e -Z_LN +Z(w).
The nuclear
where the lowest levels E& are occupied at each o. For different w, a different set of nucleon levels will be occupied, which causes different self-consistent shapes. This creates the o dependence of the HF potential Z’. However to simplify the argument, suppose we ignore the shape self-consistency requirement. Assume that the deformation is fixed at the ground-state value. Then 8, and Z become independent of w, and they have ground-state values. Consequently E&(W) is a straight line
A.L. Goodman / Hot rotating r5hYb nuclei
422
for all 0 F:,(W) =
E,
-mm,,
(26)
ma.
(27)
where the slope of the line is dE&
dw If an increase
in w does not produce
a nucleon
level crossing
at the Fermi energy,
then the same orbitals remain occupied, and the spin J does not change. However if increasing w does cause a level crossing at the Fermi energy, then a nucleon leaves
one of the crossing
orbitals
and occupies
the other
orbital,
will increase. Consequently J(w) is a stepwise function. Let the temperature This argument is now applied to ‘%J.
and the spin J
and pair gap equal
zero. Assume an oblate shape rotating around its symmetry axis. Assume deformation is spin independent, with p = 0.1. (For path b at I =40, p Then E, are the eigenvalues of e-PN +r(/3 =O.l, y = -60”). (Note that -60”) = r(p, y = ISO’).) The function J(w) is the solid stepwise path in fig. the nucleon level crossings are statistical, therefore the jumps For a collectively rotating rigid object, the spin is
that the =0.104.) r(/3, y= 10. Since
in J(w) are statistical.
J = 9rigidm 9
(28)
so that J(w) is a straight line with a slope equal to -arigid. For an oblate shape with p = 0.1 which is rotating around its symmetry axis, 4irigid= 68.2 MeV-‘. The rigid body J(w) is shown in fig. 10. The essential point is that for spins 48-60, the statistical stepwise
J(w)
to the slope
function
can be characterized by an average slope, which is similar single line, i.e., 4irigid. Although the noncollective
of the rigid body
70 60 50
(J,>
40 30 20 10 0
-0.0
0.4
0.2
w Fig. 10. The spin versus the rotational
frequency
0.6
0.8
1.0
(MeV)
for an oblate
shape
rotating
around
its symmetry
axis.
A.L. Goodman
particle
rotation
is essentially
duces an average moment
moment
/ Hot rotating
different
of inertia
“‘Yb
nuclei
from collective
423
rigid-body
for I = 48-60 which is similar
rotation,
it pro-
to the rigid-body
of inertia.
This conclusion
was first reached
by Bohr using the Fermi gas model 16). However
it is interesting that the argument seems to work for ‘?b only at the highest spins, 48-60. If the stepwise J(w) function is approximated by a straight line for the spin interval O-44, then the slope is 2163 MeV’, which is larger than $rigid by a factor of 2.4. These calculations
use one major oscillator
shell for each parity.
How would the
inclusion of higher shells affect the argument. 7 The first additional orbits which would contribute to the yrast line at high spins are the i,X,z proton and jlSjZ neutron levels. When these levels are included in the calculations, then the stepwise J(w) function is altered for I = 50-60. This change is given by the dashed stepwise line in fig. 10. Observe that inclusion of higher shells does not alter our basic conclusion. This argument regarding the moment of inertia for noncollective rotation has been given for an yrast line, which has zero temperature. At finite temperature, the nucleon level crossings no longer cause discontinuous probabilities. This is apparent from the Fermi-Dirac sequently provides
the stepwise function additional justification
jumps in orbital occupation distribution of eq. (18). Con-
J(o) becomes smooth at finite temperature. This for characterizing J(w) by an average moment of
inertia. It is instructive to consider the high temperature behaviour of 9(w’). Consider a constant temperature of 2.0 MeV. The FTHFBC equation is solved for various spins. The HFB function 9(w’) is given by the solid curve in fig. 11. There is very little variation in 4 for spins from 2 to 60. Fu,rthermore, this small variation occurs within the boundaries created by a rigid sphere and a rigid oblate shape (p = 0.1) rotating
around
its symmetry
axis.
2
l-
“0 2 2
Oblote Rigid Body ____-__------E~IO_JQ___~_____~____ 60-
Spherical
(@=O.l) -------TO---
--
Rigid Body
50-
HFB
P
T=Z.OMeV
.
40 0.0
0.2
0.4
W2 Fig. 11. The moment
of inertia
9 versus the square
0.6
0.8
1 .o
(MeV2) of the rotational
frequency
w for T= 2.0 MeV.
A.L. Goodman / Hot rotating “‘Yb
424
Our conclusion
is that oblate
rotation produces an yrast line and L~~ipid for spins 48-60. Therefore it is limit which both give 9,
a high temperature not surprising
nuclei
noncollective
that the finite temperature
This energy path is quasirotational
path b in fig. 3 also shares
at high spins, even though
this property.
the underlying
motion
is noncollective. Our discussion has been restricted to path b of figs. 2 and 3. The obvious question is whether other energy trajectories also undergo a transition from quasivibrational to quasirotational structure. We now consider several other energy paths. Fig. 2 shows three additional
paths
a, c and d, which are defined
T=(O.Ol
MeV) (Z-10))
by
(path a)
(29)
T = (0.03 MeV) (I - 10) ,
(path c)
(30)
T = (0.02 MeV) (I - 20) .
(path 4
(31)
The FTHFBC equation is solved for various energy trajectories are shown in fig. 3.
states on these paths.
The FTHFBC
Fig. 12 compares the functions 9(w’) for the four energy trajectories. The four curves in fig. 12 are remarkably similar. They all have a transition from quasivibrational structure at low spins and temperatures to quasirotational structure at high spins and moderate temperatures. Except for Z = 40, the four energy paths have similar values of 4 and w2 at each displayed spin. Fig. 13 compares the functions Z(w) for the four energy paths. Once again the similarity is striking. From figs. 12 and 13 we conclude that the existence of the quasivibrational to quasirotational transition is remarkably insensitive to the selection
of a particular
Is there a discernible
path in the energy-spin pattern
the quasivibrational-quasirotational 100 i
>
governing
plane.
the place in the energy-spin
transition
occurs?
plane where
For a particular
energy path,
r
90
“‘Yb
-
-
$ ; & W z
g Iwz
8070
-
60-
0-0
T=O.OlMeV
(I-10)
--•
T=O.O2MeV
(I-10)
=-m
T=O.O3MeV
(l-l
T=O.O2MeV
(I-20)
l
50
-
o-
P
--D
0)
I
40 ’ 0.0
0.4
0.2 u2
Fig. 12. The moment
of inertia
9 versus the square
0.6
0.8
1.0
(WP) of the rotational
frequency
w for paths a, b, c and d.
A.L. Goodman
/ Hot rotating
rssYb nuclei
425
60 50 z E VI
40 30 20
--
T=O.Ol
-
T=O.O2MeV
(I-10)
T=O.O3MeV
(l-l
T=O.O2MeV
(I-20)
- -
10
MeV (l-l
0)
0)
0
0.0
0.6
0.4
0.2
w
0.8
1.0
(MeV)
Fig. 13. The spin I versus the rotational
frequency
o for paths a, b, c and d.
this transition is not sharp. It does not occur at a specific spin. Consequently we cannot give a precise answer to our question. However, to provide a rough guideline, we could mark the end of the quasivibrational structure by the spin at which 9 stops increasing. From fig. 12 and other plots which include more spin states, one finds that this terminal spin is 38 for path a, 36 for path b, and 30 for path c. These terminal states are marked by the filled circles in fig. 3. The dotted line which connects these terminal states serves as the approximate boundary which marks the end of the quasivibrational structure. Observe that this boundary differs from the phase transition line which marks the onset of oblate noncollective rotation. For path b, the quasivibrational terminal state occurs at E = 15.1 MeV and I = 36, whereas the transition to oblate noncollective rotation happens at E = 18.4 MeV and I = 40.6. (In the cranking model the spin is a continuous variable.) Fig. 3 shows that the quasivibrational terminal states of paths b and c have similar energies but different
spins.
This implies
that for energy
paths b and c, it is the energy, quasivibrational structure. Our calculations
and
trajectories
not the spin,
in the neighbourhood
which
marks
of
the end of the
on “‘Yb are now compared to the ANL experiment on the N = 88 isotone ls4Dy. This experiment measured the quasicontinuum spectrum of ls4Dy. For a beam energy of 155 MeV, the de-excitation path of 154Dy is very similar to our path b for ‘5*Yb. For example, at I =40 path b has an excitation energy E = 18.1 MeV and a thermal excitation energy E* = 6.5 MeV, whereas the ANL de-excitation path has E = 18.9 MeV and E’ = 5.6 MeV. For Ebeam= 155 MeV, the ANL path shows a transition at spin 38. At higher spins, the E2 gamma energies are proportional to the spin, which was interpreted as indicating collective rotational structure. For spins below 38, E, is approximately independent of the spin, which suggests quasivibrational structure. By varying Ebeam, the ANL experiment studied different de-excitation paths. For these paths it was found that the change in structure
A.L.
426
occurs
at a common
consistent Finally
Goodman
energy,
E = 17 MeV,
with our calculations we consider
/ Hot rotating
rsaYb nuclei
rather
than
a common
spin.
This
is
on “‘Yb.
the B(E2)
values
and static electric
quadrupole
moment
Q.
For a rotor with large spin, these are 1
B(E2,I+Zk2)=5
Z2e2R4/32~~~Z(-y-300),
3 =p ZeR2P sin ( y - 30”) , Q (5Tr)“* where R = roA”3 ,
(34)
and r0 = 1.2 fm. Fig. 14 shows the B(E2) value for energy path b. The solid curve is the result. The B(E2) value vanishes at I = 40.6, where there is a transition noncollective rotation (y = -60”). Although the FTHFBC theory predicts of inertia and rotational frequencies which are quasirotational at high moderate
temperatures,
the
B(E2)
value
is zero,
which
suggests
FTHFBC to oblate moments spins and
an absence
of
collective rotational structure. In the ANL experiment on ‘54Dy, the average electric quadrupole moment of the initial rotational cascade was a free parameter. The value selected corresponds to B(E2) = 300 W.U. which is typical of strongly collective rotation. This is inconsistent with our HFB result for “‘Yb, which has B(E2) = 0 for I > 40. A partial resolution of this apparent contradiction is given in the next section. The quadrupole moment Q is shown in fig. 15. The FTHFBC result is the solid curve. It shows little structure. This is because the sign of Q is the same for prolate collective
and oblate
noncollective
140
rotation.
. . ..I..............,..... T=O.OZMeV
10
20
(I-10)
30
40
50
SPIN Fig. 14. The B(E2)
value versus the spin I for path b
60
A.L. Goodman / Hot rotating “*Yb nuclei
/../
/ I
/
r
T=O.O2MeV
.a I . .
10
. I..
20
(I- 10)
,
30
I...,,...,
40
427
“‘Yb 50
. 60
SPIN Fig. 15. The static electric
3. Thermal fluctuations
quadrupole
moment
Q versus the spin I for path b.
in moment of inertia and rotational
frequency
Our discussion has been restricted to FTHFBC calculations, which do not include thermal shape fluctuations. However, it is well known that thermal shape fluctuations often eliminate structural transitions which are predicted by the FTHFBC theory. Consequently we must determine whether thermal shape fluctuations wash out the transitions in the moment of inertia and rotational frequency which were discussed in sect. 2. For fixed shape,
spin, and temperature,
the FTHFBC
eqs. (12)-(18)
are iterated
to find the value of the rotational frequency which satisfies the spin constraint, eq. (22). Repeated applications of this procedure determine the function w(/3, y; Z, T). Then eq. (2) determines the moment of inertia 9(p, y; Z, T). For each shape, the FTHFBC particle density may contain rotation aligned nucleons. These rotation alignments may have a large effect upon the value of w required
to produce
a given spin. Therefore
the functions
o and 4 implicitly
depend
upon the rotation alignments. Consequently the rotation alignments will affect the dispersions in 4 and w. However it should be noted that for fixed deformation, thermal fluctuations in the occupation probabilities of aligned nucleon orbitals explicitly cause the dispersion of the rotational frequency “). We have not calculated this effect. In these calculations, the pair gaps Ap and A, are taken from the FTHFBC equilibrium state for given spin and temperature. The free energy F(/?, y; Z, T) is calculated with eqs. (19)-(21). Thermal fluctuations produce different shapes. For a particular spin and temperature, the probability that a given quadrupole deformation occurs is p(p,
y; Z, T) K eeF(B.Y:L=)lT .
(35)
A.L. Goodman / Hot rotating r5xYb nuclei
428
For a function
0(p, y; I, T), the thermal
average
of 6 is
‘W, Y; 4 T)P(P, Y; 4 T) dT @I, T)=(B)=
Two metrics
have been used to calculate
All of our calculations is defined
I
(36)
P(P, Y; 4 T) d7 thermal
averages
“,19). These are
dr, =P d/3 dy,
(37)
dr2 = p41sin 3 y[ d/3 d y .
(38)
are performed
with both metrics.
The standard
deviation
A6
by AtI’(Z, T)=[(B2)-(B)2]“2.
(39)
Thermal fluctuations are studied for path b of figs. 2 and 3. The thermal averages 3, &I and 2 are calculated for I = 10,20, 30, 36, 40, 44, 50 and 60, where T is defined by eq. (23). The function 3(m2) is given in fig. 4. The dashed line (open circles) is calculated with dr, , and the dotted line (open squares) is calculated with dT2. Even though these curves include thermal shape fluctuations, two structures remain: approximately quasivibrational for spins up to 36, and approximately quasirotational for spins above 40. The thermal fluctuations do not wash out this structural transition. Except for I = 36 and 40, the thermally averaged values of 4 and w2 are quantitatively similar to the FTHFBC values. The function line uses dr2. transition
I(W) is shown in fig. 5. The dashed Once again the structural transition
spins,
W is remarkably
similar
line uses dr, and the dotted is preserved. Except for the
to mHFR. For rotational
structures,
the
function J(w) is a straight line, as in eq. (7). The function J(G) is nearly linear over the spin interval 44-60 for dr, and 40-60 for dr2. Observe that J(w) is close to a straight line for a spin interval which is longer for (5 than for wHFB. From eq. (8) it follows
that including
the thermal
fluctuations
produces
an average
& which
is proportional to J-J,, for a spin interval which is longer than that given by the HFB E,. For both metrics, Jo= 10. Past experience with thermal shape fluctuations often led to the conclusion that these fluctuations eliminate phase transitions which are predicted by mean field theories. Why does this not happen for the quasivibrational-quasirotational transition? To answer this question, we study the standard deviations in the moment of inertia and rotational frequency. The standard deviations A$ and Aw for path b are shown in figs. 16 and 17. The solid curves use dr, and the dashed curves use dr2. As expected, A.9 and Aw increase as the temperature (spin) rises from zero. However these standard deviations reach a maximum at T = 0.5 MeV (I = 35). For higher temperatures (spins) there is a precipitous decline in A.9 and do. These standard deviations are small, except for
A.L. Goodman
10
/ Hot rotating
30
20
429
rssYb nuclei
50
40
60
SPIN Fig. 16. The standard
intermediate
deviation
in the moment
spins near 36. It follows
of inertia
A$ versus the spin I for path b.
that the thermal
averages
3, 0 and 2
must
be close to the corresponding HFB values, except for spins near 36. This argument is given in a more quantitative form by considering the fractional fluctuations
in 4 and w. These are defined
by (40)
Cd”
6”
(41)
(5 .
If the fractional
fluctuations
.
0.05,
0.04
are small compared
r
to unity
S9<1,
(42)
6wel.
(43)
. I
.
I .
.
,
.
*,
-
T=O.O2MeV
10
20
(l-l
0)
30
40
50
60
SPIN Fig. 17. The standard
deviation
in the rotational
frequency
Aw versus the spin I for path b.
A.L. Goodman / Hot rotating “‘Yb
430
then the most probable values
3 and
(i.e. FTHFBC)
W. For example,
0.8 MeV. The fractional
much smaller
the state on path b with Z = 50 and
are 69 = 0.018 and 6w = 0.021 for metric
69 = 0.024 and &I = 0.026 for metric do not destroy
values of $ and w will be close to the average
consider
fluctuations
2. The reason
the quasivibrational-quasirotational than unity
How do thermal
shape
nuclei
that thermal transition
for states on path b, except fluctuations
1, and
shape fluctuations
is that 69 and &J are
for spins near 36.
affect E,? For an E2 transition,
eq. (4) gives
E,=2w, where
w is in MeV. The standard
deviation
(44) in E, is
AE, = 2Aw, and the fractional
fluctuation
T =
(45)
in E, is (46)
For the state on path b with I = 50 and T = 0.8 MeV, SE, = 0.021 for metric
1, and
6E, = 0.026 for metric 2. The thermal shape fluctuations produce only a 2% variation in E,. Consequently the y energies remain very sharp, even when thermal shape fluctuations are included. It should be noted that there are other mechanisms, such as rotational damping 2”), which produce additional dispersion in E,. Why do the standard deviations in the moment of inertia and rotational frequency have a dramatic decline for T > 0.52 MeV (I > 36)? To answer this question, consider how 3 and w vary with deformation for fixed spin and temperature. 18 shows ,a(& y) for Z =36 constant values of 9. These
we Fig.
and T =0.52 MeV (path b). The solid contours have contours are spaced 4 MeV-’ apart. The filled circle
denotes the FTHFBC or most probable shape. The dashed lines are contours of the relative shape probability distribution P(/3, y). These are shown for P = 0.1 and 0.5. The FTHFBC state has P = 1.0 and 9 = 94 MeV-‘. Consider the shapes which are populated with over 50% relative probability, i.e, P> 0.5. These shapes have moments
of inertia
which extend
over the broad
interval
81-113 MeV-‘.
Therefore
the thermal shape fluctuations produce wide variations in 9. Compare the corresponding contour map for I = 44 and T = 0.68 MeV (path b), which is given in fig. 19. Whereas the contours in 9 are closely spaced in fig. 18, the contours in fig. 19 are far apart. Consequently the shapes with relative probability P > 0.5 have moments of inertia in the narrow interval 78-84 MeV’. So the thermal shape fluctuations produce very little variation in 4. A similar comparison is given in figs. 20 and 21 for the effect of thermal shape fluctuations on the rotational frequency. The solid lines are contours in w, which have intervals of 0.02 MeV. The dashed lines show P = 0.1 and 0.5. For I = 36 and T = 0.52 MeV, the shapes with relative probability P > 0.5 have values of w which span the broad interval 0.32 - 0.45 MeV. So there are large thermal fluctuations in
A.L. Goodman / Hot rotating ““Yb nuclei
T=0.!52MeV Moment
of
Inertia
431
’ w 0.3 -60’
Fig. 18. Contour map in the /3, y plane for ‘5XYb The solid contours have constant values of the moment of inertia 9. The dashed contours have constant values of the shape probability distribution P. The spin is 36 and the temperature is 0.52 MeV.
w. In contrast, for I = 44 and T = 0.68 MeV, the shapes with P > 0.5 have values of o confined to the narrow interval 0.53-0.57 MeV. Therefore the thermal fluctuations in w are small. We have seen that as we move along path b from the state at I = 36, T = 0.52 MeV to the state at I = 44, T = 0.68 MeV, the standard deviations A4 and Aw are drastically reduced. Is this effect caused by the increase in temperature or by the
Y O0 I I 0.1
/
I=44 T=0.68MeV Moment Inertia Fig. 19. See fig. 18. The moment
of -60’
of inertia and shape probability
distribution
for I =44and
T=0.68
MeV.
A.L. Goodman / Hot rotating “‘Yb
432
nuclei
I=36 T=0.52MeV WY) 4PA
jO”
Fig. 20. Contour map in the p, y plane for ‘ssYb. The solid contours have constant values of the rotational frequency w. The dashed contours have constant values of the shape probability distribution I? The spin is 36 and the temperature
is 0.52 MeV.
158
I=44 T=0.68MeV
PWY) 481r) Fig.
21. See fig. 20. The
rotational
-60'
frequency
and
shape
T = 0.68 MeV.
probability
distribution
for
I =44
and
A.L. Goodman
-0.0
0.2
0.4
/ Hot rotating
0.6
0.8
“‘Yb
nuclei
1.0
1.2
433
1.4
1.6
T (MeV) Fig. 22. The standard increase
in spin?
deviation
A4 versus the temperature
T for I = 40 and 50.
Fig. 22 shows A4 versus the temperature
for I = 40 and 50. The
solid curves use dT1, and the dashed curves use dT2. If I is fixed at 40, raising the temperature causes A9 to peak at T - 0.45 MeV. At higher temperatures, there is a sharp decline in A9, and A9 is nearly constant for T = 0.8-1.4 MeV. For Z = 50, the peak in A9 is less prominent. Next compare the I = 40 curves to the I = 50 curves. At all temperatures, A9 is larger for I = 40 than for I = 50. The conclusion is that increasing the temperature and increasing the spin both contribute to the decrease in the standard deviation 44, as we move up energy path b. A similar comparison for the standard deviation Aw is shown in fig. 23. If the spin is fixed at 40 or 50, then increasing the temperature produces a maximum in Au at T - 0.45 MeV, after which Aw decreases. Also, Aw is larger for I = 40 than for I = 50, if T-C 1.2 MeV. We conclude increasing
the spin both contribute
0.0
0.2
0.4
0.6
0.8
T Fig. 23. The standard
deviation
that increasing
to the decline
the temperature
in Aw on energy
path b.
1.0
1.6
1.2
1.4
(MeV)
Aw versus the temperature
T for I =40 and 50.
and
A.L.
434
Motional Normally temperature.
narrowing
should affect the damping
one expects the damping However
as the temperature based,
Goodman / Hot rotating
motional
increases, slightly
of giant resonances
width for a collective narrowing
if T is above
in part, upon the expectation
(for fixed I) increases
‘58Yb nuclei
causes
resonance
the damping
some moderate
that the dispersion
with temperature
in hot nuclei 21). to increase width
with
to decrease
value. The argument
in the rotational
2’)
Aw cc T”’ . We have shown that for moderate temperatures, a Aw which actually decreases with temperature. narrowing, excitation alignments occupation
is
frequency
(47) thermal shape fluctuations produce This should enhance the motional
and cause the rotational damping width to decrease more rapidly as the energy increases. Although our calculations implicitly permit rotation to affect Aw, we have not calculated the thermal fluctuations probabilities of aligned nucleon orbitals at fixed deformation.
in the These
fluctuations produce a dispersion in w [ref. I’)]. which might restore eq. (47). Finally, we return to the question raised near the end of sect. 2. For spins above 40, the FTHFBC state for “*Yb is oblate noncollective rotation for all temperatures, with a B( E2) value of zero. However the ANL experiment on the isotone ‘54Dy was interpreted with B( E2) = 300 W.U. for moderate temperatures and spins above 40. This apparent contradiction is partially resolved by calculating the effect of thermal shape fluctuations on the B(E2) values 22,23). In eqs. (36) and (39), let 0 = B(E2), where B( E2) is given in eq. (32). Fig. 14 shows the thermally averaged B( E2) value, as well as the standard deviation AB(E2), for energy path b. The dashed lines use metric 1, and the dotted lines use metric 2. (The two dashed curves cross one another, and so do the two dotted curves.) Although the FTHFBC state is oblate noncollective rotation ( y = -60”) relative probabilities
for I > 40, fig. 19 shows that thermal fluctuations produce high for populating prolate collective rotation (y = 0’). These fluctu-
ations create a B(E2) value. For example, at I = 50 and T = 0.8 MeV, fig. 14 shows that B( E2) = 41 W.U. for metric 1 and B( E2) = 88 W.U. for metric 2. Further increases in the B(E2) value could be achieved by including higher lying shells in the calculation. For spins above 40, the standard deviation in the B(E2) value is comparable to the average B(E2) value. The effect of thermal fluctuations on the static quadrupole fig. 15.
moment
is shown
in
4. Conclusions Various paths in the energy-spin plane of ‘%I have been studied. At high spins and moderate temperatures, mean field calculations predict quasirotational behaviour, where E, 0~ I and 4 = 4lrigid, even though the underlying structure is noncollective single particle rotation around a symmetry axis. At low spins and temperatures, there is a transition to approximately quasivibrational structure.
A.L. Goodman
When thermal to quasivibrational
shape fluctuations structure
are close to the mean-field The standard
deviations
/ Hot rotating
are included,
remains. values,
“‘Yb
nuclei
435
the transition
The thermally
from quasirotational
averaged
except at the transition
in 9 and w show a marked
values
of 9 and w
spins. decline
at moderate
tem-
peratures. For high spins and moderate temperatures, the fractional fluctuation in E, is only 2%, so that thermal shape fluctuations cause very little dispersion in E2 gamma
energies.
The author
is grateful
to T.L. Khoo for stimulating
conversations.
References 1) R. Holzmann, T.L. Khoo, W.C. Ma, I. Ahmad, B.K. Dichter, H. Emling, R.V.F. Jam&ens, M.W. Drigert, U. Garg, M.A. Quader, P.J. Daly, M. Piiparinen and W. Trzaska, Phys. Rev. Lett. 62 (1989) 520 2) C. Baktash, Y. Schutz, I.Y. Lee, F.K. McGowan, N.R. Johnson, M.L. Halbert, D.C. Hensley, M.P. Fewell, L. Courtney, A.J. Larabee, L.L. Reidinger, A.W. Sunyar, E. der Mateosian, O.C. Kistner and D.G. Sarantites, Phys. Rev. Lett. 54 (1985) 978 3) 1. Ragnarsson, T. Bengtsson, W. Nazarewicz, J. Dudek and G.A. Leander, Phys. Rev. Lett. 54 (1985) 982 4) C. Baktash, in Proc. Workshop on nuclear structure at moderate and high spin, Berkeley, 1986, ed. M.A. Deleplanque, R.M. Diamond and F.S. Stephens (Lawrence Berkeley Laboratory, Berkeley, 1986) p. 74 5) C. Baktash, in The variety of nuclear shapes, Proc. Int. Conf. on nuclear shapes, Aghia Pelaghai, Crete, 1987, ed. J. Garrett, C. Kalfas, G. Anagnostatos, E. Kossionides and R. Vlastou (World Scientific, Singapore, 1988) p. 537 6) A.L. Goodman, Nucl. Phys. A352 (1981) 30 7) K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys. A357 (1981) 20 8) M. Sano and M. Wakai, Prog. Theor. Phys. 48 (1972) 160 9) L.C. Moretto, Phys. Lett. B40 (1972) 1 10) A.L. Goodman, Phys. Rev. C29 (1984) 1887 11) J.L. Egido, C. Dorso, J.O. Rasmussen and P. Ring, Phys. Lett. B178 (1986) 139 12) A.L. Goodman, Phys. Rev. C37 (1988) 2162 13) A.L. Goodman, Phys. Rev. C39 (1989) 2478 14) M. Baranger and K. Kumar, Nucl. Phys. All0 (1968) 490 15) K.Kumar and M. Baranger, Nucl. Phys. All0 (1968) 529 16) A. Bohr, in Elementary modes of excitation in nuclei, Proc. Int. School of Physics, Enrico Fermi, ed. A. Bohr and R.A. Broglia (North-Holland, Amsterdam, 1977) p. 3 17) S. Vydrug-Vlasenko, R.A. Broglia, T. Dossing and W.E. Ormand, in Proc. Conf. on high-spin nuclear structure and novel nuclear shapes, Argonne, 1988, ed. I. Ahmad, R. Chasman, R. Janssens and T.L. Khoo (Argonne National Laboratory, Argonne, 1988) p. 156 18) M. Gallardo, M. Diebel, T. Dossing and R.A. Broglia, Nucl. Phys. A443 (1985) 415 19) S. Levit and Y. Alhassid, Nucl. Phys. A413 (1984) 439 20) B. Lauritzen, T. Dossing and R.A. Broglia, Nucl. Phys. A457 (1986) 61 21) R.A. Broglia, T. Dossing, B. Lauritzen and B.R. Mottelson, Phys. Rev. Lett. 58 (1987) 326 22) A.L. Goodman, Phys. Rev. C38 (1988) 1092 23) A.L. Goodman, Phys. Rev. C39 (1989) 2008
MOMENT OF INERTIA AND ROTATIONAL FREQUENCY IN HOT ROTATING ‘*Yb NUCLEI Alan L. GOODMAN’
Received 4 July 1989 Abstracti . The nucleus rs%% has a transition from quasivibrat~onal structure at low spins and temperatures to quasirotational structure at high spins and moderate temperatures. Thermal shape fluctuations produce fluctuations in 9 and w. However, these fluctuations do not eliminate the structural transition. The standard deviations in 9 and w decrease at moderate temperatures. Consequentfy, thermal shape fluctuations produce minimal dispersion in E2 gamma energies, for moderate temperatures and high spins.
1. Transition from ~uasiyibrationa~ to quasirotational structures The observable attributes of a nuclear state include its energy E and angular momentum I, However, to study the collective properties of a band of states, it is often more usefut to consider the moment of inertia 9 and the rotational frequency w. In the cranking model, these are defined by
(1) .%==J/w,
(2)
where J = [r(I + I)]“‘.
(3)
Although $ and o are not observable properties of nuclear states, they have been invaluable for understanding the structure of the yrast line and low-lying colfective bands. It is common to plot nuclear spectra in the form .9(w’), In hot nuclei, the level density is so high that individual bands cannot be resolved experimentally. We have relatively little experience in analysing the behavior of the moment of inertia and rotational frequency in hot nuclei. Nevertheless, a study of 9 and o can reveal structural transitions in hot nuclei, even though individual bands are not resolved. For an E2 transition, the value of Ey determines o through eq. (1)
itI=[t(r+
Et - Et-z l)]‘/’ -l(r-2)(1-I)1’“1StE,.
’ Supported in part by the National Science Foundation. ~3~5-9474/S9~$03.5~ @ Eisevier Science Publishers B.V. ~No~h-Holland Physics Pub~jshing Division)
A.L. Goodman / Hoi rotating “‘Yb
414
We first consider independent.
cold
nuclei.
Assume
From eq. (4), it follows
eq. (2), 3 increases Next consider
linearly
nuclei
a vibrational
spectrum.
that o is also independent
Then
E, is spin
of the spin. From
with J.
a rotational
spectrum,
where
E = J2/2$, and $ is constant.
From
(5)
either eq. (1) or eq. (2), it follows
to J. Then eq. (4) shows that E, is proportional As a variation, suppose I > I,,, the energy is
that o is proportional
to J.
that the state at I, is a rotational
band
head.
Then
E =R2/24,,,+E(I,)=(J-J,)2/24,,,+E(I,), where 9,,, is constant.
From eq. (l), it follows
(6)
that
J=.Yjrotw+JO.
(7)
The function J(w), or J($E,), has a slope equal Jo. The E2 transition energy is
to 4,,, and a vertical
E, = 2w = 2( J - Jo)/,a,,, which
is proportional
to J-J”.
Eq. (2) defines
moment
of inertia
is defined 9W_=$
From eqs. (7), (9) and (lo),
it follows
intercept
of
(8) moment
of inertia (9)
by
dJ aw
,
the kinematical
@“=4=J/w.
The dynamical
for
rot ’
(10)
that
s’2’/.9 ‘“=l-Jo/J.
(11)
Consider an ideal vibrator and an ideal rotor. The function ,a(,‘) is shown in fig. la. If Jo> 0, then the rotational 9 decreases with spin, in accordance with eqs. (10) and (11). Fig. lb gives the function J(w) or J(fE,). Now consider hot rotating nuclei. Suppose that for some particular path in the energy-spin plane, the functions .Y(w’) and J(w) resemble fig. 1. Then we could say that this path undergoes a transition from quasivibrational to quasirotational structure. For the quasivibrational states, E, is independent of J. For the quasirotational states, E, is proportional to J, or J -Jo.
A.L. Goodman / Hot rotating “‘Yb
nuclei
415
Rotational 3
J. -0
5 5 E
E -z e .e >
iti
w2 Fig. 1. Schematic
A recent Argonne tinuum
spectrum
National
provides
transition
from vibrational
Laboratory
evidence
(ANL)
CAJ
(Ey/2)
to rotational
spectrum.
measurement
for this structural
transition
However, the authors of this experiment raise some interesting peak which exhibits collective rotational behaviour is produced
of the E2 quasiconin hot ‘54Dy nuclei
‘).
questions. The E2 by a cascade which
flows through noncollective oblate states. What causes the rotational behaviour, where E, is proportional to the spin? The nucleus ‘s8Yb, which is an N = 88 isotone of ‘54Dy, has been studied at Oak Ridge National
Laboratory
(ORNL).
It was determined
that the positive-parity
yrust
states with spins 22-36 are quasivibrational ‘,‘). Although the yrast line is noncollective for spins above 40, increasing the temperature promotes collective structures 4,5). In this article we study ‘5xYb. We calculate 4 and w, as well as other properties, for various
paths in the energy-spin
Fock-Bogoliubov paths.
cranking
For each path,
plane.
(FTHFBC)
we find a transition
In sect. 2 the finite-temperature equation
hmX)is solved
from quasivibrational
for these structure
Hartreedifferent at low
spins and temperatures to quasirotational structure at high spins and moderate temperatures. In sect. 3 we show how thermal shape fluctuations produce fluctuations in the moment of inertia and the rotational frequency. Even though the FTHFBC theory predicts a structural transition, it is well known that thermal fluctuations can eliminate these transitions ‘-” ). So it is interesting to see whether fluctuations in 4 and w wash out the transition from quasivibrational to quasirotational structure. Normally one expects that thermal fluctuations will increase as the temperature rises. For example, if Aw is the standard deviation in rotational frequency for given spin, one anticipates that Aw would increase with temperature. However we find that at moderate temperatures, Aw and A4 exhibit a sudden decrease, rather than an increase. This may have implications for the description of motional narrowing in the damping of giant resonances in hot nuclei. A preliminary account of this work is given in ref. “).
A.t.
416
Goodman / HOI rotating rssYb rrdei
2. Finite-temperature The FTHFBC
equation
HFB cranking calculations
is
__l*)( z)=Ei( T)(__T* The Hartree-Fock
(HF)
hamilton~a~
(12)
is
X=e-&&u,N,-@,N*-&IX+~S where the sphericat and p,.
singe-parti~ie
energy
The HF and pair potentials rij
(131
is e, and the chemical
potentials
x
$
( 2%
/ ?JIjbfk
(141
9
Ai,-$C(ij[uIkZ)tk,. kl The HF and pair densities
V’(l-f)V],j,
t~j=(CjCi>={rifV*+ o~~npation
where T is the temperature and free energy are
(15)
are
pij={c;ci)=[fifu*+
The quasiparticie
probability
V’(l-f)tr],. is
and Ej is the quasiparticle
energy.
E =(H)=Tr[(e+fl‘)p+~At’], S=
are (-tp
are
The energy,
entropy,
(19)
-C[~ln~+(l-.J;)ln(l-J;)l,
F=E-TS,
(20) (21)
where H is the pa~r~ng_~~us~quadruQo~e (PPQ) ham~~tonian of Kumar and frequency w is adjusted to satisfy the spin constraint Baranger ‘47is). The rotational .T=(.JJ=[r(l+l>]‘/‘_
(221
The FTHFBC phase diagram for “‘Yb is given by the solid line in fig. 2. For low spins and temperatures, there is prolate (or slightly triaxial) collective rotation. Otherwise there is oblate noncollective rotation. Fig. 3 transforms the phase diagram to the energy-spin plane. Fig. 2 also contains several straight-line trajectories. We first study path b, which is defined by T = (0.02 MeV) (I - 10)
{Path b) .
(231
The FTHFBC equation is solved for every even spin from 10 to 60 and for the spin-dependent temperature of eq. (23). The resulting energy trajectory is path b in fig. 3. This path is very similar to the path observed in the ANL experiment on t54Dy with Ebeam = 15.5 MeV.
A.L. Goodman / Hot rotating ‘saYb nuclei
2.0
,
‘.
I ‘,
”
I “,
I “.
8.
”
417
* I
“, J
9
9
1.6-
OBIATE SPHEROID NON-COLLECTIVE ROTATION
ls8
Yb
_ /
//’ 1.2
-
0.8
-PROLATE
AND
COLLECTIVE
0
TRIAXIAL
ROTATION,,
10
20
’
30
40
50
60
SPIN for ‘58Yb. The critical
Fig. 2. Phase diagram
The function
9(w’)
temperature
r, versus the spin I.
for path b is given by the solid curve (filled circles)
This curve has two distinct
in fig. 4.
sections.
For spins 18-36, 9 increases very rapidly, and there is relatively little variation in w2. However for spins greater than 40,9 decreases somewhat, and becomes nearly constant for spins above 50. This curve bears some resemblance to fig. la. The function Z(w) for path b is given by the solid curve in fig. 5. This curve also has two distinct components. The slope of Z(w) is much larger for spins 18-36 than it is for spins above 40. This should be compared to the schematic transition in fig. lb. Schematic rotational structures were characterized in eq. (7), where J(w) is a straight
line. For spins 48-60 the HFB function
In accordance
with eqs. (7) and (lo),
40 t -
0
J(o)
is very close to a straight
the slope is $(*I = L+,,~-61
OBLATE ROTATION
10
20
30
40
50
SPIN Fig. 3. The energy
E versus the spin I for “sYb.
60
MeV-‘,
line.
and the
418
A.L. Goodman 1
Hot mating 15X Yb nuclei
vertical intercept is & = 10, For this spin-interval, eq. (8) shows that E, F=(0.033 MeVf (J - IO). Our anaIysis
of ..P(w’)and i(w) enables us to charac~~r~~~path b as a~~r~x~rnat~~y ~uasirot,a~~o~a~ for spins above 40, and ~p~roxirnat~~y quas~vibra~~o~~~ for spins below 36, This agrees with the ANL experiment on ““Dy and the QRNL experiment on rsxyb_ Why do the moment of inertia and rotational frequency experience a change in structure on path b at spins -38-40:” We first consider the reasons for the rapid rise in $(w’) for spins up to 36. Fig. 6 shows how the pair gaps d,, and A, vary along path b. The proton pair gap exhibits a gradual decline for spins 18-36. How much of the change in the moment of inertia is caused by this decline in A,?
/ Hot rotating
A. L. Goodman
“‘Yb
a
1.6
nuclei
419
u “‘Yb
1.2
-
0.6
-
SJ 2
a 0.4
h
n T=O.O2MeV
i
0.0
1 * ’
.
10
0 20
(I-
10) ’ 30
.
’
”
.
1 40
SPIN Fig. 6. The proton
For example,
pair gap A, and the neutron pair gap A, versus the spin I for path b.
suppose
that
the pairing
is omitted
from the calculations.
The
FTHFBC equation is solved again for path b with A = 0. The function 9(w’) is given by the A = 0 curve in fig. 7. This should be compared with the HFB curve in fig. 7. For the spin interval 20-36, the increase in 4 is 260% larger for the HFB curve than for the A = 0 curve. For this spin interval, the increase in w2 is 30% smaller for the HFB curve than for the A = 0 curve. Both of these factors make the HFB ,a(~‘) function much steeper (and 9(w’) function calculated with A = 0. Next we keep A fixed at the ground
hence
more
state value,
quasivibrational)
instead
of letting
than
the
A be spin and
temperature dependent. The 9(w’) function is given by the A,,, curve in fig. 7. Once again, this should be compared with the HFB curve in fig. 7. Keeping A fixed at
70 60 50
If?20
36
20
10
40
/
.-. .-. 0-o
HFB A 9.s.
A=0
T=O.O2MeV
(l-l
0)
_
10
30 0.0
0.2
0.4
O2 Fig. 7. The moment
0.6
0.6
1.0
(UeV2)
of inertia 4 versus the square of the rotational frequency w for path b. Calculations displayed use the FTHFBC A, the ground state A, and A =O.
A. L. Goodman
420
the ground-state
value
I = 36. Furthermore, rotation
to oblate
spin interval
causes
the moment
the transition noncollective
20-36,
/ Hoi rotating
of inertia
from
rotation
the increase
tssYb nuclei
prolate occurs
slightly
triaxial)
at I = 30, instead
in 4 is 230%
the A,.,. curve. The increase in 6.1~is 58% A,.,, curve. From these comparisons using fig. 7, reasons for the quasivibrational structure decline in the proton pair gap in this spin
to peak at I = 30, instead
(and larger
smaller
of
collective
of I = 36. For the
for the HFB curve than for
for the HFB curve than for the
we conclude that one of the significant of path b for spins 18-36 is the gradual interval. Either setting A = 0 or keeping
A fixed at the ground-state value significantly of path b for spins 18-36.
weakens
the quasivibrational
structure
The rapid increase in 4 for spins 18-36 is also caused, in part, by the shape transition and associated rotation alignments. The quadrupole deformation parameters /3 and y for path b are given by the solid curves in figs. 8 and 9. At spins 36-40 there is a rapid transition from nearly prolate collective rotation to oblate noncollective rotation (y = -6O”), and there is a decrease in /3. For I = 30 on path b, there are four orbitals which are rotation be the spin projection along the x rotation JcQ, are essentially iIjI2 neutrons
aligned.
axis. Then two of these orbitals,
=?)=0.97,
(&,2mX
(~uz~i,z,2m.X= y} = 0.93 , and two of the orbitals,
IcY~)and IQ>, are primarily
hI,,,z
protons
(axlh , ,/zmX = $) = 0.99 , (a,lh ,,/,m,=~)=0.96. 0.24
. . .
I
,
I
.
.
.
I
.
.
.
,
.
.
'5aYb
-
,,.._...... .
.
.
.
.
.
.
.
.
.
.
.
.
.._
,_,.._...~~~
T=O.O2MeV
0.00 -<‘. 10
’ . ’ . . ’ . * ’ * ’ . 20
30
40
(l-l
0)
.
’ ’ ’ ’ ’ ’ 50
60
SPIN Fig. 8. The quadrupole
deformation
parameter
p versus the spin I for path b.
Let m, Ia,) and
A.L. Goodman
40 I
/ Hot rotating
rs8Yb nuclei
421
Yb
15’
10
20
40
30
50
60
SPIN Fig. 9. The quadrupole
deformation
Since path b has finite temperature,
parameter
these orbitals
y versus the spin I for path b.
are not fully occupied.
The particle
occupation probabilities are p, = 0.94, p2 = 0.61, p3 =0.97 and p4= 0.96. As the nucleus makes the transition from collective to noncollective rotation, more orbitals become rotation aligned. At Z = 40 all nucleon spins are rotation aligned. Next we study the high spin (I > 40) section of path b. The functions ,a(~‘) and Z(w) indicate a structure which is approximately collective rotational. This is in spite of the fact that the phase diagram (fig. 2) shows noncollective rotation for all temperatures when the spin exceeds 40. Some insight into this apparent contradiction may be obtained by first considering the yrast line produced by noncollective rotation. (Assume that the spin Z is sufficiently large that the pairing gap has vanished.) In noncollective rotation the x rotation axis coincides with the symmetry axis of the oblate shape. Then each nucleon orbital has rn, as a good quantum number. Therefore the HF hamiltonian X and J, commute, so that the eigenvalues of 2, i.e., the single-nucleon
energies
in the rotating
frame,
are
E&(W) = E,(W) -Wn,, where m denotes spin is simply
m,, and E,(W) is the eigenvalue
(24) of e -Z_LN +Z(w).
The nuclear
where the lowest levels E& are occupied at each o. For different w, a different set of nucleon levels will be occupied, which causes different self-consistent shapes. This creates the o dependence of the HF potential Z’. However to simplify the argument, suppose we ignore the shape self-consistency requirement. Assume that the deformation is fixed at the ground-state value. Then 8, and Z become independent of w, and they have ground-state values. Consequently E&(W) is a straight line
A.L. Goodman / Hot rotating r5hYb nuclei
422
for all 0 F:,(W) =
E,
-mm,,
(26)
ma.
(27)
where the slope of the line is dE&
dw If an increase
in w does not produce
a nucleon
level crossing
at the Fermi energy,
then the same orbitals remain occupied, and the spin J does not change. However if increasing w does cause a level crossing at the Fermi energy, then a nucleon leaves
one of the crossing
orbitals
and occupies
the other
orbital,
will increase. Consequently J(w) is a stepwise function. Let the temperature This argument is now applied to ‘%J.
and the spin J
and pair gap equal
zero. Assume an oblate shape rotating around its symmetry axis. Assume deformation is spin independent, with p = 0.1. (For path b at I =40, p Then E, are the eigenvalues of e-PN +r(/3 =O.l, y = -60”). (Note that -60”) = r(p, y = ISO’).) The function J(w) is the solid stepwise path in fig. the nucleon level crossings are statistical, therefore the jumps For a collectively rotating rigid object, the spin is
that the =0.104.) r(/3, y= 10. Since
in J(w) are statistical.
J = 9rigidm 9
(28)
so that J(w) is a straight line with a slope equal to -arigid. For an oblate shape with p = 0.1 which is rotating around its symmetry axis, 4irigid= 68.2 MeV-‘. The rigid body J(w) is shown in fig. 10. The essential point is that for spins 48-60, the statistical stepwise
J(w)
to the slope
function
can be characterized by an average slope, which is similar single line, i.e., 4irigid. Although the noncollective
of the rigid body
70 60 50
(J,>
40 30 20 10 0
-0.0
0.4
0.2
w Fig. 10. The spin versus the rotational
frequency
0.6
0.8
1.0
(MeV)
for an oblate
shape
rotating
around
its symmetry
axis.
A.L. Goodman
particle
rotation
is essentially
duces an average moment
moment
/ Hot rotating
different
of inertia
“‘Yb
nuclei
from collective
423
rigid-body
for I = 48-60 which is similar
rotation,
it pro-
to the rigid-body
of inertia.
This conclusion
was first reached
by Bohr using the Fermi gas model 16). However
it is interesting that the argument seems to work for ‘?b only at the highest spins, 48-60. If the stepwise J(w) function is approximated by a straight line for the spin interval O-44, then the slope is 2163 MeV’, which is larger than $rigid by a factor of 2.4. These calculations
use one major oscillator
shell for each parity.
How would the
inclusion of higher shells affect the argument. 7 The first additional orbits which would contribute to the yrast line at high spins are the i,X,z proton and jlSjZ neutron levels. When these levels are included in the calculations, then the stepwise J(w) function is altered for I = 50-60. This change is given by the dashed stepwise line in fig. 10. Observe that inclusion of higher shells does not alter our basic conclusion. This argument regarding the moment of inertia for noncollective rotation has been given for an yrast line, which has zero temperature. At finite temperature, the nucleon level crossings no longer cause discontinuous probabilities. This is apparent from the Fermi-Dirac sequently provides
the stepwise function additional justification
jumps in orbital occupation distribution of eq. (18). Con-
J(o) becomes smooth at finite temperature. This for characterizing J(w) by an average moment of
inertia. It is instructive to consider the high temperature behaviour of 9(w’). Consider a constant temperature of 2.0 MeV. The FTHFBC equation is solved for various spins. The HFB function 9(w’) is given by the solid curve in fig. 11. There is very little variation in 4 for spins from 2 to 60. Fu,rthermore, this small variation occurs within the boundaries created by a rigid sphere and a rigid oblate shape (p = 0.1) rotating
around
its symmetry
axis.
2
l-
“0 2 2
Oblote Rigid Body ____-__------E~IO_JQ___~_____~____ 60-
Spherical
(@=O.l) -------TO---
--
Rigid Body
50-
HFB
P
T=Z.OMeV
.
40 0.0
0.2
0.4
W2 Fig. 11. The moment
of inertia
9 versus the square
0.6
0.8
1 .o
(MeV2) of the rotational
frequency
w for T= 2.0 MeV.
A.L. Goodman / Hot rotating “‘Yb
424
Our conclusion
is that oblate
rotation produces an yrast line and L~~ipid for spins 48-60. Therefore it is limit which both give 9,
a high temperature not surprising
nuclei
noncollective
that the finite temperature
This energy path is quasirotational
path b in fig. 3 also shares
at high spins, even though
this property.
the underlying
motion
is noncollective. Our discussion has been restricted to path b of figs. 2 and 3. The obvious question is whether other energy trajectories also undergo a transition from quasivibrational to quasirotational structure. We now consider several other energy paths. Fig. 2 shows three additional
paths
a, c and d, which are defined
T=(O.Ol
MeV) (Z-10))
by
(path a)
(29)
T = (0.03 MeV) (I - 10) ,
(path c)
(30)
T = (0.02 MeV) (I - 20) .
(path 4
(31)
The FTHFBC equation is solved for various energy trajectories are shown in fig. 3.
states on these paths.
The FTHFBC
Fig. 12 compares the functions 9(w’) for the four energy trajectories. The four curves in fig. 12 are remarkably similar. They all have a transition from quasivibrational structure at low spins and temperatures to quasirotational structure at high spins and moderate temperatures. Except for Z = 40, the four energy paths have similar values of 4 and w2 at each displayed spin. Fig. 13 compares the functions Z(w) for the four energy paths. Once again the similarity is striking. From figs. 12 and 13 we conclude that the existence of the quasivibrational to quasirotational transition is remarkably insensitive to the selection
of a particular
Is there a discernible
path in the energy-spin pattern
the quasivibrational-quasirotational 100 i
>
governing
plane.
the place in the energy-spin
transition
occurs?
plane where
For a particular
energy path,
r
90
“‘Yb
-
-
$ ; & W z
g Iwz
8070
-
60-
0-0
T=O.OlMeV
(I-10)
--•
T=O.O2MeV
(I-10)
=-m
T=O.O3MeV
(l-l
T=O.O2MeV
(I-20)
l
50
-
o-
P
--D
0)
I
40 ’ 0.0
0.4
0.2 u2
Fig. 12. The moment
of inertia
9 versus the square
0.6
0.8
1.0
(WP) of the rotational
frequency
w for paths a, b, c and d.
A.L. Goodman
/ Hot rotating
rssYb nuclei
425
60 50 z E VI
40 30 20
--
T=O.Ol
-
T=O.O2MeV
(I-10)
T=O.O3MeV
(l-l
T=O.O2MeV
(I-20)
- -
10
MeV (l-l
0)
0)
0
0.0
0.6
0.4
0.2
w
0.8
1.0
(MeV)
Fig. 13. The spin I versus the rotational
frequency
o for paths a, b, c and d.
this transition is not sharp. It does not occur at a specific spin. Consequently we cannot give a precise answer to our question. However, to provide a rough guideline, we could mark the end of the quasivibrational structure by the spin at which 9 stops increasing. From fig. 12 and other plots which include more spin states, one finds that this terminal spin is 38 for path a, 36 for path b, and 30 for path c. These terminal states are marked by the filled circles in fig. 3. The dotted line which connects these terminal states serves as the approximate boundary which marks the end of the quasivibrational structure. Observe that this boundary differs from the phase transition line which marks the onset of oblate noncollective rotation. For path b, the quasivibrational terminal state occurs at E = 15.1 MeV and I = 36, whereas the transition to oblate noncollective rotation happens at E = 18.4 MeV and I = 40.6. (In the cranking model the spin is a continuous variable.) Fig. 3 shows that the quasivibrational terminal states of paths b and c have similar energies but different
spins.
This implies
that for energy
paths b and c, it is the energy, quasivibrational structure. Our calculations
and
trajectories
not the spin,
in the neighbourhood
which
marks
of
the end of the
on “‘Yb are now compared to the ANL experiment on the N = 88 isotone ls4Dy. This experiment measured the quasicontinuum spectrum of ls4Dy. For a beam energy of 155 MeV, the de-excitation path of 154Dy is very similar to our path b for ‘5*Yb. For example, at I =40 path b has an excitation energy E = 18.1 MeV and a thermal excitation energy E* = 6.5 MeV, whereas the ANL de-excitation path has E = 18.9 MeV and E’ = 5.6 MeV. For Ebeam= 155 MeV, the ANL path shows a transition at spin 38. At higher spins, the E2 gamma energies are proportional to the spin, which was interpreted as indicating collective rotational structure. For spins below 38, E, is approximately independent of the spin, which suggests quasivibrational structure. By varying Ebeam, the ANL experiment studied different de-excitation paths. For these paths it was found that the change in structure
A.L.
426
occurs
at a common
consistent Finally
Goodman
energy,
E = 17 MeV,
with our calculations we consider
/ Hot rotating
rsaYb nuclei
rather
than
a common
spin.
This
is
on “‘Yb.
the B(E2)
values
and static electric
quadrupole
moment
Q.
For a rotor with large spin, these are 1
B(E2,I+Zk2)=5
Z2e2R4/32~~~Z(-y-300),
3 =p ZeR2P sin ( y - 30”) , Q (5Tr)“* where R = roA”3 ,
(34)
and r0 = 1.2 fm. Fig. 14 shows the B(E2) value for energy path b. The solid curve is the result. The B(E2) value vanishes at I = 40.6, where there is a transition noncollective rotation (y = -60”). Although the FTHFBC theory predicts of inertia and rotational frequencies which are quasirotational at high moderate
temperatures,
the
B(E2)
value
is zero,
which
suggests
FTHFBC to oblate moments spins and
an absence
of
collective rotational structure. In the ANL experiment on ‘54Dy, the average electric quadrupole moment of the initial rotational cascade was a free parameter. The value selected corresponds to B(E2) = 300 W.U. which is typical of strongly collective rotation. This is inconsistent with our HFB result for “‘Yb, which has B(E2) = 0 for I > 40. A partial resolution of this apparent contradiction is given in the next section. The quadrupole moment Q is shown in fig. 15. The FTHFBC result is the solid curve. It shows little structure. This is because the sign of Q is the same for prolate collective
and oblate
noncollective
140
rotation.
. . ..I..............,..... T=O.OZMeV
10
20
(I-10)
30
40
50
SPIN Fig. 14. The B(E2)
value versus the spin I for path b
60
A.L. Goodman / Hot rotating “*Yb nuclei
/../
/ I
/
r
T=O.O2MeV
.a I . .
10
. I..
20
(I- 10)
,
30
I...,,...,
40
427
“‘Yb 50
. 60
SPIN Fig. 15. The static electric
3. Thermal fluctuations
quadrupole
moment
Q versus the spin I for path b.
in moment of inertia and rotational
frequency
Our discussion has been restricted to FTHFBC calculations, which do not include thermal shape fluctuations. However, it is well known that thermal shape fluctuations often eliminate structural transitions which are predicted by the FTHFBC theory. Consequently we must determine whether thermal shape fluctuations wash out the transitions in the moment of inertia and rotational frequency which were discussed in sect. 2. For fixed shape,
spin, and temperature,
the FTHFBC
eqs. (12)-(18)
are iterated
to find the value of the rotational frequency which satisfies the spin constraint, eq. (22). Repeated applications of this procedure determine the function w(/3, y; Z, T). Then eq. (2) determines the moment of inertia 9(p, y; Z, T). For each shape, the FTHFBC particle density may contain rotation aligned nucleons. These rotation alignments may have a large effect upon the value of w required
to produce
a given spin. Therefore
the functions
o and 4 implicitly
depend
upon the rotation alignments. Consequently the rotation alignments will affect the dispersions in 4 and w. However it should be noted that for fixed deformation, thermal fluctuations in the occupation probabilities of aligned nucleon orbitals explicitly cause the dispersion of the rotational frequency “). We have not calculated this effect. In these calculations, the pair gaps Ap and A, are taken from the FTHFBC equilibrium state for given spin and temperature. The free energy F(/?, y; Z, T) is calculated with eqs. (19)-(21). Thermal fluctuations produce different shapes. For a particular spin and temperature, the probability that a given quadrupole deformation occurs is p(p,
y; Z, T) K eeF(B.Y:L=)lT .
(35)
A.L. Goodman / Hot rotating r5xYb nuclei
428
For a function
0(p, y; I, T), the thermal
average
of 6 is
‘W, Y; 4 T)P(P, Y; 4 T) dT @I, T)=(B)=
Two metrics
have been used to calculate
All of our calculations is defined
I
(36)
P(P, Y; 4 T) d7 thermal
averages
“,19). These are
dr, =P d/3 dy,
(37)
dr2 = p41sin 3 y[ d/3 d y .
(38)
are performed
with both metrics.
The standard
deviation
A6
by AtI’(Z, T)=[(B2)-(B)2]“2.
(39)
Thermal fluctuations are studied for path b of figs. 2 and 3. The thermal averages 3, &I and 2 are calculated for I = 10,20, 30, 36, 40, 44, 50 and 60, where T is defined by eq. (23). The function 3(m2) is given in fig. 4. The dashed line (open circles) is calculated with dr, , and the dotted line (open squares) is calculated with dT2. Even though these curves include thermal shape fluctuations, two structures remain: approximately quasivibrational for spins up to 36, and approximately quasirotational for spins above 40. The thermal fluctuations do not wash out this structural transition. Except for I = 36 and 40, the thermally averaged values of 4 and w2 are quantitatively similar to the FTHFBC values. The function line uses dr2. transition
I(W) is shown in fig. 5. The dashed Once again the structural transition
spins,
W is remarkably
similar
line uses dr, and the dotted is preserved. Except for the
to mHFR. For rotational
structures,
the
function J(w) is a straight line, as in eq. (7). The function J(G) is nearly linear over the spin interval 44-60 for dr, and 40-60 for dr2. Observe that J(w) is close to a straight line for a spin interval which is longer for (5 than for wHFB. From eq. (8) it follows
that including
the thermal
fluctuations
produces
an average
& which
is proportional to J-J,, for a spin interval which is longer than that given by the HFB E,. For both metrics, Jo= 10. Past experience with thermal shape fluctuations often led to the conclusion that these fluctuations eliminate phase transitions which are predicted by mean field theories. Why does this not happen for the quasivibrational-quasirotational transition? To answer this question, we study the standard deviations in the moment of inertia and rotational frequency. The standard deviations A$ and Aw for path b are shown in figs. 16 and 17. The solid curves use dr, and the dashed curves use dr2. As expected, A.9 and Aw increase as the temperature (spin) rises from zero. However these standard deviations reach a maximum at T = 0.5 MeV (I = 35). For higher temperatures (spins) there is a precipitous decline in A.9 and do. These standard deviations are small, except for
A.L. Goodman
10
/ Hot rotating
30
20
429
rssYb nuclei
50
40
60
SPIN Fig. 16. The standard
intermediate
deviation
in the moment
spins near 36. It follows
of inertia
A$ versus the spin I for path b.
that the thermal
averages
3, 0 and 2
must
be close to the corresponding HFB values, except for spins near 36. This argument is given in a more quantitative form by considering the fractional fluctuations
in 4 and w. These are defined
by (40)
Cd”
6”
(41)
(5 .
If the fractional
fluctuations
.
0.05,
0.04
are small compared
r
to unity
S9<1,
(42)
6wel.
(43)
. I
.
I .
.
,
.
*,
-
T=O.O2MeV
10
20
(l-l
0)
30
40
50
60
SPIN Fig. 17. The standard
deviation
in the rotational
frequency
Aw versus the spin I for path b.
A.L. Goodman / Hot rotating “‘Yb
430
then the most probable values
3 and
(i.e. FTHFBC)
W. For example,
0.8 MeV. The fractional
much smaller
the state on path b with Z = 50 and
are 69 = 0.018 and 6w = 0.021 for metric
69 = 0.024 and &I = 0.026 for metric do not destroy
values of $ and w will be close to the average
consider
fluctuations
2. The reason
the quasivibrational-quasirotational than unity
How do thermal
shape
nuclei
that thermal transition
for states on path b, except fluctuations
1, and
shape fluctuations
is that 69 and &J are
for spins near 36.
affect E,? For an E2 transition,
eq. (4) gives
E,=2w, where
w is in MeV. The standard
deviation
(44) in E, is
AE, = 2Aw, and the fractional
fluctuation
T =
(45)
in E, is (46)
For the state on path b with I = 50 and T = 0.8 MeV, SE, = 0.021 for metric
1, and
6E, = 0.026 for metric 2. The thermal shape fluctuations produce only a 2% variation in E,. Consequently the y energies remain very sharp, even when thermal shape fluctuations are included. It should be noted that there are other mechanisms, such as rotational damping 2”), which produce additional dispersion in E,. Why do the standard deviations in the moment of inertia and rotational frequency have a dramatic decline for T > 0.52 MeV (I > 36)? To answer this question, consider how 3 and w vary with deformation for fixed spin and temperature. 18 shows ,a(& y) for Z =36 constant values of 9. These
we Fig.
and T =0.52 MeV (path b). The solid contours have contours are spaced 4 MeV-’ apart. The filled circle
denotes the FTHFBC or most probable shape. The dashed lines are contours of the relative shape probability distribution P(/3, y). These are shown for P = 0.1 and 0.5. The FTHFBC state has P = 1.0 and 9 = 94 MeV-‘. Consider the shapes which are populated with over 50% relative probability, i.e, P> 0.5. These shapes have moments
of inertia
which extend
over the broad
interval
81-113 MeV-‘.
Therefore
the thermal shape fluctuations produce wide variations in 9. Compare the corresponding contour map for I = 44 and T = 0.68 MeV (path b), which is given in fig. 19. Whereas the contours in 9 are closely spaced in fig. 18, the contours in fig. 19 are far apart. Consequently the shapes with relative probability P > 0.5 have moments of inertia in the narrow interval 78-84 MeV’. So the thermal shape fluctuations produce very little variation in 4. A similar comparison is given in figs. 20 and 21 for the effect of thermal shape fluctuations on the rotational frequency. The solid lines are contours in w, which have intervals of 0.02 MeV. The dashed lines show P = 0.1 and 0.5. For I = 36 and T = 0.52 MeV, the shapes with relative probability P > 0.5 have values of w which span the broad interval 0.32 - 0.45 MeV. So there are large thermal fluctuations in
A.L. Goodman / Hot rotating ““Yb nuclei
T=0.!52MeV Moment
of
Inertia
431
’ w 0.3 -60’
Fig. 18. Contour map in the /3, y plane for ‘5XYb The solid contours have constant values of the moment of inertia 9. The dashed contours have constant values of the shape probability distribution P. The spin is 36 and the temperature is 0.52 MeV.
w. In contrast, for I = 44 and T = 0.68 MeV, the shapes with P > 0.5 have values of o confined to the narrow interval 0.53-0.57 MeV. Therefore the thermal fluctuations in w are small. We have seen that as we move along path b from the state at I = 36, T = 0.52 MeV to the state at I = 44, T = 0.68 MeV, the standard deviations A4 and Aw are drastically reduced. Is this effect caused by the increase in temperature or by the
Y O0 I I 0.1
/
I=44 T=0.68MeV Moment Inertia Fig. 19. See fig. 18. The moment
of -60’
of inertia and shape probability
distribution
for I =44and
T=0.68
MeV.
A.L. Goodman / Hot rotating “‘Yb
432
nuclei
I=36 T=0.52MeV WY) 4PA
jO”
Fig. 20. Contour map in the p, y plane for ‘ssYb. The solid contours have constant values of the rotational frequency w. The dashed contours have constant values of the shape probability distribution I? The spin is 36 and the temperature
is 0.52 MeV.
158
I=44 T=0.68MeV
PWY) 481r) Fig.
21. See fig. 20. The
rotational
-60'
frequency
and
shape
T = 0.68 MeV.
probability
distribution
for
I =44
and
A.L. Goodman
-0.0
0.2
0.4
/ Hot rotating
0.6
0.8
“‘Yb
nuclei
1.0
1.2
433
1.4
1.6
T (MeV) Fig. 22. The standard increase
in spin?
deviation
A4 versus the temperature
T for I = 40 and 50.
Fig. 22 shows A4 versus the temperature
for I = 40 and 50. The
solid curves use dT1, and the dashed curves use dT2. If I is fixed at 40, raising the temperature causes A9 to peak at T - 0.45 MeV. At higher temperatures, there is a sharp decline in A9, and A9 is nearly constant for T = 0.8-1.4 MeV. For Z = 50, the peak in A9 is less prominent. Next compare the I = 40 curves to the I = 50 curves. At all temperatures, A9 is larger for I = 40 than for I = 50. The conclusion is that increasing the temperature and increasing the spin both contribute to the decrease in the standard deviation 44, as we move up energy path b. A similar comparison for the standard deviation Aw is shown in fig. 23. If the spin is fixed at 40 or 50, then increasing the temperature produces a maximum in Au at T - 0.45 MeV, after which Aw decreases. Also, Aw is larger for I = 40 than for I = 50, if T-C 1.2 MeV. We conclude increasing
the spin both contribute
0.0
0.2
0.4
0.6
0.8
T Fig. 23. The standard
deviation
that increasing
to the decline
the temperature
in Aw on energy
path b.
1.0
1.6
1.2
1.4
(MeV)
Aw versus the temperature
T for I =40 and 50.
and
A.L.
434
Motional Normally temperature.
narrowing
should affect the damping
one expects the damping However
as the temperature based,
Goodman / Hot rotating
motional
increases, slightly
of giant resonances
width for a collective narrowing
if T is above
in part, upon the expectation
(for fixed I) increases
‘58Yb nuclei
causes
resonance
the damping
some moderate
that the dispersion
with temperature
in hot nuclei 21). to increase width
with
to decrease
value. The argument
in the rotational
2’)
Aw cc T”’ . We have shown that for moderate temperatures, a Aw which actually decreases with temperature. narrowing, excitation alignments occupation
is
frequency
(47) thermal shape fluctuations produce This should enhance the motional
and cause the rotational damping width to decrease more rapidly as the energy increases. Although our calculations implicitly permit rotation to affect Aw, we have not calculated the thermal fluctuations probabilities of aligned nucleon orbitals at fixed deformation.
in the These
fluctuations produce a dispersion in w [ref. I’)]. which might restore eq. (47). Finally, we return to the question raised near the end of sect. 2. For spins above 40, the FTHFBC state for “*Yb is oblate noncollective rotation for all temperatures, with a B( E2) value of zero. However the ANL experiment on the isotone ‘54Dy was interpreted with B( E2) = 300 W.U. for moderate temperatures and spins above 40. This apparent contradiction is partially resolved by calculating the effect of thermal shape fluctuations on the B(E2) values 22,23). In eqs. (36) and (39), let 0 = B(E2), where B( E2) is given in eq. (32). Fig. 14 shows the thermally averaged B( E2) value, as well as the standard deviation AB(E2), for energy path b. The dashed lines use metric 1, and the dotted lines use metric 2. (The two dashed curves cross one another, and so do the two dotted curves.) Although the FTHFBC state is oblate noncollective rotation ( y = -60”) relative probabilities
for I > 40, fig. 19 shows that thermal fluctuations produce high for populating prolate collective rotation (y = 0’). These fluctu-
ations create a B(E2) value. For example, at I = 50 and T = 0.8 MeV, fig. 14 shows that B( E2) = 41 W.U. for metric 1 and B( E2) = 88 W.U. for metric 2. Further increases in the B(E2) value could be achieved by including higher lying shells in the calculation. For spins above 40, the standard deviation in the B(E2) value is comparable to the average B(E2) value. The effect of thermal fluctuations on the static quadrupole fig. 15.
moment
is shown
in
4. Conclusions Various paths in the energy-spin plane of ‘%I have been studied. At high spins and moderate temperatures, mean field calculations predict quasirotational behaviour, where E, 0~ I and 4 = 4lrigid, even though the underlying structure is noncollective single particle rotation around a symmetry axis. At low spins and temperatures, there is a transition to approximately quasivibrational structure.
A.L. Goodman
When thermal to quasivibrational
shape fluctuations structure
are close to the mean-field The standard
deviations
/ Hot rotating
are included,
remains. values,
“‘Yb
nuclei
435
the transition
The thermally
from quasirotational
averaged
except at the transition
in 9 and w show a marked
values
of 9 and w
spins. decline
at moderate
tem-
peratures. For high spins and moderate temperatures, the fractional fluctuation in E, is only 2%, so that thermal shape fluctuations cause very little dispersion in E2 gamma
energies.
The author
is grateful
to T.L. Khoo for stimulating
conversations.
References 1) R. Holzmann, T.L. Khoo, W.C. Ma, I. Ahmad, B.K. Dichter, H. Emling, R.V.F. Jam&ens, M.W. Drigert, U. Garg, M.A. Quader, P.J. Daly, M. Piiparinen and W. Trzaska, Phys. Rev. Lett. 62 (1989) 520 2) C. Baktash, Y. Schutz, I.Y. Lee, F.K. McGowan, N.R. Johnson, M.L. Halbert, D.C. Hensley, M.P. Fewell, L. Courtney, A.J. Larabee, L.L. Reidinger, A.W. Sunyar, E. der Mateosian, O.C. Kistner and D.G. Sarantites, Phys. Rev. Lett. 54 (1985) 978 3) 1. Ragnarsson, T. Bengtsson, W. Nazarewicz, J. Dudek and G.A. Leander, Phys. Rev. Lett. 54 (1985) 982 4) C. Baktash, in Proc. Workshop on nuclear structure at moderate and high spin, Berkeley, 1986, ed. M.A. Deleplanque, R.M. Diamond and F.S. Stephens (Lawrence Berkeley Laboratory, Berkeley, 1986) p. 74 5) C. Baktash, in The variety of nuclear shapes, Proc. Int. Conf. on nuclear shapes, Aghia Pelaghai, Crete, 1987, ed. J. Garrett, C. Kalfas, G. Anagnostatos, E. Kossionides and R. Vlastou (World Scientific, Singapore, 1988) p. 537 6) A.L. Goodman, Nucl. Phys. A352 (1981) 30 7) K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys. A357 (1981) 20 8) M. Sano and M. Wakai, Prog. Theor. Phys. 48 (1972) 160 9) L.C. Moretto, Phys. Lett. B40 (1972) 1 10) A.L. Goodman, Phys. Rev. C29 (1984) 1887 11) J.L. Egido, C. Dorso, J.O. Rasmussen and P. Ring, Phys. Lett. B178 (1986) 139 12) A.L. Goodman, Phys. Rev. C37 (1988) 2162 13) A.L. Goodman, Phys. Rev. C39 (1989) 2478 14) M. Baranger and K. Kumar, Nucl. Phys. All0 (1968) 490 15) K.Kumar and M. Baranger, Nucl. Phys. All0 (1968) 529 16) A. Bohr, in Elementary modes of excitation in nuclei, Proc. Int. School of Physics, Enrico Fermi, ed. A. Bohr and R.A. Broglia (North-Holland, Amsterdam, 1977) p. 3 17) S. Vydrug-Vlasenko, R.A. Broglia, T. Dossing and W.E. Ormand, in Proc. Conf. on high-spin nuclear structure and novel nuclear shapes, Argonne, 1988, ed. I. Ahmad, R. Chasman, R. Janssens and T.L. Khoo (Argonne National Laboratory, Argonne, 1988) p. 156 18) M. Gallardo, M. Diebel, T. Dossing and R.A. Broglia, Nucl. Phys. A443 (1985) 415 19) S. Levit and Y. Alhassid, Nucl. Phys. A413 (1984) 439 20) B. Lauritzen, T. Dossing and R.A. Broglia, Nucl. Phys. A457 (1986) 61 21) R.A. Broglia, T. Dossing, B. Lauritzen and B.R. Mottelson, Phys. Rev. Lett. 58 (1987) 326 22) A.L. Goodman, Phys. Rev. C38 (1988) 1092 23) A.L. Goodman, Phys. Rev. C39 (1989) 2008