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J. Phys. B: At. Mol. Phys. 20 (1987) 4069-4086. Printed in the U K Effective electron-atom interactions and virial coefficients in alkali plasmas Ronald Redmert, Gerd Ropket and Roland ZimmermannS t Sektion Physik, Wilhelm-Pieck-Universitat Rostock, Universitatsplatz 3, Rostock, DDR-2500, German Democratic Republic f Zentralinstitut fur Elektronenphysik der Akademie der Wissenschaften der DDR, Hausvogteiplatz 5-6, Berlin, DDR-1086, German Democratic Republic Received 23 September 1986, in final form 11 February 1987 Abstract. Using thermodynamic Green’s functions, effective potentials and related virial coefficients for electron-atom interactions are derived within the second Born approximation. In addition to the exact results for hydrogen, improved values for the virial coefficients of electron-alkali-atom interactions are obtained. 1. Introduction Within a quantum statistical approach, the physical properties of a dense partially ionised plasma are deduced from the basic interactions between the constituents of the plasma, i.e. from the Coulomb interaction between the elementary charged particles. The thermodynamic functions and the equation of state for a partially ionised plasma are usually derived from the effective interaction potentials between the different species -electrons, ions and neutral composites (atoms)-applying the standard methods of statistical physics. However, this so-called ‘chemical picture’ of a many-particle system with bound states is only applicable in the low-density limit where the properties of the bound states are not influenced by the surrounding medium. In general, a more consistent approach valid for arbitrary densities should start from the basic Coulomb interaction between the elementary particles, i.e. electrons and ions, and the resulting effective interactions between the different species are modified due to density corrections. These in-medium effects are essential, especially near the Mott density where the bound states disappear (pressure ionisation). A quantum statistical approach to the thermodynamic properties of a partially ionised plasma using the technique of thermodynamic Green’s functions is given in the monograph of Kxaeft et al (1986) (see also references cited therein). A special problem of the description of partially ionised plasma is the derivation of the effective interaction between the electrons, ions and atoms from the basic interaction between the elementary constituents of the plasma. In this paper, a consistent approach for the interaction between electrons and neutral atoms is given for the low-density limit using the Green’s function technique. Furthermore, this quantum statistical approach also allows the inclusion of density corrections, e.g., self-energy and the Pauli exclusion principle. As a result, this effective interaction can be related to the polarisation potential Vp( R ) for which approximations of different degrees of complication exist. 0022-3700/87/164069 + 18$02.50 @ 1987 IOP Publishing Ltd 4069 4070 R Redmer, G Ropke and R Zimmermann A typical form of the interaction between electrons and neutral atoms is the polarisation potential (PP) (a,is the dipole polarisability, r, is the cut-off radius): V,’(R ) = - a1 + (R 2 r i ) 2 which was introduced by Buckingham (1937) and which has been widely used to estimate the influence of elastic electron-atom scattering and related properties in partially ionised plasmas (Gryaznov et a1 1980, Sobelman et a1 1981). Therefore, the PP (1.1) is called the ‘standard model potential’ throughout the paper. It reproduces the known reaction of a neutral particle in the presence of a charge at large distance R, which is given by (Born and Heisenberg 1924) VR(R+ a)= - c Y , R - ~Rydberg . units ( h = 1, m e = + , e 2 / 4 m o = 2) are used throughout, i.e. energies are measured in Ryd= 13.6 eV and distances are given in units of the Bohr radius a,. Different improvements of the standard model potential (1.1) were developed which account for, e.g., (i) higher-order polarisabilities (as a first extension the quadrupole polarisability a2 is taken into account (Bardsley 1974)) (ii) non-adiabatic effects (dynamic effects due to the kinetic energy of the polarising charged particle (Callaway et a1 1968, Eissa and Opik 1967, Norcross 1973, Walters 1976, Curtis 1981)) (iii) electron exchange effects (Callaway and Williams 1975, Daskhan et a1 1981). The consideration of all these effects leads to an effective PP which includes a number of parameters which must be determined independently (for some applications in atomic and molecular physics see, e.g., Bottcher 1971, Theodosiou 1984, Lombardi 1985). A systematic approach to the effective interaction between the electrons and neutral atoms using the Green’s function technique is given in 0 2 and the relevant approximations are pointed out. Within the frame of the Green’s function technique, the correct dynamical character of the interaction and, in particular, the influence of in-medium corrections can be investigated in a systematic way. The relations to the well known results for hydrogen are presented in P 3. Explicit results for the alkali metals are given in 9 4 where, parallel to the investigation of the electron-atom interaction, a corresponding virial coefficient is considered. One consequence of our approach is to give a more correct description of the influence of bound-state formation on the thermodynamic properties of partially ionised plasmas. 2. Green’s function approach to a polarisation potential (PP) The propagation of three particles embedded in a surrounding plasma consisting of electrons and ions with a temperature T and with chemical potentials pe and p i , respectively, is described by the following thermodynamic Green’s function: G3(123, 1‘2‘3‘, t + t’) 1 , = - ( ? { a ( ~ , t)a(2, t)a(3, t)a+(3’, tr)a+(2‘, t t ) a + ( l ’t’))) i3 (2.1) where at( i, t ) , a ( i, t ) denote :reation or annihilation operators, i standing for momentum, spin and species, and T is the time ordering operator. The average in (2.1) is performed by means of the grand canonical ensemble. Efective electron -atom interactions 407 1 In particular, considering two electrons and one ion, the scattering process of an electron by a neutral atom (taken as a bound state between the second electron and the ion) is also described by the three-particle propagator (2.1). Consequently, the effective interaction between an electron and an atom embedded in a surrounding plasma can be derived from the analysis of the three-particle Green’s function G,. As a first step, let us consider the formation of bound states (atoms) starting from the elementary particles. Representing the one-particle propagator G I (1) by a full line and the interaction potential V(12, 1’2’) by a broken line, the two-particle Green’s function G2(12, 1’2’, z ) is obtained from the Bethe-Salpeter equation [G21= [G;I+[G~I[KI[G21. (2.2) G; denotes the Green’s function of two non-interacting particles. In the low-density limit (which is considered throughout this paper), the general interaction kernel K is given by the interaction potential V and from (2.2) the following ladder equation is obtained. [G* i = 2 2’ + (2.3) + The resulting ladder Green’s function Gk(12, 1’2’, z ) is equal to (see, e.g., Stolz and Zimmermann 1978) 1 G,L(12, 1’2’, Z ) = +:p( 12)*nP ( 1‘2‘). 1 z- nP EnP EnP, rClnP are, respectively, the energy eigenvalue and wavefunction of the isolated two-particle cluster with internal quantum number n and total momentum P. For the treatment of the two-particle propagator G2 at arbitrary densities see, e.g., Redmer and Ropke (1985) and Kraeft et a1 (1986). The general treatment of the Bethe-Salpeter equation for G3(2.1) can be represented by an infinite series of diagrams. In order to get an idea of how to select important contributions we apply the chemical picture, i.e. bound states (composites) are considered as new species, and elementary particles and composites are treated on the same footing (cf Hohne et a1 1984). The process of interest here is the scattering of a free electron (1) at a two-particle cluster (2,3), i.e. considering the process 1 + (2,3) .+ 1 ’ + (2,3)’, which is given up to second order with respect to the bound-free interaction by the following diagrams: 2 3 u ; ; -m .w;mfm +> - (2.5) where exchange contributions are neglected. The vertex function (Ropke and Der 1979) r I I +I I-+ = Mnn,(q)= i V ( q ) I P-* CP + : ( p ) { + n , ( P ) I I I + H - + n , ( P + 4)) (2.6) describes the coupling between the free electron (1) and the neutral two-particle cluster (2,3) and is related to the form factor of the transition n + n’ and, in the limit q + 0, to the dipole matrix element dnn,= dr3 r & f ( r)Gn,(r ) . V(q) = - 8 T / q 2 denotes the Fourier transform of the bare Coulomb potential V ( R )= -2/R. 5 4072 R Redmer, G Ropke and R Zimmermann Considering equation (2.5), the two particles ( 2 , 3 ) belong to a neutral atom and the interaction with the third particle, the electron ( l ) ,is treated as a perturbation. The approximation (2.5) considers the interaction between the two particles ( 2 , 3 ) in arbitary order (ladder sum) whereas for the interaction with the third one ( 1 ) the Born approximation up to second order is applied. Further diagrams would describe exchange contributions, higher-order terms of the Born series and, in particular, in-medium effects. The evaluation of (2.5) leads to the following expression for Gk: G:( 123, 1’2’3’, z ) 3 c I):~( 12) 4L,P(lW z-EE,p-E3 S3,3’ flP + c nPn’P‘fl”P” +GP(12) Z- 1 1 M,,,,,(3”- 3’) Mfl,43 - 3”) z E,,rprfE,,, E,p - E3 As shown in appendix 1 , the Green’s function G) (2.7) can be related to the diagonal part of an effective three-particle potential K (A1.5) 3r to the diagonal part of a corresponding three-particle T matrix ( A l . 1 0 ) . Comparing (2.7) with the perturbation expansion (A1.10) for the T matrix, the following expressions are obtained: T$”(nPk, nP + qk - q, z ) = M,, ( q ) (2.8) As shown in appendix 2, important physical quantities are determined by the on-shell T-matrix elements T\’)(nPk, nP + qk - q, EnP+ E k )= V“’( nPk, nP + qk - q ) Ti’)(nPk, nP + qk - q, EnP+ E k ) = V‘( nPk, nP + qk - q ) (2.9) Ti2’(nPk, nP+ qk - q, EnP+ E k )= V”’( nPk, nP+ qk - 4). In the following the effective interaction Vp in (2.9) is evaluated further in order to define a polarisation potential (PP) and to demonstrate the connection to the standard model potential ( 1 . 1 ) . In the adiabatic approximation, i.e. neglecting the kinetic energies of the interacting particles (which are determined by the plasma temperature) compared with the bound-state energies, one obtaines from (2.8) and (2.9) with EnP= E , + h 2 P 2 / ( 2 M ) (2.10) Efective electron - atom interactions 4073 This expression is transformed into the coordinate space which yields (cf Dalgarno and Lynn 1957) (2.1 1) R, r denote the respective distances between the free and bound electron and the atomic nucleus. The second term in the matrix elements M,,,,,( R ) vanishes due to the orthogonality of the wavefunction. Expression (2.11) can also be derived by ordinary second-order perturbation theory without the sophisticated Green's function method. However, we point out once more that the general Green's function approach applies also for dense systems and provides us with possible generalisations of (2.11), for instance effects of dynamical screening can be taken into account (Ropke 1983). The matrix elements M n n , , ( Rcan ) be considered for the two limiting cases R = 0 and R -+ 00. For R = 0, the PP (2.1 1) is given by the expression (2.12) In the opposite case, for large distances R between the electron and the atom, the dipole expansion can be applied for the matrix elements M,,,,,(R) which yields (2.13) M,,,,(R -+ 00)= 2d,,./R2. Using the definition of the dipole polarisability of an atom in the internal state n (2.14) the asymptotic behaviour of the PP (2.11) V P ( R-+ 00, n ) = - ( Y ! " ) R - according ~, to Born and Heisenberg (1924), is reproduced. With regard to the finite value (2.12) of the PP at R = 0, the standard model potential (1.1) can be used as an interpolation formula between the explicit results for R = 0 and R -+ CO: V P ( R ,I S ) = V,'(R)=- + ffl ( R 2 r;)' ro= (al/l Vp(O,l ~ ) [ ) " ~ .(2.15) Then the cut-off radius ro is determined by the condition (2.12). Notice that other methods are usually applied for the determination of ro (e.g. comparison with known binding energies or scattering lengths). The discrepancies between the well defined PP (2.11) and the standard model potential (1.1) are pointed out in the next section for the case of hydrogen. In dense plasmas, the PP (2.1 1) is modified due to the effects of dynamical screening which can also be accounted for in the frame of Green's functions (Ropke 1983). The connection between the T matrix and corresponding virial coefficients Bc,ab which account for the non-ideal corrections to the equation of state due to the interaction between the species c and ab is shown in appendix 2, see equation (A2.6). According to (2.9), the virial coefficient Be,eican be split in an analogous way, i.e. 4074 R Redmer, G Ropke and R Zimmermann The different contributions in equation (2.16) are calculated in the next section for the case of hydrogen. Let us point out once more that the second Born approximation for the interaction of neutral composite particles with charged elementary ones is important in the region of partially ionised plasmas. The static field of the neutral particles decreases rapidly and yields only small contributions to the virial coefficients and the transport cross section. The main contribution for conditions of interest here arises from the second Born approximation which describes polarisation effects in the neutral composite and shows a long-range behaviour according to R-4. 3. Results for hydrogen We briefly review the case of hydrogen which can be treated rigorously using known hydrogen wavefunctions for the determination of the PP (2.13). Dalgarno and Lynn (1957) and Pan and Hameka (1968) (see also Hameka 1968) gave analytical expressions for the PP (2.13) valid for arbitrary distances R between the free electron and the atom which is considered to be in the ground state, V'(R, 1s). We will repeat their result explicitly because it is used for the calculation of related integral quantities such as, e.g., virial coefficients: V'(R, l s ) R 2 = 5 - ( 4 R 2 + 8 R + 1 0 ) e - 2 R + ( 4 R 3 + 7 R 2 + 8 R + 5 )e-4R - 2 ( R + 1)2e-2R(1+e-2R)Ei(2R)+2[(R + 1)2e2R+ ( R 2 + 2 R -3) + 4 ( R + 1) e-2R]Ei(-2R) -4( R + 1)2e-2R(1 +e-2R)(y + log(2R)) (3.1) where y = 0.577 . . . is Euler's constant and Ei( t ) = -J?, dx e-x/x. As a special case, V'(R = 0, 1s) = -1 is obtained. With respect to (2.12), the different contributions of transitions from the ground state to excited bound states (discrete) or scattering states (continuous) are of interest. The separate treatment of bound states yields a value of V'(R = 0, = -0.337 whereas the transitions into scattering states give rise to the main contribution V'( R = 0, 1s)" = -0.663. Notice, that the value for VP(R = 0, 1s) = -1 which represents all adiabatic contributions (2.11) is just cancelled considering non-adiabatic corrections which can be given in the form of a 'distortion potential' VD(R, 1s) (Callaway et a1 1968) which yields the value V"( R = 0 , l s ) = + l . Further modifications are expected if exchange processes are included (Callaway and Williams 1975, Daskhan er a1 1981). Considering the opposite case R -f cc for the exact PP (3.1), an asymptotic expansion is obtained according to Dalgarno and Stewart (1956): X a/ VP(R+W, IS)=-^ R2'+2 (Y/ = (21 + 1)!( I + 2) 1221 (3.2) where the hyperpolarisabilities occur. The dominating term in (3.2) is the dipole contribution with a , =;. A comparison between the well defined PP (3.1) according to Dalgarno and Lynn (1957) and Pan and Hameka (1968) and the standard model potential (1.1) is shown in figure 1 as function of the distance R between the free electron and the atom. The parameter ro was chosen to be ro = (Y :'4 = 1.4565 in order to reproduce the calculated Efective electron - atom interactions -1.0 0 , 1 3 2 4 4075 5 R(Qo) Figure 1. The PP (3.1) (full curve) for e-H interaction compared with the standard model 1.4565. potential ( 1 . 1 ) (broken curve), where a,= 9 / 2 and ro= value VP(R = 0, 1s) = - 1 . The standard model potential ( 1 . 1 ) deviates from the result (3.1) and the corrections to the asymptotic behaviour proportional to R-4 are even wrong in sign. This disagreement between both expressions leads to considerable differences between related quantities as, e.g., the virial coefficient (2.16). Using the two versions of the PP, we find for the corresponding contributions The second value represents the full result within the second Born approximation and has been analytically derived for the first time here. It agrees completely with the numerical results of Stolz and Zimmermann (1984) who analysed the expression (2.10) by evaluating the matrix elements M l s n ( q ) This . value was also published recently by Fehrenbach et a1 (1982) in connection with the electron-hole plasma in semiconductors. The Fourier transform Vp(q, 1s) which is of interest in transport theory is shown in figure 2 for both versions of the PP. Besides the deviations for q = 0 (3.3), the asymptotic behaviour q + CD is different. While the standard model potential ( 1 . 1 ) behaves like exp(-qr,), the numerical results for the Fourier transform of the PP (3.1) indicate a behaviour exp(-qc), c = 0.5. Besides the PP Vppin equation (2.9),the quantity V”) also gives rise to a corresponding contribution BL:2i to the virial coefficient Be,eiaccording to equation (2.16). Applying the effective potential V”’(nPk, n P + qk - q ) for hydrogen in the ground state n = Is, the following expression can be derived analogous to the treatment of the MontrollWard contribution to the thermodynamic functions (Kraeft et al 1973): (3.4) which represents the interaction between the electron and the static field of the atom (no virtual excitations) within the second Born approximation. A: = 27@h2/me is the electronic thermal wavelength and D ( x )= x e-x2j: d t exp( t’) denotes the Dawson integral. Using the hydrogen form factor Mlsls(q)= 8 n ( q 2 + 8)/(q2+4)’, the quantity 4076 K Redmer, Ci Ropke and R Zinimermann 3\ 30 I \ W O Figure 2. The Fourier transform of the PP (3.1) (full curve) for e-H interaction compared with the standard model potential (1.1) VE(q,1s) = - r 2 a , exp(-qr,)/r, (broken curve). BZ;, depends on temperature approximately according to T-', In particular, the value Bk?:,(T = 0) = - 2 5 ~ 1 8= -9.82 (3.5) is obtained for the case T = 0 which amounts to 28% of the result for B:,:, (3.3) which represents the sum of all virtual excitations n"# n. The Hartree contribution Bklk, BI,:), vanishes for neutral plasmas, i.e. ne = n, . Therefore, the full virial coefficient Be,,, for elastic electron scattering at the hydrogen ground state is given in the second Born approximation according to equations (3.3) and (3.5) by Be,el(T = 0) = -45.01. Notice that this result, for T = 0, is further modified by considering exchange contributions in the first and second Born approximations (Stolz and Zimmermann 1984). Let us briefly summarise the results of this section. Analytical results for the different contributions to the electron-atom virial coefficient Be,,, (2.16) were derived for hydrogen. The dominant contribution Be,,, is due to the polarisability of the atom. The Montroll-Ward like term B$, yields a further contribution which is not negligible compared with BZ,;,. This should also be accounted for in the case of alkali metals, which is treated in the aext section. + 4. Application to the alkai metals The calculation of the effective potentials V"), V") (2.9) and the derivation of the PP Vp requires a knowledge of the wavefunctions of the atom under consideration (consisting of N electrons). Therefore, except for the case of hydrogen, the numerical solution of the corresponding N-particle Schrodinger equation is necessary. In order to avoid expensive numerical work, appropriate approximation methods were Efective electron- atom interactions 4077 developed, such as the quantum defect theory (QDT) which provides the energy levels E,,. = -Ryd/n** of the valence electron and analytical results for the radial wavefunctions: R,,*,(r)= (2r/n*)"*exp(-r/n*) C akr-k K =O ak = ak-,n*[I ( I + 1) - (n* - k + l)(n* - k)]/2k a,= [ r ( n * + I + 1)r(n* - ~ ) ] - " ~ / n * ak = 0 for (4.1) k > n* + 1 (Bates and Damgaard 1949) on the basis of an effective one-particle Schrodinger equation (see also Burgess and Seaton 1960). This procedure is justified for the case of hydrogenic atoms, such as the alkali metals. It is the aim of this section to apply QDT to the determination of the virial coefficients Be,eirepresenting elastic electron scattering on neutral alkali metal atoms in the ground state n$. Before determining those virial coefficients, the efficiency of the QDT is demonstrated by calculating the PP at R=O (2.12), the dipole polarisability a!":' (2.14) and the quadrupole polarisability a:":': for all alkali metals and comparing the results with values for the polarisabilities which were taken from the review by Miller and Bederson (1977) (dipole polarisability a , ) and a pseudopotential approach of Maeder and Kutzelnigg (1979) (quadrupole polarisability a*). The generalised oscillator strengths in (4.2) were calculated with the help of the energy eigenvalues E,,* and the radial wavefunctions (4.1). The effective quantum numbers n* were matched to spectroscopic data (quantum defects) (see, e.g., Kondratyev 1967). The results for the quantities (4.2) within the QDT are shown in table 1 and are compared with the corresponding exact values for hydrogen and the values of Miller and Bederson (1977) and Maeder and Kutzelnigg (1979) for the alkali metals. We have applied the following method. First, the sum over all virtual excited states n* # n,* Table 1. The polarisation potential V P ( R= 0 , nt), the dipole polarisability ai";) and the quadrupole polarisability a?;' (4.1) for the alkali metals within quantum defect theory (QDT) compared with hydrogen results and those of Miller and Bederson (1977) (A); and Maeder and Kutzelnigg (1979) (B). H Li Na K Rb cs 1.0 1.8224 1.4175 1.0304 0.9960 0.95 12 4.5 162.9 158.0 280.5 306.0 382.4 164 159.3 293 319. 402 15.0 1442 1912 5 398 6 845 10 049 1383 1799 4597 5979 9478 4078 R Redmer, G Ropke and R Zimmermann (discrete as well as continuous) in (4.2) was calculated for hydrogen with the help of the usual hydrogen wavefunctions R,, which follows from the effective ones (4.1) (no quantum defects). These results are shown in the first row of table 1. The values for the alkali metals were obtained by a systematic replacement of the contributions of hydrogen transitions by the corresponding results for the alkali metal transitions which were determined within the QDT. This method yields a rapid convergence for the different quantities, which is demonstrated in figure 3 for V'( R = 0, no*) of potassium. The behaviour of the other alkali metals is similar to this example, as is the behaviour for the dipole and quadrupole polarisabilities. - bO. ' a96 '5 13 9 17 n* Figure 3. The rapid convergence of the quantities (4.2) with respect to n* is demonstrated for Vp(R = 0,4s) of K. The behaviour of the other alkali metals is similar to this example, and also for CY$":) and up;). A special characteristic of the alkali metals is the dominant contribution of the n$s-n,*p transition to the dipole polarisability ai"E' (about 99% of the total value) and the respective n$s-(n$ - l ) d transition to the quadrupole polarisability a:":' (from 79% for Li to 99% for Rb). Therefore, the transitions to the other virtual excited states yield only small corrections, except for the quantity V'( R = 0, no*),where the transitions into scattering states are not negligible. The agreement with the quoted values for the dipole and quadrupole polarisabilities of the alkali metals (Miller and Bederson 1977, Maeder and Kutzelnigg 1979) is sufficient for our purpose, the deviations being less than 6% except for a:";)of K (17%) and Rb (14%). The QDT is now applied for the calculation of the virial coefficients Be,ei(2.16) which are determined by the matrix elements M,,,(q). Besides the direct calculation of these quantities in q space which was performed by Stolz and Zimmermann (1984) for GaAs using hydrogen wavefunctions, the expansion of a plane wave can be used which leads to the following expression for the matrix elements (n,* = { n,*,0, 0): ground state, a = {n", I, m } : virtual excitation): (4.4) Efective electron-atom interactions 4079 where j , ( x ) denotes spherical Bessel functions. Therefore, the contribution Bz2i (3.4) to the total virial coefficient Be,eican be determined for all alkali metals when calculating the atomic form factor Mnl;OO,n~OO( q ) by means of the effective ground-state wavefunction Rnao(r). are shown in figure 4 as function of the temperature. The The results for quantity BL:2i decreases according to T-' and amounts, for typical plasma temperatures of about 5 x lo3 K, to values of 69% (Li) to 56% (Cs) of the zero-temperature results. Therefore, it is not negligible compared with the other contribution B E i . This dominant part Bz,Li is given by the expression r and was determined explicitly by calculating the n$s-n$p and n$s-(n$ + 1)s transitions. The amount of the other transitions was estimated from the corresponding hydrogen results. The values for the virial coefficients ( T = 5 x lo3 K) and Br,Li are given in table 2 for all alkali metals and, for comparison, for hydrogen ( 0 3). The contributions BE2i ( T = 5 x lo3 K) in the first column run from 17% (Li) to 26% (Cs) of the corresponding results for Br,Li within the QDT (column D). T (KI 4007, 2 194 3 2 3 105 2 3 i 300 100 m3 1 \ \ \ T IRyd) Figure 4. The Montroll-Ward-like contribution BLf?',(3.4) to the total electron-atom virial coefficient Be,,, for all alkali metals as a function of the energy and temperature, respectively. The arrows denote the values for T = 0 K, whereas the broken line indicates the results for T = 5000 K which are given explicitly in table 3. The results for the polarisation contribution Bz,Li within the QDT (C: n$s-(n$+ 1)s and n$s-n$p transition only, D: total value) are compared with the corresponding values of the standard model potential (l.l), i.e. B:,:il(l.l)= -7r2a!":)/ro(A, B). The choice of the parameters a l (dipole polarisability) and ro (cut-off radius) strongly affects the results for the virial coefficients which can be deduced by comparing columns A and B of table 2. 4080 R Redmer, G Ropke and R Zimmermann Table 2. The virial coefficients B$, and BE.:, representing elastic electron-alkali atom scattering within the quantum theory (QDT) compared with the hydrogen results and the standard mcdel potential (1.1) (all values in Ryd a i ) . A: standard model potential ( l . l ) , fitting ro to the electron affinities of Norcross (1974) and the dipole polarisabilities of Miller and Bederson (1977). B: standard model potential ( l . l ) , applying the calculated parameters a$";)and r,, = (ci$"G)/l VP(R = 0, n$)1)"4 of table 1 (first two columns). C: explicit results within the QDT from the n$s-(n$+l)s and n,*s-n,*p transitions. D: total virial coefficient Bz,!i (adding the remaining hydrogen contribution to C) within the QDT. H Li Na K Rb cs BL:2i (5000 K) A B C D 4.5 91.0 107.5 168.0 187.5 220.0 14.8 209.0 198.3 306.0 323.9 382.0 30.5 522.9 479.9 681.6 721.4 842.8 15.25 513.23 439.67 672.30 699.66 819.12 35.20 533.18 459.62 692.25 719.61 839.07 The parameter ro can be determined using experimental data for the dipole polarisability a$":' and the electron affinity E (e.g. Norcross 1974) of the alkali metals and supposing the relation V'( R = 0, ng) = - - a ~ " ~ '=/ -rE~ (e.g. Gryaznov et al 1980). The resulting virial coefficients Bz,:i within this method are quoted in column A and amount to 39% (Li) to 46% (Cs) of the QDT results (D). The results shown in column B are obtained using the dipole polarisabilities a\":) and the cut-off radii ro = ( a$.;)/I V'( R = 0, n,*)1)"4 of table 1 which were calculated by means of the QDT. The deviations from the more straightforward calculation which uses the direct determination of matrix elements (D) are less than 5 % (except H: ~O/O). Therefore, a complex theory like, e.g., the QDT,will lead to nearly identical results for the virial coefficients if it is applied for the direct calculation of the matrix elements (D) or related parameters (B). However, phenomenological approaches as mentioned above will fail. Furthermore, the additional contribution B!$ to the virial coefficient is only included within consistent many-body approaches such as, e.g., the Green's function technique. 5. Conclusion Starting from the Lippman-Schwinger equation for the general three-particle scattering problem, the electron-atom interaction as a special scattering channel was treated within a perturbation expansion up to the second order and the relevant effective interaction potentials V'l), V', V(*)were derived. These potentials are related to the atomic form factor ( V'") and the polarisation potential ( Vp) and lead to corresponding contributions to the virial coefficient for elastic electron-atom interaction, B$,, B$ and BE,;,, respectively. Analytical results for these quantities were considered for the case of hydrogen. Applying quantum defect theory for the calculation of transition matrix elements, the dipole and quadrupole polarisabilities of the alkali metals were obtained in satisfactory agreement with more complex approaches (cf table 1). Therefore, this method was used for the determination of the virial coefficients for the 408 1 Efective electron-atom interactions elastic electron-alkali atom interaction (cf table 2). Phenomenological approaches to the parameters of the standard model potential ( l . l ) , as described in the previous section, lead to virial coefficients B:,i which amount to less than 50% of the more precise values within quantum defect theory. Furthermore, the contributions BLfe, cannot be derived from those phenomenological statements which therefore underestimate the influence of electron-atom interaction on the thermodynamic properties of partially ionised alkali plasmas. This was shown recently considering the critical data as a special characteristic of dense alkali plasmas. In table 3 we show the results for the critical temperature T", applying different methods for the treatment of the neutral component (atoms, dimers, etc) of a partially ionised Cs plasma. Using the standard model potential (1.1) for the electron-atom interaction and determining the parameters a l ,ro phenomenologically (Richert et a1 1984), a value of 2600 K was obtained compared with the experimental value of 1925 K (Jungst et a1 1985). Taking into account Cs dimers as well, determining the parameter ro from the Schrodinger equation for the corresponding bcattering process and using the known binding energies (e.g. considering e- + Cs + Cs-, Eb(Cs-)= -0.47 eV (Norcross 1974)),Redmer and Ropke (1985) obtained a lower value of 2200 K for T". Table 3. The critical temperature T" of Cs plasma resulting from different treatments of the neutral component and the related virial coefficients Be,ei. Tcr(K) Reference Remarks (see 95) 2600 2200 2030 Richert et al (1984) Redmer and Ropke (1985) Redmer (1985) a,: from experiment, r,: phenomenological a,:from experiment, r,: fit to E h ( C s - ) 1924 Jiingst ef al (1985) Direct calculation of matrix elements within the QDT Experimental value A satisfactory agreement with the experimental value for the critical temperature is only attainable within a consistent many-body approach. Besides the well developed treatment of the charged particle interaction (see e.g. Richert et a1 1984), a rigorous treatment of the electron-atom interaction is also needed. Compared with the parametrisation of this interaction in the form of PP like ( l . l ) , this paper allows for a more basic treatment of these effects. Starting from the quantum statistical equation of state (A2.5) which defines the particle density as a function of temperature and chemical potential via the imaginary part of the thermodynamic Green's function, the present approach for the electron-atom virial coefficient Be,eican be generalised to allow for density corrections such as self-energy, dynamical screening and the Pauli exclusion principle, as shown by Redmer and Ropke (1985) for the case of the standard model potential (1.1). Replacing the standard model potential (1.1) by the better treatment for the electron-atom virial coefficient Be,eiof the present paper and taking into account the interaction between charged particles (electrons, ions) in arbitrary order, a further improvement of the critical temperature is obtained with the result of 2030 K (Redmer 1985) which deviates by only 5% from the experimental result. A more detailed representation of critical data for all alkali metals within the Green's function technique is in preparation. 4082 R Redmer, G Ropke and R Zimmermann Appendix 1 The PP will be derived as a special solution of the three-particle problem. Let us consider the following process: three particles ( 1 , 2 , 3 ) are added at time t = 0 to the system, and they are removed from the system at time t with different momenta (l’,2‘, 3’). The propagation of these three particles can be described by the thermodynamical three-particle Green’s function G3 (2.1). Within a perturbative treatment, this Green’s function can be expressed by the sum of all connected diagrams with three in- and out-going one-particle propagators. In lowest order with respect to density corrections, G3 is given by the single-frequency ladder Green’s function G3which follows from the three-particle Lippmann-Schwinger equation [ G I =[G~I+[G!l[VJ[G?I. (Al.l) G! = G:G:G: is the Green’s function for three non-interacting particles, and exchange contributions are dropped in (Al.1). V, is expressed by the interaction potential V according to v3(123, 1’2’3’)= v(12, 1’2’)83,3,+V(13, 1’3’)82,2,+V(23, 2’3’)81,1,. (A1.2) Then the following diagrammatic representation of ( A l . l ) is obtained (cf Joachain (A1.3) The general solution of this three-particle problem is obtained using the Fadeev technique. In this paper, we are interested in a perturbative treatment of the special channel which describes the scattering of free charged particles at neutral atoms. A rearrangement of the perturbation expansion (A1.3) which is appropriate to describe the formation of two-particle bound states (atoms) yields s”;i - = (Al.4) + For the definition of the ladder Green’s function G i and the vertex function M, see (2.4) and (2.6). This representation of G ) corresponds to the chemical picture where the two-particle subsystem (atom) is considered as a new entity so that (A1.4) is formally equivalent to the two-particle ladder equation (2.3). Instead of G i = G:Gy in (2.3), a new element GE,= GkG: arises in (A1.4). However, the two-particle propagator G,” is more involved than G: due to the occurrence of the internal quantum number n in addition to momentum and frequency. Therefore, (A1.4) is a matrix equation with respect to this new variable n. The solution of (A1.4) can be given in the representation G:(nPk, n‘p’k’, z ) , where k refers to the free-particle quantum state and nP to the two-particle basis. In order to solve the matrix equation (A1.4) it is useful to project out the diagonal part by introducing a new quantity K ( nPk, nP’k’, z ) according to k 7 k’ kl-q k’ nP nP‘=‘nP pn&- ‘ k + k k’ ‘I I nP &rL;,: -,*‘ (Al.5) where the internal quantum numbers like n” of the intermediate two-particle bound Efective electron- atom interactions 4083 state are different from n, i.e. n”# n. Then, the diagonal part of (A1.4) is transformed into k’ k - k k (A1.6) A corresponding approach can be given for the non-diagonal part of Gi. However, here we are only interested in the further evaluation of the diagonal part (A1.6). The new quantity K (A1.5) can be interpreted as an effective, dynamical potential for the electron-atom interaction without excitation of internal degrees of freedom in the final atomic state. This can be illustrated by considering the T matrix defined by Gk(nP1, nP’k’, z ) = GE,(nPk, z ) S ~ , ~ ~ ~ ~ ~ ~ +GE,(nPk, z ) T ( n P k ,nP’k‘, z)GEf(nP’k’,z ) so that for the diagonal part of T we have T = K representation, k i nP - q = P’ n ) q P’ n + ] m (A1.7) +K G i f T i k nP” G i nP” ’ or, in diagrammatic . (A1.8) n P’ This equation is equivalent to the well known relation between T and V for the two-particle problem, i.e. T = V + VG:T. The effective potential K can be expanded with respect to the bound-free interaction according to (A1.5): K = K , + K2 . . . . (A1.9) + Physical quantities are determined by the T matrix (see appendix 2). In order to be consistent we expand T up to second order with respect to the bound-free interaction and obtain T Ti’)+ T y )+ Ti2) 2: where T$’)= K 1 denotes the diagonal part of the atomic form factor, TI‘’ = K 2 yields the PP and Ti2)= K I G E f K 1leads to a Montroll-Ward-like contribution. Explicit expressions for these quantities are quoted in equation (2.8). For a more detailed discussion of the introduction of a PP see also Goldberger and Watson (1964). Appendix 2 The quantities V ( * )Vp, , V(2)which are defined by the T matrix according to equation (2.9) are immediately related to thermodynamical, kinetic and optical properties of partially ionised plasmas. For instance, the self-energy contribution of a charged particle due to interaction with a neutral cluster XC,ei( k , z ) is given within the second Born approximation by q e i (k,z ) = (A2.1) R Redmer, G Ropke and R Zimmermann 4084 where gab(E ) = [exp(p( E - p a - p b ) )- 1 I-' denotes the Bose like two-particle distribution function, pa is the chemical potential and a, b, c characterise the eiementary species (electrons and ions). Exchange contributions are dropped in (A2.1). The corresponding quasiparticle shift is defined via the real part of the self-energy as = x ( V'"( nPk, nPk) .t V'( nPk, nPk) + V'2'( nPk, nPk))g,,( Eflp) (A2.2) flP where the T matrix at the special variable leads to the effective potentials V (2.9). The self-energy contribution of a neutral cluster due to interaction with a charged particle is given by an analogous expression to (A2.1). One obtains for (ne x'ei,c 2)= I I -1 T(nPk, nPk, z+E,(k))f,(k) k (A2.3) wheref,(k) = {exp[P(E,(k) - p c ) ]+ l}-' denotes the Fermi distribution function. The related quasiparticle shift is given by + V"'( nPk, nPk))fc (k). (A2.4) Comparing (A2.2) and (A2.4), gei(Eflp)is replaced byf,(k) in the sense of the chemical picture. The equation of state for the description of thermodynamic properties, the particle density n, of species c as function of the inverse temperature p = (k,T)-' and the chemical potential p,, can be defined via the imaginary part of the one-particle Green's function G,( k, z) according to Stolz and Zimmerman (1978) (A2.5) where (lo is the normalisation volume. Then, with Dyson's equation G;'( k, z) = Az -E,(k) -Z, (k, z ) , the quasiparticle shifts (A2.2) and (A2.4) can be related to the corresponding linearised virial coefficients Bc,ei= d3RV+( R ) which describe the non-ideal contributions to the equation of state (A2.5) due to interaction of neutral clusters with charged particles as follows (n:: charged particle density, no= Z,zpg,,( E n p ) :neutral particle density (atoms)): 5 (A2.6) The interaction Vp is also important in kinetic theory, especially for the determination of the collision term in the Boltzmann equation. We consider the linearised Boltzmann Effective electron-atom interactions 4085 equation for the stationary case in a constant electrical field E (Ropke 1983) E . { ( R ,n J + ( R , ri,J)=C Fy,(liy,,4,) (A2.7) Y' where v = ( k , n P ) runs over the free particles ( k ) as well as the bound states ( n P ) , and R = -Xieiri is the centre of mass charge. F,, denotes the distribution function both for free particles and composites. The correlation functions (li,,,, &)= r0 J -m d t e"'(h,,(t),li,,) (A2.8) dT Tr(p&( -i h.r)ti,) can be expressed by means of thermodynamic Green's functions. In particular, the following result can be obtained within the second Born approximation for elastic scattering of free charged particles at neutral clusters: x 1 V'"( nPk, nP'k')+ V'( nPk, nP'k') + V'"( nPk, nP'k')12 8kl k ( 8 k 2 , k ' - 8kz,kl)8k+P,k'+P'. (A2.9) In both cases considered here, only the on-shell T-matrix elements occur which can be expressed by the quantities V (cf (2.9)). 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