Nuclear Scattering and Atomic Structure

These flashcards cover key terms and concepts related to nuclear scattering, atomic structure, and the methods used to analyze particle interactions in the context of nuclear physics.

Rutherford atom

An atomic model where a small, positively charged nucleus is surrounded by electrons, established through experiments showing large angle scattering of alpha particles.

Scattering experiments

Experiments that reveal information about the size, mass, and charge of atomic nuclei by analyzing how particles scatter off of them.

Cross section

A measure that quantifies the probability of collision between two particles, expressed as a function of incident beam flux and target density.

Mean free path (l)

The average distance a particle travels before undergoing a collision, defined as the reciprocal of the product of target density and total cross section.

Rutherford cross section (sR)

The cross section used to describe the Coulomb scattering of charged projectiles from a positive nucleus.

Mott cross section (sM)

An advanced cross section that accounts for relativistic effects in high-energy scattering and the magnetic moments of particles.

Critical angle (qcr)

The angle at which significant deviations from expected scattering behavior occur, influenced by experimental conditions.

Raleigh criterion

An expression that relates the wavelength of a particle to its scattering behavior and the size of the object being probed.

Woods-Saxon fit

A phenomenological model used to describe charge and mass distributions in atomic nuclei, accounting for non-uniform shapes.

Impact parameter (b)

A measure related to the distance between the trajectory of a projectile and the center of the target nucleus at closest approach.

Fourier transform in scattering

A mathematical tool used to analyze scattering data, allowing for the extraction of charge distributions from experimental results.

Mean free path ($l$) equation

$$l = \frac{1}{n \sigma_{total}}$$

Where:

• $n$ is the target density (particles per unit volume).

• $\sigma_{total}$ is the total cross section.

Rutherford differential cross section equation

$$\frac{d\sigma}{d\Omega} = \left( \frac{zZe^2}{16 \pi \epsilon<em>0 E</em>{kin}} \right)^2 \frac{1}{\sin^4(\theta/2)}$$

This describes the probability of scattering for a charged projectile off a point-like nucleus via Coulomb interaction.

Mott cross section equation

$$\sigma<em>{Mott} = \sigma</em>{Rutherford} \cdot \cos^2\left(\frac{\theta}{2}\right)$$

This specifically accounts for the spin of relativistic electrons being scattered by a point-like nucleus, assuming no recoil.

Woods-Saxon distribution equation

$$\rho(r) = \frac{\rho_0}{1 + \exp\left( \frac{r-R}{a} \right)}$$

Where:

• $R$ is the nuclear radius (where density is half of $\rho_0$).

• $a$ is the surface thickness parameter.

Impact parameter ($b$) equation

$$b = \frac{zZe^2}{8\pi\epsilon<em>0 E</em>{kin}} \cot\left(\frac{\theta}{2}\right)$$

Relates the impact parameter $b$ to the scattering angle $\theta$ for a projectile in a Coulomb field.

Form factor ($F(q)$) Fourier transform

$$F(q) = \int \rho(r) e^{i \vec{q} \cdot \vec{r} / \hbar} d^3r$$

This equation allows for the extraction of the charge density distribution $\rho(r)$ from measured scattering data.

Rayleigh criterion for resolution

$$\sin\theta \approx 1.22 \frac{\lambda}{D}$$

Relates the scattering wavelength $\lambda$ to the size of the object $D$ that can be resolved at an angle $\theta$.

Mean free path ($l$) equation

$l = \frac{1}{n \sigma<em>{total}}$ \n\nWhere: \n- $n$ is the target density (particles per unit volume). \n- $\sigma</em>{total}$ is the total cross section.

Rutherford differential cross section equation

$\frac{d\sigma}{d\Omega} = \left( \frac{zZe^2}{16 \pi \epsilon<em>0 E</em>{kin}} \right)^2 \frac{1}{\sin^4(\theta/2)}$ \n\nDescribes the probability of scattering for a charged projectile off a point-like nucleus via Coulomb interaction.

Mott cross section equation

$\sigma<em>{Mott} = \sigma</em>{Rutherford} \cdot \cos^2\left(\frac{\theta}{2}\right)$ \n\nAccounts for the spin of relativistic electrons being scattered by a point-like nucleus, assuming no recoil.

Woods-Saxon distribution equation

$\rho(r) = \frac{\rho<em>0}{1 + \exp\left( \frac{r-R}{a} \right)}$ \n\nWhere: \n- $R$ is the nuclear radius (where density is half of $\rho</em>0$). \n- $a$ is the surface thickness parameter.

Impact parameter ($b$) equation

$b = \frac{zZe^2}{8\pi\epsilon<em>0 E</em>{kin}} \cot\left(\frac{\theta}{2}\right)$ \n\nRelates the impact parameter $b$ to the scattering angle $\theta$ for a projectile in a Coulomb field.

Form factor ($F(q)$) Fourier transform

$F(q) = \int \rho(r) e^{i \vec{q} \cdot \vec{r} / \hbar} d^3r$ \n\nUsed to extract the charge density distribution $\rho(r)$ from measured scattering data.

Rayleigh criterion for resolution

$\sin\theta \approx 1.22 \frac{\lambda}{D}$ \n\nRelates the scattering wavelength $\lambda$ to the size of the object $D$ that can be resolved at an angle $$\