Dispersion and Spectra

When a light ray passes symmetrically through a prism of refracting angle A, it suffers minimum deviation 𝛿. For small angles, the deviation is given by 𝛿 = (𝜇 - 1)A, where 𝜇 is the refractive index. The angular dispersion is the difference in deviation between violet and red light: 𝛿v - 𝛿r = (𝜇v - 𝜇r)A. Dispersive power (𝜔) is defined as 𝜔 = (𝜇v - 𝜇r) / (𝜇y - 1), where 𝜇v, 𝜇r, and 𝜇y are the refractive indices for violet, red, and yellow light, respectively.

Two thin prisms placed in contact with refracting angles A and A' and dispersive powers ω and ω', respectively. The deviations produced by the two prisms are δ = (μ - 1)A and δ' = (μ' - 1)A'.

Emission Spectra: Light emitted by an object that is suitably excited. Carries information about the source material.

• Continuous Spectrum: Source emits light with continuously varying wavelengths. When dispersed, it forms a bright, continuous spectrum on a dark background without sharp boundaries. Example: Electric bulb, candle, red-hot iron.
• Line Spectrum: Atoms/molecules emit light when transitioning from an excited state to lower energy levels. Light has certain fixed wavelengths. When dispersed, it forms sharp bright lines on a dark background. Carries information about the atoms of the source.
• Band Spectrum: Molecular energy levels are grouped into bunches. Wavelengths emitted are also grouped. Spectrum looks like separate bands of varying colors.

Absorption Spectrum: When white light passes through an absorbing material, certain wavelengths are absorbed. The transmitted light, when dispersed, shows dark lines or bands at the missing wavelengths on a continuous colored background. Missing wavelengths give information about the absorbing material.

If the slit is wide, different points of the slit produce separate spectra which overlap each other. The color impression gets diffused due to the overlap. Such a spectrum is called an impure spectrum.

The spectrometer consists of a collimator, prism table, and telescope. Before use, the spectrometer needs to be adjusted: 1. **Leveling:** The axis of the collimator tube, telescope tube, and the surface of the prism table must be made horizontal using leveling screws. 2. **Collimator Adjustment:** Adjust the distance between the slit and collimating lens so that the slit is in the first focal plane of the lens, resulting in parallel rays from the collimator. 3. **Telescope Adjustment:** Focus the telescope for parallel rays (normal adjustment). **Applications:** **(a) Measuring the Angle of a Prism:** * Place the prism on the prism table with its refracting edge facing the collimator. * Rotate the telescope to receive the beam reflected from one surface (position T1). Ensure the slit image coincides with the vertical crosswire. * Read the angular position on the base using the vernier scale. * Rotate the telescope to position T2, where it receives the beam reflected from the other surface, again ensuring the slit image coincides with the crosswire. * Read the angular position again. * The difference between the two readings gives the angle rotated by the telescope, which is equal to 2A, where A is the angle of the prism. Sodium vapour lamp gives a nearly monochromatic light which makes the image identification easier. **(b) Measuring the Angle of Minimum Deviation for a Prism for a Given Wavelength:** * Adjust the spectrometer as described before. * Place the light source behind the slit. * Rotate the telescope to a position where the angle between the telescope axis and the collimator axis is large. * Place the prism on the prism table and rotate the table until the refracted beam is received by the telescope. Make the slit image coincide with the crosswire. * The angle between the collimator and telescope axes is the angle of deviation (δ). * Rotate the telescope slightly towards the collimator axis to decrease δ. Rotate the prism table to bring the slit image back to the crosswire. * Repeat the process until you reach a position where further rotation of the telescope towards the collimator axis prevents bringing the image back to the crosswire. This is the position of minimum deviation. * The angle between the collimator and telescope axes at this position is the angle of minimum deviation. * Note the reading of the vernier scale attached to the telescope. * Remove the prism and align the telescope with the collimator, so the slit image forms at the crosswire. Read the vernier scale again.

Dispersion and Spectra deals with the behavior of light as it passes through prisms, specifically focusing on dispersion (the splitting of light into its constituent colors) and the creation of achromatic combinations (prisms that produce deviation without net dispersion). The text also introduces the concept of dispersive power of a material.

Dispersion occurs because the refractive index of the prism material is different for different wavelengths (colors) of light. Violet light deviates the most, while red light deviates the least.

Angular dispersion is the difference between the deviation of violet light and the deviation of red light. A combination of prisms can be designed to achieve either no net angular dispersion or no net deviation.

When combining thin prisms, the refracting angles can be similarly directed or oppositely directed. The direction affects the total deviation and angular dispersion. The equations depend on the angles of the prisms and the refractive indices for different colors of light.