III. A special problem: The
relation between physical and
chemical picture
Surely this is one of the problems deeply
connected with the work of GR which found
applications not only in plasma physics but also
in nuclear, high energy, and astro physics
(as to be seen on this school and in his lecture!)
III.1. The 2 basic expansions
in stat thermodynamics
 p  n  B2 n  B3 n 
2
3
 p  z  b2 z  b3 z  ...;
2
3
n  z  2b2 z  3b3 z  ...;
2
3
B2  b2 , B3  2b2  4b2 ,....
2
both are equivalent , convergenc e?
Simple classical examples show:
Convergence depends on the forces
hard spheres
Coulomb attraction
Transition from FE to a
chemical picture (CP)
p  z  b2 z 2  b3 z 3  ...;
n  z  2b2 z  3b3 z  ...;
2
3
New interpreta tion as densities :
n  z , n2  b2 z , n3  b3 z ,....
*
*
2
*
3
Chemical entities : free p., pairs, triples , ...!
*
n2
 b2 (T )  mass action law for pairs
* 2
(n )
The chemical picture is an idealization
• Partly based on intuition, partly on
reinterpretation of the fugacity expansion
• The CP works if the b_k are all strictly
positive: Then the system can be considered
as a mixture of ideal quasiparticles.
• In reality the b_k have positive and
negative parts ---> the reinterpretation
meets difficulties, in particular for plasmas:
long r. int. inf. many bound states
II.2. Density expansions for quantum plasmas:
By introducing the 2-particle Slater sum into the cluster exp. we
get for the free energy of symm plasmas
(similar results follow from GFA):
3
F (V , T , N )  Fid  kTV [
 S ( 2 )  S ( 3)  ...]
12
\vspa
S
( 2)
 2  na nb {(1   ea eb )ab [Q ( ab )   ab
3
ab
 O ( 2 )}
Q ( ie )   (T ) atomic p.f. BPL
(1) 2 s
E ( ab )]
2 sa  1
The (freehand) dashed curve shows p/n_i kT on density
for 10000K and 20000 K. Much better are z-exp
Phase diagram
ne, cm-3
1028
Metallic plasma
nee3 = 1
rD = 0.84a0
H2 = 0.5
1024
rs = 1
=1
Chemical picture
1020
 = 0.5
Strongly
Ionized plasma
1016 2
10
104
106
T, K
III.3. Fugacity expansions for
quantum plasmas
• The low density expansions converge only at
very low densities, the bound state terms give
very exp big contributions, need some sum-up
of infinite series in density
• Following ideas of Harold Friedman, Bartsch
and WE developed around 1971 the concept of
FE in plasma physics, the concept was largely
extended by KKE in the “red book” and in
many works by F. Rogers (Livermore)
Analytical results for the fugacity expansion
(low density, neglecting asymmetry effects)
K3
( 2)
( 3)
p( z , T )  pid 
 S z  S z  ...
12 \vspa
Sz
( 2)
 2  z a zb {(1  ea eb K )ab [Q( ab )   ab
3
ab
Q( ab )  2   (T )  ...
(1) 2 s
E ( ab )]}
2 sa  1
Bound states at low density are well described but not their destruction !
Hydrogen at low density - summation
of some higher order z-terms
Inclusion higher order terms up to z^4 gives molecule
formation, the (freehand) dashed curve shows p/n_i kT on
density for 10000K and 20000 K. No press-ionization!!!
Conclusion on z-expansions
• Higher z-orders provide further bound states,
• Partial summations of infinite series in the
fugacity are difficult (even in the ideal term),
• Sum-ups in z do not improve very much the
situation at high density !
• The problem of desctruction of b.s. (pressure
ionization, Mott effect) remains open!
• Fugacity representations are good for low d.
bound st, bad for free states ! What to do? CP!
III.4. Several variants of CP
Alternativ es :
(i) reinterpre tation of the density
nn
free
n
bound
(ii) reinterpre tation of pressure
p p
free
p
bound
p .
int
Def degree of ioniz   n
free
/n
Chemical description within the
Greens Function Approach (GFA)
by KKRRZ
Alternativ es :
(iii) reinterpre tation of the G.F. density
n   d1G1 n
free
n
bound
 n , or n  n  n
int
f
b
(iv) reinterpre tation of G.F. pressure
p   d 2VG2  p free  p bound  p int
( v) Fully consistent CP : Check all TD relations
H-plasma ..........................
•
•
•
•
•
•
•
•
•
Typical
picturefor the
composition
of H-plasma
(adv CP EHJRR)
typical valley of ion
destruction of b.st.
at high density
excluded vol effects
Coexistence line /diss. /ioniz.
Summarize properties of CP
• Thermodynamically, the CP is based on
relations for chemical potentials (derivative
of F is zero, minima at the boundary or case
of several minima may not be included !!!)
• Problems may arise with pressure ion, Mott
effect, and PPT !!!
• In good/consistent formulations (BPL, Pade
appr etc) pressure ionization and PPT are
well described
III.5. Variational Approach (VA)
Minimize the free energy in CD: get composition and
dispersion around the minimum of the free energy
VA is more general than method of chem pot
and always consistent (Hilbert/Hache/Spahn)
H-plasma ...................
VA......................
•
•
•
•
•
•
•
•
•
pressure related
to ionic pressure
(optimal for comp)
PPT at low temp
destruction of b.st.
at high densities
important excl vol!
Sensitive to appr.
Similar to adv CP
in
Summarize variational approach
• New informations on many minima,
minima at the border, dispersion of the
minima
• Problems with pressure ionization remain:
not always well expressed minima of F,
• BPL partition f. is under question (small
consistent changes of the p.f. change the
thermodynamics !!!). Onsagers principle
may be violated !!!
III.6. New density reprentations based on partial
sum-up and Pade appr.
Start from low d. exact results (4 virial coeffcients are exactly
known) connect to known high density asymptotics
p  n  C1n 3 / 2  C2 n 2  C3 n 5 / 2 ln n  C4 n 5 / 2  C5 n 3 ln n  ...
\vspa
Ck (T , e 2 ) are exactly known.
At high density Gellman, Brueckner, Carr, ....
Partial series of bound state contributi ons C2 sum - up all terms
correspond ing to O(z 2 ) in the fugacity exp.!
Extended (summed-up) density expansions include:
(i)
full electron Fermi fct, (ii) Pade for ee-ii-ie long-range,
(iii) summation of infinite bound state contributions
1
p ( n, T )  p e
 ni k BT\vspa
[1  S (ne   )]  (uee  uie  uii )  ..
3
2
3
S ( x)  x  2 x  5 x  14 x  ...    1
Fermi

3

   s exp(  Es )  1  Es ; Es  screened b.s.
2
s
*
*
*
Recently: improved density expansions describe
bound state effects+long range Coul-int.
IV. Critical discussion
• The chemical description CP is good but not a “deus
ex machina”. Problems at high densities: the
chemical species are no more stable quasiparticles!
• CP may be improved by variational methods, make
sense only if the free energy has a “good” minimum!
• At high densities fug exp and all variants of CP are
under question, also BPL-partition ! We recommend
summed up density repres. (with Pade - without CP)
• High dens plasma consists of strongly corr class ions
and weakly corr degenerated electrons !!!
Suggestions
• Compare, if possible, DE, FE, CP, summed-up
density representations with computer simulations!
• Check which free energy is lower. depend on conv!
• At high densities the CP+BPL may be not the best
variational choice, results may depend on conv!
Sometimes the dispersion of the minima is so large
that the composition is vague!
• For extreme densities we recommend as an
alternative new summed-up density representations,
no split in bound and free states!
Thank you for attention
Please look for references, reprints etc. at
www.werner-ebeling.de,
summa.physik.hu-berlin.de/tsd
contact by email: [email protected]
Gerd !
•
HAPPY B I RTH DAY
• All best wishes to you, good health
• and happy family, and more localization !
• We express also the wish on further close
collaboration and the hope that you will further
contribute to the solutions of the many open
problems in the field !!!!