Transcript P202 Lecture 2

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 (BraggWilliams 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 Model
http://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/Fichi
ers/thesebook-chap5.pdf
Order-Disorder Transition
b
b’
http://cimewww.epfl.ch/people/cayron/Fichi
ers/thesebook-chap5.pdf
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.
http://cimewww.epfl.ch/people/cayron/Fichiers/thesebook-chap5.pdf
Renormalization Group
Block 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 Lounasmaa
The experimental entropy
curves for several materials
actually used for this
technique.
Carnot Cycle
What happens if you run this
cycle backwards?
Carnot Cycle
A
B
D
C
What 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.
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 Lounasmaa
The experimental entropy
curves for several materials
actually used for this
technique.
Third law of Thermodynamics
1. Nernst: It is impossible by any procedure, however idealized, to reduce the
temperature of any system to absolute zero in a finite number of operations
2. 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 methods
1. 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 neutrons
g) Cooling of the proton beam in IUCF’s “cooler”
2. Evaporation
a) This is how you keep cool in the summer
b) 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
Lines of constant
Enthalpy
Max Inversion Temps:
Nitrogen: 640 K
Hydrogen: 193K
Helium: 51K
From NASA tech.
note D-6807 (1972)
Pomeranchuk Cooling
Why does the liquid have lower entropy
than the solid in this case (and why is this
not the case for 4He)?
From:
http://ultracold.uchicago.edu/homepage/gustavo.pdf
H e V a p o r P re s s u re
10000
4H e
1000
3H e
P (T o rr)
100
10
1
0 .1
0 .0 1
0
1
2
3
T (K )
4
5
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 Refrigerator
From Lounasmaa “Experimental Principles
and Methods below 1 K.”
1. 4He at about 1 K is used to condense the
mixture
2. The mixture fills the “mixing chamber”
“still” and heat exchangers, and phaseseparates 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 Refrigerator
http://na47sun05.cern.ch/target/outline/dilref.html
1. The mixture fills the “mixing chamber”
“still” and heat exchangers, and phaseseparates 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
Phys. Today Oct.
1990 Phillips,
Cohen-Tannoudji
Optical Molasses demonstration and public explanations at:
http://nobelprize.org/nobel_prizes/physics/laureates/1997/illpres/trapping.html
http://www.colorado.edu/physics/2000/bec/lascool4.html
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
Phys. Today Oct.
1990 Phillips,
Cohen-Tannoudji
Time –of-flight technique used to
more accurately measure the
temperature of the cooled atoms at
NIST.
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”)
Laser Cooling:
Sisyphus effect
Phys. Today Oct.
1990 Phillips,
Cohen-Tannoudji
Note the temperatures
reached !!
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.
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
Sample question from Final last
year