Density of modes

In problems addressed in this text, solids appear not as the system of principal interest but as an environment, a host, of our system. We therefore focus on those properties of solids that are associated with their effect on a molecular guest. One such property is the density of modes, a function g a ) defined such that the number of modes in any frequency interval )i a a 2 is dcoglco ). As a formal definition we may write 

In fact, this function dominates also thermal and optical properties of the solids themselves because experimental probes do not address individual normal modes but rather collective mode motions that manifest themselves through the mode density. For example, the vibrational energy of a harmonic solid is given by 

The density of modes is seen to be the only solid property needed for a complete evaluation of these thennodynamic quantities. In what follows we consider this function within the one-dimensional model of Section 4.2.2. 

The difference between these two forms stems from the fact that in one dimension there are two values of k for a given A . The density of modes in frequency space is obtained from the requirement that the number of modes in a given interval of I A I is the same as in the corresponding interval of a . 

In the long wavelength limit m = ck c = a oa, g((zi) = N/naiQ. For larger k, that is, larger a , gfco) depends on m and becomes singular at the Brillouin zone boundary k = ja, (y = 2(uo- 

Consider the one-dimensional solid analyzed in Section 4.2.2. From the expression for the allowed value of A = 2k /Na)l, I = 0, l. we find that the number of possible k values in the interval A. A -b AA is Na/2K)Ak, so the density of modes per unit interval in k is 

---

For longitudinal modes, we can therefore stale that the number of modes with frequency less than a> is 2 w/waY atom-pair, or a density of modes of 6w lwy, modes per atom-pair per frequency-range. Similarly, construct the spectrum for transverse modes and plot the total on the same abscissa as in Fig. 9-6 so that comparison can be made. (That histogram did not have a normalized scale on the ordinate, so you need not worry about the ordinate.) The principal discrepancies are understandable by comparison of the Debye approximation to the spectrum shown in Fig. 9-2. 

In an unentangled melt, the number density of modes relaxing with rate e corresponds to the number of chain sections containing K Kuhn segments such that... 

In many applications we encounter such sums of contributions from different modes, and because in the limit Q oo the spectrum of modes is continuous, such sums are converted to integrals where the density of modes enters as a weight function. An important attribute of the radiation field is therefore the density of modes per unit volume in k-space, Pk, per unit frequency range, p, or per unit energy, ps (E = hoP). We find (see Appendix 3A)... 

Furthermore, as in the free particle case, Eq. (3.54) implies that the density of modes (i.e. the number, per unit interval along the A -axis, of possible values of k) is L/ln. Obviously, different solution are orthogonal to each other, b kx)bi kx)dx = <5/,//, and the choice of normalization can be arbitrary because the amplitude of the mode is determined by the solution of Eq. (3.52). 

This model assumes that all the normal mode frequencies are the same. Taking Eq. (4.42) into account the density of modes then takes the fonn... 

Fig. 4.1 Density of modes in lead. Full line - numerical calculation based on lattice geometry and interatomic potential of lead. Dashed line The Debye model fitted to the density and speed of sound of lead. (From the Cornell Solid State Simulation site, R.H. Silbee, http //www.physics.comell.edu/sss/.)...

It should be emphasized that although this success of the Debye model has made it a standard starting point for qualitative discussions of solid properties associated with lattice vibrations, it is only a qualitative model with little resemblance to real normal-mode spectra of solids. Figure 4.1 shows the numerically calculated density of modes of lead in comparison with the Debye model for this metal as obtained from the experimental speed of sound. Table 4.1 list the Debye temperature for a few selected solids. 

This bath is characterized by the density of modes function, g(u)), defined such that g(ai)dcei is the number of modes whose frequency lies in the interval Cf),...,a> + day. Let... 

The function is defined as a sum of delta-fLinctions, however for macroscopic systems this sum can be handled as a continuous fiinction of co in the same way that the density of modes, g(co) = 5 (co — a>j ) is. - Defining the coupling density by... 

The spectral density, Eqs (6.90) and (6.92) is seen to be a weighted density of modes that includes as weights the coupling strengths c lco). The harmonic frequencies atj... 

Consider Eq. (6.84). This result was obtained for a harmonic system of identical and equivalent atoms. We could however reverse our reasoning and define a vibrational spectrum for a dense atomic system from the velocity autocorrelation function according to Eq. (6.84). Since this function can be computed for all systems, including liquids and disordered solids, we may use (6.84) as a definition of a spectrum that may be interpreted as density of modes fimction for such media. We can then use it in expressions such as (4.33), and (6.92). Is this approach to dynamics in condensed phases any good ... 

The two functions I a (< ) andZ4 (co) are seen to convey the same physical information and their coexistence in the literature just reflects traditions of different scientific communities. The important thing is to understand their physical contents we have found that the power spectrum is a function that associates the dynamics of an observable A with the dynamics of a reference harmonic system with density of modes given by (7.81) and (7.79). 

This rate has two remarkable properties First, it does not depend on the temperature and second, it is proportional to the bath density of modes g(ct>) and therefore vanishes when the oscillator frequency is larger than the bath cutoff frequency (Debye frequency). Both features were already encountered (Eq. (9.57)) in a somewhat simpler vibrational relaxation model based on bilinear coupling and the rotating wave approximation. Note that temperature independence is a property of the energy relaxation rate obtained in this model. The inter-level transition rate, Eq. (13.19), satisfies (cf. Eq, (13.26)) k = k (l — and does depend on temperature. 

One way to establish a stricter conservation rule is to replace transmission by the density of modes (DOM) introduced according to [3] as... 

It must be noted here, however, that the use of the term density of modes for expression (3) is a subject of actual discussion, and the strict concept of the DOM for such systems (possessing continuous or quasidiscrete rather than discrete spectra) is yet to be introduced. However, the quantity in (3) does possess some properties of the DOM and can be taken as a phenomenological definition. 

In 1900 Rayleigh introduced density of electromagnetic modes in the theory of equilibrium electromagnetic radiation [16]. In 1916 Einstein showed that the ratio of spontaneous to stimulated emission coefficients was /zta3/ji2c3. Then in 1927 Dirac [18] introduced the quantization of electromagnetic field and showed that for the Einstein relationship to be fulfilled the spontaneous emission rate should be proportional to the number of modes available for light quanta to be emitted. Later, in solid state theory concept of the density of modes was developed with respect to electrons and other elementary excitations and evolved towards a consistent density of states (DOS) inherent in every quantum particle of matter. The notion of local density of states was introduced in complex solids. 

It is well-known that the spontaneous lifetime of an excited atom is proportional to the density of modes of the electromagnetic field p(u) ) about the atomic transition frequency u . Specifically, the Weiss-kopf-Wigner spontaneous lifetime is given by 125]... 

Consider a cubic cavity of length L and quality factor Q. Owing to the finite Q, the cavity exhibits some losses which yield a cavity linewidth Au) = w/Q. For a Lorentzian lineshape, the density of modes around a cavity mode of frequency and mode volume is given by... 

In Eq. (12.25), 6 is the angle between the polarization of the incident radiation (A ) and the direction of propagation of the scattered wave k ), R is the position of the detector, is the dynamic polarizability of the segment, and p iv) is the density of modes of the incident radiation at frequency v (Eqs. 12.9,12.12, and B12.1.14). The factor sin(5)/IAI is the same factor that determines the amplitude of the field from an oscillating electric dipole (Figs. 3.1 and 3.2), and the fluorescence from an excited molecule whose transition dipole is oriented along a fixed axis (Sect. 5.9). The polarizability Uaa can be obtained from the difference between the dielectric constant of the solution and that of the pure solvent. 

First, it appears that all fluctuations vanish in the thermodynamic limit V 00. However, this is not the case. Although the fluctuation amplitude for each mode decreases as V increases, the density of modes increases such that the sum of all fluctuation amplitudes remains constant. This is simply demonstrated by calculating the average fluctuation amplitude in real space ... 

---