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Electromagnetic decay of hot rotating nuclei
Volume 208, number l
PHYSICS LETTERS B
7 July 1988
E L E C T R O M A G N E T I C DECAY O F H O T R O T A T I N G N U C L E I ~ J.L. E G I D O Departamento de Fisica Te6rica C-XI, Universidad Aut6noma de Madrid, t?-28049 Madrid, Spain
and Hans A. W E I D E N M O L L E R Max-Planck-Institut J~r Kernphysik, D-6900 Heidelberg, Fed. Rep. Germany
Received 18 February 1988
A theory of strength functions for electromagnetic multipole emission is developed. It combines the finite-temperature cranked Hartree-Fock-Bogoliubov formalism with the random-phase approximation, and is free of spurious contributions. Numerical results for 164Erbased on the Kumar-Baranger hamiltonian are presented. They display sizeable fluctuations of the matrix elements for the statistical transitions (contrary to the statistical model), and underline the important role of M1 transitions for good rotors.
Experimental investigations o f the quasicontinuum y-rays emitted by a c o m p o u n d nucleus formed in a (HI, xn) reaction have opened the door to studying high-spin states far above the yrast line. One o f the open problems in the interpretation o f the data relates to the path in the (E, I) plane on which a welldeformed nucleus with excitation energy E and spin h i reaches the ground state via ~/-emission. This is the problem which we address in the present letter. More precisely, we wish to answer the following question. At each point in the (E, I) plane, statistical El, M1, and E2 decays compete with the collective E2 decay within an excited rotational band roughly parallel to the yrast line. What are the transition strengths for these various decays? Simple models [ 1--3 ] to answer this question unfortunately carry considerable uncertainties. In the statistical model, for instance, the energy dependence of the level density is crucial but known only poorly. The same applies to the ratio of average E1 and E2 matrix elements [3]. On the other hand, an exact calculation, taking account of the decay properties o f all states with spin h i in the energy interval Work supported in part by CAICyT, Spain, project AE-26. 58
between E and E + d E , is clearly out o f the question in the continuum region. In view of these difficulties, we have employed the following microscopic method to calculate the transition rates: We have combined the random-phase approximation ( R P A ) with the finite-temperature (FT) cranked Hartree-Fock-Bogoliubov ( C H F B ) approach in the formalism of the grand canonical ensemble. This is done by first solving the F T C H F B equations for given values o f cranking frequency and temperature. Small oscillations around the minima so determined are subsequently taken into account in terms of the FTRPA. The HFB approximation has been used in nearly all microscopic investigations of states near the yrast line [4,5 ]. There it works very well. The extension to finite temperatures has likewise been very successful [ 6 - 8 ] . In principle, this approximation provides us with the information necessary to calculate transition strengths as functions of (E, I). We have added the R P A for two reasons. First, the mean-field approximation is not very suitable for describing collective states while the RPA shifts the collective strengths of the low-lying vibrations and of the giant resonances to their proper places. Second, in the RPA the 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 208, number 1
PHYSICS LETTERS B
sition probabilities [ 10 ]. As it stands, eq. ( 1 ) holds in the laboratory system. Writing it in the rotating system requires two changes [ 10,11 ]. (i) The axis of quantization must be chosen along the rotation axis. Then, the negative of the m-quantum number of the tensor operator Q coincides with the difference (If-l~) between final and initial spins. (ii) The energy argument E in Ree must be shifted by (Ir-I~)o9, where o9 is the cranking frequency. The response function RQe in eq. ( 1 ) is obtained by contracting the general response function ~twice with the operator Q in the space of RPA configurations, and 9 is given by the Lippmann-Schwinger equation
Goldstone modes stemming from the symmetry broken in the mean-field approximation, separate exactly [9] from the normal modes. The Goldstone modes can be artificially shifted far away in energy; the shift removes the problem of spurious contributions to the transition strength [ 10 ]. Finally, the use of the grand canonical ensemble is justified because the level density is very high, already 1 or 2 MeV above the yrast line. In this framework, the strength function S o (E) for a given operator Q is given by the response function RoQ for Q, -1 SQ ( E ) = ~ [ 1 - e x p ( f l E ) ] ImRQQ (E)
7 July 1988
1)
~= ~o+~o.#~.~.
with fl the inverse temperature. Eq. (1) has to be modified slightly to yield an expression for the tran-
The matrix "#lakes into account the residual inter-
HFB ( T = 0 . 4
MEV )
io-5
lO - 5
io-S
10-6
10-6
i0- 6
i0-7
lO-7
10 - 8
,-
i0-9
' 0.4
'
'
, 1.6
,
i0- 7
r
10-8
E1
w=0.075
i0-8 r I
-::w=0~275 t 218.
10-9
I
I
I
0_5
i0-I
10-I
i0-2
10-2 ~
I
I
2.
I
I
[
10-6 i=
w=0.075 I
2
0.4
l
I
I
1.6
l
E1
~=0.575 I
01.4 11'.21
2.
0_4
2.
i0-I
.
i0-2 10-3 10- 4
' ~
I0-4
r
10-5
i0- 9
3.5
10-3 i0 - 4
(2)
10. 5 ~ " M l ~ l ' ~ l i
10. 5
10--6 ~ : ? 0 : 2 ; 5 1 ~ 1 ~
lO-6
I
2.8
10 ~ E2 10 w=0,075 1 ,, 10-1 ~ ~ I 10-2 10-3
0.5 10 2 ! ~ 10
2.
10 2 E2
[ ll~ w=0.275
1
10- I 10- 2 10-3
i0 - 4
10 - 4
10-5
10- 5
:/
3.5
\\
IO
~
[
E2
]~=0.575
10-1 [ i0--2
1.2
// /
\\ -.
10-3 i0 - 4
10-5 0.5 2. 3.5 0.4 1.2 2. Fig. 1. H FB strength functions (in w.u. ) versus 7-ray energy (in MeV) for 164Erat T= 0.4 MeV as explained in the text. 0.4
1.6
2.8
59
Volume 208, number 1
PHYSICS LETTERS B
AE) this choice of q is obviously unacceptable. Choosing ~/finite is tantamount to smearing each state over a finite energy interval of width F = 2 0. Such a choice is indicated by theories beyond the RPA which identify F with a spreading width, and is therefore physically reasonable. Choosing F>> AE we are sure to pick up all the strength numerically. In the calculations, the condition F>> AE had to be put in balance with two further requirements: (i) CPU time obviously grows strongly with decreasing AE; this puts a lower bound in AE. (ii) Values of F in excess of several tens of keV are physically unrealistic; they would smear the strength (especially that from the numerous dense-lying states at higher excitation energy) over too wide a region. [We recall that ~o in eq. (3) has lorentzian shape and a very soft cut-offin the tails. ] The RPA scheme just described allows for the cal-
action not contained in the mean-field approximation. The free response function ~o is given explicitly in terms of the quasiparticle energies E u and the temperature-dependent occupation probabilitiesfu by
2([. - L ) ~ o
= E-E--E--~4-i~/o~,,, O~,,.
7 July 1988
(3)
We note in passing that by putting "W=0, and by thereby replacing ~ in eq. ( 1 ) by ~ o we obtain the strength functions (and transition probabilities) in the HFB approximation.
In the framework of the RPA, the parameter q in eq. (3) ought to be put equal to zero. This would produce poles in R° and R at the positions of the HFB and of the RPA transition energies, respectively, and a corresponding spike structure of delta-function type in the strength function ( 1 ). For the numerical calculations (where we proceed in finite energy steps
RPA ( T = 0 . 4
MEV )
lO-5
10- 5
I0-5
10- 6
10- 6
10- 6
10- 7
10. 7
lO--7
10. 8
10- 8
10.9
10- 9
10.8 !
~=0 075 v
10-9
I
I
I
04
I
I
I
16
~=0 I
28
05
2
10-1
10-1
10-2
10. 2
10- 2
I0 -3
lO. 3
10 - 3
10-4.
lO. 4
10- 5
10- 5
10. 6
10- 6 r
r
w=O 075
= I
04 10
I
I
I
16
2 E2
10 1
lO-1 10- 2
\\
~
I
I
I
~
1o-4 ~1o-5 0.4
10. 5 10- 6
=o27s I
'
2
'
'
2.8
lO 1o
,
1
/t
I
a)=O 5 7 5 I
2.
10' E2
E2
lO
0=0.275
1
/
10-I 10-2 10-3 10-4
%"
10 -5
0.5
2.
.
Fig. 2. Same as fig. 1 but with RPA.
60
M1
r
2
10 - 5
1.6
r
01.4 1 1 1 . 2 . 1
'
35
2
10-I / \x I0--2 ~ ... 10-3 10-4
10 .3
i 2.
lO. 4
r
015. I
t
12 i I.
lO-1
t
28
014 i
35
575
0.4
1.2
2.
7 July 1988
PHYSICS LETTERS B
Volume 208, n u m b e r 1
HFB (
T=0.7
MEV
)
10- 4
10- 4
10- 4
10- 5
10- 5
lO- 5
10- 6
10- 6
lO- 6
10- 7
10- 7
10- 7
10- 8
10- 8
10- 9
I
I
1
I
I
3
I
I
5
I
10- 9
I
7
]o - 8
r= w = 0 . 2 7 5 I
I
1
1
3
I
I
5
I
I
I
10- 9
7
lO-1
10- 1
10- 2
10- 2
lO- 3
10- 3
10- 4
10- 4
10-1 10- 2 10-3 10- 4
10_ 5
10- 5
10- 5
10- 6
10- 6
10- 6
10- 7
10- 7
10- 7
10- 8
lO- 8 lO- 9
lO- 8 10 - 9
I
1
I
I
3
I
I
5
I
l
7
2
1° I \
10-4 10- 5 10- 6 1
10- 9
9
1
3
5
3
5
7
2 10 10 1 10- 1 10- 2 10- 3 10- 4 10- 5 10- 6
1
3
5
7
I
I
I
I
I
I
I
1
2
3
4
5
6
7
r
w=0.575 r
7
2 10 10 1 10- 1 10- 2 10- 3 10- 4 10- 5 10- 6
10 1 10- 1
10-2
l~ri" ~ I
r= w = 0 . 5 7 5
I
I
I
I
I
I
1
2
3
4
5
6
!
/ \
E2
.....
I
I
I
I
I
t
I
1
2
3
4
5
6
7
I
Fig. 3. Same as fig. 1 but for T = 0 . 7 MeV.
culation of all transition strengths but one: The collective intraband transition. Indeed, within the RPA the latter has the form of a transition from ground state to ground state. But the response function formalism describes only transitions connecting the ground state with an excited state. Fortunately, this difficulty is easily removed. The collectivity of the transition in question is basically due to deformation. Therefore, it can reliably be calculated in the HFB approximation. This is what we have done throughout. We have applied this scheme to the calculation of El, M1, and E2 transitions in the deformed heavy n u c l e u s 164Er. Except for one modification mentioned below, we have used the same hamiltonian and the same configuration space as Kumar and Baranger [ 12 ]. This hamiltonian is known to reproduce both
ground-state deformations and gap parameters in the rare earth region very well. It has also been used to investigate properties of nuclei in this region at finite temperature [ 5,13 ]. In the Kumar-Baranger hamiltonian, the residual interaction # i s separable. This facilitates solving the Lippmann-Schwinger equation (2). We have modified the Kumar-Baranger hamiltonian by adding a dipole-dipole term. The strength of this term was chosen so that the giant dipole resonance in spherical nuclei is close to the experimental value [ 11 ]. Adding such a term does not change the self-consistent solutions of the original hamiltonian since the mean-field expectation value of the dipole operator vanishes. For a given point in the (E, I) plane, our calculations yield 11 strength functions ( 2 L + 1 functions for each operator of multipolarity L), each a function of 61
Volume 208, n u m b e r 1
PHYSICS LETTERS B
RPA
T=O,7
MEV
)
lO-4
10 - 4
10-5
10- 6
10. 7 10. 8
E1
=
10. 9 5
7
M"I-~~j , ,
,
,
1
10- 8
10- 8
,
,
,
5
3
,
7
10- 2 10- 3
E2
I
I
I
I
3
t
I
5
I
I
10-4
.......
I
I
3
I
I
I 3
I
I 5
I
I 7
I
5
7
9
I0-9 i0 10
I
I
I
I
I
3
4
5
6
7
I
1
I
I
I
I
I
1
2
3
4
5
6
7
2 I Ii
1
7~
/I \
I0-I 10- 2
-
E2
. . . . . . .
lO- 3 lO- 4
E2
.....
10 - 5 10 - 6 I
I ~i ~lli'l'
f
2
i0- 8 I I
lO- 6 I 1
I
1
10- 7
1o-5
i-
10- 9
7
10- 1 10- 2 10- 3 10- 4 10- 5 10- 6
9
2 10 10 1 10- I 10- 2 10- 3 -
I
I
10- 7 10- 8 i0 - 9 F
1o-I
lO- 5 lO- 6
10-7
10 - 6
2
10- 4
10- 7
10-1 10- 2 10- 3 10- 4 10- 5
io-2 io-3
10- 7 10- 8 10- 9
10-6
9
10 - 1
1o- 5 10- 6
10-5 2
10- 9 3
1
10 - 4
lO- 4
10- 5
r
10 . 6
10 10 1
(
7 July 1988
I
1
' 3
'
' 5
'
"llllII~ 1 " 7
[ I
I
I
I
I
I
I
1
2
3
4
5
6
7
Fig. 4. Same as fig. 2 but for T=0.7 MeV. the energy Ev of the emitted "/-ray. Since space is limited, we display only the emission strength functions (always in Weisskopf units) for stretched transitions ( I ~ I - 1 for El and M1 and 1--,1-2 for E2), in all figures plotted versus E v (in MeV). The maximum ,/-ray energy is always determined by the excitation energy E* of the emitting nucleus (the energy above the yrast line). HFB results are shown in figs. 1 and 3, RPA results in figs. 2 and 4. In figs. 1 and 2 (3 and 4), the temperature is T = 0 . 4 MeV ( T = 0 . 7 MeV). Strength functions for El, M1, and E2 transitions are shown on a logarithmic scale in the first, second, and third rows, respectively. Also shown in the graphs in the third rows are the total E2 strengths, located at E = 2hco. To facilitate the comparison between this E2 strength and the other strength functions, we have 62
smeared it out with a Breit-Wigner function, using the same value for the w i d t h , / ' = 30 keV, as in the rest of the calculation where we also used NE= l 0 keV. In all four figures, the cranking frequencies con for the nth columns ( n = 1, 2, 3) coincide; the corresponding values of m, (in MeV) are (in this sequence) 0.075, 0.275, and 0.575. For T=0.4 MeV, these figures correspond to (E*, I) values (with E* in MeV) of (3.13, 6.1), (3.84, 27.6) and (2.53, 56.19). The excitation energy for m=0.075 is smaller than for 0.275 due to the change in the moment of inertia. (For m= 0.075 the pairing correlations are still large for T= 0 and T = 0.4 MeV whereas for m = 0.275 they are large for T = 0 but they vanish for T=0.4 causing the corresponding rise in the moment of inertia. ) The corresponding figures for T=0.7 are (9.15, 7.3), (8.96, 27.6), and (7.58, 56.0).
PHYSICSLETTERSB
Volume208, number 1
I~,~
1o
F-
El/U1
lO 1
~~
El/E2
1°- 1
Jl
,]1 . r
M1/E2
10-210_ 410-'-5)10 i
.
W=0.075
RPA ( T = 0 . 7
I
1
[
[
3
I
[
5
[
I
7
I
MEV )
I0 2
~o 1 10-1 10- 2
10 2
~ F
El/M1 El/E2 M1/E2
~ --
P
,a t~
~
lO 1
.r
~--
,~
10_ 10-4
10-5
10- 5
1
-
El/M1 El/E2 M1/E2
-
10- 2
10-4
I
I~ [ ~r
10-1 ~'-i
lO-6 r ~j'V
10- 6
7 July 1988
I
w=0.275 I
3
I
I
5
10- 6 I
i
7
I
I
I
I
l
I
I
I
1
2
3
4
5
6
7
Fig. 5. The RPA strength-function ratios S( E l )/S( E2 ), S(E 1 )/S( M ] ) and S( M 1 )/S( E2 ) versus y-ray energy ( in MeV ) on a logarithmic scalefor T=0.7 MeVas explainedin the text.
A feature clearly visible in some of the figures, especially those for the E1 strength, is an increase of the high-energy parts of the strength functions (relative to the low-energy) parts as oJ (or 1) increase. We ascribe this to increased mixing of levels due to the Coriolis force. Almost the opposite trend is displayed by the M 1 strength. This is because for larger spins, strong alignment takes place, and the magnetic properties are known to be sensitive to such effects. We also notice that for high spins, the spikes in the M 1 strength get broader and are shifted to higher energies. Comparing the E1 strengths in figs. 1 and 2 (or those in figs. 3 and 4), we note that the RPA results are typically one order of magnitude smaller than those for the HFB. This is because the RPA shifts E 1 strength to the giant dipole resonance. (More realistic results could be obtained by fitting the strength of the dipole-dipole term to the giant resonance in 164Er which we have not done.) No such difference exists for the M1 transition because for 164Er and the excitation energies considered, there is no magnetic vibrational collective. The collective E2 strength (upper curve) is much bigger than the statistical one. This is expected for a good rotor. However, the statistical E2 RPA is much increased over its HFB value. It appears that one transition at around 1 MeV for o)=0.575 MeV, becomes particularly important. It probably comes from a collective surface vibration and might be caused by a soft potential surface at this cranking frequency and temperature.
The shape of 164Eris known to change little with T as long as T< 1 MeV [ 11 ]. It is consistent with this finding that the strength functions do net change much as Tis raised from 0.4 MeV to 0.7 MeV (figs. 3, 4). Noteworthy features are: The E1 strength for fixed E increases with increasing T by roughly one order of magnitude. The M 1 and E2 strengths are also somewhat larger on average. This is the natural consequence of the growth of the level density with T; this growth does not affect the collective vibrational E2 strength, as we expect. The broad peak in the HFB E1 strength around 6-7 MeV is shifted in RPA to higher energies and is probably the HFB precursor of the giant resonance. To compare with the statistical model, we display in fig. 5 ratios of RPA strength functions versus E for T=0.7 MeV and for the same three rotational frequencies as in figs. 1-4. To smooth out the strong fluctuations of the RPA solutions with E we have used a larger value of F ( 100 keV) than before. We note that the dependence on the level density largely cancels, so that the ratios are also indicative of the relative size of the transition matrix elements and, in the case of the E 1/M1 ratio, of the ratio of the transition probabilities (the energy dependence of dipole emission is independent of the parity of the radiation) except for an overall factor of about 100. Taking into account this factor, we see that for the dipole modes at he)= 0.075 MeV, magnetic emission dominates for E < 3 MeV while it is the more important for almost 63
Volume 208, number 1
PHYSICS LETTERS B
all energies at hrn = 0.575 MeV. Both the E 1/M 1 and E 1/E2 ratios show a strong secular increase with Ev over several orders of magnitude. We ascribe this to the tails of the giant dipole resonance, but superimposed on this trend, and clearly visible also in the M1/E2 ratios, are very sizeable fluctuations easily amounting to 1 or 2 orders of magnitude. It is well to remember that in the pure statistical model, all three ratios should be straight horizontal lines; for the E l / M 1 and E l / E 2 ratios this behaviour would change if the giant E1 resonance were accounted for. In conclusion, we have set up and solved the linear response theory at finite temperature with both HFB and RPA eigenstates. In our opinion, this approach is the most realistic one presently available to describe the electromagnetic decay of hot rotating nuclei. Our calculations show that it can also be implemented. Results of the type displayed above can be calculated for an entire grid of points in the (E, I) plane. In this way, the question posed at the beginning of this letter can be answered. Such results, and a detailed description of our procedure, form the content of a future publication [9]. From the results shown above, we can already conclude that the transition matrix elements fluctuate strongly (this is at
64
7 July 1988
variance with the statistical model), and that for good rotors the M 1 transitions play an important role.
References [1] R.J. Lioua and R.A Sorensen, Nucl. Phys. A 297 (1978) 136. [2] G. Leander, Y.S. Chen and B.S. Nilsson, Phys. Scr. 24 (1981) 164. [3] M. Wakai and A. Faessler, Nucl. Phys. A 307 (1978) 349. [4] A.L. Goodman, Advances in Nuclear Physics, Vol. 11, eds. J. Negele and E. Voigt (Plenum, New York, 1979). [ 5 ] J.L. Egido, H.J. Mang and P. Ring, Nucl. Phys. A 339 (1980) 390. [6]A.L. Goodman, Nucl. Phys. A 369 (1981) 365; A 370 (1981) 90. [ 7 ] K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys. A357 (1981) 20,45. 18 ] J.L. Egido, P. Ring and H.J. Mang, Nucl. Phys. A 451 ( 1986 ) 77. [9] J.L. Egido and J.O. Rasmussen, Nucl. Phys. A 476 (1988) 48. [ 10l J.L. Egido and H.A. Weidenmueller, to be published. [ 11 ] P. Ring, L.M. Robledo, J.L. Egido and M. Faber, Nucl. Phys. A 419 (1984) 261. [ 12] B. Baranger and K. Kumar, Nucl. Phys. A 1 l0 (1968) 490. [ 13 ] A. Goodman, Phys. Rev. C 34 (1986) 1942.
PHYSICS LETTERS B
7 July 1988
E L E C T R O M A G N E T I C DECAY O F H O T R O T A T I N G N U C L E I ~ J.L. E G I D O Departamento de Fisica Te6rica C-XI, Universidad Aut6noma de Madrid, t?-28049 Madrid, Spain
and Hans A. W E I D E N M O L L E R Max-Planck-Institut J~r Kernphysik, D-6900 Heidelberg, Fed. Rep. Germany
Received 18 February 1988
A theory of strength functions for electromagnetic multipole emission is developed. It combines the finite-temperature cranked Hartree-Fock-Bogoliubov formalism with the random-phase approximation, and is free of spurious contributions. Numerical results for 164Erbased on the Kumar-Baranger hamiltonian are presented. They display sizeable fluctuations of the matrix elements for the statistical transitions (contrary to the statistical model), and underline the important role of M1 transitions for good rotors.
Experimental investigations o f the quasicontinuum y-rays emitted by a c o m p o u n d nucleus formed in a (HI, xn) reaction have opened the door to studying high-spin states far above the yrast line. One o f the open problems in the interpretation o f the data relates to the path in the (E, I) plane on which a welldeformed nucleus with excitation energy E and spin h i reaches the ground state via ~/-emission. This is the problem which we address in the present letter. More precisely, we wish to answer the following question. At each point in the (E, I) plane, statistical El, M1, and E2 decays compete with the collective E2 decay within an excited rotational band roughly parallel to the yrast line. What are the transition strengths for these various decays? Simple models [ 1--3 ] to answer this question unfortunately carry considerable uncertainties. In the statistical model, for instance, the energy dependence of the level density is crucial but known only poorly. The same applies to the ratio of average E1 and E2 matrix elements [3]. On the other hand, an exact calculation, taking account of the decay properties o f all states with spin h i in the energy interval Work supported in part by CAICyT, Spain, project AE-26. 58
between E and E + d E , is clearly out o f the question in the continuum region. In view of these difficulties, we have employed the following microscopic method to calculate the transition rates: We have combined the random-phase approximation ( R P A ) with the finite-temperature (FT) cranked Hartree-Fock-Bogoliubov ( C H F B ) approach in the formalism of the grand canonical ensemble. This is done by first solving the F T C H F B equations for given values o f cranking frequency and temperature. Small oscillations around the minima so determined are subsequently taken into account in terms of the FTRPA. The HFB approximation has been used in nearly all microscopic investigations of states near the yrast line [4,5 ]. There it works very well. The extension to finite temperatures has likewise been very successful [ 6 - 8 ] . In principle, this approximation provides us with the information necessary to calculate transition strengths as functions of (E, I). We have added the R P A for two reasons. First, the mean-field approximation is not very suitable for describing collective states while the RPA shifts the collective strengths of the low-lying vibrations and of the giant resonances to their proper places. Second, in the RPA the 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 208, number 1
PHYSICS LETTERS B
sition probabilities [ 10 ]. As it stands, eq. ( 1 ) holds in the laboratory system. Writing it in the rotating system requires two changes [ 10,11 ]. (i) The axis of quantization must be chosen along the rotation axis. Then, the negative of the m-quantum number of the tensor operator Q coincides with the difference (If-l~) between final and initial spins. (ii) The energy argument E in Ree must be shifted by (Ir-I~)o9, where o9 is the cranking frequency. The response function RQe in eq. ( 1 ) is obtained by contracting the general response function ~twice with the operator Q in the space of RPA configurations, and 9 is given by the Lippmann-Schwinger equation
Goldstone modes stemming from the symmetry broken in the mean-field approximation, separate exactly [9] from the normal modes. The Goldstone modes can be artificially shifted far away in energy; the shift removes the problem of spurious contributions to the transition strength [ 10 ]. Finally, the use of the grand canonical ensemble is justified because the level density is very high, already 1 or 2 MeV above the yrast line. In this framework, the strength function S o (E) for a given operator Q is given by the response function RoQ for Q, -1 SQ ( E ) = ~ [ 1 - e x p ( f l E ) ] ImRQQ (E)
7 July 1988
1)
~= ~o+~o.#~.~.
with fl the inverse temperature. Eq. (1) has to be modified slightly to yield an expression for the tran-
The matrix "#lakes into account the residual inter-
HFB ( T = 0 . 4
MEV )
io-5
lO - 5
io-S
10-6
10-6
i0- 6
i0-7
lO-7
10 - 8
,-
i0-9
' 0.4
'
'
, 1.6
,
i0- 7
r
10-8
E1
w=0.075
i0-8 r I
-::w=0~275 t 218.
10-9
I
I
I
0_5
i0-I
10-I
i0-2
10-2 ~
I
I
2.
I
I
[
10-6 i=
w=0.075 I
2
0.4
l
I
I
1.6
l
E1
~=0.575 I
01.4 11'.21
2.
0_4
2.
i0-I
.
i0-2 10-3 10- 4
' ~
I0-4
r
10-5
i0- 9
3.5
10-3 i0 - 4
(2)
10. 5 ~ " M l ~ l ' ~ l i
10. 5
10--6 ~ : ? 0 : 2 ; 5 1 ~ 1 ~
lO-6
I
2.8
10 ~ E2 10 w=0,075 1 ,, 10-1 ~ ~ I 10-2 10-3
0.5 10 2 ! ~ 10
2.
10 2 E2
[ ll~ w=0.275
1
10- I 10- 2 10-3
i0 - 4
10 - 4
10-5
10- 5
:/
3.5
\\
IO
~
[
E2
]~=0.575
10-1 [ i0--2
1.2
// /
\\ -.
10-3 i0 - 4
10-5 0.5 2. 3.5 0.4 1.2 2. Fig. 1. H FB strength functions (in w.u. ) versus 7-ray energy (in MeV) for 164Erat T= 0.4 MeV as explained in the text. 0.4
1.6
2.8
59
Volume 208, number 1
PHYSICS LETTERS B
AE) this choice of q is obviously unacceptable. Choosing ~/finite is tantamount to smearing each state over a finite energy interval of width F = 2 0. Such a choice is indicated by theories beyond the RPA which identify F with a spreading width, and is therefore physically reasonable. Choosing F>> AE we are sure to pick up all the strength numerically. In the calculations, the condition F>> AE had to be put in balance with two further requirements: (i) CPU time obviously grows strongly with decreasing AE; this puts a lower bound in AE. (ii) Values of F in excess of several tens of keV are physically unrealistic; they would smear the strength (especially that from the numerous dense-lying states at higher excitation energy) over too wide a region. [We recall that ~o in eq. (3) has lorentzian shape and a very soft cut-offin the tails. ] The RPA scheme just described allows for the cal-
action not contained in the mean-field approximation. The free response function ~o is given explicitly in terms of the quasiparticle energies E u and the temperature-dependent occupation probabilitiesfu by
2([. - L ) ~ o
= E-E--E--~4-i~/o~,,, O~,,.
7 July 1988
(3)
We note in passing that by putting "W=0, and by thereby replacing ~ in eq. ( 1 ) by ~ o we obtain the strength functions (and transition probabilities) in the HFB approximation.
In the framework of the RPA, the parameter q in eq. (3) ought to be put equal to zero. This would produce poles in R° and R at the positions of the HFB and of the RPA transition energies, respectively, and a corresponding spike structure of delta-function type in the strength function ( 1 ). For the numerical calculations (where we proceed in finite energy steps
RPA ( T = 0 . 4
MEV )
lO-5
10- 5
I0-5
10- 6
10- 6
10- 6
10- 7
10. 7
lO--7
10. 8
10- 8
10.9
10- 9
10.8 !
~=0 075 v
10-9
I
I
I
04
I
I
I
16
~=0 I
28
05
2
10-1
10-1
10-2
10. 2
10- 2
I0 -3
lO. 3
10 - 3
10-4.
lO. 4
10- 5
10- 5
10. 6
10- 6 r
r
w=O 075
= I
04 10
I
I
I
16
2 E2
10 1
lO-1 10- 2
\\
~
I
I
I
~
1o-4 ~1o-5 0.4
10. 5 10- 6
=o27s I
'
2
'
'
2.8
lO 1o
,
1
/t
I
a)=O 5 7 5 I
2.
10' E2
E2
lO
0=0.275
1
/
10-I 10-2 10-3 10-4
%"
10 -5
0.5
2.
.
Fig. 2. Same as fig. 1 but with RPA.
60
M1
r
2
10 - 5
1.6
r
01.4 1 1 1 . 2 . 1
'
35
2
10-I / \x I0--2 ~ ... 10-3 10-4
10 .3
i 2.
lO. 4
r
015. I
t
12 i I.
lO-1
t
28
014 i
35
575
0.4
1.2
2.
7 July 1988
PHYSICS LETTERS B
Volume 208, n u m b e r 1
HFB (
T=0.7
MEV
)
10- 4
10- 4
10- 4
10- 5
10- 5
lO- 5
10- 6
10- 6
lO- 6
10- 7
10- 7
10- 7
10- 8
10- 8
10- 9
I
I
1
I
I
3
I
I
5
I
10- 9
I
7
]o - 8
r= w = 0 . 2 7 5 I
I
1
1
3
I
I
5
I
I
I
10- 9
7
lO-1
10- 1
10- 2
10- 2
lO- 3
10- 3
10- 4
10- 4
10-1 10- 2 10-3 10- 4
10_ 5
10- 5
10- 5
10- 6
10- 6
10- 6
10- 7
10- 7
10- 7
10- 8
lO- 8 lO- 9
lO- 8 10 - 9
I
1
I
I
3
I
I
5
I
l
7
2
1° I \
10-4 10- 5 10- 6 1
10- 9
9
1
3
5
3
5
7
2 10 10 1 10- 1 10- 2 10- 3 10- 4 10- 5 10- 6
1
3
5
7
I
I
I
I
I
I
I
1
2
3
4
5
6
7
r
w=0.575 r
7
2 10 10 1 10- 1 10- 2 10- 3 10- 4 10- 5 10- 6
10 1 10- 1
10-2
l~ri" ~ I
r= w = 0 . 5 7 5
I
I
I
I
I
I
1
2
3
4
5
6
!
/ \
E2
.....
I
I
I
I
I
t
I
1
2
3
4
5
6
7
I
Fig. 3. Same as fig. 1 but for T = 0 . 7 MeV.
culation of all transition strengths but one: The collective intraband transition. Indeed, within the RPA the latter has the form of a transition from ground state to ground state. But the response function formalism describes only transitions connecting the ground state with an excited state. Fortunately, this difficulty is easily removed. The collectivity of the transition in question is basically due to deformation. Therefore, it can reliably be calculated in the HFB approximation. This is what we have done throughout. We have applied this scheme to the calculation of El, M1, and E2 transitions in the deformed heavy n u c l e u s 164Er. Except for one modification mentioned below, we have used the same hamiltonian and the same configuration space as Kumar and Baranger [ 12 ]. This hamiltonian is known to reproduce both
ground-state deformations and gap parameters in the rare earth region very well. It has also been used to investigate properties of nuclei in this region at finite temperature [ 5,13 ]. In the Kumar-Baranger hamiltonian, the residual interaction # i s separable. This facilitates solving the Lippmann-Schwinger equation (2). We have modified the Kumar-Baranger hamiltonian by adding a dipole-dipole term. The strength of this term was chosen so that the giant dipole resonance in spherical nuclei is close to the experimental value [ 11 ]. Adding such a term does not change the self-consistent solutions of the original hamiltonian since the mean-field expectation value of the dipole operator vanishes. For a given point in the (E, I) plane, our calculations yield 11 strength functions ( 2 L + 1 functions for each operator of multipolarity L), each a function of 61
Volume 208, n u m b e r 1
PHYSICS LETTERS B
RPA
T=O,7
MEV
)
lO-4
10 - 4
10-5
10- 6
10. 7 10. 8
E1
=
10. 9 5
7
M"I-~~j , ,
,
,
1
10- 8
10- 8
,
,
,
5
3
,
7
10- 2 10- 3
E2
I
I
I
I
3
t
I
5
I
I
10-4
.......
I
I
3
I
I
I 3
I
I 5
I
I 7
I
5
7
9
I0-9 i0 10
I
I
I
I
I
3
4
5
6
7
I
1
I
I
I
I
I
1
2
3
4
5
6
7
2 I Ii
1
7~
/I \
I0-I 10- 2
-
E2
. . . . . . .
lO- 3 lO- 4
E2
.....
10 - 5 10 - 6 I
I ~i ~lli'l'
f
2
i0- 8 I I
lO- 6 I 1
I
1
10- 7
1o-5
i-
10- 9
7
10- 1 10- 2 10- 3 10- 4 10- 5 10- 6
9
2 10 10 1 10- I 10- 2 10- 3 -
I
I
10- 7 10- 8 i0 - 9 F
1o-I
lO- 5 lO- 6
10-7
10 - 6
2
10- 4
10- 7
10-1 10- 2 10- 3 10- 4 10- 5
io-2 io-3
10- 7 10- 8 10- 9
10-6
9
10 - 1
1o- 5 10- 6
10-5 2
10- 9 3
1
10 - 4
lO- 4
10- 5
r
10 . 6
10 10 1
(
7 July 1988
I
1
' 3
'
' 5
'
"llllII~ 1 " 7
[ I
I
I
I
I
I
I
1
2
3
4
5
6
7
Fig. 4. Same as fig. 2 but for T=0.7 MeV. the energy Ev of the emitted "/-ray. Since space is limited, we display only the emission strength functions (always in Weisskopf units) for stretched transitions ( I ~ I - 1 for El and M1 and 1--,1-2 for E2), in all figures plotted versus E v (in MeV). The maximum ,/-ray energy is always determined by the excitation energy E* of the emitting nucleus (the energy above the yrast line). HFB results are shown in figs. 1 and 3, RPA results in figs. 2 and 4. In figs. 1 and 2 (3 and 4), the temperature is T = 0 . 4 MeV ( T = 0 . 7 MeV). Strength functions for El, M1, and E2 transitions are shown on a logarithmic scale in the first, second, and third rows, respectively. Also shown in the graphs in the third rows are the total E2 strengths, located at E = 2hco. To facilitate the comparison between this E2 strength and the other strength functions, we have 62
smeared it out with a Breit-Wigner function, using the same value for the w i d t h , / ' = 30 keV, as in the rest of the calculation where we also used NE= l 0 keV. In all four figures, the cranking frequencies con for the nth columns ( n = 1, 2, 3) coincide; the corresponding values of m, (in MeV) are (in this sequence) 0.075, 0.275, and 0.575. For T=0.4 MeV, these figures correspond to (E*, I) values (with E* in MeV) of (3.13, 6.1), (3.84, 27.6) and (2.53, 56.19). The excitation energy for m=0.075 is smaller than for 0.275 due to the change in the moment of inertia. (For m= 0.075 the pairing correlations are still large for T= 0 and T = 0.4 MeV whereas for m = 0.275 they are large for T = 0 but they vanish for T=0.4 causing the corresponding rise in the moment of inertia. ) The corresponding figures for T=0.7 are (9.15, 7.3), (8.96, 27.6), and (7.58, 56.0).
PHYSICSLETTERSB
Volume208, number 1
I~,~
1o
F-
El/U1
lO 1
~~
El/E2
1°- 1
Jl
,]1 . r
M1/E2
10-210_ 410-'-5)10 i
.
W=0.075
RPA ( T = 0 . 7
I
1
[
[
3
I
[
5
[
I
7
I
MEV )
I0 2
~o 1 10-1 10- 2
10 2
~ F
El/M1 El/E2 M1/E2
~ --
P
,a t~
~
lO 1
.r
~--
,~
10_ 10-4
10-5
10- 5
1
-
El/M1 El/E2 M1/E2
-
10- 2
10-4
I
I~ [ ~r
10-1 ~'-i
lO-6 r ~j'V
10- 6
7 July 1988
I
w=0.275 I
3
I
I
5
10- 6 I
i
7
I
I
I
I
l
I
I
I
1
2
3
4
5
6
7
Fig. 5. The RPA strength-function ratios S( E l )/S( E2 ), S(E 1 )/S( M ] ) and S( M 1 )/S( E2 ) versus y-ray energy ( in MeV ) on a logarithmic scalefor T=0.7 MeVas explainedin the text.
A feature clearly visible in some of the figures, especially those for the E1 strength, is an increase of the high-energy parts of the strength functions (relative to the low-energy) parts as oJ (or 1) increase. We ascribe this to increased mixing of levels due to the Coriolis force. Almost the opposite trend is displayed by the M 1 strength. This is because for larger spins, strong alignment takes place, and the magnetic properties are known to be sensitive to such effects. We also notice that for high spins, the spikes in the M 1 strength get broader and are shifted to higher energies. Comparing the E1 strengths in figs. 1 and 2 (or those in figs. 3 and 4), we note that the RPA results are typically one order of magnitude smaller than those for the HFB. This is because the RPA shifts E 1 strength to the giant dipole resonance. (More realistic results could be obtained by fitting the strength of the dipole-dipole term to the giant resonance in 164Er which we have not done.) No such difference exists for the M1 transition because for 164Er and the excitation energies considered, there is no magnetic vibrational collective. The collective E2 strength (upper curve) is much bigger than the statistical one. This is expected for a good rotor. However, the statistical E2 RPA is much increased over its HFB value. It appears that one transition at around 1 MeV for o)=0.575 MeV, becomes particularly important. It probably comes from a collective surface vibration and might be caused by a soft potential surface at this cranking frequency and temperature.
The shape of 164Eris known to change little with T as long as T< 1 MeV [ 11 ]. It is consistent with this finding that the strength functions do net change much as Tis raised from 0.4 MeV to 0.7 MeV (figs. 3, 4). Noteworthy features are: The E1 strength for fixed E increases with increasing T by roughly one order of magnitude. The M 1 and E2 strengths are also somewhat larger on average. This is the natural consequence of the growth of the level density with T; this growth does not affect the collective vibrational E2 strength, as we expect. The broad peak in the HFB E1 strength around 6-7 MeV is shifted in RPA to higher energies and is probably the HFB precursor of the giant resonance. To compare with the statistical model, we display in fig. 5 ratios of RPA strength functions versus E for T=0.7 MeV and for the same three rotational frequencies as in figs. 1-4. To smooth out the strong fluctuations of the RPA solutions with E we have used a larger value of F ( 100 keV) than before. We note that the dependence on the level density largely cancels, so that the ratios are also indicative of the relative size of the transition matrix elements and, in the case of the E 1/M1 ratio, of the ratio of the transition probabilities (the energy dependence of dipole emission is independent of the parity of the radiation) except for an overall factor of about 100. Taking into account this factor, we see that for the dipole modes at he)= 0.075 MeV, magnetic emission dominates for E < 3 MeV while it is the more important for almost 63
Volume 208, number 1
PHYSICS LETTERS B
all energies at hrn = 0.575 MeV. Both the E 1/M 1 and E 1/E2 ratios show a strong secular increase with Ev over several orders of magnitude. We ascribe this to the tails of the giant dipole resonance, but superimposed on this trend, and clearly visible also in the M1/E2 ratios, are very sizeable fluctuations easily amounting to 1 or 2 orders of magnitude. It is well to remember that in the pure statistical model, all three ratios should be straight horizontal lines; for the E l / M 1 and E l / E 2 ratios this behaviour would change if the giant E1 resonance were accounted for. In conclusion, we have set up and solved the linear response theory at finite temperature with both HFB and RPA eigenstates. In our opinion, this approach is the most realistic one presently available to describe the electromagnetic decay of hot rotating nuclei. Our calculations show that it can also be implemented. Results of the type displayed above can be calculated for an entire grid of points in the (E, I) plane. In this way, the question posed at the beginning of this letter can be answered. Such results, and a detailed description of our procedure, form the content of a future publication [9]. From the results shown above, we can already conclude that the transition matrix elements fluctuate strongly (this is at
64
7 July 1988
variance with the statistical model), and that for good rotors the M 1 transitions play an important role.
References [1] R.J. Lioua and R.A Sorensen, Nucl. Phys. A 297 (1978) 136. [2] G. Leander, Y.S. Chen and B.S. Nilsson, Phys. Scr. 24 (1981) 164. [3] M. Wakai and A. Faessler, Nucl. Phys. A 307 (1978) 349. [4] A.L. Goodman, Advances in Nuclear Physics, Vol. 11, eds. J. Negele and E. Voigt (Plenum, New York, 1979). [ 5 ] J.L. Egido, H.J. Mang and P. Ring, Nucl. Phys. A 339 (1980) 390. [6]A.L. Goodman, Nucl. Phys. A 369 (1981) 365; A 370 (1981) 90. [ 7 ] K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys. A357 (1981) 20,45. 18 ] J.L. Egido, P. Ring and H.J. Mang, Nucl. Phys. A 451 ( 1986 ) 77. [9] J.L. Egido and J.O. Rasmussen, Nucl. Phys. A 476 (1988) 48. [ 10l J.L. Egido and H.A. Weidenmueller, to be published. [ 11 ] P. Ring, L.M. Robledo, J.L. Egido and M. Faber, Nucl. Phys. A 419 (1984) 261. [ 12] B. Baranger and K. Kumar, Nucl. Phys. A 1 l0 (1968) 490. [ 13 ] A. Goodman, Phys. Rev. C 34 (1986) 1942.