CRITICALPHENOMENA

IO.T THERMODYNAMICS IN THE NEIGHBORHOOD OF THE CRITICAL POINT

The entire structure of thermodynamics,as describedin the preceding ters, appearedat mid-century to be logically complete,but the re founderedon one ostensiblyminor detail.That "detail" had to with the propertiesof systemsin the neighborhoodof the critical point. ical thermodynamicscorrectly predicted that various "generalized :ptibilities" (heat capacities,compressibilities, magnetic susceptibili- etc.) should divergeat the critical point, and the gerreralstruciure of ssicalthermodynamics strongly suggested the analytic lbrm (or "shape") those divergences.The generalizedsusceptibilities do diverge,but the rlytic form of the divergencesis not as expected. In addition the yergencesexhibit regularities indicative of an underlying integrative inciple inexplicableby classicalthermodynamics. Observationsof the enormousfluctuations at critical points date back 1869,when T. Andrewsrreported the "critical opalescenceo'of fluids. scattering of light by the huge density fluctuations renders water ilky" and opaqueat or very near the critical temperatureand pressure .29 K, 22.09 MPa). Warming or cooling the water a fraction of a in restoresit to its normaltransparent state. Similarly, the magnetic susceptibility divergesfor a magnetic system r its critical transition, and again the fluctuations in the magnetic t are divergent. A variety of other types of systemsexhibit critical or "second-order" msitions; severalare listed in Table 10.1 along with the corresponding parameter" (the thermodynamicquantity that exhibits divergent ions,analogous to the magneticmoment).

rT. Andrews, Phil. Trans. Royal Soc.f59, 575 (1869).

255 256 critical Phenomena

TABLE 10.1 Exarnplesof Critical Points and Their Order Parameters*

Crilical Point Order Parameter Example Tr, (K)

Liquid-gas Molar volume HrO 647.05 Ferromagnetic Magneticmoment Fe 1044.0 Antiferromagnetic Sublatticemagnetic FeF, 78.26 moment )\-line in aHe aHe quantummechanical aHe 1.8-2.1 amplitude Superconductivity Electron pair amplitude Pb 7.L9 Binary fluid mixture Fractional segregation ccl4-c7Fr4 301.78 of components Binary alloy Fraction of one atomic Cu-Zn 739 specieson one sublattice Ferroelectric Electric dipole moment Triglycinesulfate 322.5

'Adapted from Shang-Keng lvla, Modern Theory of Critical Phenomeza (Addison-Wesley Advanced Book Program, CA,1976. Used by permission)

In order to fix thesepreliminary ideasin a specificway we focus on the gas-liquid transition in a fluid. Consider first a point P, Z on the coexistencecurve; two local minima of the underlying Gibbs potential then compete,as in Fig. 10.1(page 205). If the point of interestwere to move off the coexistencecurve in either direction then one or the other of the two minima would becomethe lower. The two physical states,corre- sponding to the two minima, have very different valuesof molar volume,

? -----> FIGURE IO.T Competition of two minima of the underlying Gibbs potential near the coexistence curve. Thermodynamicsin the Neighborhood of the Critical Point 257

{.t

[:/

T-> FIGURE IO.2 Coalescenceof minima of the underlying Gibbs potential as the critical point is approached. molar entropy, and so forth. Thesetwo statescorrespond, of course,to the two phasesthat competein the first-orderphase transition. Supposethe point P,T on the coexistencecurve to be chosencloser to the critidal point. As the point approachesthe critical T and P the two minimaof the underlyingGibbs potential coalesce (Fig. 10.2). For all points beyondthe critical point (on the extendedor extrapolated coexistencecurve) the minimum is single and normal (Fig. 10.3).As the critical point is reached(moving inward toward the physical coexistence curve) the single minimum developsa flat bottom, which in turn develops e "bump" dividing the broadenedminimum into two separateminima. The singleminimum "bifurcates" at the critical point. The flattening of the minimum of the Gibbs potential in the region of the critical stateimplies the absenceof a "restoring force" for fluctuations eway from the critical state (at least to leading order)-hence the diver- gent fluctuations. This classicalconception of the developmentof phase transitions was formulated by Lev Landau,z and extended and generalizedby Laszlo Tisza,' to form the standardclassical theory of critical phenomena.The cssential idea of that theory is to expand the appropriate underlying thermodynamic potential (conventionallyreferred to as the "free energy functional") in a power seriesin Z - 7", the deviation of the temperature from its value 7"(P) on the coexistencecurye. The qualitative features described here then determine the relative signs of the first several

2cf . L. D. Landau and E. M. Lifshitz, StatisticalPhysics, MIT Press,Cambridge, Massachuserts end London, 1966. 3cf . L. Tisza, GeneralizedThermodynamics, MIT Press, Cambridge, Massachusettsand London, 1966 (see particularly papers 3 and 4). 258 Critical Phenomena li v' \? 4\ "l\J \' l \v/

\\ i9 Gl \^/

\L / \ v/

T ----+ FIGURE 10 3 The classicalpicrure of the developmentof a first-orderphase transition. The dotted curve is the qxtrapolated(non-physical) coexistence curve.

coefficients, and these terms in turn permit calculation of the analytic behavior of the susceptibilitiesas T" approachesthe critical temperature 7,,. A completely analogoustreatment of a simple mechanicalanalogue model is given in the Example at the end of this section,and an explicit thermodynamic calculation will be carried out in Section 10.4. Ar this point it is sufficient to recognize that the Landau theory is simple, straightforward, and deeply rooted in the postulates of macroscopic thermodynamics;it is basedonly on thosepostulates plus the reasonable assumptionof analyticity of the free energyfunctional. However,a direct comparison of the theoreticalpredictions with experimentalobservations was long bedeviledby the extremedifficulty of accuratelymeasuring and controlling temperature in systemsthat are incipiently unstable, with gigantic fluctuations. ln 1944Lars Onsageraproduced the flrst rigorousstatistical mechanical solution for a nontrivial model (the "two-dimensional lsing model',), and it exhibited a type of divergencevery different from that expected.The scientific community was at first loath to accept this disquieting fact, particularly as the model was two-dimensional(rather than three-dimen- sional), and furthermore as it was a highly idealized construct bearing little resemblanceto real physical systems.ln 1945 E. A. Guggenheims

aL. Onsager, Phys. Reu.65,177 (1944). sE. A Guggenheim, "I. Chem Phys.l3,253 (1945) Thermodynamics in the Neighborhood of the Critical Point 259 observedthat the shapeof the coexistencecurve of fluid systemsalso cast doubt on classicalpredictions, but it was not until the early 1960sthat precisemeasurements6 forced confrontation of the failure of the classical Landau theory and initiated the painful reconstructiod that occupiedthe decadesof the 1960sand the 1970s. Deeply probing insightsinto the nature of critical fluctuationswere developedby a number of theoreticians,including Leo Kadanoff, Michael Fischer,G. S. Rushbrooke,C. Domb, B. Widom, and many others.8'eThe construction of a powerful analytical theory ("renormalization theory") wasaccomplished by KennethWilson, a high-energytheorist interested in statistical mechanicsas a simpler analogue to similar difficulties that plaguedquantum field theory The sourceof the failure of classicalLandau theorycan be understood relativelyeasily, although it dependsupon statisticalmechanical concepts yet to be developedin this text. Neverthelesswe shall be able in Section 10.5 to anticipatethose results sufficiently to describethe origin of the difficulty in pictorial terms.The correction of the theory by renormaliza- tion theory unfortunatelylies beyond the scopeof this book, and we shall simply describethe generalthermodynamic consequences of the Wilson theory. But first we must developa frameworkfor the descriptionof the analytic form of divergen\ quantities,and we must review both the classicalexpectations and t\e (very different)experimental observations. To all of this the following mechanicalanalogue is a simpleand explicit introduction.

Example The mechanicalanalogue of Section9.1 providesinstructive insights into the flatteningof theminimum of thethermodynamic potential at thecritical point as that minimum bifurcatesinto two competingminima below {.. We again considera length of pipe bent into a semicircle,closed at both ends,standing verticallyon a tablein the shapeof an invertedU, containingan internalpiston. On either side of the piston there is 1 mole of a monatomicideal gas.The metal balls that were insertedin Section9.1 in order to break the symmetry(and thereby to produce a first-order rather than a second-ordertransition) are not present. If d is the angle of the piston with respectto the vertical, ^ft is the radius of curvature of the pipe section, and Mg is the weight of the piston (we neglect gravitational effectson the gas itself), then the potential energyof the piston is

'cl. P Heller and G B. Benedek, Phys. Reu. Let.8,428 (1962) 1cf . H. E. Stanley, Introduction to Phase Transitions und Critical Phenomena, Oxford Univ. Press, New York and Oxford, 1971 *./. H. E. Stanley, Ibid. eP. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and Critical Phenomenu, John Wiley and Sons, NY 1977. 260 Critical Phenomena

(MgR)cosl, and the Helmholtz potential is

F = IJ _ ZS: (USR)cos0+ FL+ FR

The Helmholtz potentials F. and .F^of the gasesin the left-hand and right-hand sectionsof the pipe are given by (recall Problem5.3-1)

Ft.n: F'(T)-Rrh( +\ \vol where F'(T) is a function of ronly. The volumesare determinedby the position d of the piston

n,: (r-r)r", ,^: (t*r)n, where we have taken zo as half the total volumeof the pipe. It follows then that, for small 0,

F(0,7):Usnft -T+u*"'l0204.| +2F,(r).^'[(+)' .+(+)' . : - [usn+ zF,(r)l.(#^, lusn)e, *(fiusa. .. *^r) o4+

The coefficientof 0a is intrinsically positive,but the coefficientof 02 changessign at a temperature7",

,tf2 r",:fu(Msfu)

For r > 7", thereis then only a singleminimum; the piston residesat the apex of the pipe and the two gaseshave equal volumes. < : _ For 7 7", the state d 0 is a maximum of the Helmholtz potential and there are two symmetric minima at

T _T 0: +{6n----!---=- 247 * tzT,, Diuergenceand Stahitity 261

For 7 : 7,, the Helmholtz potential has a very flat minimum, arising only from the fourth-order terms. Spontaneousfluctuations thereby experienceonly weak restoring forces.The "Brownian motion" (fluctuation)of the position of the piston is correspondinglylarge. Furthermore,even a trivially small force applied to the piston would induce a very large displacement;the "generalizedsuscept- ibility" diverges. Although we have now seen the manner in which this model develops a bifurcating Helmholtz functional at the critical temperature,it may be instructive also to reflect on the manner in which a first-order transition occurs at lower temperatures.For this purposesome additional parametersmust be introduced, to bias one minimum of ln relative to the other. We might simply tilt the table slightly, thereby inducing a first-order transition from one minimum to the other. Alternatively, and more familiarly, a first-orderphase transition can be thermally induced. In Section 9.1 this possibility was built into the model by inclusion of two metal ball-bearingsof different coefficientsof thermal expansion; a more appealingmodel would be one in which the two gasesare differently nonideal. Although this example employs a rather artificial system, the fundamental equation mimics that of homogeneousthermodynamic systems, and the analysis given above anticipates many features of the classical Landau theory to be describedin Section10.4.

IG2 DIVERGENCE AND STABILITY

The descriptive picture of the origin of divergences at the critical point, rs alluded to in the preceding section, is cast into an illuminating p€rspective by the stability criteria (equation 8.15 and Problem 8.2-3)

0rg \ I 'o (#).'o (10.1) aT2IP

(#),(#).-(#h)"0 (10.2)

stability criteria expressthe concavity requirementsof the Gibbs tial. The "flattening" of the Gibbs potential at the critical point to a failure of theseconcavity requirements. In fact all three the stability criteria fail simultaneously, and a, Kr, arLd c, diverge ether. Further perspective is provided by a physical, rathdr than a l, point of view. Consider a particular point P, Z on the coexistence 262 Criticul Phenomena

FIGURE10 4 Schematic isotherms of a two-phase sys- tem.

curveof a two-phasesystem. The isothermsof the systemare qualitatively similar to thoseshown in Fig. 10.4(recall Fig.9.12, althoughthe van der Waals equationof statemay not be quantitatiuelyrelevant). In particular, the isothermshave a flat portion in the P-T plane On this flat portion the systemis a mixture of two phases,in accordancewith the "lever rule" (Section 9.4). The volume can be increasedat constant pressureand temperature,the systemresponding simply by alteringthe mole fraction in each of the two coexistentphases. Thus, formally, the isothermal compressibilityrr: -u-t( 0u/0P), diverges. Again consideringthis same systemin the mixed two-phasestate, supposethat a small quantity of heat Q (: 7AS) is injected.The heat suppliesthe heat of transition (the heat of vaporizationor the heat of melting) and a smallquantity of mattertransforms from one phaseto the other. The temperatureremains constant. Thus co: T(0s/07.), diverges. The divergenceof r, and of c" existsformally all along the coexistence locus.Across the coexistencelocus in the P-T planeboth r, and co are ( discontinuous,jumping from one finite value to another by pa-ssing ( throughan intermediateinfinity (in the mixed-phasestate), see Fig. 10.5. I As the point of crossingof the coexistencecurve is chosencloser to the I critical point, classicalLandau theory predicts that the "jump" of rc, I should decreasebut that the intermediate infinity should remain. This ( s 7 I I

A I h / a FIGURE IO 5 ,$ Discontinuity and divergence of gener- ) nlized susceptibilities across a coexis- tence locus. The abscissacan be either T or P, along a line crossing the coexis- TorP + tence locus in the 7 - P plane. Order Parameters and Critical Exponents 263

description is correct except very close to the critical point, in which region nonclassicalbehavior dominated by the fluctuations intervenes. Nevertheless,the qualitativebehavior remains similar-a divergenceof r. at the critical point, albeit of an alteredfunctional form. The heat capacity behavessomewhat differently. As we shall seelater, Landau theory predicts that as the critical point is approachedboth the jump in the heat capacityand the intermediate divergenceshould fade away. In fact the divergenceremains, though it is a weaker divergence than that of xr.

IO.3 ORDER PARAMETERS AND CRITICAL EXPONENTS

Although Landau's classicaltheory of critical transitions was not quantitatively successful,it did introduce several pivotal concepts. A particularly crucial observationof Landau was that in any phasetransi- tion thereexists an "order parameter"that can be so definedthat it is zero in the high-temperaturephase and nonzeroin the low-temperaturephase. Order parametersfor varioussecond-order transitions are listed in Table 10.1.The simplestcase, and the prototypicalexample, is providedby the paramagnetic to ferromagnetictransition (or its electric analogue).An appropriateorder parameteris the magneticmoment, which measuresthe cooperativealignment of the atomic or moleculardipole moments. Another simple and instructive transition is the binary alloy "order-disorder" transition that occurs, for instance,in copper-zinc (Cu-Zn) alloy. The crystal structureof this material is "body-centered cubic," which can be visualizedas being composed of two interpenetrating simplecubic lattices.For conveniencewe refer to one of the sublatticesas the I sublatticeand to the other as the B sublattice.At high temperatures the Cu and Zn atoms of the alloy are randomly located, so that any particular lattice point is equally likely to be populated by a zinc or by a copper atom. As the temperatureis lowered,a phasetransition occurs such that the copper atomspreferentially populate one sublatticeand the zinc atoms preferentiallypopulate the other sublattice.Immediately below the transition temperaturethis preferenceis very slight,but with decreas- ing temperaturethe sublatticesegregation increases. At zero temperature one of the sublatticesis entirely occupiedby copper atoms and the other sublattice is entirely occupied by zinc atoms. An appropriate order parameteris (N/" - Nt)/No, or the differencebetween the fraction of I sites occupied by zinc atoms and the fraction occupied by copper atoms. Above the transition temperaturethe order parameteris zero; rt becomesnonzero at the transition temperature;and it becomeseither * 1 or -1 at T:0. As in the order-disorder transition, the order parametercan alwaysbe chosen to have unit magnitude at zero temperatl.rre;it is then "normal- 264 Critical Phenomenu

ized." In the ferromagnetic case the normalized order parameter is \f)/I(O); whereas the extensive parameter is the magnitic moment IQ). In passingwe recall the discussionin section 3.8 on unconstrainable variables. As was pointed out, it sometimeshappens that a formally defined intensive plrameter does not have a ptrysical realization. The copper-zinc alloy systemis such a case.In contrast to the ferromagnetic case (in which the order parameteris the magnetic moment 1 anl the intensive parameter au/al is the magneticfield B"), the order parameter for the copper-zinc alloy is (N/" - Nd,) but the intensiveparameter has no physical reality. Thus the thermodynamictreatment of the cu-zn systemrequires that the intensiveparameter always be assignedthe value zero. similarly th9 intensiveparameter conjugate to the order parameter aHe of the superfluid transition must be ta[en as zero. Identification of the order parameter, and recognition that various generalized susceptibilitiesdiverge at the critical point, motivates the definition of a set of "critical exponents" that desCribethe behavior of thesequantities in the critical region. _ !n tle thermodynamiccontext there are four basic critical exponents, defined as follows. The molar heat capacity(c, in the fluid caseor c, in the magneticcase) divergesat the critical point with exponentsc abovl 7", and i below 7",

cuor cB,- Q - 7",)-" (T , 7",) (10.3)

cuotcB"- (7",- T)-"' (7.7,,) (10.4)

The "generaliz".,9susceptibilities", fc, : -(|u/lp)r/u in the fluid case or Xr: po(AI/AB")r/u in the magneticcase, diverge with expo- nents f ot "l'.

Krorxr-Q-T",)lt (T > 7,,) (10.5)

Kr or x7.- (Tr,- T)-' (7.7,,) (10.6)

Along the coexistencecurve the order parametervaries as (?:, - T)p

AuorI-(Tc,-DP (r.7,,) (10.7) and, of course, the order parameter vanishesfor Z , 7,,. Note that a grime indicates T . T* for the exponentsa, and 7,; whereasB can be defined only for , I 7", so that a prime is superfluous. C lassical Theory in the Critical Region: Landau Theory

Finally, on the critical isotherm (i.e., for T : 7,,) the order parameter and its correspondingintensive parameter satisfy the relation

I - BrtD or AD - (P - p",)'/u (10.8)

which define the exponent 6. In addition there are several critical exponents defined in terms of *atistical mechanicalconcepts lying outside the domain of macroscopic thermodynamics.Perhaps the most significant of these additional expo- nents describesthe range of fluctuations, or the size of the correlated regions within the system.The long wavelengthfluctuations dominate near the critical point, and the rangeof the correlatedregions diverges. This onset of long-rangecorrelated behavior is the key to the statistical mechanical(or "renormalizationgroup") solution to the problem. Because large regions are so closelycorrelated, the details of the particular atomic ttructure of the specific material becomeof secondaryimportance! The etomic structure is so masked by the long-rangecorrelation that large families of materialsbehave similarly-a phenomenonknown as " univer- rality," to which we shall return subsequently.

CLASSICAL THEORY IN THE CRITICAL REGION: LANDAU THEORY

The classicaltheory of Landau, which evaluatesthe critical exponents, ides the standard of expectation to which we can contrast both observationsand the results of renormalizationgroup the-

We consider a systemin which the unnormalizedorder parameteris f. have in mind, perhaps,the magnetizationof a uniaxial crystal (in ich the dipoles are equally probably " up" or "down" above the ition temperature),or the binary Cu-Zn alloy. The Gibbs potential is a function of T, P,g, Np N2,.. ., N, (10.e)

the immediatevicinity of the critical point the order parameteris small, ing a seriesexpansion in powersof 6

G : Go* Gro* Grq'* Grqt+ ... (10.10)

Gs,GDGr,. . . are functionsof Z, P, Ny - . ., N,. For the magnetic or binary alloy the symmetry of the problems immediately pre- the odd terms, requiring that the Gibbs potential be even-in p; is no a priori difference between spin up and spin down, or between

Ifi IHOVIIAMAT.FYZ. FAKULTY lhihovnaFr Zavr$Ry(tyz. odo.) Ke Kartovu3 266 Critical Phenomena

theA and B sublattices.(This reasoning is a precursorand a prototypeof l more elaboratesymmetry arguments in morecomplex systems.) I i G(7,P,0,AL,...,4):Go* Gre,+ Goeo+ ... (10.11) I Each of the expansion coefficients is a function of T, p, and the N,'s; G,,: Gn(7,P,Np...,N.). We now concentrateour attention on ihe extrapolated coexistencecurve-the dotted curve in Fig. 10.3. Along this locus P is a function of. T, and all mole numbers are constant, so that each of the expansiol coefficients G, is effectively a function of ? only. Correspondingly, G is effectively a function only of Z and e. Th-" shgle of G(7,0) as a function of g, for small g, is shown in Fig. 10.6 for the four possiblecombinations of signsof G, and Go. G(r,0) G(r, Q)

6+ Q+

G(r,6) G(r,6) 1 6 t ( e

6----> 6 -----> FIGURE106 I Possible shapes of G(7,+) for various signs of the expansion coefficients. t A point on the extrapolated coexistencecurve ("beyond" the critical 4 b tr

negative (Fig. 10.6). The function Go(T) normally remains positive. The crit,ical temperature is uiewedsimply as the temperature at which G, happens to haue a zero. I The change of sign of G, at the critical point implies that a series - fi expansion of G, in powers of (" {) has the form n GrIT, P(T)] : (r - T",)Gl + termsof order(T - 7,,)'+ . . . (10.12) Classical Theory in the Critical Region: lttndauTheory 267

Now, let the intensive parameter conjugate to f have the value zero. In the magnetic case, in which { is the normalized magnetic moment, this ies that thereis no externalmagnetic field, whereasin the binary alloy intensiveparameter is automaticallyzero. Then, in either type of case

-:AG - ao z(T 7,,)G:++ 4G4e3+ . -. : 0 (10.13)

equation has different solutionsabcrve and below {,. For T > 7,, only real solutionis 4 : g.

0:0 (forZ>7,,) (10.14)

7", the solution + : 0 correspondsto a maximum rather than a um value of G (recall Fig. 10.6),but there are two real solutions lng to fiunrma

0: 'l,t(r,,-nl' (r s r",) (10.15)

is the basic conclusionof the classicaltheory of critical points. The parameter (magneticmoment, difference in zinc and copper occupa- ln of the I sublattice, etc.) spontaneouslybecomes nonzero and grows as ',, - T)'/' for temperaturesbelow 7,,. The critical exponent B, defined in ion 10.7, therebyis eualuatedclassically to hauethe ualuel.

B(classical): 7/2 (tO.t6)

contrast, experiment indicates that for various ferromagnets or fluids value of B is in the neighborhood of 0.3 to 0.4. In equationL0.L3 we assumedthat the intensiveparameter conjugate to is zero; this was dictatedby our interestin the spontaneousvalue of 4 7,,. We now seek the behavior of the "susceptibility" 1, for turesjust above7,,, Xr beingdefined by

x;l :.(#),*-, (10.17)

the magneticcase X;l is equal to N(AB"/ADr.r+o so that g,01. is the niliar molar magnetic susceptibility (but in the present context we shall be concerned with the constant factor po). Then

:2(T - r,,)Gl+ 72G4Q2+ ... (10.18) *r;' 268 Criticul Phenomenu

or taking 0 - 0 accordingto the definition10.17, : - #";' z(r r,,)Gt+.'. r 2 r,, (ro.rs)

This result evaluatesthe classicalvalue of the exponent y (equation 10.5) as unrty

y(classical): 1 (10.20)

Again, for ferromagnetsand for fluids the measuredvalues of y are in the regionof 1.2 to 1.4. . For^!_< 7,, t\e order parameterf becomesnonzero. Inserting equa_ tion 10.15for {(f) into equation10.18

-r)+... *r;':z(r-r,,)G|+r2G4"l+H. :4(7,,-r)G!+... (10.21) we therefore conclude that the classicalvalue of f, is unity (recall equation 10.6).Again this doesnot agreewith experiment,which yields valuesof y' in the regionof 1.0to 1.2. The valuesof the criticalexponents that follow from the Landautheory are listed,for convenience,in Table 10.2. TABLE10.2 critical Exponentqclassical values and ApproximateRange of observedvalues

Approximate range of Exponent Classical ualue obserued ualues d. 0 -0.2

I 1 1.2

6 J 4<6<5

Example It is instructive to calculate the classical values of the critical exponents for a system with a given, definite fundamental equation, thereby corroborating the more general Landau analysis. Calculate the critical indices for a system de- scribed by the van der Waals equation of state. Classical Theory in the Critical Region: Landau Theory 269

Solution From Example1 of Section9.4, the van der Waalsequation of statecan be written in "reducedvariables": 8T -F_ _3 3a-l a2 whereP = P/P- and similarlyfor f and D.Then, defining p=P-L 0=fr-1 e=i-l end multiplyingthe van der Waalsequation by (1 + 0)z we obtairfo 2p(t + lo + to2 + +0t): -3u3 + 8(1+ 20 + 02)

- lo3+ e(+ 60+ 902+ ... ) + ... If e:0 (that is ?: [,) then 0 is proportionalto (-p)i, so that the critical cxponent 6 is identified as 6 : 3. To evaluatey we calculate

r;r:._r -V(!:\.\0vlr : -,(H)": 6De* ...

f: l' :1. To calculate p we recall that 0(tr) : 0(t), where 0(n) is defined by the last luationin Example2,page24l. eG) =tn(3r- 1)-(3t - 1)-' + st(4uT). : ln(3t+ 2\-(3i + 2\-t+ ?(r+ r)-'('+ 1)-' : ln2+ 1+ &it + fe(1 r e-ri- ie - t|+ ...) , from 0@s): d(Dr)we find i(03+ oso/+o?)+ c - c2- e(os+ a/): o p(}r): p(0), whichgives o?+ opu+or* 4e - 6c(or+o/) : o :se latter two equationsconstitute two equationsin the two unknowns 0, and Eliminating (% + 0/) we are left with a singleequation in 0, - Dr; we find

ot- L: a(-e;i* ... identifiesthe criticalexponent B as i. The remaining critical exponentsare a and c', referring to the heat capacity. van der Waals equationof statealone doesnot determinethe heat capacity, we can turn to the "ideal van der Waalsfluid" definedin Section3.5. For that

roH. Stanley, Introduction to Phase Transitiotts and Critical Phenomena, Oxford Univ. Press, New and Oxford, 1971.(sect. 5.5). 270 Critical Phenomena system the heat capacity cu is a constant, with no divergence at the critical point, anda:a':0. rO.5 ROOTS OF THE CRITICAL POINT PROBLEM

The reader may well ask how so simple, direct, and general an argumen! as that of the preceding section can possibly lead to incorrect resulti. Does the error lie within the argument itself, or does it lie deeper, at the very foundations of thermodynamics?That puzzlement was shired by thermo- dynamicists for three decades.Although we cannot enter here into the

F P P d s o b p

p a t p d v tt c c, o rl n Rootsof the Critical Point Problem 271

4.1 I

(7",-T) +

0i 6ro 6ro 62 RE107 lity distributions, average,and most probable values for the fluctuating order . The temperaturesare T2 S Tr 5 {. The probability distributions are shownas ,, curves.The classicalor most probablevalues are fie and Of *td thesecoincide the minima of G. The averageor observablevalues are 01 utd Qi.Th" rate of change the averagevalues is more rapid than the rate of changeof the most probable values of the asvmmetrvof the curvesfor ?i. This is more consistentwith a critical index = t rather than l, as shownin the small figure.

(showndotted) is correspondinglybroad and asymmetric.The value Q! of 0 is shifted to the left of the mostprobable value Qip. a temperatute Tz further removedfrom the critical temperaturethe tial well is almost symmetricnear its minimum, and the probability ty is almost symmetric.The averagevalue Qi and the most probable glp are then almost identical. As the temperaturechanges from Zt T, the classicallypredicted change in the order parameteris +fp - +T' the statisticalmechanical prediction is E! - {!. Thus we seethat thermodynamicsincorrectly predicts the temperaturedependence the order parameteras the critical temperatureis approached,and that failure is connected with the shallow and asymmetric nature of the of the potential. - To extend the reasoning slightly further, we observe that $ip QTo is ler than +i - Qi (Fig. 10.7).That is, the classicalthermodynamic iction of the shift in f (for a given temperaturechange) is smaller the true shift (i.e., than the shift in the auerage value of f). This is istent with the classical prediction of B : I rather than the true value - *, ur indicatedin the insertin Fig. 10.7. This discussionprovides, at best, a pictorial insight as to the origin of failure of classical Landau theory. It gives no hint of the incredible th and beauty of "renormalization-group theory," about which we shall have only a few observations to make. 272 Critical Phenomena.

10.6 SCALING AND UNTVERSALITY

As mentionedin the last paragraphof Sectionr0.3, the dominant effect that emergesin the renormalizationgroup theory is the onsetof long-range correlated behavior in the vicinity of the critical point. This lccuis because the long wave length excitations are most easily excited. As fluctu-ations grow the very long wave length fluctuations grow most rapidly, and they dominate the properties in the critical region. Two effectsresult from the dominanceof long rangecorrelated fluctuations. The first class of effectsis describedby the term scaling.Specifically, the divergenceof the susceptibilitiesand the growth of the order parame- ter are linked to the divergenceof the rangeof the correlatedfluctuations. Rather than reflecting the full atomic comprexity of the system, the diverse critical phenomenaall scaleto the range of the divergentcorrela- S tions and thence to each other. This interrelation among the critical 10.: exponentsis most economicallystated in the "scaling hypothesis,,,the fundamental result of renormalization-group theory. thii result states that the dominant term in the Gibbs potential (or anotherthermodynamic potential, as appropriateto the critical transition considered)in the reeion of the critical point, is of the form - - - tl| pt+r,t6 \ G, lT 7,,1,-"f # l, (T -- 7",) (10.22) \ lr - T,l'-"I we here use the magnetic notation for convenience,but B" can be whr interpreted generally as the intensive parameter conjugate to ihe order assr parameter f. The detailed functional form of the Gibbs potential is its , discontinuous acrossthe coexistencecurve, as expected,and ihis discon- 1 x;-the tinuity in form is indicatedby the notation y function /+ applies eva for T j Tu.gld the (different) function applies for z I T"-,.Fuither- /- i'regular" ft more the Gibbs potential may have additional termi, the terms written in equation 10.22being only the dominant part of the Gibbs potential in the limit of approachto the critical point. The essential content- of equation l0.zz is that the quantity Gr4T : Tr,)'-".is not a functionof both T and,B, separately,but only of the singlevariable n!+rt61r - T,,l'-o.It can equaliywell-be written as a function of the squareof this compositevariable,- or of any other power. We shall later write it as a function of B"/(T - 7,,)(2-a)t7\r+t's. The scaling property expressedin equation 10.22 refates all other critical-ex_ponentsby uniuersalrelationships to the two exponentsa and 6, as we shall now demonstrate.The procedureis straightfoiward; we simply evaluate each of the critical exponentsfrom the fundamental equation 10.22. we first evaluatethe critical index c, to corroboratethat the symbol a appearing in equation 10.22does have its expecredsignificance. For this Scaling and Unioersality 273 1(x) purpose we take B": 0. The functions / are assumedto be well t(0) behavedin the region of x : 0, with / being finite constants.Then the heat capacity is

a2G,(8":o) cB. - - (2- - d)V - T",l-"ft(0) (10.23) aT2 "x1

Hence the critical index for the heat capacity, both above and below 7,,, is identified as equal to the parameter a in G", whence

a.' : d. (70.24)

Similarly, the equationof state I : I(7, B") is obtained from equation 10.22by differentiation

Br +r/6 r- -ft -tr-T,,l'-"f't \_u IiT - T",l'-d J AB"

(10.2s)

*(x) i(x). 't(0) where f denotes(d/dx)f Again the functions f are assumedfinite, and we have thereforecorroborated that the symbol 6 has its expectedsignificance (as definedin equation 10.8). To focus on the temperaturedependence of 1 and of 1, in order to evaluate the critical exponentsB and y, it is most convenientto rewrite t - f ut a function gt of B"/(T 7,,)(2-d)6/(L+E). - - G, lr r,,|,-os'(r,_ ,"ftrrrr-r) tto.rul

B" (to.zt) - a)6/(r+6) lT T",l('-

2-a (10.28) B:1+6 274 Critical Phenomena

Also

AI s',(r- 8l/(r+6\ J B" - 'a:rlT l(2- ot, x: lro lI D d)6/(r + 8) 64- (rT_ Tc)Q-

(10.2e)

whence

1-6 y: y, : (a\ _ 2\:_ (10.30) /1+6

Thus all the critical indiceshave been evaluatedin terms of a and 6. The observedvalues of the critical indices of various systemsare, of course, consistentwith theserelationships. As has been stated earlier, there are two primary consequencesof the dominance of long range correlatedfluctuations. One of these is the scaling of critical propertiesto the rangeof the correlations,giving rise to the scaling relations among the critical exponents.The secondconse- quenceis that the numericalvalues of the exponentsdo not dependon the detailed atomic characteristicsof the particular material, but are again determinedby very generalproperties of the divergentfluctuations. Re- normalization group theory demonstratesthat the numericalvalues of the exponents of large classesof materials are identical; the values are determined primarily by the dimensionalityof the system and by the dimensionalityof the orderparameter. The dimensionalityof the systemis a fairly self-evidentconcept. Most thermodynamicsystems are three-dimensional.However it is possibleto study two-dimensionalsystems such as monomolecular layers adsorbed on crystallinesubstrates. Or one-dimensionalpolymer chains can be studied. An even greater range of dimensionsis available to theorists,who can (and do) construct statistical mechanicalmodel systemsin four, flve, or more dimensions(and evenin fractionalnumbers of dimensions!). The dimensionality of the order parameterrefers to the scalar,vector, or tensorial nature of the order parameter.The order parameter of the binary alloy discussedin Section10.3 is one-dimensional(scalar). The order parameter of a ferromagnet,which is the magnetic moment, is a vector and is of dimensionalitythree. The order parameterof a supercon- ductor, or of superfluidaHe, is a complexnumber; having independent real and imaginarycomponents it is consideredas two-dimensional.And again theoreticalmodels can be devisedwith other dimensionalitiesof the order parameters. 275

Systems* with the same spatial dimensionality and with the same imensionality of their order parametersare said to be in the same universality class." And systemsin the sameuniversality classhave the valuesof their criticalexponents.

l. Show that the following identitieshold among the critical indices a+ 2B + y:2 ("Rushbrooke'sscalinglaw") Y:B(6-1) (" Widom's scalinglaw")

Are the classical values of the critical exponents consistent with the ng relations?

is assumed that the interatomic forces in the system are not of infinite ranee.