Rutherford scattering

The Rutherford scattering describes the scattering of charged particles in a charged scattering center. In the initial experiment, the scattering of was alpha particles of gold - atomic nuclei studied. The resulting particle trajectories are hyperbolas . The distribution of the scattered particles allows conclusions to be drawn about the structure of the scattering center. This led to the realization that the positive charge in the atoms is concentrated in a small space in the atomic center. Until then, JJ Thomson's model was valid , in which the positive charge of the atom is homogeneously distributed in a sphere ( Thomson's atomic model ). Hans Geiger , Ernest Marsden and Ernest Rutherford were involved in these experiments . Looking at the measurement results, which indicate that the mass of the atom is concentrated in a small nucleus, Rutherford is reported to have said: "This is as unlikely as shooting a cotton ball with a pistol and the bullet rebounding."

Rutherford scattering test (Manchester, 1909–1913)

Setup and implementation of the experiment

Experimental set-up: 1: radioactive radium, 2: lead jacket for shielding, 3: alpha particle beam, 4: fluorescent screen or photography screen 5: gold foil 6: point at which the rays hit the foil, 7: particle beam hits the screen, only a few Particles are deflected.

A radioactive substance that emits radiation is placed in a lead block with an opening on one side : alpha , beta and gamma radiation . The rays emerging from the opening in the lead block are passed through an electric field to separate them from one another. Characterized the negative are electrons (beta radiation) to the positive pole and the positive helium - atomic nuclei (alpha rays) deflected to the negative pole, while the direction of uncharged photons (gamma-rays) remains unchanged. The alpha radiation is directed vertically onto a gold foil only 0.5 μm thick (approx. 1000 atoms in a row). The radiation emerging from the foil can then be made visible with a fluorescent screen or a film attached to it. Gold was used because even then it could be processed into very thin layers using simple mechanical means and has a high atomic mass. This is where the name gold foil experiment comes from .

observation

Left half: test result as would be expected according to the Thomson model . Right half: Result obtained and illustration with the Rutherford model .

• Almost all alpha particles can pass through the gold foil unhindered.
• About every 10,000th alpha particle is deflected by 90 degrees or more.
• The larger the scattering angle, the less often this deflection occurs.
• Some alpha particles are backscattered.

For the observed distribution, Rutherford developed the below-described scattering formula.

interpretation

The extremely rare deflection of the alpha particles and their angular distribution can be understood from the fact that the atoms only have a very small center of mass that is positively charged. This center of mass is called the atomic nucleus . Since most of the particles pass the gold foil unhindered, there must be a large space between the cores. This result led to Rutherford's atomic model . The electrons, which move around the nucleus in the huge empty space relative to the nucleus diameter, shield the concentrated positive nuclear charge so that the atom appears neutral to the outside.

Rutherford's scattering formula

The Rutherford scattering formula specifies the so-called differential scattering cross-section (also known as the effective cross-section) as a function of the scattering angle in the center of gravity system : ${\displaystyle \vartheta }$

${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}=\left({\frac {1}{4\pi \varepsilon _{0}}}{\frac {Z_{1}Z_{2}e^{2}}{4E_{0}}}\right)^{2}{\frac {1}{\sin ^{4}\left({\frac {\vartheta }{2}}\right)}}}$

The same formula in physically meaningful units:

${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}[\mathrm {barn} ]\approx 1{,}3\cdot 10^{-3}\left({\frac {Z_{1}Z_{2}}{E_{0}[\mathrm {MeV} ]}}\right)^{2}{\frac {1}{\sin ^{4}\left({\frac {\vartheta }{2}}\right)}}}$

This describes the probability that scattered particles will hit the solid angle after being deflected by the angle . ${\displaystyle \vartheta }$ ${\displaystyle \mathrm {d} \Omega =2\pi \sin(\vartheta )\,\mathrm {d} \vartheta }$

The following quantities are still used in the formula:

Electric field constant (dielectric constant)	${\displaystyle \varepsilon _{0}=8{,}854\cdot 10^{-12}\,{\frac {\mathrm {C} }{\mathrm {Vm} }}}$
Charge of the scattered particle	${\displaystyle Z_{1}e}$
Charge of the atomic nucleus	${\displaystyle Z_{2}e}$
Elemental charge	${\displaystyle e=1{,}602\cdot 10^{-19}\,\mathrm {C} }$
Initial energy of the scattered particle	${\displaystyle E_{0}}$

The prefactor can be obtained by using the following quantities:

Fine structure constant	${\displaystyle \alpha \ =\ {\frac {1}{4\pi \varepsilon _{0}}}\;{\frac {e^{2}}{\hbar c}}\approx 1/137}$
Unit for the cross section	${\displaystyle 1\,\mathrm {barn} =100\,\mathrm {fm} ^{2}}$
	${\displaystyle \hbar c=197\,\mathrm {MeV} \cdot \mathrm {fm} }$

Rutherford derived the Rutherford scattering formula from classical physics. A complete quantum mechanical treatment of the problem with the help of the Born approximation shows that the Rutherford scattering formula is correct in the first order and quantum mechanical effects only represent small corrections. Another problem of the Rutherford formula is the limit case , for which the differential cross section becomes infinitely large. However, small angles correspond to a large impact parameter. In the case of very large collision parameters, however, the atomic electrons shield the core. The only way to have very small angles with small impact parameters is to increase the energy of the alpha particles. For very high energies, however, the charge distribution of the atomic nucleus can no longer be assumed to be point-like. Then the form factor of the charge distribution is also included in the scattering formula. In addition, with high projectile energies one can no longer assume that the scattering occurs only through electromagnetic interaction . If both nuclei approach each other up to a contact radius, the strong interaction plays a greater role. ${\displaystyle \vartheta =0}$

Plausibility examination of the dependencies

According to Feynman's rules , the scattering of one particle of the charge on a second particle results in the charge for the matrix element ${\displaystyle Z_{1}e}$ ${\displaystyle Z_{2}e}$

${\displaystyle M_{fi}\sim (Z_{1}e)\cdot (Z_{2}e)\;{\textrm {,}}}$

whereby the propagator was neglected. After Fermi's golden rule applies

${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\sim |M_{fi}|^{2}\;,}$

with which it follows that

${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}\sim (Z_{1}e)^{2}\cdot (Z_{2}e)^{2}=(Z_{1}Z_{2}e^{2})^{2}.}$

Derivation of the Rutherford scattering formula

Due to the repulsive effect of the Coulomb force , the orbit of the alpha particle ( ) becomes a hyperbola. ${\displaystyle F={\frac {Z_{1}Z_{2}e^{2}}{4\pi \varepsilon _{0}r^{2}}}}$ ${\displaystyle Z_{1}=2}$

Rutherford scattering from an atomic point of view

The major semi-axis a of the hyperbola can be seen from the approach

${\displaystyle E_{\mathrm {kin} }=e\Phi _{c}={\frac {Z_{1}Z_{2}e^{2}}{4\pi \varepsilon _{0}2a}}={\frac {Z_{1}Z_{2}e^{2}}{8\pi \varepsilon _{0}a}}}$

determine, where is the minimum distance of the alpha particle when it collides centrally with the core. depends on the kinetic energy and can also be used for collisions that are not central. The impact parameter is the minimum distance between the alpha particle and the core if it were to continue flying in a straight line. In fact, the alpha particle is scattered around the angle . The following equations are obtained from the geometry of the hyperbola: ${\displaystyle 2a}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \vartheta }$

${\displaystyle \tan \left(\alpha \right)={\frac {b}{a}}=\tan \left(90^{\circ }-{\frac {\vartheta }{2}}\right)=\cot \left({\frac {\vartheta }{2}}\right)}$,

there and with it ${\displaystyle 2\alpha +\vartheta =180^{\circ }}$

${\displaystyle \cot {\frac {\vartheta }{2}}={\frac {b}{a}}={\frac {8\pi \varepsilon _{0}E_{\mathrm {kin} }}{Z_{1}Z_{2}e^{2}}}b}$.

By deriving the last formula, you get a relationship between the width of a hollow cone and the corresponding width of the deflection angle . ${\displaystyle \mathrm {d} b}$ ${\displaystyle \mathrm {d} \vartheta }$ ${\displaystyle \vartheta }$

${\displaystyle -{\frac {1}{2\sin ^{2}{\frac {\vartheta }{2}}}}\mathrm {d} \vartheta ={\frac {8\pi \varepsilon _{0}E_{\mathrm {kin} }}{Z_{1}Z_{2}e^{2}}}\mathrm {d} b}$

Cross-section when the alpha particles pass through the film

Let the particle density ( atoms per volume ) of the scattering material and the thickness of the film be the average cross-sectional area per atom that the alpha particle experiences when passing through the film. is also called the cross section. ${\displaystyle z={\frac {n}{V}}}$ ${\displaystyle n}$ ${\displaystyle V}$ ${\displaystyle x}$ ${\displaystyle \sigma ={\frac {A}{n}}={\frac {\frac {V}{x}}{n}}={\frac {1}{zx}}}$ ${\displaystyle \sigma }$

The probability of landing in the ring of the hollow cylinder then results from ${\displaystyle P(\vartheta )\mathrm {d} \vartheta }$

${\displaystyle P(\vartheta )\mathrm {d} \vartheta ={\frac {A_{b}}{\sigma }}={\frac {2\pi bdb}{\frac {1}{zx}}}=zx2\pi b\mathrm {d} b}$.

Scatter cone in the Rutherford experiment

Of particles are scattered into the hollow cone. The likelihood of this is ${\displaystyle N}$ ${\displaystyle \mathrm {d} N'}$

${\displaystyle {\begin{aligned}P(\vartheta )\vartheta &={\frac {\mathrm {d} N'}{N}}=zx2\pi b\mathrm {d} b\\&=zx2\pi {\frac {Z_{1}Z_{2}e^{2}}{8\pi \varepsilon _{0}E_{\mathrm {kin} }}}\cot {\frac {\vartheta }{2}}\cdot {\frac {Z_{1}Z_{2}e^{2}}{8\pi \varepsilon _{0}E_{\mathrm {kin} }}}\cdot {\frac {1}{2\sin ^{2}{\frac {\vartheta }{2}}}}\mathrm {d} \vartheta \\&=zx{\frac {Z_{1}^{2}Z_{2}^{2}e^{4}}{64\pi \varepsilon _{0}^{2}E_{\mathrm {kin} }^{2}}}\cdot {\frac {\cos {\frac {\vartheta }{2}}}{\sin ^{3}{\frac {\vartheta }{2}}}}\mathrm {d} \vartheta \end{aligned}}}$

${\displaystyle \mathrm {d} N}$gives the number of particles in the solid angle . ${\displaystyle \mathrm {d} \Omega }$

${\displaystyle \mathrm {d} \Omega =2\pi \sin(\vartheta )d\vartheta =4\pi \sin {\frac {\vartheta }{2}}\cos {\frac {\vartheta }{2}}\mathrm {d} \vartheta }$

It follows:

${\displaystyle \mathrm {d} \vartheta ={\frac {1}{4\pi \sin {\frac {\vartheta }{2}}\cos {\frac {\vartheta }{2}}}}\mathrm {d} \Omega }$.

So results for the probability

${\displaystyle {\frac {\mathrm {d} N}{N}}=zx{\frac {Z_{1}^{2}Z_{2}^{2}e^{4}}{256\pi ^{2}\varepsilon _{0}^{2}E_{\mathrm {kin} }^{2}}}\cdot {\frac {1}{\sin ^{4}{\frac {\vartheta }{2}}}}\mathrm {d} \Omega }$

This is the Rutherford scattering formula. It indicates how high the probability is for a particle to be scattered in the solid angle . ${\displaystyle \mathrm {d} \Omega }$

The scattering formula is often given with the help of the differential cross section . It is a measure of the same probability. ${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}}$

It applies

${\displaystyle {\frac {\mathrm {d} N}{N}}={\frac {\mathrm {d} \sigma }{\sigma }}=z\cdot x\cdot \mathrm {d} \sigma }$

and thus

${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}={\frac {Z_{1}^{2}Z_{2}^{2}e^{4}}{256\pi ^{2}\varepsilon _{0}^{2}E_{\mathrm {kin} }^{2}}}\cdot {\frac {1}{\sin ^{4}{\frac {\vartheta }{2}}}}=\left({\frac {Z_{1}Z_{2}e^{2}}{4\pi \varepsilon _{0}\cdot 4\cdot E_{\mathrm {kin} }}}\right)^{2}\cdot {\frac {1}{\sin ^{4}{\frac {\vartheta }{2}}}}.}$

Remarks

1. ${\displaystyle \vartheta =0}$is not defined as there is a minimum deflection angle . This is assumed when the alpha particle moves at a distance from the atom, i.e. at the edge of the circular cross-sectional area. For a larger impact parameter , the alpha particle is in the stray field of the neighboring atom and the deflection angle increases again. The following applies: ${\displaystyle \vartheta _{\mathrm {min} }}$ ${\displaystyle b=b_{\mathrm {max} }}$ ${\displaystyle b}$
   

   ${\displaystyle \sigma ={\frac {A}{n}}=b_{\mathrm {max} }^{2}\cdot \pi }$
   and

   ${\displaystyle \tan {\frac {\vartheta _{\mathrm {min} }}{2}}={\frac {Z_{1}Z_{2}e^{2}}{8\pi \varepsilon _{0}E_{\mathrm {kin} }\cdot b_{\mathrm {max} }}}}$.

2. The integral over the probability distribution gives 1 ${\displaystyle P(\vartheta )\mathrm {d} \vartheta }$

   ${\displaystyle \int _{\vartheta _{\mathrm {min} }}^{\pi }{P(\vartheta )\mathrm {d} \vartheta }=1}$

3. The same applies to the area integrals

   ${\displaystyle \int _{\vartheta \geqslant \vartheta _{\mathrm {min} }}{zx{\frac {Z_{1}^{2}Z_{2}^{2}e^{4}}{256\pi ^{2}\varepsilon _{0}^{2}E_{\mathrm {kin} }^{2}}}\cdot {\frac {1}{\sin ^{4}{\frac {\vartheta }{2}}}}\mathrm {d} \Omega }=1}$
   and

   ${\displaystyle \int _{\vartheta \geqslant \vartheta _{\mathrm {min} }}\left({\frac {Z_{1}Z_{2}e^{2}}{4\pi \varepsilon _{0}\cdot 4\cdot E_{\mathrm {kin} }}}\right)^{2}\cdot {\frac {1}{\sin ^{4}{\frac {\vartheta }{2}}}}\mathrm {d} \Omega =\sigma }$

See also

• Rutherford Backscattering Spectrometry

literature

• E. Rutherford, The Scattering of α and β Particles by Matter and the Structure of the Atom , Philosophical Magazine. Series 6, 21 (May 1911) p. 669–688 ( PDF , short version)
• H. Geiger and E. Marsden , On a Diffuse Reflection of the α-Particles ( Memento of April 24, 2008 in the Internet Archive ), Proceedings of the Royal Society 82A (1909), p. 495-500
• H. Geiger and E. Marsden , LXI. The Laws of Deflexion of a Particles through Large Angles , Philosophical Magazine 25 (1913), p. 604-623
• Gerthsen, Kneser, Vogel: Physik , 16th edition, pp. 630-633, Springer-Verlag

Web links

• LP - Rutherford's scattering experiment (including sketches, photos, video and references)
• Rutherford experiment at student level ( LEIFIphysik )

Individual evidence

1. ↑ Wolfgang Demtröder: Experimental Physics 3: Atoms, Molecules and Solids . Springer Berlin Heidelberg, June 13, 2016, ISBN 978-3-662-49094-5 , p. 64.