Using the method of the second quantization to investigate the phenomenon of superfluidity in asymmetric nuclear matter

Abstract. In this paper we use the method of the second quantization to investigate possibility of superfluidity formation in the asymmetric nuclear matter. Using quasi-spin formalism to study the problem of deuteron condensation and applying the Hatree - Fock - Bogoliubov approximation to investigate the exceeding neutron system in the deuteron condensation, we obtain the total energy E and state vector ψ of the initial asymmetric nuclear matter. After that, we investigate the onset of superfluidity in the exceeding neutron system. Finally, these results of numerical calculation of the gap function ∆, the excited energy of a quasi-particle pair, the critical temperature TC have been discussed and compared with that of other authors.

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JOURNAL OF SCIENCE OF HNUE Natural Sci., 2010, Vol. 55, No. 6, pp. 44-56 USING THE METHOD OF THE SECOND QUANTIZATION TO INVESTIGATE THE PHENOMENON OF SUPERFLUIDITY IN ASYMMETRIC NUCLEAR MATTER Le Cong Tuong(∗), Pham Thi Nu Hanoi National University of Education Vo Hanh Phuc, Tran Duy Khuong Institute of Nuclear Science and Technology (∗)E-mail: tuonglc@hnue.edu.vn Abstract. In this paper we use the method of the second quantization to investigate possibility of superfluidity formation in the asymmetric nu- clear matter. Using quasi-spin formalism to study the problem of deuteron condensation and applying the Hatree - Fock - Bogoliubov approximation to investigate the exceeding neutron system in the deuteron condensation, we obtain the total energy E and state vector ψ of the initial asymmetric nuclear matter. After that, we investigate the onset of superfluidity in the exceeding neutron system. Finally, these results of numerical calculation of the gap function ∆, the excited energy of a quasi-particle pair, the criti- cal temperature TC have been discussed and compared with that of other authors. Keywords: nuclear matter, superfluidity. 1. Introduction The possible onset of superfluidity in nuclear matter was investigated soon after the formulation of the Bardeen, Cooper and Schrieffer (BCS) theory of super- conductivity [1] and the pairing correlation theory of atomic nuclei [1]. According to BCS approach [3], applied to nuclear matter, one showed that a superfluid phase could exist for a wide class of nucleon-nucleon potentials [4]. At low temperatures as it is well known in nuclei, nuclear matter, and neutron matter (neutron stars) superfluidity can arise in the isospin triplet channel [5]. In the traditional understanding, superfluidity of finite nuclear matter is re- garded as superfluidity of the two independent systems of protons and neutrons, in which the protons and neutrons form pp and nn Cooper pairs. In some recent researches on the superfluidity in symmetric nuclear matter [6, 7], one assumed protons and neutrons forming np Cooper pairs. The issue of superfluidity in nuclear matter was regarded for a long time until Midgal pointed out that a possible laboratory for the superfluidity of, at least, 44 Using the method of the second quantization to investigate the phenomenon... neutron matter could be the neutron star phenomenology, where pairing could bring about many interesting macroscopic phenomena. It was shown in neutron star physics that the presence of neutron and proton superfluid phase has been invoked to explain the dynamical and thermal evolution of a neutron star [8]. We treat the asymetric nuclear matter, there are more neutrons than protons. In nature, all heavy and stable nuclei have more neutrons than protons, so that from the solution of the superfluidity problem of asymetric nuclear matter, we can investigate the superfluidity of the finite nuclear matter (superfluidity of heavy nu- clei). We assume that all protons pairs with neutrons forming deuteron with total momentum 2Ko. This is acceptable because deuteron is only a bound state of nu- cleon in nature. At low densities, deuterons can be regarded as quasi-bosons, and they would concentrate at the same energy level. In this article, we suppose that the proton-neutron interaction is nuclear interaction, so deuteron system can only exist in the condensed phase and does not exist in superfluid phase. We now consider the deuteron system which has some exceeding neutrons. As we have shown that the deuteron system can be regarded as a condensed tank with respect to the exceeding neutron system. The exceeding neutrons move in the external field which is established by the deuteron system. These neutrons interact with each other and with the condensed deuteron system by this external field. We assume that the number of exceeding neutrons are even, and all these neutrons form Cooper pairs. We use Hartree-Fock Bogoliubov approximation to calculate the pairing interaction. Therefore, in the frame work of some approximations, the superfluid problem of the asymmetric proton-neuton system can be simplified as the following problem: investigate the possibility to form the superfluidity of the exceeding neutron pairing system, above the condensed deuteron system. In this paper, we use the second quantization method, particularly using the quasi-spin formular [2, 3], to investigate the onset of superfluidity of pairing neutrons, based on BCS mechanism, over the deuteron condensed system. 2. Content 2.1. Hamiltonian of Asymmetric Nuclear System Hamiltonian of asymmetric nuclear systems can be written as H = HD +HN (2.1) where HD describes the deuteron condensation system, HN describes the exceed- ing neutron system pairing completely, by BCS mechanism, above the deuteron condensation system. HD = −D ·ND = −D ·D+2K0 ·D2K0 = − D Ω · ∑ q,q′ a+K0+q,↑a + K0−q,↓ aK0−q′,↓aK0+q′,↑ (2.2) 45 Le Cong Tuong, Pham Thi Nu, Vo Hanh Phuc and Tran Duy Khuong HN = ∑ k (k − µ) · (a+k,↑ak,↑ + a+−k,↓a−k,↓) + ∑ k,k′ V (k, k′)a+k′,↑a + −k′,↓a−k,↓ak,↑ (2.3) Here −D is a constant, and plays a role of the deuteron binding energy; Ω is the number of plane waves forming deuteron wave packet; k is energy of an exceeding neutron moving in the mean-field which is established by the exceeding neutrons system and the external field- by the deuterons system. If we denote HoN as the Hamiltonian describing the exceeding neutron system moving in the condensed deuteron tank, k -eigenvalue of H o N corresponding to the eigenfunctions ϕk, and ignore the BCS pairing interaction, the Schro¨dinger equation reads HoN = kϕk (2.4) k = (~ · k)2 2mN + VD + VN = (~ · k)2 2mN + Veff (2.5) where VD and VN are external fields formed by the deutrons system and the mean- field of the exceeding neutrons system respectively, Veff is the total field of them. Veff is the phenomenological potential and its concrete form will be chosen later. At the low-density, deuteron behaves as a quasi-boson. It means that deuteron operators commute with fermion operators. In other words, HD can be regarded as commutating with HN . So the wave function describing our system can be written in the form of product of two wave functions ψ = ψD · ψN (2.6) where ψD describes the deuteron condensation system, ψN describes the exceeding neutron system pairing completely, induced by BCS mechanism. The Schro¨dinger equation is written as follows H · ψ = (HD +HN)ψDψN = HDψDψN +HNψDψN = (ED + EN )ψDψN = EψDψN (2.7) A such our problem can be divided into two small problems: ∗ Find the eigenfunction ψD and the eigenvalue ED of operator HD, which describes the deuterons in condensed state. ∗ Find the eigenfunction ψN and the eigenvalue EN of operator HN , which describes the exceeding neutron pairing system, induced by BCS mechanism, in the deuteron condensation. 46 Using the method of the second quantization to investigate the phenomenon... 2.1.1. Problem 1 The wave function describes the m deuterons system with momentum 2~Ko can be written as simple as follows: ψD = (D + 2Ko )m | 0〉 (2.8) where | 0〉 is the completely vacuum state. Using the quasi-spin formalism [9], we determine that the wave function ψD is the eigenfunction of HD with the corresponding eigenvalue ED: ED = −mD · (1− m− 1 Ω ) ≈ −mD (2.9) where Ω is the number of fermi plane waves forming the deuteron wave packet. When Ω  m, the deuterons system can be regarded as the boson gas moving in its mean-field. 2.1.2. Problem 2 For convenience, the state of neutron is described by the wave vector k with upward spin as K, and by the wave vector −k with downward spin as −K. Hamil- tonian operator HN of the exceeding neutron system is given by (2.3) is written as follows: HN = ∑ K (K − µ) · (a+KaK + a+−Ka−K) + ∑ K,K ′ V (K,K ′)a+K ′a + −K ′a−KaK (2.10) We will make a transformation of operator HN to a new form, which describes the physical characteristics of the exceeding neutron system conveniently, namely the Bogoliubov transformations. Instead of the operators a+K , aK , we use the new ones: creation α+K and annihilation αK operator, in the forms:{ αK = uKaK − vKa+−K α−K = uKa−K + vKa + K (2.11) or { α+K = uKa + K − vKa−K α+−K = uKa + −K + vKaK (2.12) where the uK , vK coefficients are real and satisfy the following conditions uK = u−K , vK = −v−K , and u2K + v2K = 1. (2.13) We can perform reverse transformations 47 Le Cong Tuong, Pham Thi Nu, Vo Hanh Phuc and Tran Duy Khuong { aK = uKαK + vKα + −K a−K = uKα−K − vKα+K (2.14) or { a+K = uKα + K + vKα−K a+−K = uKα + −K − vKαK (2.15) Substitute (2.14) and (2.15) into (2.10) we write the expression of the Hamil- tonian HN as HN = H0 +H1 +H2 +H3 +H4 (2.16) where H0 = ∑ K 2(K − µ)v2K + ∑ K,K ′ V (K,K ′)uKvKuK ′vK ′ H1 = ∑ K{(K − µ)(u2K − v2K)− 2uKvK ∑ K ′ V (K,K ′)uK ′vK ′}(NK +N−K) H2 = ∑ K,K ′ V (K,K ′)uKvKuK ′vK ′(NK +N−K)(NK ′ +N−K ′) H3 = ∑ K(α + Kα + −K + α−KαK) · hK hK = 2(K − µ)uKvK + (u2K − v2K) ∑ K ′ V (K,K ′)uK ′vK ′(1−NK ′ −N−K ′) NK = α + KαK , N−K = α + −Kα−K H4 = ∑ K,K ′ V (K,K ′)(u2Kα + Kα + −K − v2Kα−KαK)(u2K ′α−K ′αK ′ − v2Kα+K ′α+−K ′) (2.17) Here NK is called the quasifermion number operator. In the ground state of the exceeding neutron no quasifermion exists, so that the wave function ψ0N describing the ground state is also the wave function describing the vacuum state of the quasi-fermions system (the vacuum in analogy). Similarity to the theory of superconductivity, the wave function describes the exceeding neutron system pairing completely, can be written as: ψ0N = ∏ K (uK + vKa + Ka + −K) | 0〉 (2.18) Note that because HD and HN is commutative, so the wave function φ0 = ψ0N · ψD also the wave function describes the vacuum state of the quasifermions system. If φK is the eigenfunction corresponding the eigenvalue nK of the NK operator, we have: NKφK = nKφK , nK = 0, 1 φK = (α + Kα + −K) nKφ0, αKφ0 = 0, α−Kφ0 = 0 (2.19) Because the system of eigenfunction φK is ortho-normalized, we have 48 Using the method of the second quantization to investigate the phenomenon... 〈φK, φK〉 = 1 〈φK, α+Kα+−KφK〉 = 0 〈φK, α−KαKφK〉 = 0 (2.20) The terms in H3 contain (α + Kα + −K+α−KαK) which is not diagonalized, we can choose uK (or vK) in such as a way to the eigenvalue of hK is zero, i.e 2(K − µ)uKvK + (u2K − v2K) ∑ K ′ V (K,K ′)uK ′vK ′(1− nK ′ − n−K ′) = 0 (2.21) here nK ′, n−K ′ = 0, 1. From condition (21) we can write: 〈φ,H3φ〉 = 0 (2.22) with φ = φ(· · · , nK , n−K , · · · , nK ′, n−K ′, · · · ) = ∏ K φK = ∏ K (α+Kα + −K) nKφ0 (2.23) is the wave function describing the excited state of the exceeding neutron system. Using conditions (2.20) and (2.21), we obtain the energy of the exceeding neutron system EN EN = E0 + E1 + E2 (2.24) where E0 = ∑ K 2(K − µ)v2K + ∑ K,K ′ V (K,K ′)uKvKuK ′vK ′ E1 = ∑ K{(K − µ)(u2K − v2K)− 2uKvK ∑ K ′ V (K,K ′)uK ′vK ′}(nK + n−K) E2 = ∑ K,K ′ V (K,K ′)uKvKuK ′vK ′(nK + n−K)(nK ′ + n−K ′) (2.25) We assume that nK is equal to n−K and then replace nK by fK which is the average number of quasi-fermion at K-state. Using some simple transformation, we can rewrite the expression of energy of the exceeding neutron system as EN = ∑ K(K − µ)− ∑ K(K − µ)(u2K − v2K)(1− 2fK) + ∑ K,K ′ V (K,K ′)uKvKuK ′vK ′(1− 2fK)(1− 2fK ′) (2.26) And the condition (2.21) can be written as follows: 2(K − µ)uKvK + (u2K − v2K) ∑ K ′ V (K,K ′)uK ′vK ′(1− 2fK ′) = 0 (2.27) 49 Le Cong Tuong, Pham Thi Nu, Vo Hanh Phuc and Tran Duy Khuong Using the formula (2.26) and the condition (2.27) we will determine the energy EN of the exceeding neutron system. We have obtained the total enery E and wave function ψ of the initial asym- metric nuclear system. In the next section, we will investigate the possibility of forming the superfluidity of the exceeding neutron system in the condensed deuteron tank. 2.2. Superfluidity of the exceeding neutron system First, we will calculate the entropi S and the free energy F = E − TS of the exceeding neutron system. From the minimum condition of the free energy ( ∂F ∂fK = 0), we obtain the dependence of the distribution function fK on temperature T . Using some simple transformation we also get expressions of coefficients uK , vK , energy of a quasi-particle ξK , energy of the exceeding neutron system EN and an equation of gap function ∆K as u2K = 1 2 {1 + (K − µ) [(K − µ)2 +∆2K ]1/2 }, v2K = 1 2 {1− (K − µ) [(K − µ)2 +∆2K ]1/2 } (2.28) ξK = [(K − µ)2 +∆2K ]1/2 (2.29) EN = ∑ K (K − µ)− ∑ K { (K − µ) 2 [(K − µ)2 +∆2K ]1/2 + ∆2K 2[(K − µ)2 +∆2K ]1/2 } · tanh[ [(K − µ) 2 +∆2K ] 1/2 2kBT ] (2.30) ∆K = −1 2 ∑ K ′ V (K,K ′) ∆K ′ [(K ′ − µ)2 +∆2K ′]1/2 tanh[ [(K ′ − µ)2 +∆2K ′]1/2 2kBT ] (2.31) The pairing function of the exceeding neutron system is defined κ(K) = 〈φ, a+Ka+−Kφ〉 = ∆K 2ξK tanh[ ξK 2kBT ] (2.32) The solution of equation (2.31) depends on parameters such as chemical poten- tial µ , temperature T , form of the potential interaction between neutrons V (K,K) and potential Veff . In the following calculation for the exceeding neutron system in S10 channel, we will use the potential interaction V (K,K) in the same form of Yamaguchi potential within the first order approximation [8] 50 Using the method of the second quantization to investigate the phenomenon... V (K,K ′) = −λν(K)ν(K ′) (2.33) where ν(K) = 1 K2+α2 , λ, α are coefficients. For the infinite nuclear matter, we can assume that the condensation deuteron tank is a spherical symmetric system. So the potential Veff can be chosen as a spherical symmetric potential with its Fourier transformation as follows Veff(K) = − V0 [(K − δ)2 + β2]2 (2.34) where V0, δ, β are coefficients. The dependence of the potental Veff on momentum is shown at Figure 1. 0 2 4 6 8 10 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 Th e ef fe ct iv e po te nt ia l V ef f [M eV ] k[1/fm] Figure 1. The dependence of the effective potental Veff on momentum The minimum energy to destroy a neutron pair on the Fermi surface is 2∆(kF ) = 2∆F . Since most physical processes involve transitions around the Fermi surface, we mainly focus on ∆F . Figure 2 shows the dependence of gap function on momentum with different values of chemical potential. The curve describing the dependence of pairing function κ(k) on momentum at temperature T = 0.1MeV , chemical potential µ = 2MeV is showed at Figure 3. The pairing function gets its peak around Fermi momentum. Figure 4 describes the dependence of the pairing function on momentum with different values of chemical potential. 51 Le Cong Tuong, Pham Thi Nu, Vo Hanh Phuc and Tran Duy Khuong 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 G ap fu nc tio n [M eV ] k[1/fm] T=0.1 MeV, Chemical potential =2 MeV T=0.1 MeV, Chemical potential =2 MeV T=0.1 MeV, Chemical potential =2 MeV Figure 2. The dependence of gap function on momentum with different values of chemical potential 0 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 Pa iri ng fu nc tio n k[1/fm] T=0.1 MeV, Chemical potential = 2MeV Figure 3. The curve describing the dependence of pairing function κ(k) on momentum at temperature T = 0.1MeV , chemical potential µ = 2MeV 52 Using the method of the second quantization to investigate the phenomenon... 0 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 Pa iri ng fu nc tio n k[1/fm] T=0.1 MeV, Chemical potential =2 MeV T=0.1 MeV, Chemical potential =10 MeV T=0.1 MeV, Chemical potential =15 MeV Figure 4. The dependence of the pairing function on momentum with different values of chemical potential 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 12 14 G ap fu nc tio n [M eV ] k F [MeV] T=0.1 MeV Figure 5. The curves describing dependence of gap function on Fermi momentum In the weak-pairing approximation, we assume that value of the gap function 53 Le Cong Tuong, Pham Thi Nu, Vo Hanh Phuc and Tran Duy Khuong at Fermi momentum ∆(kF ) = ∆F can be determined only by interaction between particles near Fermi surface k = kF . This is acceptable because the gap function gets its maximum value around the Fermi momentum (Figure 2). The curves describing dependence of gap function on Fermi momentum and chemical potential are shown at Figure 5 and Figure 6. 0 25 50 75 100 0 2 4 6 8 10 12 14 G ap fu nc tio n [M eV ] Chemical potential [MeV] T=0.1 [MeV] Figure 6. The curves describing dependence of gap function on chemical potential 0.0 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 5 6 7 8 Cr iti ca l T em pe ra tu re [M eV ] k F [MeV] Figure 7. The curves describing dependence of the critical temperature TC on Fermi momentum 54 Using the method of the second quantization to investigate the phenomenon... At the critical temperature TC , the gap function vanishes, so the equation (29) becomes: 1 = −1 2 ∑ K ′ V (K,K ′) 1 (K ′ − µ)tanh[ (K ′ − µ) 2kBT ] (2.35) Equation (2.35) allows to determine the critical temperature TC . The curves describing dependence of the critical temperature TC on Fermi momentum and chem- ical potential are shown at Figure 7 and Figure 8. The critical temperature TC > 0 when chemical potential µ decreases to zero. The critical temperature TC only vanishes if chemical potential gets a negative value. -20 0 20 40 60 80 100 0 1 2 3 4 5 6 7 8 Cr iti ca l T em pe ra tu re [M eV ] Chemical potential [MeV] Figure 8. The curves describing dependence of the critical temperature TC on chemical potential 3. Conclusion In this paper, we use the method of second quantization to investigate pos- sibility of superfluidity formation in the asymmetric nuclear matter. Using the quasi-spin formalism to study the problem of condensation deuteron and applying the Hatree Fock Bogoliubov approximation to investigate the superfluidity of the exceeding neutron system in which neutrons are completely matched in pairs due to BCS mechanism in the condensation deuteron tank, we obtain the total energy E and state vector ψ of the initial asymmetric nuclear matter. We also investigate the 55 Le Cong Tuong, Pham Thi Nu, Vo Hanh Phuc and Tran Duy Khuong possibility of superfluidity formation of the exceeding neutron system. These results of numerical calculation of the gap function ∆, the excited energy of a quasi-particle pair, the critical temperature TC are quite suitable to results of other authors. REFERENCES [1] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175 [1] A. Bohr, B. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936 [2] N.N. Bogoliubov, V.V. Tolmachev and D.V.Shirkov, 1959. New Method in the Theory of Superconductivity, Consultants Bureau, New York (1959). L.N. Cooper, R.L. Mills and A.M. Sessler, Phys. Rev. 114 (1959) 1377 [3] V.J. Emery and A.M. Sessler, Phys. Rev. 119 (1960) 248 [4] A. Bohr and B.R. Mottelson. Nuclear Structure (Benjamin, New York, 1969), Vol. I; P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer, New York, 1980). [5] S.L. Shapiro and S.A. Teukolsky, 1983. Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects (Wiley, New York, 1983); Neutron Stars, edit by D. Pines, R. Tamagaki anh S. Tsurata (Addison- Wesley, New York, 1992) [6] U. Lombardo, Superfluidity in Nuclear Matter, in Nuclear Methods and nuclear Equation of State, e