Applications of the Canonical Ensemble Simple Models of

Applications of the Canonical Ensemble: Simple Models of Paramagnetism First a Quantum System of Spins s = ½

Paramagnetic Materials • The following is a simple model to develop our understanding of Statistical Mechanics in The Canonical Ensemble. • Paramagnets contain atoms which have magnetic dipole moments ( ). These do not interact with each other but can respond to an applied external magnetic (B) field. • The dipoles can be crudely thought of as independent (atomic) bar magnets arranged on a crystal lattice.

• The magnetic dipoles can be crudely thought of as bar magnets arranged on a crystal lattice. Crude 1 Dimensional Picture: • In an external magnetic field B each dipole can exist in one of two states – aligned with the field (spin up) or anti aligned (spin down). • Spin up dipoles have energy - B, spin down + B. • We want to calculate dependence of the magnetization of the material on the temperature T & the applied field B.

Curie Model of Paramagnetism • Energy level diagram of an S = ½ spin in an external magnetic field along the z-axis Note: B = 0 H, E = g BH Typically g = 2, which corresponds to E ~ 1 cm-1 at H = 104 G

• All dipoles are independent, so we only need to look at the average properties of one dipole. We can use all the other dipoles as the heat bath in the Canonical Ensemble. • There are only 2 possible microstates & we know their energies already, so its easy to get the Partition Function Z for a single dipole.

• In the Canonical Ensemble, it’s easy to calculate the probabilities of spin up & spin down: Spin Up Probability: Spin Down Probability:

• To calculate the mean magnetic moment of an individual dipole: • So the mean energy of an individual dipole is:

• These results are for one individual dipole, but because the dipoles do not interact, all other dipoles behave similarly. • So, a solid of N dipoles has average energy: • Similarly, the mean magnetic moment (magnetization) in the direction of the applied field is: • Note that U = -MB , which is the classical E & M result

• So, the magnetization M or the magnetic moment per unit volume L has the form: • In the limit of weak B fields or high temperatures T, x<<1 tanh x x

Results

“Saturation Magnetization” “Curie’s Law” of Paramagnetism M 0 χH, χ (N 0μ 2)/(k. T) χ “Curie Susceptibility”

Curie’s “Law” • The susceptibility is the magnetization per applied field intensity which for small magnetizations is given by H = B/ 0. • This is Curie’s “Law”. It holds well for many paramagnetic materials. It works so well that it can be used for temperature calibration! • For example Cerium Magnesium Nitrate obeys Curie’s “Law” down to T = 0. 01 K!

Experimental vs. T for Cu(CH 3 CN)4(BF 4)2 1/ = T/C is a straight line of slope C-1 & intercept zero T = C is a straight line parallel to the x-axis at constant T, showing the temperature independence of the magnetic moment.

Curie’s “Law” • Because single electrons have magnetic dipole moments, if they are placed them in a magnetic field, they’ll align with the field. However the energy difference between dipoles aligned with the field & against it is much smaller than thermal energy at room temperature. 2μB << k. BT So, there are random orientations, with equal populations aligned with & against the field. • As T is lowered, this energy difference becomes more important & the population changes, with more becoming aligned anti-parallel to the field.

• To explain this behavior, Curie invented a parameter – called the “Magnetic Susceptibility”, χ which is a measure of how attracted a sample is to a magnetic field. This is normally measured as an apparent mass increase. As more electrons align anti-parallel to the field at low temperature, χ increases. In fact, χ is inversely proportional to the temperature. This is the Curie “Law”: (1/χ) = CT C = “The Curie Constant”

Magnetic Heat Capacity • The paramagnetic solid has an energy that depends on temperature. It therefore has a magnetic heat capacity. • Experimentally one measures the heat capacity at constant magnetic field intensity H. • The paramagnetic compound is weakly magnetic, so B = 0 H so B is also constant.

Magnetic Heat Capacity

• This is often called The Schottky Heat Capacity • In fact, this is a general result for the heat capacity in any two level system.

Isolated Paramagnetic Solid • Now consider a very similar problem to the one just discussed. Constrain (fix) the total energy U of the isolated system. • N total dipoles, n spin-up aligned with the applied B field: • U is obviously a function of n. A • A given energy U(n) corresponds to a given number of n spin up atoms with statistical weight: -

• So, the Entropy is given by: • For large N (~1023) Stirling’s Approximation can be used: • So The Temperature of the spin system is given by:

• Further manipulation gives: • Also, we know that: • So that: • Now, solve for n, to find the density of spin up atoms:

Negative Temperatures? !! • From the previous derivation, • Manipulate this to obtain: • Note that, if n < N/2 then more than half the dipoles are anti-parallel to the field & we also get the surprising result that T becomes negative!

What is a Negative Temperature? • First note that, as the temperature T , the populations of spin-up & spin-down particles become equal! • We just saw that if n < N/2 (more spins down than up), T becomes negative! This is called a “population inversion”. • A negative temperature state must therefore be “hotter” than T , because it is a higher energy state of the system than T ! • For a negative temperature, the entropy & statistical weight must be decreasing functions of E.

What is Negative Temperature? • For a negative temperature, the entropy and statistical weight must be decreasing functions of E. • This can happen if the system possess a state of finite maximum energy – such as the paramagnet with U = N B. • No systems exist where this happens for all particular aspects (i. e. vibrational, electronic & magnetic energies). • However, if one such aspect or subsystem is effectively decoupled from the others, so they do not interact, that subsystem may be considered to reach internal equilibrium without being in equilibrium with the others. • This is the case for magnetic systems where the relaxation times between atomic spins is much quicker than the relaxation between spins and the vibrational modes of the lattice.

Negative Temperature • In the paramagnet, the lowest possible energy is U = -N B and the highest U = +N B. These are both unique microstates so S = 0. • In between we can only reach states with positive energy with a negative temperature!!!

A More General Discussion of Negative Temperature • In physics, certain systems, such as the magnetic system just discussed can achieve a negative temperature on the Kelvin scale! • A system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature! • If a negative-temperature system and a positivetemperature system come into contact, heat will flow from the negative temperature system to the positive-temperature system.

• That a system at negative temperature is hotter than any system at positive temperature seems paradoxical if absolute temperature is interpreted as an average kinetic energy of the system. • The paradox is resolved by understanding temperature through its more rigorous definition as the tradeoff between energy and entropy, with the reciprocal of the temperature, thermodynamic parameter = [1/(k. BT)], as the more fundamental quantity. • Systems with a positive temperature will increase in entropy as one adds energy to the system. Systems with a negative temperature will decrease in entropy as one adds energy to the system. [3]

• Most familiar systems cannot achieve negative temperatures, because adding energy always increases their entropy. • The possibility of decreasing in entropy with increasing energy requires the system to "saturate" in entropy, with the number of high energy states being small. These kinds of systems, bounded by a maximum amount of energy, are generally forbidden classically.

• So, a negative temperature is strictly a quantum phenomenon. The possibility of decreasing entropy with increasing energy requires a system to “saturate” in entropy, with the number of high energy states being small. • Most such systems, such as the paramagnetic system just discussed, have a maximum amount of energy that they can hold. • So, as they approach that maximum energy their entropy actually begins to decrease.