Influence of Quantal and Statistical Fluctuations on Phase Transitions in Finite Nuclei

Transcription

1 Commun. Theor. Phys. 56 (2011) Vol. 56, No. 4, October 15, 2011 Influence of Quantal and Statistical Fluctuations on Phase Transitions in Finite Nuclei G. Kanthimathi, 1 N. Boomadevi, 2 and T.R. Rajasekaran 2 1 R. M. D. Engineering College, R. S. M. Nagar, Kavaraipettai , Tamil Nadu, India 2 Department of Physics, Manonmaniam Sundaranar University, Tirunelveli , Tamil Nadu, India (Received March 31, 2011; revised manuscript received June 15, 2011) Abstract Investigations on thermal evolution of pairing-phase transition and shape-phase transition in light nuclei are made as a function of pair gap, deformation, temperature and angular momentum using a finite temperature statistical approach with main emphasis to fluctuations. The occurrence of a peak structure in the specific heat predicted as signals of the pairing-phase and shape-phase transitions are reviewed and it is found that they are not actually true phase transitions and it is only an artifact of the mean field models. Since quantal number and spin fluctuations and statistical fluctuations in pair gap, deformation degrees of freedom and energy when incorporated, it wash out the pairing-phase transition and smooth out the shape-phase transition. Phase transitions due to collapse of pair gap and deformation is discussed and a clear picture of pairing-phase transition in light nuclei is presented in which pairing transition is reconciled. PACS numbers: Ma, k, t, Gz Key words: phase transitions, specific heat, fluctuations 1 Introduction The atomic nucleus comprises a unique system exhibiting both microscopic features and statistical aspects, which afford a major source of knowledge of nuclear properties at high excitation energies and angular momentum. From the theoretical point of view, statistical descriptions of many body quantum systems have turned out to be quite frequent in the literature [1] highlighting the finite temperature properties of nuclei. Of particular interest are the types of phase transition that arise and the critical temperatures at which they happen. Recently a renewed interest in the study of phase transition has emerged as an exciting topic. It is really a fascinating and still open question whether phase transitions do exist in finite systems of nuclei at finite temperature and signature of these transitions remains regardless of fluctuations. Until the present, there are many problems concerning the meaning of phases and phase transitions in finite nuclei. Indeed there are two types of phase transitions (i) pairing-phase transitions (superfluid to normal fluid) and (ii) shape-phase transitions (deformed to spherical shapes). There are many theoretical [2] and experimental [3] supports for the appearance of these phase transitions in finite nuclei. Evidences include the vanishing of gap parameter in superconducting nuclei [4] for pairing-phase transition and the quadrupole moment in deformed nuclei [5] for shape-phase transition. Phase transition from superfluid to normal fluid has been investigated based on finite temperature mean field theories such as Bardeen Cooper Schrieffer (BCS), Hartree Fock (HF), and Hartree Fock Bogliubov (HFB) formalisms. However, the empirical analysis of experimental data does not predict a sudden phase transition, the reason being the neglect of fluctuations in the mean field approximations. Especially in light nuclei, quantal and statistical fluctuations are inevitable in identifying phase transitions. [2] Strictly speaking, quantal and statistical fluctuations are of crucial importance since they smooth out the singularities allied with the phase transitions. Furthermore pairing is also expected to be small in light nuclei, but later it is reported that the contribution of pairing is very significant. [6] As such not much progress has been made in identifying the effects of fluctuations on pairing-phase transitions in light nuclei, it is the main aim of this work, to focus the interplay between phase transitions and fluctuations. The role of fluctuations has also become the recent topic of interest. A system at zero temperature has quantal number and spin fluctuations and at finite temperature there are statistical fluctuations in the pair gap, deformation and energy. Rigorous investigations have pointed out the major part played by various kinds of quantal and statistical fluctuations. An outstanding example is the fluctuations in the pairing strength which wash out the collapse of pairing correlations. Also of interests are the large amplitude fluctuations in the relevant order parameters. For instance, in the Landau theory of phase transition, statistical fluctuations of quadrupole deformation have enlightened the observed temperature and spin dependence of the giant dipole resonance. [7] Moretto has shown that Supported by a Project (No. F. No /2008(SR)) sanctioned by University Grants Commission, New Delhi, India c 2011 Chinese Physical Society and IOP Publishing Ltd

2 No. 4 Communications in Theoretical Physics 719 thermal fluctuations smooth out the sharp pairing phase transition in finite nuclei. [8] Goodman has also emphasized the effect of statistical fluctuations using HFB formalism and have noticed that fluctuations in the pair gap wash out the pairing-phase transition. [9] Egido et al. analyzed the effects of statistical shape fluctuations and have found the smearing out of deformed to spherical shapephase transition. [10] The relativistic Hartree-BCS theory has also pointed the role of thermal shape fluctuations on both pairing-phase and shape-phase transitions. [11] Later, the shell model Monte Carlo method shows that pairing does not vanish at the critical temperature (T c ) but still survives at T > T c. [12] Theoretical studies based on static path approximation (SPA) incorporating statistical fluctuations also predicts a non vanishing pair gap at finite temperature but are inadequate at low temperature because of the ignorance of quantal fluctuations. [13] Likewise random phase approximations (RPA) provide better analysis of phase transitions with the inclusion of quantal fluctuations except for high temperature since thermal fluctuations are not included. A microscopic description is then provided by a RPA+SPA treatment where both quantal and statistical fluctuations effects are assessed. [14] Very recently a self consistent quasiparticle RPA shows that pairing-phase transitions are indeed smoothed out with a long exponential tail that extends up to higher temperature. [15] In our recent paper, we have studied the pairing-phase transition for the hot rotating nucleus 154 Dy. [16] In the present study, we have extended our investigation of nuclear phase transitions in light nuclei with the inclusion of quantal and statistical fluctuations. Calculations are executed for the fluctuations of selected observables that convey the signals for the phase transition like particle number, spin, pair gap, deformation, and energy. To our knowledge, so far efforts have not yet been made to explain the effects of all these fluctuations in light nuclei hence the present work discusses them in detail. Several authors have scrutinized [8,11 20] the subsistence of phase transition in finite nuclei. One important tool to study the phase transition is the specific heat. Attempts have been made to observe phase transitions and experiments have shown an S-shape in the specific heat capacity as an indication of phase transition, but recent evidences indicate that number conservation produces similar S-shape in the specific heat capacity even without transition. [21] A major activity in the study of phase transitions in nuclei has been carried out by the interacting boson model (IBM). Recently Cejnar [22] used a coherent state formalism using IBM with cranking to analyze the phase transitions with respect to angular momentum and temperature. In series of papers, the appearance of phase transitions in light nuclei are discussed. [23 26] On the other hand, analogous investigations for some of the light nuclei in the s-d shell have shown that the proposed phase transition does not occur. Miller et al. have presented convincing evidences for the emergence of a prominent peak in the specific heat for Mg at temperature T 3.1 MeV as a signal for phase transition. [23] However Civitarese et al. [24] have reported particulars against the occurrence of phase transition in certain even-even nuclei belonging to the 2s-1d shell using nuclear SU 3 scheme. [25] Further, they pointed out in their calculation on specific heat that the presence of a bump at temperature of the order of 2 4 MeV is owing to finite size effect, quite to a peak structure, which substantiates a phase transition. Later work by Dukelsky et al. [26] reanalyzed the problem using an SU(3) Hamiltonian and have confirmed that there is no phase transition and the study of fluctuations is important for better analysis. Recently a spherical shell model approach analyses the nature of phase transition dependence on temperature and angular momentum in light nuclei and have found the smearing out of pairing-phase transitions due to fluctuations. [27] Thus to explore the nature of the phase transitions, we use the finite temperature statistical theory of nucleus based on microscopic approach incorporating the temperature, angular momentum, deformation degrees of freedom, collective and non-collective rotation of the system. This theory has been used in our earlier calculations pertaining to the evaluation of single particle level density parameter, [28] neutron separation energy and emission probability at high spins. [29] Moretto [8] has specified an expression for the thermodynamical potential of rotating nuclei at a finite temperature where the Lagrangian multipliers conserve the energy, particle number, and angular momentum. We have also treated the system with constant angular momentum, which is in principle possible for rather light systems within restricted configuration spaces. However, certain modifications are incorporated to the BCS Hamiltonian like pairing and the projection of angular momentum to study the impact of pairing and angular momentum on excited paired nucleus. Furthermore, BCS theory violates particle number conservation in finite nuclei and causes particle number (quantal) fluctuations even at zero temperatures. At finite temperature, statistical fluctuations also get added to the quantal one. Hence all the quantal and statistical fluctuations are taken into account in this manuscript, which is neglected by the conventional BCS approach. The method of extracting the nuclear specific heat (C v ) as a function of angular momentum and temperature is provided in this manuscript and calculations have been performed on the specific heat for the light nuclei like Ne, Mg, and 14Si. For illustrative purpose, a heavy nucleus is also considered, where the first small peak in specific heat signified as a pairing-phase transition Ne, 24 12

3 720 Communications in Theoretical Physics Vol. 56 from superfluid to normal fluid is ruled out when statistical fluctuations in the pair gap ( ) are included. Similarly, the second broad peak predicted as a signature of shape-phase transition from deformed to spherical shape, gets smooth out once statistical fluctuations in the deformation parameter (β) are included. With this basis, the presence of bump in the specific heat for some of the 2s- 1d shell nuclei are analyzed with respect to pairing-phase and shape-phase transitions. It is found that when fluctuations in the number of particles, spin, pair gap, deformation, and energy are incorporated, quantal number, spin fluctuations, and statistical fluctuations in the pair gap, deformation degree of freedom, and energy smear out the pairing-phase transition and smooth out the shape-phase transition in all the light nuclei considered. The level density is also a vital property, which has been the focus of latest publication [30] is also discussed. There have been several attempts to derive level density from microscopic models. [31] At higher excitation energies the nuclear level density increases so rapidly that it is practically impossible to study the transition between the levels and hence a statistical description becomes adequate. Most calculations of nuclear level densities are extensions and modifications of Fermi gas model to which pairing and shell effects are added semiempirically. [32] The angular momentum and temperature dependence of level densities affords a rigorous test for nuclear models. Alhassid et al. have taken efforts to obtain the angular momentum dependence of the level density microscopically. [33] We have also investigated the level densities at higher excitation energies and angular momentum within the framework of statistical model. The level density parameter a which incorporates a dependence on deformation and energy is also extracted. It very well reproduces the empirical relation a A/10 or A/8. Since our main focus is on fluctuations, the complete details of level density and level density parameter are not provided in this manuscript and we would like to refer Ref. [28] and references therein. The formalism used to extract the nuclear specific heat, level density and all the fluctuations is described in Sec. 2. In Sec. 3 the results and discussions are given. Some conclusions are drawn in Sec Formalism Statistical descriptions of finite nuclear systems are generally based on grand canonical ensemble averages. One often needs a statistics with good quantum numbers like angular momentum or particle number, which requires a use of constrained ensembles. The common procedure consists in determining first the grand partition function of the system and then in restricting it in such a way as to conserve energy, number of particles, and angular momentum. The development of statistical theory has resulted in the successful application to high spin states in light nuclei. [34] The statistical properties of the system are contained in the grand canonical partition function based on BCS Hamiltonian and is given by [28] lnq = η k (ε Z k µ Z E Z k ) + k ln{1 + exp[ η(e Z k λm Z k )]} η 2 Z/G Z, (1) where Ek Z = [(εz k µ Z) Z ]1/2 are the proton quasi-particle energies, G Z is the pairing strength, and Z is the gap parameter. The quantity η is the reciprocal of the temperature (η = 1/T) and µ Z is the chemical potential. The particle number equations for protons, the equation for angular momentum M and for energy E are Z = k M Z = k E Z = k [ 1 { (ε Z k µ Z )tanh η 2 (EZ k λm Z k )/2E Z k }], (2) {m Z k /[1 + expη(ez k λmz k )]}, (3) ε Z k { 1 (ε Z k µ Z)tanh η } 2 (EZ k λmz k )/2EZ k 2 Z /G Z. (4) The gap parameter Z is obtained as a function of η, λ Z, and µ Z by solving the gap equation tanh η 2 (EZ k λmz k )/2EZ k = 2/G Z. (5) k The above equation expresses Z as an implicit function of η, λ, and µ. The dependence of pair gap upon angular momentum is given by the equation Z = 0 (1 M Z /M C ), (6) 0 denotes the pair gap at zero temperature and zero angular momentum and M C denotes the critical angular momentum above, which the pair gap vanishes. We have ω 0 = sinh(1/gg), (7) M C = gm Z 0, (8) where g is the level density. Here we assume that pairing correlations extends over an energy interval ±ω above and below the Fermi surface. At this condition, the Coriolis

4 No. 4 Communications in Theoretical Physics 721 force creates a perturbation of the order of the pairing gap. The entropy S Z of the system is then determined as S Z = k + k ln { 1 + exp[ η(e Z k λmz k )]} η[(e Z k λm Z k )/(1 + expη(e Z k λm Z k ))].(9) A similar set of equations for neutrons also exists. The total energy E(M, T), total angular momentum M and total entropy S are obtained as E(M, T) = E N (M, T) + E Z (M, T), (10) M = M N + M Z, (11) S = S N + S Z. (12) The single particle levels for protons ε Z k with spin projections m Z k and neutrons εn k with spin projections mn k are obtained from the Nilsson Hamiltonian. [35] The excitation energy E of the system is E (M, T) = E(M, T) E 0, (13) with E 0 the ground state energy of the system. The specific heat C v as a function of angular momentum M and temperature T is extracted using the equation C v (M, T) = de (M, T) dt. (14) According to Snover, [36] the nuclear level density for various excitation energies and angular momentum is given by [37] ρ(e ) = ( 2 /2θ) 3/2 (2M + 1) a exp(2 ae ) 12(E + T) 2, (15) where θ is the rigid body moment of inertia, E is the excitation energy of the system and a is the single particle level density parameter. The single particle level density parameter can be extracted from the relation predicted in the Fermi gas model a(m, T) = S2 (M, T) 4E (M, T). (16) The free energy of the hot rotating system contains all the thermodynamic information and is computed as F(M, T) = E(M, T) TS. (17) The mean square fluctuation of any observable O is given as (δo) 2 = O 2 O 2. (18) We have considered various fluctuations like particle number, spin, pair gap, deformation, and energy and we have the following relations. Particle number fluctuations are very important at T = 0, thus having contributions from quantal and statistical fluctuations. (δn) 2 = N 2 N 2. (19) where The average pair gap ( ) is defined as P( )d 0 = P( )d, (20) 0 P( ) e F( )/kt. (21) The average deformation β for a given angular momenta M is given as β exp(s(β, γ, T, M))dβdγ 0 β = exp(s(β, γ, T, M))dβdγ. (22) 0 A system at constant temperature has fluctuations in energy represented as (δe) 2 = E 2 E 2. (23) Likewise the fluctuations in the angular momentum is expressed as (δm) 2 = M 2 M 2. (24) In our calculations, all the parameters like total energy, excitation energy, specific heat, level density, and level density parameter are evaluated as a function of angular momentum M from 0 to 16, temperature T from 0 MeV to 5 MeV and deformation parameter β from 0.6 to +0.6 (in steps of 0.1) and the shape parameter γ takes 180 (non-collective oblate) and 120 (collective prolate). The most probable values are obtained after minimizing the free energy. In this method only the z component M of the total angular momentum I is considered. The single particle spin projection (m k ) is a good quantum number only for axially symmetric shapes in nuclei. For triaxial deformation, the single particle spin projection (m k ) ceases to be a good quntum number since the matrix elements for triaxially deformed system connects states of different (m k ). However, Moretto [8] has exemplified that the laboratory fixed z-axis can be made to coincide with the body fixed z -axis and it is possible to identify and substitute M for the total angular momentum I. In the quantum mechanical limit, the z component M of the total angular momentum is M I(I + 1) = I + 1/2. An alternative way such as cranking perpendicular to the symmetry axis (triaxial cranking) can solve this problem. 3 Results and Discussion Phase transitions in light nuclei are explored with the inclusion of fluctuations in the relevant order parameters. For instance, the foremost order parameter for the superfluid to normal fluid pairing-phase transition is the pair gap and for the spherical to deformed shape-phase transition is the deformation parameter. Hence we have calculated the most probable value of deformation parameters, pair gap, specific heat, level density for some of the 2s-1d shell light nuclei and efforts have been made to understand

5 722 Communications in Theoretical Physics Vol. 56 the nature of phase transitions allied with pairing energy and quadrupole shape. The fluctuations in the order parameters are also considered and the expectation values of these parameters are extracted for each angular momentum state and temperature by the statistical average over the order parameters. of neutron and proton pair gap and they indicate the pairing-phase transition. Further increase in temperature shows a broad bump at T 2.7 MeV, which denotes the shape-phase transition. Note that all these are only artifact of the mean field approximations and in reality one cannot find any pronounced signature of phase transitions because of strong fluctuations at finite temperature. Actually by considering fluctuations into account, these discontinuities disappear and we obtain a smooth curve without any phase transition. Fig. 1 Nuclear pair gap ( ) as a function of temperature (T) for 152 Gd. Fig. 3 The nuclear specific heat (C) as a function of temperature (T) for 152 Gd. Fig. 2 The quadrupole deformation parameter (β) as a function of temperature (T) for 152 Gd. In order to check the validity of the results, a heavy nucleus (Gd) is considered first excluding fluctuations. The result of pair gap ( ) as a function of temperature (T) is displayed in Fig. 1 and it is noticed that the pair gap ( ) monotonically decreases with increasing temperature and drops down to zero at T 0.6 MeV representing pairingphase transition. In addition to the pairing-phase transition, further shape-phase transition is seen in Fig. 2 where the most probable quadrupole deformation parameter (β) with respect to temperature falls to zero at T 2.7 MeV denoting the deformed to spherical shape-phase transition. Figure 3 represents the variation of specific heat as a function of temperature where the peaks specify pairingphase and shape-phase transitions. Two closely separated peaks are seen at T 0.6 MeV caused by the collapse Fig. 4 Fluctuations in pair gap ( ) as a function of temperature (T) for 152 Gd. The effect of washing out of phase transition is better demonstrated in Fig. 4 where the statistical fluctuations in pair gap ( ) is displayed as a function of temperature. Due to statistical fluctuations in, the pair gap existing at zero temperature does not vanish at the critical temperature but decreases at higher temperature. Moreover the deformed to spherical shape-phase transition seems to occur at T 2.7 MeV is obscured by the effect of statistical fluctuations in deformation parameter (β) shown in Fig. 5. Keeping this in mind, systematic investigations on light nuclei are done to explore the nature of phase transitions and the results are the following.

6 No. 4 Communications in Theoretical Physics 723 Fig. 5 Fluctuations in quadrupole deformation parameter (δβ) as a function of temperature (T) for 152 Gd. presented in Fig. 7(a) for Ne, 10Ne, 12Mg, and 14Si. All the nuclei have a general tendency to exhibit an abrupt change in the specific heat and the peak in specific heat occurs at different temperatures for different nuclei. A small peak appears at T 0.5 MeV for all the four nuclei and a larger peak appears at T 2.0 MeV for 22 10Ne, T 2.7 MeV for Ne, and at T 3.5 MeV for 12Mg and 28 14Si. The shape of these curves is analogous to the one reported in Ref. [24] for all the light nuclei. Figure 7(b) illustrates the results of level density calculations with a sudden change around the critical temperature for all the nuclei. It is obvious that the abrupt change in the level density coincides with the peaks in the specific heat for all the nuclei. Thus the peaks in the specific heat are the result of a sudden increase in the many body level density around the critical temperature. Thermal excitation induces change in the nuclear shape and the peak in specific heat appears as a result of change in level density allied with thermal excitation. Fig. 6 The excitation energy (E ) as a function of temperature (T) for various angular momentum (M) for 20 10Ne, 22 10Ne, 24 12Mg, and 28 14Si. The numbers on the curve refer to the angular momentum in units of. The numerical results of excitation energy E as a function of temperature (T) and angular momentum (M) for the nuclei Ne, 10Ne, 12Mg, and 14Si are presented in Fig. 6 and it shows a rapid change of slope for M = 0 at T 0.5 MeV and T MeV for all the four nuclei. This consents moderately with a canonical ensemble average of energy obtained in FTHF calculations of Miller et al. [38] They have noticed a change in the slope at T 0.5 MeV due to the contributions of ground state rotational band and the change at T 3.0 MeV as a signal for a deformed to spherical phase transition. [38] The reason for this transition has been interpreted as the tendency of the system to become less deformed as temperature increases and at a specific critical temperature it experiences a deformed to spherical phase transition. To verify further the nature of this transition, the nuclear specific heat C versus temperature for M = 0 is Fig. 7 (a) The nuclear specific heat (C) as a function of temperature (T) for 20 10Ne, 22 10Ne, 24 12Mg, and 28 14Si. (b) The nuclear level density (ρ) as a function of temperature (T) for 20 10Ne, 22 10Ne, 24 12Mg, and 28 14Si. Miller et al. have also computed the specific heat for 20 10Ne using the canonical ensemble from the eigenstates of different effective interactions [23] and they have pointed out similar peaks in specific heat and in analogy to the excitation energy plots, the small peak at T 0.5 MeV is related to the ground state rotational band and the other one at T 2.1 MeV to the deformed to spherical shapephase transition. This is comparable to the mean field results where a sudden pairing-phase transition occurs at a particular temperature. It is worth mentioning here

7 724 Communications in Theoretical Physics Vol. 56 that the influence of pairing correlations and the effects of quantal and statistical fluctuations are completely not discussed in their papers. Indeed exact analysis does not predict a sharp pairing-phase transition owing to fluctuations. Essentially pairing correlation is clearly observed to play an outstanding role in light nuclei and recent shell model studies of pairing correlation for light nuclei like Ne, 14Si represent a well defined smooth decrease around T = 2 MeV in the pair energy vs. temperature plot which can be understood as the sharp wash out of pairing-phase transition. [27] Hence fluctuations in appears to be a powerful statistical tool to explore pairing-phase transitions in light nuclei and it will be interesting to notice how these fluctuations affect the existence of phase transition. field, which are more established than the mean field itself. The decrease in the pair gap is associated with multiple breaking of Cooper pairs and the crossing of aligned configurations with the ground state band. The average of these crossing leads to a smooth drop and hence for high spin states pairing will not vanish due to strong antipairing effect. It is also evidenced from the Fig. 9 which shows the variation of deformation parameter (β) with respect to temperature with and without fluctuations. The deviation of the most probable quadrupole deformation parameter (β) as a function of temperature without fluctuations falls zero at T 3.5 MeV but once fluctuations are included it is not approaching zero and the signature of deformed to spherical shape-phase transition is suppressed. Fig. 9 The continuous curve corresponds to the variation of the most probable quadrupole deformation parameter (β) for various temperature (T) for 22 10Ne. The dotted curve represents to the fluctuations in quadrupole deformation parameter (δβ) as a function of temperature (T) for 22 10Ne. Fig. 8 (a) Pair gap ( ) as a function of temperature (T) for 22 10Ne. (b) Fluctuations in pair gap ( ) as a function of temperature (T) for 22 10Ne. For light nuclei, both quantum as well as statistical fluctuations are important. Of particular importance are the statistical fluctuations in the pair gap ( ), which are prominent at finite temperature. Calculations performed for all the four light nuclei considered exhibits the same behavior and so for illustrative purpose, we have presented the results for 22 10Ne. The temperature dependence of pair gap ( ) for 22 10Ne with and without fluctuations is depicted in Fig. 8 and it shows the monotonous decrease in pair gap with increasing temperature and at one particular temperature pairing has got completely vanished representing a phase transition from paired to unpaired configuration, but statistical fluctuations in pair gap when considered become non zero even at high temperature thus the occurrence of pairing-phase transition is safely ruled out. This is because in very small systems, most of the pair correlations are generated by the fluctuations of the pair Correspondingly, quantal number and spin fluctuations are presented in Fig. 10. At T = 0, δn = 0 and it shows that fluctuation increases with temperature without any transition. Similar effects are displayed in Fig. 11 where it is obvious that the energy fluctuations completely wash out the phase transitions. Hence we conclude that fluctuation effects exclude the sharp disappearance in pairing and smooth out many significant statistical features such as individual pair breaking and deformation collapse. Thus the interplay between phase transitions and fluctuations along with pairing correlations and deformation forces are significant for a better understanding of phase transitions. Nevertheless, recent article shows that the pair correlations, which are quenched at zero temperature and high rotational frequency reappear at higher temperature. [39] But the present result in light nuclei does not show such a remarkable rise since in the s-d shell nuclei it is not possible to have very large angular momentum. Moreover, the experimental information about pairing correlations at high spin should be considered as very limited today.

8 No. 4 Communications in Theoretical Physics 725 Further investigations are needed for the complete understanding of the nature of phase transitions. In our model, the grand canonical ensemble is used which is perhaps the most popular in finite temperature calculations but fluctuations in particle number is very significant in a system with small number of nucleons. For calculating finite temperature properties of light nuclei, canonical ensemble is physically most relevant. On the other hand, the well known argument that nuclear temperature should be derived from micro canonical ensemble is quite often under debate. Also, our modified BCS theory does not include particle number projection and the canonical partition function can be obtained by performing the number projection, which is the future scope of this work. Fig. 11 Energy fluctuations (δe) as a function of temperature (T) for 22 10Ne. Fig. 10 (a) Number fluctuations (δn) as a function of temperature (T) for 22 10Ne. (b) Spin fluctuations (δm) as a function of angular momentum (M) for 22 10Ne. 4 Conclusion Calculation on the temperature and angular momentum dependence of deformation parameters, pair gap, specific heat, level density are made for some of the 2s-1d shell light nuclei mainly with respect to their ability to describe the phase transition and the results are discussed with the inclusion of fluctuations. Among them, the influence of statistical fluctuation that wash out the collapse of pairing correlation has been investigated in detail and the effect of washing out of the pairing-phase transition and smoothening of shape-phase transition due to fluctuations in pair gap and deformation is clearly demonstrated. For all the nuclei considered the effect of washing out the sharp phase transition from superfluid to normal driven by temperature is obvious and so we conclude that there is no true phase transitions and this may be due to the finiteness of space involved in the calculation. References [1] V. S. Ramamurthy, S.S. Kapoor, and S.S. Kataria, Phys. Rev. Lett. 25 (1970) 386; D. Mocelj, et al., Phys. Rev. C 75 (2007) [2] S. Levit and Y. Alhassid, Nucl. Phys. A 413 (1984) 439; J.L. Egido and P. Ring, J. Phys. G: Nucl. Part. Phys. 19 (1993) 1. [3] W. Dilg, W. Schatl, H. Vonach, and M. Uhl, Nucl. Phys. A 217 (1973) 269. [4] A.L. Goodman, Nucl. Phys. A 352 (1981) 30. [5] H.G. Miller, R.M. Quick, and G. Bozzolo, and J.P. Vary, Phys. Lett. B 168 (1986) 13. [6] C. Gregoire, T.T.S. Kuo, and D.B. Stout, Nucl. Phys. A 530 (1991) 94. [7] L.D. Landau and E.M. Lifshitz, Statistical Physics, Pergamon, Oxford (1980). [8] L.G. Moretto, Nucl. Phys. A l82 (1972) 641; A 216 (1973) 1; A l85 (1972) 145; A 180 (1972) 337. [9] A.L. Goodman, Phys. Rev. C 37 (1988) [10] J.L. Egido, C. Dorso, J.O. Rasmussen, and P. Ring, Phys. Lett. B 178 (1986) 139. [11] B.K. Agrawal, Tapas Sil, J.N. De, and S.K. Samaddar, Phys. Rev. C 62 (2000) [12] V. Zelevinsky, B.A. Brown, N. Frazier, and M. Horoi, Phys. Rep. 276 (1996) 85. [13] R. Rossignoli and P. Ring, Ann. Phys. (N.Y.) 235 (1994) 350. [14] R. Rossignoli, N. Canosa, and P. Ring, Phys. Rev. Lett. 80 (1998) 1853.

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