Download - Equation Of State (EOS)
1
Deg
ener
ate
Elec
tron
Gas
Equa
tion
Of S
tate
(EO
S)
Deg
ener
ate
Elec
tron
Gas
Bol
tzm
ann
(B-s
tatis
tics)
dis
trib
utio
n (c
lass
ical
pic
ture
)
num
ber d
Ne
of fr
ee e
lect
rons
in d
Van
d sp
heric
al s
hell
is:
Tp m
axto
sm
alle
r pva
lues
for
= co
nsta
nt.
B-st
atis
tics
viol
ates
Pau
li’s
excl
usio
n pr
inci
ple.
-ele
ctro
ns a
re fe
rmio
ns (s
pin
½).
-
we
cons
ider
gas
of h
igh
pres
sure
with
dV
bein
g fu
lly p
ress
ure
ioni
zed.
- -
,
Deg
ener
ate
Elec
tron
Gas
W. P
auli:
eac
h qu
antu
m c
ell (
dpx
dpy
dpz
dx d
ydz
) can
hol
d on
ly 2
ele
ctro
ns.
quan
tum
cel
l vol
ume:
dp x
dpyd
p zdV
= h3
.
num
ber o
f ele
ctro
ns in
she
ll
:
.
Paul
i:
.
Viol
atio
n al
so fo
r T=
cons
tant
and
hig
h de
nsiti
es, s
ince
f(p)
dp~
n e .
need
to in
clud
e qu
antu
m e
ffect
s if
eith
erT
too
low
or e
lect
ron
dens
ity to
o hi
gh, i
.e.
if el
ectro
ns b
ecom
e de
gene
rate
.
Tp m
axto
sm
alle
r pva
lues
for
= co
nsta
nt.
Deg
ener
ate
Elec
tron
Gas
The
com
plet
ely
dege
nera
te e
lect
ron
gas
@ T
=0
All e
lect
rons
hav
e lo
wes
t ene
rgy
with
out v
iola
ting
Paul
i’s p
rinci
ple,
i.e.
all
phas
e ce
lls u
p to
pF
are
allo
cate
d by
2 e
lect
rons
, all
othe
r cel
ls a
bove
pF
are
empt
y.
T=
0 K
tota
l num
ber N
eof
ele
ctro
ns:
(non
-rela
tivis
tic)
2
Deg
ener
ate
Elec
tron
Gas
non-
rela
tivis
tic:
If n e
is s
uffic
ient
ly la
rge
p Fca
n be
com
e so
hig
h th
at e
lect
ron
v~
spee
d of
ligh
t c
rela
tivis
tic(L
anda
u &
Lifs
chitz
vol
.2):
rest
ene
rgy
(rest
mas
s)
kin.
ene
rg.
The
com
plet
ely
dege
nera
te e
lect
ron
gas
@ T
=0
mom
entu
m fl
ux in
dire
ctio
n n
of e
- mov
ing
into
sol
id-a
ngle
ele
men
t d
s:
Deg
ener
ate
Elec
tron
Gas
Pres
sure
= fl
ux o
f mom
entu
m (t
hrou
gh u
nit s
urfa
ce a
nd s
econ
d).
The
com
plet
ely
dege
nera
te e
lect
ron
gas
@ T
=0
tota
l flu
x in
dire
ctio
n n
by in
tegr
atio
n ov
er a
ll s
of a
hem
isph
ere:
;
inte
rnal
ene
rgy
of e
lect
ron
gas
per u
nit v
olum
e:
; .
Deg
ener
ate
Elec
tron
Gas
The
com
plet
ely
dege
nera
te e
lect
ron
gas
@ T
=0
with
and
impo
rtanc
e of
rela
tivis
tic e
ffect
s...
..
Deg
ener
ate
Elec
tron
Gas
Tota
l num
ber o
f ele
ctro
ns:
Pres
sure
of e
lect
ron
gas:
does
not
dep
end
on T
The
com
plet
ely
dege
nera
te e
lect
ron
gas
@ T
=0
rela
tivity
par
amet
er...
..
Not
e:
3
Deg
ener
ate
Elec
tron
Gas
Non
-rela
tivis
tic li
mit
(asy
mpt
otic
beh
avio
ur) :
Inde
pend
ent
of T
The
com
plet
ely
dege
nera
te e
lect
ron
gas
@ T
=0
:
.
Deg
ener
ate
Elec
tron
Gas
Extre
me
rela
tivis
tic li
mit:
The
com
plet
ely
dege
nera
te e
lect
ron
gas
@ T
=0
:
.
Inde
pend
ent
of T
Deg
ener
ate
Elec
tron
Gas
Parti
alde
gene
racy
of e
lect
ron
gas
For f
inite
Tno
t all
elec
trons
are
den
sely
pac
ked
in m
omen
tum
spa
ce.
For h
igh
Tw
e ex
pect
them
to h
ave
a Bo
ltzm
ann
dist
ribut
ion.
We
furth
er e
xpec
t a s
moo
th tr
ansi
tion
from
com
plet
ely-
to n
on-d
egen
erat
e ca
se.
Mos
t pro
babl
e oc
cupa
tion
in m
omen
tum
spa
ce d
escr
ibed
by
Ferm
i-Dira
c(F
-D) s
tatis
tics:
…. d
egen
erac
y pa
ram
eter
(de
term
ines
deg
ree
of p
artia
l deg
ener
acy)
max
. allo
wed
occ
upat
ion
in s
hell
fillin
g fa
ctor
<1:
frac
tion
of o
ccup
ied
cells
Deg
ener
ate
Elec
tron
Gas
Parti
alde
gene
racy
of e
lect
ron
gas
Ferm
i-Dira
c(F
-D) d
istri
butio
n fu
nctio
n f(p
) for
par
tially
deg
ener
ated
ele
ctro
n ga
s:
T =
1.9
x 10
7K
= 10
n e=
1028
cm-3
4
Deg
ener
ate
Elec
tron
Gas
Part
ial d
egen
erac
y of
ele
ctro
n ga
s
F-D
:
whe
re
…. d
egen
erac
y pa
ram
eter
.
Non
-rela
tivis
tic (e
-den
sity
ne)
:
Deg
ener
ate
Elec
tron
Gas
Part
ial d
egen
erac
y of
ele
ctro
n ga
s
usin
g
Non
-rela
tivis
tic (e
-den
sity
ne)
:
F-D
:
&
Ferm
i-Dira
c in
tegr
als:
Deg
ener
ate
Elec
tron
Gas
Part
ial d
egen
erac
y of
ele
ctro
n ga
sD
egen
erat
e El
ectr
on G
asPa
rtial
dege
nera
cy o
f ele
ctro
n ga
s
Appr
oxim
atio
ns to
F-D
inte
gral
s:
(e- b
ehav
e al
mos
t lik
e an
idea
l gas
)
with
(stro
ng d
egen
erac
y)
5
Deg
ener
ate
Elec
tron
Gas
Parti
alde
gene
racy
of e
lect
ron
gas
Ferm
i-Dira
c(F
-D) d
istri
butio
n fu
nctio
n f(p
) for
par
tially
deg
ener
ated
ele
ctro
n ga
s:
T =
1.9
x 10
7K
= 10
n e=
1028
cm-3
the
larg
er T
the
smoo
ther
, i.e
. th
e Bo
ltzm
ann-
like
the
trans
ition
ab
out p
F.
Deg
ener
ate
Elec
tron
Gas
Parti
alde
gene
racy
of e
lect
ron
gas
Non
-rela
tivis
tic (e
lect
ron
pres
sure
):
with
with Th
is e
quat
ion
toge
ther
with
equ
atio
n fo
r ne
form
EO
S to
obt
ain
Pe(
n e, T
).
Deg
ener
ate
Elec
tron
Gas
Parti
alde
gene
racy
of e
lect
ron
gas
EOS
for n
on-r
elat
ivis
tic(x
0) e
lect
ron
gas
(sum
mar
y):
For g
iven
Tan
d n e
or
(1) p
rovi
des
a
nd w
ith (2
) Pe
is o
btai
ned
(and
als
o U
e)
(1)
(2)
Deg
ener
ate
Elec
tron
Gas
Parti
alde
gene
racy
of e
lect
ron
gas
EOS
for e
xtre
me-
rela
tivis
tic(x
) ele
ctro
n ga
s (s
umm
ary)
:
For g
iven
Tan
d n e
or
(1) p
rovi
des
a
nd w
ith (2
) Pe
is o
btai
ned
(and
als
o U
e)
(1)
(2)
∞
6
The
equa
tion
of s
tate
of s
tella
r mat
ter
The
ion
gas
(non
-deg
ener
ate)
:
of
sam
e or
der a
s
.
If io
ns o
f fer
mio
ns ty
pe (e
.g. p
roto
ns, H
e3) t
hey
may
bec
ome
dege
nera
te li
ke e
lect
rons
use
sam
e eq
uatio
ns a
s fo
r ele
ctro
n ga
s w
ith m
ere
plac
ed b
y m
ion
i.e.,
eq. f
or n
:(n
on-re
lativ
ist.)
Supp
ose
e-ga
s ha
s ce
rtain
than
the
ion
gas
with
sam
e
w
ill ha
ve th
e sa
me
if:
But:
ions
are
mai
n co
ntrib
utor
to d
ensi
ty
beca
use
e-ar
e de
gene
rate
(hig
her
mom
entu
m) a
lread
y at
low
er d
ensi
ties
The
equa
tion
of s
tate
of s
tella
r mat
ter
EOS
for (
norm
al s
tella
r mat
ter)
and
all d
egre
es o
f deg
ener
acy
and
rela
tivis
tic e
ffect
s:
For g
iven
a
nd c
hem
ical
com
posi
tion
:
from
(2)
(1)
(2)
(1)
Inte
rnal
ene
rgy
pe
r uni
t mas
s:
.
The
equa
tion
of s
tate
of s
tella
r mat
ter
Bord
er (
=co
nst.)
: ide
al –
dege
nera
te (n
on-re
l.)
com
pl. d
egen
. (no
n-re
l.) e
-:
whe
re d
oes
dege
nera
cy b
ecom
e im
porta
nt?
The
equa
tion
of s
tate
of s
tella
r mat
ter
Bord
er: n
on-re
lat.
and
rela
t. (d
egen
erat
e)
rel.
para
met
er
whe
re d
o re
lativ
istic
effe
cts
beco
me
impo
rtant
?
7
The
equa
tion
of s
tate
of s
tella
r mat
ter
Bord
er: i
deal
–de
gene
rate
(rel
ativ
istic
)
com
pl. d
egen
. (re
l.) e
-:
The
equa
tion
of s
tate
of s
tella
r mat
ter
Bord
er: i
deal
gas
and
radi
atio
n pr
essu
re
whe
re d
oes
radi
atio
n pr
essu
re b
ecom
e im
porta
nt?
The
equa
tion
of s
tate
of s
tella
r mat
ter
Ther
mod
ynam
ic q
uant
ities
, e.g
.
, for
som
e lim
iting
cas
es (P
rad=
0, id
eal i
ons)
:
com
plet
e de
gene
racy
& n
on-re
lativ
.:
com
plet
e de
gene
racy
& re
lativ
istic
:
stro
ng d
egen
erac
y
& n
on-re
lativ
.:
for s
mal
l
co
ntrib
utio
n:
usin
g
The
equa
tion
of s
tate
of s
tella
r mat
ter
Extre
mel
y re
lativ
istic
, x,(a
nd c
ompl
. deg
en. e
-) ga
s:
∞
Ther
mod
ynam
ic q
uant
ities
, e.g
.
, for
som
e lim
iting
cas
es (P
rad=
0, id
eal i
ons)
:
8
The
equa
tion
of s
tate
of s
tella
r mat
ter
Non
-rela
tivis
tic(d
egen
erat
e e-
):
(as
in a
n id
eal g
as)
Extre
me-
rela
tivis
tic (s
trong
deg
en.):
Ther
mod
ynam
ic q
uant
ities
, e.g
.
, for
som
e lim
iting
cas
es (P
rad=
0, id
eal i
ons)
:
inde
pend
ent o
f !
(non
-deg
. ion
s)
The
equa
tion
of s
tate
of s
tella
r mat
ter
Cry
stal
lizat
ion
-so
far a
ny in
tera
ctio
n be
twee
n io
ns w
ere
negl
ecte
d ( =
idea
l gas
)-n
ot v
alid
for h
igh
and
low
T.
… m
ean
sepa
ratio
n be
twee
n io
ns
… io
n ch
arge
-if th
erm
al k
inet
ic e
nerg
ykT
beco
mes
sim
ilar t
o el
ectro
stat
ic (p
oten
tial)
bind
ing
ener
gy(C
oulo
mb
ener
gy) i
ons
tend
to fo
rm a
rigi
d la
ttice
-> m
inim
izes
thei
r tot
al e
nerg
y
Def
.: c
oupl
ing
para
met
er
pote
ntia
l (C
oulo
mb)
bin
ding
ene
rgy
(ther
mal
) kin
etic
ene
rgy
… io
ns h
ave
B-di
strib
utio
n
… io
ns tr
y to
form
a c
ryst
alth
at h
as a
low
er e
nerg
y
The
equa
tion
of s
tate
of s
tella
r mat
ter
Cry
stal
lizat
ion
Crit
ical
val
ue fo
r tra
nsiti
on (S
hapi
ro &
Teu
kols
ky 1
983)
:
with
w
e ob
tain
est
imat
e fo
r crit
ical
(mel
ting)
tem
pera
ture
Tm
:
Such
con
ditio
ns a
re fo
und
in
cool
ing
whi
te d
war
fs
The
equa
tion
of s
tate
of s
tella
r mat
ter
Neu
tron
izat
ion
high
-ene
rgy
e-ca
n co
mbi
ne w
ith p
roto
ns to
form
neu
trons
if to
tal e
- ene
rgy
is:
At re
lativ
ely
low
t
hene
utro
n w
illde
cay
with
in 1
1 m
in to
pro
duce
pro
ton-
e-pa
irw
ith th
e e-
havi
ng e
nerg
y
.
.
How
ever
, for
com
plet
e de
gene
racy
Ferm
i ene
rgy
coul
d
a
nd re
leas
ede-
have
not
eno
ugh
ener
gy to
find
em
pty
cell
in p
hase
spa
ce
neu
tron
cann
ot d
ecay
Ferm
i sea
of e
-sta
biliz
es n
eutro
ns if
.
Usi
ng
and
&&
prot
on-e
- gas
n
eutro
n ga
s.i.e
. for
9
usin
g
and
&&
neut
ron
drip
(rele
ase
of fr
ee
neut
rons
incr
ease
sl
ope
agai
n)
Hae
nsel
, Pot
ekhi
n, Y
akol
ev (2
007)
rela
tivis
tic e
-in
crea
se o
ffre
e ne
utro
ns(d
egen
erat
e
)
prot
on-e
- gas
n
eutro
n ga
s.i.e
. for
e-ca
ptur
e:
The
equa
tion
of s
tate
of s
tella
r mat
ter
Neu
tron
izat
ion
In s
tars
situ
atio
n is
mor
e co
mpl
icat
ed:
at h
igh ,
pla
sma
cont
ains
hea
vy n
ucle
i,w
hich
cap
ture
e-(
“inve
rse
deca
y") t
o be
com
e ne
utro
n-ric
h is
otop
es
e
-en
ergy
nee
ds to
be
high
er th
an E
F.
If nu
clei
bec
ome
too
n-ric
h, th
ey b
reak
up
& re
leas
e n(
s)
n
eutro
n dr
ip.
The
equa
tion
of s
tate
of s
tella
r mat
ter
A se
lf-co
nsis
tent
app
roxi
mat
e ap
proa
ch
Idea
: fin
d a
sing
le e
xpre
ssio
n fo
r the
EO
S fro
m w
hich
all
ther
mod
ynam
ic q
uant
ities
e.g.
, , U
, cp,
, e
tc, a
re c
onsi
sten
tly d
eriv
ed fo
r giv
en P
, Tan
d X
i
Ansa
tz:
use
TD p
oten
tial o
f fre
e en
ergy
F(T
,V,{N
i}) =
U -
TSan
d fin
d re
actio
n eq
uilib
rium
by s
elec
ting
thos
e {N
} tha
t min
imiz
es F
(max
imiz
es e
ntro
py S
) for
giv
en T
,V,
subj
ect t
o co
nditi
on th
at to
tal n
umbe
rs o
f fre
e e-
and
any
nucl
ei a
re c
onst
ant.
From
min
imiz
ed fr
ee e
nerg
y F(
T,V
,{Ni})
all
TD q
uant
ities
can
be
deriv
ed, e
.g.
The
equa
tion
of s
tate
of s
tella
r mat
ter
A se
lf-co
nsis
tent
app
roxi
mat
e ap
proa
ch
Star
t fro
m c
anon
ical
par
titio
n fu
nctio
n (Z
usta
ndss
umm
e) Z
.
Con
side
r phy
sica
l sys
tem
(with
Ham
ilton
ian
H) c
onfin
ed in
a b
ox o
f vol
ume
Vin
con
tact
with
a h
eat r
eser
voir
at te
mpe
ratu
re T
:
… s
um o
ver a
ll di
agon
al te
rms
of H
amilt
on o
pera
tor,
whi
chin
clud
es th
e su
m o
ver a
ll in
tern
al e
xcita
tion
stat
es j
e.g.
, of
spe
cies
i
The
free
ener
gy F
(T,V
,{Ni})
is th
en o
btai
ned
from
:
()
Stat
istic
al m
echa
nics
-th
erm
odyn
amic
s
Prob
abilit
y
Parti
tion
func
tion
(can
onic
al)
Ludw
ig B
oltz
man
n(1
844
-190
6)
Hel
mho
ltz fr
ee e
nerg
y F
10
The
equa
tion
of s
tate
of s
tella
r mat
ter
A se
lf-co
nsis
tent
app
roxi
mat
e ap
proa
ch
Parti
tion
func
tion:
Free
ene
rgy:
The
equa
tion
of s
tate
of s
tella
r mat
ter
A se
lf-co
nsis
tent
app
roxi
mat
e ap
proa
ch
Saha
equ
atio
nca
n be
der
ived
from
min
imiz
ing
free
ener
gy F
(T,V
,{Ni})
(e.g
. Däp
pen
& G
uzik
(200
0)).
Addi
tiona
l ‘co
rrect
ions
’, su
ch a
s th
e el
ectro
n ch
emic
al p
oten
tial,
, can
than
ea
sily
and
con
sist
ently
be
adde
d to
Fby
.
The
equa
tion
of s
tate
of s
tella
r mat
ter
A se
lf-co
nsis
tent
app
roxi
mat
e ap
proa
ch
Tack
ling
the
prob
lem
of t
he d
iver
gent
par
titio
n fu
nctio
n Z i
nt
k…
. nr.
of e
lem
ents
j…. n
r. of
ioni
zatio
n st
ates
of e
ach
elem
ent
i…. n
r. of
bou
nd (e
nerg
y) s
tate
s of
eac
h el
emen
t
…. n
ewly
intro
duce
d w
eigh
ts d
escr
ibin
g pr
obab
ility
that
sta
te e
xist
s(M
HD
EO
S; M
ihal
as, H
umm
er, D
äppe
n 19
88)
The
equa
tion
of s
tate
of s
tella
r mat
ter
A se
lf-co
nsis
tent
app
roxi
mat
e ap
proa
ch
…. F
inite
vol
ume
of a
tom
s an
d io
ns
“
pres
sure
(den
sity
) ion
izat
ion”
…. D
ebye
-Hüc
kel a
ppro
xim
atio
n fo
r Col
oum
b ef
fect
s (s
cree
ning
effe
ct th
roug
h el
ectro
stat
ic p
oten
tial o
f ion
s) …. D
ebye
leng
th