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Landau-Ginzburg theory of phase transitions in heated rotating nuclei
Nuclear Physics A406 (1983) 94-108 0 North-Holland Publishing Company
LANDAU-GINZBURG THEORY OF PHASE TRANSITIONS IN HEATED ROTATING NUCLEI ALAN L. GOODMANi Physics Department and Quantum Theory Group, Tulane University, New Orleans, Louisiana 10118, USA
Received 1 February 1983 (Revised 5 April 1983) Abstract: The L~dau-Ginsburg theory is appIied to heated rotating suited systems. It is demonstrated that the free energy F’ is an even function of the order parameter A even when external fields are present. The energy surfaces F’(d) are calculated for various rotational frequencies and temperatures. These surfaces provide clear illustrations of first-order and secondorder phase transitions, as well as a critical point.
1. Introduction
The Landau-Ginzburg (LG) theory 1-4) is a mean-field theory which describes a wide variety of phase transitions. The essential idea is to express the free energy of a system as a power series in the order parameter. Knowledge of the signs of the first few coefficients. in this expansion determines whether the system will experience a phase transition as well as the order of the transition. The LG theory constructs the free energy as a function of the order parameter. These energy surfaces have minima and maxima, which give the values of the order parameter when the system is in therma equilibrium. Points on these surfaces which are not extrema are not equilibrium states of the system. This is similar to those nuclear collective models which construct potential energy surfaces in deformation space. The deformation j3 is an order parameter for rotations. The LG theory will be applied to heated rotating nuclei in this article. The order parameter is chosen as the pair gap A. For each rotational frequency COand temperature T, the energy surface F’(d) will be constructed, where 8“ is the free energy in a rotating frame. These surfaces will provide vivid illustrations of firstorder and second-order phase transitions, as well as critical points. The LG theory supplements previous descriptions 5*6) of phase transitions in heated rotating nuclei which refer only to equilibrium states and not to energy surfaces.
7 Supported in part by the US National Science Foundation. 94
A. L. Goodman / Landau-Ginzburg theory 2.
Landau-Giuzburg
For a system at constant temperature
95
theory of rotating nuclei
and volume the Helmholtz free energy,
F = E-TS,
(2.1)
where S is the entropy, is minimized. The LG theory expands F in powers of some order parameter u:
WI, T) = :
C,(T)y"
(2.2)
n=O
where N is 4 for second-order transitions and 6 for first-order transitions. If there are no external fields, then generally C, = 0 for odd IZ.The condition for a secondorder transition at To is that near To
C,(T) = C-T,),
(2.3)
where a is a positive constant, and C, > 0. The condition for a first-order transition is that C, < 0 and C, > 0. A critical point occurs at T, if C,(T,) = C,(T,) = 0 and C6 > 0. At each temperature the equilibrium value of yeis determined by (aF/iYr]), = 0. If there is an external field, then the expansion (2.2) generally contains a linear term. For example, consider a system at constant temperature and volume in the presence of an external magnetic field H. Then, the magnetic Gibbs free energy, G = E-TS-HM,
where M is the magnetization, LG expansion is
is minimized. The order parameter
G(M, T, H) = c CJT, H)M”. n
(2.4)
is M, and the (2.5)
It is clear from eq. (2.4) that the expansion (2.5) should include the term which is linear in M. For a nucleus with a given temperature and rotational frequency, the Helmholtz free energy in a rotating frame, F’ = E-
TS-ol,
(2.6)
where I is the angular momentum, is minimized. The usual analogy between the effect of magnetic fields on superconductors and the effect of rotations on nuclei
96
A. L. Goodman / Landau-Ginzburg theory
suggests that the order parameter should be chosen as I. Then, the LG expansion F’(f, T, 0) =
c c&r,
w)I”,
II
(2.7)
would contain the term which is linear in I. However, the order parameter is here chosen as the pair gap A, rather than the spin I. Then, the LG expansion is F’(A, T, Co)=
c C,(T,
0)A”.
(2.8)
n
Does this expansion contain odd powers of A? When a nucleus is not rotating, the pair field is time-reversal even. However, rotations induce a time-reversal odd component in the pair field ‘). So it might be expected that F’ contains odd powers of A. It will now be demonstrated that F’ is an even function of A: F’( - A) = F’(A),
(2.9)
so that there are no odd powers in the expansion (2.8). Consider the HFB equation 8,9) ~cX,=L: EiX,,
(2.10) (2.11)
where X is the Hartree-Fock (HF) hamiltonian in a rotating frame and Ei is the quasiparticle energy. Define the unitary transformation
A=
1
0
( > 0
e-i+
’
(2.12)
where Cpis an arbitrary phase angle. From eq. (2.10) it follows that (Ax-4
1)(AX,) = E,(AX,).
(2.13)
Substitute eq. (2.12) into eq. (2.13), with the result that (2.14)
If A produces the quasiparticle transformation (Y, 6) and quasiparticle energy E, then e’@bdleads to the transformation (Vi, e-l@vi) and the same quasiparticle
energy. Treating the phase a&e
4 as a parameter, we define d(#) =
At finite temperature
el”d(O),
(2.15)
V+> = U(O),
(2.16)
V(f$) = e-W(O),
(2.17)
K(4) = NO)”
(2.18)
tke FIF density p and the pair density r are 10-12) p = ~jx*-i_ t =
V(l -f,P,
(2.19f
Dfv*+ V’(1 -f)U,
(2.20)
where (2.21) and ,B= l/kT. From eqs. (2.16)-(2.21) it follows that (2.22) r(f#J)= e%{O).
(2,231
The internal energy E = (H) contains terms with pp and l*t, so that
E(4) = E(O).
(2.24)
The entropy onfy contains the quasiparticle energies (2.18), and the spin is linear in Q, so that
Consequently, the free energy in a rotating frame (2.6) is invariant with respect to the transformative (2.15) F’(c)) = F(O),
(2.27)
or Ff@LI > = F(d )*
(2.2s)
98
) ~~~~-~i~z~~r~
A. L. ~~~~
ffimy
For 6, = 7~ eq. (2.28) reduces to eq. (2.9)$ which proves that F’(d j is an even function. The cssentid point in the derivation is that F’ contains no terms which are linear in the pair density 1. From the definition of the pair potential, (2.29)
it follows that eqs. (U5) and (2.23) are consistent, so the traRsformatio~ A of eq. (2.12) provides a self-consistent symmetry. The HFB equation determines the order parameter and free energy when the system is in ~~li~~b~urn. However, the LG expansion cakufates F’ at all valaes of A, not just the ~quilibr~~~mvalues. Nevertheless, the co~clu~~o~ that F’ is even in d remains valid for any value of 8. Consider any general 3ogoliubo~~ transformation
Us=I
t:
(U,jCf
+
j
f$jCj),
(2.30)
and any set of quasiparticle energies E+ The numbers U, I/ and E need not be a self-consistent solution of the HFB equation. From U, V and E one can calcula$e F and A. Next, apply the transformation ~2.1~)-~2.~~) to U, V and E with (b = 7~. Then, recalculate F’ and d, As described above, f;” is invariant and d changes sign. So F’ is even in d for any density operator of the HFB &ass. Therefore the LG expansioa f2.8) is
F’th T, ~1 = C,(T,O)+C,(T,W)~~-~_~,(II‘,~)~~+~,~~,~)~~~
(2.31)
As usual the expansion is truncated at sixth order. For each temperature and rotational frequency, the equilibrium value of d is determined by the condition
The HFB theory considers a system maintaked at constant r, o and & where p is the chemical potential. Then, the minimized quantity is not F:, but the grand ~ot~n~~~~in a rotating frame 52 = F’-jLN,
(2.33)
where N is the particle number. Therefore one should expand Q’ in even powers of d: (234)
99
As an illustration of the LG theory of heated rotating systems, we consider the two-level model of Krumlinde aad Szymanski with no rotor 13714),In this model the chemical potential is zero, so that f2’ = F’. The procedure is as follows: Choose a density operator, and use it to calculate F’ and d, Repeat for many choices of the density operator. The function F’(d) is thereby mapped out. In the WFB a~~~o~~ma~on the many-bay density operator is
where the quasiparticle number operator is
and Q’ is defined by eq. (2.30). The qu~si~artjc~e occn~ation ~r~b~~i~i~y is fi. Any unitary transfo~a~on U, 9’ and any J such that 0 S d 5 1, define a possible HFB density operator. The choice U, r&i,j’ need not be the se~~cons~stent m~ni~nm which solves the HFB equation. The fo~~o~~~~~algorithm is used to construct a set of density operators: First, choose values for 7’ and CO.Second, in the basis 1, 2, 3, 4 of fig. 1, choose U, V and f its
o-
0 0
0
B0
4
3,
0
0
A-
f;+
-B+
0
-B_
1 p
f3.3)
(3.5)
100
A. L. Good~u~ / ~duu-Gi~zburg
theory
where
Pil = $2 -&I+, f*
E, =
(3.7)
= [l +eaE+)-L,
(3.8)
[E’ i- (S&&O)~]“,
(3.9)
and @= 1,&T, E is the single-particle splitting shown in fig. 1, A, and B, are positive, and B_ is positive when 6 > $0 and negative when 6 < fo. For a given I, T, trf then U, V, f are determined by the arbitrary constant 6, which ranges from zero to infinity. So the density operator DnFB(6, T, ~0) is a continuous function of the parameter 6. Third, for each 6, T, w use DWFBto calculate d and F’. The results are A@, T, cc))= jGQ
F
1 +
6-k) tanh &?E+)+ -$--
1
tanh (#E _ ) ,
F’(6, T,w) = E-TS-cd,
(3.1 I)
where E(6, T, co) = --Lk2
s(kT,~)=
tanh (#E+) E
(3.10)
+ tanh ($E_) E-
+
1--
A2
(3.12)
G’
(3.13)
-2kQr_CE [flnJ+(l-j)In(l-f)], f
tanh (#E,)
6-gf3
- r
tanh (#E_)
1 ,
(3.14)
and G is the number of identical sets of four states shown in fig. 1. For a given 6, T, w eqs. (3.10) and (3.11) produce one point in the F’(A) plane. By varying 6, T, w the function F’(d, T, o) is mapped out. In general, 6 is not equal to A. Then, the state DWFB(8,T, o) is not a state of thermal equilibrium and it is not a solution of the HFB equation. However, for the few choices of 6 where 6 is equal to A, then DHFBis an equilibrium solution of the HFB equation. Also, it is only when 6 = A that E, can be interpreted as a quasiparticle energy. Figs. 2-6 depict the free energy in a rotating frame versus the pair potential for
101
kT=O
-14
G=O.l ,Q=lO c=o.t
II 0
I
0.2
II
I
0.4
11
0.6
I
I
0.8
L!!A
Fig. 2. Free energy in the rotating frame I? versus the pair gap A for various rotational frequencies CO. The temperature T is zero. The dots identify the minima and maxima. The quantities F’, A, Aw, kT, G and E have units of MeV.
various rotational frequencies and temperatures. As an example of a first-order phase transition, consider fig. 2, which shows F’(d) at T = 0. At o = 0 the equilibrium state is given by the ~nimum at d = 0.995 MeV. The normal state at d = 0 is a maximum, and is therefore a state of unstable equilibrium. A large potential barrier separates the superfluid and normal states. As 03 increases, this barrier decreases. At hw = 0.8 MeV the state at d = 0.092 MeV is still the absolute minimum. The normal state at d = 0 changes from a maximum to a relative minimum. There is a new maximum at d = 0.375 MeV. So at ho = 0.8 MeV there are three extrema, and each gives a self-consistent solution to the HFB equation. For fixed T, CLIthe maximum is unstable, and it describes a state in the middle of the backbending region. It should be emphasized that this instability is in the function F’(d, T, 0). Chu et al. have shown that for the function F(d, T, I) the instability disappears IS). When tie is increased to 1.0 MeV, the superffuid state at
A. L. Goodman / Landau-Ginzburg theory
102
A = 0.989 MeV and the normal state at A = 0 are degenerate. At hw = 1.2 MeV the normal state becomes the absolute minimum. Consequently, when hw is increased past 1.0 MeV the equilibrium state discontinuously jumps from A = 0.989 MeV to A = 0. Whenever there is a discontinuous change in the order parameter caused by an infinitesimal change in an intensive variable, such as o, the phase transition is first order. Finally, at ho = 1.6 MeV the superfluid extrema disappear, and only the normal state remains. To summarize the lesson of fig. 2, if F’(A) contains two minima, and if increasing w shifts the absolute minimum from one of these potential wells to the other, then there is a first-order phase transition. Fig. 3 shows F’(A) at kT = 0.2 MeV. The qualitative features are the same as at T = 0. However, raising the temperature from 0 to 0.2 MeV substantially reduces the barrier at III = 0 between the superfluid state and the normal state, and almost eliminates the barrier at ho = 1.0 MeV between the superfluid minimum and the superfluid maximum. Yet the phase transition is still first order. Fig. 4 shows the potential surfaces at kT = 0.3 MeV. These surfaces contain only one minimum. Therefore the transition from superfluid to normal is not first order at this temperature. For Ro = 0.9 MeV the potential surface is flat over the range 0 < A < 0.7 MeV. This is the characteristic feature of a critical point, which is a point on a phase equilibrium line ~(7’) which divides the line into two segments, one corresponding to first-order transitions and the other to second-
0
I
I
0.2
I
I
0.4
I
n
I
0.6
I
I
0.8
I
Fig. 3. See fig. 2. The temperature is 0.2 MeV/k.
J
1.0
103
order transitions. The critical point is kT, = 0.3 MeV, &co= 0.9 MeV. At this point the mean-field approximation should be supplemented by Iarge fluctuations in d. The potential surfaces at kT = 0.4 MeV are given in fig. 5. As the rotational frequency is increased from zero, the minimum shifts continuously toward the left, so that the equilibrium value of d decreases continuously from 0.703 MeV to 0. This continuous decrease of the order parameter is characteristic of a second-order phase transition. Fig. 6 shows that at kT = 0.5 MeV the normal state is the absolute minimum even when the system is not rotating. There is no su~~u~dity at this te.m~ra~ure_
Fig. 5. See fig. 2 T%e ttxnperatur~ is 0.4 MeV/k.
104
-18 0
I
I 0.2
I
II 0.8
1.0
Fig. 6. See fig. 2. The temperature is 0.5 MeV/k.
Consider heating the non-rotating system at o = 0. Figs. 2-6 show that the equilibrium value of d decreases continuously from 0.995 MeV to 0 as the temperature is increased from zero. There is only one minimum in the potential surface. So heating the non-rotating super~uid induces a second-order phase transition. The LG expansion of eq. (2.31) will now be applied to the potential surfaces in figs. 2-6. Each curve has a given T, w. The function F’(d) =
C, + C,d2+ C4A4 -t C6d6,
(3.15)
is fit to each curve in figs. 2-6, where Co, C,, C,, C, are determined by a leastsquares fit. Higher-order terms are not needed since these curves contain no more than two inflection points. First, consider the system at zero temperature. The LG coefficients C,(o), C,(m), C,(o) are given in fig. 7. Observe that Cz changes sign at Ao, = 0.55 MeV, so that near o0 C,(w) ‘1: &J-w,),
(3.16)
where a is a positive constant, and C, < 0, Cs > 0. These are the LG conditions for a first-order phase transition. Recall that the transition occurs at ho, = 1.0 MeV, so that o0 < 0,. The functions C,(w), C,(W), C,(o) are almost linear for ho > 0.4 MeV. For the temperature kT = 0.3 MeV the LG coefficients are shown in fig. 8. Again Cz changes sign and C, < 0, C, > 0. However, for the flat energy surface at
A. L. Goodman / Landau-Ginzburg
I
24.
’
’
p
I
105
theory
’
-
20 16
12.
-24
-32 R=lO
-36 -400
0.2
0.4
0.6
0.8
1.0 1.2 1.4 1.6
fiw Fig. 7. The Landau-Ginzburg coefficients C,, C, and C6 versus the rotatlonal frequency w. The temperature T is zero. The coefficient C, has units of (MeI’)‘-“. The quantities hw, kT, G and E have units of MeV.
-'*O
0.2
0.4 0.6
Fig. 8. See fig.7. The
0.8 tiw
1.0 1.2 1.4
temperature is 0.3 MeV/k.
1.6
106 hw =
A. L. G#o~~u~ / beau-Ginsburg
theory
0.9 Me%‘, an excellent fit can also be obtained with C, = C, = 0, so that
F’(d) = c, -l-C&P,
(3.17)
where C6 > 0. This is the LG condition for a critical point, which divides regimes of first-order and second-order phase transitions. The critical point is kT, = 0.3 MeV, hw, = 0.9 MeV. Finally, the energy surfaces of the non-rotating system (o = 0) are fit with the expansion (3.18)
F’(d) = C,tC,d2+C,d?
The A6 term is not needed, since these curves contain only one inflection point. The coefficients C,(T) and C,(T) are given in fig. 9. Observe that C, changes sign at kT, = 0.55 MeV, so that near To C,(T) 2: a(T-
(3.19)
T,),
where cx is a positive constant, and C, > 0. These are the LG conditions second-order phase transition.
for a
6 kT Fig. 9. The Landau-Gmzburg
coefficients C, and C, versus the temperature 7’. The rotational frequency w is zero. See fig. 7 for units.
107
4. Cooclusions Motivated by the Landau-Ginzburg theory, the free-energy surface has been generated as a function of the order parameter d for various rotational frequencies and temperatures. These surfaces provide clear illustratjons of phase transitions. For spin zero, these surfaces contain only one minimum. When the system is rotated at low temperatures, a second minimum evolves. As the rotational frequency increases the absolute minimum jumps from the first potential well to the second, producing a discontinuity in d. This is the signature of a first-order phase transition. For states with a fixed 7; cc) the HFB solutions in the backbending region are maxima on the free-energy surfaces, rather than minima. At the critical point T,, w, the energy surface is flat over a wide range of d. For T > T, the potentials contain only one minimum, and d decreases to zero continuously as w is increased. Similarly, for o = 0, the potentials have only one minimum, and A goes to zero continuously as T is increased. These are examples of second-order phase transitions. Since our system contains only Z&2= 20 particles, one should consider whether fluctuations alter these conclusions. There are two kinds of fluctuations. The first is created by the approximate nature of the wave functions implicit in the mean-field approximation. Since particle number and angular momentum are not conserved, there are fluctuations in these quantities. Number fluctuations wih wash out the sharp second-order phase transitions ‘). However, the first-order transitions should not be affected by number non-conservation. After all, a first-order transition simply means backbending, which many real nuclei actually exhibit. Angular momentum fluctuations in the two-level model were calculated in ref. 14), The relevant quantity is not the spin ~uctuation, but rather the spin fluctuation energy. The calculations show that this energy becomes less important as either the temperature or the spin increases. Even at T = 0 and hw, = 1.0 MeV (I = 5), the spin fluctuation energy is only 0.25 MeV at the potential energy minimum, whereas the height of the potential energy barrier at T = 0 and ho, = 1.0 MeV (see fig. 2) is 1.65 MeV. Consequently, spin fluctuations do not wash out this first-order transition. The second type of ~uctuation is statistical, which would occur even for exact wave functions. Since d is chosen as the order parameter, there wilf be statistical fluctuations in A. The probability that the system has a given value of A is p(A)
K
e-F’bWkT_
(4.1)
The mean-field approximation determines the most probable value of A, which is given by the maximum in P(d). The most probable A vanishes as the temperature is increased at cu = 0, creating a second-order transition. However, Moretto 16) has shown that the average value of A, which is determined by the distribution P(d),
108
A. L. Gurney
/ ~~d~u-Gj~~urg
theory
does not vanish as the temperature is increased at u = 0. consequently, statistical fluctuations in d wash out the second-order phase transition at CJ = 0. However, these fluctuations are small at low temperatures and vanish at T = 0. Therefore they do not remove the first-order transition described above. To summarize, both types of fluctuations wash out the second-order phase transitions, but not the first-order phase transitions. The author is grateful to R. Gitmore, D. H. Feng and P. Ring for stim~ll~t~ng conversations.
References I) L. D. Landau and E. M. Lifshitz, Statistical physrcs (Pergamon press, Oxford, 1980) 2) 3) 4) 5) 6) 7) 8)
R. Gilmore, Catastrophe theory for scientists and engineers (Wiley, New York, 1981) C. Kittel and II. Kroemer, Thermal physics (Freeman, San Francisco, 1980) M. Tinkham, Xntroduction to superconductivtty (Krieger, Huntington, New York, 1980) A. L. Goodman, Nucl. Phys. A369 (1981) 365 K. Tanabe and K. Sugawara-Tanabe, Nucl. Phys. A390 (1982) 385 A. L. Goodman. Nucl. Phys. A265 (1976) 113 A. L. Goodman, in Advances in nuclear physics, ed. J. Negeie and E. Vogt, voi. Ii (Plenum, New York, t979j 9) P. Ring and P. Schuck, The m&ear many-body problem (Springer, New York, 1980) 10) A. L. Goodman, Nucl. Phys. A352 (LY81) 30 If) K, Tanabe, K. Sugawara-Teak and H. J. Mang, Nucl. Phys. A?57 (1981) 20 12) M. Sano and M. Wakal, Prog. Theor. Phys. 48 (1972) 160 13) J. Krumhnde and Z. Szymanski, Ann. of Phys. 79 (1973) 201 14) A. L. Goodman, Nucl. Phys. A352 (1981) 45 1.5) S. Y. Chu, E. R. Marshalek, P. Ring, J. Krumhnde and J. 0. Rasmussen, Phys. Rev. C12 (1975) 1017 16) L. C. Moretto, Phys. Lett. 40B (1972) I
LANDAU-GINZBURG THEORY OF PHASE TRANSITIONS IN HEATED ROTATING NUCLEI ALAN L. GOODMANi Physics Department and Quantum Theory Group, Tulane University, New Orleans, Louisiana 10118, USA
Received 1 February 1983 (Revised 5 April 1983) Abstract: The L~dau-Ginsburg theory is appIied to heated rotating suited systems. It is demonstrated that the free energy F’ is an even function of the order parameter A even when external fields are present. The energy surfaces F’(d) are calculated for various rotational frequencies and temperatures. These surfaces provide clear illustrations of first-order and secondorder phase transitions, as well as a critical point.
1. Introduction
The Landau-Ginzburg (LG) theory 1-4) is a mean-field theory which describes a wide variety of phase transitions. The essential idea is to express the free energy of a system as a power series in the order parameter. Knowledge of the signs of the first few coefficients. in this expansion determines whether the system will experience a phase transition as well as the order of the transition. The LG theory constructs the free energy as a function of the order parameter. These energy surfaces have minima and maxima, which give the values of the order parameter when the system is in therma equilibrium. Points on these surfaces which are not extrema are not equilibrium states of the system. This is similar to those nuclear collective models which construct potential energy surfaces in deformation space. The deformation j3 is an order parameter for rotations. The LG theory will be applied to heated rotating nuclei in this article. The order parameter is chosen as the pair gap A. For each rotational frequency COand temperature T, the energy surface F’(d) will be constructed, where 8“ is the free energy in a rotating frame. These surfaces will provide vivid illustrations of firstorder and second-order phase transitions, as well as critical points. The LG theory supplements previous descriptions 5*6) of phase transitions in heated rotating nuclei which refer only to equilibrium states and not to energy surfaces.
7 Supported in part by the US National Science Foundation. 94
A. L. Goodman / Landau-Ginzburg theory 2.
Landau-Giuzburg
For a system at constant temperature
95
theory of rotating nuclei
and volume the Helmholtz free energy,
F = E-TS,
(2.1)
where S is the entropy, is minimized. The LG theory expands F in powers of some order parameter u:
WI, T) = :
C,(T)y"
(2.2)
n=O
where N is 4 for second-order transitions and 6 for first-order transitions. If there are no external fields, then generally C, = 0 for odd IZ.The condition for a secondorder transition at To is that near To
C,(T) = C-T,),
(2.3)
where a is a positive constant, and C, > 0. The condition for a first-order transition is that C, < 0 and C, > 0. A critical point occurs at T, if C,(T,) = C,(T,) = 0 and C6 > 0. At each temperature the equilibrium value of yeis determined by (aF/iYr]), = 0. If there is an external field, then the expansion (2.2) generally contains a linear term. For example, consider a system at constant temperature and volume in the presence of an external magnetic field H. Then, the magnetic Gibbs free energy, G = E-TS-HM,
where M is the magnetization, LG expansion is
is minimized. The order parameter
G(M, T, H) = c CJT, H)M”. n
(2.4)
is M, and the (2.5)
It is clear from eq. (2.4) that the expansion (2.5) should include the term which is linear in M. For a nucleus with a given temperature and rotational frequency, the Helmholtz free energy in a rotating frame, F’ = E-
TS-ol,
(2.6)
where I is the angular momentum, is minimized. The usual analogy between the effect of magnetic fields on superconductors and the effect of rotations on nuclei
96
A. L. Goodman / Landau-Ginzburg theory
suggests that the order parameter should be chosen as I. Then, the LG expansion F’(f, T, 0) =
c c&r,
w)I”,
II
(2.7)
would contain the term which is linear in I. However, the order parameter is here chosen as the pair gap A, rather than the spin I. Then, the LG expansion is F’(A, T, Co)=
c C,(T,
0)A”.
(2.8)
n
Does this expansion contain odd powers of A? When a nucleus is not rotating, the pair field is time-reversal even. However, rotations induce a time-reversal odd component in the pair field ‘). So it might be expected that F’ contains odd powers of A. It will now be demonstrated that F’ is an even function of A: F’( - A) = F’(A),
(2.9)
so that there are no odd powers in the expansion (2.8). Consider the HFB equation 8,9) ~cX,=L: EiX,,
(2.10) (2.11)
where X is the Hartree-Fock (HF) hamiltonian in a rotating frame and Ei is the quasiparticle energy. Define the unitary transformation
A=
1
0
( > 0
e-i+
’
(2.12)
where Cpis an arbitrary phase angle. From eq. (2.10) it follows that (Ax-4
1)(AX,) = E,(AX,).
(2.13)
Substitute eq. (2.12) into eq. (2.13), with the result that (2.14)
If A produces the quasiparticle transformation (Y, 6) and quasiparticle energy E, then e’@bdleads to the transformation (Vi, e-l@vi) and the same quasiparticle
energy. Treating the phase a&e
4 as a parameter, we define d(#) =
At finite temperature
el”d(O),
(2.15)
V+> = U(O),
(2.16)
V(f$) = e-W(O),
(2.17)
K(4) = NO)”
(2.18)
tke FIF density p and the pair density r are 10-12) p = ~jx*-i_ t =
V(l -f,P,
(2.19f
Dfv*+ V’(1 -f)U,
(2.20)
where (2.21) and ,B= l/kT. From eqs. (2.16)-(2.21) it follows that (2.22) r(f#J)= e%{O).
(2,231
The internal energy E = (H) contains terms with pp and l*t, so that
E(4) = E(O).
(2.24)
The entropy onfy contains the quasiparticle energies (2.18), and the spin is linear in Q, so that
Consequently, the free energy in a rotating frame (2.6) is invariant with respect to the transformative (2.15) F’(c)) = F(O),
(2.27)
or Ff@LI > = F(d )*
(2.2s)
98
) ~~~~-~i~z~~r~
A. L. ~~~~
ffimy
For 6, = 7~ eq. (2.28) reduces to eq. (2.9)$ which proves that F’(d j is an even function. The cssentid point in the derivation is that F’ contains no terms which are linear in the pair density 1. From the definition of the pair potential, (2.29)
it follows that eqs. (U5) and (2.23) are consistent, so the traRsformatio~ A of eq. (2.12) provides a self-consistent symmetry. The HFB equation determines the order parameter and free energy when the system is in ~~li~~b~urn. However, the LG expansion cakufates F’ at all valaes of A, not just the ~quilibr~~~mvalues. Nevertheless, the co~clu~~o~ that F’ is even in d remains valid for any value of 8. Consider any general 3ogoliubo~~ transformation
Us=I
t:
(U,jCf
+
j
f$jCj),
(2.30)
and any set of quasiparticle energies E+ The numbers U, I/ and E need not be a self-consistent solution of the HFB equation. From U, V and E one can calcula$e F and A. Next, apply the transformation ~2.1~)-~2.~~) to U, V and E with (b = 7~. Then, recalculate F’ and d, As described above, f;” is invariant and d changes sign. So F’ is even in d for any density operator of the HFB &ass. Therefore the LG expansioa f2.8) is
F’th T, ~1 = C,(T,O)+C,(T,W)~~-~_~,(II‘,~)~~+~,~~,~)~~~
(2.31)
As usual the expansion is truncated at sixth order. For each temperature and rotational frequency, the equilibrium value of d is determined by the condition
The HFB theory considers a system maintaked at constant r, o and & where p is the chemical potential. Then, the minimized quantity is not F:, but the grand ~ot~n~~~~in a rotating frame 52 = F’-jLN,
(2.33)
where N is the particle number. Therefore one should expand Q’ in even powers of d: (234)
99
As an illustration of the LG theory of heated rotating systems, we consider the two-level model of Krumlinde aad Szymanski with no rotor 13714),In this model the chemical potential is zero, so that f2’ = F’. The procedure is as follows: Choose a density operator, and use it to calculate F’ and d, Repeat for many choices of the density operator. The function F’(d) is thereby mapped out. In the WFB a~~~o~~ma~on the many-bay density operator is
where the quasiparticle number operator is
and Q’ is defined by eq. (2.30). The qu~si~artjc~e occn~ation ~r~b~~i~i~y is fi. Any unitary transfo~a~on U, 9’ and any J such that 0 S d 5 1, define a possible HFB density operator. The choice U, r&i,j’ need not be the se~~cons~stent m~ni~nm which solves the HFB equation. The fo~~o~~~~~algorithm is used to construct a set of density operators: First, choose values for 7’ and CO.Second, in the basis 1, 2, 3, 4 of fig. 1, choose U, V and f its
o-
0 0
0
B0
4
3,
0
0
A-
f;+
-B+
0
-B_
1 p
f3.3)
(3.5)
100
A. L. Good~u~ / ~duu-Gi~zburg
theory
where
Pil = $2 -&I+, f*
E, =
(3.7)
= [l +eaE+)-L,
(3.8)
[E’ i- (S&&O)~]“,
(3.9)
and @= 1,&T, E is the single-particle splitting shown in fig. 1, A, and B, are positive, and B_ is positive when 6 > $0 and negative when 6 < fo. For a given I, T, trf then U, V, f are determined by the arbitrary constant 6, which ranges from zero to infinity. So the density operator DnFB(6, T, ~0) is a continuous function of the parameter 6. Third, for each 6, T, w use DWFBto calculate d and F’. The results are A@, T, cc))= jGQ
F
1 +
6-k) tanh &?E+)+ -$--
1
tanh (#E _ ) ,
F’(6, T,w) = E-TS-cd,
(3.1 I)
where E(6, T, co) = --Lk2
s(kT,~)=
tanh (#E+) E
(3.10)
+ tanh ($E_) E-
+
1--
A2
(3.12)
G’
(3.13)
-2kQr_CE [flnJ+(l-j)In(l-f)], f
tanh (#E,)
6-gf3
- r
tanh (#E_)
1 ,
(3.14)
and G is the number of identical sets of four states shown in fig. 1. For a given 6, T, w eqs. (3.10) and (3.11) produce one point in the F’(A) plane. By varying 6, T, w the function F’(d, T, o) is mapped out. In general, 6 is not equal to A. Then, the state DWFB(8,T, o) is not a state of thermal equilibrium and it is not a solution of the HFB equation. However, for the few choices of 6 where 6 is equal to A, then DHFBis an equilibrium solution of the HFB equation. Also, it is only when 6 = A that E, can be interpreted as a quasiparticle energy. Figs. 2-6 depict the free energy in a rotating frame versus the pair potential for
101
kT=O
-14
G=O.l ,Q=lO c=o.t
II 0
I
0.2
II
I
0.4
11
0.6
I
I
0.8
L!!A
Fig. 2. Free energy in the rotating frame I? versus the pair gap A for various rotational frequencies CO. The temperature T is zero. The dots identify the minima and maxima. The quantities F’, A, Aw, kT, G and E have units of MeV.
various rotational frequencies and temperatures. As an example of a first-order phase transition, consider fig. 2, which shows F’(d) at T = 0. At o = 0 the equilibrium state is given by the ~nimum at d = 0.995 MeV. The normal state at d = 0 is a maximum, and is therefore a state of unstable equilibrium. A large potential barrier separates the superfluid and normal states. As 03 increases, this barrier decreases. At hw = 0.8 MeV the state at d = 0.092 MeV is still the absolute minimum. The normal state at d = 0 changes from a maximum to a relative minimum. There is a new maximum at d = 0.375 MeV. So at ho = 0.8 MeV there are three extrema, and each gives a self-consistent solution to the HFB equation. For fixed T, CLIthe maximum is unstable, and it describes a state in the middle of the backbending region. It should be emphasized that this instability is in the function F’(d, T, 0). Chu et al. have shown that for the function F(d, T, I) the instability disappears IS). When tie is increased to 1.0 MeV, the superffuid state at
A. L. Goodman / Landau-Ginzburg theory
102
A = 0.989 MeV and the normal state at A = 0 are degenerate. At hw = 1.2 MeV the normal state becomes the absolute minimum. Consequently, when hw is increased past 1.0 MeV the equilibrium state discontinuously jumps from A = 0.989 MeV to A = 0. Whenever there is a discontinuous change in the order parameter caused by an infinitesimal change in an intensive variable, such as o, the phase transition is first order. Finally, at ho = 1.6 MeV the superfluid extrema disappear, and only the normal state remains. To summarize the lesson of fig. 2, if F’(A) contains two minima, and if increasing w shifts the absolute minimum from one of these potential wells to the other, then there is a first-order phase transition. Fig. 3 shows F’(A) at kT = 0.2 MeV. The qualitative features are the same as at T = 0. However, raising the temperature from 0 to 0.2 MeV substantially reduces the barrier at III = 0 between the superfluid state and the normal state, and almost eliminates the barrier at ho = 1.0 MeV between the superfluid minimum and the superfluid maximum. Yet the phase transition is still first order. Fig. 4 shows the potential surfaces at kT = 0.3 MeV. These surfaces contain only one minimum. Therefore the transition from superfluid to normal is not first order at this temperature. For Ro = 0.9 MeV the potential surface is flat over the range 0 < A < 0.7 MeV. This is the characteristic feature of a critical point, which is a point on a phase equilibrium line ~(7’) which divides the line into two segments, one corresponding to first-order transitions and the other to second-
0
I
I
0.2
I
I
0.4
I
n
I
0.6
I
I
0.8
I
Fig. 3. See fig. 2. The temperature is 0.2 MeV/k.
J
1.0
103
order transitions. The critical point is kT, = 0.3 MeV, &co= 0.9 MeV. At this point the mean-field approximation should be supplemented by Iarge fluctuations in d. The potential surfaces at kT = 0.4 MeV are given in fig. 5. As the rotational frequency is increased from zero, the minimum shifts continuously toward the left, so that the equilibrium value of d decreases continuously from 0.703 MeV to 0. This continuous decrease of the order parameter is characteristic of a second-order phase transition. Fig. 6 shows that at kT = 0.5 MeV the normal state is the absolute minimum even when the system is not rotating. There is no su~~u~dity at this te.m~ra~ure_
Fig. 5. See fig. 2 T%e ttxnperatur~ is 0.4 MeV/k.
104
-18 0
I
I 0.2
I
II 0.8
1.0
Fig. 6. See fig. 2. The temperature is 0.5 MeV/k.
Consider heating the non-rotating system at o = 0. Figs. 2-6 show that the equilibrium value of d decreases continuously from 0.995 MeV to 0 as the temperature is increased from zero. There is only one minimum in the potential surface. So heating the non-rotating super~uid induces a second-order phase transition. The LG expansion of eq. (2.31) will now be applied to the potential surfaces in figs. 2-6. Each curve has a given T, w. The function F’(d) =
C, + C,d2+ C4A4 -t C6d6,
(3.15)
is fit to each curve in figs. 2-6, where Co, C,, C,, C, are determined by a leastsquares fit. Higher-order terms are not needed since these curves contain no more than two inflection points. First, consider the system at zero temperature. The LG coefficients C,(o), C,(m), C,(o) are given in fig. 7. Observe that Cz changes sign at Ao, = 0.55 MeV, so that near o0 C,(w) ‘1: &J-w,),
(3.16)
where a is a positive constant, and C, < 0, Cs > 0. These are the LG conditions for a first-order phase transition. Recall that the transition occurs at ho, = 1.0 MeV, so that o0 < 0,. The functions C,(w), C,(W), C,(o) are almost linear for ho > 0.4 MeV. For the temperature kT = 0.3 MeV the LG coefficients are shown in fig. 8. Again Cz changes sign and C, < 0, C, > 0. However, for the flat energy surface at
A. L. Goodman / Landau-Ginzburg
I
24.
’
’
p
I
105
theory
’
-
20 16
12.
-24
-32 R=lO
-36 -400
0.2
0.4
0.6
0.8
1.0 1.2 1.4 1.6
fiw Fig. 7. The Landau-Ginzburg coefficients C,, C, and C6 versus the rotatlonal frequency w. The temperature T is zero. The coefficient C, has units of (MeI’)‘-“. The quantities hw, kT, G and E have units of MeV.
-'*O
0.2
0.4 0.6
Fig. 8. See fig.7. The
0.8 tiw
1.0 1.2 1.4
temperature is 0.3 MeV/k.
1.6
106 hw =
A. L. G#o~~u~ / beau-Ginsburg
theory
0.9 Me%‘, an excellent fit can also be obtained with C, = C, = 0, so that
F’(d) = c, -l-C&P,
(3.17)
where C6 > 0. This is the LG condition for a critical point, which divides regimes of first-order and second-order phase transitions. The critical point is kT, = 0.3 MeV, hw, = 0.9 MeV. Finally, the energy surfaces of the non-rotating system (o = 0) are fit with the expansion (3.18)
F’(d) = C,tC,d2+C,d?
The A6 term is not needed, since these curves contain only one inflection point. The coefficients C,(T) and C,(T) are given in fig. 9. Observe that C, changes sign at kT, = 0.55 MeV, so that near To C,(T) 2: a(T-
(3.19)
T,),
where cx is a positive constant, and C, > 0. These are the LG conditions second-order phase transition.
for a
6 kT Fig. 9. The Landau-Gmzburg
coefficients C, and C, versus the temperature 7’. The rotational frequency w is zero. See fig. 7 for units.
107
4. Cooclusions Motivated by the Landau-Ginzburg theory, the free-energy surface has been generated as a function of the order parameter d for various rotational frequencies and temperatures. These surfaces provide clear illustratjons of phase transitions. For spin zero, these surfaces contain only one minimum. When the system is rotated at low temperatures, a second minimum evolves. As the rotational frequency increases the absolute minimum jumps from the first potential well to the second, producing a discontinuity in d. This is the signature of a first-order phase transition. For states with a fixed 7; cc) the HFB solutions in the backbending region are maxima on the free-energy surfaces, rather than minima. At the critical point T,, w, the energy surface is flat over a wide range of d. For T > T, the potentials contain only one minimum, and d decreases to zero continuously as w is increased. Similarly, for o = 0, the potentials have only one minimum, and A goes to zero continuously as T is increased. These are examples of second-order phase transitions. Since our system contains only Z&2= 20 particles, one should consider whether fluctuations alter these conclusions. There are two kinds of fluctuations. The first is created by the approximate nature of the wave functions implicit in the mean-field approximation. Since particle number and angular momentum are not conserved, there are fluctuations in these quantities. Number fluctuations wih wash out the sharp second-order phase transitions ‘). However, the first-order transitions should not be affected by number non-conservation. After all, a first-order transition simply means backbending, which many real nuclei actually exhibit. Angular momentum fluctuations in the two-level model were calculated in ref. 14), The relevant quantity is not the spin ~uctuation, but rather the spin fluctuation energy. The calculations show that this energy becomes less important as either the temperature or the spin increases. Even at T = 0 and hw, = 1.0 MeV (I = 5), the spin fluctuation energy is only 0.25 MeV at the potential energy minimum, whereas the height of the potential energy barrier at T = 0 and ho, = 1.0 MeV (see fig. 2) is 1.65 MeV. Consequently, spin fluctuations do not wash out this first-order transition. The second type of ~uctuation is statistical, which would occur even for exact wave functions. Since d is chosen as the order parameter, there wilf be statistical fluctuations in A. The probability that the system has a given value of A is p(A)
K
e-F’bWkT_
(4.1)
The mean-field approximation determines the most probable value of A, which is given by the maximum in P(d). The most probable A vanishes as the temperature is increased at cu = 0, creating a second-order transition. However, Moretto 16) has shown that the average value of A, which is determined by the distribution P(d),
108
A. L. Gurney
/ ~~d~u-Gj~~urg
theory
does not vanish as the temperature is increased at u = 0. consequently, statistical fluctuations in d wash out the second-order phase transition at CJ = 0. However, these fluctuations are small at low temperatures and vanish at T = 0. Therefore they do not remove the first-order transition described above. To summarize, both types of fluctuations wash out the second-order phase transitions, but not the first-order phase transitions. The author is grateful to R. Gitmore, D. H. Feng and P. Ring for stim~ll~t~ng conversations.
References I) L. D. Landau and E. M. Lifshitz, Statistical physrcs (Pergamon press, Oxford, 1980) 2) 3) 4) 5) 6) 7) 8)
R. Gilmore, Catastrophe theory for scientists and engineers (Wiley, New York, 1981) C. Kittel and II. Kroemer, Thermal physics (Freeman, San Francisco, 1980) M. Tinkham, Xntroduction to superconductivtty (Krieger, Huntington, New York, 1980) A. L. Goodman, Nucl. Phys. A369 (1981) 365 K. Tanabe and K. Sugawara-Tanabe, Nucl. Phys. A390 (1982) 385 A. L. Goodman. Nucl. Phys. A265 (1976) 113 A. L. Goodman, in Advances in nuclear physics, ed. J. Negeie and E. Vogt, voi. Ii (Plenum, New York, t979j 9) P. Ring and P. Schuck, The m&ear many-body problem (Springer, New York, 1980) 10) A. L. Goodman, Nucl. Phys. A352 (LY81) 30 If) K, Tanabe, K. Sugawara-Teak and H. J. Mang, Nucl. Phys. A?57 (1981) 20 12) M. Sano and M. Wakal, Prog. Theor. Phys. 48 (1972) 160 13) J. Krumhnde and Z. Szymanski, Ann. of Phys. 79 (1973) 201 14) A. L. Goodman, Nucl. Phys. A352 (1981) 45 1.5) S. Y. Chu, E. R. Marshalek, P. Ring, J. Krumhnde and J. 0. Rasmussen, Phys. Rev. C12 (1975) 1017 16) L. C. Moretto, Phys. Lett. 40B (1972) I