V ( r i, r j ) U[ρ( r i )]. Then particle i feels an average potential which depends on the particle density. ρ( r) = j. - PDF

Chapter 8 Mean Field Theory The concept of mean field theory is widely used for the description of interacting many-body systems in physics. The idea behind is to treat the many-body system not by summing

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Chapter 8 Mean Field Theory The concept of mean field theory is widely used for the description of interacting many-body systems in physics. The idea behind is to treat the many-body system not by summing up all mutual two-body interactions of the particles but to describe the interaction of one particle with the remaining ones by an average potential created by the other particles. Let s consider particle i at position r i which feels the potential U created by particles j U( r i ) = j V ( r i, r j ) U[ρ( r i )]. Then particle i feels an average potential which depends on the particle density ρ( r) = j Ψ j( r)ψ j ( r) at its position. The problem is now treated in terms of a mean field potential U[ρ] which is a functional of the density ρ. Mean field theory is very efficient for the description of manybody systems like finite nuclei or infinite nuclear matter, respectively neutron matter which occurs in the interior of neutron stars. The task is now to find the correct density functional U[ρ] which minimises the many-body Hamiltonian. That such a density functional exists can be proven within density functional theory (Kohn-Sham Theory [2, 1]), how it looks like depends on the problem. To find a functional which comes as close as possible to the - in principle existing - exact solution is the task to be solved what is in general very difficult. An additional caveat for nuclear systems is thereby that the existence theorem, the Hohenberg-Kohn theorem [3] has been proven for particles in an external field, e.g. atoms in the electromagnetic field. That this theorem holds as well for self-bound systems, such as nuclei, is assumed but has not been proven explicitly. In the following we discuss well established and very successful relativistic mean field models for nuclear systems. However, before coming to the models in detail, we briefly sketch some basic features of nuclear many-body systems. 39 40 CHAPTER 8. MEAN FIELD THEORY k z k x Figure 8.1: Schematic representation of the local momentum distribution of infinite isospin symmetric nuclear matter (Fermi sphere) where each momentum state is four times degenerate. 8.1 Basic features of infinite nuclear matter The simplest, however, already highly non-trivial system, is infinite nuclear matter or infinite neutron matter. In this case the infinite system is homogeneous and isotropic and therefore the wave functions are given by plane waves e i k x. The particle density is obtained by the sum over all occupied states inside the phase space volume (2π) 3 ρ = N/V = k,λ Ψ k,λ ( r)ψ k,λ( r) γ (2π) 3 d 3 k n( k). (8.1) In the continuum limit the sum in (8.1) is replaced by the integral over the momentum distribution n( k). Since we deal with fermions the quantum states inside the volume (2π) 3 have to be different. Therefore all states k are occupied up to the Fermi momentum k F. Therefore the distribution of occupied states if given by the Fermi sphere with radius k F n( k) = Θ(k F k ). Evaluating (8.1) leads to the following relation between density and Fermi momentum ρ = γ 6π 2k3 F (8.2) where γ is a degeneracy factor. For isospin symmetric nuclear matter with equal number of protons and neutrons is γ = 4 which means that each momentum state k can be occupied 8.1. BASIC FEATURES OF INFINITE NUCLEAR MATTER 41 0 E/A [MeV] ρ/ρ 0 Figure 8.2: Schematic representation of the equation of state of isospin symmetric nuclear matter. by four states (proton and neutron, both with spin up/down). Consequently, for pure neutron matter the spin-isospin degeneracy is γ = 2. The key quantity which describes the properties of infinite matter is the Equation of State (EOS), i.e. the energy density ǫ as a function of particle density. The energy density is just the sum of kinetic energy and the mean field ǫ(ρ) = γ d 3 k k 2 (2π) 3 2M n( k) + U[ρ] One can also characterise the equation of state in terms of pressure density instead of energy density which is usually done in hydrodynamics when ideal fluids or gases are described. Both descriptions are equivalent since energy density and pressure are related by thermodynamical relations. In nuclear physics, however, it is more practical to use the energy density or even the energy per particle E/A = ǫ/ρ = 3k2 F + U[ρ]/ρ (8.3) 10M to characterise nuclear matter. In (8.3) we evaluated the integral for the kinetic energy and it should be noted that this is the non-relativistic expression for the energy per particle. From the existence of stable nuclei it follows that the energy per particle E/A must have a minimum. This point is called the saturation point. The value, i.e. the binding energy per nucleon, can be derived from the Weizäcker mass formula for finite nuclei E = a 1 A + a } {{ } 2 A 2 3 } {{ } volume surface + a 3 Z 2 A 1/3 } {{ } coulomb + a 4 (A Z) 2 A } {{ } symmetry + λ a 5 A 3/4 } {{ } pairing energy. (8.4) Just as a reminder, the Weizäcker mass formula contains five terms and with these five terms it provides a rather accurate fit to the periodic table of stable nuclei. The contributions are: the volume term describing the nuclear bulk properties, i.e. the conditions 42 CHAPTER 8. MEAN FIELD THEORY in the interior of a heavy nucleus. The surface term accounts for the surface tension as it exists also in a liquid drop. The symmetry term arises from the isospin dependence of the nuclear forces (exchange of isovector mesons) and it scales with the difference of proton and neutron number. The pairing term is due to the phenomenon of super-fluidity which exists also in nuclear systems and will be discussed in detail in Chapter XXX. Since we consider infinite, isotropic and homogeneous nuclear matter only the volume term contributes and the Weizäcker mass formula gives us a value of E/A 16 MeV at the saturation point. Also the density where this minimum occurs is known from electron scattering on finite nuclei. It is the density in the interior of heavy nuclei, e.g. in 208 Pb. This so-called saturation density is about ρ fm 3. Thus a key requirement for a realistic density functional is to meet the nuclear saturation properties: ρ fm 3 ; k F fm 1 = 260 MeV; and E/A(ρ 0 ) 16 MeV. Figure 8.3: Nuclear matter at supra-normal densities does not exists in nature except in neutron stars and supernovae explosions (left: Chandra X-Ray space telescope picture of the Crab nebula with a radio pulsar neutron star in the centre, the remnant of a supernova explosion of a star with 8 12 solar masses in the year 1054.). For short times superdense matter can be created on earth by means of relativistic heavy ion collisions (right: simulation of a relativistic heavy ion collision of Au+Au at the Relativistic Heavy Ion Collider RHIC at Brookhaven/USA). Intuitively the value of the saturation density is easy to understand: with a radius of about 1.2 fm the volume occupied by a nucleon is about 8 fm 3 and hence the density where the nucleons start to touch is 1/8 fm 3. From the Van der Waal s like behaviour of the nuclear forces (see Fig. 7.3) it is intuitively clear that the configuration where the nucleons 8.1. BASIC FEATURES OF INFINITE NUCLEAR MATTER 43 start to touch is the energetically most favourable one. Here the contribution from the strong intermediate range attraction is maximal. If the matter is further compressed the nucleons start to feel the repulsive hard core of the potential. This is the reason why under normal conditions nuclear systems do not exist in nature at densities which exceed the saturation density ρ 0. However, there exists one exception and these are neutron stars. In neutron stars the gravitational pressure is able to compress nuclear matter up to five to ten times saturation density as model calculations show. In such model calculations the neutron star properties depend crucially on the high density behaviour of the nuclear EOS. The investigation of such highly compressed dense matter is therefore a hot topic of present research, both theoretically and experimentally. From the experimental side the only way to create supra-normal densities are energetic heavy ion collisions where two ions are accelerated and shot on top of each other. One believes that in such reactions densities between two up to ten times saturation density are reached, depending on the bombarding energy. The problem is, however, that the dense system exists only for a very short period before it explodes. 44 CHAPTER 8. MEAN FIELD THEORY 8.2 The σω-model The σω-model is a very transparent and efficient model, describing nuclear matter, neutron stars and finite nuclei. The σω-model contains only two mesons, namely the well known scalar σ and vector ω meson. Based on only these two mesons it is the simplest version of Quantum Hadron Dynamics (QHD) which contains, however, already at that level all relevant aspects of relativistic nuclear dynamics. Sometimes the σω-model is also called Walecka model, who developed the first version of QHD in 1974 [4]. However, the original idea of an effective scalar and vector exchange goes even back to the year of 1956 (Dürr 1956 [5]). Originally the attempt of QHD was to formulate a renormalizable meson theory of strong interactions. In the meantime it is, however, considered as an effective theory which should only be applied at the mean field level. It is effective in the sense that the coupling constants of the σ and ω mesons with the nucleon are not determined from free nucleon-nucleon scattering as in the case of the boson-exchange potential discussed in the previous chapter but are treated as free parameters which can be adjusted to the properties of nuclear matter, in particular the nuclear saturation point. In this sense the σω-model provides the simplest form of a relativistic density functional for the nuclear EOS (8.3). First we give some literature concerning the σω-model: J.D. Walecka, Theoretical Nuclear and Subnuclear Physics, Oxford 1995 B.D. Serot & J.D. Walecka, Advances in Nuclear Phys., Calderon Press 1986 We start with a short overview of the degrees of freedom: nucleon scalar σ-meson (iso-scalar) vector ω-meson (iso-scalar) (vector ρ-meson) (iso-vector) In the following we neglect the ρ-meson which is of importance for a very accurate description of finite nuclei properties (single particle spectra, neutron skins etc.) but does not contribute in infinite nuclear matter. Looking at the list of degrees of freedom, one might now ask: where is the pion? In the previous section we learned that the pion plays a very important role for the interactions in NN scattering. The reason is here that the pion exchange does not contribute to the potential at the mean field level but only by exchange terms (Fock-diagrams). The philosophy of an effective model like the σω-model is to treat contributions which are beyond the approximation scheme of the model not explicitly but to absorb them in some way into the model parameters, i.e. into the coupling constants. Therefore the pion is not included as an explicit degree of freedom. 8.2. THE σω-model Lagrange density and field equations Before deriving the field equations for nucleons and mesons, the Lagrangian density has to be introduced L = 1 4 F µνf µν m2 ω ω µω µ { ( µ σ) 2 m 2 σ σ2} + ψγ µ i µ ψ + M ψψ g ω ψγµ ψω µ + g σ ψψσ (8.5) Next, we will identify the different terms in (8.5). The first line of Eq. (8.5) contains the Lagrangian of free nucleons and mesons: the first and the third term represent the kinetic energy of mesons, whereas the second and 4th term stand for the meson rest energy. Correspondingly, the other two terms are the nucleon kinetic energy ψ µ γ µ ψ and rest energy M ψψ. The second line contains the interaction part ψ ( γ µ g ω ω µ + g σ σ)ψ. We are familiar with the field-strength-tensor, defined by and we saw earlier that F µν is antisymmetric F µν = µ ω ν ν ω µ (8.6) F µν = F νµ. The field equations result from the Euler-Lagrange equations x µ where Φ can be substituted by ψ, ψ, σ, ω µ. We obtain the Dirac equation in the medium: L ( Φ x µ ) L Φ [γ µ (i µ g ω ω µ ) (M g σ σ)]ψ = 0. (8.7) The meson-fields enter into the Dirac equation as one would expect from minimal substitution known from electrodynamics. The vector field goes into the derivative whereas the scalar field which has no analog in electrodynamics goes into the mass: scalar-field mass vector-field appears in the derivative µ D µ = µ + ig ω ω µ Now, the meson-field equations read: 1. The Klein-Gordon equation with a source-term: [ ] µ µ + m 2 σ σ = gσ ψψ (8.8) 2. The Proca equation with a source-term massive Maxwell equation: ν F µν + m 2 ω ω µ = g ω ψγµ ψ (8.9) 46 CHAPTER 8. MEAN FIELD THEORY Both source terms are proportional to the corresponding densities where one has to distinguish between the scalar density ˆρ S = ψψ and the four-vector baryon current ĵ µ = ψγ µ ψ = (ˆρ B, ˆ j). Here, ˆρ B is the baryon density, whereasˆ j represents the vector current. The distinction between the scalar density ˆρ S and the time-like baryon density ˆρ B is a novel and essential feature of a relativistic description. This has severe consequences for the entire dynamics, as we will see later on. The energy-momentum-tensor is given by T µν = Lδ µν L ( µ Φ) νφ. For isotropic and homogeneous systems, T µν takes the form T µν = (ǫ + P)u µ u ν Pg µν with the four-velocity u µ = (γ, γ v), which becomes (1, 0) in the local rest frame of the matter. ǫ stands for the energy density and P for the pressure Mean field theory To introduce the Mean-Field Theory, we consider a volume V, filled with N nucleons. Consequently, we have a baryon density of ρ B = N. If V 0, the baryon density V increases correspondingly when N is kept constant. For this reason, the source terms in (8.8) and (8.9) become huge compared to quantum fluctuations. As a consequence, the meson fields and the corresponding source terms can be substituted by their classical expectation values. More precisely, only the nucleon fields ψ and ψ are quantised while the meson fields are treated as classical fields. ˆσ ˆσ ψ ˆω µ ˆω µ V µ Considering nuclear matter in the rest frame (the natural one), the spacial components of the baryon current vanish rf ĵµ = j µ = (ρ B, 0). Now we can derive the mean field potentials. Using (8.9) and (8.8) respectively, we get 8.2. THE σω-model 47 V µ = V 0 δ µ0 = g ω ρ m 2 B ω and Φ = g σ ρ m 2 S. σ Thus, in momentum space the Dirac equation reads in mean-field approximation ( α k ) + βm u( k) = (E g ω V 0 ) u( k) (8.10) where the energy eigenvalues are given by E = ± k 2 + M 2 + g ω V 0 (8.11) (± stands again for particles and anti-particles). The effective mass is The baryon field is equal to its expression in vacuum k λ M = M g σ Φ. (8.12) ˆψ( x) = 1 [ u λ ( k) â k λ e i k x + v λ ( ] k)ˆb V k λ e i k x where â and ˆb are particle and anti-particle annihilation operators, â and ˆb the corresponding creation operators. However, now the nucleons are so-called quasi-particles which are dressed by the interaction with the medium. This is reflected by the fact that the nucleons carry an effective mass (8.12) ū( k)u( k) = M k 2 + M 2 u ( k) u( k) but the structure of the nucleon spinors inside the medium is identical to those in the vacuum with the replacement M M. If we further introduce an effective energy E = k 2 + M 2 we can write the equations in the medium in a very transparent way. (For the sake of covariance we distinguish for the moment between k and k although the spatial components of the vector field vanish in mean field theory of nuclear matter, i.e. V = 0 and both momenta are actually equal k = k.) The energy eigenvalues (8.11) are given by E = ±E + g ω V 0. Multiplying the Dirac equation (8.28) from the left with γ 0 we can rewrite it in the covariant form with the effective four-momentum (γ µ k µ M ) u( k) = 0 (8.13) k µ = k µ g ω V µ = (E, k ). 48 CHAPTER 8. MEAN FIELD THEORY The comparison with (8.7) shows that the effective derivative D µ transforms in momentum space to the effective momentum k µ id µ = i µ g ω ω µ k µ = k µ g ω V µ. From these considerations follows that the particles obey now a new mass-shell condition, namely k 2 M 2 = 0 k 2 M 2 = 0. The difference between the in-medium mass-shell condition and that in free space can be expressed in terms of the optical potential. After some simple algebra one finds with 0 = k 2 M 2 = k 2 M 2 2MU opt, U opt (ρ, k) = g σ Φ + k µg ω V µ M + (g σφ) 2 (g ω V µ ) 2 (8.14) 2M The optical potential U opt given by (8.14) is covariant and a Lorentz scalar. It is also called the Schrödinger-equivalent optical potential since it is exactly that potential which occurs when the non-relativistic Schrödinger equation is derived from the in-medium Dirac equation (8.7). If we insert the single particle energy k 0 = E from (8.11) into (8.14) we obtain the optical potential to leading order in the fields (where quadratic terms and terms which go with with 1/M have been neglected): U opt g σ Φ + g ω V 0. (8.15) Remark: baryon-density Computing the baryon-density one has to take into account the contribution from antiparticles. As we learned in XXX in the case of the electrons, in the vacuum all anti-particle states are occupied (filled Dirac sea). Using Wick s theorem (see Chapter XXX) these contributions can be separated ˆρ B = 1 V a k λ a k λ } {{ } k λ particle b k λ b k λ } {{ } anti-particle =: ˆψ ( x) ˆψ( x) normal-ordered product = ˆψ ( x) ˆψ( x) 0 ˆψ ( x) ˆψ( x) 0 } {{ } vacuum state = filled Dirac-sea 8.3. NUCLEAR AND NEUTRON MATTER Nuclear and Neutron Matter The Equation of State In this subsection we deal with infinite nuclear matter and neutron matter. Nuclear matter consists of an equal number of protons and neutrons, i.e. it is an isospin symmetric system. All states are filled up to the Fermi momentum k F. The Fermi energy is the energy of a nucleon sitting at the Fermi surface, i.e. that of a nucleon with momentum k = k F and it is given by E F = EF + g ω V 0 = kf 2 + M 2 + g ω V 0. (8.16) Both, protons and neutrons have spin-up and spin-down states. Consequently, each protonneutron-state is four times degenerate. Therefore, we have γ = 4 in the following equations, describing the energy density ǫ and ρ B. ǫ(k F ) = g2 ω ρ 2 2m 2 B + m2 σ (M M ) 2 + γ k F d 3 k k2 + M ω 2gσ 2 (2π) 2 (8.17) 3 0 ρ B = γ (2π) 3 k F 0 d 3 k = γ 6π 2k3 F. The equations for the scalar density and the effective mass are given by ρ S = γ (2π) 3 k F 0 M d 3 k k2 + M 2 [ = γ 4π M k F kf 2 + M 2 M ln k F + ] kf 2 + M 2 M (8.18) M = M g σ Φ = M g2 σ ρ m 2 S. (8.19) σ These two equations cannot be decoupled. In such a case one speaks about self-consistent equations which have to be solved iteratively. This means that one has to solve this set of coupled equations numerically. Remark: self-consistent equations Self-consistent equations are typical for problems which cannot be solved within perturbation theory. The solution techniques are, however, not difficult. One chooses a starting value for M, usually the free mass, which is inserted into the equation for ρ S. The 50 CHAPTER 8. MEAN FIELD THEORY value for ρ S is then used to calculate the new M and this value is reinserted into the equation for ρ S and so on. This procedure is repeated until convergence is received nuclear matter neutron matter 0.8 effective mass E/A-M [MeV] EOS M*/M ρ [fm -3 ] ρ [fm -3 ] Figure 8.4: Equation of state (left) and effective mass (right) for nuclear and neutron matter in the σω-model (QHD-I). The yellow area in the left panel indicates the empirical region of saturation. To continue we evaluate as a next step the integral for the kinetic en
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