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X-ray and Neutron Scattering - Coggle Diagram
X-ray and Neutron Scattering
The scattering is characterized by the resultant change in its momentum
P
and energy,
E
Elastic Scattering
\( E = 0 \Leftrightarrow \omega = 0\)
In x-ray scattering, the photon is massless and the reason for the previous statement is
Dispersion relation: \(c=\frac{\omega}{k}\) where c is the speed of light and k = |
k
|
\( E_i = \hbar \omega_i = \hbar c |k_i|\) and \( E_f = \hbar \omega_f = \hbar c |k_f|\)
\( |k_i| = |k_f| = \frac{2\pi}{\lambda} \)
Trigonometry
\( Q = \frac{4\pi sin(\theta)}{\lambda}\)
For neutron scattering, the neutrons have a mass \(m_n\) and the energy transfer is given by the change in its kinetic energy
\( E = \frac{|\hbar k_i|^2}{2m_n} - \frac{|\hbar k_f|^2}{2m_n}\)
Inelastic scattering
\( E = \hbar\omega\) where \( \omega = \omega_i - \omega_f\)
\( \omega = 2\pi\nu\)
\( \nu \) is the velocity of the neutron
The differential scattering cross section, \( \frac{d\sigma}{d\Omega}\)
For elastic scattering the energy has been integrated out
For
X-RAYS
The number of particles of wavelength \(\lambda\) deflected in the direction of \(2\theta\) and \(\phi\) per unit time and area is given by the modulus squared
\(|\phi_f|^2 = \phi_f\phi^*_f = \frac{\Phi}{r^2}|f(\lambda,\theta)|^2\)
\(\sigma(\lambda) = 2\pi \int^{\pi}_{2\theta=0} |f(\lambda,\theta)|^2 \text{sin}s\theta d2\theta\)
For
NEUTRONS
\(\sigma = 4\pi|b|^2\)
For both x-rays and neutrons, the incoming particles can be described as a plane wave
\( \psi_i = \psi_0e^{ikr}\)
After interacting with the atom the particles move radially outwards. The scattered particles can hence be described as a spherical wave.
Single atom: \( \psi_f = \psi_0f(\lambda,\theta)\frac{e^{ikr}}{r}\)
Multiple atoms: \( \psi_f = \psi_0 e^{ik_ir_j}\sum^{N}_{j=1}f_j(\lambda,\theta)\frac{e^{iQr_j}}{|r-r_j|}\)
The distance to the detectors, where the measurements are taken, is much larger than the typical size of a sample, This means that, to a very good approximation: \(|r-r_j| = |r| = r\)
By taking the square modulus, replacing \(|r-r_j|\) with \(r\) and moving is out from the sum, the differential cross section can be related to the structure of the sample:
\( \left(\frac{d\sigma}{d\Omega}\right)_{el} \propto \left| \sum^{N}_{j=1} f_j(\lambda,\theta) e^{iQr_j}\right|^2 \)
For
NEUTRONS
\(f(\lambda,\theta) = -b\)
b is referred to as the scattering length and is a measure of how strong the interaction between the incident neutron and the nuclei is.
The negative sign is only a matter of convention and determines whether the incident and outgoing waves are in or (180 degrees) out of phase.
The scattering lengths are in principle complex numbers but the imaginary part is in general so small so that b can be treated a real.
Depends on the makeup of the nucleus and the orientation of its spin (if not zero) relative to that of the neutron.
The scattering lengths are isotope specific
The scattering lengths have two different values for nuclei with a non-zero spin
Average value and standard deviation: \(b = \langle b \rangle \pm \Delta b\)
This is also true for a sample with a natural mixture of isotopes for a given atom.
The variance can be written as: \(\langle b^2 \rangle = \langle b \rangle^2 + (\Delta b)^2\)
This enables the average scattering cross section to be written as the sum of the coherent and incoherent cross sections
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The scattering lengths do not vary with atomic number in a simple or monotonic way
For
X-RAYS
In x-ray scattering the interaction is long-range and electromagnetic with the orbital electrons
\( f(\lambda,\theta)\) diminishes monotonically with increasing \(\theta\) and decreasing \(\lambda\)
Depends on the element
It is a function of \(sin(\theta)/\lambda\)
This decay is known as the atomic form factor: \(f(\lambda,\theta) = Zg(Q)r_e\)
\( f(\lambda,\theta)\) has the same sign for all elements
\( f(\lambda,\theta)\) has a magnitude proportional to the atomic number Z
The characteristics of \(f(\lambda,\theta)\) are determined by:
Size
Shape
Can be quantified by the scattering length density (SLD) function \(\beta(x,y,z)\) of the appropriate material: \(f(\lambda,\theta) = \beta(r_j)dV\)
\( \left(\frac{d\sigma}{d\Omega}\right)_{el} \propto \left| \int\int\int \beta(R) e^{iQR} d^3R\right|^2 \propto S_{el(Q)} \)
The elastic differential cross-section is related to the structure of the sample through the Fourier transform of its SLD function.
X-rays and synchrotron sources
The simplest way to produce x-rays is through the generation of very high temperatures. All objects give our electromagnetic waves, called
black body
radiation, but the bulk of these emissions occur at a wavelength, \(\lambda_{max}\), which depends inversely on its temperature, T.
Another way is through the use of thermal energy
An electron can be accelerated towards a metal target by the application of a positive voltage.
Reactors and pulsed neutron sources