Resonant scattering length

For the case of 2 -pole resonance, the resonant scattering length is given by [41]... 

The expression (1.5) for the resonant scattering length can be expanded in powers of the unit vector m that defines the magnetic quantization axis of the atom in the sample. The resonant scattering length for an electric dipole transition (L= I) gets the form [42]... 

Fig. 8.10 Laser-induced Feshbach resonance scattering length a (in terms of the Bohr radius ao) (solid line) and inelastic-collision rate coefficient Ainei (dashed line) as a function of the detiming of the laser from the photoassociation resonance for typical parameters of Inset schematic diagram of the optical coupling of the scattering unbound state A with the excited bound state B. (Reprinted from Theis et al. 2004 with courtesy and permission of the American Physical Society.)...

Figure 2 Variations in the neutron scattering amplitude or scattering length as a function of the atomic weight. The irregularities arise from the superposition of resonance scattering on a slowly increasing potential scattering. For comparison the scattering amplitudes for X rays under two different conditions are shown. Unlike neutrons, the X-ray case exhibits a monotonic increase as a function of atomic weight.

However, a series of factors introduce losses in the system namely, the reflectivities of the mirrors (RiandR2) on the figure, which reflect only a fraction, Ri and R2, of the intensity. Additional losses can be produced by absorption in the windows of the cell that contains the active medium (if this is the case), diffraction by apertures, and scattering due to particles or imperfect surfaces. All of those losses can be included in a loss factor per trip, expressed as e. Thus, considering both amplification and intensity decrease per round trip, the intensity after a single round trip through a resonator of length d is... 

In the X-ray anomalous scattering method we make use of the fact that the scattering length of an element changes appreciably near an absorption resonance and is given by ... 

Resonance scattering means that neighbours and isotopes can have very different scattering lengths, e.g. two nickel isotopes, Ni (0.87) and Ni (1.44), or the transition elements Fe (0.95), Co (0.25) and Ni (1.02) (all X 10 m), whereas for X-rays, scattering would be closely proportional to the atomic numbers 26, 27 and 28, respectively. Neutron diffraction is therefore particularly useful in the study of alloys. 

Another type of application for synchrotron radiation is based upon the possibility to continuously vary the wave-length and perform diffraction experiments close to the absorption edge of a specific element. This approach, termed anomalous dispersion or resonant scattering is in principle well suited to obtain spatial information on certain groups of atoms within a membrane assembly, notably bound metal ions, but has so far not found broader use in this field. For a review on this technique, see the chapter by Stuhrmann in this volume. 

The spin and parity assignments shown are based on the analysis of resonance reactions for the 7 MeV levels these levels have comparable reduced widths. The 6 MeV levels must be formed by s-nucleons if the large thermal neutron scattering length, and the lv proton absorption in Li are to be simply explained ... 

As we have seen above the observed low energy scattering lengths may nevertheless be determined by the proximity of sharp neutron resonances. Adair S... 

The scattering length R may also be determined from direct measurements of the total cross section in thick material. In this case, the neutrons with energies corresponding to the resonance energies are removed by absorption and the measured transmission depends mainly on those neutrons whose energies lies between resonances. These neutrons are attenuated effectively only by potential scattering. 

In the region between resonances the net contribution of distant resonances may not cancel for scattering, but may give a systematic modification of the effective scattering length. The consequences of this point are explored in greater detail in the following section in connection with the theory of Feshbach, Porter, and Weisskopf. 

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