THE UNIVERSITY OF BRITISH COLUMBIA

PHYSICS 455

Lecture # 7 :

Mon. 20 Jan. 1997

Several Simple Systems

I. Recapitulate TWO-STATE SYSTEMS Generally

II. ONE SPIN-½ PARTICLE in a MAGNETIC FIELD:

• The difference in energy between the two states (spin up and spin down) is = mB.

• Thus U₁ = -mB tanh(mB/).

• Thus for N non-interacting spin-½ particles in a magnetic field B we have U = -NmB tanh(mB/) = -MB.

• The magnetization is therefore M = Nm tanh(mB/).

• This appears to differ significantly from the simply Curie law we derived in the limit of mB: M = Nm²B/.

  The version derived from the partition function is ``correct'' in that it holds for all temperature including 0 where M Nm. However, the exact formula does reduce to the Curie law in the appropriate high-temperature limit: as mB/ 0, exp(mB/) 1 mB/ and tanh(mB/) mB/. Thus M Nm²B/ [Curie law].

Magnetization as a function of temperature: dashed line represents the Curie law (valid in the high temperature limit). At low the spin polarization levels off at 100%.

III. The QUANTUM OSCILLATOR:

• Characteristic feature: evenly spaced levels: = where ² = k/M is the resonant frequency of the mass M on a ``spring'' of spring constant k.

• Important distinction from previous examples: no upper limit on _n. We therefore expect the thermal average energy of one oscillator, U₁, and its entropy ₁ to both increase monotonically with . The heat capacity will not decrease at high temperature as it did with the two-state system.

• Zero of energy: It is inconvenient to do the following calculations with the extra term _o = ½ included, so we just measure _n from _o for now.

• Partition function: Let x and y e^-x < 1, giving our familiar sum Z = yⁿ = 1/(1-y) or Z = 1/(1-exp[-/]).

• Mean thermal energy: U₁ = ² (log Z)/ = / (exp[/] - 1).

• Now we can put back the zero suppression if we like, adding _o to U₁. But since we can only measure energy on a relative scale, it hardly matters where we pick the zero.

• Heat capacity: C_V = (U/) in dimensionless units (multiply by k_B if you prefer conventional units). A little calculus yields C_V = (/)² exp[/] / (exp[/] - 1)².

  This approaches unity as , indicating that new ``degrees of freedom'' (higher energy levels) are thermally activated at a constant rate as the temperature increases.

• All this has been for a single SHO. A system of N such independent oscillators will, on average, have just N times the energy and heat capacity of one. A 3D version (e.g. a simple crystal lattice) will have three vibrational degrees of freedom (x, y and z) for each mass, giving a heat capacity (dimensionless version) of 3N at high temperature. Expressed in conventional units this would be 3N k_B; or, in terms of the heat capacity per mole, c_V 3R as , where R N_o k_B and N_o = 6.02^.10²³ is Avogadro's number.

  This is known as the Law of Dulong & Petit and describes the heat capacity of many solids quite well - namely, those crystals which are built up from a basis of dissimilar atoms which vibrate against each other at a well-defined frequency . (Such vibrations are known as optical phonons because their frequencies are high, comparable with infrared photons.) In simpler crystals the lowest energy vibrational modes involve many atoms moving together, the so-called ``acoustic phonon modes;'' we shall treat that case in some detail later on.

• This description in terms of a single is known as the Einstein model (old Albert sure did get around!) and the characteristic temperature below which optical phonon modes ``freeze out'' is also given his name: the Einstein temperature is _E /k_B.