Class 12 > Unit # 27: Nuclear Physics > Mass Spectrograph

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Principle, Construction and Working of Mass Spectrograph

Video Lecture: Mass Spectrograph

Watch the complete physics lecture explaining the detailed working and mathematical derivation of the mass spectrograph:

Watch directly on YouTube (https://youtu.be/XA6dZGtykqo)

Mass Spectrograph & Principle

Mass Spectrograph: A mass spectrograph is an analytical device used to separate isotopes of an element based on their masses. It works by accelerating charged particles and routing them through electric and magnetic fields, where the resulting deflection determines the particle mass.

Principle

The Bainbridge mass spectrometer operates on the principle of separating charged particles (ions) based on their specific mass-to-charge ratio by passing them through a velocity selector and subsequently a uniform magnetic field. It is primarily used for the accurate determination of isotopic masses and relative abundances.

Construction

1. Ion Source: Produces positively charged ions from a gas sample. These ions are accelerated and collimated through slits ($s_1, s_2$) to form a fine beam of charged particles.
2. Velocity Selector: Contains a perpendicular electric field ($E$) and magnetic field ($B_1$). Only ions possessing a specific velocity ($v = E / B_1$) pass straight through undeflected.
3. Deflection Chamber (Magnetic Analyzer): Contains a second uniform magnetic field ($B_2$) which forces the ions to move in circular trajectories of varying radii depending on their mass, hitting a photographic plate or detector at specific focal points.

Figure: Schematic layout of the Bainbridge Mass Spectrograph showing the Ion Source, Velocity Selector, and Magnetic Deflection Chamber.

Working & Mathematical Derivation

Ions pass through the velocity selector with a unique speed and then enter the magnetic analyzer where they are deflected along circular paths with different radii, recorded on the photographic plate.

Step 1: Velocity Selector Condition

In the velocity selector, the electric force acting on the ion is balanced by the magnetic force:

$F_e = F_m \quad \text{--- (i)}$

Electric force is given by $F_e = qE$ (where $q = e$) and magnetic force is $F_m = qvB_1$. Substituting into equation (i):

$eE = evB_1 \implies v = \frac{E}{B_1}$

Step 2: Acceleration through Potential Difference

Ions are accelerated through a known potential difference $V$, gaining kinetic energy:

$\text{Kinetic Energy} = qV \implies \frac{1}{2}mv^2 = eV$

Rearranging for $v^2$:

$v^2 = \frac{2eV}{m}$

Step 3: Magnetic Analyzer & Mass Calculation

Inside the magnetic analyzer, the magnetic field $B_2$ provides the necessary centripetal force for circular motion:

$\frac{mv^2}{r} = evB_2 \implies mv = eB_2 r \implies m = \frac{eB_2 r}{v}$

Substituting the velocity $v = \frac{E}{B_1}$ into the mass equation:

$m = \frac{eB_2 r}{\left(\frac{E}{B_1}\right)} = \frac{e B_1 B_2 r}{E}$

Squaring both sides:

$m^2 = \frac{e^2 B_1^2 B_2^2 r^2}{E^2}$

Alternatively, substituting $v^2$ from the acceleration step into the magnetic deflection relation ($\frac{m^2 v^2}{r^2} = e^2 B_2^2$):

$m = \frac{e B_2^2 r^2}{2V}$

This expression shows that for fixed values of magnetic field and potential difference, the mass of an ion is directly proportional to the square of the radius ($r^2$) of the path it follows.

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