# (p.268) (p.269) B On the theory of phase transitions

This appendix is a supplement to Sections 15.1 and 15.2 supposed to enhance the understanding of the physico-chemical background of phase transitions in the solid state. However, this is only a brief glimpse into the theory of phase transitions.

In Section 15.1.1
(page 198), EHRENFEST'S definition of first‐ and second-order phase transitions is presented. The determining factors for the classification are the derivatives of the Gibbs free energy with respect to temperature, pressure, or other variables of state. The role of the derivatives is illustrated in Fig. B.1
. In a schematic way, it is shown how the free enthalpies *G*
_{1} and *G*
_{2} of two phases depend on temperature. A first-order phase transition takes place if the two curves intersect, the intersection point being the point of transition *T _{c}
*. The more stable phase is the one having the lower

*G*value; the one with the higher

*G*value can exist as a metastable phase. When switching from the curve

*G*

_{1}to

*G*

_{2}at the point

*T*, the slope of the curve is changed abruptly; the first derivative switches discontinuously from one value to another.

_{c}A second-order phase transition entails no metastable phases. The dotted curve *G*
_{2} represents the hypothetical (computed or extrapolated) course for the high-temperature modification below *T _{c}
*. The curve

*G*

_{1}ends at

*T*, where it merges with the curve

_{c}*G*

_{2}with the same slope, but with a different curvature. Mathematically, curvature corresponds to the second derivative; it changes dis-continuously at

*T*.

_{c}During a first-order transition both phases coexist. Properties that refer to the whole specimen, for example, the uptake of heat or the magnetization, result from the superposition of the properties from both phases, corresponding to their constituent amounts. Measurement yields a hysteresis curve (Fig. B.2
). Upon heating, phase 1 (stable below *T _{c}
*) at first lingers on in a metastable state above

*T*(dotted line in Fig. B.2 ). The more

_{c}*T*has surpassed

*T*, the more nuclei of phase 2 are formed, and the more they grow, the more their properties dominate until finally only phase 2 is present. Heat uptake follows the right branch of the curve and shows a sluggish progression of the phase transition at

_{c}*T 〉 T*. Upon cooling, the same happens in the opposite direction, with sluggish heat release at

_{c}*T < T*. The width of the hysteresis loop is not a specific (p.270)

_{c}The order of a phase transition can differ depending on conditions. For example, pressure causes crystalline cerium to undergo a first-order phase transition γ‐Ce → α‐Ce; it is associated with a significant, discontinuous change of volume (−13 % at room temperature) and an electron transition 4*f* → 5*d*
[391]. The structures of both modifications, γ‐Ce and α‐Ce, are cubic-closest packings of spheres *(Fm*3̄*m*, atoms at position 4*a)*. The higher the temperature, the lesser is the change of volume, and finally it becomes *ΔV* = 0 at the critical point [392]. Above the critical temperature (485 K) or the critical
(p.271)
pressure (1.8 GPa), there no longer exists a difference between γ‐Ce and α‐Ce (Fig. B.3
).

*I*4̄2

*d*, the H atoms are misordered within hydrogen bonds:

In the ferroelectric phase, space group *Fdd*2, they are ordered. At ambient pressure, a first-order transition takes place at 122 K with a small volume discontinuity *(ΔV* approx. −0.5 %). At pressures above 0.28 GPa and temperatures below 108 K (the ‘tricritical’ point; Fig. B.4
) it is of second order [395,396].

In Section 15.1.1
, after EHRENFEST's definition, a newer definition is mentioned according to which the continuous change of an order parameter is decisive. At first glance, this seems to be a completely different criterion as compared to EHRENFEST's classification. Actually, there is not so much of a difference. Phase transitions of *n*‐th order in terms of EHRENFEST having *n 〉* 2 are not really observed. The case *n* = ∞ does not describe a phase transition, but a continuous change of properties (e.g. of thermal dilatation; this is not to be confused with a continuous phase transition). Even at a continuous phase transition, invariably some thermodynamic function experiences a discontinuity. That embraces more than just a sudden change of a second derivative of the Gibbs free energy, but any kind of discontinuity or singularity. The change of the order parameter η as a function of some variable of state (e.g. temperature *T)* exhibits a singularity at the point of transition: on the one side *(T < T _{c})*, an analytical dependence according to a power law holds (eqn (15.2)
, page 204), but at the point of transition

*(T = T*the order parameter vanishes (it becomes

_{c})*η*= 0 for

*T*≥

*T*Fig. 15.4 top left, page 204).

_{c};In the following explanations, we repeatedly resort to the example of the continuous phase transition of CaCl_{2} (Section 15.2
, page 200). Let the Gibbs free energy *G* of a crystal have the value *G*
_{0} at the point of transition. For small values of the order parameter *η*, the change of
Gibbs free energy relative to *G*
_{0} can be expressed by a Taylor series. All terms with odd powers have been omitted from the series, since *G* has to remain unchanged when the sign of *η* is changed (it makes no difference whether the octahedra of CaCl_{2} have been turned to one or the other side; Figs. 15.2, 15.3
). This proposition is universally valid according to Landau theory: For a continuous phase transition, only even powers may appear in the Taylor series.

As long as *η* is small, the Taylor series can be truncated after a few terms, whereby the term with the highest power has to have a positive coefficient; this is important, because otherwise *G* would become more and more negative with increasing *η*, which would be an unstable state. The coefficients *a*
_{2}, *a*
_{4},… depend on the variables of state *T* and *p*. Essential is the dependence of *a*
_{2} on
(p.272)
*T* and *p*. For a temperature-dependent phase transition, *a _{2} =* 0 must hold at the point of transition, above it is

*a*0 and below

_{2}〉*a*0, while

_{2}<*a*4,

*a*

_{6},… remain approximately constant. If, in accordance with experience,

*a*changes linearly with temperature close to

_{2}*T*, one has eqn (B.2) . The effect of temperature on the Gibbs free energy is shown in Fig. B.5 . At temperatures above

_{c}*T*has a minimum at

_{c}, G*α*= 0; the structure of CaCl

_{2}is tetragonal. At the point of transition

*T = T*, there is also a minimum at

_{c}*α*= 0, with a gentle curvature close to

*α*= 0. Below

*T*, the curve has a maximum at

_{c}*α*= 0 and two minima. The more the temperature is below

*T*, the deeper are the minima and the more distant they are from η = 0. At

_{c}*T*<

*T*the structure is no longer tetragonal and the octahedra have been turned to one side or the other, corresponding to the

_{c}*η*value of one of the minima.

If *a*
_{4} 〉 0, the Taylor series
(B.1)
can be truncated after the fourth-power term. From the first derivative with respect to η one obtains condition
(B.3)
for the minima: If *T* < *T _{c}
* and thus

*a*

_{2}< 0, the positions of the minima are at α

_{1,2}. Taking $A=\pm \sqrt{k{T}_{c}/{a}_{4}}$, the order parameter

*η*of the stable structure then follows the power law given in eqn (B.5) .

*A* is a constant and *β* is the *critical exponent*. The derived value of *α* = ½ is valid under the assumptions made that *G* can be expressed by a Taylor series neglecting powers higher than 4, that *a _{2}
* depends linearly on the temperature difference

*T*−

*T*, and that

_{c}*a*

_{4}〉 0 is (nearly) independent of temperature.

In addition, it is assumed that the order parameter does not fluctuate. This, however, cannot be taken for granted close to the point of transition. Just below *T _{c}
*, the two minima of the curve for

*G*(Fig. B.5 ,

*T*≲

*T*are shallow and the maximum in between is low. The energy barrier to be surmounted to shift from one minimum to the other is less than the thermal energy, and

_{c})*η*can fluctuate easily from one minimum to the other. At the point of transition itself (curve

*T = T*in Fig. B.5 ) and slightly above, the curvature is so small that fluctuations of η induced by temperature have nearly no influence on the Gibbs free energy. The local fluctuations can be different in different regions of the crystal.

_{c}_{2}is long; all of the octahedra (or a large number of them) would have to participate in the fluctuations. For long-range interactions, Landau theory is satisfied and the critical exponent amounts to

*β*≈ ½.

If *T 〉 T _{c}
*, the minimum for

*G*is at

*η*= 0, and at this value one has the first and second derivatives of the Gibbs free energy with respect to temperature as mentioned in the margin.

If *T < T _{c}
*, one obtains the value of

*G*corresponding to the minima by inserting eqns (B.2) and (B.4) into eqn (B.1) , truncated after the fourth power:

From this, we obtain the first and second derivatives with respect to temperature for *T < T _{c}
*:

When one approaches the point of transition from the side of low temperatures, one has *(T _{c} − T) →* 0 and η → 0, and

*∂G/∂T*merges at

*T*with the value

_{c}*∂G*

_{0}

*/∂T*that it also adopts above

*T*. However, below

_{c}*T*has a value that is smaller by ½

_{c}, ∂^{2}G/∂T^{2}*k*

^{2}/a_{4}than at

*T 〉 T*. The first derivative of

_{c}*G*thus does not experience a sudden change at

*T*, but the second derivative does, just as expected for a second-order transition according to EHRENFEST's definition.

_{c}Displacing an octahedron of CaCl_{2} from its equilibrium position requires a force *F*. Force is the first derivative of energy with respect to displacement; for our purposes that is the first derivative of the Gibbs free energy with respect to *η, F = ∂G/∂η*. The force constant *f* is the first derivative of *F* with respect to *η*; actually, it is only a constant for a harmonic oscillator, then one has *F* = *fη* (Hooke's law; a harmonic oscillator is one for which the Taylor series contains only terms of powers zero and two). For the curves *T 〉 T _{c}
* and

*T < T*in Fig. B.5 , Hooke's law is only an approximation close to the minima. The second derivatives of the Taylor series (B.1), truncated after the fourth power, corresponds to eqn (B.6) . If

_{c}*T 〉 T*, with the minimum at

_{c}*η*= 0, the force constant is thus

*f*≈

*a*. If

_{2}*T < T*, the minima are at ${\eta}_{1,2}=\pm \sqrt{-{a}_{2}/{a}_{4}}$, eqn (B.4) . By substitution of these values into eqn (B.6) , one obtains the force constant in proximity to these minima.

_{c}The square *v*
^{2} of a vibrational frequency is proportional to the force constant. Therefore, the frequency can be chosen as an order parameter according to eqn (B.5)
with **v** instead of *η*. When approaching the temperature of transition, *f* → 0 and thus *v* → 0. The vibration becomes a soft mode.

This is valid for phase transitions in both directions. If one starts at a high temperature, from tetragonal CaCl_{2}, the octahedra perform rotational vibrations
(p.274)
about the equilibrium position *η* = 0. Below *T _{c}
*, after the phase transition, the octahedra vibrate about the equilibrium positions

*η*

_{1}〉 0 or

*η*

_{2}< 0; these shift increasingly away from

*η*= 0 the more the temperature difference

*T*−

_{c}*T*increases. Simultaneously, the square of the vibrational frequency

*v*

^{2}again increases proportional to (

*T*). According to eqn (B.8), at

_{c}− T*T < T*one expects a temperature dependence of the squared frequency

_{c}*v*

^{2}twice as large as when

*T 〉 T*.

_{c}
Equations (B.6)
and (B.8) are not valid in close proximity to the point of transition. Hooke's law is not then valid. In fact, the frequency of the soft mode of CaCl_{2} does not decrease down to 0 cm^{−1}, but only to 14 cm^{−1}. In addition, the measured temperature dependence of its squared frequency at *T < T _{c}
* is not twice as large, but 6.45 times larger than at

*T 〉 T*(Fig.15.4 , page 204; [280]).

_{c}Landau theory shows good qualitative agreement with experimental observations. However, there are quantitative discrepancies, and these are quite substantial in the case of short-range interactions, for example, magnetic interactions. Especially, this applies in proximity to the point of transition. An improvement has been achieved by K. G. WILSON'S
*renormalization-group theory* [397, 398].
^{1}

Landau theory is a ‘mean-field theory’ that assumes equal conditions in the whole crystal (also called molecular field theory in the case of magnetic phase transitions). This is not fulfilled for short-range atomic interactions. In this case, fluctuations cannot be neglected.

Consider a substance like EuO as an example. It exhibits a second-order paramagnetic-ferromagnetic phase transition at *T _{c}
* = 69.2 K. The spins of the Eu atoms tend to line up in the same direction. If the spin of an Eu atom has the orientation ↑, it affects neighbouring Eu atoms to adopt this same ↑ spin. Due to the short range of the magnetic interaction only the nearest-neighbour spins are coupled to each other. Indirectly, however, farther atoms are indeed being influenced. If the spin ↑ of the first atom has induced an ↑ alignment of the second atom, this atom will cause a next (third) atom to adopt an ↑ alignment as well, etc. As a consequence, the spins of all atoms are correlated. However, thermal motion counteracts correlation, such that a spin is occasionally reversed and adopts the ‘wrong’ orientation ↓. The lower the temperature, the less probable is this reorientation.

Competition between the tendency toward a uniform spin orientation and the thermal introduction of disorder has the consequence that the correlation can only be detected up to a certain distance, the *correlation length*.

At high temperatures, the correlation length is nearly zero. The spins are oriented at random, and they frequently change their orientation. In the mean, the number of ↑ and ↓ spins is balanced, and the material is paramagnetic. As the temperature falls the correlation length increases. Domains appear in which spins mostly point in the same direction. The overall magnetization is still zero because there are domains with either orientation (Fig. B.6 ). The domains are (p.275)

As the temperature approaches the critical temperature *T _{c}
*, the Curie temperature, the correlation length grows rapidly. The domains become larger. At the Curie temperature the correlation length becomes infinite. Any two spins are correlated, no matter what the distance between them is. Nevertheless, fluctuations persist, and the material remains unmagnetized. However, it is exquisitely sensitive to small perturbations. Holding the spin of a single atom in one direction creates a disturbance that spreads throughout the lattice and gives the entire material a magnetization. Below

*T*a ferromagnetic order of the spins arises.

_{c}The method of treating the outlined model mathematically is *renormaliza-tion*. Let *p* be the probability of finding another ↑ spin next to an ↑ spin. The lattice is divided into blocks of a few spins each. Using the probability *p*, the number of spins of each orientation in the block is calculated. If the majority of spins in the block is ↑, an ↑ spin is assigned to the whole block (thick arrows in Fig. B.6
). The effect is to eliminate all fluctuations in spin direction whose scale of length is smaller than the block size. In the same way, in several steps of iteration, the blocks are collected into larger and even larger blocks. The effects of interactions at small scales of length are thus extended to large scales.

The model is independent of the kind of interaction between the particles (it is not restricted to spin-spin interactions). According to the *universality hypothesis* by GRIFFITHS
[399], the laws that control continuous phase transitions, in contrast to discontinuous ones, depend only on the range of interaction and the number of dimensions *d* and *n*. *d* is the number of space dimensions in which the interactions are active; *n* is the number of ‘dimensions’ of the order parameter, i.e. the number of components needed for its description (e.g. the number of vector components to define the spin vector). A short-range interaction decreases by more than *r*
^{−(d+2)} as a function of distance *r*; a long-range interaction decreases according to *r*
^{−(d+σ)}, with *σ < d/2*.

From the theory, for short-range interactions, one obtains a value of *β* = ⅛ for the critical exponent if *d* = 2; this applies, for example, to ferromagnets
(p.276)
whose magnetic interactions are restricted to planes and to adsorbed films. If *d* = 3, depending on *n*, critical exponents of *β* = 0.302 (*n* = 0) to *β* = 0.368 (*n* = 3) are obtained. Experimental values for EuO (*d* = 3, *n* = 3) yield *β* = 0.36. For long-range interactions the results are consistent with Landau theory (i.e. *β* = 0.5).

Finally, let us consider what are the consequences if odd powers appear in the Taylor series, eqn (B. 1). A power of one, *η ^{1}
*, is not possible because the order parameter has to be zero at the transition temperature;

*G*as a function of

*η*has to have a minimum at

*T 〉 T*, the first derivative must be zero at

_{c}*η*= 0, which implies

*a*= 0. Including a third power, the series up to the fourth power corresponds to eqn (B.9) . If we again assume a linear dependence of

_{1}*a*on temperature,

_{2}*a*

_{2}= k(T**‐**T_{0}), and assume that

*a*0 and

_{3}〉*a*

_{4}〉 0 are nearly independent of temperature, we obtain curves for the Gibbs free energy as shown in Fig. B.7 (

*T*

_{0}is not the transition temperature).

At high temperatures there is only one minimum at *η* = 0; there exists one stable high-temperature modification. As temperature falls, a second minimum appears at a negative *η* value; at first, this minimum is above the minimum at *η* = 0 (curve *T 〉 T _{c}
* in Fig. B.7
). This means that a metastable phase can exist at this value of

*η*. When the temperature is lowered further, such that both minima are at the same height, their

*G*values are equal and

*ΔG*= 0; this is the condition of a state of equilibrium, the corresponding temperature is the transition temperature

*T = T*. If

_{c}*T < T*, the minimum at

_{c}*η*< 0 is below that at

*η*= 0. Therefore, there exists a stable phase at

*T < T*with a negative value of η and a metastable phase at

_{c}*η*= 0.

^{2}

These are the conditions for a discontinuous phase transition. The order parameter *η* corresponds to the value of *η* of the lower minimum. Starting from

*η*= 0, at decreasing temperatures a metastable phase appears at a certain temperature; below

*T*this phase becomes stable and it has an order parameter

_{c}*η*that is quite different from zero. There is no continuous change of the structure with a continuous change of

*η*, but below

*T*the second structure is suddenly the more stable one; the order parameter experiences a sudden change from

_{c}*η*= 0 to

*η*= 0 involving a discontinuous change of structure.

The appearance of a metastable phase entails hysteresis. As the temperature falls, the structure at first persists in being metastable in the high-temperature form with *η* = 0 even below *T _{c}
*; the transformation does not set in before the new, lower minimum has become low enough. The lower the new minimum at

*η*= 0 is, the lower is, viewed from

*η*= 0, the maximum between the two minima; the energy barrier to be surmounted from the minimum at

*η*= 0 to the minimum at

*η*= 0 becomes lower. That means decreasing energy of activation and thus a faster transformation the further

*T*is below the point of transition

*T*.

_{c}Such a model is appropriate only for the description of the thermodynamic conditions of discontinuous phase transitions if both structures are similar, so that an order parameter can be found whose (small, but discontinuous) change results in a straightforward conversion of one structure to the other. The model is applicable to discontinuous displacive phase transitions, but hardly to reconstructive phase transitions. It should also be pointed out that this is a purely thermodynamic point of view by which the Gibbs free energies of two structures are compared with the aid of an order parameter. There is no reference to any mechanism as to how the phase transition actually proceeds. The sudden change of the numerical value of *η*, which states the shift of the thermodynamic stability from one structure to the other, does not mean that the atoms actually execute a simple displacement in the structure. Mechanism and kinetics rather are a matter of nucleation and growth. The atoms cannot all be displaced simultaneously, otherwise there would be no hysteresis.

## Notes:

(
^{1}
)
This is not a branch of normal group theory. The renormalization group is not a group in terms of group theory.