Download Effective electron-atom interactions and virial coefficients in alkali

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
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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
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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.
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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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