Rutherford scattering ~ INFOSCOPE

Ernest Rutherford

Rutherford scattering is a phenomenon in physics that was explained by Ernest Rutherford in 1911, which led to the development of the Rutherford model (planetary model) of the atom, and eventually to the Bohr model. It is now exploited by the materials analytical technique Rutherford backscattering. Rutherford scattering is also sometimes referred to as Coulomb scattering because it relies only upon static electric (Coulomb) forces, and the minimal distance between particles is set only by this potential. The classical Rutherford scattering of alpha particles against gold nuclei is an example of "elastic scattering" because the energy and velocity of the outgoing scattered particle is the same as that with which it began.

Rutherford also later analyzed inelastic scattering when he projected alpha particles against hydrogen nuclei (protons) ; however this latter process is not referred to as "Rutherford scattering", although Rutherford was first to observe it. At the end of such processes, non-coulombic forces come into play. These forces, and also energy gained from the scattering particle by the lighter target, change the scattering results in fundamental ways which suggest structural information about the target. A similar process probed the insides of nuclei in the 1960s, and is called deep inelastic scattering.

The initial discovery was made by Hans Geiger and Ernest Marsden in 1909 when they performed the gold foil experiment under the direction of Rutherford, in which they fired a beam of alpha particles (helium nuclei) at layers of gold leaf only a few atoms thick. At the time of the experiment, the atom was thought to be analogous to a plum pudding (as proposed by J.J. Thomson), with the negative charges (the plums) found throughout a positive sphere (the pudding). If the plum-pudding model were correct, the positive "pudding", being more spread out than in the current model of a concentrated nucleus, would not be able to exert such large coulombic forces, and the alpha particles should only be deflected by small angles as they pass through.

However, the intriguing results showed that around 1 in 8000 alpha particles were deflected by very large angles (over 90°), while the rest passed straight through with little or no deflection. From this, Rutherford concluded that the majority of the mass was concentrated in a minute, positively charged region (the nucleus/ central charge) surrounded by electrons. When a (positive) alpha particle approached sufficiently close to the nucleus, it was repelled strongly enough to rebound at high angles. The small size of the nucleus explained the small number of alpha particles that were repelled in this way. Rutherford showed, using the method below, that the size of the nucleus was less than about 10−14 m (how much less than this size, Rutherford could not tell from this experiment alone; see more below on this problem of lowest possible size).

Derivation

The differential cross section can be derived from the equations of motion for a particle interacting with a central potential. In general, the equations of motion describing two particles interacting under a central force can be decoupled into the center of mass and the motion of the particles relative to one another. For the case of light alpha particles scattering off heavy nuclei, as in the experiment performed by Rutherford, the reduced mass is essentially the mass of the alpha particle and the nucleus off of which it scatters is essentially stationary in the lab frame.

Substituting into the Binet equation yields the equation of trajectory

$\frac{d^{2}u}{d\theta^{2}}+u=-\frac{Z_{1}Z_{2}e^{2}}{4\pi\epsilon_{0}mv_{0}^{2}b^{2}}=-\kappa,$

where $u={1 \over r}$ , $v_0$ is the speed at infinity, and $b$ is the impact parameter.

The general solution of the above differential equation is

$u=u_{0}\cos(\theta-\theta_{0})-\kappa,$

and the boundary condition is

$u\to 0 \quad r\sin\theta\to b \quad(\theta\to\pi).$

If we choose

$\theta_{0}=\frac{\pi}{2}+\arctan b\kappa.$

then the deflection angle Θ can be seen from solving $u\to 0$ as

$\Theta=2\theta_{0}-\pi=2\arctan b\kappa=2\arctan\frac{Z_{1}Z_{2}e^{2}}{4\pi\epsilon_{0}mv_{0}^{2}b}.$

b can be solved to give

$b=\frac{Z_{1}Z_{2}e^{2}}{4\pi\epsilon_{0}mv_{0}^{2}}\cot\frac{\Theta}{2}.$

To find the scattering cross section from this result consider its definition

${\frac{d\sigma}{d\Omega}(\Omega) d \Omega}={\hbox{number of particles scattered into solid angle d} \Omega \hbox{ per unit time} \over \hbox{incident intensity}}$

Since the scattering angle is uniquely determined for a given $E$ and $b$ , the number of particles scattered into an angle between $\Theta$ and $\Theta+d\Theta$ must be the same as the number of particles with associated impact parameters between $b$ and $b+db$ . For an incident intensity $I$ , this implies the following equality

$2\pi I b \left|db\right| = I \frac{d\sigma}{d\Omega} d\Omega$

For a radially symmetric scattering potential, as in the case of the Coulombic potential, $d\Omega = 2\pi\sin{\Theta}d\Theta$ , yielding the expression for the scattering cross section

$\frac{d\sigma}{d\Omega} = \frac{b}{\sin{\Theta}} \left|\frac{db}{d\Theta}\right|$

Finally, plugging in the previously derived expression for the impact parameter $b(\Theta)$ we find the Rutherford scattering cross section

$\frac{d\sigma}{d\Omega} =\left(\frac{ Z_1 Z_2 e^2}{8\pi\epsilon_0 m v_0^2}\right)^2 \csc^4{\left(\frac{\Theta}{2}\right)}.$

Extension to situations with relativistic particles and target recoil

The extension of Rutherford-type scattering to energy regions in which the incoming particle has spin and magnetic moment, and is traveling at relativistic energies, and there is enough momentum-transfer that the struck particle recoils with some of the incoming particle's energy (so the process is inelastic rather than elastic), is called Mott scattering.

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

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Sunday, 2 February 2014

Rutherford scattering

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