Basic Theoretical Physics: A Concise Overview P38 potx

53.2 Some Semiquantitative Considerations on the Development of Stars 387
For free electrons with spin g-factor 2 we have:
U
0
=2V
k
F

0
d
3
k
(2π)
3

2
k
2
2m
.
Using
d
3
k =4πk
2
dk
we obtain the non-relativistic zero-point pressure
p
0
=
2U

0
3V
≡
2
15π
2
ε
F
k
3
F
, with ε
F
=

2
k
2
F
2m
.
In the result for a classical ideal gas,
p =
N
V
k
B
T,
one thus has to replace not only the thermal energy k
B

T by ε
F
but also the
number density
N
V
by k
3
F
(i.e. essentially by the reciprocal of the third power of the separation of two
electrons at the Fermi energy). In this way one again obtains the correct
result, apart from dimensionless constants.
The zero-point pressure of an electron gas (also referred to as degeneracy
pressure) is the phenomenon preventing the negatively charged electrons in
metals from bonding directly with the positively charged atomic nuclei. This
degeneracy pressure also plays an important part in the following section.
53.2 Some Semiquantitative Considerations on the
Development of Stars
The sun is a typical main sequence star with a radius R ≈ 10
6
km and
a surface temperature T ≈ 6000 K; the mass of the sun is denoted below
by M
0
. In contrast to the main sequence stars, so-called white dwarfs, e.g.,
Sirius B, typically have a mass M ≤ 1.4 M
0
of the same order as the sun,
but radii about two orders of magnitude smaller, with R ≈ 10
4

km. So-called
neutron stars have somewhat larger mass, M ≥ 1.4 M
0
, but R ≈ 10 km, and
so-called black holes, which (roughly put) “suck in” all surrounding matter
and radiation below a critical distance, have a mass M ≥ (3 to 7) M
0
or even
 M
0
.Theattractive force due to gravitation is opposed in the interior of
the star by a corresponding repulsive force or internal pressure(see below).
Equilibrium between this internal pressure and the gravitational attrac-
tion in a spherical shell between r and r +dr is found in general from the
388 53 Applications I: Fermions, Bosons, Condensation Phenomena
following identity:
4πr
2
· [p(r +dr) −p(r)] = −γ
M(r)
r
2
4πr
2
dr · 
M
(r) .
Here 
M
(r) is the mass density, from which we obtain, after omitting the

index M, the differential equation:
dp
dr
= −γ
M(r)(r)
r
,
where M(r) is the total mass up to a radius r:
M(r)=
r

0
d˜r4π˜r
2
(˜r) .
Main sequence stars lie on a diagonal line of negative slope in a so-called
Hertzsprung-Russel diagram, this being a plot of luminosity L versus mass
M. For these stars the pressure p is determined from the ideal gas equation:
p =
N
V
k
B
T.
The behavior of the function M(r) can be roughly characterized as follows:
In a small core region around the centre of a star, which we are not interested
in at present, temperatures are extremely high (10
7
K and higher) due to
fusion processes (hydrogen is “burnt” to form helium), whereas outside the

core in the remainder of the star including its surface region there is a roughly
constant
3
, relatively moderate temperature, e.g., T ≈ 6000 K. Using this
rough approximation we can replace the above differential equation by an
average relationship between pressure p, particle density
n
V
:=
N
V
and temperature T where
¯p =¯n
V
·k
B
T.
4
Thus,
dp
dr
≈
¯p
R
.
3
This is a crude approximation, which is nonetheless essentially true for our prob-
lem.
4
In the following the “bar” indicating an average will usually be omitted.

53.2 Some Semiquantitative Considerations on the Development of Stars 389
Writing
¯ =¯n
V
·m
proton
we obtain the following sequence of equations:
¯p
¯ ·c
2
=
k
B
T
m
proton
· c
2
!
= γ
M/c
2
R
=:
R(M)
R
≈ 10
−6
,
i.e. the so-called Schwarzschild radius of the sun,

R(M
0
)=γM
0
/c
2
,
has an approximate value of only 1 km.
For main sequence stars the ratio
¯p
¯c
2
=
R(M)
R
tells us how small general relativistic space curvature effects are, i.e. O(10
−6
).
(We shall see later that in the case of white dwarfs such effects are also small:
O(10
−4
); not until we come to neutron stars do the effects reach the order of
magnitude of 1.)
After exhausting the original nuclear fuel (i.e., when hydrogen in the
core region has been fully converted into helium) an accelerating sequence
of processes occurs, commencing with the conversion of helium into heav-
ier elements and ending with iron. During this sequence the temperature T
decreases gradually, and as a consequence, as we see from the above series
of equations, the radius of the star and its luminosity increase, whereas the
total mass remains approximately constant, since the core represents only

a small fraction of the total mass of the star. As a consequence, a so-called
red giant is formed, which is a type of star such as the bright twinkling
red star of Betelgeuze in the upper left part of the constellation of Orion.
Finally a so-called supernova explosion occurs, where the gas cloud of the
star is almost completely repelled and the remaining rest mass collapses
into a) a white dwarf for M
<
∼
1.4M
0
, b) a neutron star for M
>
∼
1.4M
0
or c) a black hole for M
>
∼
(3 to 7)M
0
or  M
0
5
. (These numbers must be
regarded as only very approximate, especially with regard to black holes.
The point is, however, that the gravitational attraction can be counter-
acted by the degeneracy pressure of the electron gas in a white dwarf or
the neutron gas in a neutron star, but no longer in the case of a black
hole
6

.)
5
e.g., 10
6
M
0
6
or perhaps by quantum fluctuations
390 53 Applications I: Fermions, Bosons, Condensation Phenomena
a) For white dwarfs we may write (using a non-relativistic approach):
p = p
(e)
0
=
2U
(e)
0
3V
≈ n
V
ε
(e)
F
,
M
= m
p
·n
V
,

ε
(e)
F
≈

2
2m
e

1
d
2
1,2

2
≈

2
2m
e

n
(e)
V

2
3
.
The indices
(e)

and
(p)
(see below) refer to electrons and protons respec-
tively. Now define the mixed density

(p,e)
0
:=
m
p


m
e
c

3
.
The proton mass m
p
occurs in the numerator of this expression. However,
in the denominator we have the third power of the Compton wavelength
of the electron, since the electrons determine the pressure and density in
a white dwarf, whereas the protons determine its mass. Thus for a white
dwarf we have (apart from numerical factors of the order of unity):
p

M
c
2

≈
m
e
m
p


M

(p,e)
0

2
3
!
=
R(M)
R
≈ 10
−4
, since R ≈ 10
4
km .
Incidentally the surface temperature of a white dwarf is possibly very
much higher than for main sequence stars, e.g., T ≈ 27000 K in a standard
case. However, for the degenerate electron gas theory to be applicable, it
is only important that the condition ε
F
 k
B

T nevertheless holds well:
ε
F
≈
m
p
c
2
2000
≈ 10
5
eV , ˆ=10
9
K .
Here we have used the fact that the proton mass is approximately 2000
times that of an electron, i.e., m
p
c
2
≈ 931 MeV, whereas m
e
c
2
is only
0.511 MeV.
b) Neutron stars: If the imploding main-sequence star is heavier than about
1.4 times the mass of the sun, the electrons can no longer withstand the
gravitational attraction, not even using a relativistic calculation. But the
electrons react with the equally abundant protons in an inverse β-process
to become neutrons. (Normal β-decay is indeed n → p + e +¯ν

e
,withan
electron-antineutrino, ¯ν
e
. However, the theory of equilibrium in a chemical
reaction, which we shall go into later, also allows transitions primarily
to occur in the opposite direction when an electron and proton become
squashed to a separation of the order of 10
−13
cm, i.e., p + e → n +ν
e
). In
1987 a supernova in the Large Magellanic Cloud occurred accompanied
by a neutrino shower that scientists in Japan were able to observe. For
53.3 Bose-Einstein Condensation 391
such an explosion the remaining mass collapses to an object with a radius
of only about 10 km to form a neutron star. Since the moment of inertia
J =
2
5
MR
2
has decreased by, say, 10 orders of magnitude, the angular velocity ω
correspondingly increases by as many orders of magnitude, due to the
law of conservation of angular momentum. Finally one thus observes the
star’s remains as a so-called pulsar with enormously high values of ω
and correspondingly large magnetic field fluctuations, which periodically
recur like a cosmic beacon as the pulsating star rotates. In any case, since
neutron and proton have approximately the same mass, we may write:
p

(n−star)

M
c
2
=


M

(n)
0

2
3
!
=
R(M)
R
≈ 0.1to1.
The curvature of space, which Albert Einstein predicted in his general
theory of relativity, then becomes important. In calculating 
0
one must
now insert the Compton wavelength of the neutron, and not that of the
electron. This is indicated by the indices
(n−star)
and
(n)
.

c) Black holes:ForM  M
0
even the degeneracy pressure of the neutrons is
not sufficient to compensate for gravitational attraction, and a so-called
black hole forms. In this instance the Schwarzschild radius
R(M)=γ ·
M
c
2
has the meaning of an event horizon, which we shall not go into here.
Instead we refer you to the little RoRoRo-volume by Roman and Han-
nelore Sexl, “White Dwarfs – Black Holes”, [42], in which the relation-
ships are excellently shown in a semi-quantitative way at high-school or
undergraduate level.
7
53.3 Bose-Einstein Condensation
After having considered an ideal Fermi gas we shall now deal with an ideal
(i.e. interaction-free) Bose gas. We have
N(≡N
T,μ
)=

j=0,1,2,
n
j

T,μ
,
7
The book by Sexl and Urbantke, [43], is more advanced. Black holes are treated

particularly thoroughly in the very “ fat ” book by Misner, Thorne and Wheeler,
[44].
392 53 Applications I: Fermions, Bosons, Condensation Phenomena
where the index j =0, 1, 2, refers to single particle modes, and
n
j

T,μ
=
1
e
ε
j
−μ
k
B
T
− 1
is valid. Here, ε
j
are the single-particle energies, where the lowest energy is
given by ε
j=0
≡ 0. Dividing by the very large, but finite volume of the system
V ,weobtain
n
V
(T,V,μ):≡
N
V

=
V
−1
e
−
μ
k
B
T
− 1
+
∞

0
+
dε˜g(ε)
e
ε−μ
k
B
T
− 1
. (53.3)
0
+
is an arbitrarily small positive number. The integral on the right-hand-side
of the equation replaces the sum

j=1,2,
n

j

T,μ
in the so-called thermodynamic limit V →∞, whereas the first term on the
right, which belongs to j = 0, gives zero in this limit, as long as the chemical
potential μ is still negative. The quantity
˜g(ε):=
g(ε)
V
remains finite in the thermodynamic limit; and for Bose particles (with inte-
gral spin s) the following is valid:
˜g(ε)dε =(2s +1)·
d
3
k
(2π)
3
.
With
d
3
k =4πk
2
dk and ε(k):=

2
k
2
2M
B

,
where M
B
is the mass of the Bose particle, we obtain a result in the form
˜g(ε)=(2s +1)·c
M
· ε
1
2
,
where c
M
is a constant with dimensions,
c
M
=
M
3
2
2
1
2
π
2

3
.
Thus,
n
V

=
V
−1
e
−
μ
k
B
T
− 1
+(2s +1)· c
M
·
∞

0
+
ε
x
dε
e
ε−μ
k
B
T
− 1
, with x :=
1
2
.

53.3 Bose-Einstein Condensation 393
We shall now consider the limit μ → 0 for negative μ. The following is
strictly valid:
∞

0
ε
x
dε
e
ε−μ
k
B
T
− 1
≤
∞

0
ε
x
dε
e
−μ
k
B
T
− 1
≡ Γ (x +1)·ζ(x +1)· (k
B

T )
x+1
. (53.4)
If x>(−1), we have the gamma function
Γ (x +1)=
∞

0
t
x
e
−t
dt, and ζ(s) , for s>1 ,
the so-called Riemann zeta function
ζ(s)=
∞

n=1
1
n
s
;
(ζ(
3
2
)is2.612 ).
Thus, as long as the density remains below the critical limit n
c
(T ), which
results at a given temperature from the above inequality for μ → 0, every-

thing is “normal”, i.e. the first term on the right-hand-side of (53.3) can be
neglected in the thermodynamic limit, and μ (< 0) is determined from the
equation
n
V
(T,μ)=(2s +1)· c
M
·
∞

0
ε
1
2
dε
e
ε−μ
k
B
T
− 1
.
The critical density given above is ∝ T
x+1
. On the other hand, at a given
density one thus has a critical temperature T
c
(n
V
).

However, if at a given temperature the critical density n
c
(T )isexceeded
or at a given density n
V
the temperature is below T
c
, i.e.,
n
V
≡ n
c
(T )+Δn
V
, with Δn
V
> 0 ,
then the chemical potential remains “held” constant at zero,
μ(T,n
V
) ≡ 0 , ∀T ≤ T
c
.
Also,
Δn
V
=
V
−1
e

−
μ
k
B
T
− 1
, i.e. ≈
k
B
T
−V ·μ
, and
−μ =
k
B
T
VΔn
V
→ 0forV →∞.
394 53 Applications I: Fermions, Bosons, Condensation Phenomena
Thus in the thermodynamic limit the behavior is not smooth at T
c
,but
shows a discontinuity in the derivative
dμ(T )
dT
.
One can easily ascertain the order of magnitude of the critical tempera-
ture: Whereas for metals (Fermi gases) the following relation holds between
Fermi energy and density (apart from a factor of the order of unity)

ε
F
≈

2
2m
e
n
2
3
V
,
for Bose particles we have
k
B
T
c
≈

2
2M
B
n
2
3
V
(more exactly: k
B
T
c

=

2
2M
B
n
2
3
V
2.612
). Both expressions correspond to each
other in the substitution ε
F
→ k
B
T
c
with simultaneous replacement of the
particle mass m
e
→ M
B
.Thefactor
n
2
3
V
is therefore common to both, because, e.g.,

2

2m
e
n
2
3
V
gives the characteristic value of the kinetic energy of the electrons in the re-
gion of the Fermi energy. These are the same semi-quantitative considerations
as in the previous section on star development.
The first term on the right-hand-side of equation (53.3) can be assigned
to the superfluid component. It relates to condensed particles in their ground
state. The fraction of this so-called condensate is 100% at T = 0 K, decreasing
continuously to 0% as T → T
c
. The second term is the normal fluid.It
distributes itself over the single-particle excited states corresponding to grand
canonical Boltzmann-Gibbs statistics for bosons.
At normal pressure He
4
becomes liquid at 4.2 K and superfluid at 2.17 K.
The fact that the superfluid component possesses no internal friction can be
experimentally demonstrated by the well-known “fountain effect” and other
similar effects. However, if one calculates the critical temperature T
c
from
the above exact formula, one obtains 3.5 K instead of 2.17 K, and at very low
temperatures only 8% of the liquid is condensed, not 100%: The reason for
these quantitative discrepancies lies in interaction effects which are neglected
in the theory of an ideal Bose gas. Pure Bose-Einstein condensation, i.e.,
53.4 Ginzburg-Landau Theory of Superconductivity 395

with negligible particle interaction, has only recently
8
been found at ultralow
temperatures (T
c
<
∼
10
−7
K) in alkali gases.
The other noble gases (Ar, Kr, Xe) do not show superfluid behavior,
because they first become solid. He on the other hand remains liquid at
normal pressure even at very low temperatures, because the kinetic energy
of the atoms is too large for solidification to occur. In contrast to He
4
the
He
3
isotope is a fermion, not a boson. Therefore, it was a great surprise
when in 1972 Osheroff et al., [26], found superfluidityy in He
3
.Allthree
authors of that paper were awarded the physics Nobel prize of 1996 honoring
their detection that also He
3
becomes superfluid, however, at temperatures
about three orders of magnitude smaller than He
4
: T
c

=2.6mK
9
for He
3
.
As we shall see later, this occurs by the formation of so-called Cooper pairs
each consisting of two fermions, which themselves form a pair condensate
10
.
Legget was able to interpret the experiments of Osheroff et al. theoretically
and was awarded the Nobel prize in 2003 for this achievement, together with
Ginzburg and Abrikosov, who were rewarded for their work in the field of
superconductivity (see next section).
53.4 Ginzburg-Landau Theory of Superconductivity
The phenomena of superconductivity and superfluidity are in fact very closely
connected: a superconducting system can be thought of as a charged super-
fluid (see below), though the charge carriers are not of the elementary value
q
e
= e, as one had believed up to the introduction of the BCS theory in
1957
11
. Instead, they correspond to q
e
=2e,i.e.toCooperpairs,whichwere
proposed just before the BCS theory, [25]. However, aside from that, a phe-
nomenological theory of superconductivity had already been proposed in 1950
by Ginzburg and Landau, [46], which proved to be very fruitful and correct in
all details, and which lead amongst other things to the flux line lattice theory
of Abrikosov, [47], being established, for which – as already mentioned – the

Nobel prize in 2003 was awarded (Abrikosov, Ginzburg, Legget).
In the following section we shall describe Ginzburg and Landau’s theory
of superconductivity: In this theory the superconducting condensate is de-
scribed by a complex so-called order parameter function Ψ(r,t). The name
of this function reminds one of quantum mechanics; however, the capital letter
8
Cornell, Ketterle and Wiemann were awarded the Nobel prize in 2001 for work
they had performed on the Bose-Einstein condensation of ultracold gases of alkali
atoms in 1995.
9
These are low temperatures, but not yet the ultralow ones mentioned above.
10
The complexity of the order parameter in He
3
is described in a comprehensive
book by Vollhardt and W¨olfle, [45].
11
Named after Bardeen, Cooper und Schrieffer. The BCS theory, see [25], was
proposed in 1957, almost half a century after the experimental discovery of the
phenomenon (1911) by Kammerlingh-Onnes in Leiden.
396 53 Applications I: Fermions, Bosons, Condensation Phenomena
suggests that Ψ is not to be thought of as probability amplitude, but rather
as a classical quantity. For stationary states the time dependence will not be
explicitly mentioned. n
s
(r)=|Ψ(r)|
2
is the density of the superconducting
condensate (“pair density”) and
j

s
(r)=
q
e
m
eff
·Re {Ψ
∗
(−i∇−q
e
A)Ψ}
=
q
e
m
eff
·


2i
(Ψ
∗
∇Ψ −Ψ∇Ψ
∗
) − q
e
n
s
A


is the supercurrent density. Separating Ψ (r) into modulus and phase,
Ψ = |Ψ(r)|·e
iφ(r)
,
we have
j
s
= q
e
·n
s
(r) ·
∇−q
e
A
m
eff
φ(r) ,
an expression, whose gauge-invariance
12
can be explicitly seen; m
eff
is the
effective mass of the carriers of the superconductivity.
The free energy F (T,V)iswrittenasapowerseriesin|Ψ|
2
. Neglecting
terms which do not influence the onset of superconductivity we have:
F (T,V)= min
Ψ,Ψ

∗
,A
⎧
⎨
⎩

V
d
3
r
2

1
2m
eff
|(−i∇−q
e
A) Ψ|
2
+ α ·(T − T
0
) ·|Ψ|
2
β
2
|Ψ|
4
+

+


R
3
d
3
r
(curlA)
2
2μ
0
⎫
⎬
⎭
. (53.5)
Here, μ
0
is the permeability of free space; α and β are positive constants, and
differentiation should be carried out independently with respect to Ψ and Ψ
∗
(i.e. with respect to the real part and imaginary parts of Ψ(r)), as well as
with respect to rotation of A, i.e. with respect to the magnetic induction
B =curlA .
In (53.5), the last integral over R
3
is the magnetic field energy, whereas the
first integral (delimited by square brackets) represents the free energy of the
condensate. The important term,
α ·(T − T
0
)|Ψ|

2
,
which shows a change of sign at the critical temperature T
0
, has been intro-
duced by the authors in an ad hoc way, and is justified by the results which
follow (see below).
12
Invariance w.r.t. gauge transformations, A → A+∇f(r); Ψ → Ψ ·exp(iq
e
f(r)/),
simultaneously and for any f(r).