when we have a scalar effective mass me we may express t in the form &amp;formula; . the index s in the equation t = AE-s is therefore 1/2 . for a non-degenerate semiconductor with s = 1/2 we have from &amp;symbol; 10.6 equation ( 83 ) , &amp;formula; since &amp;formula; and &amp;formula; . thus we have &amp;formula; . the mobility me when we have only lattice scattering by the acoustical modes of vibration is therefore given by &amp;formula; . this gives the well known T-3/2 law for the variation of mobility in pure semiconductors at high temperatures ; it is unfortunately not very well obeyed for most semiconductors since both anisotropy and other scattering mechanisms tend to modify the mobility to an appreciable extent . we defer the calculation of the constant a to &amp;symbol; 13.4.4 . 10.10 . low-mobility semiconductors . although the theory of electrical conductivity which we have given seems to be applicable to metals and to normal semiconductors having a high electron and hole mobility , we may readily show that it can not be applied without serious modification to materials for which the mobility is low . there are many such materials which appear to have mobilities of about 1 cm2 &amp;sol; V sec . for such materials the relaxation time t would have a value of about 6 x 10-16 sec if we have me = m and a smaller value still if me &lt; m . now because of collisions the value of the energy can not be precisely stated for a time much greater than t so that , if dE is the uncertainty in energy , we must have ( cf &amp;symbol; 1.4 , equation ( 67 ) ) &amp;formula; . thus if &amp;formula; , &amp;formula; . the allocation of energy levels in a band therefore becomes meaningless , particularly if the energy spread of the carriers , as in a non-degenerate semiconductor , is only of the order of kT . for a semiconductor like Ge , on the other hand , &amp;formula; for low-mobility semiconductors the band theory of conduction must be abandoned and we must regard conduction by electrons as a form of field-assisted tunnelling between adjacent atoms of the crystal . this process has been discussed extensively by A F Joff&amp;eacute; and has been applied by him to the study of liquid and amorphous semiconductors . the limitations of the band theory , particularly as applied to narrow bands with high effective mass have also been recently discussed by H Fro&quot;hlich and G L Sewell . the theory of conduction by jumping from site to site has also been used by N F Mott to discuss conduction by impurities in semiconductors at low temperatures , the so-called impurity band conduction . the full details of the theory of this type of conduction have not yet been worked out to anything like the same extent as for conduction in materials having a high electron or hole mobility . 11 . the effective-mass approximation . 11.1 . the quasi-classical approximation . we have shown in chapter 8 that , to a high degree of approximation , an electron moving in a perfect crystalline lattice in an external field of force F may be regarded as a particle moving classically in the field , the particle having a tensorial effective mass ; the equations of motion were derived in &amp;symbol; 8.8 . these may be expressed in terms of the wave vector k or crystal momentum P by means of the vector equation &amp;formula; . this equation may be transformed into an acceleration equation , giving the rate of change of the averaged velocity &amp;formula; , in the form &amp;formula; where 1/Me represents the effective-mass tensor whose Cartesian components 1/mrs are given by &amp;formula; . when the energy E may be expressed in the quadratic form &amp;formula; where the quantities Ars are constants , and Ars = Asr and the components of the effective-mass tensor are constants . the various simplifications of equation ( 2 ) which may be made when some of the quantities Ars are zero or equal have been discussed in &amp;symbol; 8.8 ; in particular , when &amp;formula; so that the effective mass is a scalar me , the equation of motion reduces to the simple classical form &amp;formula; . we shall refer to this , and the more general form ( 2 ) , as the quasi-classical approximation . it should be clearly appreciated that these equations are in no sense based on classical mechanics - their derivation depends essentially on the quantum theory of electron waves in crystals as shown in &amp;symbol; 8.8 . the term quasi-classical is used to indicate that their form is classical . once they have been derived , however , they may be used to describe the motions of the conduction electrons in the crystal by treating the electrons as classical particles . in the derivation of equation ( 1 ) we pointed out that there were certain restrictions under which it could be applied . in particular , the force F must be slowly varying , i.e it must change very slightly between neighbouring cells in the crystal . we have applied equations of this form to discuss the motion of electrons under external electric and magnetic fields and have found that this description leads to results in excellent agreement with experiment when the fields are not too strong . we have also used the idea of effective mass in &amp;symbol; 9.3.6 to discuss the motion of an electron in the Coulomb field of an impurity atom in a semiconductor . here , however , we have a rather paradoxical state of affairs in that , while we regarded the electron as a particle of mass me , we used wave mechanics to derive the energy levels of the impurity centre , quoting the well-known result for a hydrogen atom . indeed we may readily see that the quasi-classical approximation only holds provided the wavelength le of the quasi particle is short compared with the distance over which the field varies appreciably ; this is the well-known criterion for the application of classical mechanics to the motion of a particle in a field of force . for the motion of a free particle of mass m in a field of force given by a potential function V(r) the classical equation of motion &amp;formula; is replaced by Schro&quot;dinger &apos;s equation &amp;formula; or more generally by the equation &amp;formula; where H ( p , r ) is the classical Hamiltonian expressed in terms of the momentum p . equations ( 7 ) or ( 8 ) determine the stationary-state wave function ps associated with the energy E . because of the similarity of equations ( 5 ) and ( 6 ) it would seem not unreasonable to replace equation ( 5 ) , when we are dealing with an external field of force in a crystal to which classical mechanics can not be applied , by an equation of the form &amp;formula; where V(r) is the potential which determines the force . this is effectively what we have done in discussing the energy levels of an impurity centre in a semiconductor in &amp;symbol; 9.3.6 . we shall devote most of the present chapter to proving that such an equation can indeed be used to describe the motion . some thought will have to be given to the interpretation of the wave function ps . it is clearly not the same as the wave functions used to describe the motion of the electron in the perfect crystal ; as we shall see , it is not the whole wave function but may be interpreted as a slowly varying amplitude . the extension of equation ( 9 ) to the case when the effective mass is tensorial may be expected to follow in the same way as equation ( 8 ) is an extension of equation ( 7 ) , the Hamiltonian H being the sum of the energy of the electron in the crystal as a function of the crystal momentum P ( which we should expect to replace the momentum p of a free particle ) and the potential energy V(r) being derived from the external force . we might reasonably therefore expect the equation which determines the motion of an electron in a crystal under an external force to be &amp;formula; where Ep(P) is the energy of an electron in the perfect crystal given as a function of the crystal momentum P . in terms the wave vector k , equation ( 10 ) may be written in the form &amp;formula; where Ek(k) is the energy of electrons in the perfect crystal as a function of the wave vector k . for slowly varying fields it is well known that equations such as ( 7 ) and ( 8 ) give the same results as classical mechanics ; similarly , equations ( 9 ) and ( 10 ) will give the same results as the quasi-classical approximation when this is applicable . equations ( 10 ) and ( 10a ) clearly reduce to equation ( 9 ) when we have a scalar effective mass me ; they represent a higher degree of approximation than equation ( 2 ) . so far , we have only given plausible arguments for their form ; we shall now proceed to derive them using the quantum theory of the motion of electrons in a crystal . 11.2 . quantum theory of the effective-mass approximation . the wave equation describing the motion of electrons in a crystal in a perturbing field of force may be derived in a number of ways . an elegant derivation , which also shows up well the physical principles involved , originally given by G Wannier , has been developed by J C Slater , and we shall first of all follow this method of derivation . in order to use Wannier &apos;s method we shall have to introduce some wave functions which he used and which are generally known as Wannier functions ; they are built up from the Bloch wave functions which we have already used in our discussion of the motion of electrons in a perfect lattice . these functions are particularly well suited to this kind of problem , whereas for many other problems the Bloch functions are to be preferred . as we shall see , the Wannier functions are localised , whereas , the Bloch functions are spread throughout the whole crystal ; the latter are therefore appropriate for the discussion of problems in which we do not require to specify the position of an electron closely , while the former are useful when discussing problems associated with a definite point in the crystal such as an isolated impurity centre . it was indeed in order to obtain a localised wave function that the Wannier functions were first introduced . we know that the Bloch functions bk(r) defined by &amp;formula; are solutions of the wave equation for the perfect crystal and hence that a wave function representing a solution of the wave equation may be expanded as a series of such functions . if we restrict the values of k to the first Brillouin zone there will be N such allowed values corresponding to each energy band , where N is the number of unit cells in the crystal . in order to obtain an exact expansion of the wave function we should require to use Bloch functions bkn(r) corresponding to all bands . however , when we have a substantial gap between the bands it appears that we may obtain , under certain conditions , a good approximation by using Bloch functions only from the band in which we are interested , and these we shall denote by bk(r) . we then have for the expansion of the wave function ps &amp;formula; . 11.2.1 . the Wannier functions . the expansion given in equation ( 12 ) is not very easily interpreted physically if a number of coefficients An(k) are required to give an accurate description of the wave function representing the motion of an electron in the perturbing field of force . to overcome this difficulty Wannier ( loc cit ) introduced a new set of wave functions , derived from the Bloch wave functions , which have the property of being localised . consider the wave function &amp;formula; where the constants an are at our disposal , the sum being taken over the N allowed values of k . in the first unit cell of the crystal we may choose the constants an to make all the functions bn(k) add . we shall assume that the Bloch functions are normalised for a volume V containing N unit cells , and we have already seen that they are orthogonal , so that we have &amp;formula; . in the definition of the Bloch functions there is an arbitrary phase term and we use the constants an which may be written in the form exp ( ibn ) to take out this phase term . indeed we may assume that the Bloch functions are so defined that they add to give the maximum contribution in the first unit cell so that we may take &amp;formula; for all values of n . 