Rutherford Scattering Formula

Before 1909, the only image that one had in his mind about an atom was that of plums in a pudding. However, in 1909, a revolutionary experiment took place. Undoubtedly, this experiment was powerful enough to give a jolt to the known atomic structure. Yeah! I'm talking about the epic Gold foil experiment. In addition to introducing a new member to the atomic structure, the Nucleus, this also set the stage for the profound Bohr's Model of atom. So, Our fourteenth article in the Series "Month of Equation" is dedicated to the mathematical explanation of this experiment given by Ernest Rutherford in 1911, the "Rutherford Scattering formula".

The Rutherford Scattering formula basically gives the relationship between the number of particles deflected by a heavy mass and the angle of deflection. In simple words,

For a detector at a specific angle with respect to the incident beam, the number of particles per unit area striking the detector is what Rutherford formula gives !

Mathematical Expression:

The mathematical expression for the formula is as follow :

{ \bf{N(\theta) = \frac{N_{i} n L Z^{2} k^{2} e^{4}}{4 r^{2} E^{2} \sin^4(\theta/2)}}}

  • Here, N(θ) represents the number of alpha particles scattered corresponding to the Scattering angle θ ,
  • {N_{i}} is the number of incident alpha particles
  • n is the number of atoms per unit volume in target
  • L is the thickness of target
  • Z is the atomic number of target
  • e is the electronic charge
  • k is the Coulomb's constant
  • r is distance between target and detector
  • E is the incident energy of incident alpha particles
  • θ is the scattering/deflection angle between the detector and direction of incident beam

So, this formula says the the number of particles that would be get deflected by an angle θ due to scattering from a fixed nuclei is inversely proportional to the fourth power of the sine function of one half the angle of deflection. Rutherford allowed a beam of alpha particles to impinge on a very thin gold foil. Some particles passed without any deflection, some got deflected at small angle, some at large and some even back scattered.Moreover, this experiment was also performed with different thickness of gold foil as well and observations were made. Thus, this formula was basically given to generalise the distribution of the deflected alpha particles.

Rutherford scattering formula has been derived by considering the coulomb force between the alpha particles and target nuclei, the laws of conservation of momentum, trajectory of incident particles, etc.

Rutherford Scattering
Setup of Rutherford scattering experiment
Courtesy: Pinterest

Relation between Impact parameter and Scattering angle:

Rutherford scattering formula suggests that the number of particles detected by the detector depend on the deflection/scattering angle. However, the angle of deflection further depends upon several factors like Impact parameter. Okay! First of all let's try to understand what is Impact parameter. Impact parameter (b) is the perpendicular distance between the path of a projectile and the horizontal where the target is placed. It is clearly shown in the image below. Large deflections occur for small impact parameters. The scattering of the alpha particle by the central repulsive Coulomb force leads to a hyperbolic trajectory. From the scattering angle and momentum, the impact parameter can be calculated. Another version of Rutherford scattering formula is also there in terms of Cross section.

Rutherford Scattering
Impact parameter and Scattering angle

Also read: The entry of Bohr and Rutherford in Quantum Physics

Author's Message:

Rutherford scattering experiment and the formula set some benchmarks in the field of research. It became the first method ever used to measure the size of nuclei. Moreover, these days, it is highly exploited in field of material sciences. So, whether you are an Experimentalist or a theoretical physicist, you will encounter this formula for sure. Although, in this article, I have not explained the mathematical steps behind the derivation of this formula. But, if anyone is interested, you can easily find them in several books.

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