2. 1 Lecture 7 Gaseous systems composed of molecules with internal motion. Monatomic molecules. Diatomic molecules. Fermi gas. Electron gas. Heat capacity of electron gas.

3. 2 Gaseous systems composed of molecules with internal motion In most of our studies so far we have consider only the translation part of the molecular motion. internal motion Though this aspect of motion is invariably present in a gaseous system, other aspects, which are essentially concerned with the internal motion of the molecules, also exist. It is only natural that in the calculation of the physical properties of such a system, contributions arising from these motions are also taken into account. In doing so, we shall assume here that a) the effects of the intermolecular interactions are negligible and

4. 3 (7.1) is fulfilled; effectively, this makes our system an ideal, Boltzmannian gas. b) the nondegeneracy criterion partition function Under these assumptions, which hold sufficiently well in a large number of practical applications, the partition function of the system is given by (7.2) where (7.3)

5. 4 The factor in brackets is the transitional partition function of a molecule, while the factor j(T) is supposed to be the partition function corresponding to the internal motions. The latter may be written as (7.4) where  i is the molecular energy associated with an internal state of motion (which characterized by the quantum numbers i ), while g i represents the degeneracy of that state. The contributions made by the internal motions of the molecules to the various thermodynamic quantities of the system follow straightforwardly from the function j(T). We obtain

6. 5 (7.5) (7.6) (7.7) (7.8) F int = - N kT lnj  int = - kT lnj E int =NkT 2 (7.9)

7. 6 Thus the central problem in this study consists of deriving an explicit expression for the function j(T) from a knowledge of the internal states of the molecules. For this purpose, we note that the internal state of a molecule is determined by: electronic state, state of nuclei, vibrational state and rotational state. Rigorously speaking, these four modes of excitation mutually interact; in many cases, however, they can be treated independently of one another. We then write j(T)=j elec (T) j nuc (T) j vib (T) j rot (T) (7.10) with the result that the net contribution made by the internal motions to the various thermodynamic quantities of the system is given by a simple sum of the four respective contributions.

8. 7 Monatomic molecules kT E ion T<<E ion /k  10 4 -10 5 o K. At the very outset we should note that we cannot consider a monatomic gas except at temperatures such that the thermal energy kT is small in comparison with the ionization energy E ion ; for different atoms, this amounts to the condition: T<<E ion /k  10 4 -10 5 o K. quite insignificant At these temperatures the number of ionized atoms in the gas would be quite insignificant. The same would be true for atoms in excited states, for the reason that separation of any of the excited states from the ground state of the atom is generally comparable to the ionization energy itself. Thus, we may regard all the atoms of the gas to be in their (electronic) ground state.

9. 8 L=S=0 Now, there is a well-known class of atoms, namely He, Ne, A,..., which, in their ground state, possess neither orbital angular momentum nor spin ( L=S=0 ). g e =1 Their (electronic) ground state is clearly a singlet: g e =1. hyperfine structure in the electronic state The nucleus, however, possesses a degeneracy, which arises from the possibility of different orientations of the nuclear spin. (As is well known, the presence of the nuclear spin gives rise to the so-called hyperfine structure in the electronic state. kTS n g n =2S n +1 vibrational or rotational states However the intervals of this structure are such that for practically all temperatures of interest they are small in comparison with kT.) If the value of this spin is S n, the corresponding degeneracy factor g n =2S n +1. Moreover, a monatomic molecule is incapable of having any vibrational or rotational states

10. 9 The internal partition function (7.10) of such a molecule is therefore given by j(T)=g gr.st. =g e  g n =2S n +1 (7.11) the internal motions chemical potential and the entropy of the gas Equations (7.4-7.9) then tell us that the internal motions in this case contribute only towards properties such as the chemical potential and the entropy of the gas; they do not make contribution towards properties such as the internal energy and the specific heat. L  0,S  0 - In other cases, the ground state of the atom may possess both orbital angular momentum and spin ( L  0,S  0 - as, for example, in the case of alkali atoms), the ground state would then possess a definite fine structure. F int = - N kT lnj  int = - kT lnj E int =NkT 2

11. 10 (7.12) The intervals of this structure are in general, comparable with kT ; hence, in the evaluation of the partition function, the energies of the various components of the fine structure must be taken into account. Since these components differ from one another in the value of the total angular momentum J, the relevant partition function may be written as The forgoing expressions simplifies considerably in the following limiting cases: kT >> all  J ; then (7.13)

12. 11 kT<< all  J ; then (7.14) where J 0 is the total angular momentum, and  0 the energy of the atom in the lowest state. In their case, the electronic motion makes no contribution towards the specific heat of the gas. And, in view of the fact that both at high temperatures the specific heat tends to be equal to the translational value 3/2 Nk, it must be passing through a maximum at a temperature comparable to the separation of the fine levels. Needless to say, the multiplicity ( 2S n +1 ) introduced by the nuclear spin must be taken into account in each case.

13. 12 Diatomic molecules kT kT T<<E diss /k  10 4 -10 5 o K Now, just as we could not consider a monatomic gas except at temperatures for which kT is small compared with the energy of ionization, for similar reasons one may not consider a diatomic gas except at temperatures for which kT is small compared with the energy of dissociation; for different molecules this amounts once again to the condition: T<<E diss /k  10 4 -10 5 o K. At this temperatures the number of dissociated molecules in the gas would be quite insignificant.

14. 13 C v =(C v ) elec +(C v ) vib +(C v ) rot (7.15) The heat capacitance of the diatomic gas is consist from three parts At the same time, in most cases, there would be practically no molecules in the excited states as well, for the separation of any of these states from the ground state of the molecule is in general comparable to the dissociation energy itself. Let us consider them consequently. (7.16) In the case of electron contribution the electronic partition function can be written as follows

15. 14 g 0 g 1  where g 0 and g 1 are degeneracy factors of the two components while  is their separation energy. (7.17) T >  /k  /k We note that this contribution vanishes both for T >  /k and has a maximum value for a certain temperature  /k ; cf. the corresponding situation in the case of monatomic atom. The contribution made by (7.16) towards the various thermodynamic properties of the gas can be readily calculated with the help of the formula (7.4-7.9). In particular we obtain for the contribution towards specific heat

16. 15 10 3 o K Let us now consider the effect of vibrational states of the molecules on the thermodynamic properties of the gas. To have an idea of the temperature range, over which this effect would be significant, we note that the magnitude of the corresponding quantum of energy, namely, for different diatomic gases is of order of 10 3 o K. 10 4 o K 10 2 o K Thus we would obtain full contributions (consistent with the dictates of the equipartition theorem) at temperatures of the order of 10 4 o K or more, and practically no contribution at temperatures of the order of 10 2 o K or less. We assume, however, that the temperature is not high enough to excite vibrational states of large energy; the oscillations of the nuclei are then small in amplitude and hence harmonic.

17. 16 (7.18) T>>  v Nk T >  v the vibrational specific heat is very nearly equal to the equipatition value Nk ; otherwise, it is always less than Nk. In particular, for T<<  v, the specific heat tends to zero (see Figure 7.1); the vibrational degrees of freedom are then said to be "frozen".  (n+1/2) h  /2  j vib (T) n=  The energy levels for a mode of frequency  are then given by the well-known expression (n+1/2) h  /2 . The evaluation of the vibrational partition function j vib (T) is quite elementary. In view of the rapid convergence of the series involved, the summation may formally be extended to n= . The corresponding contributions towards the various thermodynamic properties of the system are given by eqn.(4.64 -4.69). In particular, we have

18. 17 Figure 7.1 The vibrational specific heat of a gas of diatomic molecules. At T=  v the specific heat is already about 93 % of the equipartition value.

19. 18 Finally, we consider the effect of the states of the nuclei and the rotational states of the molecule: wherever necessary, we shall take into account the mutual interaction of these modes. heternuclear homonuclear This interaction is on no relevance in the case of the heternuclear molecules, such as AB; it is, however, important in the case of homonuclear molecules, such as AA. In the case of heternuclear molecules the states of the nuclei may be treated separately from the rotational states of the molecule. Proceeding in the same manner as for the monatomic molecules we conclude that the effect of the nuclear states is adequately taken care of through degeneracy factor g n. Denoting the spins of the two nuclei by S A and S B, this factor is given by

20. 19 g n = (2S A +1)(2S B +1) (7.19) As before, we obtain a finite contribution towards the chemical potential and the entropy of the gas but none towards the internal energy and specific heat. Now, the rotational levels of a linear "rigid" with two degrees of freedom (for the axis of rotation) and the principle moments of inertia ( I, I, 0 ), are given by (7.20) here I=M(r 0 ) 2, where M=m 1 m 2 /(m 1 +m 2 ) is the reduced mass of the nuclei and r 0 the equilibrium distance between them. The rotational partition function of the molecule is then given by

21. 20 (7.21) For T>>  r the spectrum of the rotational states may be approximated by a continuum. The summation (7.21) is the replaced by integration : (7.22)

22. 21 The rotational specific heat is the given by (C V ) rot =Nk (7.23) which is indeed consistent with equipartition theorem. A better evaluation of the sum (7.21) can be made with the help of the Euler-Maclaurin formula Putting one can obtain (7.24)

23. 22 Mulholland's formula which is the so-called Mulholland's formula; as expected, the main term of this formula is identical with the classical partition function (7.22). The corresponding result for the specific heat is (7.25) Nk which shows that at high temperatures the rotational specific heat decreases with temperatures and ultimately tends to the classical value Nk.

24. 23 Thus, at high (but finite) temperatures the rotational specific heat of diatomic gas is greater than the classical value. On the other hand, it must got to zero as T  0. We, therefore, conclude that it must pass through at least one maximum. (See Figure 7.2) Fig.7.2. The rotational specific heat of a gas of heteronuclear diatomic molecules.

25. 24 T<<  r In the opposite limiting case, namely for T<<  r, one may retain only the first few terms of the sum (7.21); then (7.26) whence one obtains, in the lowest approximation (7.27) T  0 "frozen". Thus, as T  0, the specific heat drops exponentially to zero (Fig. 7.2). Now we can conclude that at low temperatures the rotational degrees of freedom of the molecules are also "frozen".

26. 25 Fermi gas Fermi function Let us consider the perfect gas composed of fermions. Let us consider in this case the behavior of the Fermi function given by equation (5.46)  F (0). for the case when the assembly is at the absolute zero of temperature and the Fermi energy is  F (0).  >  F (0),{  -  F (0)}/kT=  for  >  F (0), {  -  F (0)}/kT=  while  <  F (0), {  -  F (0)}/kT=-  for  <  F (0), {  -  F (0)}/kT=- . T=0 {  -  F (0)}/kT When T=0 the quantity {  -  F (0)}/kT has two possible values:

27. 26 for  <  F (0), for  >  F (0), (7.28) There are therefore two possible values of the Fermi function: Figure 7.3. Fermi-Dirac distribution function plotted at absolute zero and at a low temperature kT<< . The Fermi level  o at T=0 is shown.

28. 27   F (0) f(  )T=0 Equation (7.28) implies that, at the absolute zero of temperature, the probability that a state with energy   F (0) will be empty. The form of f(  ) at T=0 is shown as a function of energy in Figure 7.3. one fermion allowed per state Fermi level Thus with only one fermion allowed per state, all the lowest states will be occupied until the fermions are all accommodated. The Fermi level, in this case, is simply the highest occupied state and above this energy level the states are unoccupied. fermions occupy the lowest available energy states. This behavior may be explained as following. At the absolute zero of temperature, the fermions will necessarily occupy the lowest available energy states.

29. 28 Figure 7.3. Fermi-Dirac distribution function plotted at absolute zero and at a low temperature kT<< . The Fermi level  o at T=0 is shown. T<<T F  /k  F /k Fermi-Dirac distribution T F  F  F  F  (0 For the temperatures T<<T F  /k  F /k the Fermi-Dirac distribution behavior is shown in the Fig. 7.3 by bold line. The Fermi temperature T F and the Fermi energy  F are defined by the indicated identities. The Fermi energy  F is defined as the value of the chemical potential at the absolute temperature  F  (0).

30. 29 f=1/2  =  T=0  =  kT  -  >>kT We note that f=1/2 when  = . The distribution for T=0 cuts off abruptly at  = , but at a finite temperature the distribution fuzzes out over a width of the order of several kT. At high energies  -  >>kT the distribution has a classical form.  The value of the chemical potential is a function of temperature, although at low temperature for an ideal Fermi gas the temperature dependence of  may often be neglected.  (  )  The determination of  (  ) is often the most tedious stage of a statistical problem, particularly in ionization problems. We note that  is essentially a normalization parameter and that the value must be chosen to make the total number of particles come out properly.

31. 30 f -df/d   (x-a): An important analytic property of f at low temperatures is that -df/d  is approximately a delta function. We recall the central property of the Dirac delta function  (x-a): (7.29) Now consider the integral -df/d  F(  ) F(  ) At low temperatures -df/d  is very large for  and is small elsewhere. Unless F(  ) is rapidly varying in this neighborhood we may replace it by F(  ) and the integral becomes

32. 31 (7.30) f(0)  1 But at low temperatures f(0)  1, so that (7.31) a result similar to (7.29).

33. 32 Electron gas half-integral spin Fermi-Dirac statistics The conducting electrons in a metal may be considered as nearly free, moving in a constant potential field like the particles of an ideal gas. Electrons have half-integral spin, and hence the Fermi-Dirac statistics are applicable to an ideal gas of electrons. f k =f(  k -  ) (7.32) (7.33) k  k  f k FD k We use k to specify the state of the electron, and  k its energy. Let  be the chemical potential of the electron. Each state can accommodate at most one electron. Let f k be the FD average population of state k.

34. 33 The energy distribution of the states is an important property. Let (7.34)  (T=0) (7.35) g(  )  0 g(  ) The function g(  ) is the energy distribution of the states per unit volume, which we simply call density of states, and  is the energy with respect to  0. The calculation of g(  ) gives (7.36)

35. 34 The thermodynamic properties of this model can be largely expressed in terms of f and g, e.g. the density N/V of the electrons and the energy density E/V are (7.37) (7.38) If T=0, then all the low energy states are filled up to the Fermi surface. Above this surface all the states are empty. The energy at the Fermi surface is  0, i.e the chemical potential when T=0, and is always denoted by  F :  F  (T=0)=  0 (7.39) The Fermi surface can be thought of as spherical surface in the momentum space of the electrons. The radius of the sphere p F is called Fermi momentum (7.40)

36. 35 N There are N states with energy less than  F : (7.41) volume of sphere in momentum spacevolumespin state (=2 (h) 3 h/2  =1 i.e (volume of sphere in momentum space)  (volume)  (spin state (=2))  (h) 3, with h/2  =1. Hence p F =(3  2 n) 1/3 (7.42) n  N/Va=h/(2  p F ) where n  N/V. Let a=h/(2  p F ) is approximately the average distance between the electrons, hence: (7.43) ah/ (2  a). is approximately the zero-point energy of each electron. This zero-point energy is a result of the wave nature of the electron or a necessary result of the uncertainty principle. To fix an electron to within a space of size a, its momentum would have to be of order h/ (2  a).

37. 36 10 -8 cm In most metals, the distance between electrons is about 10 -8 cm and  F ~1 eV ~10 4 o K T<<  F  F ~1 eV ~10 4 o K (See Table 7.1). Therefore, at ordinary temperatures, T<<  F, i.e. the temperature is very low, only electrons very close to the Fermi surface can be excited and most of the electrons remain inside the sphere experiencing no changes. (7.44) (7.45) (7.46) (7.47)

38. 37 Table 7.1 Properties of the electron gas at the Fermi surface.

39. 38 Heat Capacity Only a very small portion of the electrons is influenced by temperature. Hence the concept of holes appears naturally. an empty state a hole The states below the Fermi surface are nearly filled, and empty states are rare. We shall call an empty state a hole. new mixed gas of holes together with electrons above Fermi surface Now this model becomes a new mixed gas of holes together with electrons above Fermi surface. (We shall call these outer electrons.) p F p F TT=0 The momentum of a hole is less than p F, while that of the outer electrons is larger than p F. The lower the temperature T, the more dilute the gas is. At T=0, this gas disappears.

40. 39 T=0 -  '  '. At T=0 the total energy is zero, i.e. there are no holes or outer electrons. The holes are also fermions because each state has at most one hole. Hence, a state of energy -  ' can produce a hole of energy  '. The average population of the hole is (for each state): 1-f(-  '-  )= f(  '+  ) (7.48)  =0 Now the origin of the energy is shifted to the energy at the Fermi surface, i.e.  =0. The energy of a hole is the energy required taking an electron from inside the Fermi surface to the outside. T<<  F T g(  ) As T<<  F, the energy of the holes or the outer electrons cannot exceed T by too much. In this interval of energy, g(  ) is essentially unchanged, i.e.

41. 40 g(  )  g(0)=mp F /  2 (7.49)  0 Hence the energy distribution is the same for the holes or the outer electrons. Therefore,  0. All the calculations can now be considerably simplified, e.g. the total energy is (7.50) where 2g(0) is the density of states of the holes plus the outer electrons.  F  F >>T The energy of a hole cannot exceed  F, but  F >>T and so the upper limit of the integral in (7.50) can be taken to be . This integration is easy:

42. 41 (7.51) Substituting in (7.50), and differentiating once, we get heat capacity (7.52) C v =3/2 N This result is completely different from that of the ideal gas in which C v =3/2 N. 3/2 NT/  F In that case each gas molecule contribute a heat capacity of 3/2. Now only a small portion of the electrons is involved in motion and the number of active electrons is about NT/  F

43. 42 (7.53) C v ~N(T/  F ). Hence C v ~N(T/  F ). From (7.52) we get (7.54) 1 C Each active electron contribute approximately 1 to heat capacity C.

44. 43 Figure 7.4 Electron (hole) energy versus momentum.