Effect of isobars at finite temperature in pion condensation in 16O

Volume 120B, number 1,2,3 PHYSICS LETTERS 6 January 1983

EFFECT OF ISOBARS AT FINITE TEMPERATURE IN PION CONDENSATION IN 160 ¢r

R.K. TRIPATHI 1 and Amand FAESSLER Institut ffir Theoretische Physik, Universita't Tfibingen, D-7400 Tu'bingen, West Germany

Received 2 June 1982 Revised manuscript received 6 September 1982

The role of A (1232) at f'mite temperature is considered in pion condensation in 160. Taking into account the effect of the ~'s the free energy of the spin-isospin phase, where the spins of all the protons are oriented one way and those of neutrons in the opposite direction, is compared with the free energy of the normal phase. For higher temperatures (T > 15 MeV) the critical density of the phase transition increases with temperature. Smaller temperatures are found to contribute to the development of the condensate. The A's get increasingly more important with higher density and temperature.

The consequences of pion condensation [1,2] in nuclear physics and astrophysics are tremendous. It is now agreed that the density o f the phase transition when (and if) the condensate sets in would be larger than (say) twice the normal nuclear matter density. One may hope to achieve such high densities in high energy heavy ion collisions. Such densities would certainly be produced at non zero temperatures. Conse- quently it is but natural to study the effect of temperature in the condensate phase transition.

In nuclear mat ter ffmite temperature effects have been investigated [ 3 - 5 ] by looking into pion self-energy propagator at finite temperatures. In infinite nuclear matter [6] pion condensation is equivalent to a phase transition from normal phase to spin- isospin density wave phase o f laminated structure of protons with spin up and neutrons with spin down alternating with protons with spin down and neutrons with spin up. A f'mite nucleus is too small to accommodate this periodic structure. It has been suggested [7,8] that one may realize this sp in- isospin density phase in finite nuclei by having all protons spin aligned in one direction and all neutrons spin aligned in the opposite direction. This description does not run into the diffi- culties o f divergences [9] which one f'mds in the usual t reatment [1,2]. It is known that the A's do play a

Supported by BMFT. 1 Permanent address: Institute of Physics, Bhubaneswax, India.

very important role in the pion condensation at zero temperature. It is the purpose of this article to investi- gate the influence of the A particle at f'mite temperatures within the above model.

The quanti ty of interest in the case of finite temperature T is the free energy,

F = E - TS, (1)

where E is the energy and S is the entropy. The free energy for the condensed phase F C, and the free energy for the normal phase F N are calculated and the difference ( F C - F N ) is minimized. It should be emphasized that our main aim is to study the influence of the A's at finite temperature within the context of the above model and we would only draw conclusions which are independent of the finer details of the interaction. Consequently, we would restrict ourselves to the simpler form of the interaction. The single particle potential is assumed to be an harmonic oscillator and the residual interaction is taken to be one-pion and one p-meson exchange. The IINN and pNN coupling con- stants are taken to be f2/4rr = 0.08 and f2/47r = 5.0. We have included monopole type form factors with the cut off masses A~ = 1000 MeV and A 0 = 2000 MeV. For the temperature dependent H a r t r e e - F o c k approximation, the total energy is

E= +1- G (2) a 2 c~O

54 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland


and the entropy of the system is given by

S = - k ~ [ f lnf~ +(1 - f a ) l n ( 1 - f~) l , (3) tX

where fa is the occupation probability for the level Is),

fa = { 1 + exp [ ( e -- I.t)/kT] }- 1. (4)

The single-particle energy

= + ~ ( ~ l V + p l a ~ ) f ~ (5) e e a

and ea is the oscillator energy. It should be pointed out that in the above equation the single-particle energy is needed (in fa ) in order to calculate the single-particle energy itself. This implies that the single-particle energy has to be calculated selfconsistently. The chemical potential is determined by the number constraint,

~ f a = N . (6) ot

It is good to remark that eqs. (5) and (6) warrant the double selfconsistencies which would have to be satis- fied with respect to the single-particle energies and the chemical potential. The method of solution is as fol- lows. For a given nucleus and temperature T, one makes initial guesses for the single-particle energies. Eq. (6) is then solved by varying the chemical potential until one gets the right number of nucleons N. Eq. (5) is then used to get the new guess for the single-particle energies. The process is repeated until one gets the same value of the single-particle energies and chemical potential as fed into eqs. (5) and (6) thus satisfy- ing both the selfconsistencies.

The level Is) are harmonic oscillator wave functions in cartesian representation INxNyN z rzpz). For the condensed phase at T = 0 each of the levels (N x,Ny, Nz): (0, 0,0), (1,0, 0), (0, 1,0), (1, 1,0), (2, 0,0), (0,2, 0), (3,0, 0), (0, 3, 0) contains (pf , n¢). For T > 0 all levels with N x + Ny ~< 6 and N z = 0 were included. This allows admixtures of 3 unoccupied T = 0 shells. The admixture of/X-isobar was allowed by taking the single-particle state,

Is) = { I N ) +AIAa))/(1 +A2) 1/2, (7)

where A is the mixing amplitude for the/X-isobar. Cer- tainly it would be preferable to have different mixing amplitude for each single particle state. This would,

however, bring in a lot of additional variational param- eters in our calculation. Consequently to keep the calculation to a managable size, we made the approximation and assumed the same value for the mixing amplitude for each state. This approximation can certainly be improved upon. However, we believe that this does describe the influence of the A's in an average way. Besides we would like to emphasize that we would draw only those conclusions which would be independent to the finer details of this approximation. The in- clusion of the/X-isobar means that we have also to calculate the transition matrix element: NA--NA, NN- AA, NA-AA, NN-NA and AA--AA.

The normal phase is four fold degenerate and at T = 0 occupies (0, 0,0), (1,0,0),(0, 1,0), (0, 0, 1) levels. For T > 0 admixture of 3 unoccupied T = 0 shells was allowed here as well.

Volume conservation is provided by

3 bxbyb z = b s , (8)

and the ratio of the densities is defined as

Plpo = (Ro/R)3, (9) with

R2= b (N (10) OL

and P0 is the density at T = 0 with b s = 1.7 fm. Fig. 1 shows a plot of (F C - FN) against density ra-

tio (P/Po) for different values of temperature. For each value of temperature there is density Pc beyond which F C > F N and the nucleus undergoes phase transition to pion condensed phase. The phase diagram is shown in fig. 2, where the solid line (line 2) represents the results of the present calculation. If the level densities of the two phases would be the same and there would be no shell structures then one would expect TC oc (p/p0)2 for lower temperatures T < eF, where e F is the Fermi energy. This result can be obtained quite simply by treating the system to be a degenerate fermion system and making the expansion of the fermion distribution function for lower temperatures [10]. Similarly for higher temperatures T > e F one would expect the linear dependence T C c¢ (P/Po)" This result can again be seen by making expansion of the Fermi distribution function but now for the nondegen- crate case [10]. The expected result is shown by a dashed-dot ted-dot ted curve (line 3) in fig. 2. One

55


i kT : 100 10 ~\ I

5

:$

%

~ o I

1.~_ u

-5

80 \ \

\ \ \ \

I I L78

( P/Po )

Fig. 1. Difference o f the free energies of the condensed phase and normal phase versus the density ratio for various values o f temperature.

notices that the behaviour obtained by the present calculation (line 2 )a t higher temperatures T > 15 MeV is as expected and can be understood in terms of the entropy effect. For higher temperatures the entropy of the normal phase is expected to be twice the entropy of the condensed phase, since the normal phase is four fold degenerate while the condensed phase is doubly degenerate. In order to look into the importance of the entropy term we have also shown the phase diagram one would get by leaving out the entropy term in the free energy. This means one looks into the difference of the temperature dependent energies only. This is shown by the dashed-dot ted line (line 4) in fig. 2. One sees that the entropy term gets quite important for T > 7.5 MeV and is mainly respon- sible for the linear temperature dependence of the phase diagram at higher temperatures. A comparison of the present result (line 2) with the case where one takes only nucleons into account [11] (line 1) shows that the slope of the two curves is different in the two cases. This is due to two reasons: first the slope of the linear behaviour depends on the interaction used. Sec- ondly, as the temperature and density increase the A content by the wave function increases as well. This point is made more explicit later on.

56

100 -

9 0 -

8 0 -

70--

6 0 -

50- -

3 0 - F- -

2 0 -

Normal Phase

/

/ / /

/ // / '

/ / // / /'

/ ' /

/ /

/ / /

/

6 / ," 1 0 - ~ '

0 - - I . . . . . . . . . . . . . . . . . . . . . . . . . . . I 1 l 2 3 4 5 6

( P/Po )

Fig. 2. Phase diagram of 160. Solid line (line 2) with A's and dotted line (line l) for nucleons only. See text for the description of other curves.

/

/ /

J

Pion . -" C°ndensed ' . , " Phase

/

I

1 l J 7 8

It is interesting to note that for k T < 7.5 MeV in- creasing temperature assists in condensation. This is quite different from the expected curve (line 3) and is due to shell effect or level density effect at T = 0. Since 160 is a closed shell nucleus the level density at the Fermi surface at T = 0 for the normal shell is quite low. On the contrary the Fermi surface o f the condensed phase at T = 0 is in the middle of the N = 3 shell and the level density is higher. Hence at low temperature T ~ hw thermal excitation of the normal phase is negligible but the thermal excitation o f the condensed phase is significant. Consequently for lower temperatures the entropy of the condensed phase is higher, thus favouring the condensation. One also notices that some of the thermal excitation in the present case goes on to increase the content o f the A's as well. This point is made more explicit later in the text. The phase curve is triple valued. At T = 0 the nucleus is nor. mal, undergoes a phase transition to the condensed phase and then at higher temperatures goes to normal phase. It may be remarked that should the level densities for the two phases at T = 0 be reversed as is the case for 12C, then one would get a curve for the tern-


perature dependence which would be mirror image of the curve obtained here at lower temperatures i.e. rais- ing the temperature would drastically hinder the condensation. Since the condensed phase is expected to be produced at finite temperatures, our calculations indi- cate that 12C from this point of view alone is most un- favourable case to look for the pion condensation. Na- turally, should the level densities of the two phases at T = 0 be the same then one would get the curve line 3. Hence in order to observe the thermally assisted pion condensation one should select a nucleus like 160 where the level density at T = 0 is low for normal phase but it is high for the condensed phase.

Fig. 3 shows the mixing amplitude against density ratio as a function of temperature. As the density increases for a fixed temperature the A admixture increases. Also at fixed density the A admixture increases with temperature. It is of interest to look into the A admixture at normal density. Fig. 3 gives A ~ 0.03 in general qualitative agreement with the value recently obtained by Morris et al. [12]. Also it is in good general agreement with the estimates predicted by Bohr and Mottelson [13] at T = 0. This is particularly interesting keeping in view the simplicity of our model. In a re-

6

80,100

~0

4 2O

0

x

2 -

1

0 J J 1 I ~ I I I 1 2 3 4 5 6 7 B

( 0 / 0 o )

Fig. 3. Mixing ampli tude of the ,x's versus densi ty for vaxious values o f temperature.

cent calculation [14] of Ar + Ca at 0.5 GeV/nucl. the system was found to move along a trajectory from p

3.8 00, T= 0 MeV to p ~ 2.0 00 at T ~ 10 MeV. This is surprisingly close to our estimates found in fig. 2.

It should be emphasized that as the temperature increases one should take into account the thermal production of the pions and the renormalization of the field due to their presence. We note that the pions thus produced are not free and in the real situation interact very strongly with the nucleons. In the region of interest here their mean free path is so short that pions will be spending most of their time "resonating" with the nucleons, their scattering being dominated by the formation of the A's. Consequently, we have in- eluded the renormalization of the field due to thermal excitations of pions in the form of A resonance. This has the advantage that it automatically includes the major part of the pion-nucleon interaction. That this method does include the renormalization of the pion field due to the thermal excitations of the pions can be checked very simply by estimating the relative popula- tion of the A's with respect to nucleons. This is given by ~4 exp [-(Mzx - MN)C2/kT], (apart from the effects due to the single-particle energies, which we have left out in this expression for the simplicity of the dis- cussion). This factor is very naturally built in our eqs. (7), (4) and (5). It is gratifying to note that the number of pions calculated this way is in complete agreement with the number of pions calculated using the Bose distribution for the pions.

Our calculation indicates that the A-degrees of freedom and renormalizations due to thermal excitations get increasingly important with the increase of temperature in pion condensation in finite nuclei. The formalism presented here seems to provide a reason- able description of this effect. After the completion of the present work we learnt that in a very recent calculation in neutron matter using a different formalism Baym and Kolehmainen [15] have also emphasized the importance of the thermal excitations in pion condensation in neutron matter. They include the pion production using the Bose distribution for pions and then put on the pion-nucleon interactions using a lagrangian formalism in different models subsequently. Our conclusions are in complete general agreement with their observations. The formalism presented here is an alternate and seemingly simpler way of describing

57


the renormalization of the pion field due to thermal

excitations.

Extremely valuable extensive discussions with

Prof. H.H. Miather are acknowledged.

References

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H.H. M~ither, Phys. Rev. C23 (1981) 1154. [10] S. Chandrasekhar, Introduction to the study of stellar

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107B (1981) 341. [12] C.L. Morris et al., Phys. Lett. 108B (1982) 172. [13] A. Bohr and B.R. Mottelson, Phys. Lett. 100B (1981) 10. [14] K.K. Gudima and V.D. Toneev, Sov. J. Nucl. Phys. 31

(1980) 755. [15] G. Baym and K. Kolehmainen, Nucl. Phys. A382 (1982)

528.

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Effect of isobars at finite temperature in pion condensation in 16O