Critical Exponents

Critical Exponents

From P. Chaikin and T Lubensky“Principles of Condensed Matter Physics”

Notice that convention allows for different exponents on either side of the transition, but often these are found to be the same.

Universality Classes

From P. Chaikin and T Lubensky “Principles of Condensed Matter Physics”

Theory suggests that the class (i.e. set of exponents) depends on spatial dimensionality, symmetry of the order parameter and interaction (and range of the latter as well) but not on the detailed form or strength of the interactions

Ising Model• Consider a lattice on which each site is occupied by either a + or a – (up

or down spin to model magnetism, A or B element to model a binary alloy etc.).

• Label each such state as si (for site I, two possible values).• We assume ONLY nearest-neighbor interactions, and describe that

interaction with a single energy scale J.• The total configurational energy is then: E = -J Snn(si sj)• In this model J>0 suggests like neighbors are preferred (lower energy if si

and sj are of the same sign)• Exact solutions have been found for 1 and 2 dimensions, not yet for 3

dimensions.• Applications:

• Magnetism (both ferromagnetism and antiferromagnetism)• Binary alloys while assuming random arrangements of atoms (Bragg-

Williams model) shows phase separation for J>0.• Binary alloys with correlations between bonding and configuration

treated via the law of mass action (i.e. bonds forming and breaking; the “Quasi-chemical” approximation) can show order-disorder transitions as well as phase separation etc.

Ising Modelhttp://webphysics.davidson.edu/applets/ising/default.html

For an interesting view of a dynamic realization of the Ising model look at: http://physics.ucsc.edu/~peter/ising/ising.html

or: http://www.pha.jhu.edu/~javalab/ising/ising.html

Order-Disorder Transition: Brass

http://cimewww.epfl.ch/people/cayron/Fichiers/thesebook-chap5.pdf

Order-Disorder Transition


In this case, once any site has a slightly higher probability for having Cu (or Zn) rather than the other, the symmetry is broken, new Bragg peaks start to appear, and the order parameter is non-zero.


b b’

Renormalization GroupBlock Spins (Kadanoff)

In CMP, you are usually interested in this sort of procedure to connect interactions at microscopic length scales to phenomena at larger length scales (say near critical points).In HEP, you use similar procedures to connect phenomena observed at large length scales to fundamental interactions at much smaller length scales (to connect dressed electrons to fundamental electrons with vacuum fluctuations providing the “dressing”

http://en.wikipedia.org/wiki/Renormalization_group

Spin systems at Temperature

A

B

Which of the two above spin systems is at the lower temperature?

Spin systems at Temperature

A

B

Which of the two above spin systems is at the lower temperature?

You can’t say, it depends on what the applied field is (or more precisely, on whether the two are in the same applied field). The picture itself can only tell you which is at the greater value of mB/kBT !

Adiabatic Demagnetization

From Zemansky(a schematic representation of A.D)

From LounasmaaThe experimental entropy curves for several materials actually used for this technique.

Carnot CycleWhat happens if you run this cycle backwards?

Carnot CycleWhat happens if you run this cycle backwards?A. Compress at TH, forcing heat

out into the hot reservoir and reducing the entropy of the working fluid.

B. Cool at constant entropy by expanding slightly adiabatically

C. Expand in contact with the cold object, decreasing its entropy and increasing that of the working fluid.

D. Compress adiabatically, increasing the temperature back to TH.

A

B

C

D

Essentially, you are pumping entropy from the lower temperature reservoir to the higher temperature reservoir, by adding enough work (extra energy) to satisfy the second law.

Adiabatic Demagnetization

From Zemansky(a schematic representation of A.D)

From LounasmaaThe experimental entropy curves for several materials actually used for this technique.

Third law of Thermodynamics1. Nernst: It is impossible by any procedure, however idealized, to reduce the

temperature of any system to absolute zero in a finite number of operations2. The entropy of any system as the property that its limit as T->0+, S->So

where So is a constant independent of all parameters of that particular system.

3. The entropy goes to zero as T-> 0.

The last of these clearly follows from the definition of entropy if we can assume that the quantum mechanical ground state of the system is non-degenerate. There are some systems (e.g. glasses, and some other systems), where it is not clear that this is true. The second form is often of more practical use in terms of limiting the behavior of a system’s properties; e.g. in trying to cool a system down there may be lots of residual entropy in the nuclear spin system but it may be completely decoupled from the properties you are interested in (so in this case, the practical limit of So may be non-zero).

A collection of cooling methods1. Contact with something cold:

a) Ice baths (273 K for pure water, colder for salt baths etc.).b) Dry Ice (195 K at atmospheric pressure)c) Liquid nitrogen (77 K at atmospheric pressure)d) Liquid Helium (4.2 K at atmospheric pressure)e) Closed-cycle refrigerators (down to 1.5K now).f) Moderation of neutronsg) Cooling of the proton beam in IUCF’s “cooler”

2. Evaporationa) This is how you keep cool in the summerb) Pumping on any of the baths mentioned above will reduce the temperature

(why?) For He, the base temperature reached is about 1 to 1.5 K depending on how you design the cryostat and the size of your pump. For 3He, the base temperature is about 350 mK. (why?)

c) This is used to get to nK in laser cooled rarefied gas experiments.3. Joule-Thomson expansion.4. Adiabatic Demag (see above; mK with atomic moments, <mK with nuclear)5. Dilution refrigerator

a) This uses phase separation in 3He/4He mixtures to “evaporate” concentrated 3He into a dilute 3He/4He phase at temperatures as low as a few mK.

6. Laser cooling

This is a very important process in the history of low-temperature physics, as it is the primary technique used to liquify gases (Linde and Air Liquide use it a lot!!). A short argument that we won’t bother going through will convince you that the idealized process is one that takes place at constant ENTHALPY. What change in temperature is effected for an ideal gas as a result of a given change in pressure through this process?

Inversion curves in JT-expansion

From NASA tech. note D-6807 (1972)

Lines of constant Enthalpy

Max Inversion Temps:Nitrogen: 640 KHydrogen: 193KHelium: 51K

Pomeranchuk Cooling

From:http://ultracold.uchicago.edu/homepage/gustavo.pdf

Why does the liquid have lower entropy than the solid in this case (and why is this not the case for 4He)?

He Vapor Pressure

T (K)0 1 2 3 4 5

P (T

orr)

0.01

0.1

1

10

100

1000

10000

4He3He

3He/4He Dilution Refrigerator

From Lounasmaa “Experimental Principles and Methods below 1 K.”NOTE: The mixture case has significant cooling power at temperatures much below the limit for a 3He fridge alone (figure on left). This is because the dilution fridge does not rely on liberating liquid 3He atoms into the gas phase; rather they get “liberated” into a dilute phase of 3He atoms in a solvent of liquid 4He (see the phase diagram on the right).

3He/4He Dilution RefrigeratorFrom Lounasmaa “Experimental Principles and Methods below 1 K.”1. 4He at about 1 K is used to condense the

mixture2. The mixture fills the “mixing chamber”

“still” and heat exchangers, and phase-separates once the temperature falls below about 500 mK (reached by pumping on the mixture).

3. The dilute phase concentrates at the bottom (since it is more dense than pure 3He).

4. You then HEAT up the still raising the vapour pressure of 3He in the dilute phase to the point where you can pump a reasonable amount of it.

5. You then circulate 3He as the refrigerant, and the mixing chamber is cooled much like an evaporation refrigerator or dissolving salt into water for making ice cream.

3He/4He Dilution Refrigeratorhttp://na47sun05.cern.ch/target/outline/dilref.html

1. The mixture fills the “mixing chamber” “still” and heat exchangers, and phase-separates once the temperature falls below about 500 mK (reached by pumping on the mixture).

2. The dilute phase concentrates at the bottom (since it is more dense than pure 3He).

3. You then HEAT up the still raising the vapour pressure of 3He in the dilute phase to the point where you can pump a reasonable amount of it.

4. You then circulate 3He as the refrigerant, and the mixing chamber is cooled much like an evaporation refrigerator or dissolving salt into water for making ice cream.

Laser Cooling: Optical Molasses

Optical Molasses demonstration and public explanations at:http://nobelprize.org/nobel_prizes/physics/laureates/1997/illpres/trapping.htmlhttp://www.colorado.edu/physics/2000/bec/lascool4.html

Phys. Today Oct. 1990 Phillips, Cohen-Tannoudji

Naïve expectations predicted a minimum temperature to be reached the “Doppler minimum” of somewhere around 200 mK (depending on the atom etc., but independent of the laser power (at least for low enough power)

Laser Cooling: Sisyphus effect

Even before 1990 this effect had been used to achieve temperatures as low as 2.6 mK in Cs (almost 2 orders of magnitude below the “Doppler limit”)


Time –of-flight technique used to more accurately measure the temperature of the cooled atoms at NIST.

Laser Cooling: Sisyphus effect

This linear dependence of the ultimate temperature on the ratio of the laser intensity to the “detuning” demonstrated rather conclusively that this sisyphus mechanism was involved.


Note the temperatures reached !!

Review for the Final• The Final Exam will be next Friday 6 May at 12:30 to 2:30 PM in SW 218.• OFFICE HOURS:

• Wednesday: 1:30 – 3:00• Thursday: 1:30 – 3:30• Friday: 9:30-11:00

• The exam will be comprehensive, but with an emphasis on material covered since exam II• Roughly 40% of the points will involve material from the last 3.5 weeks

(chpts 11, 12, 14, 16 and stuff we added to these in class). 30% from each of the other two sections of the course (covered in exams I and II).

• There will probably be 7 to 9 questions, with the requirement being that you answer 6 to 7 of these (or something like that).

• Look at lecture notes for weeks 6 and 11 to see review sessions for Exam I and II. Those should still apply for the relevant sections of the Final.

• Chapter 11• Chemical reactions• The Law of Mass Action (equilibrium constants, stoichiometry coeffs.)• Internal partition functions (degeneracy, electronic excitations, rotations,

vibrations, coupling between these etc.)

Review for the Final• Chapter 12/16

• Phase Equilibria• The importance of the chemical potential• First order phase transitions

• Latent heat• Nucleation/metastability• Hysteresis

• Classius/Clapeyron equation• Gibbs/Duhem relation• Van der Waals equation of state• Two-phase regions • Continuous phase transitions• Critical phenomena; the importance of fluctuations• Universality classes and critical exponents• Shannon entropy

• Chapter 14 (and 12.6)• Refrigerators as entropy pumps• Techniques: heat engines in reverse, Adiabatic Demag, evaporative

cooling, laser cooling, dilution fridges, Pomeranchuk, J-T expansion, etc.• Third Law and its consequences• Reminder of the real meaning of temperature

Sample question from Final last year

Critical Exponents

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