ABSTRACT
Energy E of the system as a function of a generalized coordinate x, measuring position along a line connecting two nearby local minima of E.
appropriate position coordinate x for such an atom, or set of atoms, the energy E (x) will then have two local minima, separated by a barrier, as shown in the figure. The figure is a section of the 3N -dimensional configuration space along the coordinate connecting two such local mm1ma. The atoms of interest for the specific heat will be those for which the energy barrier is sufficiently great so that resonant tmmelling between the two local minima does not occur, but sufficiently small so that tunnelling between the two levels can take place and thermal equilibration can occur during the time span of the specific heat experiment (say I0-10 sec < t < 103 sec). Of the atoms with barriers in the acceptable range, those contributing to the specific heat at a low temperature T (T is of the order ofO·l-l0°K) will be atoms for ·which the energies of the two local minima are accidentally degenerate to within an amount of order kT. We argue that this must lead to a specific heat proportional to T. In general, the energies of the two levels will be more or less random quantities, dependent on such factors as the particular configuration of atoms surrounding the two minima, on local strains, etc. Inasmuch as the positions of the two minima are spatially separated, we would expect that the probability distribution of the energy difference l:iE between the two levels will vary on the scale of these random energies (perhaps O·l ev to lev in a typical glass) and will be smooth on the scale of kT. In particular, n( l:iE), the density of levels per unit volume and per unit l:iE, with energy difference l:iE and with tunnelling time in the acceptable range, should be non-zero and continuous in the vicinity of l:iE = 0. The density of such levels with
To make our argument clearer, let us consider a model Hamiltonian which describes the two-level system consisting of the ground states in the two local energy wells. We write
where E 1 and E 2 are the energies of the local minima in the potential energy E(x), liw1 and 7iw2 are the energies of zero-point motion about these minima, hw0 is an energy of the order of the zero-point energy, and the factor exp (-.\)represents the overlap between the wave functions for the two potential wells. For the potential of the figure we have
( 2mV)I/2 .\~t hF D.x, (3)
where m is the mass of the tunnelling atom or group of atoms. Let
be the difference between the two diagonal elements of H. (We shall arbitrarily choose the indices I and 2 so that D-E~ 0.) The correction to the eigenvalues of the Hamiltonian due to the off-diagonal element will be negligible provided
If condition (5) is satisfied, then transitions between the two energy levels can occur only by a process such as phonon-assisted tunnelling, with emission or absorption of a phonon necessary to conserve energy. The rate of such transitions can be written roughly in the form
where we estimate r 0 to be of the order of I012 sec-1• A more accurate form for r will be discussed below. The condition r-1 < t can be written
Since we have a situation where r 0 is of the order of w 0 , and t ~ nr 0 -I/2D,E-1i2, there will be a substantial range of .\ in the acceptable region between -'min and -'max-Furthermore, the limits (5) and (7) of this region depend only weakly (i.e. logarithmically) on the value of D.E when D-E~ kT, so that this range may be considered constant for our purposes. If m is the
4 P. \V. Anderson et al. on
It should be emphasized that the density of levels with ,\ in the 'acceptable range ' is much smaller than the total density of modes with level splitting ~E. There are a vast number of ?lodes having small ~E which are inaccessible because their energy barriers are too big for tunnelling to occur, or because they require the cooperative motion of too many atoms. The fact that glasses are in a metastable state to begin with implies that there are large numbers of states with arbitrary ~E (both positive and negative relative to the occupied state of the system) which are only inaccessible because they are separated from the occupied states by large energy barriers.
