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Dependence of the first saddle-point energy on temperature and spin in superdeformed rare-earth nuclei
Volume 213, number 2
PHYSICS LETTERS B
20 October 1988
D E P E N D E N C E OF THE FIRST SADDLE-POINT ENERGY ON TEMPERATURE A N D SPIN IN S U P E R D E F O R M E D RARE-EARTH NUCLEI ~ J. D U D E K a b c d
a,b, T.
W E R N E R b,c and L.L. R I E D I N G E R d
Joint Institute for Heavy Ion Research, HolifieldHeavy Ion Research Facility, Oak Ridge, TN 37831, USA Centre de Recherches Nuclbaires, 1:-67037 Strasbourg Cedex, France Institute o f Theoretical Physics, Hoza 69, PL-O0 681 Warsaw, Poland University of Tennessee, Knoxville, TN 37996-1200, USA
Received 28 April 1988; revised manuscript received 25 July 1988
The first saddle-point energies (i.e., the heights of the barriers separating "'super-" and "normal-" deformed minima) are calculated as functions of spin and nuclear temperature in a few dozen rare-earth nuclei. Calculations predict systematic growth in the barriers especially at low (I ~ 20 ) spins when Z decreases, the strongest effect occurring in samarium and neodymium nuclei with 84 < N < 88. Consequences for observability of superdeformed states are discussed.
Nuclear structure effects related to the superdeformed configurations at high angular m o m e n t a are presently an exciting new focus of attention in nuclear physics. There are only four nuclei known at present where discrete transitions within rotational bands built on superdeformed configurations have been observed: 151"152Dy [1,2], and 148~149Gd[3,4]. Only for 149Gd [4] and J52Dy [5] has there been direct experimental confirmation of the presence of large nuclear elongations via lifetime measurements. Yet there have been many unsuccessful attempts to find superdeformed nuclei at high spins, and it is of great importance to study the conditions for the occurrence and stability of superdeformed states. Such a study is particularly needed for the relatively hot nuclei produced in heavy-ion induced fusion reactions. The population of a superdeformed band depends first of all on the height and width of the potential barrier between the normal and superdeformed states as well as the elevations of the superdeformed minima over the yrast states. However, different calculations produce different results for the first-saddle elevations above the superdeformed m i n i m u m energy, visible particularly at the lower spin limit where transitions out of the superdeformed band take place. While calculations with the Nilsson potential [6] 120
predict ~ 2 MeV barriers for 1 ~ 2 0 h in 152Dy, the Woods-Saxon model gives lower, nearly vanishing barriers in this nucleus [7-9 ]. It is one of the two main subjects of this paper to discuss the systematic dependence of the barrier heights as a function of Z, N a n d / , thereby providing definite predictions which can be used to test the model dependence. The second goal, strongly related to the first, is to estimate the behavior of the barrier heights with temperature, which is believed to have importance for the "trapping" of nuclear decay in a superdeformed minimum. The following discussion is based on the freerouthian calculations (free-energy in the rotating coordinate frame) using the deformed Woods-Saxon potential [10] ~ with the cranking approach and thermal excitations (cf. e.g. ref. [ 1 2 ] ) . The freerouthian is defined as usual by F'°(def, T) =R'O(def, T) - TS'O (def, T ) ,
( 1)
with the entropy satisfying relations S'°(def, T) = - ~ n~' in n~~ tJ
-Z
(1-n~')ln(1 -n~'),
(2)
v
~ For technical details, cf. refs. [ 7,8 ], or see also ref. [ 11 ].
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V, ( North-Holland Physics Publishing Division)
Volume 213, number 2
PHYSICS LETTERS B
where the probability occupation coefficients n~to- nT(def, T ) = { l + e x p [ ( e + f l - 2 ) / T ] } -~
(3)
are normalized in a standard way to the proton and neutron numbers separately wherefrom the Fermi levels 2 are calculated. The single-particle routhians, e~7, satisfy the cranking equation - e v ~v,
H~°=Hws-COjx,
RO'(def, T)=R'°(def, T = 0 ) + U°+(def, T ) ,
(5)
R'°(def, T = 0 ) = E ~o=o ~=o ...... (def) +Emicro(def) + ~ [e~(def) - e y = ° ( d e f ) ],
(6)
v
according to standard definitions and notation. The validity of the above simple expression for the Woods-Saxon type potentials can be proven in the following way. (We omit the obvious " d e f " argument for simplicity.) According to Strutinsky's approach to
to
R =E
to
. . . . . + E m i c r o --O)I
--b7 ~°=0 --- I -~~m l c.r.o . . -- ~macro
17092
The first term in large brackets equals Emicro°J=° (cf. eq. (8)), while the second, for the Woods-Saxon type potentials is to a very good approximation equal to ½Jw 2. Substituting (9) into (7) gives relation (6). Finally, the excitation energy, U+°(def, T) is calculated as U°~(def, T ) = ~ n~'(def, T)e~'(def) p
(4)
where Hws is the deformed Woods-Saxon potential with the "universal" parameters as defined in ref. [ 10 ]. Using Strutinsky's ansatz of splitting the final expression into the macroscopic and microscopic terms, the total routhian satisfies [ 12 ]:
(7)
where the routhian, R% and energy, E +, are related via canonical relation and the classical rotational energy ½J¢o 2 is used explicitly (J denotes here the classical (rigid body) moment of inertia). The E~i++o term, satisfying by definition
20 October 1988
-~
nT(def, T=0)e~'(def).
In the above equations, the symbol " d e f " , if not omitted for simplicity, denotes all the deformation degrees of freedom and T is the nuclear temperature. Details of this variant of the general approach can be found in ref. [ 13 ] and references cited there. Let us begin by presenting the results for 152Dyand noticing how simple the calculated dependence of the free-routhian barrier on temperature is, cf. fig. 1 (The barrier height is defined as the elevation above the superdeformed minimum.) For all calculated spin values the barriers decrease essentially linearly when temperature increases. We recall that in l SZDythe superdeformed states were observed via discrete transitions first. Thus the results of fig. 1 may be used as a "reference" for other nuclei. Consequently, if for a given nucleus a larger potential barrier is found compared t o l S2Dy, one may expect that the changes of "trapping" this nucleus in superdeformed states are enhanced provided the elevations of the superdeBARRIERS (MeV) FOR '~DY86
Mz
,
+
+ -++ ev
Emicro
-
+)
ev
-
,
(8)
p
(10)
v
]
Ir ........
°+
i+
0
;"+
I",.
,
++
.
+
'
.
.
.
.
.
where the symbol < > denotes the Strutinsky averaging, can be rewritten as follows: 60
--
CO
Emicro -- 2 ev -- E e~ ' = ° v
+(~ eg=°- (~ eg=°))
+((~
+%%%+~
+%%%~,+
+%%%%7,
+%~%+,+
TEMPERATURE
TEMPERATURE
TEMPERATURE
TEMPERATURE
1.
e~'=°)-- ( ~ e + ) ) .
(9)
Fig. 1. The tSZDy free-energy harrier heights versus temperature for increasing spin values as indicated. The barrier heights are defined with respect to the superdeformed minima. Note the nearly linear decrease of the barriers with increasing temperature. Also, the slopes of the lines practically do not change with spin.
121
Volume 213, number 2
PHYSICS LETTERS B
f o r m e d m i n i m a in both c o m p a r e d nuclei are similar. Such a situation occurs according to our calculations for instance in 152Dy a n d ~5°Gd; the relative elevations have been discussed in ref. [ 7 ]. T r a p p e d nuclei are expected to emit fast collective 7-rays resulting in extended rotational bands built on the superdeformed shape• The simplicity of the barrier (first-saddle) dependence on t e m p e r a t u r e at a given spin holds not only for ~52Dy (fig. 1 ) but also for other nuclei in this region, and this can be utilized to present the results for m a n y nuclei in a compact way. Fig. 2 illustrates similar information using the phase diagram (T, I ) representation defined as follows. F o r each value o f spin we increase the t e m p e r a t u r e until the free-routhian barrier reduces to a small value, Blimit ( T, I). This value is arbitrarily set equal to 300 keV as representative o f a small barrier which can be penetrated by the quantal system with high probability• The conclusions of this work do not depend on this exact value chosen; setting Blimit (T, I ) = 0 will slightly move upward all the phase curves represented in fig. 2. In the figure, barriers larger than 300 keV exist for all the (T, I ) points situated between the phase curve and PHASE DIAGRAM 20 18 16 14 12 10 0.8 06 0.4 0.2 0.0
Dy ..............ijiill
N=80
the zero reference line. Using the fact that the barrier d e p e n d e n c e on t e m p e r a t u r e (fig. 1 ) can be approxim a t e d by a straight line allows us to construct, for each spin value and for each nucleus, a representation like the one in fig. 1. F o r this purpose it is sufficient to read the (T, I ) value from the phase d i a g r a m for which Blimit = 300 keV, place the corresponding point in the B versus T representation similar to that in fig. 1, and draw a straight line with the slope equal to the average slope represented in fig. 1. It becomes clear that the barrier conditions for D y nuclei with 80 ~ 40. Thus, the possibilities for the superdeformed m i n i m u m trapping in e r b i u m nuclei are predicted to be inferior with respect to those in dysprosium• The results for gadolinium a n d s a m a r i u m isotopes are shown in figs. 4 and 5, respectively• These cases seem much more favorable, as there is a systematic growth in the barrier height when Z decreases, especially at the lowest spins• In comparing the curves for various elements, a trend o f the t e m p e r a t u r e dependence in the barriers at high spin, e.g. at I = 60, is evident. While the strongest barriers disappear at T ~ 500
N-~o ....... ..' .."
/
/
.;~,__. /
_N-9. 8_
PHASE DIAGRAM
'
20 30 40 50 60 70 80 SPIN
Fig. 2. Phase diagram for the Dy isotopes indicated. The curves are composed of ( T, I) ---(temperature, spin ) points at which the barrier heights are equal to 300 keV (see text). For any given isotope, the points lying above the curves indicate the "no barrier zone" where trapping the nucleus in a superdeformed minimum is unlikely. On the contrary, ( T,/)-points below the curves indicate increasing barriers for decreasing temperature. The systematic change in the curve character with increasing neutron number deserves noting. The curves corresponding to 80
20 October 1988
2.0 1.8 >~ 1.6 1.4 1.2 1.0 0.8 ~ 0.6 F~ 0.4 0.2 0.0
Er N=84 : ' i ; ~ '
.:t # ::s
.., .~, t
.. ..-
; i~
,'z I
.........- Z/ 20
30
40
50
,:,'-" 60
70
N=88 N=90
• s~ I
....~"
.N=86 ..... • •
80
~_-_9_z _N-?4 90 100
SPIN
Fig. 3. Similar to that in fig. 2 but for the erbium nuclei. This diagram is equivalent to 360 single-frame diagrams of fig. 1, as explained in the text.
Volume 213, number 2
PHYSICS LETTERS B P H A S E DIAGRAM
2.0 1.8
~"
1.s
~
1.4
Gd N=84
1.2 E~
1.0
~_, 0.6
~
~-?_o ~.?..9.~ N-_9.4_
0.8
0.4 0.2 0.0
..... ...... .-"
,- :'
.~;-"" 20
30
40
,_#
,/./' 50
60 70 SPIN
80
90
Fig. 4. Similar to that in fig. 3 but for gadolinum nuclei (compare with figs. 2 and 3 ). Note the increasing barrier stability with respect to nuclear temperature increase, especially at lower spins. keV for erbium nuclei, they survive up to the T ~ 800 keV range in G d and Sm isotopes at •=60 (cf. figs. 2 - 5 ) . Results for n e o d y m i u m nuclei are similar to those for samarium. In conclusion, the calculations presented here may lead to important implications concerning the population of the superdeformed bands. The following qualitative but systematic predictions related to the first saddle-point energies can be made. ( 1 ) The barriers grow when the proton n u m b e r decreases from Z = 68 down to Z = 60 for practically the whole spin range. Thus the trapping in the superdeformed m i n i m u m should be more favorable in rareearth nuclei o f lower Z. O f course, the population of the superdeformed structure depends on the actual PHASE DIAGRAM 2.0 1.8 " ~ 1.6 ~-~ 1.4
Sm N=84
~-.~8 ~ ~
0.8 0.6 0.4 0.2 0.0
......
.." ........ I " ;';" ;i,i I ,/i
•<::;5.." ,/,,~
20
30
40
.N=9o ..... N=92
~_=._9._4
50
60
70
80
90
100
SPIN
Fig. 5. Similar to that in fig. 3 but for samarium nuclei (representative qualitatively also for neodymium nuclei with the same neutron numbers).
20 October 1988
level density corresponding to this and to the competing minima, which in turn depend on relative elevations of the minima (cf. ref. [ 7 ] for the latter). (2) Since the barriers are predicted to be systematically larger in nuclei of lower Z, also at low spins, the decay out of the superdeformed bands in Z ~ 5862 isotones should occur at spins systematically lower as compared to 152Dy" This conclusion applies for superdeformed bands at a relatively broad temperature
range. (3) Unless the increased pairing correlations act to increase the barrier penetrabilites at low frequencies (cf. refs. [ 13,14 ] ) the trend mentioned above should apply as well at low spins and low-temperature limit ( T - , 0 ) . This possible increase can be viewed as a result of the following WKB estimate of the barrier penetrability factor: P ~ e x p { ( 2 / h)fAn~/2M[ V(x)-E]dx] where the effective inertia satisfies [ 13 ]
g. (An 2
m~~-
+
g, (An 2
Z t.A,S},,'
(11)
with g= and g= being proportionality constants, A~,= the pairing gaps and (An/Afl),,, the numbers of single-particle level crossings under the barrier stretch Aft. For 3~,~ increasing at low frequencies the penetrability P may increase dramatically. (4) It is visible from eq. ( 11 ) that the probability of trapping the superdeformed nucleus depends on the structural differences between the superdeformed and normal states (the number of level crossings in terms of the mean field approximations, (An/ Aft) ). Qualitative estimates of this factor can be done in the context o f the present article using e.g. diagram 3 ofref. [15]. (5) Predictions ( 1 ) - (3) may also be influenced systematically if other characteristics of the barriers (e.g. barrier thickness) varied significantly from a nucleus to nucleus. This does not seem to be the main factor for Z ~ 62-66, N ~ 84-88 nuclei where the superdeformations change relatively slowly, but for quantitative estimates further studies are needed. It is therefore suggested that the experimental search for superdeformed bands in Z < 62 nuclei may lead to important new information on nuclear behavior in those exotic configurations. The Joint Institute for Heavy Ion Research has as 123
Volume 213, number 2
PHYSICS LETTERS B
member institutions the University of Tennessee, Vanderbilt University, and the Oak Ridge National L a b o r a t o r y ; it is s u p p o r t e d b y t h e m e m b e r s a n d b y the Department of Energy through Contract Number DE-AS05-76ER0-4936 with the University of Tennessee. This work was supported in part by the Polish Ministry of Science and Education within the p r o j e c t C P B P 01.09.
References [ 1 ] G.E. Rathke, R.V.F. Janssens, M.W. Drigert, I. Ahmed, K. Beard, V. Garg, M. Hass, T.L. Khoo, H.J. K6rner, W.C. Ma, S. Pilotte, P. Taras and F.L.H. Wolfs, to be published. [2] P.J. Twin, B.M. Nyako, A.H. Nelson, J. Simpson, M.A. Bentley, H.W. Cranmer-Gordon, F.D. Forsyth, D. Howe, A.R. Mokhtar, J.D. Morrison, J.F. Sharpey-Schafer and G. Sletten, Phys. Rev. Lett. 57 (1986) 811. [ 3 ] M.A. Deleplanque, C. Beausang, J. Burde, R.M. Diamond, J.E. Draper, C. Duyar, A.O. Machiavelli, R.J. McDonald and F.S. Stephens, submitted for publication. [4] B. Haas, P. Taras, S. Flibotte, F. Banville, J, Gascon, S. Cournoyer, S. Monaro, N. Nadon, D. Prevost, D. Thibault, J.K. Johansson, D.M. Tucker, J.C. Waddington, H.R.
124
20 October 1988
Andrews, G.C. Ball, D. Horn, D.C. Radford, D. Ward, C.St. Pierre and J. Dudek, Phys. Rev. Left. 60 (1988) 505. [5]M.A. Bentley, G.C. Ball, H.W. Cranmer-Gordon, P.D. Forsyth, D. Howe, A.R. Mokhtar, J.D. Morrison, J.F. Sharpey-Schafer, P.J. Twin, B. Fant, C.A. Kalfas, A.M. Nelson, J. Simpson and G. Sletten, Phys. Rev. Lett. 59 (1987) 2141. [ 6 ] I. Ragnarsson, T. Bengtsson, G. Leander and S. Aberg, Nucl. Phys. A 347 (1980) 287. [ 7 ] J. Dudek and W. Nazarewicz, Phys. Rev. C 31 ( 1985 ) 298. [ 8 ] J. Dudek, Proc. Intern. Winter Meeting on Nuclear physics (Bormio, Italy, 1987), ed. I. Iori (Milan, 1987). [9] V.M. Strutinsky, Soy. J. Nucl. Phys. 45 (1987) 1117. [10]S. Cwiok, J. Dudek, W. Nazarewicz, J. Skalski and T. Werner, Comput. Phys. Commun. 46 (1987) 379, and references therein. [ 11 ] J. Dudek, W. Nazarewicz and N. Rowley, Phys. Rev. C 35 (1987) 1489. [12]A.V. Ignatyuk, I.M. Mikhailov, L.H. Molina, R.G. Nazmitdinov and K. Pomorski, Nucl. Phys. A 346 (1980) 191. [13] J. Dudek, Proc. Intern. Conf. on Nuclear shapes (Crete, Greece, I987), ed. J. Garrett. [14]B. Herskind, B. Lauritzen, K. Schiffer, R.A. Broglia, F. Barranco, N. Gallardo, J. Dudek and E. Vigezzi, Phys. Rev. Lett. 59 (1987) 2416. [ 15 ] J. Dudek, W. Nazarewicz, Z. Szymanski and G.A. Leander, Phys. Rev. Len. 59 (1987) 1405.
PHYSICS LETTERS B
20 October 1988
D E P E N D E N C E OF THE FIRST SADDLE-POINT ENERGY ON TEMPERATURE A N D SPIN IN S U P E R D E F O R M E D RARE-EARTH NUCLEI ~ J. D U D E K a b c d
a,b, T.
W E R N E R b,c and L.L. R I E D I N G E R d
Joint Institute for Heavy Ion Research, HolifieldHeavy Ion Research Facility, Oak Ridge, TN 37831, USA Centre de Recherches Nuclbaires, 1:-67037 Strasbourg Cedex, France Institute o f Theoretical Physics, Hoza 69, PL-O0 681 Warsaw, Poland University of Tennessee, Knoxville, TN 37996-1200, USA
Received 28 April 1988; revised manuscript received 25 July 1988
The first saddle-point energies (i.e., the heights of the barriers separating "'super-" and "normal-" deformed minima) are calculated as functions of spin and nuclear temperature in a few dozen rare-earth nuclei. Calculations predict systematic growth in the barriers especially at low (I ~ 20 ) spins when Z decreases, the strongest effect occurring in samarium and neodymium nuclei with 84 < N < 88. Consequences for observability of superdeformed states are discussed.
Nuclear structure effects related to the superdeformed configurations at high angular m o m e n t a are presently an exciting new focus of attention in nuclear physics. There are only four nuclei known at present where discrete transitions within rotational bands built on superdeformed configurations have been observed: 151"152Dy [1,2], and 148~149Gd[3,4]. Only for 149Gd [4] and J52Dy [5] has there been direct experimental confirmation of the presence of large nuclear elongations via lifetime measurements. Yet there have been many unsuccessful attempts to find superdeformed nuclei at high spins, and it is of great importance to study the conditions for the occurrence and stability of superdeformed states. Such a study is particularly needed for the relatively hot nuclei produced in heavy-ion induced fusion reactions. The population of a superdeformed band depends first of all on the height and width of the potential barrier between the normal and superdeformed states as well as the elevations of the superdeformed minima over the yrast states. However, different calculations produce different results for the first-saddle elevations above the superdeformed m i n i m u m energy, visible particularly at the lower spin limit where transitions out of the superdeformed band take place. While calculations with the Nilsson potential [6] 120
predict ~ 2 MeV barriers for 1 ~ 2 0 h in 152Dy, the Woods-Saxon model gives lower, nearly vanishing barriers in this nucleus [7-9 ]. It is one of the two main subjects of this paper to discuss the systematic dependence of the barrier heights as a function of Z, N a n d / , thereby providing definite predictions which can be used to test the model dependence. The second goal, strongly related to the first, is to estimate the behavior of the barrier heights with temperature, which is believed to have importance for the "trapping" of nuclear decay in a superdeformed minimum. The following discussion is based on the freerouthian calculations (free-energy in the rotating coordinate frame) using the deformed Woods-Saxon potential [10] ~ with the cranking approach and thermal excitations (cf. e.g. ref. [ 1 2 ] ) . The freerouthian is defined as usual by F'°(def, T) =R'O(def, T) - TS'O (def, T ) ,
( 1)
with the entropy satisfying relations S'°(def, T) = - ~ n~' in n~~ tJ
-Z
(1-n~')ln(1 -n~'),
(2)
v
~ For technical details, cf. refs. [ 7,8 ], or see also ref. [ 11 ].
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V, ( North-Holland Physics Publishing Division)
Volume 213, number 2
PHYSICS LETTERS B
where the probability occupation coefficients n~to- nT(def, T ) = { l + e x p [ ( e + f l - 2 ) / T ] } -~
(3)
are normalized in a standard way to the proton and neutron numbers separately wherefrom the Fermi levels 2 are calculated. The single-particle routhians, e~7, satisfy the cranking equation - e v ~v,
H~°=Hws-COjx,
RO'(def, T)=R'°(def, T = 0 ) + U°+(def, T ) ,
(5)
R'°(def, T = 0 ) = E ~o=o ~=o ...... (def) +Emicro(def) + ~ [e~(def) - e y = ° ( d e f ) ],
(6)
v
according to standard definitions and notation. The validity of the above simple expression for the Woods-Saxon type potentials can be proven in the following way. (We omit the obvious " d e f " argument for simplicity.) According to Strutinsky's approach to
to
R =E
to
. . . . . + E m i c r o --O)I
--b7 ~°=0 --- I -~~m l c.r.o . . -- ~macro
17092
The first term in large brackets equals Emicro°J=° (cf. eq. (8)), while the second, for the Woods-Saxon type potentials is to a very good approximation equal to ½Jw 2. Substituting (9) into (7) gives relation (6). Finally, the excitation energy, U+°(def, T) is calculated as U°~(def, T ) = ~ n~'(def, T)e~'(def) p
(4)
where Hws is the deformed Woods-Saxon potential with the "universal" parameters as defined in ref. [ 10 ]. Using Strutinsky's ansatz of splitting the final expression into the macroscopic and microscopic terms, the total routhian satisfies [ 12 ]:
(7)
where the routhian, R% and energy, E +, are related via canonical relation and the classical rotational energy ½J¢o 2 is used explicitly (J denotes here the classical (rigid body) moment of inertia). The E~i++o term, satisfying by definition
20 October 1988
-~
nT(def, T=0)e~'(def).
In the above equations, the symbol " d e f " , if not omitted for simplicity, denotes all the deformation degrees of freedom and T is the nuclear temperature. Details of this variant of the general approach can be found in ref. [ 13 ] and references cited there. Let us begin by presenting the results for 152Dyand noticing how simple the calculated dependence of the free-routhian barrier on temperature is, cf. fig. 1 (The barrier height is defined as the elevation above the superdeformed minimum.) For all calculated spin values the barriers decrease essentially linearly when temperature increases. We recall that in l SZDythe superdeformed states were observed via discrete transitions first. Thus the results of fig. 1 may be used as a "reference" for other nuclei. Consequently, if for a given nucleus a larger potential barrier is found compared t o l S2Dy, one may expect that the changes of "trapping" this nucleus in superdeformed states are enhanced provided the elevations of the superdeBARRIERS (MeV) FOR '~DY86
Mz
,
+
+ -++ ev
Emicro
-
+)
ev
-
,
(8)
p
(10)
v
]
Ir ........
°+
i+
0
;"+
I",.
,
++
.
+
'
.
.
.
.
.
where the symbol < > denotes the Strutinsky averaging, can be rewritten as follows: 60
--
CO
Emicro -- 2 ev -- E e~ ' = ° v
+(~ eg=°- (~ eg=°))
+((~
+%%%+~
+%%%~,+
+%%%%7,
+%~%+,+
TEMPERATURE
TEMPERATURE
TEMPERATURE
TEMPERATURE
1.
e~'=°)-- ( ~ e + ) ) .
(9)
Fig. 1. The tSZDy free-energy harrier heights versus temperature for increasing spin values as indicated. The barrier heights are defined with respect to the superdeformed minima. Note the nearly linear decrease of the barriers with increasing temperature. Also, the slopes of the lines practically do not change with spin.
121
Volume 213, number 2
PHYSICS LETTERS B
f o r m e d m i n i m a in both c o m p a r e d nuclei are similar. Such a situation occurs according to our calculations for instance in 152Dy a n d ~5°Gd; the relative elevations have been discussed in ref. [ 7 ]. T r a p p e d nuclei are expected to emit fast collective 7-rays resulting in extended rotational bands built on the superdeformed shape• The simplicity of the barrier (first-saddle) dependence on t e m p e r a t u r e at a given spin holds not only for ~52Dy (fig. 1 ) but also for other nuclei in this region, and this can be utilized to present the results for m a n y nuclei in a compact way. Fig. 2 illustrates similar information using the phase diagram (T, I ) representation defined as follows. F o r each value o f spin we increase the t e m p e r a t u r e until the free-routhian barrier reduces to a small value, Blimit ( T, I). This value is arbitrarily set equal to 300 keV as representative o f a small barrier which can be penetrated by the quantal system with high probability• The conclusions of this work do not depend on this exact value chosen; setting Blimit (T, I ) = 0 will slightly move upward all the phase curves represented in fig. 2. In the figure, barriers larger than 300 keV exist for all the (T, I ) points situated between the phase curve and PHASE DIAGRAM 20 18 16 14 12 10 0.8 06 0.4 0.2 0.0
Dy ..............ijiill
N=80
the zero reference line. Using the fact that the barrier d e p e n d e n c e on t e m p e r a t u r e (fig. 1 ) can be approxim a t e d by a straight line allows us to construct, for each spin value and for each nucleus, a representation like the one in fig. 1. F o r this purpose it is sufficient to read the (T, I ) value from the phase d i a g r a m for which Blimit = 300 keV, place the corresponding point in the B versus T representation similar to that in fig. 1, and draw a straight line with the slope equal to the average slope represented in fig. 1. It becomes clear that the barrier conditions for D y nuclei with 80 ~
N-~o ....... ..' .."
/
/
.;~,__. /
_N-9. 8_
PHASE DIAGRAM
'
20 30 40 50 60 70 80 SPIN
Fig. 2. Phase diagram for the Dy isotopes indicated. The curves are composed of ( T, I) ---(temperature, spin ) points at which the barrier heights are equal to 300 keV (see text). For any given isotope, the points lying above the curves indicate the "no barrier zone" where trapping the nucleus in a superdeformed minimum is unlikely. On the contrary, ( T,/)-points below the curves indicate increasing barriers for decreasing temperature. The systematic change in the curve character with increasing neutron number deserves noting. The curves corresponding to 80
20 October 1988
2.0 1.8 >~ 1.6 1.4 1.2 1.0 0.8 ~ 0.6 F~ 0.4 0.2 0.0
Er N=84 : ' i ; ~ '
.:t # ::s
.., .~, t
.. ..-
; i~
,'z I
.........- Z/ 20
30
40
50
,:,'-" 60
70
N=88 N=90
• s~ I
....~"
.N=86 ..... • •
80
~_-_9_z _N-?4 90 100
SPIN
Fig. 3. Similar to that in fig. 2 but for the erbium nuclei. This diagram is equivalent to 360 single-frame diagrams of fig. 1, as explained in the text.
Volume 213, number 2
PHYSICS LETTERS B P H A S E DIAGRAM
2.0 1.8
~"
1.s
~
1.4
Gd N=84
1.2 E~
1.0
~_, 0.6
~
~-?_o ~.?..9.~ N-_9.4_
0.8
0.4 0.2 0.0
..... ...... .-"
,- :'
.~;-"" 20
30
40
,_#
,/./' 50
60 70 SPIN
80
90
Fig. 4. Similar to that in fig. 3 but for gadolinum nuclei (compare with figs. 2 and 3 ). Note the increasing barrier stability with respect to nuclear temperature increase, especially at lower spins. keV for erbium nuclei, they survive up to the T ~ 800 keV range in G d and Sm isotopes at •=60 (cf. figs. 2 - 5 ) . Results for n e o d y m i u m nuclei are similar to those for samarium. In conclusion, the calculations presented here may lead to important implications concerning the population of the superdeformed bands. The following qualitative but systematic predictions related to the first saddle-point energies can be made. ( 1 ) The barriers grow when the proton n u m b e r decreases from Z = 68 down to Z = 60 for practically the whole spin range. Thus the trapping in the superdeformed m i n i m u m should be more favorable in rareearth nuclei o f lower Z. O f course, the population of the superdeformed structure depends on the actual PHASE DIAGRAM 2.0 1.8 " ~ 1.6 ~-~ 1.4
Sm N=84
~-.~8 ~ ~
0.8 0.6 0.4 0.2 0.0
......
.." ........ I " ;';" ;i,i I ,/i
•<::;5.." ,/,,~
20
30
40
.N=9o ..... N=92
~_=._9._4
50
60
70
80
90
100
SPIN
Fig. 5. Similar to that in fig. 3 but for samarium nuclei (representative qualitatively also for neodymium nuclei with the same neutron numbers).
20 October 1988
level density corresponding to this and to the competing minima, which in turn depend on relative elevations of the minima (cf. ref. [ 7 ] for the latter). (2) Since the barriers are predicted to be systematically larger in nuclei of lower Z, also at low spins, the decay out of the superdeformed bands in Z ~ 5862 isotones should occur at spins systematically lower as compared to 152Dy" This conclusion applies for superdeformed bands at a relatively broad temperature
range. (3) Unless the increased pairing correlations act to increase the barrier penetrabilites at low frequencies (cf. refs. [ 13,14 ] ) the trend mentioned above should apply as well at low spins and low-temperature limit ( T - , 0 ) . This possible increase can be viewed as a result of the following WKB estimate of the barrier penetrability factor: P ~ e x p { ( 2 / h)fAn~/2M[ V(x)-E]dx] where the effective inertia satisfies [ 13 ]
g. (An 2
m~~-
+
g, (An 2
Z t.A,S},,'
(11)
with g= and g= being proportionality constants, A~,= the pairing gaps and (An/Afl),,, the numbers of single-particle level crossings under the barrier stretch Aft. For 3~,~ increasing at low frequencies the penetrability P may increase dramatically. (4) It is visible from eq. ( 11 ) that the probability of trapping the superdeformed nucleus depends on the structural differences between the superdeformed and normal states (the number of level crossings in terms of the mean field approximations, (An/ Aft) ). Qualitative estimates of this factor can be done in the context o f the present article using e.g. diagram 3 ofref. [15]. (5) Predictions ( 1 ) - (3) may also be influenced systematically if other characteristics of the barriers (e.g. barrier thickness) varied significantly from a nucleus to nucleus. This does not seem to be the main factor for Z ~ 62-66, N ~ 84-88 nuclei where the superdeformations change relatively slowly, but for quantitative estimates further studies are needed. It is therefore suggested that the experimental search for superdeformed bands in Z < 62 nuclei may lead to important new information on nuclear behavior in those exotic configurations. The Joint Institute for Heavy Ion Research has as 123
Volume 213, number 2
PHYSICS LETTERS B
member institutions the University of Tennessee, Vanderbilt University, and the Oak Ridge National L a b o r a t o r y ; it is s u p p o r t e d b y t h e m e m b e r s a n d b y the Department of Energy through Contract Number DE-AS05-76ER0-4936 with the University of Tennessee. This work was supported in part by the Polish Ministry of Science and Education within the p r o j e c t C P B P 01.09.
References [ 1 ] G.E. Rathke, R.V.F. Janssens, M.W. Drigert, I. Ahmed, K. Beard, V. Garg, M. Hass, T.L. Khoo, H.J. K6rner, W.C. Ma, S. Pilotte, P. Taras and F.L.H. Wolfs, to be published. [2] P.J. Twin, B.M. Nyako, A.H. Nelson, J. Simpson, M.A. Bentley, H.W. Cranmer-Gordon, F.D. Forsyth, D. Howe, A.R. Mokhtar, J.D. Morrison, J.F. Sharpey-Schafer and G. Sletten, Phys. Rev. Lett. 57 (1986) 811. [ 3 ] M.A. Deleplanque, C. Beausang, J. Burde, R.M. Diamond, J.E. Draper, C. Duyar, A.O. Machiavelli, R.J. McDonald and F.S. Stephens, submitted for publication. [4] B. Haas, P. Taras, S. Flibotte, F. Banville, J, Gascon, S. Cournoyer, S. Monaro, N. Nadon, D. Prevost, D. Thibault, J.K. Johansson, D.M. Tucker, J.C. Waddington, H.R.
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Andrews, G.C. Ball, D. Horn, D.C. Radford, D. Ward, C.St. Pierre and J. Dudek, Phys. Rev. Left. 60 (1988) 505. [5]M.A. Bentley, G.C. Ball, H.W. Cranmer-Gordon, P.D. Forsyth, D. Howe, A.R. Mokhtar, J.D. Morrison, J.F. Sharpey-Schafer, P.J. Twin, B. Fant, C.A. Kalfas, A.M. Nelson, J. Simpson and G. Sletten, Phys. Rev. Lett. 59 (1987) 2141. [ 6 ] I. Ragnarsson, T. Bengtsson, G. Leander and S. Aberg, Nucl. Phys. A 347 (1980) 287. [ 7 ] J. Dudek and W. Nazarewicz, Phys. Rev. C 31 ( 1985 ) 298. [ 8 ] J. Dudek, Proc. Intern. Winter Meeting on Nuclear physics (Bormio, Italy, 1987), ed. I. Iori (Milan, 1987). [9] V.M. Strutinsky, Soy. J. Nucl. Phys. 45 (1987) 1117. [10]S. Cwiok, J. Dudek, W. Nazarewicz, J. Skalski and T. Werner, Comput. Phys. Commun. 46 (1987) 379, and references therein. [ 11 ] J. Dudek, W. Nazarewicz and N. Rowley, Phys. Rev. C 35 (1987) 1489. [12]A.V. Ignatyuk, I.M. Mikhailov, L.H. Molina, R.G. Nazmitdinov and K. Pomorski, Nucl. Phys. A 346 (1980) 191. [13] J. Dudek, Proc. Intern. Conf. on Nuclear shapes (Crete, Greece, I987), ed. J. Garrett. [14]B. Herskind, B. Lauritzen, K. Schiffer, R.A. Broglia, F. Barranco, N. Gallardo, J. Dudek and E. Vigezzi, Phys. Rev. Lett. 59 (1987) 2416. [ 15 ] J. Dudek, W. Nazarewicz, Z. Szymanski and G.A. Leander, Phys. Rev. Len. 59 (1987) 1405.