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