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Effect of Landau Quantization on the Equations of State in Dense Plasmas with Strong Magnetic Fields

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Effect of Landau Quantization on the Equations of State in Dense Plasmas with Strong Magnetic Fields High Power Laser Programme – Theory and Computation Effect of Landau quantization on the equations of state in dense plasmas with strong magnetic fields S Eliezera, P A Norreys, J T Mendonçab, K L Lancaster Central Laser Facility, CCLRC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon., OX11 0QX, UK a on leave from Soreq NRC, Yavne 81800, Israel bon leave from Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Main contact email address: [email protected] Introduction where Γ is phenomenological coefficient and TD is the Debye The equations of state (EOS) are the fundamental relation temperature. A very useful phenomenological EOS for a solid 2) between the macroscopically quantities describing a physical is given by the Gruneisen EOS , 1) system in equilibrium . The EOS relates all thermodynamic γ VE P = i quantities, such as density, pressure, energy, entropy, etc. i (3) Knowledge of the EOS is required in order to solve V 3 α V hydrodynamic equations in specific physical situations, such as γ = plasma physics associated with laser interaction with matter, κ c V shock wave physics, astrophysical objects etc. The properties of matter are summarised in the EOS. where the quantities on the right hand side of the second equation can be measured experimentally: α = linear expansion The concept of Landau quantization in the presence of strong coefficient, κ = isothermal compressibility, cV = specific heat at magnetic fields is presented in the rest of this section and, in the constant volume. There are more sophisticated EOS for next section, the electron EOS in the presence of strong ions3),4), however in this report we do not consider further the magnetic fields is calculated and presented for non-relativistic ion contributions. Since the magnetic field affects mainly the plasmas for both zero and finite temperatures. Finally, the electrons, the electronic contribution is discussed in this report. regime of applicability to laser-plasma interactions is discussed. The short space in this report prevents any detailed description Magnetic fields of a few GG appear to be within reach of the 5) of these concepts – the interested reader is directed to our recent existing petawatt lasers . In this report, the EOS are calculated 1) in the presence of these magnetic fields. Since only the paper where Landau quantisation is discussed in depth . electrons are influenced by the magnetic field under These EOS were previously scattered throughout the consideration, only the electronic equation of state is astrophysical literature and were caste in terms of densities of reconsidered and analysed. For the ion part of the EOS one can 6 -3 12 10 gcm and magnetic fields >10 G, which are, of course, take the Gruneisen EOS or any other appropriate model. appropriate for those expected in neutron star atmospheres. The role of this report is to reformulate the EOS in terms of those that can potentially be realised in laboratory plasmas. By doing so, we have two aims in mind: first, to alert the experimental Landau Levels laser-plasma physics community to the potential of realising The motion of a charged particle in a magnetic field is Landau quantisation in the laboratory for the first time since the quantized6). In particular, the motion of an electron (mass m theory was first formulated; second, to indicate that there are and charge –e,) perpendicular to a constant magnetic field (B) is significant differences in the EOS when these strong magnetic quantized (the vectors are denoted by bold letters). The kinetic fields are present, as discussed in some detail our recent paper energy of the transverse motion is quantized into Landau levels for the interested reader. with a quantum number nL. The Landau energy levels are defined by7-9) If one neglects the electron-phonon interactions then the thermodynamic functions can be expressed as a superposition of the electrons and the atoms (or ions) of the medium under consideration. For example, one can write the energy E and the pressure P in the following form 2 1 2 Π⊥ ⎛ 1 ⎞ ε ⊥ = mv ⊥ = = ⎜n′L + ⎟hω B ; n′L = 0,1,2,3,... 2 2m ⎝ 2 ⎠ E = E + E eA dr (4) e i (1) Π = p + = mv = m c dt P = Pe + Pi p = −ih∇ ; Β = ∇ × Α where Ee and Pe are the electron energy and pressure accordingly while Ei and Pi are the contributions of the atom (or eB ion in a plasma medium) vibrations to the energy and pressure. ωB = For example, the Debye EOS for ions is given by2) mc ħ = h/2π, h is the Planck constant, r and v are the electron ⎛T ⎞ E = 3Nk TD⎜ D ⎟ (2) i B ⎝ T ⎠ position in space and velocity accordingly, Π is the mechanical momentum, p is the canonical momentum and A is the vector 3ΓNk BT ⎛T ⎞ P = D⎜ D ⎟ i V ⎝ T ⎠ potential. The classical equation of motion for an electron at a position r and its solution in the perpendicular direction are 3 x y3dy D()x ≡ 3 ∫ y given by x 0 e −1 99 Central Laser Facility Annual Report 2004/2005 High HighPower Power Laser Laser Programme Programme – Theory – Theory and Computation and Computation dΠ ⎛ − e ⎞⎛ dr ⎞ B = α 2 B = ⎜ ⎟⎜ ⎟ × B (5) 0 rel (12) dt ⎝ c ⎠⎝ dt ⎠ e 2 α = c cΠ h r − R = ⊥ = r ⊥ c eB B 1 2 (6) EOS in Strong Magnetic Fields, the Non-Relativistic Case mcv⊥ 1 ⎛ hc ⎞ rB = = ()2n′L + 1 2 ⎜ ⎟ eB ⎝ eB ⎠ In this section we consider the non-relativistic EOS of a free electron gas in strong magnetic field at finite temperature7-9). ⎛ c ⎞ As is described in introduction, the electron motion R c = r⊥ + ⎜ ⎟B × Π ⊥ ⎝ eB2 ⎠ perpendicular to the magnetic field is determined by the magnetic field and the electron energy levels are quantized into Rc is the position vector of the guiding centre of the electron Landau states. The energy spectrum of one electron is given by gyromotion, while the Landau quantization (4) has been used in 2 deriving the quantized radius of gyration r . p B ε = ε + ε = n ω + z (13) ⊥ z Lh B 2m The energy of the electron (E) is given by n L = n′L +1/ 2 + s / 2 = 0,1,2,...;n′L = 0,1,2...s = ±1. 2 pz ε = n LhωB + (7) pz is the electron momentum along the magnetic field and is 2m continuous. n = n′ + 1 + 1 s ; n = 0,1,2...;n′ = 0,1,2...;s = ±1. L L 2 2 L L A most important difference between zero and non-zero (or rather small or large) magnetic field is “the counting of the The spin energy εs of the electron is included in the energy equation number of states”. In particular, 1 εs = hωBs ; s = ±1 (8) number of states in phase space (B = 0) 2 ∆x∆y∆z∆p x ∆p y ∆p z V∆p x ∆p y ∆p z For the ground level (n’L = 0) s = -1 the degeneracy is 1, for = 3 = 3 (14) exited levels s = -1 or =1 and therefore the degeneracy is 2. For h h extremely large magnetic fields number of states in phase space [B = (0,0, B)] V∆p x ∆p y ∆p z V ()2πp ∆p ∆p ()2πm∆ε ∆p m 2c3 ⊥ ⊥ z ⊥ z 2 13 (9) = 3 = 3 = 3 hωB ≥ mc ⇒ B ≥ Brel = = 4.41⋅10 [G] h h h eh VeB ∆p = z the transverse motion of the electron becomes relativistic, and 2 h c in this case the following energy of the electron is obtained from the Dirac equation (10) This degeneracy has to be multiplied by the quantum degeneracy gnL =1 for nL = 0 and gnL = 2 for nL = 1,2,3…. In 1 ε = m2c4 + p 2c2 + 2n mc2 ω 2 = calculating the number of states, with a magnetic field in the z ()z L h B direction, the Landau quantization of Equation (14) has been 1 (10) used. ⎡ ⎛ 2n B ⎞⎤ 2 p 2c2 m2c4 ⎜1 L ⎟ ⎢ z + ⎜ + ⎟⎥ The starting point in calculating the EOS without or with ⎣⎢ ⎝ Brel ⎠⎦⎥ magnetic field is the grand partition function for Fermi-Dirac particles (in this case, an electron gas). However, in the case of a large B one has to use the electron energy given in (13), and The relativistic motion of the electrons [Equation (10)] is when changing the sums with integrals, starting with the grand strongly influenced by the extremely high magnetic fields (Brel). partition function, we obtain in this case the following equations For the laser plasma interactions we shall define the high of state (the details of which can be found in reference 1)): magnetic fields by 1 VeB()mk T 2 2 ∞ ⎛µ−n ω ⎞ ∞ e 2 m 2e3c N = nV= B g F ⎜ Lh B ⎟ ≡ N 9 2 ∑ nL −1 ⎜ ⎟ ∑ nL ωB ≥ ⇒ B ≥ B0 = ≈ 2.35 ⋅10 [G] h c 2 k T h 3 (11) nL=0 ⎝ B ⎠ nL=0 a 0 h 3 1 (15) 2 eB()k T 2 m 2 2 2 ∞ ⎛µ−n ω ⎞ h −9 P = B g F ⎜ Lh B ⎟ a 0 = ≈ 5.29 ⋅10 [cm] ∑ nL 1 ⎜ ⎟ 2 2 2 k T me h c nL=0 ⎝ B ⎠ where a is the Bohr radius. For B much larger than B , the PV ∞ 0 0 E = + ω n N h B ∑ L nL electron cyclotron energy ħωB is larger than the typical 2 nL=0 Coulomb energy, and therefore the usual perturbation in laser E+ PV−µN 3PV ∞ µN plasma interaction of the magnetic field effects relative to the S = = + ω n N − T 2 h B ∑ L nL T Coulomb interactions is not permitted anymore.
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