Physics of Scale Activities


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Apr 1965 - Jul 1966

 Neutron diffraction of beta-brass
Dietrich and Als-Nielsen employed neutron-diffraction techniques to study beta-brass (a mixture of copper and zinc). Since each type of atom has different scattering properties, interesting critical scattering characteristics can be observed in the vicinity of the critical temperature. At the time of the Washington conference (April 1965), they indicated excellent agreement with Orstein-Zernike classical theory and "slight but definite disagreement with the Ising model" for short-range order. A year later the focus had shifted to compatibility with the Ising model. For an appropriate long-range order parameter, Dietrich and Als-Nielsen found that the temperature dependence indicated a critical exponent beta = 0.305 +/- 0.005. That result was consistent with what one would expect in a three-dimensional Ising model, the best fit to date for a real physical system.

Primary: O. W. Dietrich and J. Als-Nielsen, "Critical scattering from beta-brass," in in M. Green and J. Sengers, eds., Critical Phenomena: Proceedings of a Conference (Washington, D.C.: National Bureau of Standards, 1966), 144- 149.

J. Als-Nielsen and O. W. Dietrich, "Pair-correlation function in disordered beta-brass as studied by neutron diffraction," Phys. Rev. 153 (1967): 706-710.

O. W. Dietrich and J. Als-Nielsen, "Temperature dependence of short-range order in beta-brass," Phys. Rev. 153 (1967): 711-717.

J. Als-Nielsen and O. W. Dietrich, "Long-range order and critical scattering of neutrons below the transition temperature in beta-brass," Phys. Rev. 153 (1967): 717-721.

Secondary: Heller 1967, 819.
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Apr 21, 1965 

 Narrowing the gap between mean field theory and the Ising model
The central assumption of mean field theory, that one can expand the thermodynamic functions in a series about the critical temperature, falls apart if the functions at the critical point are mathematical singularities. The singularity evident in the rigorous two-dimensional Ising model remained the strongest grounds for rejecting mean field theory near the critical point. Vaks and Larkin suggested that under conditions of sufficient generality, both binary alloys and a Bose gas exhibit the same kinds of singularities that one might expect for a three-dimensional Ising model.

In a subsequent paper with Pikin, they also showed how self- consistent field methods could be used to generate correction terms to the phenomenological theory.

Primary: V. G. Vaks and A. I. Larkin, "On phase transitions of second order," Soviet Physics JETP 22 (1966): 678-687; "O fazovykh perekhodakh vtorogo roda," ZhETF 49 (1965): 975- 989. V. G. Vaks, A. I. Larkin, and S. A. Pikin, "Self-consistent field method for the description of phase transitions," Soviet Physics JETP 24 (1967): 240-249; "O metode samosoglasovannogo polia pri opisanii fazovykh perekhodov," ZhETF 51 (1966): 361-375.

Secondary: Brush 1983, 201. ter Haar 1969, 5.
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Jun 21, 1965 - 1967

 Non-classical equation of state: beginnings of scaling
In 1965 three different sets of authors put forth candidates for a non-classical equation of state valid near the critical point. Domb and Hunter began with a high- temperature expansion of the Ising ferromagnet as a function of magnetic field. Patashinskii and Pokrovskii achieved similar results expressed in terms of multiple spin correlations. Widom found an equation of state for liquids, concluding that the most general form of specific heat singularity would be a logarithmic infinity symmetric with respect to the critical temperature from either side. Within two years the three approaches were shown to be equivalent by Griffiths.

Widom conjectured that a more general relation was obeyed by the relevant variables, provided one expressed the equation of state as a homogeneous function. In Griffiths' formulation for the ferromagnetic case, the magnetization, temperature, and magnetic field may then be equated by a function of a single variable (rather than two). The critical exponents then have definite relations which turn out to hold across diverse phenomenological regimes.

Primary: B. Widom, "Surface tension and molecular correlations near the critical point," J. Chem. Phys. 43 (1965): 3892-3897. B. Widom, "Equation of state in the neighborhood of the critical point," J. Chem. Phys. 43 (1965): 3898-3905. A. Z. Patashinskii and V. L. Pokrovskii, "Behavior of ordered systems near the transition point," Soviet Physics JETP 23 (1966): 292-297; "O povedenii uporiadochivaiushchikhsia sistem vblizi tochki fazovogo perekhoda," ZhETF 50 (1966): 439-447. C. Domb and D. L. Hunter, "On the critical behaviour of ferromagnets," Proc. Phys. Soc. 86 (1965): 1147-1151. R. B. Griffiths, "Thermodynamic functions for fluids and ferromagnets near the critical point," Phys. Rev. 158 (1967): 176-187.

Secondary: Domb 1996, 219. Hoddeson et al. 1992, 582. Patashinskii and Pokrovskii 1979, 60.
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Jan 1966 - Feb 1966

 Kadanoff and scaling laws
Contemporaneously with Widom, Domb and Hunter, and Patashinskii and Pokrovskii, Kadanoff develops similar ideas about the relations among the critical exponents. What had been for Widom, in a manner of speaking, plausibility arguments, were elevated by Kadanoff into the so-called "scaling laws."

Primary: L. P. Kadanoff, "Spin-spin correlations in the two- dimensional Ising model," Nuovo Cimento 44B (1966): 276-305. L. P. Kadanoff, "Scaling laws for Ising models near T[sub c]," Physics 2 (1966): 263-272.

Secondary: Brush 1983, 257. Fisher 1967, 675. Kadanoff et al. 1967, 403.
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Jun 20, 1966 - Jul 29, 1966

 Brandeis Summer School
Several of the key players in the interdisciplinary dialogue on critical phenomena give extended lecture series in June-July 1966 at the ninth Brandeis Summer Institute in Theoretical Physics. Lecturers include Benedek, Brout, Chase, Dyson, Kac, Kadanoff, Lax, Montroll, and Vinen.

M. Chrétien, E. P. Gross and S. Deser, eds., Statistical Physics, Phase Transitions and Superfluidity (Brandeis University Summer Institute in Theoretical Physics, 1966), 2 vols. (New York: Gordon and Breach, 1968).
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1967 

 Quantizing gauge fields: functional integration method
Popov and Faddeev provide rules for quantizing gauge fields in the functional integration method.

V.N. Popov and L.D. Faddeev, Preprint, Institute of Theoretical Physics, Academy of Sciences of the Ukrainian SSR, Kiev; Phys. Lett. 85 (1967): 30.
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1969 

 Faddeev extends Dirac's generalized Hamiltonian dynamics
Faddeev develops a general quantization method by means of the functional integration technique for the generalized Hamiltonian dynamics developed previously by Dirac.

L. D. Faddeev, "The Feynman integral for singular Lagrangians," Teor. Mat. Fiz. 1 (1969): 3; Theor. Math. Phys. (USSR) 1 (1969): 1.
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1971 - Jan 01, 1971

 Poincare group to supersymmetry group
Golfand and Likhtman elaborate algebra of the supersymmetry group as an expansion of the algebra of the Poincare group.

Ia. A. Gol'fand and E. P. Likhtman, Pis'ma ZhETF 13 (1971): 452; JETP Letters 13 (1971): 323.
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1971 

 Regularization by higher-order derivatives
Slavnov introduces regularization by means of higher-order derivatives.

A.A. Slavnov, "Invariant regularization of non-linear chiral theories," Nucl. Phys. B31 (1971): 301-315.
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1972 - Jan 01, 1972

 Generalized Ward identities
Slavnov renormalizes the Yang-Mills theory incorporating the case of a spontaneously broken symmetry, describing generalized Ward identities in the process.

A.A. Slavnov, Teor. Mat. Fiz. 10 (1972): 153; 13 (1972): 174; Theor. Math. Phys. USSR 10 (1972): 99; 13 (1972): 1064.
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1974 

 Monopoles
't Hooft and Poliakov independently discuss monopoles.

't Hooft, Nucl. Phys. B79 (1974): 276.

A.M. Poliakov, Pis'ma ZhETF 20 (1974): 430; "Particle spectrum in quantum field theory," JETP Lett. 20 (1974): 194-195.
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1975 - Jan 01, 1975

 Instantons
Who names the instanton?

A.A. Belavin, A.M. Poliakov, A.S. Schwartz, Iu.S. Tiupkin, Phys. Lett. B59 (1975): 85.
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1983 - Jan 01, 1983

 Electrical neutrality of matter and GUTs
Okun, Voloshin, and Zakharov argue that the electrical neutrality of matter follows in a natural way from grand unification.

L.B. Okun, M.B. Voloshin, and V.I. Zakharov, Phys. Lett. B138 (1983): 115.
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