378 51 General Statistical Physics Statistical Operator; Trace FormalismIn addition there is a Hermitian density operator ˆ ator or statistical operator, whose eigenvalues are just the p
Trang 150 Production of Low and Ultralow Temperatures; Third Law 375
At ultralow temperatures of this order of magnitude the phenomenon of Bose-Einstein condensation comes into play (see Sect 53.3)
In this case the third law only plays a part at even lower temperatures where quantization of the translational energy would become noticeable, i.e.,
at temperatures
k B T < ∼
2
2MeffR2 , where R describes the size scale of the sample and Meff may be the mass
of a Na atom in the normal state above the Bose condensation (or in the condensed phase: an effective mass) Anyhow, we should remind ourselves that at these low temperatures and the corresponding low particle numbers (some 104 to 105 instead of 1023) one should work with the microcanonical ensemble, not the canonical or grand canonical ones However one should also remember that the basic temperature definition
M ·6v s27
= 3kT
is also valid for microcanonical ensembles.
Trang 251 General Statistical Physics
(Formal Completion; the Statistical Operator; Trace Formalism)
Is it really necessary to diagonalize the Hamilton operator ˆH of the system
( ˆH → E j), if one “just” wants to calculate the partition function
Z(T ) =
j
e− Ej
kB T
of the system and obtain the thermodynamic potentials or thermal expecta-tion values
1 ˆ
A
2
T
j
e− Ej
kB T
Z(T ) ψ j | ˆ Aψ j ? The answer to this rhetorical question is of course negative Instead of di-agonalizing the Hamiltonian we can make use of the so-called trace formalism.
This approach is based on the definition:
trace ˆA :=
j
ψ j | ˆ Aψ j ,
which is valid for every complete orthonormal1basis It is easy to show that the expression on the r.h.s of this equation, the sum of the diagonal elements
of the matrix
A i,j :=ψ i | ˆ Aψ j ,
is invariant with respect to a base change It therefore follows, for example, that
Z(T ) = trace e −β ˆ H ,
where
e−β ˆ H
is the operator, which has in the base of the eigenfunctions of ˆH a matrix
representation with diagonal elements
e−βE j
(In another base it can also be defined by the power series ∞
n=0
(−β) n
n! Hˆn.)
1 Orthonormality is not even necessary On the other hand, operators for which the trace exists, belong to a class of their own (trace class)
Trang 3378 51 General Statistical Physics (Statistical Operator; Trace Formalism)
In addition there is a Hermitian density operator ˆ
ator or statistical operator, whose eigenvalues are just the probabilities p j (ψ j
are the corresponding eigenstates) For example, one can write the relation
S
k B
=−
j
p j ln p j
abstractly as
S
k B =
or even more abstractly:
S
just as one also uses, instead of the formula
ˆ A :=
j
p j ψ j | ˆ Aψ j ,
the more abstract formula
ˆ A ˆ:= traceˆA
However, one must realize that for this additionally gained freedom of avoid-ing diagonalizavoid-ing the Hamiltonian there is the penalty of more complicated calculations2 For example, it is easy to calculate the partition function if one has already diagonalized ˆH, whereas without diagonalization of it,
calcula-tion of
−β ˆ H
trace e−β ˆ H
becomes very difficult Indeed, the trace of a matrix product involves a double sum, e.g.,
traceˆA =
j,k j,k A k,j
2
A type of “conservation law for effort” holds here.
Trang 452 Ideal Bose and Fermi Gases
In the following section we shall consider identical particles, such as
elemen-tary particles and compound particles, as well as quasi-particles which are similar to light quanta (photons, electromagnetic waves): e.g., sound quanta (phonons, elastic waves) and spin-wave quanta (magnons) These particles or
quasi-particles are either
a) fermions (particles or quasi-particles with spin s = 1/2, 3/2, in units
of), such as electrons, protons, neutrons and He3, as well as quarks, from
which nucleons are formed (nucleons are compound particles made up of three quarks), or
b) bosons (particles or quasi-particles with spin s = 0, 1, 2, in units of
), such as, for example, the pion, which is an elementary particle of rest mass m0 ≈ 273 MeV/c2 consisting of two quarks; or a He4 particle; or one of the above-named quasi-particles which all possess zero rest mass and, as a result, vanishing chemical potential
Fermionic quasi-particles also exist in solids For example, in polar
semi-conductors there are the so-called polarons, which are electrons accompanied
by an attached phonon cloud This fermionic quasi-particle possesses a
non-negligible rest mass
If these particles or quasi-particles do not interact with each other (or only interact weakly), the energy levels are given by:
E n1,n2, =
nmax
n1 =0
.
nmax
n f=0
(n1ε1+ n2ε2+ ) ,
and the number of particles is:
N n1,n2, =
nmax
n1=0
.
nmax
n f=0
(n1+ n2+ )
There are thus n1 particles or quasi-particles in single-particle states of
en-ergy ε1, etc For fermions, nmax = 1, while for bosons, nmax = ∞ These statements are fundamental to quantum mechanics (viz Pauli’s exclusion
principle) In addition, if there is no particle interaction (i.e., in a dilute
Trang 5380 52 Ideal Bose and Fermi Gases
Bose or Fermi gas), the partition function can be factorized as follows:
Ztot(β, μ) = Z1(β, μ) · Z2(β, μ) · · Z f (β, μ) · ,
where β and μ are reciprocal temperature and chemical potential which
de-termine the mean values of energy and particle number respectively
It is therefore sufficient to calculate the partition function for a single factor, i.e., a single one-particle level For example,
Z f (β, μ) :=
nmax
n f=0
e−β·(ε f −μ)n f
For fermions the sum consists of only two terms (nmax = 1); a convergence
problem does not arise here For bosons on the other hand, since nmax is without an upper limit1, there is an infinite geometric series, which converges
if μ < ε f In both cases it then follows that
Z f (β, μ) =
1± e −β(ε f −μ)±1
For the±-terms we have a plus sign for fermions and a minus sign for bosons.
Only a simple calculation is now required to determine lnZ f and the expec-tation value
n f T := d
d(βμ)lnZ f
One then obtains the fundamental expression
n f T,μ= 1
where the plus and minus signs refer to fermions and bosons respectively For
a given temperature
k B β and average particle number N , the chemical potential μ is determined from
the auxiliary condition:
j
n j T,μ ,
1
An analogy from everyday life: With regard to the problem of buying a dress,
French women are essentially fermionically inclined, because no two French
women would buy the same dress, irrespective of the cost On the other hand,
German women are bosonically inclined, since they would all buy the same dress,
provided it is the least expensive
2 The limiting case μ → 0 − is treated below in the subsection on Bose-Einstein
condensation
Trang 652 Ideal Bose and Fermi Gases 381
as long as for all j we have ε j < μ (In the boson case, generally μ ≤ 0, if the lowest single-particle energy is zero As already mentioned, the case μ = 0 is
treated below in the chapter on Bose-Einstein condensation.)
The classical result of Boltzmann statistics is obtained in (52.2) when the
exponential term dominates the denominator, i.e., formally by replacing the term ±1 by zero It is often stated that different statistics are required for
fermions as opposed to bosons or classical particles, but this is not really the case, since the derivation of (52.2) is made entirely within the framework of the grand canonical Boltzmann-Gibbs statistics, and everything is derived together until the difference between fermions and bosons is finally expressed
by the value of nmax prescribed by the Pauli principle (see above), which depends on the fact that the spin angular momentum (in units of ) is an
integer for bosons ( → nmax=∞) and a half-integer for fermions (→ nmax=
1) In this respect the (non-classical) property of spin is crucial (We have
already seen in quantum mechanics how the Pauli principle is responsible for atomic structure and the periodic table of elements.) It is important to make this clear in school and undergraduate university physics and not to disguise the difficulties in the theory.3
3
e.g., one should mention that spin with all its unusual properties is a consequence
of relativistic quantum theory and that one does not even expect a graduate
physicist to be able to understand it fully
Trang 753 Applications I: Fermions, Bosons,
Condensation Phenomena
In the following sections we shall consider several applications of phenomeno-logical thermodynamics and statistical physics; firstly the Sommerfeld theory
of electrons in metals as an important application of the Fermi gas formal-ism, see [40] Actually, we are not dealing here with a dilute Fermi gas, as
prescribed by the above introduction, but at best with a Fermi liquid, since the particle separations are as small as in a typical liquid metal However,
the essential aspect of the formalism of the previous chapter – which is that interactions between particles can be neglected – is still valid to a good ap-proximation, because electrons avoid each other due to the Pauli principle As
a result, Coulomb interactions are normally relatively unimportant, as long
as the possibility of avoidance is not prevented, e.g., in a transverse direction
or in d = 1 dimension or by a magnetic field.
53.1 Electrons in Metals (Sommerfeld Formalism)
a) The internal energy U (T , V, N ) of such an electron system can be written
U (T , V, N ) =
∞
0
dε · ε · g(ε) · n(ε) T,μ , (53.1)
where g(ε) is the single-particle density; furthermore, dεg(ε) is equal to the number of single-particle energies ε f with values in the interval dε
(i.e., this quantity is∝ V ).
b) Similarly, for the number of particles N :
N =
∞
0
dεg(ε) · n(ε) T,μ (53.2)
The value of the chemical potential at T = 0 is usually referred to as the Fermi energy ε F , i.e μ(T = 0) = ε F Depending on whether we are dealing
at T = 0 with a non-relativistic electron gas or an ultrarelativistic electron gas (ε F m e c2 or e c2, where m e ≈ 0.5 MeV/c2 is the electron mass),
Trang 8384 53 Applications I: Fermions, Bosons, Condensation Phenomena
we have from the Bernoulli pressure formula either
p = 2U 3V or p =
U 3V . For electrons in metals, typically ε F =O(5) eV, so that at room temperature
we are dealing with the non-relativistic case1
An approximation attributable to Sommerfeld, [40], will now be described Firstly,
n(ε) T,μ for T = 0
is given by a step function, i.e
n(ε) T →0,μ= 0 for ε > ε F and = 1 for all ε < ε F
(neglecting exponentially small errors) Furthermore we can write the inte-grals (53.1) and (53.2), again neglecting exponentially small errors, in the form
∞
0
dε dF
dε · n(ε) T,μ , where F (ε) are stem functions,
F (ε) =
ε
0
dεf (ε) ,
of the factors
f (ε) := ε · g(ε) and f(ε) := g(ε)
appearing in the integrands of equations (53.1) and (53.2) Compared to
n(ε) T,μ ,
a function whose negative slope behaves in the vicinity of ε = μ as a (slightly smoothed) Dirac δ function:
−d
4k B T ·cosh ε −μ
2k B T
2(≈ δ(ε − μ)) ,
the functions F (ε), including their derivatives, can be regarded at ε ≈ μ as
approximately constant On partial integration one then obtains2:
∞
0
dε dF
dε · n(ε) T,μ=
∞
0
dεF (ε) ·
−d
dε n(ε) T,μ
,
1
The next chapter considers ultrarelativistic applications
2 The contributions which have been integrated out disappear, since F (0) = 0,
Trang 953.1 Electrons in Metals (Sommerfeld Formalism) 385
where one inserts for F (ε) the Taylor expansion
F (ε) = F (μ) + (ε − μ) · F (μ) + (ε − μ)2
2 · F (μ) +
On integration, the second, odd term gives zero (again neglecting exponen-tially small termsO(e −βε F)) The third gives
π2
6 (k B T )
2· F ,
so that, for example, from (53.2) the result
N =
μ
0
dεg(ε) + π
2(k B T )2
(μ) +
follows, where as usual the terms denoted by dots are negligible The integral gives
μ
0
dεg(ε) = N + (μ(T ) − ε F)· g(ε F ) + ,
so that:
μ(T ) = ε F − π2(k B T )2
6 · g (ε F)
g(ε F) + From (53.1) we also obtain
U (T , V, N ) =
μ
0
dεεg(ε) + π
2(k B T )2
6 · [εg(ε)]
|ε=μ +
Inserting the result for μ(T ), after a short calculation we thus obtain (with
U0:=
ε3F
0
dεε · g(ε)):
U (T , V, N ) = U0+π
2(k B T )2
6 g(ε F ) +
By differentiating U with respect to T it follows that
electrons in a metal give a contribution to the heat capacity:
C V =∂U
∂T = γk B T , which is linear in T and where the coefficient γ is proportional to the density
of states at the Fermi energy ε F :
γ ∝ g(ε )
Trang 10386 53 Applications I: Fermions, Bosons, Condensation Phenomena
For free electrons
g(ε) ∝ ε1
,
so that
g (ε
F)
g(ε F) =
1
2ε2F , and μ(T ) = ε F ·
1− π2(k B T )2
12(ε F)2 +
,
with negligible terms + This corresponds to quite a small reduction in μ(T ) with increasing temperature T This is quite small because at room
temperature the ratio
k B T
ε F
2
itself is only of the order of (10−2)2, as ε F ≈ 3 eVcorresponds to a temperature
of 3·104K (i.e., hundred times larger than room temperature) In some metals such as Ni and Pd,
(k B T )2g
g
is indeed of the same order of magnitude, but in these metals g (ε F) has
a negative sign – in contrast to the case of free electrons – so that at room
temperature μ(T ) is here slightly larger than at T = 0 (However, in some compounds γ is larger by several orders of magnitude than usual (even at zero
temperature) so that one speaks of “heavy fermions” in these compounds.)
Electrons in conventional metals behave at room temperature (or generally for (k B T )2 ε2
F ) as a so-called degenerate Fermi gas.
The essential results of the previous paragraphs, apart from factors of the order ofO(1), can be obtained by adopting the following simplified picture: Only the small fraction
k B T
ε F
of electrons with energies around ε = μ ≈ ε F are at all thermally active.
Thus, multiplying the classical result for the heat capacity
C V = 3
2N k B by
k B T
ε F
,
we obtain the above linear dependence of the heat capacity on temperature for electrons in a metal, apart from factors of the order ofO(1) In particular g(ε F) can be approximated by
N
ε F
.
In order to calculate the zero-point energy U0 one must be somewhat more careful Indeed,
U0=
ε F
dεε · g(ε)