General Theory of Diffraction[1]p 35

                For the derivation of the scattering amplitudes we make use of Fig. 3.1

Bob  The “derivation” here assumes that an electrons at R and P are driven by the incoming electromagnetic wave.  As a result of the acceleration of the electron it emits a spherical outgoing wave.  The scattering cross section for this can be derived for input radiation of the form  to be [2].  In this w₀ is the

 

 

natural frequency of the electron, gamma is its damping constant and  is the classical radius of the electron.  The effect of polarization is ignored. The essential part of this derivation is that the radiation emitted from Q is all coherent.  It is then scattered as an outgoing spherical wave from each position P to be observed by the experiment at B.  The radiation from each P is coherent and thus the amplitudes can cancel.  The total amplitude that we observe is the sum of the coherent radiation emitted from all points P in the illuminated source.

 

Fig. 3.1

Here Q is the location of the source of radiation, P is the position of the scattering center, and B the point of observation.  As an example of a source we shall take the spherical light waves emitted in conjungtion with an electronic transition in an atom.  At sufficiently large distances from the source the sphercala waves can be approximated as plane waves.  The amplitude (of to X-rays more addurately the field strength vector) at position P and time t may thus be written

       (3.1)

… The spherical waves observed at B are therefore described by

     (3.2)

For a fixed position P, the wave vector k is in the direction R’-r. Thus …

     (3.3)

At large distances

      (3.4)

Inserting 3.1 into 3.4

 (3.5)

The total scattering amplitude is given by integration over the entire scattering region

             (3.6)

Motion of Electrons and Transport Phenomena p 191

9.1     Motion of Electrons in Bands and Effective Mass

Bob --- this section is a summary of results rather than a derivation of them.

 

This representation   

The translational motion of the wave packet is described by the group velocity  .  This generalizes to .

The correspondence principle used with the field  implies .  From the time-dependent Schroedinger equation, [this] can be shown to apply quite generally to wave packets of Bloch states, provided the electromagnetic fields are not too large compared with atomic fields and are slowly varying on an atomic length and time scale.

                Carrying these results to the velocity  

 

This equation is completely analogous to the classical equation of motion  of a point charge (-e) in a field ,  if the scalar mass m is formally replaced by the so-called effective-mass tensor .  The inverse of this mass tensor

 

is simply given by the curvature of .  Because the mass tensor  and also its inverse are symmetric, they can be transformed to principle axes.

                Bob  note that these are all inner band results. 

The E(k) near the band edge stays with the lower curve, making it parabolic for k near the band edge.

9.2    Currents in Bands and Holes p 195

A volume element of dk at k contributes to the particle current density j_n

 

since the density of states in k-space is [V/(2pi)³].  We have taken into account that spen degenerate states must be counted twice.

                Taken together, the electrons in a fully occupied band therefore make the following contribution to the electrical current density j:

.

Because the band is fully occupied  E(k)=E(-k)  integral is zero.

                For partially occupied bands

 

The total current … may thus be formally described as a current of positive particles, assigned to the unoccupied states of the band (empty k).  These quasiparticles are known as holes, and they can be shown to obey equations of motion analogous to those [for electrons].

                Holes also behave like positive particles with regard to their dynamics in an external field.  If a band is almost completely filled, then only the highest energy part in the vicinity of the maximum contains unoccupied states.  In thermodynamic equilibrium electrons always adopt the lowest energy states, so that holes are found at the upper edge of the band.  Near to the maximum, the parabolic approximation for E(k) applies

 

 indicates that we are concerned with the effective mass at the top of the band, which is negative.  … the acceleration of a hole in one of these states under the influence of an electric field is

 

The equation of motion is that of a positively charged particle with a positive effective mass, i.e,.  holes at the top of a band have a positive effective mass.

Chapter 12

This figure is from[3].  The similar 7.13 picture is from[4].  The spin-orbit interaction was not included in the above, giving rise to even more branches.  There is a m^*_l a m^*_t and two hole terms m^*_hh and m^*_lh corresponding to different band curvatures.