by the Feynman Path Integral Method. S. V. Shevkunov*. St. Petersburg State Polytechnic University, St. Petersburg, 195251 Russia. *e mail: shevk54@mail.ru.
ISSN 10637761, Journal of Experimental and Theoretical Physics, 2011, Vol. 112, No. 4, pp. 668–693. © Pleiades Publishing, Inc., 2011. Original Russian Text © S.V. Shevkunov, 2011, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2011, Vol. 139, No. 4, pp. 769–797.
STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Simulation of Thermal Ionization in a Dense Helium Plasma by the Feynman Path Integral Method S. V. Shevkunov* St. Petersburg State Polytechnic University, St. Petersburg, 195251 Russia *email: shevk54@mail.ru Received February 5, 2010
Abstract—The region of equilibrium states is studied where the quantum nature of the electron component and a strong nonideality of a plasma play a key role. The problem of negative signs in the calculation of equi librium averages a system of indistinguishable quantum particles with a spin is solved in the macroscopic limit. It is demonstrated that the calculation can be conducted up to a numerical result. The complete set of symmetrized basis wave functions is constructed based on the Young symmetry operators. The combinatorial weight coefficients of the states corresponding to different graphs of connected Feynman paths in multipar ticle systems are calculated by the method of random walk over permutation classes. The kinetic energy is cal culated using a viral estimator at a finite pressure in a statistical ensemble with flexible boundaries. Based on the methods developed in the paper, the computer simulation is performed for a dense helium plasma in the temperature range from 30000 to 40000 K. The equation of state, internal energy, ionization degree, and structural characteristic of the plasma are calculated in terms of spatial correlation functions. The parameters of a pseudopotential plasma model are estimated. DOI: 10.1134/S106377611104011X
1. STATE OF THE ART OF THE PROBLEM The degree of ionization of a plasma at tempera tures from several to several tens of electronvolts and densities comparable to the density of gases under nor mal conditions strongly depends on temperature and can change by an order of magnitude with temperature varied in a relatively narrow range. This leads to a strong dependence of the thermal, electric, and spec tral properties of the plasma on external conditions. Plasma of this type can be used as a tool in a number of technological applications. In particular, such con ditions are produced in a double electric layer during explosive electron emission. Explosive emission allows one to obtain highcurrent lowenergy electron beams and can be initiated both in vacuum and in a diode filled with helium [1–4]. High electric field strengths near a cathode surface and giant emission current den sities are achieved under these conditions due to for mation of a double plasma layer. The double electric layer in an expanding plasma bunch is formed because of a strong difference in the kinetics of the electron and ion components. The plasma bunch is produced due to Joule heating by the emission current from microscopic roughnesses on the cathode surface. The microstructure and interactions between the electron and ion components at the atomic level play a key role under these conditions [5]. In the region of states considered above, pressure in the plasma achieves a few thousands atmospheres. Because of this, such a plasma exists in technological
processes in the form of shortliving shock bunches with the lifetime determined by the rate of relaxation processes and dependent on the inhomogeneity size. The longest lifetimes of this plasma in terrestrial con ditions are achieved during the explosion of an atomic charge. After a plasma bunch produced at the explo sion center cools below 5 × 104 K, the electron shells of ions are recovered and a dense lowtemperature plasma is formed. The properties of this plasma strongly depend on temperature and its emission and absorption spectra exhibit narrow lines. Because the relaxation time of the electron sub system is two orders of magnitude shorter than that of the ion subsystem, the ionization–recombination equilibrium corresponding to the heating temperature is established before the plasma bunch is expanded and the plasma density decreases. Here, the same effect of inertial confinement takes place as in experiments on controlled thermonuclear fusion, however, at much lower temperatures. The comparatively low tempera ture provides the incomplete destruction of ion shells and the preservation of discrete optical spectra. This circumstance allows one to produce the inverse popu lation of energy levels and to use such a plasma as an active medium for lasers. Partially ionized gases are used as active media in nuclearpumped lasers [6–10]. There exist systems with gasphase radioactive sources in the form of a thin foil placed on the walls of a laser cell, systems using an aerosol fuel, and systems pumped by slow neutrons produced by an external nuclear reactor.
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SIMULATION OF THERMAL IONIZATION IN A DENSE HELIUM PLASMA
Such devices provide the direct conversion of nuclear energy to coherent electromagnetic radiation energy [11–13]. However, lasers of this type can emit only in the infrared and visible spectral regions. To generate shortwavelength coherent radiation, the highdensity pump energy is needed, which can be obtained only in a nuclear explosion transforming the laser medium to a plasma. A dense thermally ionized plasma in high power singleshot pulsed setups is prepared as a laser medium upon heating by the nuclear explosion energy and is pumped by the energy of nuclear fission debris, gamma rays, or thermonuclear fusion products [14– 17]. Such devices can generate superpower coherent Xray pulses. Plasma used as a laser medium should contain ions with electron shells, i.e., it should exist at relatively low temperatures. The ions of helium atoms with a mass close to the neutron mass can provide the most rapid energy absorption and transfer to the laser medium from a fast neutron flux produced in the nuclear reaction. In experiments with the explosion of nuclear charges based on the fission of plutonium isotopes [18, 19], a laser medium is prepared by rapid heating spe cial solid rods to produce plasma, which is then rapidly cooled with the recovery of the electron shells of the ions. A plasma bunch has no time to change its shape and density during cooling. As a result, the relatively dense plasma is formed at temperatures about a few electronvolts. Unlike the hightemperature plasma obtained in nuclear fusion, the theoretical description of plasmas at temperatures on the order of a few electronvolts and densities comparable to gas densities under normal conditions requires the consideration of strong spatial correlations and quantum effects in the electron com ponent. Strongly nonideal systems of quantum parti cles can be successfully described by using computer simulations, as in the case of systems obeying the clas sical statistics. Computer simulations can be most conveniently performed by using the formulation of quantum mechanics in terms of path integrals pro posed by Feynman in the 1960s [20]. Based on this approach in conjunction with Monte Carlo calcula tions of multidimensional integrals, the modern theo retical method is being developed for the ab initio description of strongly nonideal systems of quantum particles at finite temperatures. Along with quantita tive results that can be obtained by the path integral method, of special interest is the refinement of the method itself. The first attempts to use the path integral method to plasma studies were made in the mid1970s [21– 27]. In [28, 29], this method was applied for investiga tions of the thermal stability of the electron shell of the hydrogen atom. In [30, 31], the principally exact pro cedure of including the exchange symmetry to the path integral formalism taking the spin symmetry into account was formulated for a twoelectron system. This procedure was applied to the general case of mul
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tielectron systems in [32–35]. The path integral method was used to simulate an electron pair in the problem of electrides [36, 37], scattering of positrons by helium atom clusters [38], thermal destruction of the electron shell of beryllium ions [39], dense hydro gen plasma [40–42], and dense electron gas [43, 44]. In this paper, we used the path integral method to study a dense helium plasma in the region of parame ters corresponding to a substantially quantum behav ior of the electron component and a strong tempera ture dependence of the degree of ionization. We obtained some quantitative results for plasma, esti mated the possibilities of this method and analyzed its theoretical substantiation and directions of its improvement. 2. CALCULATION METHOD 2.1. Basis Functions Plasma is considered as a disordered system con sisting of the interacting positive nuclei and negative electrons involved in the thermal motion. The nuclei are treated as classical particles and electrons as indis tinguishable quantum particles with a spin 1/2. They are described by the quantum statistics methods in terms of Feynman path integrals. The system is sup plemented with periodic boundary conditions [45] or a cell with movable boundaries containing a nucleus and two electrons is simulated. The equilibrium averages should be calculated in the representation of a complete set of linearly inde pendent wave functions. The procedure of symmetri zation over permutations violates the linear indepen dence of functions. However, it can be shown [46] that the use of the symmetrization procedure based on the Young operators allows one to construct a complete set of orthogonal symmetrized functions for indistin guishable particles with a spin. Each Young operator generates the basis of the irreducible representation of a permutation group and the corresponding subspace of symmetrized functions. The combination of all these subspaces possesses completeness and their basis functions are mutually orthogonal [46]. 2.2. Spin and Exchange Although the Hamiltonian of a nonrelativistic sys tem does not act directly on spin variables, the energy of the system strongly depends on its spin state via the permutation symmetry of the coordinate part of the wave function. Because fermions are indistinguish able, the total wave function should be antisymmetric upon simultaneous permutations in a pair of spin and coordinate variables. In this case, the wave function can have, separately for spin and coordinate variables, a symmetry of a complex type which cannot be reduced to a simple symmetry or antisymmetry. It is shown [46] that all functions in the form of bilinear combinations of spin and coordinate functions
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obtained by symmetrization using Young operators with mutually transposed schemes satisfy this require ment. In the case of particles with spin 1/2, the state, which is the eigenfunction of the operator of the squared spin, uniquely corresponds to a certain Young scheme of the spin part of the wave function and to a dual Young scheme of the coordinate part of the wave function. The result of the Hamiltonian action on coordinate variables substantially depends on the type of the permutation symmetry of the coordinate part, which depends in turn on the spin state of the system. 2.3. Matrix Elements
ψ ( S, { m i }, { r i }; { σ i }, { x i } ).
∑ ( 2S + 1 ) ∑ W ( Pˆ ) S
n
(2)
n
ˆ ) |P ˆ { r }〉, × d r 〈{ r i }| exp ( – βH n i
∫
N
ˆ ) are combinatorial coefficients. Accord where WS( P n ing to Feynman [20], we can write matrix elements ˆ ) |P ˆ { r }〉 in (2) in the form of continual 〈{ r i }| exp(–β H n i ˜ (t) in the 3Ndimensional integrals over virtual paths R ˆ {r }: space connecting points R(1) = {r } and R(n) = P n
i
Z=
∑ S
R
×
(1)
⎛i
˜
⎞ ˜
∫ dR ∫ exp ⎝ បΦ ( [ R ( t ) ] )⎠ DR ( t ) R
(1)
Mm 0 ⎞ 3NM/2 = lim ⎛ dR ( 1 ) dR ( 2 ) M → ∞⎝ 2πβប 2⎠
∫
∫
(4)
…dR ( M ) exp ⎛ iΦ ( R ( 1 ), …, R ( M ) )⎞ , ⎝ប ⎠ ˜ ( kt/M ), R ( k ) ≡ { ri ( k ) } = R
k = 1, 2, 3, …, M, N
dR = dr 1 dr 2 dr 3 …dr M ≡ d r, and the action Φ(R(1), R(2), …, R(M)) is calculated on a broken line connecting points R(1), R(2), …, R(M). 2.4. Partition Function A complete set of linearly independent permuta tionsymmetrized wave functions is constructed by representing the total wave functions of nonrelativistic particles in the form of a bilinear combination of spin, n k χ J (S, {mi}; {σi}), coordinate, f J (S, {ri}; {xi}), and wave functions symmetrized by Young operators with mutu ally transposed Young schemes and different n and k arrangements of arguments in cells of the Young scheme [48]. By summing the diagonal matrix elements N d r 〈S, { m }, { r }|Fˆ |S, { m }, { r }〉
∫
i
= N!
i
∑
i
i
χˆ ( S, { m i }, { σ i } )
{ σ i = ± 1/2 }
×
i
∑b
n nk χ J ( S,
(5) { m i }; { σ i } )
n, k
ˆ ) WS ( P n
k N × d r d xf ( { r i }; { x i } )Fˆ f J ( S, { r i }; { x i } )
∫ ∫ N
n
(3)
(n)
∫ R
∫ ∑
( 2S + 1 ) dR
(n)
(1)
It can be shown [32, 33] that summation of the diago nal elements of the matrix and the summation over the eigenvalues {mi}, {ri} S(S + 1) that are not connected by permutations in sequences {mi}, {ri} leads to the partition function in the form
S
R
where
The total wave function of a system of N indistin guishable quantum particles depends on coordinate, {xi}, and spin, {σi}, variables. The eigenfunction of the coordinate, spin projection, and squared spin opera tors is determined by the corresponding sets of the eigenvalues {ri}, {mi}, and S(S + 1):
Z=
The continual integral is defined as a limiting mul tidimensional integral [20, 47]
˜ ( t ) ] )⎞ DR ˜ ( t ), exp ⎛ iΦ ( [ R ⎝ប ⎠
˜ ( t ) ] ) is the action functional on the path where Φ ( [ R ˜ (t) up to the formal replacement of time t by the R –iបβ. The end of imaginary inverse temperature: t any threedimensional virtual path ri(t) is the begin ning of another path, so that a circular structure is formed which is topologically similar to the permuta ˆ [32, 33]. tion graph P n
over {ri} and {mi} which are not connected by permu tations and S, we obtain the trace of operator Fˆ = ˆ ), i.e., the partition function of the system exp(–β H [32, 33] Z=
⎛
∑ ( 2S + 1 ) ⎜⎝ ∑
{ σ i = ± 1/2 }
S
×
∑b nk
χ˜ ( S, { m i }; { σ i } )
n nk χ J ( S,
⎞ { m i }; { σ i } ) ⎟ G ( ˆJ ( S ) ) ⎠
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=
∑ ( 2S + 1 )W˜ G ( ˆJ ( S ) ),
ω ( { νi } ) =
S
S
∑
χˆ ( S, { m i }; { σ i } )
{ σ i = ± 1/2 }
×
∑
(7)
n b nk χ J ( S,
{ m i }; { σ i } )
2
1 h Z = lim ⎛ ⎞ M → ∞ N n! ⎝ 2πm n k B T⎠
are the weight coefficients, and ˆ ) = d N r d N xf * ( { r }; { x } ) G(P k i i
∫ ∫
ˆ { x } ). × Fˆ f ( { r i }; P k i
(8)
m e M ⎞ 3MN/2 1 ⎛ × N! ⎝ 2πβប 2⎠
The partition function can be written in the form [32, 33] 1 Z = N!
∑ ( –1 ) ∫ d c(n)
S
S
N
β – M
ˆ ) |P ˆ { r }〉, r 〈{ r i }| exp ( – βH n i
M
N
j k
j=1
S
S
i
{ νi }
N ˆ ) |Π ˆ { r }〉 × d r 〈{ r i }| exp ( – βH { νi } i
∫
1 = N!
∑
{ νi }
∫
N
ω ( { νi } ) d r
(10)
k=1
Nn
e j j r – rl i
