Lorentzian energy

Long-range order theory, 35 4-5 Looper s walk capping, 32 438-445 Lorentzian energy averaging, 34 217 Lorentzian function, energy-dependent, 34 243 Losod, 33 215, 224, 258 Low-coordinated transition-metal ions, 34 133 Low energy... 

Fig. 1.5 The Lorentzian energy distribution of the source recoilless radiation.

A simple example showing this is a pure Lorentzian energy density,... 

K.M. Sluis, E. A. Gislason, Decay of a quantum-mechanical state described by a truncated Lorentzian energy distribution, Phys. Rev. A 43 (1991) 4581. 

In Mhssbauer spectroscopy (see Chap. 25 in Vol. 3), the density function of Cauchy distribution is called a Lorentzian curve. In nuclear physics, the same function is also called the Breit-Wigner curve (Lyons 1986). (See also Chap. 2 in this Volume.) This curve is characteristic of the energy uncertainty of excited (nuclear) states, which follows from the fact that excited states have exponential distribution with a finite mean life t. The natural linewidth T, i.e., the FWHM of the Lorentzian energy density, is twice of the parameter y(F = 2y). 

Thus the reflection coefficient for scattering from a diffusing particle has a Lorentzian energy shape with a width of... 

The energy spectrum of the resonance states will be quasi-discrete it consists of a series of broadened levels with Lorentzian lineshapes whose full-width at half-maximum T is related to the lifetime by F = Fn. The resonances are said to be isolated if the widths of their levels are small compared with the distances (spacings) between them, that is... 

The exponential decay of the A population corresponds to a Lorentzian line shape for the absorption (or emission) cross section, a, as a fiinction of energy E. The lineshape is centred around its maximum at E. The fiill-width at half-maximum (F) is proportional to... 

In these Lorentzian lines, the parameter x deseribes the kinetie deeay lifetime of the moleeule. One says that the speetral lines have been lifetime or Heisenberg broadened by an amount proportional to 1/x. The latter terminology arises beeause the finite lifetime of the moleeular states ean be viewed as produeing, via the Heisenberg uneertainty relation AEAt >fe, states whose energy is "uneertain" to within an amount AE. 

Lorentzian shape centered at the original energy of the adsorbate. The choice of the energy zero is the same as in the subsequent two figures, but is irrelevant in this case. 

We consider the same atom as in Case 1, with a valence electron at an orbital energy of = 12.0 eV above the bottom of the sp band, when the atom is far from the surface. This level is narrow, like a delta function. When approaching the surface the adsorbate level broadens into a Lorentzian shape for the same reasons as described above, and falls in energy to a new position at 10.3 eV. From Eq. (73) for Wa(e) we see that the maximum occurs for e = -i- A(e), i.e. when the line described... 

As a result, the energy E of photons emitted by an ensemble of identical nuclei, rigidly fixed in space, upon transition from their excited states (e) to their ground states (g), scatters around the mean energy Eq = E. Eg. The intensity distribution of the radiation as a function of the energy E, the emission line, is a Lorentzian curve as given by the Breit-Wigner equation [1] ... 

In the following, we consider the shape and the width of the Mdssbauer velocity spectrum in more detail. We assume that the source is moving with velocity u, and the emission line is an unsplit Lorentzian according to (2.2) with natural width E. If we denote the total number of y-quanta emitted by the source per time unit toward the detector by Nq, the number N E)AE of recoU-free emitted y-rays with energy y in the range to -f dE is given by ([1] in Chap. 1)... 

In a Mdssbauer transmission experiment, the absorber containing the stable Mdssbauer isotope is placed between the source and the detector (cf. Fig. 2.6). For the absorber, we assume the same mean energy q between nuclear excited and ground states as for the source, but with an additional intrinsic shift A due to chemical influence. The absorption Une, or resonant absorption cross-section cr( ), has the same Lorentzian shape as the emission line and if we assume also the same half width , cr( ) can be expressed as ([1] in Chap. 1)... 

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