Scattering cross sections

In ion-solid interactions, it is customary to describe the number of particles scattered through different angles, 9, in terms of a quantity called the angular differential scattering cross-section. Imagine the experiment depicted in Fig. 4.1, where a beam of ions is incident on a thin foil and is scattered into a detector of area Aa at a polar angle between 9 and 9 + A9. Each of the ions in the incident 

We now define da(d ), the differential scattering cross-section, to be given by da(0) 1 dn  

On examining Fig. 4.2, we see that all incident particles with impact parameter b are headed in a direction to strike the rim of the circle drawn aroimd the target nucleus and will be deflected by an angle 0c- The area of this circle is and any particle with a trajectory that strikes anywhere within this area will be deflected by an angle greater than 6c- The target area defined by the impact parameter is called the total cross-section, cr 9c)  

For projectiles moving with small values of b, the cross-section defined by (4.3) will be small, but, due to the interaction forces, the scattering angle will be large. Thus, b is proportional to cr 9, while b and a 9 are inversely related to 9c. From this discussion we see that b = b 9.  

In addition to the total cross-section, there is the differential cross-section, dcr (0c), and its relationship to b. As shown in Fig. 4.3, particles incident with 

Let a beam of particles strike a particle which is at rest in a laboratory system. One may then have coordinates in which the fixed particles is the center. One may also, as we shall do here, choose center-of-mass coordinates for the system composed of the struck and striking particles, where the center of mass of the two particles is at rest at the center of the coordinates. If is the mass of the struck particle and Wj is the mass of the striking particle, this is mathematically equivalent to a fictitious particle of reduced mass 

Suppose now t t we have one scattering center and an incident flux of N particles per unit area per unit time, with the flux small enough for there to be no interference between particles in the beam. Then the number of particles scattered per unit time into a solid angle dCl is Na d, ) dCl. Here 0 and f are defined with respect to the fixed center of mass, and is a proportionality factor known as the differential scattering cross-section. The total scattering cross-section is given as 

It should be mentioned that the scattering cross-section refers to the asymptotic behavior of the scattered particles, i.e., the behavior after traversing a comparatively large distance from the place of collision. However, one does not usually obtain the asymptotic form without consideration of the general solution. 

We shall have instances of this procedure in the ensuing sections. Now, we wish only to find the relationship between the transition probability and the differential scattering cross-section for a particular example. 

Assuming our particles are without internal structure and that the functions ipj in Eq. (127) are plane waves normalized to unity in a large cubical box, L , the initial state is then given as 

Let s assume that we have scatterer (e.g. a metal nanoparticle) inside the vacuum which is excited by an incident uniform plane-wave with wavevector k, c = kk G The electric and magnetic incident fields are (see Sec. 1.1.1)  

The total (incident plus scattering) electromagnetic field can be written as [15]  

If we define a surface far away around the nanoparticle, the averaged power absorbed will be  

Note that all the sources of the field are outside the surface A, and thus Pabs T can be positive or zero (if the imaginary part of dielectric constant of the nanoparticle is zero). 

These equations have been simply obtained using Eqs. (1.275, 1.276) in Eq. (1.41). 

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We have seen that the strength of Raman scattered radiation is directly related to the Raman scattering cross-section (Oj ). The fact that this cross-section for Raman scattering is typically much weaker than that for absorption (oj limits conventional SR as a sensitive analytical tool compared to (Imear) absorption... 

Neutron scattering depends upon nuclear properties, which are related to fluctuations in the neutron scattering cross section a between the scatterer and the surroundings. The scattered amplitude from a collection of scatterers can thus be written as (similar to (B 1.9.29)) ... 

The ratio of elastically to inelastically scattered electrons and, thus, their importance for imaging or analytical work, can be calculated from basic physical principles consider the differential elastic scattering cross section... 

The intensity of SS /. from an element in the solid angle AD is proportional to the initial beam intensity 7q, the concentration of the scattering element N., the neutralization probability P-, the differential scattering cross section da(0)/dD, the shadowing coefficient. (a, 5j ) and the blocking coefficient(a,5 ) for the th component on the surface ... 

The identity of target elements is established by the energy of the scattered particles after an elastic collision. The number of atoms per unit area, N, is found from the number of detected particles (called the yield, Y) for a given number Q of particles incident on the target. The connection is given by the scattering cross section a(9) by... 

This is shown schematically in figure B 1.24.3. In the simplest approximation the scattering cross section a is given by... 

Figure Bl.24.13. A thin film of LaCaMn03 on an LaA103 substrate is characterized for oxygen content with 3.05 MeV helium ions. The sharp peak in the backscattering signal at chaimel 160 is due to the resonance in the scattering cross section for oxygen. The solid line is a simulation that includes the resonance scattering cross section and was obtained with RUMP [3]. Data from E B Nyeanchi, National Accelerator Centre, Fame, South Africa.

This is the Rutherford scattering cross section. It is interesting to note that Bom and classical theory also reproduce this cross section. Moreover,... 

The classical scattering cross section for a given process is simply... 

The peak absorption (scattering) cross sections are thus useful comparative measures of detectivity because the latter is a product of the line strength and the practical line resolution. 

When the exciting frequency is nonresonant (distant from any electronic transition), the differential scattering cross section at wavelength X is as in equation 8 ... 

Beryllium has a high x-ray permeabiUty approximately seventeen times greater than that of aluminum. Natural beryUium contains 100% of the Be isotope. The principal isotopes and respective half-life are Be, 0.4 s Be, 53 d Be, 10 5 Be, stable Be, 2.5 x 10 yr. Beryllium can serve as a neutron source through either the (Oi,n) or (n,2n) reactions. Beryllium has alow (9 x 10 ° m°) absorption cross-section and a high (6 x 10 ° m°) scatter cross-section for thermal neutrons making it useful as a moderator and reflector in nuclear reactors (qv). Such appHcation has been limited, however, because of gas-producing reactions and the reactivity of beryUium toward high temperature water. 

If the displacements of the atoms are given in terms of the harmonic normal modes of vibration for the crystal, the coherent one-phonon inelastic neutron scattering cross section can be analytically expressed in terms of the eigenvectors and eigenvalues of the hannonic analysis, as described in Ref. 1. 

While the underlying mechanisms of HREELS are pretty well understood, many important details relating to selection rules and scattering cross sections remain unknown. 

The relative number of particles backscattered from a target atom into a given solid angle for a given number of incident particles is related to the differential scattering cross section ... 

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