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ANNUAL REPO RT,ONR CONTRACT NO00 4 79 C 0073,INVESTIGATION. NONIDEAL PLASMA PROPERTIES,Prepared for OFFICE OF NAVAL RESEARCH. Covering Period I November 1979 31 December 1980,Principal Investigator H E Wilhelm. 1 May 1981,Department of Engineering Sciences,University of Florida. Gainesville Florida 32611,i L w J L t,I INTRODUCTION 1.
II CONDUCTIVITY OF NONIDEAL PLASMAS WITH MANY PARTICLE INTERACTIONS 4. III CONDUCTIVITY OF NONIDEAL CLASSICAL AND QUANTUM PLASMAS 19. IV CONDUCTIVITY OF NONIDEAL QUASI METALLIC PLASMAS 39. V ANOMALOUS DIFFUSION ACROSS MAGNETIC FIELDS IN PLASMAS 84. VI COLLECTIVE MICROFIELD DISTRIBUTION IN THERMAL PLASMAS 96. VII FREE ENERGY OF NONIDEAL CLASSICAL AND DEGENERATE PLASMAS 115. VIII FREE ENERGY OF RANDOM SOUND OSCILLATIONS 140,I INTRODUCTION. This report contains theoretical investigations on the electrical conduc. tivity and thermodynamic properties of nonideal plasmas which were carried. through in the period from 1 November 1979 to 31 December 1980 under ONR Contract. N00014 79 C 0073 In addition the theoretical results were compared with experi. mental data for nonideal plasmas These comparisons are of a preliminary nature. since the experimental conductivities for nonideal plasmas differ not only quant. itatively but also qualitatively in the literature. CHAPTER II The dependence of the electrical conductivity a of dense non. degenerate plasmas on the nonideality parameters y Ze2 n 3 KT was evaluated. by summing the probabilities for v body interactions v 2 3 4 of the con. duction electrons It is shown that a is noticeably smaller than the binary con. ductivity a 2 for y 10 The theoretical decrease of u with increasing y is. confirmed however only by some experimental data while other experimental data. indicate an increase of a with increasing y for the same pressure. CHAPTER III Based on the classical and quantum Boltzmann equations the. electrical conductivities of classical and degenerate nonideal plasmas were evalu. ated Although in this kinetic approach many body interactions are taken into. account only through an exponentially shielded Coulomb potential in which the. electron ion scattering occurs the results give in agreement with the experimen. tal data conductivities which are by about one order of magnitude smaller than the. Spitzer conductivity for ideal plasmas The increase of the dimensionless conduc. 1 2e2 KT 3 2, tivity a m e a KT with increasing y is confirmed by some experimental. data but not by all of them The new Coulomb logarithm does no longer go to zero. for large y values as in the Spitzer theory but is well behaved for large elect. ron densities and even for solid state densities due to the consideration of. electron degeneracy, CHAPTER IV With the help of quantum field theoretical methods from the. theory of metals the electrical conductivity of nonideal plasmas was calculated. under consideration of electron scattering by low frequency plasmons ion waves. and high frequency plasmons electron waves for classical and degenerate conditions. The resulting conductivity formulas agree with the Spitzer theory for y 0 and ex. hibit numerical values which are considerably smaller than the Spitzer values but. are still larger than the theoretical conductivities obtained in III for increasins. y The numerical values a agree with the experimental data qualitatively but are. somewhat too high, CHAPTER V The possibility of anomalous diffusion and conduction transverse. to magnetic fields B was studied since large charged particle transport across. magnetic fields is of interest for MHD generators For weakly nonideal plasmas. the anomalous transverse conductivity was shown to be a w2p 4w 2w where w. ne2 Eom 1 2 is the plasma frequency and w eBo m is the gyration frequency. of the electrons e m This formula agrees with experimental data for weakly non. ideal plasmas but should be also correct qualitatively for nonideal plasmas There. are however no experimental data available on anomalous diffusion and conduction. in magnetic fields for nonideal plasmas, CHAPTER VI In connection with the electric current transport in the elect.
ric field fluctuations produced collectively by the electrons and ions in random. thermal motion the electric microfield distribution of thermal plasmas was deri. ved by equilibrium statistical mechanics Comparison with the resulting tempe. rature dependent microfield distribution with the classical T independent Holts. mark distribution and its later extensions indicates that the latter theories are. approximately applicable to strongly nonideal plasmas but are invalid for ideal. plasmas to which they are usually applied in literature. CHAPTERS VII VIII By means of Bose statistics the contribution of the. thermally excited longitudinal electron and ion waves to the free energy of. nonideal classical and quantum plasmas was calculated It is shown that the ran. dom low frequency ion oscillations contribute more to the free energy than the. high frequency electron oscillations The free energy of the random ion waves. is quantitatively comparable to the free energy of the thermal non collective. ion motions for high densities n 1023cm 3 and standard plasma temperatures. T 106 0 K Similar calculations were performed for dense gases in which the. random sound oscillations lead however only to a small correction of the free. The theoretical research on nonideal plasmas needs further clarifications. by experiments In particular more reliable conductivity data for nonideal al. kali and noble gas plasmas are needed This is a preliminary report of research. results which will be communicated later in form of publications. II CONOUCTIVITY OF NHOtIDEAL PLASMAS IITH MNY PARTICLE INTERACTIONS. H E Wilhelm, The dependence of the electrical conductivity of nondegenerate. dense plasmas on the nonideality parameter y Ze2 n 3 KT ratio of. Coulomb interaction and thermal energies is derived by summing the. probabilities for v body interactions v 2 3 4 of the elec. trons As an application the dimensionless probability coefficients. for binary and triple Coulomb interactions are calculated by means. of simple physical models and a conductivity formula for moderately. nonideal plasmas 0 y 1 is derived in which all parameters are. known The theory is shown to agree with recent experimental data. INTRODUCTION, High pressure plasmas 101 bar P 106 bar produced by shock wave. compression are now of considerable technical interest A large number of. publications 1 1 0 are concerned with the measurement of the anomalous electrical. conductivity of proper nonideal plasmas 10 1 y 1 Theoretically however. only the conductivity of ideal y 0 and weakly nonideal y 1 plasmas is. adequately understood 1 1 1 2 The degree of nonideality of a fully ionized. plasma is defined by the interaction parameter y which represents the ratio of. average Coulomb interaction Ze2 n 1 3 and thermal KT energies n electron. density Z ion charge number e elementary charge,y Ze2n1 3 KT 1 670 x 10 3Zn 3T e s u 1. The conductivity theories of ideal and weakly nonideal 0 y 1 plasmas. break down for y 1071 since the Debye radius,D Z 47r l Z 1 2 y 1 2 n 1 3 Y 1 2 n 1 3 2. loses its physical meaning as an electric shielding and Coulomb interaction. length This is seen from the number of electrons ND in the Debye sphere of a. scattering ion which is no longer large compared with one for y 10. ND 4n 3 Z 4w l Z 3 2 Y 3, For strongly nonideal conditions n 102 0 cm 3 and T 104 K we have.
y 0 775 D 4 881 x 10 8 cm and N D 4 87 x 10 2 Another reason for the. inapplicability of the conductivity theory of ideal and weakly nonideal plasmas. to proper nonideal plasmas is the standard assumption of shielded binary. Coulomb collisions v 2 whereas in reality the conductivity is determined. by many particle interactions v 2 3 4 for y 10, The many body inLeraction is one of the classical unsolved problems of. physics For this reason we calculate the conductivity of nonideal plasmas. and the probabilities for many particle interactions by means of dimensional. theory 1 3 14 This approach gives the exact dependence on the relevant dimen. sional plasma parameters 1 3 1 4 and numerically correct results up to a dimen. sionless coefficient 113 14,which is in general of the order 100 The plasma. is assumed to be fully ionized and nondegenerate i e. 2 3 2 15T 3 2,n Zni n2 2rmKT h 4 828 x 10lST3 4,ELECTRICAL CONDUCTIVITY. In a system of reference in which magnetic fields are absent a linear. electric current response j oE exists provided that the generating electric. field is weaker than the critical plasma field for electron heating For any. gaseous liquid or solid plasma the electrical conductivity a is. o ne2 m T 5, since the electrons of mass m M dominate the electric current transport in. plasmas The interaction frequency T of the electrons is the sum of the. interaction frequencies T for the v particle interactions. since the probabilities frequencies T1 for many particle interactions of the. order v are additive v 2 for binary v 3 for ternary etc N is related. to the total number N of charged particles of the system by N N I 1. For physical reasons the conductivity o sec 1I of a fully ionized. classical plasma can depend only on the dimensional plasma parameters. 3 2 1 2 l1 3 2 2, e cm gr sec m gr n cm 1 KT gr cm sec and the characteristic.
dimensionless constant Z n n KT thermal energy The conductivity a and. the parameters e m n and KT have the dimensions V L dimension of length. T dimension of time M dimension of mass,V e L 3 2MI 2T I. 1 m M D n L 3 D KT ML2T 2, Dimensional theory is based on the axioms of Dupr Accordingly the. secondary quantity a is given in terms of the primary quantities e m n and. o CzeNmN2nN3 KT N4 8, C is a dimensionless coefficient which depends on dimensionless parameters. such as Z n n1 and can only be determined by means of a detailed physical. model C is either a true constant of order of magnitude one Cz 1 or It. is a slowly varying function quasi constant Comparison of the powers of the. independent dimensions L M and T Eq 7 in Eq 8 gives the compatibility. N 3N 2N 0 N N N 0 N 2N l 9,21 3 4 21 2 4 1 4 1, These are three independent equations since only three independent dimensions. L M and T exist which determine three of the four powers Ni in terms of the. N 1 1 2N N N N N 0,22 2 3 2 3 4 0, Combining Eqs 8 and 10 yields a conductivity expression a 0.
contains a still undetermined power N,2 1 2 2 1 3 N. G CZN ne m e n KT 11, In order to understand the physical meaning of Eq 11 it is rewritten in. the form of Eq 5,a ne 2 mT I,N 3 3 2 12,1 ne2 1 2l 21l 3 v 3 2. T C 1 ne m 1 e n v 2 3 4 N 13, It is now seen that Eq 11 or the equivalent Eqs 12 and 13 represent the. conductivity of a hypothetical plasma in which each electron experiences only. many body interactions of a fixed order v since the probability for a v body. interaction has the n dependence,T 1 2 3 4 N 14, For example for a hypothetical plasma with 2 body interactions only so called.
ideal plasma Eqs 12 and 13 give,2 I 1 1e 4 1 2 3 2. a2 ne m2 2,2 m n KT 15, where the dimensionless coefficient is known from the kinetic theory of binary. collisions 11 Cz2 3 4 2 r 2 Z in A i e CZ2 is a quasi constant which. varies only slightly with n via the Coulomb logarithm in A. Since the probabilities T V for the individual v body interactions are. additive Eqs 12 and 13 result in the following formulas for the inter. action frequency T E and the conductivity a ne mT of actual plasmas. in which v 2 3 4 body interactions take place,1 N 1 30v 3 2. T C p y 16,a ne m i 3 3 2 17,Zv 4 1 2 v v 2 3 4 N 18. One2 m 1 2,w 4n i y Ze 1Ze2n 3 KT 19, are dimensionless coefficients the plasma frequency and the nonideality.
parameter respectively, In order to expose the many body effects v 3 of the nonideal plasma. for comparison with the corresponding formulas of binary kinetic theory v 2. Eqs 16 and 17 are rewritten as,T 1 m 1 2e4n KT 3 2 Cz. 12 N Cz1 y Z 3 v 2 20,m 1 2e 2 KT 3 2 CZ1 N CZ1 y Z 3 v 2 21. Equations 20 and 21 give the interaction frequency and conductivity of. nonideal plasmas in terms of a series in y which converges rapidly for. 0 y I and converges for any y 1 since it is finite 1 N. N 1 6 1 9 1 3 N 2,Czl V z Z3 Y Z CZ4 y Z CZ 5 Y Z 22. For ideal plasmas y 0 Eqs 20 and 21 reduce to T1 T2 1 and a a2. in accordance with the kinetic theory of binary interactions For weakly. nonideal plasmas y 1 Eqs 20 and 21 show that T Z T21 and a a 2. For moderately nonideal 1,y 100 and strongly nonideal 100 Y 7.
plasmas the electron interaction frequency T increases and the conductivity. decreases considerably and by orders of magnitude respectively These. theoretical results are in agreement with measurements on nonideal plasmas. which exhibit considerably smaller conductivities than expected from binary. collision theory, It should be noted that Eqs 16 and 17 or 20 and 21 are applicable. to nondegenerate plasmas only i e to densities n n or interaction parameters. 3 Ze2 h 2 1 2,yf 2 KT 2m 172 2 823 x 10 ZT 23, For a physical interpretation of the above results the partial collision. frequency in Eq 13 for the v body interaction of a conduction electron with. random sound oscillations lead however only to a small correction of the free energy The theoretical research on nonideal plasmas needs further clarifications by experiments In particular more reliable conductivity data for nonideal al kali and noble gas plasmas are needed This is a preliminary report of research

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