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Similarities and contrasts between critical point behavior of heavy fluid alkalis and d-dimensional Ising model
Physics Letters A 378 (2014) 254–256
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Similarities and contrasts between critical point behavior of heavy fluid alkalis and d-dimensional Ising model N.H. March a,b,c,∗ a b c
Abdus Salam International Centre for Theoretical Physics, Trieste, Italy Department of Physics, University of Antwerp, Antwerp, Belgium University College, Oxford University, Oxford, England
a r t i c l e
i n f o
Article history: Received 15 October 2013 Accepted 22 October 2013 Available online 25 October 2013 Communicated by V.M. Agranovich
a b s t r a c t A brief summary is first given of recent progress in establishing the near-critical point behavior of the fluid alkalis Rb and Cs. Departure from the law of Rectilinear Diameters is emphasized, along with its consequences for theories emphasizing homogeneity and scaling. The behavior as the critical points of Rb and Cs are approached is compared and contrasted with the d-dimensional Ising model. © 2013 Elsevier B.V. All rights reserved.
Keywords: Critical exponents Heavy fluid alkalis Ising model
ζ =1+
1. Background Though most of the attention will be focused on the behavior of the heavy fluid alkalis Rb and Cs near their critical points, we shall also make contact with critical point properties of the d-dimensional Ising model. Starting therefore with fluid alkali metals Rb and Cs, we note the unorthodox behavior of the liquid–vapor coexistence curve (LVCC). Here, the key experimental data comes from Jüngst et al. [1]. We follow Leys et al. [2], see also [3], in plotting the measurements of Jüngst et al. [1], where the reduced difference density is defined by
=
ρl − ρ v ρc
(1)
where ρl , ρ v and ρc respectively represent the density in liquid, vapor and at the critical point. The scaled average density is defined by
ζ=
ρl + ρ v 2ρc
(2)
which implies ζ at the critical point equals unity. As Leys et al. [2,3] stress already, the curve labeled 2 is accurately linear. Motivated by early work of Guggenheim [4], March et al. [5] obtained the relation
*
Correspondence to: Department of Physics, University of Antwerp, Antwerp, Belgium. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.10.030
6 243
3
(3)
for a number of insulating liquids. Fig. 8.1 of Leys [3] reproduces the plot of Leys et al. [2] of ζ vs 3 for Rb and, as these authors stress, the shape found by Guggenheim [4] is not reproduced for metallic Rb. This was also found to be the case from the available experimental data of Jüngst et al. [1] on Cs. Leys et al. [2,3] therefore wrote a generalization of Eq. (3) having the form
ζ = 1 + κ Γ .
(4)
As discussed above, Leys et al. [2,3] found the correct choice of Γ was 2 for Rb and Cs, whereas for the insulating fluids Γ = 3 works well. Generalizing the work of March et al. [5], it is easy to verify that both the above results for the heavy alkalis and the insulating fluids can be subsumed into the differential equation
μ
d2 dζ 2
2 d + (Γ − 1) =0 dζ
(5)
with the appropriate choice of Γ as discussed above. Leys et al. [2,3] then examined first the way tends to zero at the critical point, and found empirically that the data for Cs was well fitted by the form
1 T 3 = const × 1 − . Tc
(6)
From experiments on insulating liquids an exponent having the value (0.35 ± 0.02) was found (see, e.g. [6]). Therefore, with account taken of experimental error, the way the difference density
N.H. March / Physics Letters A 378 (2014) 254–256
varies as the critical temperature is approached is not significantly changed in going from insulators to the metallic fluids Rb and Cs. As the departure from the law of Rectilinear Diameters (see, e.g. [7]) is a major point stressed below for Rb and Cs, with Γ = 2 inserted into Eq. (4) one is then led to the prediction for ζ [2,3]
ζ = 1 + const × 1 −
T
23 (7)
Tc
while the ‘law’ of Rectilinear Diameters implies that
T . ζ = 1 + const × 1 −
(8)
Tc
As we shall emphasize in the following section, this affects important issues like homogeneity and scaling in currently formulated versions of the theory of approach to the critical point, such as discussed, for example, in the elegant study of Hubbard and Schofield [8]. As Hubbard and Schofield stated [8], the result (Eq. (4) in [8]) obtained by applying the Wilson theory to a liquid– vapor critical point implies the law of Rectilinear Diameters. However, these results only hold for small enough for -expansions; their application to the interesting case d = 3, = 1 is problematic. 2. Long-range forces: do they alter scaling laws for critical exponents or only cause departures from the law of rectilinear diameters? Section 1 has presented a summary of existing knowledge on the behavior of the heavy fluid alkalis Rb and Cs, near their critical points. The purpose of this section is to point out the need to generalize the Hubbard–Schofield homogeneity relation proposed [8] about the liquid–vapor critical point. As these authors emphasize at the outset of their presentation, their treatment is appropriate to treat identical classical nuclei interacting by short-range pair potentials. Therefore, in writing their prediction for the density ρ in their Eq. (4), they note that they have assimilated the Law of Rectilinear Diameters (LRD), which we have stated explicitly in the previous section. But for fluid Rb and Cs, we have emphasized above that this Law is not obeyed. This is doubtless because, in such alkali metals, as was emphasized in the theoretical work of Corless and March [9] and of Worster and March [10] the range of the pair potential φ(r ) is very long-range because of the sharp spherical Fermi surface of radius kF . Then at sufficiently large r, one expects from the above studies that
φ(r ) ∼
A cos(2k F r + χ ) r3
(9)
where χ is a phase shift. This violates the criterion Hubbard and Schofield [8] set down for their theory to apply. Not by any means all of their liquid critical-point theory is affected. However, after mentioning the LRD, they derive from their density proposal, among other things, the scaling relation
γ = (2 − η)ν .
(10)
We single this out, especially by appealing to the book of Cardy [11] in which he stresses that care is needed as to the regime of applicability of Eq. (10). The relation may then be inappropriate for liquid Rb and Cs near their critical points. Unfortunately, at the present time this line of reasoning does not enable us to make even approximate predictions for the critical exponents of liquid metals Rb and Cs. Therefore, we have been led to invoke again the so-called one-parameter model of
255
Table 1 Critical exponents for ILM model [12] for β chosen as 1/3 and taking νI ≡ νILM = 3/10, Zhang’s predictions [13] for 3d Ising model and Fisher’s series expansion [16] for 3d Ising model.
ILM model for β chosen as 1/3 and taking νI = 3/10 yielding Zhang: 3d Ising Fisher’s series expansion: 3d Ising
α
β
γ
δ
η
ν
1/3
1/3
1
4
1/5
5/9
0 1/8
3/8 5/16
5/4 5/4
13/3 5
1/8 0
2/3 5/8
Iqbal, Liu and Mezei (ILM, below) [12], derived using AdS/CFT duality. Recently, we have discussed this model in relation to the d-dimensional Ising case [13]. We showed there that γ was always equal to unity – the mean-field value – whatever the choice of the one-parameter, denoted νI below to distinguish it from the critical exponent ν . However, the variation of νI alters the other critical exponents, and we shall therefore make a choice of β . Jüngst et al. obtained β = 0.355 ± 0.01 for Cs, and β = 0.360 ± 0.01 for Rb by fitting their data by Eqs. (1) and (2) in their paper [1], which are close to Zhang’s prediction β = 3/8 [13,14]. They thought that this is the 2β anomaly, since it differs with the theoretically predicted exponent values β = 0.327 (obtained by Renormalization Group or Monte Carlo). This anomaly was first attributed to difficulty to deduce an exact exponent value from their experimental data, because of the severe experimental problems connected with the high critical temperatures of metals. Then they attempted to fit their data with the theoretically predicted exponent values β = 0.327. Although it is also with excellent fits, they immediately stated that this procedure is insufficient to rule out the 2β anomaly. In our opinion, the fitting of the experimental data has been misled by such the theoretically predicted exponent value (β = 0.327), not only in Jüngst et al.’s paper [1], but also in many other works. The so-called 2β anomaly is not an anomaly, but accurate experimental result close to the exact solution. We suggest checking carefully all the experimental data with such 2β anomaly reported in literatures and fitting them by using Zhang’s predictions. In what follows, we used β = 0.327 ≈ 1/3, and to find from the ILM relation that
β=
1 2
− νILM 2νILM
(11)
that νI for this choice is in fact 3/10, which is recorded in the first row of Table 1. δ is then obtained from the ILM result that
δ=
1 2 1 2
+ νILM − νILM
(12)
the result δ = 4 being also entered in Table 1. This, combined with the entries in Table 1, shows that the usual scaling relation
γ = (δ − 1)β
(13)
is indeed satisfied. If we invoke the Hubbard–Schofield relation [8]
δ=
d + (2 − η) d − (2 − η)
(14)
(see also [15]), putting d = 3, we readily find η as 1/5. To get ν , we invoke the Josephson relation ν = (2 − α )/d and then we obtain ν = 5/9 for d = 3 which completes the first row of Table 1. For comparison, row 2 shows Zhang’s predictions of the 3d Ising model [13]. For comparison, we also list the critical exponents obtained
256
N.H. March / Physics Letters A 378 (2014) 254–256
by Fisher’s series expansion [16] in the third row of Table 1. We have shown in [15] that Onsager’s exact exponents for the 2d Ising model and Zhang’s predictions for the critical exponents of the 3d Ising model all satisfy this Hubbard–Schofield relation. In summary, we have added to present knowledge on the behavior of Rb and Cs near their critical point and proposed the critical exponents recorded in row 1 of Table 1. It will be an interesting matter for the future to see experimentally how the critical exponents for Rb and Cs compare with these three rows in Table 1. Acknowledgements The author’s thanks are due to Dr. Z.D. Zhang for much invaluable collaboration in this general area. However, the present author has the sole responsibility for the present Letter. NHM wishes to thank Professors D. Lamoen and C. Van Alsenoy for making possible his continuing affiliation with the University of Antwerp.
References [1] S. Jüngst, B. Knuth, F. Hensel, Phys. Rev. Lett. 55 (1985) 2160. [2] F.E. Leys, N.H. March, G.G.N. Angilella, M.L. Zhang, Phys. Rev. B 66 (2002) 073314. [3] F.E. Leys, Inhomogeneous electron liquid theory applied to metallic phases and nanostructures, Doctoral Thesis, University of Antwerp, 2003. [4] E.A. Guggenheim, Thermodynamics, North-Holland, Amsterdam, 1967. [5] N.H. March, M.P. Tosi, R.G. Chapman, Phys. Chem. Liq. 18 (1988) 195. [6] N.H. March, Liquid Metals: Concepts and Theory, Cambridge University Press, Cambridge, 1990. [7] J.S. Rowlinson, Nature 319 (1986) 362. [8] J. Hubbard, P. Schofield, Phys. Lett. A 40 (1972) 245. [9] G.K. Corless, N.H. March, Philos. Mag. 6 (1961) 1285. [10] J. Worster, N.H. March, J. Phys. Chem. Solids 24 (1963) 1305. [11] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1990. [12] N. Iqbal, H. Liu, M. Mezei, in: M. Dine, T. Banks, S. Sachdev (Eds.), String Theory and Its Applications, World Scientific, Singapore, 2012, p. 707, Chapter 13. [13] Z.D. Zhang, Philos. Mag. 87 (2007) 5309. [14] Z.D. Zhang, Chin. Phys. B 22 (2013) 030513. [15] Z.D. Zhang, N.H. March, Phys. Chem. Liq. 51 (2013) 261. [16] M.E. Fisher, Rep. Prog. Phys. 30 (1967) 615.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Similarities and contrasts between critical point behavior of heavy fluid alkalis and d-dimensional Ising model N.H. March a,b,c,∗ a b c
Abdus Salam International Centre for Theoretical Physics, Trieste, Italy Department of Physics, University of Antwerp, Antwerp, Belgium University College, Oxford University, Oxford, England
a r t i c l e
i n f o
Article history: Received 15 October 2013 Accepted 22 October 2013 Available online 25 October 2013 Communicated by V.M. Agranovich
a b s t r a c t A brief summary is first given of recent progress in establishing the near-critical point behavior of the fluid alkalis Rb and Cs. Departure from the law of Rectilinear Diameters is emphasized, along with its consequences for theories emphasizing homogeneity and scaling. The behavior as the critical points of Rb and Cs are approached is compared and contrasted with the d-dimensional Ising model. © 2013 Elsevier B.V. All rights reserved.
Keywords: Critical exponents Heavy fluid alkalis Ising model
ζ =1+
1. Background Though most of the attention will be focused on the behavior of the heavy fluid alkalis Rb and Cs near their critical points, we shall also make contact with critical point properties of the d-dimensional Ising model. Starting therefore with fluid alkali metals Rb and Cs, we note the unorthodox behavior of the liquid–vapor coexistence curve (LVCC). Here, the key experimental data comes from Jüngst et al. [1]. We follow Leys et al. [2], see also [3], in plotting the measurements of Jüngst et al. [1], where the reduced difference density is defined by
=
ρl − ρ v ρc
(1)
where ρl , ρ v and ρc respectively represent the density in liquid, vapor and at the critical point. The scaled average density is defined by
ζ=
ρl + ρ v 2ρc
(2)
which implies ζ at the critical point equals unity. As Leys et al. [2,3] stress already, the curve labeled 2 is accurately linear. Motivated by early work of Guggenheim [4], March et al. [5] obtained the relation
*
Correspondence to: Department of Physics, University of Antwerp, Antwerp, Belgium. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.10.030
6 243
3
(3)
for a number of insulating liquids. Fig. 8.1 of Leys [3] reproduces the plot of Leys et al. [2] of ζ vs 3 for Rb and, as these authors stress, the shape found by Guggenheim [4] is not reproduced for metallic Rb. This was also found to be the case from the available experimental data of Jüngst et al. [1] on Cs. Leys et al. [2,3] therefore wrote a generalization of Eq. (3) having the form
ζ = 1 + κ Γ .
(4)
As discussed above, Leys et al. [2,3] found the correct choice of Γ was 2 for Rb and Cs, whereas for the insulating fluids Γ = 3 works well. Generalizing the work of March et al. [5], it is easy to verify that both the above results for the heavy alkalis and the insulating fluids can be subsumed into the differential equation
μ
d2 dζ 2
2 d + (Γ − 1) =0 dζ
(5)
with the appropriate choice of Γ as discussed above. Leys et al. [2,3] then examined first the way tends to zero at the critical point, and found empirically that the data for Cs was well fitted by the form
1 T 3 = const × 1 − . Tc
(6)
From experiments on insulating liquids an exponent having the value (0.35 ± 0.02) was found (see, e.g. [6]). Therefore, with account taken of experimental error, the way the difference density
N.H. March / Physics Letters A 378 (2014) 254–256
varies as the critical temperature is approached is not significantly changed in going from insulators to the metallic fluids Rb and Cs. As the departure from the law of Rectilinear Diameters (see, e.g. [7]) is a major point stressed below for Rb and Cs, with Γ = 2 inserted into Eq. (4) one is then led to the prediction for ζ [2,3]
ζ = 1 + const × 1 −
T
23 (7)
Tc
while the ‘law’ of Rectilinear Diameters implies that
T . ζ = 1 + const × 1 −
(8)
Tc
As we shall emphasize in the following section, this affects important issues like homogeneity and scaling in currently formulated versions of the theory of approach to the critical point, such as discussed, for example, in the elegant study of Hubbard and Schofield [8]. As Hubbard and Schofield stated [8], the result (Eq. (4) in [8]) obtained by applying the Wilson theory to a liquid– vapor critical point implies the law of Rectilinear Diameters. However, these results only hold for small enough for -expansions; their application to the interesting case d = 3, = 1 is problematic. 2. Long-range forces: do they alter scaling laws for critical exponents or only cause departures from the law of rectilinear diameters? Section 1 has presented a summary of existing knowledge on the behavior of the heavy fluid alkalis Rb and Cs, near their critical points. The purpose of this section is to point out the need to generalize the Hubbard–Schofield homogeneity relation proposed [8] about the liquid–vapor critical point. As these authors emphasize at the outset of their presentation, their treatment is appropriate to treat identical classical nuclei interacting by short-range pair potentials. Therefore, in writing their prediction for the density ρ in their Eq. (4), they note that they have assimilated the Law of Rectilinear Diameters (LRD), which we have stated explicitly in the previous section. But for fluid Rb and Cs, we have emphasized above that this Law is not obeyed. This is doubtless because, in such alkali metals, as was emphasized in the theoretical work of Corless and March [9] and of Worster and March [10] the range of the pair potential φ(r ) is very long-range because of the sharp spherical Fermi surface of radius kF . Then at sufficiently large r, one expects from the above studies that
φ(r ) ∼
A cos(2k F r + χ ) r3
(9)
where χ is a phase shift. This violates the criterion Hubbard and Schofield [8] set down for their theory to apply. Not by any means all of their liquid critical-point theory is affected. However, after mentioning the LRD, they derive from their density proposal, among other things, the scaling relation
γ = (2 − η)ν .
(10)
We single this out, especially by appealing to the book of Cardy [11] in which he stresses that care is needed as to the regime of applicability of Eq. (10). The relation may then be inappropriate for liquid Rb and Cs near their critical points. Unfortunately, at the present time this line of reasoning does not enable us to make even approximate predictions for the critical exponents of liquid metals Rb and Cs. Therefore, we have been led to invoke again the so-called one-parameter model of
255
Table 1 Critical exponents for ILM model [12] for β chosen as 1/3 and taking νI ≡ νILM = 3/10, Zhang’s predictions [13] for 3d Ising model and Fisher’s series expansion [16] for 3d Ising model.
ILM model for β chosen as 1/3 and taking νI = 3/10 yielding Zhang: 3d Ising Fisher’s series expansion: 3d Ising
α
β
γ
δ
η
ν
1/3
1/3
1
4
1/5
5/9
0 1/8
3/8 5/16
5/4 5/4
13/3 5
1/8 0
2/3 5/8
Iqbal, Liu and Mezei (ILM, below) [12], derived using AdS/CFT duality. Recently, we have discussed this model in relation to the d-dimensional Ising case [13]. We showed there that γ was always equal to unity – the mean-field value – whatever the choice of the one-parameter, denoted νI below to distinguish it from the critical exponent ν . However, the variation of νI alters the other critical exponents, and we shall therefore make a choice of β . Jüngst et al. obtained β = 0.355 ± 0.01 for Cs, and β = 0.360 ± 0.01 for Rb by fitting their data by Eqs. (1) and (2) in their paper [1], which are close to Zhang’s prediction β = 3/8 [13,14]. They thought that this is the 2β anomaly, since it differs with the theoretically predicted exponent values β = 0.327 (obtained by Renormalization Group or Monte Carlo). This anomaly was first attributed to difficulty to deduce an exact exponent value from their experimental data, because of the severe experimental problems connected with the high critical temperatures of metals. Then they attempted to fit their data with the theoretically predicted exponent values β = 0.327. Although it is also with excellent fits, they immediately stated that this procedure is insufficient to rule out the 2β anomaly. In our opinion, the fitting of the experimental data has been misled by such the theoretically predicted exponent value (β = 0.327), not only in Jüngst et al.’s paper [1], but also in many other works. The so-called 2β anomaly is not an anomaly, but accurate experimental result close to the exact solution. We suggest checking carefully all the experimental data with such 2β anomaly reported in literatures and fitting them by using Zhang’s predictions. In what follows, we used β = 0.327 ≈ 1/3, and to find from the ILM relation that
β=
1 2
− νILM 2νILM
(11)
that νI for this choice is in fact 3/10, which is recorded in the first row of Table 1. δ is then obtained from the ILM result that
δ=
1 2 1 2
+ νILM − νILM
(12)
the result δ = 4 being also entered in Table 1. This, combined with the entries in Table 1, shows that the usual scaling relation
γ = (δ − 1)β
(13)
is indeed satisfied. If we invoke the Hubbard–Schofield relation [8]
δ=
d + (2 − η) d − (2 − η)
(14)
(see also [15]), putting d = 3, we readily find η as 1/5. To get ν , we invoke the Josephson relation ν = (2 − α )/d and then we obtain ν = 5/9 for d = 3 which completes the first row of Table 1. For comparison, row 2 shows Zhang’s predictions of the 3d Ising model [13]. For comparison, we also list the critical exponents obtained
256
N.H. March / Physics Letters A 378 (2014) 254–256
by Fisher’s series expansion [16] in the third row of Table 1. We have shown in [15] that Onsager’s exact exponents for the 2d Ising model and Zhang’s predictions for the critical exponents of the 3d Ising model all satisfy this Hubbard–Schofield relation. In summary, we have added to present knowledge on the behavior of Rb and Cs near their critical point and proposed the critical exponents recorded in row 1 of Table 1. It will be an interesting matter for the future to see experimentally how the critical exponents for Rb and Cs compare with these three rows in Table 1. Acknowledgements The author’s thanks are due to Dr. Z.D. Zhang for much invaluable collaboration in this general area. However, the present author has the sole responsibility for the present Letter. NHM wishes to thank Professors D. Lamoen and C. Van Alsenoy for making possible his continuing affiliation with the University of Antwerp.
References [1] S. Jüngst, B. Knuth, F. Hensel, Phys. Rev. Lett. 55 (1985) 2160. [2] F.E. Leys, N.H. March, G.G.N. Angilella, M.L. Zhang, Phys. Rev. B 66 (2002) 073314. [3] F.E. Leys, Inhomogeneous electron liquid theory applied to metallic phases and nanostructures, Doctoral Thesis, University of Antwerp, 2003. [4] E.A. Guggenheim, Thermodynamics, North-Holland, Amsterdam, 1967. [5] N.H. March, M.P. Tosi, R.G. Chapman, Phys. Chem. Liq. 18 (1988) 195. [6] N.H. March, Liquid Metals: Concepts and Theory, Cambridge University Press, Cambridge, 1990. [7] J.S. Rowlinson, Nature 319 (1986) 362. [8] J. Hubbard, P. Schofield, Phys. Lett. A 40 (1972) 245. [9] G.K. Corless, N.H. March, Philos. Mag. 6 (1961) 1285. [10] J. Worster, N.H. March, J. Phys. Chem. Solids 24 (1963) 1305. [11] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1990. [12] N. Iqbal, H. Liu, M. Mezei, in: M. Dine, T. Banks, S. Sachdev (Eds.), String Theory and Its Applications, World Scientific, Singapore, 2012, p. 707, Chapter 13. [13] Z.D. Zhang, Philos. Mag. 87 (2007) 5309. [14] Z.D. Zhang, Chin. Phys. B 22 (2013) 030513. [15] Z.D. Zhang, N.H. March, Phys. Chem. Liq. 51 (2013) 261. [16] M.E. Fisher, Rep. Prog. Phys. 30 (1967) 615.