Debye temperature: using

The relative intensity of a certain LEED diffraction spot is 0.25 at 300 K and 0.050 at 570 K using 390-eV electrons. Calculate the Debye temperature of the crystalline surface (in this case of Ru metal). 

Thermal Properties at Low Temperatures For sohds, the Debye model developed with the aid of statistical mechanics and quantum theoiy gives a satisfactoiy representation of the specific heat with temperature. Procedures for calculating values of d, ihe Debye characteristic temperature, using either elastic constants, the compressibility, the melting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ed., Oxford University Press, 1985, pp 24-29). 

The Debye temperatures of stages two and one were determined by inelastic neutron scattering measurements [33], The total entropy variation using equation 8 is in the order of about 2 J/(mol.K). Although smaller in value, such variation accounts for 10-15% of the total entropy and should not be neglected. We are currently carrying on calculations of the vibrational entropy from the phonon density of states in LixC6 phases. 

Although the Debye model reproduces the essential features of the low- and high-temperature behaviour of crystals, the model has its limitations. A temperature-dependent Debye temperature, d(F), can be calculated by reproducing the heat capacity at each single temperature using the equation... 

An alternative to deriving the Debye temperature from experimental heat capacities is to derive an entropy-based Debye temperature by calculation of the s that reproduces the observed entropy for each single temperature using... 

In summary, LEED is most often used to verify the structure and quality of single crystal surfaces, to study the structure of ordered adsorbates and to study surface reconstructions. In more sophisticated uses of LEED one also determines exact positions of atoms, the nature of defects and the morphology of steps, as well as Debye temperatures of the surface. 

The fact that the surface Debye temperature is lower than that of the bulk has two consequences. First, the surface is always a weaker scatterer than the bulk. Second, the intensity of the surface signal decreases faster with increasing temperature than the intensity of the bulk signal. Sometimes one can use this property to recognize surface behavior from measurements with bulk sensitive techniques [12]. 

The recoilless fraction, /, has been calculated (13) for monotomic lattices using the Debye approximation. When the specific heat Debye temperatures of the alkali iodides are inserted in the Debye-Waller factor, a large variation of f follows (from 0.79 in Lil to 0.15/xCsI). It is not... 

Recoilless Optical Absorption in Alkali Halides. Recently Fitchen et al (JO) have observed zero phonon transitions of color centers in the alkali halides using optical absorption techniques. They have measured the temperature dependence of the intensity of the zero phonon line, and from this have determined the characteristic temperatures for the process. In contrast to the Mossbauer results, they have found characteristic temperatures not too different from the alkali halide Debye temperatures. 

It may of course be unnecessary to consider all these terms and the equation is much simplified in the absence of magnetism and multiple electronic states. In the case of Ti, it is possible to deduce values of the Debye temperature and the electronic specific heat for each structure the pressure term is also available and lambda transitions do not seem to be present. Kaufman and Bernstein (1970) therefore used Eq. (6.2), which yields the results shown in Fig. 6.1(c). 

Other phases are then characterised relative to this ground state, using the best approximation to Eq. (6.1) that is appropriate to the available data. For instance, if die electronic specific heats are reasonably similar, there are no lambda transitions and T 6o, then the entropy difference between two phases can be expressed just as a function of the difference in their Debye temperatures (Domb 1958) ... 

By contrast, in an earlier assessment, Weiss and Tauer (1958) decided to use the measured magnetisation values as a primary input over the whole temperature range, but then had to depart from the experimentally observed Debye temperatures. Interestingly, their treatment led to the conclusion that the G curves for / Mn and 7-Mn would also intersect twice so that the behaviour of Fe could no longer be... 

The thermal properties of AU55 are treated in Sect. 3, using especially the results of MES measurements [24,25,42]. These are discussed in connection with the concept of bulk versus surface modes in small particles. An explanation of the temperature dependence of the MES [42] absorption intensities and the Cv results [25] on the basis of a model using the site coordination and the center-of-mass motion are briefly reviewed. The consequences of the Mossbauer results for surface Debye temperatures and for the melting temperature of small gold particles are also discussed. 

The four sites of Aujj exhibit different line intensities, and from the relative site occupations, the Mossbauer f-factors for the different sites could be calculated [24], using standard techniques [91]. These, in turn, could be related to effective Einstein (or Debye) temperatures 0 (or 0 ) associated with the vibrations of the individual sites. An unexpected consequence was that the three surface sites could not be described by a single meaning that the use of... 

The experimental observation that one has different Debye temperatures for the three distinct surface sites of the AU55 cluster makes the use of a continuum-model picture for discussing the thermal behavior questionable. Indeed, for such small particle sizes, where the surface structure is so manifest, the use of the concept of surface modes becomes dubious, and is certainly inadequate to explain the observed temperature dependence of the f-factors. None the less, it has proven possible to describe the low temperature specific heat of AU55 quite well using such a continuum-model, when the center-of-mass motion is taken into account [99],... 

Similar expressions can be generated for holes simply by letting coc - — relaxation time xB needs justification, which will not be attempted here. Suffice it to say that this assumption is not bad for elastic scattering processes, which include most of the important mechanisms. A well-known exception is polar optical-phonon scattering, at temperatures below the Debye temperature (Putley, 1968, p. 138). We have further assumed here that t is independent of energy, although this condition will be relaxed later. 

Fig. 74. Linear thermal expansion of Gdo.sSro.sMnOj in zero field and under applied fields of 6 T and 12 T. The dashed curve represents a fit of the high-temperature linear thermal expansion using a Griineisen law and a Debye temperature p = 300 K. After Garcfa-Landa et al. (1998a, 1998b).

Procedures used are described in detail elsewhere (5). The effect of matrix scattering is again approximated by modifying the atomic scattering factors according to expression ( 1). Random fluctuations in atomic positions parallel to the axis of the helix are allowed for by multiplying f (D) by the term exp(- sB Zl) where B is a Debye temperature factor. 

We note that it is the lower crossover temperature Tcl that is usually measured in experiments. The above simple analysis shows that this temperature is determined by the intermolecular vibrational frequencies rather than by the properties of the reaction complex or by the static barrier. It is not surprising then, that in most solid-state reactions the observed value of Tcl is of order of the Debye temperature of the crystal. Although the result (2.81) was obtained using the approximation w, exponential term turns out to be exact for arbitrary values of the two frequencies. It is instructive to compare (2.81) with (2.27) and see that friction slows tunneling down, while the q mode promotes it. 

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