31. January: During the first day we more or less came through the material from Chapter 6 which is specified on the syllabus (page 135-150 in 1.ed. or  page 155-171 in 2. ed., see the "Course description"). To leading order, the heat capacity of a Fermi gas is linear in temperature and is proportional to the density of states at the Fermi surface. This term is derived in Kittel, and is here derived by applying the so-called "Sommerfeld expansion". The notes by Henrik Smith and me (in Danish) Noter og opgaver i statistisk fysik may be used as a supplement to Marder's presentation.

Tomorrow I will discuss "Bloch's theorem" (Chapter 7), "the nearly free electron model" and "the  tight-binding model" (Chapter 8) from 13-15 in D411. On Friday, 9-12 in D317, we are going to use all three hours on the exercises: Problems 6.3, 6.4, and 6.5 in Chapter 6  and 7.1 and 7.3 in Chapter 7.

1. February: I came through the most of the pages mentioned in the syllabus for the Chapters 7 and 8. All three hours on Friday and the first hour on Monday are used for the exercises. In the next two hours on Monday (10-12) we shall discuss Cohesion of Solids (Chapter 11) and make a start on the "Phonons" discussed in Chapter 13.

4. February: We got started on the exercises. Monday morning (9.15-10 in D317) I am going to hand out my solutions to the problems of the first week. Anders has offered to present his own solution to problem 6.5 on the blackboard. The remaining time will be used for a short discussion of the solutions to the other problems.

7. February: This Monday I discussed cohesion (Chapter 11) and phonons (Chapter 13). The most important part in the chapter on cohesion is the discussion of metals, and I used the opportunity to tell you about the Hartree, Hartree-Fock, and pseudo-potential approximations (section 11.4.1). Tomorrow I will discuss the thermodynamic behaviour of phonons and then begin on the topic of next week, i.e. "classical theories of magnetism and ordering" in Chapter 24. The text to problem "HS.1" (or HS's problem 1) on Friday is found in the "Notes to Condensed Matter Physics 2" on page 11.

8. February: I finished the discussion of the "phonons" by telling you about the statistics of phonons - the Einstein and the Debye models. I got started on Chapter 24, where I told about spontaneous magnetization of ferro-, ferri, and antiferromagnets. I also did introduce the Ising model, which we are going to analyze in terms of the MF (mean-field) approximation on next Monday. The section 24.4.1 about domains is going to be replaced by the discussion of the electric (which applies as well for magnetic) dipole moment in the notes ("..to condensed matter physics 2"), and I will also go through the next section in the notes on "Magnetic energy and domains".

The table on the back cover of Marder is missing one fundamental constant, which we need for calculating, for instance, the mass density of silicon: The (unified) atomic mass unit = 1 u = m(¹²C)/12 = 1.66053878(8) 10^-27kg, which you may also find via my link to "Fundamental Physical Constants" placed in the folder with "Links".

14. February: The topic of this week is "Ordering", and the mean field solution of the Ising model was presented today. We also began the discussion of the formation of magnetic domains. The domains are created in order to reduce the "demagnetization field". The case considered by Marder, eqs. (24.58) - (24.61), is somewhat unrealistic, and he defines y to be the easy axis in (24.57), but uses the opposite choice  in (24.58) - (24.61) and in Fig. 24.8 (A). The anisotropy is of decisive importance for the width of the domain walls, not really for the width of the domains. The size of the domains (of the order of μm) is determined by a complex interplay between how well the demagnetization field is averaged out (as many domains as possible) and the energy cost involved in the creation of the domain walls. Marder's discussion of the domain effects in section 24.4.1 should be replace by my version presented in the section "Magnetic energy and domains" in "Notes to Condensed Matter Physics 2". Tomorrow I will return to Marder and discuss the order-disorder transition in β-brass alloy (bcc crystal with 50% Zn and 50% Cu), the Landau theory of second-order phase transitions, and present the more advanced theories for describing phase transitions (scaling theories and Wilson's renormalization group theory). If we have more time tomorrow I will go on with the discussion of domains and the dipole-dipole interaction, else the finishing of the two first section in the notes has to wait until Monday in next week.
 

15. February: Today we discussed the sublattice ordering of the β-brass alloy and found that it may be mapped on the (antiferromagnetic) Ising model. I also discussed the critical behaviour of a system close to a continuous second-order phase transition. In the critical regime, the fluctuations dominate the system and cover all length scales at the critical point. The "self-similarity" of the system at any length scale - the "scale  invariance" - provides the basis for Kadanoff's scaling and Wilson's renormalization group theories. On Friday we are going to work on the two exercises 24.4 and 24.6, and I will also add a few more words on the  "universality" of second-order phase transitions. That the behaviour of a system near a second-order phase transition, i.e. the critical exponents, is determined by only a few essential factors (the dimension of the space and the dimension of the order parameter) .

Correction of Marder's discussion of the Landau expansion. The free energy density, as a function of the thermodynamical variables H and T, is given by Eq. (24.75), if +HM is replaced by -HM. This follows from the free energy derived in "Notes to condensed matter physics 2": dF = - SdT - MdH, requiring that the partial  derivative of F = F(H,T) with respect to H should be -M.  [note that M = M(H,T), but at equilibrium F is at a minimum with respect to a variation of the parameter M, i.e. the derivative of F with respect to M is zero]. In all the following equations (24.77), (24.78), (24.84), and (24.87), H should be replaced by -H. The sign of the susceptibility in (24.86) then becomes positive both above and below the transition.

The correct expressions for the interaction parameters in (24.64) and (24.67) are going to appear in the answer to this week's excecises.

21. February: The self-consistent equation determining the order  parameter in Problem 24.6 can only be solved numerically, and you may find a Mathematical program for doing that here. Today I finished the discussion of the two first sections in the notes on the dipole interaction in solids and magnetic domains. We got started on Chapter 25, on magnetism of atoms and electrons. Tomorrow we will finish this chapter and go on with the case where the magnetic moments are interacting with each other (Chapter 26).
 

22. February: The topic of this week is "Magnetism". In chapter 25 Marder discusses the creation of magnetic moments, either the localized ones due to the atomic core electrons in unfilled shells (L = S = 0 for all filled shells) or the itinerant moments due to the band electrons of a metal. I am not so happy for Figure 25.3, illustrating the Pauli paramagnetism, and I refer you instead to page 8 in "Notes to Condensed Matter Physics 2". The next  chapter 26 contains much material. One essential issue is that the strong Coulomb interaction, combined with the Pauli principle, gives rise to interactions between the atomic or the itinerant moments proportional to the "exchange integral". In many systems, the leading order term is isotropic in "spin space", i.e. is of the form of the Heisenberg exchange interaction. The derivation in section 26.3 of the Heisenberg Hamiltonian using "second quantization", should only be read in a cursory way. In the case where the Heisenberg interaction J (nearest neighbours) is positive, the ground state is the ferromagnet, where all spins are parallel and of maximum magnitude, <S_z(R)> = S for all R.

We may decide to use some of the spare time in the last week for discussing the excitations appearing in a ferromagnet, i.e. the spin waves, but as for now you should cancel the part of Chapter 26 that starts with the section "26.3.3 Spin Waves" (on p. 753/805) from the syllabus - and the problem [26.4] in the list of exercises.  On Friday I will rather quickly go through the solutions to the problems 25.2 and 26.2, so that you may concentrate your efforts on the two more important problems 25.4 and HS.4.  Next week is free of schedule, and on Monday, the 7. of March, we start on the next topic - transport properties.

8. March: In Chapter 16 the classical Drude model and the "semiclassical" extension of the equations-of-motion are introduced. The semiclassical model includes the most important quantum mechanical modifications of the Drude model, i.e. that the velocity of an electron is the "group velocity" of the corresponding wave packet, and that the time derivative of the velocity is the inverse of an effective mass tensor times the time derivative of the crystal momentum. The "Limitations of Semiclassical Dynamics" are summarized on page 429 in Marder.

In Chapter 17, the semiclassical  equations-of-motion are built into the Boltzmann equation determining  the probability or "occupation number" function g(r,k,t), i.e. the occupation number of electrons in the state (n,k) at position r at time t.  Using the "relaxation time approximation" the Boltzmann equation leads to results for the charge and heat currents due to the electrons, which are similar to those derived by Drude except that the mass of the electrons is being replaced by an average of the effective mass (or band mass) over the occupied states.

Marder has made a few mistakes in his discussion of the Hall effect as discussed in "Notes on Condensed Matter Physics 2".

On  Friday we are going to do one exercise on AC conductivity 17.4, one on thermal current 17.5 and afterwards three different exercises on the Hall effect, 17.8, 17.9 and the one by Henrik Smith (HS. 3). Notice that the problem numbers in the second edition are increased by 1 and are 17.5, 17.6, 17.9 and 17.10. The topic of next week is "superconductivity" and we are going to cover the first part of Chapters 27.

14. March: The last topic of the course is  "Superconductivity". We came through the sections on the phenomenological theory, the London equation and the Ginzburg-Landau  theory. The magnetic free energy is handled more easily than done by Marder if using Eq. (2.5) in "Notes on Condensed Matter Physics 2" as the starting equation. In this case B²/8π is being replaced by 2πM² in the Ginzburg-Landau equation (27.27), see page 10 in the notes.

Corrections  to Marder: The equations (27.3)-(27.6) are only valid for μ = 1 (delete μ in all these equations). Use (24.3) rather than (24.5) for deriving (27.3) and that the external current density is constant in time. Equation (27.28) is an "indirect" consequence of minimizing (27.27). I think it is better to say that the minimizing of (27.27) predicts the right magnetic energy, if M is considered to be determined by the current given by (27.29a). Notice that the curl of B in (27.28) is equal to 4π times the curl of M according to the Maxwell's equations (since the curl of H is zero within the sample). In other words you should replace Eq. (27.28) by the Maxwell equation, c times curl of M = j, and that the minimization of (27.27) shows that the current in this equation is the one given by (27.29a).

Tomorrow I will tell about flux quantization, type II superconductors and the Josephson effect.

15. March: Next week is the last one in the course. The first hour on Monday are going to be used on a discussion of this week's problems on superconductors. Afterwards I will present the spin wave theory for the ferro- and antiferromagnetic Heisenberg system. I expect I will use a major part of the two hours on that (including a discussion of problem 26.4). Else the idea is that we use the remaining time in next week for discussing previous exam problems from the candidate course "Solid State Physics II" (you will get the problems on Monday), and whatever issues of general interest you may bring up. On Monday, we are also going to decide which topic you should present at the oral exam (taking place on the first of April). We have 12 different topics, and I will ask you to choose a number between 1 and 12, which is then being translated to one of the 12 topics below from a random sequence made beforehand. The 12 topics are:

1)     Bloch's theorem and nearly free electrons.
2)     Lattice vibrations. Einstein and Debye model.
3)     Dynamics of Bloch electrons.
4)     Transport phenomena (Boltzmann equation).
5)     Mean-field theory and the Ising model.
6)     Order-disorder transition of an alloy.
7)     Second-order phase transition. Landau free energy.
8)     Atomic magnetism.
9)     Heisenberg model.
10)    Landau-Ginzburg theory of superconductivity.
11)    Flux quantization in type II superconductor.
12)    The Josephson effect.
 

Those who do not turn up at the lectures on Monday or Tuesday, I am going to contact via email.

21. March: Today I discussed the Heisenberg ferromagnet - its ground state and the spin wave excitations. At low temperatures (compared with the Curie temperature T_C), the elementary excitations of the ferromagnet are "the spin waves", where the different spins are precesing around the equilibrium (z) direction with a phase, which changes from one site to the next as prescribed by the wave vector. The Heisenberg Hamiltonian may be transformed into a Hamiltonian of N independent harmonic oscillator, one at each value of k, by the use of the Holstein-Primakoff transformation for the local spin operators S_i. The excitations of this Hamiltonian are non-conserved bosons (the chemical potential μ = 0), which are named "magnons". These are similar to the quantized elastic waves, the phonons, except that there is only one polarization, one magnon, for each value of k. The spin waves are well-defined (infinite lifetime) in the zero temperature limit of the ferromagnet. At non-zero temperatures the occurrence of fourth and higher-order products of the Bose operators (deriving from the expansion of the square root) implies that the magnons are interacting with each other, resulting in a finite lifetime of the magnons and a change of the excitation energies, and also the possibility that "bound states" of two (or more) magnons may occur, deriving from that the energy of a two magnon state may possibly be smaller than the sum of the two individual magon energies. The influences of the magnon bound states are anticipated to be very weak, both on the thermodynamic and on the resonance properties of the system. In the antiferromagnetic case, the assumed Néel state is not the true ground state, and the harmonic-oscillator approximation gives rise to a correction of the ground state energy and to a reduction of the sublattice magnetization at zero temperature, |<S_z>| < S, see the answer to exercise 26.4. - Notice that Marder in  this Chapter 26 uses a J_ll', which is half the coupling between the neighbours. The indices in (26.47) run over all sites giving rise to a factor of 2 in (26.49) or in (26.56), when the sum is restricted to the distinct pairs.

Below I repeat the list of topics for the oral exam. The names of you who are going to present which one of the topics is included.

1)     Bloch's theorem and nearly free electrons. - Erik
2)     Lattice vibrations. Einstein and Debye model. - Logi
3)     Dynamics of Bloch electrons. - Kevin
4)     Transport phenomena (Boltzmann equation). - Mikolaj
5)     Mean-field theory and the Ising model.
6)     Order-disorder transition of an alloy. - Gitte
7)     Second-order phase transition. Landau free energy. - Maria
8)     Atomic magnetism. - Olivier
9)     Heisenberg model.
10)    Landau-Ginzburg theory of superconductivity.
11)    Flux quantization in type II superconductor. - Simon
12)    The Josephson effect. - Anders Simonsen

22. March: We got started on solving the problems from previous exams in "Solid State Physics II". You may find the problems in the folder with course material. Later on, I will also place the solutions in that folder. On Friday we are going to continue with Problem 2008 and 2010, (and 2009 if we have any time left).

25. March: All information concerning the oral and the written exam is placed in the folder named "Exam".