Einstein frequency

To simplify the analysis, the Einstein single frequency (coe) model is used. The Einstein frequency is given by ... 

This model, the Einstein model for heat capacity, predicts that the heat capacity is reduced on cooling and that the heat capacity becomes zero at 0 K. At high temperatures the constant-volume heat capacity approaches the classical value 3R. The Einstein model represented a substantial improvement compared with the classical models. The experimental heat capacity of copper at constant pressure is compared in Figure 8.3 to Cy m calculated using the Einstein model with 0g = 244 K. The insert to the figure shows the Einstein frequency of Cu. All 3L vibrational modes have the same frequency, v = 32 THz. However, whereas Cy m is observed experimentally to vary proportionally with T3 at low temperatures, the Einstein heat capacity decreases more rapidly it is proportional to exp(0E IT) at low temperatures. In order to reproduce the observed low temperature behaviour qualitatively, one more essential factor must be taken into account the lattice vibrations of each individual atom are not independent of each other - collective lattice vibrations must be considered. 

Both the Einstein and Debye theories show a clear relationship between apparently unrelated properties heat capacity and elastic properties. The Einstein temperature for copper is 244 K and corresponds to a vibrational frequency of 32 THz. Assuming that the elastic properties are due to the sum of the forces acting between two atoms this frequency can be calculated from the Young s modulus of copper, E = 13 x 1010 N m-2. The force constant K is obtained by dividing E by the number of atoms in a plane per m2 and by the distance between two neighbouring planes of atoms. K thus obtained is 14.4 N m-1 and the Einstein frequency, obtained using the mass of a copper atom into account, 18 THz, is in reasonable agreement with that deduced from the calorimetric Einstein temperature. 

The exact initial value of the friction is known and is given by the well-known Einstein frequency coq ... 

With the above-mentioned radial distribution function, expression for the Einstein frequency after performing the angular integration Eq. (114) reduces to... 

Here 2 is the Einstein frequency, is the energy of the medium reorganization, I It,n is ihe Bessel function of imaginary argument, and pj- is the final-state density. 

Positions and velocities of all particles are calculated at every step of the molecular dynamics simulation, producing a complete time evolution of the system. In order for this time evolution to be accurate, the integration time step St has to be much smaller than the shortest characteristic time of the system (the reciprocal Einstein frequency of the Lennard-Jones crys-... 

Here the subscript 1 denotes the local mode and the other modes are represented by the Einstein frequency of the order of the solvent Debye frequency, and we have assumed that a>i > a>2. 

Consider now one of these variable and its contribution to the potential energy, z(r) = 27rg 2(7Xz(r)2. This is the potential energy of a three-dimensional isotropic harmonic oscillator. The total potential energy, Eq. (16.82) is essentially a sum over such contributions. This additive form indicates that these oscillators are independent of each other. Furthermore, all oscillators are characterized by the same force constant. We now also assume that all masses associated with these oscillators are the same, namely we postulate the existence of a single frequency Ms., sometimes referred to as the Einstein frequency of the solvent polarization fluctuations, and Ws are related as usual by the force constant... 

To be more quantitative, we assign an internal energy Efee to the fee phase, Ebee to the bcc phase, and associated Einstein frequencies cofee and cobec- As a result, the free energies (neglecting anharmonicity and the electronic contribution) may be written as... 

To make the elements of this model more concrete, we consider the way in which the transition temperature depends upon the difference in the Einstein frequencies of the two structural competitors. To do so, it is convenient to choose the variables a> = (cob + coa)/2 and Aco = cob — coa) 2, where u>a and cob are the Einstein frequencies of the two structural competitors. If we now further measure the difference in the two frequencies, Aco, in units of the mean frequency m according to the relation Aco = fco, then the difference in free energy between the two phases may be written as... 

Fig. 6.15. Dependence of the critical temperature, here reported as /3c = 1 /kTc, on the difference in Einstein frequencies of the competing phases.

Whenever the external vibrations produee large anisotropic displacement parameter values for the scattering atoms it will exaggerate the impact of any given value of Q. The phonon wing envelope will move to even higher frequencies and the response will broaden. Only two characteristics of a sample bear on its anisotropic displacement parameter (with samples at low temperatures), the effective molecular mass, Hef[, and the Einstein frequency, see ( 2.6.2.1). The lighter the... 

These INS studies of the alkali metal hydrides provide an excellent example of the careful analysis of INS experimental data. It includes the application of corrections for multiple scattering, neutron absorption and heavy-ion scattering the extraction of quantities related to the hydrogen dynamics (the hydrogen mean square displacement, mean kinetic energy and the hydrogen Einstein frequency) and provided the density of vibrational states for each type of atom, shown individually and ab initio modelling of the full INS spectra. 

Toe 2.57 single weighted mean fi-equency (the Einstein frequency) representative of all the external vibrations cm" (d)... 

In this expression K(0) is the so-called Einstein frequency , which can be calculated exactly from the information of the site-site intermolecular potential as well as of the density pair-correlation functions. K 0) is also an equilibrium quantity but with the three-particle correlation functions, whose approximate expression has been derived in [91]. 

In case, Er is smaller than the characteristic phonon energy ( 10-2 gv for solids), E vib causes a change in the vibrational energy of the oscillators by integral multiples of phonon energy i.e. 0, 1 fta E, 2 fttus etc. (where h = h./2 K and cue is the Einstein frequency i.e. frequency with which each atom vibrates). 

The first person to make serious headway with this approach was Albert Einstein. In 1907, Einstein proposed to understand the motions of the atoms in the crystal using Planck s idea of quantized energy. A crystal is composed of N atoms, say. These N atoms can vibrate within their crystal lattice in the x, the y, or the z direction, giving a total of 3N possible vibrational motions. Einstein assumed that the frequencies of the vibrations were the same, some frequency labeled Vg, or the Einstein frequency. If this were the case, and we are only considering vibration-type motions of the atoms in the crystal, then the heat capacity of the crystal can be determined by applying the vibrational part only of the heat capacity from the vibrational partition function ... 

---