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Statistical fluctuations and first-order phase transitions in heated rotating nuclei
Volume 131B, n u m b e r 1,2,3
STATISTICAL FLUCTUATIONS
PHYSICS L E T T E R S
10 N o v e m b e r 1983
AND FIRST-ORDER PHASE TRANSITIONS
IN H E A T E D R O T A T I N G N U C L E I A l a n L. G O O D M A N ~ Physics Department and Quantum Theory Group, Tulane University, New Orleans, LA 70118, USA Received 20 June 1983 For any temperature below 0.20 MeV, the finite-temperature H a r t r e e - F o c k - B o g o l i u b o v cranking theory predicts a sharp first-order phase transition in the i13/2 model. However when statistical fluctuations in the pair gap are included, these phase transitions are s m o o t h e d out when the temperature is as low as 0.05-0.10 MeV.
Many rare-earth nuclei have yrast lines which backbend. That is, the zero-temperature equation of state to(I), where to is the rotational frequency and I is the spin, displays the cubic behavior which is the classical signature of a first-order phase transition. Recent calculations [1,2] with the finite-temperature H a r t r e e - F o c k Bogoliubov cranking ( F T H F B C ) theory [3-5] show that the isothermal equation of state to(I) retains the backbending characteristic up to a critical t e m p e r a t u r e To. Consequently increasing to while T is fixed induces a sharp first-order phase transition as long as T < To. H o w e v e r since nuclei are finite systems, one should consider whether statistical fluctuations smooth out these phase transitions. It should be emphasized that for any finite system with a given temperature, statistical fluctuations would exist even if one knew the exact wave functions and eigenenergies of the system. These fluctuations do not result from the approximate nature of the F T H F B C wave functions. For a given T and to the F T H F B C theory minimizes the free energy in a rotating frame F' = E -
T S - o21.
(1)
We choose the pair gap A as the order p a r a m e t e r of the system. Since the F F H F B C theory neglects statistical fluctuations, at each T and to the system is considered to have a sharp 1 Supported in part by the U S National Science Foundation.
value of A, which is the value that minimizes F'(A) at the given T and to. Now let us go beyond F T H F B C by permitting statistical fluctuations in A. Then the relative probability that the system has a given value of A is [6] P(A) =
exp[-F'(a)/kT].
(2)
So P(A) is maximized where F'(A) is minimized To calculate p ( a ) for a given T and to, construct a set of density operators of the F T H F B C class. For each density operator calculate F ' and h. This generates the function F'(A), and consequently the function P(A). As an illustration consider a system of six neutrons in the i13/2 shell. The hamiltonian contains single-particle energies appropriate to an axially symmetric potential with/3 = +0.24 plus a m o n o p o l e pairing term. For this system the critical t e m p e r a t u r e k T c is 0.20 MeV. This implies that for any T < Tc there exists a critical frequency toe(T) for which F'(A) has two degenerate minima. Examples are shown in fig. 1. Observe that the top two curves each have two degenerate minima, one with A = 0 (normal state) and one with A ¢ 0 (superfluid state). These curves also contain one relative maximum. The three extrema on each curve are self-consistent solutions of the F T H F B C equations corresponding to different I at the same T and toc. If to is reduced below toc, then the superfluid minimum
0 031-9163/83/0000-0000/$03.00 © 1983 North-Holland
5
Volume 131B, number 1,2,3 "10"44 /
I
I
PHYSICS L E T T E R S I
I
I
!
10 November 1983 I
I
I
I
I
I
i
1
I
I
A: T 0.02,6,2c=0.~57
-10.46
L
D
1.0
T = O.02,(-Oe = 0 . 1 5 7
B: T - 0 . 0 5 , c o c -
0.~57
C: T = 0.10.coo - 0 . 1 5 6 O: T = 0.20,COc = 0.1505
C
0.8 -10.48 0.6
P
-10.50
j=13/2
0.4
N=6 K =2.0 0~2
-10.52
G = 0r448
0,2
-lO.54 -
0.6
0.8A1.0
1,2
i
I
I
1.4
1.6
1.8
2.0
Fig. 2. The relative probability P versus the pair gap & for various temperatures T and critical rotational frequencies ~oc. See fig. 1.
-10.56
-10.58
-10.60
j=13/2 T:
/
0.20,OJc =0.1
N: 6
505
K= 2.0
-10.62 -10.64
0.4
T = 0.10,r.~c = 0 . 1 5 6
G = 0.448 --
0
I 0.2
I o.4
1 0.6~
I 0.8
I
1.0
I 1.2
1.4
Fig. 1. Free energy in a rotating frame F ' versus pair gap A for various temperatures T and critical rotational frequencies tot. The quantities F', A, K, (3, k T and h~c have units of MeV.
drops below the normal minimum. If co is increased above co¢, then the normal minimum becomes lower than the superfluid minimum. Hence precisely at co¢ the absolute minimum jumps from one potential well to the other, producing a discontinuity in the order p a r a m e t e r A. This is the signature of a sharp first-order phase transition. At the critical point kT~ = 0.20 MeV, hco~ = 0.1505 MeV, the three extrema merge, and F'(A) is flat over a wide range of A. For T > To, F'(A) contains only one minimum, and the phase transition induced by rotations is secondorder. The statistical fluctuations in the order p a r a m e t e r A are calculated with eq. (2) and the energy surfaces of fig. 1. The result p(A) is shown in fig. 2. For a t e m p e r a t u r e of 0.02 MeV, p(A) has two well-defined peaks at values of A given by the minima of fig. 1. Between these
two peaks the relative probability falls to a minimum of 0.05. Consequently for k T = 0.02 MeV, the barrier in F'(A) which separates the two minima is sufficiently high so that statistical fluctuations do not wash out the effect of the energy barrier. If k T is reduced below 0.02 MeV, then the two peaks in P(&) become narrower. As T ~ 0, the statistical fluctuations disappear, and the width of both peaks in p(A) approaches zero. For k T = 0.05 MeV, the two peaks in p(A) b e c o m e wider and the relative minimum in p(A) increases to 0.35. For k T = 0.10 MeV, the two peaks in p(A) are barely discernible. The relative probability P that the system occupies the relative m a x i m u m in F ' is 0.78. Therefore the fluctuations in A severely reduce the effect of the barrier in F ' which separates the two degenerate minima. From eq. (2) and fig. 1 it is seen that there are two reasons for this p h e n o m e n o n . First the barrier in F ' decreases in height as the t e m p e r a t u r e increases. Second even if the barrier height were independent of the temperature, the effect of the barrier would be diminished by increasing the temperature. At the critical point kTc = 0.20 MeV, h~oc = 0.1505 M e V the system has an equal probability P for A to be anywhere from 0 to 0.7 MeV. This illustrates the maxim that a system at a critical point is characterized by unusually large fluctuations in the order parameter. From figs. 1 and 2 we may draw the following conclusions. In the absence of statistical
Volume 131B, number 1,2,3
PHYSICS LETTERS
fluctuations, rotations induce a sharp first-order phase transition for any temperature below 0.20 MeV. However when statistical fluctuations in A are included, the potential barrier in F ' which separates the degenerate minima is penetrated. Then the phase transition from one minimum to the other is no longer sharp, but is considerably smoothed out. For temperatures as low as 0.05--0.10 MeV, the effect of the potential barrier is severely reduced. Therefore it may be improper to speak of a first-order phase transition for temperatures between 0.10 MeV and 0.20 MeV. For a non-rotating i13/2 system, increasing the temperature in the F T H F B C approximation induces a sharp second-order phase transition
10 November 1983
from superfluid to normal. However including fluctuations in A smooths out this transition. This result was first found by Moretto for the non-rotating uniform model [7].
References [1] A.L. Goodman, Nucl. Phys. A352 (1981) 45. [2] A.L. Goodman, Nucl. Phys. A369 (1981) 365. [3] M. Sano and M. Wakai, Prog. Theor. Phys. 48 (1972) 160. [4] A.L. Goodman, Nucl. Phys. A352 (1981) 30. [5] K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys. A357 (1981) 20. [6] E.M. Lifshitz and L.P. Pitaevskii, Statistical physics, Vol. 1 (Pergamon, Oxford, 1980) p. 472. [7] L.G. Moretto, Phys. Lett. 40B (1972) 1.
STATISTICAL FLUCTUATIONS
PHYSICS L E T T E R S
10 N o v e m b e r 1983
AND FIRST-ORDER PHASE TRANSITIONS
IN H E A T E D R O T A T I N G N U C L E I A l a n L. G O O D M A N ~ Physics Department and Quantum Theory Group, Tulane University, New Orleans, LA 70118, USA Received 20 June 1983 For any temperature below 0.20 MeV, the finite-temperature H a r t r e e - F o c k - B o g o l i u b o v cranking theory predicts a sharp first-order phase transition in the i13/2 model. However when statistical fluctuations in the pair gap are included, these phase transitions are s m o o t h e d out when the temperature is as low as 0.05-0.10 MeV.
Many rare-earth nuclei have yrast lines which backbend. That is, the zero-temperature equation of state to(I), where to is the rotational frequency and I is the spin, displays the cubic behavior which is the classical signature of a first-order phase transition. Recent calculations [1,2] with the finite-temperature H a r t r e e - F o c k Bogoliubov cranking ( F T H F B C ) theory [3-5] show that the isothermal equation of state to(I) retains the backbending characteristic up to a critical t e m p e r a t u r e To. Consequently increasing to while T is fixed induces a sharp first-order phase transition as long as T < To. H o w e v e r since nuclei are finite systems, one should consider whether statistical fluctuations smooth out these phase transitions. It should be emphasized that for any finite system with a given temperature, statistical fluctuations would exist even if one knew the exact wave functions and eigenenergies of the system. These fluctuations do not result from the approximate nature of the F T H F B C wave functions. For a given T and to the F T H F B C theory minimizes the free energy in a rotating frame F' = E -
T S - o21.
(1)
We choose the pair gap A as the order p a r a m e t e r of the system. Since the F F H F B C theory neglects statistical fluctuations, at each T and to the system is considered to have a sharp 1 Supported in part by the U S National Science Foundation.
value of A, which is the value that minimizes F'(A) at the given T and to. Now let us go beyond F T H F B C by permitting statistical fluctuations in A. Then the relative probability that the system has a given value of A is [6] P(A) =
exp[-F'(a)/kT].
(2)
So P(A) is maximized where F'(A) is minimized To calculate p ( a ) for a given T and to, construct a set of density operators of the F T H F B C class. For each density operator calculate F ' and h. This generates the function F'(A), and consequently the function P(A). As an illustration consider a system of six neutrons in the i13/2 shell. The hamiltonian contains single-particle energies appropriate to an axially symmetric potential with/3 = +0.24 plus a m o n o p o l e pairing term. For this system the critical t e m p e r a t u r e k T c is 0.20 MeV. This implies that for any T < Tc there exists a critical frequency toe(T) for which F'(A) has two degenerate minima. Examples are shown in fig. 1. Observe that the top two curves each have two degenerate minima, one with A = 0 (normal state) and one with A ¢ 0 (superfluid state). These curves also contain one relative maximum. The three extrema on each curve are self-consistent solutions of the F T H F B C equations corresponding to different I at the same T and toc. If to is reduced below toc, then the superfluid minimum
0 031-9163/83/0000-0000/$03.00 © 1983 North-Holland
5
Volume 131B, number 1,2,3 "10"44 /
I
I
PHYSICS L E T T E R S I
I
I
!
10 November 1983 I
I
I
I
I
I
i
1
I
I
A: T 0.02,6,2c=0.~57
-10.46
L
D
1.0
T = O.02,(-Oe = 0 . 1 5 7
B: T - 0 . 0 5 , c o c -
0.~57
C: T = 0.10.coo - 0 . 1 5 6 O: T = 0.20,COc = 0.1505
C
0.8 -10.48 0.6
P
-10.50
j=13/2
0.4
N=6 K =2.0 0~2
-10.52
G = 0r448
0,2
-lO.54 -
0.6
0.8A1.0
1,2
i
I
I
1.4
1.6
1.8
2.0
Fig. 2. The relative probability P versus the pair gap & for various temperatures T and critical rotational frequencies ~oc. See fig. 1.
-10.56
-10.58
-10.60
j=13/2 T:
/
0.20,OJc =0.1
N: 6
505
K= 2.0
-10.62 -10.64
0.4
T = 0.10,r.~c = 0 . 1 5 6
G = 0.448 --
0
I 0.2
I o.4
1 0.6~
I 0.8
I
1.0
I 1.2
1.4
Fig. 1. Free energy in a rotating frame F ' versus pair gap A for various temperatures T and critical rotational frequencies tot. The quantities F', A, K, (3, k T and h~c have units of MeV.
drops below the normal minimum. If co is increased above co¢, then the normal minimum becomes lower than the superfluid minimum. Hence precisely at co¢ the absolute minimum jumps from one potential well to the other, producing a discontinuity in the order p a r a m e t e r A. This is the signature of a sharp first-order phase transition. At the critical point kT~ = 0.20 MeV, hco~ = 0.1505 MeV, the three extrema merge, and F'(A) is flat over a wide range of A. For T > To, F'(A) contains only one minimum, and the phase transition induced by rotations is secondorder. The statistical fluctuations in the order p a r a m e t e r A are calculated with eq. (2) and the energy surfaces of fig. 1. The result p(A) is shown in fig. 2. For a t e m p e r a t u r e of 0.02 MeV, p(A) has two well-defined peaks at values of A given by the minima of fig. 1. Between these
two peaks the relative probability falls to a minimum of 0.05. Consequently for k T = 0.02 MeV, the barrier in F'(A) which separates the two minima is sufficiently high so that statistical fluctuations do not wash out the effect of the energy barrier. If k T is reduced below 0.02 MeV, then the two peaks in P(&) become narrower. As T ~ 0, the statistical fluctuations disappear, and the width of both peaks in p(A) approaches zero. For k T = 0.05 MeV, the two peaks in p(A) b e c o m e wider and the relative minimum in p(A) increases to 0.35. For k T = 0.10 MeV, the two peaks in p(A) are barely discernible. The relative probability P that the system occupies the relative m a x i m u m in F ' is 0.78. Therefore the fluctuations in A severely reduce the effect of the barrier in F ' which separates the two degenerate minima. From eq. (2) and fig. 1 it is seen that there are two reasons for this p h e n o m e n o n . First the barrier in F ' decreases in height as the t e m p e r a t u r e increases. Second even if the barrier height were independent of the temperature, the effect of the barrier would be diminished by increasing the temperature. At the critical point kTc = 0.20 MeV, h~oc = 0.1505 M e V the system has an equal probability P for A to be anywhere from 0 to 0.7 MeV. This illustrates the maxim that a system at a critical point is characterized by unusually large fluctuations in the order parameter. From figs. 1 and 2 we may draw the following conclusions. In the absence of statistical
Volume 131B, number 1,2,3
PHYSICS LETTERS
fluctuations, rotations induce a sharp first-order phase transition for any temperature below 0.20 MeV. However when statistical fluctuations in A are included, the potential barrier in F ' which separates the degenerate minima is penetrated. Then the phase transition from one minimum to the other is no longer sharp, but is considerably smoothed out. For temperatures as low as 0.05--0.10 MeV, the effect of the potential barrier is severely reduced. Therefore it may be improper to speak of a first-order phase transition for temperatures between 0.10 MeV and 0.20 MeV. For a non-rotating i13/2 system, increasing the temperature in the F T H F B C approximation induces a sharp second-order phase transition
10 November 1983
from superfluid to normal. However including fluctuations in A smooths out this transition. This result was first found by Moretto for the non-rotating uniform model [7].
References [1] A.L. Goodman, Nucl. Phys. A352 (1981) 45. [2] A.L. Goodman, Nucl. Phys. A369 (1981) 365. [3] M. Sano and M. Wakai, Prog. Theor. Phys. 48 (1972) 160. [4] A.L. Goodman, Nucl. Phys. A352 (1981) 30. [5] K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, Nucl. Phys. A357 (1981) 20. [6] E.M. Lifshitz and L.P. Pitaevskii, Statistical physics, Vol. 1 (Pergamon, Oxford, 1980) p. 472. [7] L.G. Moretto, Phys. Lett. 40B (1972) 1.