N
1
V
V
where F ideal is the free energy of the ideal gas (i.e. the same gas but with U = 0), given by Eq. (16).
I believe that Eq. (89) is a very convincing demonstration of the enormous power of statistical physics methods. Instead of trying to solve an impossibly complex problem of classical dynamics of N
>> 1 (think of N ~ 1023) interacting particles, and only then calculating appropriate ensemble averages, the Gibbs approach reduces finding the free energy (and then, from thermodynamic relations, all other thermodynamic variables) to the calculation of just one integral on its right-hand side of Eq. (89). Still, this integral is 3 N-dimensional and may be worked out analytically only if the particle interactions are weak in some sense. Indeed, the last form of Eq. (89) makes it especially evident that if U 0
everywhere, the term in the parentheses under the integral vanishes, and so does the integral itself, and hence the addition to F ideal.
Now let us see what would this integral yield for the simplest, short-range interactions, in which the potential U is substantial only when the mutual distance r kk’ r k – r k’ between the centers of two particles is smaller than certain value 2 r 0, where r 0 may be interpreted as the particle’s radius. If the gas is sufficiently dilute, so that the radius r 0 is much smaller than the average distance r ave between the particles, the integral in the last form of Eq. (89) is of the order of (2 r 0)3 N, i.e. much smaller than ( r ave)3 N
= VN. Then we may expand the logarithm in that expression into the Taylor series with respect to the small second term in the square brackets, and keep only its first non-zero term:
T
F F
U T
d r d r e
.
(3.90)
ideal
N 3 ... 3
1
N
/
1
V
Moreover, if the gas density is so low, the chances for three or more particles to come close to each other and interact (collide) simultaneously are typically very small, so that pair collisions are the most important. In this case, we may recast the integral in Eq. (90) as a sum of N( N – 1)/2 N 2/2
similar terms describing such pair interactions, each of the type
N 2
U (r )/ T
V
kk
e
'
1
d 3 r d 3 r
.
(3.91)
k
k'
It is convenient to think about the r kk’ r k – r k’ as the radius-vector of the particle number k in the reference frame with the origin placed at the center of the particle number k’ – see Fig. 6a.
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(a)
(b)
"
r
r r
"
r '
kk'
Fig. 3.6. The definition of the
particle k
'
r
interparticle distance vectors
at their (a) pair and (b) triple
interactions.
particle k’
Then in Eq. (91), we may first calculate the integral over r k’, while keeping the distance vector r kk’, and hence U(r kk’), constant, getting one more factor V. Moreover, since all particle pairs are similar, in the remaining integral over r kk’ we may drop the radius-vector’s index, so that Eq. (90) becomes 2
T N
N
U r T
T
1
( ) /
F F
V
e
d r F
N B T
(3.92)
ideal
N
1 3
2
( ),
V
2
ideal
V
where the function B( T), called the second virial coefficient,42 has an especially simple form for spherically-symmetric interactions:
Second
1
r
3
1
virial
B T
( )
U ( ) / T
1 e
d r 4 r 2
dr
U( r)/ T
1 e
.
(3.93)
coefficient
2
2 0
From Eq. (92), and the second of the thermodynamic relations (1.35), we already know something particular about the equation of state P( V, T):
F
2
N T
2
N
N
P
P
B( T ) T
B( T )
.
(3.94)
ideal
2
2
V
V
V
V
T , N
We see that at a fixed gas density n = N/ V, the pair interaction creates additional pressure, proportional to ( N/ V)2 = n 2 and a function of temperature, B( T) T.
Let us calculate B( T) for a few simple models of particle interactions. The solid curve in Fig. 7
shows (schematically) a typical form of the interaction potential between electrically neutral atoms/molecules. At large distances the interaction of particles that do not their own permanent electrical dipole moment p, is dominated by the attraction (the so-called London dispersion force) between the correlated components of the spontaneously induced dipole moments, giving U( r) r-6 at r
.43 At closer distances the potential is repulsive, growing very fast at r 0, but its quantitative 42 The term “virial”, from Latin viris (meaning “force”), was introduced to molecular physics by R. Clausius. The motivation for the adjective “second” for B( T) is evident from the last form of Eq. (94), with the “first virial coefficient”, standing before the N/ V ratio and sometimes denoted A( T), equal to 1 – see also Eq. (100) below.
43 Indeed, independent fluctuation-induced components p( t) and p ’( t) of dipole moments of two particles have random mutual orientation, so that the time average of their interaction energy, proportional to p( t)p ’( t)/ r 3, vanishes. However, the electric field E of each dipole p, proportional to r-3, induces a correlated component of p ’, also proportional to r-3, giving interaction energy proportional to p ’E r-6, with a non-zero statistical average.
Quantitative discussions of this effect, within several models, may be found, for example, in QM Chapters 3, 5, and 6.
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form is specific for particular atoms/molecules.44 The crudest description of such repulsion is given by the so-called hardball model:
,
0
for r 2 r ,
U ( r)
0
(3.95)
,
0
2
for
r r ,
0
– see the dashed line and the inset in Fig. 7.
U ( r)
Fig. 3.7. Pair interactions of particles.
2 r 0
Solid line: a typical interaction potential;
dashed line: its hardball model (95);
0
r
dash-dotted line: the improved model
(97) – all schematically. The inset
U min
illustrates the hardball model’s physics.
2 r 0
As Eq. (93) shows, in this model the second virial coefficient is temperature-independent: 2
1 0 r
2
4
2
B( T ) b
4 r
dr
2 r 4 V ,
where V
r
,
(3.96)
0 3
3
0
0
0
2
3
3
0
so that the equation of state (94) still gives a linear dependence of pressure on temperature.
A correction to this result may be obtained by the following approximate account of the long-range attraction (see the dash-dotted line in Fig. 7):45
,
0
for r 2 r ,
U ( r)
0
(3.97)
U ( r with
,
)
U T
2
for
,
r r .
0
For this improved model, Eq. (93) yields:
1
2
U ( r)
a
B( T ) b
4 r
dr
b ,
with a 2
2
r dr U ( r)
.
(3.98)
2
T
T
2 0
r
2 0
r
In this model, the equation of state (94) acquires a temperature-independent term:
44 Note that the particular form of the first term in the approximation U( r) = a/ r 12 – b/ r 6 (called either the Lennard-Jones potential or the “12-6 potential”), that had been suggested in 1924, lacks physical justification, and in professional physics was soon replaced with other approximations, including the so-called exp-6 model, which fits most experimental data much better. However, the Lennard-Jones potential still keeps creeping from one undergraduate textbook to another one, apparently for a not better reason than enabling a simple analytical calculation of the equilibrium distance between the particles at T 0.
45 The strong inequality U << T in this model is necessary not only to make the calculations simpler. A deeper reason is that if (– U min) becomes comparable with T, particles may become trapped in this potential well, forming a different phase – a liquid or a solid. In such phases, the probability of finding more than two particles interacting simultaneously is high, so that Eq. (92), on which Eqs. (93)-(94) and Eqs. (98)-(99) are based, becomes invalid.
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2
2
2
N
N
a
N
N
N
P T b T b a .
(3.99)
V V
T
V
V
V
Still, the correction to the ideal-gas pressure is proportional to ( N/ V)2 and has to be relatively small for this result to be valid.
Generally, the right-hand side of Eq. (99) may be considered as the sum of two leading terms in the general expansion of P into the Taylor series in the density n = N/ V of the gas: Pressure:
2
3
N
N
N
virial
P T B( T
)
C( T
)
... ,
(3.100)
expansion
V
V
V
where C( T) is called the third virial coefficient. It is natural to ask how can we calculate C( T) and the higher virial coefficients. This may be done, first of all, just by a careful direct analysis of Eq. (90),46 but I would like to use this occasion to demonstrate a different, very interesting and counter-intuitive approach, called the cluster expansion method,47 which allows streamlining such calculations.
Let us apply to our system, with the energy given by Eq. (87), the grand canonical distribution.
(Just as in Sec. 2, we may argue that if the average number N of particles in a member of a grand canonical ensemble, with fixed and T, is much larger than 1, the relative fluctuations of N are small, so that all its thermodynamic properties should be similar to those when N is exactly fixed.) For our current case, Eq. (2.109) takes the form
N
2
E
/ T
N T
p
T
ln Z ,
with
/
m, N
Z e
e
,
k
E
U ( r ,..., r ) . (3.101)
N
N
m, N
N
N 0
m
k 2
1
1
m
(Notice that here, as at all discussions of the grand canonical distribution, N means a particular rather than the average number of particles.) Now let us try to forget for a minute that in real systems of interest the number of particles is extremely large, and start to calculate, one by one, the first terms ZN.
In the term with N = 0, both contributions to Em,N vanish, and so does the factor N/ T, so that Z 1. In the next term, with N = 1, the interaction term vanishes, so that E
0
m,1 is reduced to the kinetic
energy of one particle, giving
2
/ T
p
Z e
.
(3.102)
1
exp k
k
2 mT
Making the usual transition from the summation to integration, we may write
2
T
gV
p
Z ZI ,
where
/
3
Z e
d p
I .
(3.103)
1
1
2
exp
,
and
1
3
2
1
mT
This is the same simple (Gaussian) integral as in Eq. (6), giving
46 L. Boltzmann has used that way to calculate the 3rd and 4th virial coefficients for the hardball model – as much as can be done analytically.
47 This method was developed in 1937-38 by J. Mayer and collaborators for the classical gas, and generalized to quantum systems in 1938 by B. Kahn and G. Uhlenbeck.
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3 / 2
T
gV
T
mT
/
Z e
2 mT3/2
/
e
gV
.
(3.104)
2
3
2
2
Now let us explore the next term, with N = 2, which describes, in particular, pair interactions U =
U(r), with r = r – r ’. Due to the assumed particle indistinguishability, this term needs the “correct Boltzmann counting” factor 1/2! – cf. Eqs. (12) and (88):
1
2 /
p 2
p 2
Z
T
e
(r) /
exp
.
(3.105)
2
k k' U T
e
!
2 k, k'
2 mT
2 mT
Since U is coordinate-dependent, here the transfer from the summation to integration should be done more carefully than in the first term – cf. Eqs. (24) and (88):
2
2
2
2 / T 1 ( gV )
p
1
3
p'
Z e
3
U
(r)/ T 3
exp
exp
.
(3.106)
2
!
2 2
d p
d p'
e
d r
6
2 mT
2 mT
V
Comparing this expression with the Eq. (104) for the parameter Z, we get
Z 2
1
Z
I ,
where I
e U (r)/ T d 3 r .
(3.107)
2
2
2
!
2
V
Acting absolutely similarly, for the third term of the grand canonical sum we may get
Z 3
1
Z
I ,
where I
e U ( ' r, "
r ) / T d 3 r'd 3 r" ,
(3.108)
3
3
3
!
3
V 2
where r ’ and r ” are the vectors characterizing the mutual positions of 3 particles – see Fig. 6b.
These results may be extended by induction to an arbitrary N. Plugging the expression for ZN
into the first of Eqs. (101) and recalling that = – PV, we get the equation of state of the gas in the form T
2
3
Z
Z
P
ln1 ZI
I
I ... .
(3.109)
1
2
3
V
!
2
!
3
As a sanity check: at U = 0, all integrals IN are equal to 1, and the expression under the logarithm in just the Taylor expansion of the function eZ, giving P = TZ/ V, and = – PV = – TZ. In this case, according to the last of Eqs. (1.62), the average number of particles of particles in the system is N = –(/) T, V =
Z, because since Z exp{/ T}, Z/ = Z/ T.48 Thus, in this limit, we have happily recovered the equation of state of the ideal gas.
Returning to the general case of non-zero interactions, let us assume that the logarithm in Eq.
(109) may be also represented as a direct Taylor expansion in Z:
T
J
Cluster
l
l
P
Z .
(3.110) expansion:
V
pressure
l1 l!
48 Actually, the fact that in that case Z = N could have been noted earlier – just by comparing Eq. (104) with Eq.
(32).
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(The lower limit of the sum reflects the fact that according to Eq. (109), at Z = 0, P = ( T/ V) ln1 = 0, so that the coefficient J 0 in a more complete version of Eq. (110) would equal 0 anyway.) According to Eq, (1.60), this expansion corresponds to the grand potential
J
l
l
PV T
Z .
(3.111)
l1 l!
Again using the last of Eqs. (1.62), and the definition (104) of the parameter Z, we get J
Cluster
l
l
N
expansion:
Z .
(3.112)
1 (
l
l
)!
1
N
This equation may be used for finding Z for the given N, and hence for the calculation of the equation of state from Eq. (110). The only remaining conceptual action item is to express the coefficients Jl via the integrals IN participating in the expansion (109). This may be done using the well-known Taylor expansion of the logarithm function, 49
l
ln 1
( )
l
1
1
.
(3.113)
l1
l
Using it together with Eq. (109), we get a Taylor series in Z, starting as
T
2
3
Z
Z
P
Z
( I )
1
I
I
.
(3.114)
2
( )1 (3 )1
3
2
...
V
!
2
!
3
Comparing this expression with Eq. (110), we see that
J ,
1
1
1
J I 1
U
(r)/
e
T d r
2
2
1 3 ,
V
(3.115)
J ( I )
1 (
3 I )
1
3
3
2
1
U
( '
r , "
r ) / T
U
( '
r ) / T
U
( "
r ) / T
U
( '''
r ) /
e
e
e
e
T d r'd r"
2
2 3
3
, ...
V
where '''
r '
r "
r - see Fig. 6b. The expression of J 2, describing the pair interactions of particles, is (besides a different numerical factor) equal to the second virial coefficient B( T) – see Eq. (93). As a reminder, the subtraction of 1 from the integral I 2 in the second of Eqs. (115) makes the contribution of each elementary 3D volume d 3 r into the integral J 2 different from zero only if at this r two particles interact ( U 0). Very similarly, in the last of Eqs. (115), the subtraction of three pair-interaction terms from ( I 3 – 1) makes the contribution from an elementary 6D volume d 3 r’d 3 r” into the integral J 3
different from zero only if at that mutual location of particles, all three of them interact simultaneously, etc.
49 Looking at Eq. (109), one may think that since = Z + Z2 I 2/2 +… is of the order of at least Z ~ N >> 1, the expansion (113), which converges only if < 1, is illegitimate. However, the expansion is justified by its result (114), in which the n th term is of the order of N n( V 0/ V) n-1/ n!, so that the series does converge if the gas density is sufficiently low: N/ V << 1/ V 0, i.e. r ave >> r 0. This is the very beauty of the cluster expansion, whose few first terms, rather unexpectedly, give good approximation even for a gas with N >> 1 particles.
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In order to illustrate the cluster expansion method at work, let us eliminate the factor Z from the system of equations (110) and (112), with accuracy to terms O( Z 2). For that, let us spell out these equations up to the terms O( Z 3):
PV
J
J
2
2
3
3
J Z
Z
Z ..., .
(3.116)
1
T
2
6
J
2
3
3
N J Z J Z
Z ... ,
(3.117)
1
2
2
and then divide these two expressions. We get the following result:
2
2
PV
1 ( J / 2 J ) Z ( J / 6 J ) Z ...
J
J
J
2
1
3
1
2
2
3
2
1
Z
Z ,
(3.118)
2
2
N T
1 ( J / J ) Z ( J / 2 J ) Z ...
2 J
2 J
3 J
2
1
3
1
1
1
1
whose final form is accurate to terms O( Z 2). In this approximation, we may again use Eq. (117), now solved for Z with the same accuracy:
J
2
2
Z N
N .
(3.119)
J 1
Plugging this expression into Eq. (118), we get the virial expansion (100) with
2
J
J
J
2nd and 3rd
2
2
3
2
B( T )
V ,
C( T )
V
.
(3.120) virial
2
2 J
J
3 J
coefficients
1
1
1
The first of these relations, combined with the first two of Eqs. (115), yields for the 2nd virial coefficient the same Eq. (96), B( T) = 4 V 0, that was obtained from the Gibbs distribution. The second of these relations enables the calculation of the 3rd virial coefficient C( T). (Let me leave the calculation of J 3 and C( T), for the hardball model, for the reader’s exercise.) Evidently, a more complete solution of Eqs. (114), (116), and (117) may be used to calculate an arbitrary virial coefficient, though starting from the 5th coefficient, such calculations may be completed only numerically even in the simplest hardball model.
3.6. Exercise problems
3.1. Use the Maxwell distribution for an alternative (statistical) calculation of the mechanical work performed by the Szilard engine discussed in Sec. 2.3.
Hint: You may assume the simplest geometry of the engine – see Fig. 2.4.
3.2. Use the Maxwell distribution to calculate the drag
A
coefficient –F/ u, where F is the force exerted by an ideal classical gas on a piston moving with a low velocity u, in the simplest
u
geometry shown in the figure on the right, assuming that collisions of
gas particles with the piston are elastic.
3.3. Derive the equation of state of the ideal classical gas from the grand canonical distribution.
3.4. Prove that Eq. (22),
Chapter 3
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V V
V V
1
2
1
2
S
N ln
N ln
,
1
2
V
V
1
2
derived for the change of entropy at mixing of two ideal classical gases of completely distinguishable particles (that initially had equal densities N/ V and temperatures T), is also valid if particles in each of the initial volumes are indistinguishable from each other but different from those in the counterpart volume. For simplicity, you may assume that masses and internal degeneracy factors of all particles are equal.
3.5. A round cylinder of radius R and length L, containing an ideal classical gas of N >> 1
particles of mass m each, is rotated about its symmetry axis with angular velocity . Assuming that the gas as the whole rotates with the cylinder, and is in thermal equilibrium at temperature T, (i) calculate the gas pressure distribution along its radius, and analyze its temperature dependence, and
(ii) neglecting the internal degrees of freedom of the particles, calculate the total energy of the gas and its heat capacity in the high- and low-temperature limits.
3.6. N >> 1 classical, non-interacting, indistinguishable particles of mass m are confined in a parabolic, spherically-symmetric 3D potential well U(r) = r 2/2. Use two different approaches to calculate all major thermodynamic characteristics of the system, in thermal equilibrium at temperature T, including its heat capacity. Which of the results should be changed if the particles are distinguishable, and how?
Hint: Suggest a replacement of the notions of volume and pressure, appropriate for this system.
3.7. In the simplest model of thermodynamic equilibrium between the liquid and gas phases of the same molecules, temperature and pressure do not affect the molecule's condensation energy .
Calculate the concentration and pressure of such saturated vapor, assuming that it behaves as an ideal gas of classical particles.
3.8. An ideal classical gas of N >> 1 particles is confined in a container of volume V and wall surface area A. The particles may condense on container walls, releasing energy per particle, and forming an ideal 2D gas. Calculate the equilibrium number of condensed particles and the gas pressure, and discuss their temperature dependences.
3.9. The inner surfaces of the walls of a closed container of volume V, filled with N >> 1
particles, have N S >> 1 similar traps (small potential wells). Each trap can hold only one particle, at potential energy – < 0. Assuming that the gas of the particles in the volume is ideal and classical, derive an equation for the chemical potential of the system in equilibrium, and use it to calculate the potential and the gas pressure in the limits of small and large values of the N/ N S ratio.
3.10. Calculate the magnetic response (the Pauli paramagnetism) of a degenerate ideal gas of spin-½ particles to a weak external magnetic field, due to a partial spin alignment with the field.
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3.11. Calculate the magnetic response (the Landau diamagnetism) of a degenerate ideal gas of electrically charged fermions to a weak external magnetic field, due to their orbital motion.
3.12.* Explore the Thomas-Fermi model of a heavy atom, with nuclear charge Q = Ze >> e, in which the electrons are treated as a degenerate Fermi gas, interacting with each other only via their contribution to the common electrostatic potential (r). In particular, derive the ordinary differential equation obeyed by the radial distribution of the potential, and use it to estimate the effective radius of the atom.50
3.13.* Use the Thomas-Fermi model, explored in the previous problem, to calculate the total binding energy of a heavy atom. Compare the result with that for a simpler model, in that the Coulomb electron-electron interaction of electrons is completely ignored.
3.14. Calculate the characteristic Thomas-Fermi length TF of weak electric field’s screening by conduction electrons in a metal, modeling their ensemble as an ideal, degenerate, isotropic Fermi gas.
Hint: Assume that TF is much larger than the Bohr radius r B.
3.15. For a degenerate ideal 3D Fermi gas of N particles, confined in a rigid-wall box of volume V, calculate the temperature dependencies of its pressure P and the heat capacity difference ( CP – CV), in the leading approximation in T << F. Compare the results with those for the ideal classical gas.
Hint: You may like to use the solution of Problem 1.9.
3.16. How would the Fermi statistics of an ideal gas affect the barometric formula (28)?
3.17. Derive general expressions for the energy E and the chemical potential of a uniform Fermi gas of N >> 1 non-interacting, indistinguishable, ultra-relativistic particles.51 Calculate E, and also the gas pressure P explicitly in the degenerate gas limit T 0. In particular, is Eq. (48) valid in this case?
3.18. Use Eq. (49) to calculate the pressure of an ideal gas of ultra-relativistic, indistinguishable quantum particles, for an arbitrary temperature, as a function of the total energy E of the gas, and its volume V. Compare the result with the corresponding relations for the electromagnetic blackbody radiation and for an ideal gas of non-relativistic particles.
3.19.* Calculate the speed of sound in an ideal gas of ultra-relativistic fermions of density n at negligible temperature.
50 Since this problem, and the next one, are important for atomic physics, and at their solution, thermal effects may be ignored, they were given in Chapter 8 of the QM part of the series as well, for the benefit of readers who would not take this SM part. Note, however, that the argumentation in their solutions may be streamlined by using the notion of the chemical potential , which was introduced only in this course.
51 This is, for example, an approximate but reasonable model for electrons in white dwarf stars, whose Coulomb interaction is mostly compensated by the charge of nuclei of fully ionized helium atoms.
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3.20. Calculate basic thermodynamic characteristics, including all relevant thermodynamic potentials, specific heat, and the surface tension of a uniform, non-relativistic 2D electron gas with given areal density n N/ A:
(i) at T = 0, and
(ii) at low temperatures (in the lowest order in T/F << 1, giving a nonzero result), neglecting the Coulomb interaction effects.52
3.21. Calculate the effective latent heat ef ≡ – N( Q/ N 0) N,V of evaporation of the spatially-uniform Bose-Einstein condensate as a function of temperature T. Here Q is the heat absorbed by the (condensate + gas) system of N >> 1 particles as a whole, while N 0 is the number of particles in the condensate alone.
3.22.* For an ideal, spatially-uniform Bose gas, calculate the law of the chemical potential’s disappearance at T T c, and use the result to prove that the heat capacity CV is a continuous function of temperature at the critical point T = T c.
3.23. In Chapter 1 of these notes, several thermodynamic relations involving entropy have been discussed, including the first of Eqs. (1.39):
S G
/ T
.
P
If we combine this expression with Eq. (1.56), G = N, it looks like that for the Bose-Einstein condensate, whose chemical potential equals zero at temperatures below the critical point T c, the entropy should vanish as well. On the other hand, dividing both parts of Eq. (1.19) by dT, and assuming that at this temperature change the volume is kept constant, we get
C T S
T
V
/ . V
(This equality was also mentioned in Chapter 1.) If the CV is known as a function of temperature, the last relation may be integrated over T to calculate S:
C ( T )
S
V
dT
.
const
T
V const
According to Eq. (80), the specific heat for the Bose-Einstein condensate is proportional to T 3/2, so that the integration gives a non-zero entropy S T 3/2. Resolve this apparent contradiction, and calculate the genuine entropy at T = T c.
3.24. The standard analysis of the Bose-Einstein condensation, outlined in Sec. 4, may seem to ignore the energy quantization of the particles confined in volume V. Use the particular case of a cubic confining volume V = a a a with rigid walls to analyze whether the main conclusions of the standard theory, in particular Eq. (71) for the critical temperature of the system of N >> 1 particles, are affected by such quantization.
52 This condition may be approached reasonably well, for example, in 2D electron gases formed in semiconductor heterostructures (see, e.g., the discussion in QM Sec. 1.6, and the solution of Problem 3.2 of that course), due to the electron field’s compensation by background ionized atoms, and its screening by highly doped semiconductor bulk.
Chapter 3
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3.25.* N >> 1 non-interacting bosons are confined in a soft, spherically-symmetric potential well U(r) = m2 r 2/2. Develop the theory of the Bose-Einstein condensation in this system; in particular, prove Eq. (74b), and calculate the critical temperature T *
c . Looking at the solution, what is the most
straightforward way to detect the condensation in experiment?
3.26. Calculate the chemical potential of an ideal, uniform 2D gas of spin-0 Bose particles as a function of its areal density n (the number of particles per unit area) , and find out whether such gas can condense at low temperatures. Review your result for the case of a large ( N >> 1) but finite number of particles.
3.27. Can the Bose-Einstein condensation be achieved in a 2D system of N >> 1 non-interacting bosons placed into a soft, axially-symmetric potential well, whose potential may be approximated as U(r) = m22/2, where 2 x 2 + y 2, and { x, y} are the Cartesian coordinates in the particle confinement plane? If yes, calculate the critical temperature of the condensation.
3.28. Use Eqs. (115) and (120) to calculate the third virial coefficient C( T) for the hardball model of particle interactions.
3.29. Assuming the hardball model, with volume V 0 per molecule, for the liquid phase, describe how the results of Problem 3.7 change if the liquid forms spherical drops of radius R >> V 1/3
0
. Briefly
discuss the implications of the result for water cloud formation.
Hint: Surface effects in macroscopic volumes of liquids may be well described by attributing an additional energy (equal to the surface tension) to the unit surface area.53
53 See, e.g., CM Sec. 8.2.
Chapter 3
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Chapter 4. Phase Transitions
This chapter gives a rather brief discussion of coexistence between different states (“phases”) of collections of many similar particles, and transitions between these phases. Due to the complexity of these phenomena, which involve particle interactions, quantitative analytical results in this field have been obtained only for a few very simple models, typically giving only a very approximate description of real systems.
4.1. First-order phase transitions
From our everyday experience, say with water ice, liquid water, and water vapor, we know that one chemical substance (i.e. a set of many similar particles) may exist in different stable states – phases.
A typical substance may have:
(i) a dense solid phase, in which interatomic forces keep all atoms/molecules in virtually fixed relative positions, with just small thermal fluctuations about them;
(ii) a liquid phase, of comparable density, in which the relative distances between atoms or molecules are almost constant, but the particles are virtually free to move around each other, and (iii) a gas phase, typically of a much lower density, in which the molecules are virtually free to move all around the containing volume.1
Experience also tells us that at certain conditions, two different phases may be in thermal and chemical equilibrium – say, ice floating on water with the freezing-point temperature. Actually, in Sec.
3.4 we already discussed a qualitative theory of one such equilibrium: the Bose-Einstein condensate’s coexistence with the uncondensed “vapor” of similar particles. However, this is a rather exceptional case when the phase coexistence is due to the quantum nature of the particles (bosons) that may not interact directly. Much more frequently, the formation of different phases, and transitions between them, are due to particle repulsive and attractive interactions, briefly discussed in Sec. 3.5.
Phase transitions are sometimes classified by their order.2 I will start their discussion with the so-called first-order phase transitions that feature non-zero latent heat – the amount of heat that is necessary to turn one phase into another phase completely, even if temperature and pressure are kept constant.3 Unfortunately, even the simplest “microscopic” models of particle interaction, such as those discussed in Sec. 3.5, give rather complex equations of state. (As a reminder, even the simplest hardball model leads to the series (3.100), whose higher virial coefficients defy analytical calculation.) This is 1 The plasma phase, in which atoms are partly or completely ionized, is frequently mentioned on one more phase, on equal footing with the three phases listed above, but one has to remember that in contrast to them, a typical electroneutral plasma consists of particles of two very different sorts – positive ions and electrons.
2 Such classification schemes, started by Paul Ehrenfest in the early 1930s, have been repeatedly modified to accommodate new results for particular systems, and by now only the “first-order phase transition” is still a generally accepted term, but with a definition different from the original one.
3 For example, for water the latent heat of vaporization at the ambient pressure is as high as ~2.2106 J/kg, i.e. ~
0.4 eV per molecule, making this ubiquitous liquid indispensable for effective fire fighting. (The latent heat of water ice’s melting is an order of magnitude lower.)
© K. Likharev
SM: Statistical Mechanics
why I will follow the tradition to discuss the first-order phase transitions using a simple phenomenological model suggested in 1873 by Johannes Diderik van der Waals.
For its introduction, it is useful to recall that in Sec. 3.5 we have derived Eq. (3.99) – the equation of state for a classical gas of weakly interacting particles, which takes into account (albeit approximately) both interaction components necessary for a realistic description of gas condensation/liquefaction: the long-range attraction of the particles and their short-range repulsion. Let us rewrite that result as follows:
N 2
NT
Nb
P a
1
.
(4.1)
V 2
V
V
As we saw at the derivation of this formula, the physical meaning of the constant b is the effective volume of space taken by a particle pair collision – see Eq. (3.96). The relation (1) is quantitatively valid only if the second term in the parentheses is small, Nb << V, i.e. if the total volume excluded from particles’ free motion because of their collisions is much smaller than the whole volume V. In order to describe the condensed phase (which I will call “liquid” 4), we need to generalize this relation to the case Nb ~ V. Since the effective volume left for particles’ motion is V – Nb, it is very natural to make the following replacement: V V – Nb, in the equation of state of the ideal gas. If we also keep on the left-hand side the term aN 2/ V 2, which describes the long-range attraction of particles, we get the van der Waals equation of state:
N 2
NT
Van der
P a
.
(4.2)
Waals
V 2
V Nb
equation
One advantage of this simple model is that in the rare gas limit, Nb << V, it reduces back to the microscopically-justified Eq. (1). (To verify this, it is sufficient to Taylor-expand the right-hand side of Eq. (2) in small Nb/ V << 1, and retain only two leading terms.) Let us explore the basic properties of this model.
It is frequently convenient to discuss any equation of state in terms of its isotherms, i.e. the P( V) curves plotted at constant T. As Eq. (2) shows, in the van der Waals model such a plot depends on four parameters: a, b, N, and T, complicating general analysis of the model. To simplify the task, it is convenient to introduce dimensionless variables: pressure p P/ P c, volume v V/ V c, and temperature t
T/ T c, normalized to their so-called critical values,
1 a
8 a
P
, V 3 Nb, T
,
(4.3)
c
27 b 2
c
c
27 b
whose meaning will be clear in a minute. In this notation, Eq. (2) acquires the following form, 3
8 t
p
,
(4.4)
2
v
3 v 1
so that the normalized isotherms p( v) depend on only one parameter, the normalized temperature t – see Fig. 1.
4 Due to the phenomenological character of the van der Waals model, one cannot say for sure whether the condensed phase it predicts corresponds to a liquid or a solid. However, in most real substances at ambient conditions, gas coexists with liquid, hence the name.
Chapter 4
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SM: Statistical Mechanics
2
2
.
1
1
.
1
P
p
P c
t 0
.
1
1
Fig. 4.1. The van der Waals equation of
9
.
0
state, plotted on the [ p, v] plane for several
values of the reduced temperature t T / T c.
8
.
0
Shading shows the single-phase instability
0
range in that ( P/ V)
0
1
2
T > 0.
v V / Vc
The most important property of these plots is that the isotherms have qualitatively different shapes in two temperature regions. At t > 1, i.e. T > T c, pressure increases monotonically at gas compression (qualitatively, as in an ideal classical gas, with P = NT/ V, to which the van der Waals system tends at T >> T c), i.e. with ( P/ V) T < 0 at all points of the isotherm.5 However, below the critical temperature T c, any isotherm features a segment with ( P/ V) T >0. It is easy to understand that, as least in a constant-pressure experiment (see, for example, Fig. 1.5),6 these segments describe a mechanically unstable equilibrium. Indeed, if due to a random fluctuation, the volume deviated upward from the equilibrium value, the pressure would also increase, forcing the environment (say, the heavy piston in Fig. 1.5) to allow further expansion of the system, leading to even higher pressure, etc. A similar deviation of volume downward would lead to a similar avalanche-like decrease of the volume.
Such avalanche instability would develop further and further until the system has reached one of the stable branches with a negative slope ( P/ V) T. In the range where the single-phase equilibrium state is unstable, the system as a whole may be stable only if it consists of the two phases (one with a smaller, and another with a higher density n = N/ V) that are described by the two stable branches – see Fig. 2.
P
stable liquid
phase
liquid and gas
in equilibrium
2'
A u
P ( T )
1
2
0
unstable
A
stable gaseous
d
branch
phase
Fig. 4.2. Phase equilibrium
'
1
0
at T < T
V
c (schematically).
5 The special choice of numerical coefficients in Eq. (3) makes the border between these two regions to take place exactly at t = 1, i.e. at the temperature equal to T c, with the critical point’s coordinates equal to P c and V c.
6 Actually, this assumption is not crucial for our analysis of mechanical stability, because if a fluctuation takes place in a small part of the total volume V, its other parts play the role of pressure-fixing environment.
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In order to understand the basic properties of this two-phase system, let us recall the general conditions of the thermodynamic equilibrium of two systems, which have been discussed in Chapter 1: T T (thermal equilibrium),
(4.5)
1
2
(“chemical” equilibrium),
(4.6)
1
2
Phase
the latter condition meaning that the average energy of a single (“probe”) particle in both systems has to equilibrium conditions
be the same. To those, we should add the evident condition of mechanical equilibrium,
P P (mechanical equilibrium),
(4.7)
1
2
which immediately follows from the balance of normal forces exerted on an inter-phase boundary.
If we discuss isotherms, Eq. (5) is fulfilled automatically, while Eq. (7) means that the effective isotherm P( V) describing a two-phase system should be a horizontal line – see Fig. 2: P P ( T ) .
(4.8)
0
Along this line,7 internal properties of each phase do not change; only the particle distribution is: it evolves gradually from all particles being in the liquid phase at point 1 to all particles being in the gas phase at point 2.8 In particular, according to Eq. (6), the chemical potentials of the phases should be equal at each point of the horizontal line (8). This fact enables us to find the line’s position: it has to connect points 1 and 2 in that the chemical potentials of the two phases are equal to each other. Let us recast this condition as
2
2
d ,0
i.e. dG 0 ,
(4.9)
1
1
where the integral may be taken along the single-phase isotherm. (For this mathematical calculation, the mechanical instability of states on some part of this curve is not important.) By its construction, along that curve, N = const and T = const, so that according to Eq. (1.53c), dG = – SdT + VdP + dN, for a slow (reversible) change, dG = VdP. Hence Eq. (9) yields
2
VdP 0.
(4.10)
1
This equality means that in Fig. 2, the shaded areas A d and A u should be equal. 9
7 Frequently, P 0( T) is called the saturated vapor pressure.
8 A natural question: is the two-phase state with P = P 0( T) the only state existing between points 1 and 2? Indeed, the branches 1-1 ’ and 2-2 ’ of the single-phase isotherm also have negative derivatives ( P/ V) T and hence are mechanically stable with respect to small perturbations. However, these branches are actually metastable, i.e.
have larger Gibbs energy per particle (i.e. ) than the counterpart phase and are hence unstable to larger perturbations – such as foreign microparticles (say, dust), protrusions on the confining walls, etc. In very controlled conditions, these single-phase “superheated” and “supercooled” states can survive almost all the way to the zero-derivative points 1 ’ and 2 ’, leading to sudden jumps of the system into the counterpart phase. (At fixed pressure, such jumps go as shown by dashed lines in Fig. 2.) In particular, at the atmospheric pressure, purified water may be supercooled to almost –50C, and superheated to nearly +270C. However, at more realistic conditions, perturbations result in the two-phase coexistence formation close to points 1 and 2.
9 This Maxwell equal-area rule (also called “Maxwell’s construct”) was suggested by J. C. Maxwell in 1875
using more complex reasoning.
Chapter 4
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As the same Fig. 2 figure shows, the Maxwell rule may be rewritten in a different form,
Maxwell
2
equal-area
P P ( T ) dV
0 .
(4.11)
0
rule
1
which is more convenient for analytical calculations than Eq. (10) if the equation of state may be explicitly solved for P – as it is in the van der Waals model (2). Such calculation (left for the reader’s exercise) shows that for that model, the temperature dependence of the saturated vapor pressure at low T
is exponential,10
a
27
P ( T ) P exp
,
with
T , for T T ,
(4.12)
0
c
T
b
8 c
c
corresponding very well to the physical picture of particle’s thermal activation from a potential well of depth .
The signature parameter of a first-order phase transition, the latent heat of evaporation 2
Latent
heat:
definition
dQ,
(4.13)
1
may also be found by a similar integration along the single-phase isotherm. Indeed, using Eq. (1.19), dQ
= TdS, we get
2
TdS T ( S S )
.
(4.14)
2
1
1
Let us express the right-hand side of Eq. (14) via the equation of state. For that, let us take the full derivative of both sides of Eq. (6) over temperature, considering the value of G = N for each phase as a function of P and T, and taking into account that according to Eq. (7), P 1 = P 2 = P 0( T):
G
G
dP
G
G
dP
1
1
0
2
2
0
.
(4.15)
T
P
dT T
P
dT
P
T
P
T
According to the first of Eqs. (1.39), the partial derivative ( G/ T) P is just minus the entropy, while according to the second of those equalities, ( G/ P) T is the volume. Thus Eq. (15) becomes dP
dP
S V
0 S V
0 .
(4.16)
1
1 dT
2
2 dT
Solving this equation for ( S 2 – S 1), and plugging the result into Eq. (14), we get the following Clapeyron-Clausius formula:
Clapeyron-
dP
Clausius
T V
(
V
0
)
.
(4.17)
2
1
formula
dT
For the van der Waals model, this formula may be readily used for the analytical calculation of in two limits: T << T c and ( T c – T) << T c – the exercises left for the reader. In the latter limit, ( T c – T)1/2, naturally vanishing at the critical temperature.
10 It is fascinating how well is this Arrhenius exponent hidden in the polynomial van der Waals equation (2)!
Chapter 4
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Finally, some important properties of the van der Waals’ model may be revealed more easily by looking at the set of its isochores P = P( T) for V = const, rather than at the isotherms. Indeed, as Eq. (2) shows, all single-phase isochores are straight lines. However, if we interrupt these lines at the points when the single phase becomes metastable, and complement them with the (very nonlinear!) dependence P 0( T), we get the pattern (called the phase diagram) shown schematically in Fig. 3a.
(a)
(b)
P
V V
V V
c
c
P
critical
V V c
points
P
c
liquid
gas
liquid
solid
P t
gas
0
T
0
T
c
T
t
T
Fig. 4.3. (a) Van der Waals model’s isochores, the saturated gas pressure diagram, and the critical point, and (b) the phase diagram of a typical three-phase system (all schematically).
At this plot, one more meaning of the critical point { P c, T c} becomes very vivid. At fixed pressure P < P c, the liquid and gaseous phases are clearly separated by the saturated pressure line P 0( T), so if we achieve the transition between the phases just by changing temperature (see the red horizontal line in Fig. 3a), we have to pass through the phase equilibrium point, being delayed there to either give to the system the latent heat or take it out. However, if we perform the transition between the same initial and final points by changing both the pressure and temperature, going around the critical point (see the blue line in Fig. 3a), no definite point of transition may be observed: the substance stays in a single phase, and it is a subjective judgment of the observer in which region that phase should be called the liquid, and in which region – the gas. For water, the critical point corresponds to the temperature of 647 K (374C), and P c 22.1 MPa (i.e. ~200 bars), so that a lecture demonstration of its critical behavior would require substantial safety precautions. This is why such demonstrations are typically carried out with other substances such as either diethyl ether,11 with its much lower T c (194C) and P c (3.6 MPa), or the now-infamous carbon dioxide CO2, with even lower T c (31.1C), though higher P c (7.4
MPa). Though these substances are colorless and clear in both gas and liquid phases, their separation (by gravity) is still visible, due to small differences in the optical refraction coefficient, at P < P c, but not above P c.12
Thus, in the van der Waals model, two phases may coexist, though only at certain conditions – in particular, T < T c. Now a natural, more general question is whether the coexistence of more than two 11 (CH3-CH2)-O-(CH2-CH3), historically the first popular general anesthetic.
12 It is interesting that very close to the critical point the substance suddenly becomes opaque – in the case of ether, whitish. The qualitative explanation of this effect, called the critical opalescence, is simple: at this point, the difference of the Gibbs energies per particle (i.e. the chemical potentials) of the two phases becomes so small that unavoidable thermal fluctuations lead to spontaneous appearance and disappearance of relatively large (a-few-m-scale) single-phase regions in all the volume. A large concentration of boundaries of such randomly-shaped regions leads to strong light scattering.
Chapter 4
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phases of the same substance is possible. For example, can the water ice, the liquid water, and the water vapor (steam) all be in thermodynamic equilibrium? The answer is essentially given by Eq. (6). From thermodynamics, we know that for a uniform system (i.e. a single phase), pressure and temperature completely define the chemical potential ( P, T). Hence, dealing with two phases, we had to satisfy just one chemical equilibrium condition (6) for two common arguments P and T. Evidently, this leaves us with one extra degree of freedom, so that the two-phase equilibrium is possible within a certain range of P at fixed T (or vice versa) – see again the horizontal line in Fig. 2 and the bold line in Fig. 3a. Now, if we want three phases to be in equilibrium, we need to satisfy two equations for these variables:
( P, T ) ( P, T ) ( P, T ) .
(4.18)
1
2
3
Typically, the functions ( P, T) are monotonic, so that the two equations (18) have just one solution, the so-called triple point { P t, T t}. Of course, the triple point { P t, T t} of equilibrium between three phases should not be confused with the critical points { P c, T c} of transitions between each of two-phase pairs.
Fig. 3b shows, very schematically, their relation for a typical three-phase system solid-liquid-gas. For example, water, ice, and water vapor are at equilibrium at a triple point corresponding to P t 0.612
kPa13 and T t = 273.16 K. The practical importance of this particular temperature point is that by an international agreement it has been accepted for the definition of not only the Kelvin temperature scale, but also of the Celsius scale’s reference, as 0.01C, so that the absolute temperature zero corresponds to exactly –273.15C.14 More generally, triple points of purified simple substances (such as H2, N2, O2, Ar, Hg, and H2O) are broadly used for thermometer calibration, defining the so-called international temperature scales including the currently accepted scale ITS-90.
This analysis may be readily generalized to multi-component systems consisting of particles of several (say, L) sorts.15 If such a mixed system is in a single phase, i.e. is macroscopically uniform, its chemical potential may be defined by a natural generalization of Eq. (1.53c):
L
l
l
dG SdT VdP dN .
(4.19)
l1
The last term reflects the fact that usually, each single phase is not a pure chemical substance, but has certain concentrations of all other components, so that ( l) may depend not only on P and T but also on the concentrations c( l) N( l)/ N of particles of each sort. If the total number N of particles is fixed, the number of independent concentrations is ( L – 1). For the chemical equilibrium of R phases, all R values of ( l)
( l)
( l)
( l)
( l)
r ( r = 1, 2, …, R) have to be equal for particles of each sort: 1 = 2 = … = R , with each r depending on ( L – 1) concentrations c ( l)
r , and also on P and T. This requirement gives L( R – 1) equations for ( L –1) R concentrations c ( l)
r , plus two common arguments P and T, i.e. for [( L –1) R + 2] independent variables. This means that the number of phases has to satisfy the limitation
Gibbs
phase
L( R )
1 ( L )
1 R ,
2
i.e. R L 2 ,
(4.20)
rule
13 Please note that for water, P t is much lower than the normal atmospheric pressure (101.325 kPa).
14 Note the recent (2018) re-definition of the “legal” kelvin via joule (see, appendix CA: Selected Physical Constants); however, the new definition is compatible, within experimental accuracy, with that mentioned above.
15 Perhaps the most practically important example is the air/water system. For its detailed discussion, based on Eq.
(19), the reader may be referred, e.g., to Sec. 3.9 in F. Schwabl, Statistical Mechanics, Springer (2000). Other important applications include liquid solutions, and metallic alloys – solid solutions of metal elements.
Chapter 4
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Essential Graduate Physics
SM: Statistical Mechanics
where the equality sign may be reached in just one point in the whole parameter space. This is the Gibbs phase rule. As a sanity check, for a single-component system, L = 1, the rule yields R 3 – exactly the result we have already discussed.
4.2. Continuous phase transitions
As Fig. 2 illustrates, if we fix pressure P in a system with a first-order phase transition, and start changing its temperature, then the complete crossing of the transition-point line, defined by the equation P 0( T) = P, requires the insertion (or extraction) some non-zero latent heat . Eqs. (14) and (17) show that is directly related to non-zero differences between the entropies and volumes of the two phases (at the same pressure). As we know from Chapter 1, both S and V may be represented as the first derivatives of appropriate thermodynamic potentials. This is why P. Ehrenfest called such transitions, involving jumps of potentials’ first derivatives, the first-order phase transitions.
On the other hand, there are phase transitions that have no first derivative jumps at the transition temperature T c, so that the temperature point may be clearly marked, for example, by a jump of the second derivative of a thermodynamic potential – for example, the derivative C/ T which, according to Eq. (1.24), equals to 2 E/ T 2. In the initial Ehrenfest classification, this was an example of a second-order phase transition. However, most features of such phase transitions are also pertinent to some systems in which the second derivatives of potentials are continuous as well. Due to this reason, I will use a more recent terminology (suggested in 1967 by M. Fisher), in which all phase transitions with =
0 are called continuous.
Most (though not all) continuous phase transitions result from particle interactions. Here are some representative examples:
(i) At temperatures above ~490 K, the crystal lattice of barium titanate (BaTiO3) is cubic, with a Ba ion in the center of each Ti-cornered cube (or vice versa) – see Fig. 4a. However, as the temperature is being lowered below that critical value, the sublattice of Ba ions starts moving along one of six sides of the TiO3 sublattice, leading to a small deformation of both lattices – which become tetragonal. This is a typical example of a structural transition, in this particular case combined with a ferroelectric transition, because (due to the positive electric charge of the Ba ions) below the critical temperature the BaTiO3 crystal acquires a spontaneous electric polarization even in the absence of external electric field.
(a)
(b)
Cu
Ba Ti O
Zn
Fig. 4.4. Single cells of
crystal lattices of (a)
BaTiO3 and (b) CuZn.
(ii) A different kind of phase transition happens, for example, in Cu x Zn1- x alloys – so-called brasses. Their crystal lattice is always cubic, but above certain critical temperature T c (which depends on x) any of its nodes may be occupied by either a copper or a zinc atom, at random. At T < T c, a trend toward ordered atom alternation arises, and at low temperatures, the atoms are fully ordered, as shown in Fig. 4b for the stoichiometric case x = 0.5. This is a good example of an order-disorder transition.
Chapter 4
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(iii) At ferromagnetic transitions (such as the one taking place, for example, in Fe at 1,388 K) and antiferromagnetic transitions (e.g., in MnO at 116 K), lowering of temperature below the critical value16 does not change atom positions substantially, but results in a partial ordering of atomic spins, eventually leading to their full ordering (Fig. 5).
(a)
(b)
Fig. 4.5. Classical images
of fully ordered phases: (a)
a ferromagnet, and (b) an
antiferromagnet.
Note that, as it follows from Eqs. (1.1)-(1.3), at ferroelectric transitions the role of pressure is played by the external electric field E, and at the ferromagnetic transitions, by the external magnetic field H. As we will see very soon, even in systems with continuous phase transitions, a gradual change of such an external field, at a fixed temperature, may induce jumps between metastable states, similar to those in systems with first-order phase transitions (see, e.g., the dashed arrows in Fig. 2), with non-zero decreases of the appropriate free energy.
Besides these standard examples, some other threshold phenomena, such as the formation of a coherent optical field in a laser, and even the self-excitation of oscillators with negative damping (see, e.g., CM Sec. 5.4), may be treated, at certain conditions, as continuous phase transitions.17
The general feature of all these transitions is the gradual formation, at T < T c, of certain ordering, which may be characterized by some order parameter 0. The simplest example of such an order parameter is the magnetization at the ferromagnetic transitions, and this is why the continuous phase transitions are usually discussed on certain models of ferromagnetism. (I will follow this tradition, while mentioning in passing other important cases that require a substantial modification of the theory.) Most of such models are defined on an infinite 3D cubic lattice (see, e.g., Fig. 5), with evident generalizations to lower dimensions. For example, the Heisenberg model of a ferromagnet (suggested in 1928) is defined by the following Hamiltonian:
Heisenberg
model
H ˆ J σˆ σˆ h σˆ ,
(4.21)
k
k '
k
k, k '
k
where σîs the Pauli vector operator18 acting on the k th spin, and h is the normalized external magnetic k
field:
16 For ferromagnets, this point is usually referred to at the Curie temperature, and for antiferromagnets, as the Néel temperature.
17 Unfortunately, I will have no time/space for these interesting (and practically important) generalizations, and have to refer the interested reader to the famous monograph by R. Stratonovich, Topics in the Theory of Random Noise, in 2 vols., Gordon and Breach, 1963 and 1967, and/or the influential review by H. Haken, Ferstkörperprobleme 10, 351 (1970).
Chapter 4
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SM: Statistical Mechanics
h m
.
(4.22)
0
0H
(Here m0 is the magnitude of the spin’s magnetic moment; for the Heisenberg model to be realistic, it should be of the order of the Bohr magneton B e/2 m e 0.92710-23 J/T.) The figure brackets { j, j’}
in Eq. (21) denote the summation over the pairs of adjacent lattice sites, so that the magnitude of the constant J may be interpreted as the maximum coupling energy per “bond” between two adjacent particles. At J > 0, the coupling tries to keep spins aligned, i.e. to install the ferromagnetic ordering.19
The second term in Eq. (21) describes the effect of the external magnetic field, which tries to orient all spin magnetic moments along its direction.20
However, even the Heisenberg model, while being rather approximate (in particular because its standard form (21) is only valid for spins-½), is still rather complex for analysis. This is why most theoretical results have been obtained for its classical twin, the Ising model:21
E J
s s
h
s .
(4.23) Ising
m
k k '
k
model
k, k '
k
Here Em are the particular values of the system’s energy in each of its 2 N possible states with all possible combinations of the binary classical variables sk = 1, while h is the normalized external magnetic field’s magnitude – see Eq. (22). (Despite its classical character, the variable sk, modeling the field-oriented Cartesian component of the real spin, is usually called “spin” for brevity, and I will follow this tradition.) Somewhat shockingly, even for this toy model, no exact analytical 3D solution that would be valid at arbitrary temperature has been found yet, and the solution of its 2D version by L. Onsager in 1944 (see Sec. 5 below) is still considered one of the top intellectual achievements of statistical physics.
Still, Eq. (23) is very useful for the introduction of basic notions of continuous phase transitions, and methods of their analysis, so that for my brief discussion I will mostly use this model.22
Evidently, if T = 0 and h = 0, the lowest possible energy,
E
JNd ,
(4.24)
min
where d is the lattice dimensionality, is achieved in the “ferromagnetic” phase in which all spins sk are equal to either +1 or –1, so that sk = 1 as well. On the other hand, at J = 0, the spins are independent, and if h = 0 as well, all sk are completely random, with the 50% probability to take either of values 1, so that sk = 0. Hence in the general case (with arbitrary J and h), we may use the average Ising
s
(4.25) model:
k
order
parameter
18 See, e.g., QM Sec. 4.4.
19 At J < 0, the first term of Eq. (21) gives a reasonable model of an antiferromagnet, but in this case, the external magnetic field effects are more subtle; I will not have time to discuss them.
20 See, e.g., QM Eq. (4.163).
21 Named after Ernst Ising who explored the 1D version of the model in detail in 1925, though a similar model was discussed earlier (in 1920) by Wilhelm Lenz.
22 For more detailed discussions of phase transition theories (including other popular models of the ferromagnetic phase transition, e.g., the Potts model), see, e.g., either H. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford U. Press, 1971; or A. Patashinskii and V. Pokrovskii, Fluctuation Theory of Phase Transitions, Pergamon, 1979; or B. McCoy, Advanced Statistical Mechanics, Oxford U. Press, 2010. For a very concise text, I can recommend J. Yeomans, Statistical Mechanics of Phase Transitions, Clarendon, 1992.
Chapter 4
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as a good measure of spin ordering, i.e. as the order parameter. Since in a real ferromagnet, each spin carries a magnetic moment, the order parameter is proportional to the Cartesian component of the system’s magnetization, in the direction of the applied magnetic field.
Now that the Ising model gave us a very clear illustration of the order parameter, let me use this notion for quantitative characterization of continuous phase transitions. Due to the difficulty of theoretical analyses of most models of the transitions at arbitrary temperatures, their theoretical discussions are focused mostly on a close vicinity of the critical point T c. Both experiment and theory show that in the absence of an external field, the function ( T) is close to a certain power,
,
for τ
i.e.
,
0
T T ,
(4.26)
c
of the small deviation from the critical temperature – which is conveniently normalized as T T
c
.
(4.27)
T c
Remarkably, most other key variables follow a similar temperature behavior, with critical exponents being the same for both signs of . In particular, the heat capacity at a fixed magnetic field behaves as23
c
.
(4.28)
h
Similarly, the (normalized) low-field susceptibility 24
.
(4.29)
h0
h
Two other important critical exponents, and , describe the temperature behavior of the correlation function sksk’, whose dependence on the distance rkk’ between two spins may be well fitted by the following law,
1
r
s s
exp
k k'
kk' ,
d 2
r
(4.30)
kk
r c
'
with the correlation radius
r
.
(4.31)
c
Finally, three more critical exponents, usually denoted , , and , describe the external field dependences of, respectively, c, , and r c at > 0. For example, is defined as
1
h
.
(4.32)
(Other field exponents are used less frequently, and for their discussion, the interested reader is referred to the special literature that was cited above.)
The leftmost column of Table 1 shows the ranges of experimental values of the critical exponents for various 3D physical systems featuring continuous phase transitions. One can see that their values vary from system to system, leaving no hope for a universal theory that would describe them all 23 The forms of this and other functions of are selected to make all critical exponents non-negative.
24 In most models of ferromagnetic phase transitions, this variable is proportional to the genuine low-field magnetic susceptibility m of the material – see, e.g., EM Eq. (5.111).
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exactly. However, certain combinations of the exponents are much more reproducible – see the four bottom lines of the table.
Table 4.1. Major critical exponents of continuous phase transitions
Exponents and
Experimental
Landau’s
2D Ising
3D Ising
3D Heisenberg
combinations
range (3D)(a)
theory
model
model
Model(d)
0 – 0.14
0(b)
(c)
0.12
–0.14
0.32 – 0.39
1/2
1/8
0.31
0.3
1.3 – 1.4
1
7/4
1.25
1.4
4-5
3
15
5
?
0.6 – 0.7
1/2
1
0.64
0.7
0.05
0
1/4
0.05
0.04
( + 2 + )/2
1.00 0.005
1
1
1
1
– /
0.93 0.08
1
1
1
?
(2 – )/
1.02 0.05
1
1
1
1
(2 – )/ d
?
4/ d
1
1
1
(a)
Experimental data are from the monograph by A. Patashinskii and V. Pokrovskii, cited above.
(b) Discontinuity at = 0 – see below.
(c) Instead of following Eq. (28), in this case ch diverges as ln.
(d) With the order parameter defined as jB/B.
Historically the first (and perhaps the most fundamental) of these universal relations was derived in 1963 by J. Essam and M. Fisher:
2 2 .
(4.33)
It may be proved, for example, by finding the temperature dependence of the magnetic field value, h, that changes the order parameter by the same amount as a finite temperature deviation > 0 gives at h =
0. Comparing Eqs. (26) and (29), we get
h
.
(4.34)
By the physical sense of h, we may expect that such a field has to affect the system’s free energy F by an amount comparable to the effect of a bare temperature change . Ensemble-averaging the last term of Eq. (23) and using the definition (25) of the order parameter , we see that the change of F (per particle) due to the field equals – h and, according to Eq. (26), scales as h (2 + ).25
25 As was already discussed in Secs. 1.4 and 2.4, there is some dichotomy of terminology for free energies in literature. In models (21) and (23), the magnetic field effects are accounted for at the microscopic level, by the inclusion of the corresponding term into each particular value Em. From this point of view, the list of macroscopic variables in these systems does not include either P and V or their magnetic analogs, so that we may take G F +
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In order to estimate the thermal effect on F, let me first elaborate a bit more on the useful thermodynamic formula already mentioned in Sec. 1.3:
S
C T
,
(4.35)
X
T
X
where X means the variable(s) maintained constant at the temperature variation. In the standard “P-V”
thermodynamics, we may use Eqs. (1.35) for X = V, and Eqs. (1.39) for X = P, to write 2
2
S
F
S
G
C T
T
,
C T
T
.
(4.36)
V
2
P
2
T
T
T
T
V , N
P,
V ,
N
N
P, N
As was just discussed, in the ferromagnetic models of the type (21) or (23), at a constant field h, the role of G is played by F, so that Eq. (35) yields
S
2
F
C T
T
.
(4.37)
h
T
T 2
h, N
h, N
The last form of this relation means that F may be found by double integration of (– Ch/ T) over temperature. With Eq. (28) for ch Ch, this means that near T c, the free energy scales as the double integral of ch – over . In the limit << 1, the factor T may be treated as a constant; as a result, the change of F due to > 0 alone scales as (2 – ). Requiring this change to be proportional to the same power of as the field-induced part of the energy, we finally get the Essam-Fisher relation (33).
Using similar reasoning, it is straightforward to derive a few other universal relations of critical exponents, including the Widom relation,
1,
(4.38)
very similar relations for other high-field exponents and (which I do not have time to discuss), and the Fisher relation
2 .
(4.39)
A slightly more complex reasoning, involving the so-called scaling hypothesis, yields the following dimensionality-dependent Josephson relation
d 2 .
(4.40)
The second column of Table 1 shows that at least three of these relations are in a very reasonable agreement with experiment, so that we may use their set as a testbed for various theoretical approaches to continuous phase transitions.
4.3. Landau’s mean-field theory
The highest-level approach to continuous phase transitions, formally not based on any particular microscopic model (though in fact implying either the Ising model (23) or one of its siblings), is the mean-field theory developed in 1937 by L. Landau, on the basis of prior ideas by P. Weiss – to be PV = F + const, and the equilibrium (at fixed h, T and N) corresponds to the minimum of the Helmholtz free energy F.
Chapter 4
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discussed in the next section. The main idea of this phenomenological approach is to represent the free energy’s change F at the phase transition as an explicit function of the order parameter (25). Since at T T c, the order parameter has to tend to zero, this change,
