Q.Discuss the action of a Bainbridge mass spectrometer to determine the isotopic masses.

The Bainbridge mass spectrometer combines a velocity selector (crossed electric and magnetic fields) with a separate uniform magnetic-field region; ions of a given charge and velocity trace semicircular paths whose radius depends only on their mass, letting isotopic masses be measured very precisely.

Construction

The apparatus has three main parts: (i) an ion source that produces positive ions of the element/isotopes to be studied (usually by ionising a vapour and accelerating the ions through a potential difference);

(ii) a velocity selector, a region with a uniform electric field $\vec E$ and a uniform magnetic field $\vec B_1$ arranged perpendicular to each other and to the ion's initial velocity;

(iii) a separate deflecting chamber with a uniform magnetic field $\vec B_2$, with a photographic plate or detector at the far end.

Working

Step 1 — Velocity selection: The ions, of charge $q$, enter the velocity selector, where they experience an electric force $qE$ and a magnetic force $qvB_1$ (from $\vec v \times \vec B_1$). Only ions moving with the particular speed for which these two forces exactly balance travel straight through the narrow exit slit; all others are deflected and blocked. The balance condition is

$qE = qvB_1 \;\Rightarrow\; v = \dfrac{E}{B_1}$

Thus every ion that emerges from the selector — regardless of its mass — has the same known speed $v$.

Step 2 — Circular deflection: These selected ions then enter the second region, where only the uniform magnetic field $B_2$ acts (perpendicular to $\vec v$). The magnetic force provides the centripetal force for circular motion:

$qvB_2 = \dfrac{mv^2}{r} \;\Rightarrow\; r = \dfrac{mv}{qB_2}$

The ion travels a semicircular path of radius $r$ and strikes a photographic plate at a distance $2r$ from the entry slit.

Step 3 — Determining the mass: Since $v = E/B_1$, substituting gives

$r = \dfrac{m}{qB_2}\cdot\dfrac{E}{B_1} \;\Rightarrow\; m = \dfrac{qB_1B_2 r}{E}$

All quantities on the right ($q$, $B_1$, $B_2$, $E$) are known/set by the apparatus, and $r$ is measured from the position of the trace on the photographic plate. Hence the mass $m$ of each isotope can be calculated very precisely.

Determining isotopic masses …