High temperature approximation

Frequently we are interested in a region of temperature in which the energy levels are small compared with the thermal energy 

The partition function expanded in powers of T, then becomes [1] 

Here TV is the total number of energy levels , and the expanding coefficients are 

This formula is further simplified if the origin of the energy is chosen in such a way that 

The advantage of this high-temperature expansion is that we need not determine the individual energy levels but can use the theorem of the invariance of the trace of a matrix. For example, to find the value of e2 we square the Hamiltonian and pick out only the diagonal terms. On summing these terms over all possible values, many of the sums cancel. For example, the zero-field splitting Hamiltonian of the form 

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In the presence of a potential function U(x,y), the density matrix in the high-temperature approximation has the fomi... 

The experiment starts at equilibrium. In the high-temperature approximation, the equilibrium density operator is proportional to the sum of the operators, which will be called F. If there are multiple exchanging sites with unequal populations, p-, the sum is a weighted one, as in equation (B2.4.31). 

The production of allyl chloride could be effected by direct chlorination of propylene at high temperatures (approximately 500°C and one atmosphere). The reaction substitutes an allylic hydrogen with a chlorine atom. Hydrogen chloride is a by-product from this reaction ... 

This universality is peculiar for the high-temperature approximation, which is valid for //J < 1 only. For sufficiently high temperature the quantum theory confirms the classical Langevin theory result of J-diffusion, also giving xj = 2xE (see Chapter 1). This relation results from the assumed non-adiabaticity of collisions and small change of rotational energy in each of them ... 

NFS spectra of the molecular glass former ferrocene/dibutylphthalate (FC/DBP) recorded at 170 and 202 K are shown in Fig. 9.12a [31]. It is clear that the pattern of the dynamical beats changes drastically within this relatively narrow temperature range. The analysis of these and other NFS spectra between 100 and 200 K provides/factors, the temperature dependence of which is shown in Fig. 9.12b [31]. Up to about 150 K,/(T) follows the high-temperature approximation of the Debye model (straight line within the log scale in Fig. 9.12b), yielding a Debye tempera-ture 6x) = 41 K. For higher temperatures, a square-root term / v/(r, - T)/T,... 

Using the wave functions of the harmonic oscillator in each potential well of the proton, we can estimate the total effect of the inertia on the transition probability in the high-temperature approximation for the medium67 ... 

A spin system, initially in thermal equilibrium, can be described by a density operator density operator cr(t) in the rotating frame with a speed a)tf=2nftf can be written as3,4... 

An estimate of the VPIE of monatomics can be obtained from the first quantum correction using the Wigner high temperature approximation (appropriate because the level spacing in the quantized intermolecular well is small compared to the thermal energy, hv/kT 1, see Chapters 3 and 4)... 

The second group contains the low frequencies, m < 1, which are to be treated in the high temperature approximation. This, we have seen, accounts for excitation into upper levels by expanding I n(s/s ) 1 in even powers of u (see Chapter 4)... 

We start at equilibrium. In the high-temperature approximation, the equilibrium density operator is proportional to the weighted sum of the operators, which we will call 4- We assume that a simple, non-selective pulse has been used at the start of the experiment. This rotates the equilibrium 2 magnetization onto the x axis. After the pulse the density matrix is therefore given by 4, and it will evolve as in equation (7) or (8). If we substitute (8) into (10), we get the NMR signal as a function of time t, as given by (11). The detector sees each spin (but not each coherence ) equally well. 

In Ref. 38 expressions for t T) are given without making a high-temperature approximation. In the following, however, we assume that 2kBT > hoi. When the inhomogeneous width is very small, t (T) becomes (in the high-temperature and impulsive limits)... 

Further Mossbauer effect studies (304 — 12 K) and magnetic susceptibility measurements (301 - 1K) on the neutral complex [Fe(papt)2 ] have been performed recently 1S9 The magnetic data are shown in Fig. 34. The values of - In f(s T2) and - In f(J Aj) have been found to follow the high-temperature approximation of the Debye model above 105 K and 140 K, respectively, if anharmonic corrections have been introduced. No simple model is available at present, which would be capable to account for the complete temperature dependence of the Debye-Waller factors in this crossover system. 

Even for the largest magnetic fields available for NMR, the energy levels are separated only by mi/Iicalories, and the argument in the exponential is very small except at extremely low temperature. Hence, the high temperature approximation e x 555 1 — x may be employed to show that the fractional excess population in the lower level is... 

On is the vibrational Raman transition frequency, and Yn is the vibrational dephasing constant Within the high temperature approximation, Eq.(22) is expressed as... 

Table IV also lists, for 298.1° and 500°K., values of the characteristic frequency 7 necessary to make the modified high-temperature approximation, or y method, reproduce the exact (s2/si)f values. The "y corresponding to these fit-producing 7 were calculated according to...

Low-temperature approximation. Zero-point-energy approximation. High-temperature approximation. 

Reactive electrode submerged arc (RESA) is a totally new process for making pow-ders. RESA produces extremely high temperatures (approximately 10,000 K) with a pressure of 1 atm H2O (possibly more in the nanoenvironment). It allows one to change liquids very easily. Figure 1.13 shows the apparatus to produce powders. 

The evaluated emulsions stability versus time at high temperature (approximately 70 C) is about 3 days. 

As M becomes large, each matrix element appearing in Eq. (4.6) is equivalent to the density matrix evaluated at a high-temperature MT. In the large M limit the matrix elements can be replaced by high-temperature approximations which become exact for infinite M. A number of such approximations have been used and a discussion of the various forms along with a discussion of their relative merits can be found elsewhere. For the purposes of this discussion we take the form... 

In the physical picture ion-pairs are just consequences of large values of the Mayer /-functions that describe the ion distribution [22], The technical consequence, however, is a major complication of the theory the high-temperature approximations of the /-functions applied, e.g. in the mean spherical approximation (MSA) or the Percus-Yevick approximation (PY) [25], suffice in simple fluids but not in ionic systems. 

A solution to these difficulties is a blend of the chemical picture in which clustered ion configurations are described by the mass action law, while the interactions between the various entities are treated by methods applying the high-temperature approximations of the /-functions, e.g. by the MSA. The Debye-Hiickel (DH) theory [26], although derived from classical electrostatics, is also a high-temperature approximation, whose range of applicability can be extended by supplementing a mass action law for ion pair formation [27],... 

Transport Coefficients of Qnantum-Classical Systems 543 where we used the high temperature approximation... 

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