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On light scattering near phase-transition points in the solid state
Volume 47A, number 4
PHYSICS LETTERS
8 April 1974
ON LIGHT SCATTERING NEAR PHASE-TRANSITION POINTS IN T H E S O L I D S T A T E V.L. GINZBURG P.N. Lebedev Physical Institute Acad. Sci. Moscow, USSR
A.P. LEVANYUK A. V. Shubnikov Institute of Crystallography Acad. ScL Moscow, USSR
Received 24 January 1974 The question of fight scattering near phase transitions and, particularly, near a tricritical point in a solid state is discussed. The question of light scattering near phase-transition points in the solid state and, particularly, of the possibility to observe critical opalescence in this case has been discussed for about twenty years already [ 1-12]. Unfortunately, this problem has been investigated rather slowly both due to experimental difficulties and as a result of a rather peculiar situation. The point is that the light scattering theory which takes no account of shear deformations [ 1 , 3 , 4 - 7 ] leads, particularly, to the conclusion that very strong scattering appears near a tricritieal point. Since, as it seemed this very effect has been observed [2] for the a ~ # transi. tion in quartz and no further experiments have followed for a long time, a stimulus for developing a more detailed theory of scattering in solids has also been missing. Meanwhile the experiments [8] as well as some others give a certain reason to believe that scattering by static inhomogeneities (by the domain boundaries of twins [8], boundaries between a and phases [12] or by some other inhomogeneities - this is not yet clear) rather than a critical opalescence was in fact observed in [2]. Thus, evidently, a suspicion arose if critical opalescence is possible in a solid at all (see particularly [1 1, 12] ). On the other hand, there can be no doubt that the order parameter fluctuations increase near a tricritical point and, particularly, near the a ~ 13transition in quartz X-ray scattering is observed [ 13] and that neutron scattering [ 14] taking place apparently, by thermal fluctuations (this is probably referred also to the light scattering near a transition poiht in SrTiO 3 [10] ). Thus, a question arises about the reason for the difference between our theory [1,4--7] and observations.
It was rather long ago that we noticed that such a reason is apparently the fact that in refs. [1,4-7] account was not taken of the role of shear deformations [15], but more detailed calculations for different specific cases and mainly the comparison of the theory with experiment has not yet completed. However in connection with the appearance of the above-mentioned critical papers We think it reasonable to illuminate the principal side of the problem here. For this purpose, as in our previous papers, let us consider a phase transition described by one order parameter ~/and without an explicit account taken of a crystal anisotropy but taking into account deformations (strain tensor #ik). We shall use here a self-consistent field theory (of the Landau type) which in the zero the approximation does not take fluctuations into account (the character of such an approximation is clear, for example, from refs. [16-18]. Then the expression for the thermodynamical potential has the form 4- °t~2 +13 ')' 6 + r ~ 2 U l l ~=~--]., -~,~4 + -~tl
K 2
1
(1)
1 6(grad ~/)2
+ -~ u II + I't(Uik - ~ ~ik U,lI )2 + -2
where K is the bulk modulus and/a is the shear modulus. In equilibrium (here 131 = 13- 2r2/K; it is assumed that the stresses Oik = a~[aUik = O)
%2 =-131 + (13 27
(2)
Near a tricritical point Ttc, the coefficients ¢~and 131 345
Volume 47A, number 4
PHYSICS LETTERS
are in our approximation proportional to ( T t c - T ) , while 3' = const. From this, rather close to Ttc, we have
*/2 ~ (Ttc_T)l/2.
8 April 1974
quasistatic optical inhomogeneities which are of particular importance also only near a phase transition point.
The average square of the Fourier - component */q, is equal to (for analogous calculations see ref. [4] ;
g
References
2=____~kT{ 2 2 2%(/31-
1~Tql
4otT)l/2+ 16/'w2./2 ~-1 3Kg + 8 q 2 / .(3)
If/a = 0 and T + Ttc, then I*/q 12 ~ ( T t c - T ) -1, the term 8q 2 being ignored; if # :/: 0, then I*/q 12 ( T t e - T) -1/2. At the same time for the case when the connection between Ae and A*/is quadratic the intensity J o f the "first order" scattering [4] in an ordered phase is proportional to ,1o2 I*/q 12 • Under these conditions at/a = 0 the intensity J "~ (Ttc - T) -1/2 but when/a :/: 0 and approaching Ttc the intensity increase is finite. This indeed must take place in quartz. If from symmetry considerations the change in the permeability Ae ~/x*/(see ref. [3] ), then J ~ l*/t/12 ~ (Ttc - T) -1/2 even at # :/:0. Besides, the "second order" scattering, which exists already in a disordered (high-temperature) phase [4], may prove noticeable. The scattering in metastable phases near spinodals, i.e. lines corresponding to the stability limits [15] and the scattering in the presence of stresses (a "clamped" crystal etc) and of an electric field require special consideration. Thus there exist a lot of possibilities, though o f course we cannot, generally speaking, put ~ ---0 in a solid (in distinction, say, from liquid crystals) and such an assumption as a whole leads to an increase in the critical opalescence as compared with that estimated for/a :/: 0. It is an, other question that in this or that case, this opalescence, particularly for first order transitions (though they are close to Ttc) can be masked by the scattering by such
346
[1] V.L. Ginzburg, Dokl. Akad. Nauk SSSR 105 (1955) 240. [2] I.A. Iakovlev, T.S. Velichkina and L.F. Mikheeva. Dokl. Akad. Nauk SSSR 107 (1956) 675; Kristallographiya 1 (1956) 123. Usp. Fiz. Nauk 69 (1957) 411. [3] M.A. Kdvoglaz and S.A. Ribak, Zh. Exp. Teor. Fiz. 33 (1957) 139. [4] V.L. Ginzburg and A.P. Levanyuk, J. Phys. Chem. Solids 6 (1958) 51; Memorial Volume to G.S. Landsberg, Acad. Sci. Publ. House. p. 104, Moscow (1959). [5] V.L. Ginzburg and A.P. Levanyuk, Zh. exp. teor. fiz. 39 (1960) 192. SOy.Phys. JETP 12 (1961) 138. [6] V.L. Ginzburg, Usp. Fiz. Nauk 77 (1962) 621; SOy. Phys. Uspekhi 5 (1963) 649. [7 ] A.P. Levanyuk and Sobyanin, Zh. Exp. Teor. Fiz. 53 (1967) 1024. [8] S.M. Shapiro and H.Z. Cummins, Phys. Rev. Lett 21 (1968) 1578; 19 (1967) 361. [9] I.J. Fritz and H.Z. Cummins, Phys. Rev. Lett. 28 (1972) 96. [10] E.F. Steigmeier, H. Auderset and G. Harbeke, Solid State Commun. 12 (1973) 1077. [11] F.J. Bartis, Phys. Lott. 43A (1973) 61. [121 J. Bartis, J. Phys. C, Solid State Phys. 6 (1973) L295. [13] K. Ishida and G. Honjo, J. Phys. See. Japan 26 (1969) 1558. [14] J.D. Axe and G. Shirane, Phys. Rev. B1 ~1970) 342. [15] A.P. Levanyuk, Zh. exp. teor. fiz., to be published. [16] V.L. Ginzburg, Fiz. Tver. Tela 2 (1960) 2031. Soy. Phys. Solid. State 2 (1960) 1824. [17] E.K. Riedd and F.J. Wegner, Phys. Rev. Lett. 29 (1972) 349. A.M. Goldman, Phys. Rev. Lett. 30 (1973) 1038. [18] R. Bausch, Z. Physik 254 (1972) 81; 258 (1973) 423.
A.A.
PHYSICS LETTERS
8 April 1974
ON LIGHT SCATTERING NEAR PHASE-TRANSITION POINTS IN T H E S O L I D S T A T E V.L. GINZBURG P.N. Lebedev Physical Institute Acad. Sci. Moscow, USSR
A.P. LEVANYUK A. V. Shubnikov Institute of Crystallography Acad. ScL Moscow, USSR
Received 24 January 1974 The question of fight scattering near phase transitions and, particularly, near a tricritical point in a solid state is discussed. The question of light scattering near phase-transition points in the solid state and, particularly, of the possibility to observe critical opalescence in this case has been discussed for about twenty years already [ 1-12]. Unfortunately, this problem has been investigated rather slowly both due to experimental difficulties and as a result of a rather peculiar situation. The point is that the light scattering theory which takes no account of shear deformations [ 1 , 3 , 4 - 7 ] leads, particularly, to the conclusion that very strong scattering appears near a tricritieal point. Since, as it seemed this very effect has been observed [2] for the a ~ # transi. tion in quartz and no further experiments have followed for a long time, a stimulus for developing a more detailed theory of scattering in solids has also been missing. Meanwhile the experiments [8] as well as some others give a certain reason to believe that scattering by static inhomogeneities (by the domain boundaries of twins [8], boundaries between a and phases [12] or by some other inhomogeneities - this is not yet clear) rather than a critical opalescence was in fact observed in [2]. Thus, evidently, a suspicion arose if critical opalescence is possible in a solid at all (see particularly [1 1, 12] ). On the other hand, there can be no doubt that the order parameter fluctuations increase near a tricritical point and, particularly, near the a ~ 13transition in quartz X-ray scattering is observed [ 13] and that neutron scattering [ 14] taking place apparently, by thermal fluctuations (this is probably referred also to the light scattering near a transition poiht in SrTiO 3 [10] ). Thus, a question arises about the reason for the difference between our theory [1,4--7] and observations.
It was rather long ago that we noticed that such a reason is apparently the fact that in refs. [1,4-7] account was not taken of the role of shear deformations [15], but more detailed calculations for different specific cases and mainly the comparison of the theory with experiment has not yet completed. However in connection with the appearance of the above-mentioned critical papers We think it reasonable to illuminate the principal side of the problem here. For this purpose, as in our previous papers, let us consider a phase transition described by one order parameter ~/and without an explicit account taken of a crystal anisotropy but taking into account deformations (strain tensor #ik). We shall use here a self-consistent field theory (of the Landau type) which in the zero the approximation does not take fluctuations into account (the character of such an approximation is clear, for example, from refs. [16-18]. Then the expression for the thermodynamical potential has the form 4- °t~2 +13 ')' 6 + r ~ 2 U l l ~=~--]., -~,~4 + -~tl
K 2
1
(1)
1 6(grad ~/)2
+ -~ u II + I't(Uik - ~ ~ik U,lI )2 + -2
where K is the bulk modulus and/a is the shear modulus. In equilibrium (here 131 = 13- 2r2/K; it is assumed that the stresses Oik = a~[aUik = O)
%2 =-131 + (13 27
(2)
Near a tricritical point Ttc, the coefficients ¢~and 131 345
Volume 47A, number 4
PHYSICS LETTERS
are in our approximation proportional to ( T t c - T ) , while 3' = const. From this, rather close to Ttc, we have
*/2 ~ (Ttc_T)l/2.
8 April 1974
quasistatic optical inhomogeneities which are of particular importance also only near a phase transition point.
The average square of the Fourier - component */q, is equal to (for analogous calculations see ref. [4] ;
g
References
2=____~kT{ 2 2 2%(/31-
1~Tql
4otT)l/2+ 16/'w2./2 ~-1 3Kg + 8 q 2 / .(3)
If/a = 0 and T + Ttc, then I*/q 12 ~ ( T t c - T ) -1, the term 8q 2 being ignored; if # :/: 0, then I*/q 12 ( T t e - T) -1/2. At the same time for the case when the connection between Ae and A*/is quadratic the intensity J o f the "first order" scattering [4] in an ordered phase is proportional to ,1o2 I*/q 12 • Under these conditions at/a = 0 the intensity J "~ (Ttc - T) -1/2 but when/a :/: 0 and approaching Ttc the intensity increase is finite. This indeed must take place in quartz. If from symmetry considerations the change in the permeability Ae ~/x*/(see ref. [3] ), then J ~ l*/t/12 ~ (Ttc - T) -1/2 even at # :/:0. Besides, the "second order" scattering, which exists already in a disordered (high-temperature) phase [4], may prove noticeable. The scattering in metastable phases near spinodals, i.e. lines corresponding to the stability limits [15] and the scattering in the presence of stresses (a "clamped" crystal etc) and of an electric field require special consideration. Thus there exist a lot of possibilities, though o f course we cannot, generally speaking, put ~ ---0 in a solid (in distinction, say, from liquid crystals) and such an assumption as a whole leads to an increase in the critical opalescence as compared with that estimated for/a :/: 0. It is an, other question that in this or that case, this opalescence, particularly for first order transitions (though they are close to Ttc) can be masked by the scattering by such
346
[1] V.L. Ginzburg, Dokl. Akad. Nauk SSSR 105 (1955) 240. [2] I.A. Iakovlev, T.S. Velichkina and L.F. Mikheeva. Dokl. Akad. Nauk SSSR 107 (1956) 675; Kristallographiya 1 (1956) 123. Usp. Fiz. Nauk 69 (1957) 411. [3] M.A. Kdvoglaz and S.A. Ribak, Zh. Exp. Teor. Fiz. 33 (1957) 139. [4] V.L. Ginzburg and A.P. Levanyuk, J. Phys. Chem. Solids 6 (1958) 51; Memorial Volume to G.S. Landsberg, Acad. Sci. Publ. House. p. 104, Moscow (1959). [5] V.L. Ginzburg and A.P. Levanyuk, Zh. exp. teor. fiz. 39 (1960) 192. SOy.Phys. JETP 12 (1961) 138. [6] V.L. Ginzburg, Usp. Fiz. Nauk 77 (1962) 621; SOy. Phys. Uspekhi 5 (1963) 649. [7 ] A.P. Levanyuk and Sobyanin, Zh. Exp. Teor. Fiz. 53 (1967) 1024. [8] S.M. Shapiro and H.Z. Cummins, Phys. Rev. Lett 21 (1968) 1578; 19 (1967) 361. [9] I.J. Fritz and H.Z. Cummins, Phys. Rev. Lett. 28 (1972) 96. [10] E.F. Steigmeier, H. Auderset and G. Harbeke, Solid State Commun. 12 (1973) 1077. [11] F.J. Bartis, Phys. Lott. 43A (1973) 61. [121 J. Bartis, J. Phys. C, Solid State Phys. 6 (1973) L295. [13] K. Ishida and G. Honjo, J. Phys. See. Japan 26 (1969) 1558. [14] J.D. Axe and G. Shirane, Phys. Rev. B1 ~1970) 342. [15] A.P. Levanyuk, Zh. exp. teor. fiz., to be published. [16] V.L. Ginzburg, Fiz. Tver. Tela 2 (1960) 2031. Soy. Phys. Solid. State 2 (1960) 1824. [17] E.K. Riedd and F.J. Wegner, Phys. Rev. Lett. 29 (1972) 349. A.M. Goldman, Phys. Rev. Lett. 30 (1973) 1038. [18] R. Bausch, Z. Physik 254 (1972) 81; 258 (1973) 423.
A.A.