a r X i v :c o n d -m a t /0209470v 1 [c o n d -m a t .s t a t -m e c h ] 19 S e p 2002

Critical phenomena and critical relations

M.Matlak ?and B.Grabiec

Institute of Physics,Silesian University,4Uniwersytecka,

40-007Katowice,Poland

February 1,2008

Abstract

We consider systems which exhibit typical critical dependence of the

speci?c heat:?c ∝(T C ?T )?γ(T

(T >T C )where γ,γ′are critical exponents (γ=αfor ?c =?c p,N ,γ=

3

,uniaxial ferroelectrics;a =1,liquid He 4

).Starting from the critical behaviour of the speci?c heat we ?nd the Gibbs (Helmholtz)potential in the vicinity of the critical point for each case separately.We derive in this way many exact critical relations in the limit T →T C which remain the same for each considered case.They de?ne a new class of universal critical relations independent from the underlying microscopic mechanism and the symmetry of these systems.The derived relations are valid for a very broad class of magnetic,ferroelectric and superconducting materials,as well as,for liquid He 4.

Subject classi?cation:05.70.Fh,05.70.Jk,74.25.Bt,67

1Introduction

Critical phenomena in solids,connected with second order phase transitions,belong to very attractive and intriguing problems in physics.Many theoretical models and methods have been applied to resolve these interesting questions (cf e.g.Refs [1]-[5]for a review and the original papers cited therein),how-ever,the most important,general and successfull method,applied in this area of physics,seems to be still the phenomenological approach,initiated by the pioneer works of Landau,Refs [6],[7].This approach has succsessfully been applied to superconducting systems resulting in many useful critical relations,derived in Ref.[8].For many other systems (magnetic,ferroelectric,liquid He 4,etc.)this approach unfortunately fails because of critical exponents (cf e.g.Refs [1]-[5]).We can,however,improve this approach and calculate the

Gibbs(Helmholtz)potential near the critical point from the observed speci?c heat anomaly(TT C).Similarly to Ref.[8]we include into the con-siderations the critical behaviour of the chemical potential as indicator of phase transitions in a more general sense(see Ref.[8]and papers cited therein).We derive in this way many interesting thermodynamical relations(cross-relations) at the critical point in the limit T→T C.These exact relations seem to be of fundamental importance for critical phenomena in solids and should generally be valid for a very broad class of magnetic or ferroelectric systems,as well as, for superconducting systems as a special case.Surprisingly,they are also valid in the case of logarithmic divergence of the speci?c heat(uniaxial ferroelectrics and liquid He4).The derived relations possess very similar form to the critical relations presented in Ref.[8]and de?ne a new class of exact relations at the critical point independent from the underlying microscopic mechanism leading to second order phase transitions and independent from the symmetry of the system.

2Phenomenological Gibbs(Helmholtz)poten-tials and critical relations

Performing experimental measurements of the temperature dependence of the speci?c heat c p,N at constant pressure p and constant number of particles N we can?nd that for magnetic,ferroelectric or superconducting systems c p,N ex-hibits a pronounced anomaly at the critical point T=T C(cf e.g.Refs[1]-[5]). Independently from the considered system(here:magnetic,ferroelectric or su-perconducting)it is convenient to introduce a reference system which possesses exactly the same structure(symmetry),atomic masses,etc.,the speci?c heat

of this system c(0)

p,N exhibits,however,no anomaly at T=T C.Let us introduce

an auxiliary quantity

?c p,N(T)≡c p,N(T)?c(0)

p,N

(T).(1) A very broad class of materials(magnetic,ferroelectric or superconducting) shows a general,critical behaviour(see e.g.[1]-[5])of the type1

?c p,N(T)∝ (T C?T)?α,T

(T?T C)?α′,T>T C

(2)

whereα,α′are critical exponents,depending on the material(α,α′>0).For magnetic and ferroelectric systems?c p,N usually diverges at the critical point, for superconducting ones?c p,N exhibits only a?nite jump.For simplicity,we

restrict ourselves to the case T

?c p,N(T)=A(p,N)(T C?T)?α(3) where A(p,N)is a parameter.The critical temperature T C should also be con-sidered as a function of p and N(T C=T C(p,N)),as suggested by experimental results.The thermodynamics of the system in the vicinity of the critical point (T

Z(T,p,N)=Z0(T,p,N)+?Z(4) where

?Z=Z?Z0,(5) Z0is the Gibbs potential of the reference system and c(0)p,N=?T ?2Z0

?T2 p,N(6) reproduces the critical behaviour of?c p,N,given by the formula(3).It is easy to?nd that the leading term of?Z should have the general form2

?Z=D(p,N)(T C?T)2?α(7) where D(p,N)is a parameter.Really,when applying(6)and(7)we obtain ?c p,N=?T(2?α)(1?α)D(p,N)(T C?T)?α

≈?T C(2?α)(1?α)D(p,N)(T C?T)?α

=A(p,N)(T C?T)?α(8) where

A(p,N)=?T C(2?α)(1?α)D(p,N).(9) Thus,the asymptotic form of the Gibbs potential(4)in the vicinity of the critical point(T

Z(T,p,N)=Z0(T,p,N)+D(p,N)(T C?T)2?α.(10)

The phenomenological relations(3)and(10),resulting from the experimental, critical behaviour of a broad class of systems can easily be used to derive many

critical relations.The chemical potentialμ=(?Z

?N )T,p)and the

volume of the system V=(?Z

?p )T,N)can easily be found from

(10).We obtain

μ=μ0+ ?D

?N p(11) and

V=V0+ ?D

?p N(12) where we have taken into account that T C=T C(p,N).With the use of(11) and(12)we obtain

? ?μ?N p(T C?T)

+(1?α)D· ?T C

?T p,N=?(T C?T)?α(2?α)[ ?D

?p N],(14)? ?μ?p?N(T C?T)2

+(2?α)(T C?T) ?D?p N

+ ?D?N p+D· ?2T C

?p N

?T C

?

?μ

?N 2

p

(T C ?T )2

+(2?α)(T C ?T ) 2

?D

?N p

+D ·

?2T C

?N

p

2

(16)

and ?

?V ?p 2

N

(T C ?T )2

+(2?α)(T C ?T ) 2

?D

?p

N

+D ·

?2T C

?p

N

2

(17)

where ?X ≡X (T )?X 0(T )has exactly the same interpretation as in the formula (1).It is interesting to note that for magnetic (ferroelectric)systems

?c p,N ,?(?μ?T )p,N ,?(?μ?N )T,p and ?(

?V

?T )p,N

?N

p

,(18)

lim T →T ?C

?(?V

?c p,N =

?ln T C

?p )T,N

?N

p

?T C ?N )T,p

?N

p

?T C

?p )T,N

?p

N

?T C

3The

additional terms to ?Z (see Appendix A)modify only the expressions (11)-(17).By

forming the quotients (18)-(22)in the limit T →T ?

C their contribution is equal to zero (see Appendix A)and therefore the critical relations (18)-(22)are,in fact,exact.

5

and therefore we can drop the lim

T→T?

C in the expressions(18)-(22)and we obtain

the formulae(29)-(33)from Ref.[8].In orther words the formulae(18)-(22)are valid for a broad class of magnetic,ferroelectric and superconducting systems. Because of Maxwell relations ?S?T p,N, ?S?T p,N and ?V?p T,N there are only?ve independent relations of the type

(13)-(17).Because? ?S?T p,N,? ?S?T p,N and ? ?V?p T,N the corresponding critical relation with the presence of? ?S?N T,p and? ?V

?T

)p,N

?T )p,N

=

(?T C

(?T C

?N

)p,(?T C

?p T,N?T

p,N?T p,N ?N T,p?T

p,N

?p T,N?T p,N

α,T

(T?T C)?

whereα′are critical exponents.We restrict ourselves again to the vicinity of the critical point for T

?c V,N=α(28) where

D(V,N)(T C?T)2?

D(V,N)as a parameter and T C=T C(V,N).Here F0can be interpreted as the Helmholtz potential of the reference system.Repeating the calculations with the use of(29),quite similar to the presented above with the use of the Gibbs potential(cf(11)-(17)),we can?nd the exact relations similar to(18)-(22)4.We obtain

lim

T→T?

C ?(?μ

?c V,N

= ?ln T C

?T

)V,N

?V N,(31)

lim

T→T?

C ?(?μ

?c V,N

=? ?ln T C?V N,(32)

lim

T→T?

C ?(?μ

?c V,N

=? ?ln T C?N V,(33)

and

lim

T→T?

C ?(?p

?c V,N

= ?ln T C?V N.(34)

There are again only?ve exact expressions of this type because of the Maxwell relations ?S?T V,N, ?S?T V,N and ?p

?V T,N.There exists also a freedom to form another quotiens what re-sults in many other expressions easy to obtain(we don’t write them explicitely here).The analogous cross-relations,similar to(24)-(26)have the form

lim

T→T?

C ? ?μ?c V,N=T C lim T→T?

C

? ?μ?c V,N · lim T→T?

C

? ?p?c V,N ,(35)

lim

T→T?

C ? ?μ?c V,N=?T C lim T→T?

C

? ?μ?c V,N 2(36)

and

lim

T→T?

C ? ?p?c V,N=T C lim T→T?

C

? ?p?c V,N 2.(37)

The other cross-relations are also very easy to?nd.In the case of supercon-

ducting systems we can drop the lim

T→T?

C in these expressions and we obtain the

formulae(46)-(53)from Ref.[8].Similar calculations can also be repeated in the vicinity of the critical point for the case T>T C,using the form of the spe-ci?c heat for this case(see(2)and(27)).It leads,however,to the same critical

expressions(18)-(22),(24)-(26)and(30)-(37)where the limit T→T?

C should

be replaced by T→T+C.In other words,we can replace in all these expressions the limit T→T?C by a more general form lim

T→T C

.

3Logarithmic anomaly of the speci?c heat

It is interesting to see whether the derived critical relations(18)-(22),(24)-(26) and(30)-(37)are also valid in the case of logarithmic divergence of the speci?c heat,when

?c p,N(T)∝(ln(T C?T))a(38) where a=1

2

D(p,N)(T C?T)2[ln(T C?T)]a(40)

where D(p,N)is a parameter.Applying the formulae(5)and(6)we obtain ?c p,N(T)=T D(p,N){[ln(T C?T)]a

+3a

2

[ln(T C?T)]a?2}.(41)

In the vicinity of the critical point the second and third term in the parenthesis of(41)can completely be neglected(a=1

?N

)T,p(μ0=(?Z0

?p

)T,N(V0=(?Z0

2 ?D

2

[ln(T C?T)]a?1} ?T C

2 ?D

2

[ln(T C?T)]a?1} ?T C

?T p,N= ?D2[ln(T C?T)]a?1} + ?T C

2

[ln(T C?T)]a?1+

a(a?1)

?

?V

?p

N

(T C ?T ){[ln(T C ?T )]a +a

?p

N

D {[ln(T C ?T )]a

+

3a

2

[ln(T C ?T )]a ?2},

(47)

?

?μ

2

?2D ?N

p

?T C

2[ln(T C ?T )]a ?1}?D

?T C

?N

p {[ln(T C ?T )]a +

3a

2

[ln(T C ?T )]a ?2},(48)

?

?μ

2

?2D ?N

p

?T C

2[ln(T C ?T )]a ?1}?D [

?T C

2

[ln(T C ?T )]a ?1

+a (a ?1)

and

? ?V2 ?2D

?p N ?T C

2

[ln(T C?T)]a?1}

?D[ ?T C2[ln(T C?T)]a?1 +

a(a?1)

?T p,N(A.1)

11

we can write

?S= ?c p,N

T C

?1is ful?lled and therefore we can write

1

T C?(T C?T)=

1

1?(T C?T)T C

∞

l=0(T C?T)l

T C ∞ l=01

T C ∞

l=0(T C?T)l?α+1

T C ∞

l=0(T C?T)l?α+2

T C

(T C?T)2?α

T l+1

C

(l?α+2)(l?α+1)

(A.6)

and we see that the?rst term(leading term)in(A.6)has the form(7)(see also (9)).This term properly reproduces the critical behaviour of the speci?c heat (see formula(8))and therefore we have neglected in further considerations(see (11)-(17))the second term in(A.6).It is,of course,possible to take into account the full expression(A.6)for?Z but it is completely irrelevant.The presence of the second term in(A.6)modi?es the relations(11)-(17)producing very long expresions.It has,however,absolutely no in?uence on the critical relations (18)-(22)because the additional terms in(11)-(17)make no contributions in the limit T→T?

C

when forming the quotients(18)-(22).

12

6Appendix B

Starting from the expression(39)and using(A.3)we?nd

?S= ?c p,N

T C [ln(T C?T)]a

∞

l=0(T C?T)l+1

l+1

,...,a l,m=(?1)m a(a?1)·....·(a?m+1)

3,1).Because?S should vanish at T=T C(S=S0)the

arbitrary constant C1(p,N)should also be zero.Finaly we?nd that the exact expression for?Z has the form

?Z=?A(p,N)

T l C (l+1)(l+2)

∞

m=0a l,m [ln(T C?T)]?m∞ n=0b l,m,n[ln(T C?T)]?n

+C2(p,N)

(B.4) where

b l,m,0=1,b l,m,1=?a?m

(l+2)n

(B.5) and C2(p,N)is an arbitrary constant.Because the?rst term in(B.4)vanishes in the limit T→T?C the arbitrary constant C2(p,N)should be equal to zero (?Z=0(Z=Z0)at T=T C).The expression(B.4)can be rewritten in the short form to be

?Z=?

A(p,N)

(40)where D(p,N)is related to A(p,N)by the formula(43).It is,in principle, possible to perform the calculation(formulae(44)-(50))starting from the full expression(B.6).It,however,leads to extremely long formulae and therefore we have presented the results using only the leading term.The terms,generated by R(T C?T)in(B.6)produces additionally extremely long expressions in(44)-(50)which are completely irrelevant when forming the quotients(18)-(22)and

taking the limit T→T?

C because their contribution to the expressions(18)-(22)

is equal to zero in this limit.Therefore the critical relation(18)-(22)are also exact in the case of the logarithmic divergence of the speci?c heat(39). References

[1]H.E.Stanley,Introduction to Phase Transitions and Critical Phenomena,

Clarendon Press,Oxford1971.

[2]M.Ausloos and R.I.Elliot(Eds),Magnetic Phase Transitions,vol.48,

Springer-Verlag,Berlin1983.

[3]J.J.Biney,N.J.Dowrick,A.J.Fisher and M.E.J.Newman,The Theory of Crit-

ical Phenomena.An Introduction to the Renormalization Group,Clarendon Press,Oxford1992.

[4]T.Mitsui,I.Tatsuzaki and E.Nakamura,An Introduction to the Physics of

Ferroelectrics,Gordon and Breach Science Publishers,N.York,London, Paris1976.

[5]E.M.Lines,A.M.Glass,Principles and Applications of Ferroelectric and Re-

lated Materials,Clarendon Press,Oxford1977.

[6]https://www.wendangku.net/doc/3c17862026.html,ndau,Phys.Z.Sovjet.11,545(1937).

[7]W.L.Ginzburg and https://www.wendangku.net/doc/3c17862026.html,ndau,Sovjet Phys.-JETP20,1064(1950).

[8]B.Grabiec and M.Matlak,phys.stat.sol.(b)231,299(2002).

[9]M.J.Buckingham and W.M.Fairbank,Progress in Low Temperature Physics,

Ed.C.J.Gorter,vol.3,p.80,North-Holland,Amsterdam1961.

14

HTML标签以及各个标签属性大全(网页制作必备).

取消换行和
相反 总类（所有HTML文件都有的） -------------------------------------------------------------------------------- 文件类型（放在档案的开头与结尾） 文件主题（必须放在「文头」区块内） 文头（描述性资料，像是「主题」） 文体（文件本体） 结构性定义（由浏览器控制的显示风格） -------------------------------------------------------------------------------- 标题（？表示从1到6，有六层选择）数值越高字越小 标题对齐左对齐、居中对齐、右对齐 区分