As part of his discovery of the neutron in 1932, James...

As part of his discovery of the neutron in $1932,$ James Chadwick determined the mass of the newly identified particle by firing a beam of fast neutrons, all having the same speed, at two different targets and measuring the maximum recoil speeds of the target nuclei. The maximum speeds arise when an elastic head-on collision occurs between a neutron and a stationary target nucleus. (a) Represent the masses and final speeds of the two target nuclei as $m_{1}, v_{1}, m_{2},$ and $v_{2}$ and assume Newtonian mechanics applies. Show that the neutron mass can be calculated from the equation $$m_{n}=\frac{m_{1} v_{1}-m_{2} v_{2}}{v_{2}-v_{1}}$$ (b) Chadwick directed a beam of neutrons (produced from a nuclear reaction) on paraffin, which contains hydrogen. The maximum speed of the protons ejected was found to be $3.3 \times 10^{7} \mathrm{m} / \mathrm{s}$. since the velocity of the neutrons could not be determined directly, a second experiment was performed using neutrons from the same source and nitrogen nuclei as the target. The maximum recoil speed of the nitrogen nuclei was found to be $4.7 \times 10^{6} \mathrm{m} / \mathrm{s}$. The masses of a proton and a nitrogen nucleus were taken as $1 \mathrm{u}$ and $14 \mathrm{u}$, respectively. What was Chadwick's value for the neutron mass?

Key Concepts

Newtonian Mechanics

Newtonian mechanics is the classical framework used to analyze the motion of objects moving at speeds much less than the speed of light. It underpins the methods used in collision analysis through its fundamental laws of motion, ensuring that the conservation equations for momentum and kinetic energy can be applied accurately.

Conservation of Kinetic Energy

In elastic collisions, not only is momentum conserved, but the total kinetic energy of the system is also conserved. This additional conservation law provides a second equation that can be used alongside momentum conservation to determine unknown quantities like particle masses or recoil velocities.

Experimental Determination of Particle Mass

This concept involves using observable quantities, such as recoil speeds from collisions, along with conservation principles to calculate the mass of an unknown particle. By setting up experiments where different target masses yield measurable recoil velocities, one can derive equations that lead to the determination of the unknown particle's mass.

Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. This fundamental concept in classical mechanics allows for the derivation of relationships between the masses and velocities of colliding particles in a head-on collision, which is essential in analyzing the outcomes of particle interactions.

Conservation of Momentum

The conservation of momentum principle states that in an isolated system, the total momentum before a collision equals the total momentum after the collision. This concept is critical for formulating the equations that relate the initial and final velocities of interacting particles and is widely used in physics to solve problems involving collisions.

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