Dispersion and Spectra

Continuous Emission Spectrum:

• Appearance: Continuous distribution of colors without gaps
• Origin: Objects with closely packed energy levels (solids, liquids, dense gases)
• Examples: Incandescent bulbs, candle flames, heated solids
• Physics: Thermal excitation causes transitions between numerous overlapping energy levels

Line Emission Spectrum:

• Appearance: Discrete bright lines on dark background
• Origin: Individual atoms making specific quantum transitions between energy levels
• Examples: Sodium vapor lamps (589.0nm & 589.6nm yellow lines)
• Physics: Excited atoms emit photons of precise energies when electrons transition between discrete energy states

Band Emission Spectrum:

• Appearance: Groups of closely spaced lines appearing as bands
• Origin: Molecules with closely spaced vibrational and rotational energy sublevels
• Examples: Molecular gases under electrical discharge
• Physics: Transitions between energy level bunches that are widely separated from other bunches

Manifestations of Dispersion:

1. Wavelength-dependent refraction causing separation of colors
2. Different focal points for different wavelengths
3. Color fringing at boundaries of images

Problems in Imaging Applications:

1. Chromatic Aberration:

   • Longitudinal: Different wavelengths focus at different distances
   • Lateral: Different wavelengths focus at different heights from optical axis

2. Color Fringing: Colored edges around high-contrast boundaries

3. Reduced Resolution: Blurring due to overlapping color focal points

4. Reduced Contrast: Especially in systems requiring precise focus

5. Spectral Artifacts: False colors or missing wavelength information

Solutions include achromatic lenses (crown+flint combinations), apochromatic designs (correcting for three wavelengths), and reflective optics (mirrors that avoid dispersion entirely).

Line Absorption Spectrum:

• Appearance: Sharp, discrete dark lines on bright continuous background
• Origin: Atomic transitions where specific wavelengths are absorbed when electrons move from lower to higher energy states
• Examples: Fraunhofer lines in solar spectrum, stellar spectra
• Applications:
  • Identifying chemical elements in astronomical objects
  • Measuring relative velocities via Doppler shifts
  • Determining stellar compositions and temperatures

Band Absorption Spectrum:

• Appearance: Dark bands (groups of closely spaced lines) on bright continuous background
• Origin: Molecular transitions involving vibrational and rotational energy states
• Examples: Absorption by hydrogen gas at moderate temperature, organic compounds in solution
• Applications:
  • Identifying molecular compounds in samples
  • Atmospheric analysis and pollution monitoring
  • Biochemical analysis of organic solutions
  • Pharmaceutical quality control

Key Distinction: Line spectra reveal atomic composition while band spectra reveal molecular structure and bonding characteristics.

Principles of Dispersion Without Average Deviation:

1. When two prisms with angles $A$ and $A'$ are arranged in reverse orientation:

   • Net deviation: $\delta = (\mu_y - 1)A - (\mu'_y - 1)A'$
   • Angular dispersion: $\delta_v - \delta_r = (\mu_v - \mu_r)A - (\mu'_v - \mu'_r)A'$

2. For zero average deviation: $(\mu_y - 1)A = (\mu'_y - 1)A'$

3. For non-zero dispersion: $(\mu_v - \mu_r)A ≠ (\mu'_v - \mu'_r)A'$

Application in Achromatic Lens Design:

1. Combine crown glass (low dispersion) and flint glass (high dispersion) lenses
2. Make one lens converging and one diverging
3. Choose powers to satisfy: $P_{\text{crown}} + P_{\text{flint}} = P_{\text{desired}}$
4. Choose materials and powers so dispersion of one cancels the other
5. Result: Two wavelengths focus at exactly the same point

This principle allows lenses to form images without color fringing, critical for microscopes, telescopes, and precision optical instruments.

• Refractive index decreases as wavelength increases (approximately follows Cauchy's equation: μ = μ₀ + A/λ²)
• Materials with higher average refractive indices (like Silicate Flint Glass) show steeper curves, meaning greater dispersion
• Materials with lower average refractive indices (like Fluorite) show flatter curves, meaning less dispersion
• Violet light (shorter wavelength, ~400nm) experiences higher refraction than red light (longer wavelength, ~700nm)
• This relationship explains why prisms separate white light into a spectrum of colors

Example: Silicate Flint Glass has a refractive index of ~1.66 at 400nm and ~1.61 at 700nm, while Fluorite ranges from ~1.44 to ~1.43 across the same range.

Pure Spectrum Requirements:

• Incident light beam must be parallel before reaching the dispersing element
• All rays of a particular wavelength must be collected at one specific point
• Narrow slit width to prevent multiple overlapping spectra

Impure Spectrum Causes:

• Wide slit causing separate overlapping spectra from different points
• Non-parallel incident light causing wavelength mixing
• Improper focusing allowing wavelengths to spread across the focal plane

Key Conditions for Purity:

1. Slit must lie precisely in the focal plane of the collimating lens
2. Collimated rays must be perfectly parallel when striking the dispersing element
3. The focusing lens must have proper focal length to collect parallel rays of each wavelength
4. Proper optical alignment throughout the system

A pure spectrum shows distinct color boundaries with no overlap, while an impure spectrum shows diffused color impressions with gradual transitions.

The different focal positions result from wavelength-dependent refraction, explained by:

1. Refractive Index Variation: The refractive index (μ) varies with wavelength according to Cauchy's equation: μ = μ₀ + A/λ² where A is Cauchy's constant and λ is wavelength

2. Dispersion Mechanism:

   • Violet light (shorter wavelength) has higher refractive index than red light (longer wavelength)
   • Higher refractive index causes greater deviation when passing through the dispersing element
   • This creates angular separation between different wavelengths

3. Focal Position Effect:

   • The focusing lens collects parallel rays of each wavelength at different points in its focal plane
   • Violet light focuses closer to the lens than red light due to its greater deviation
   • The separation between focal points depends on the dispersive power of the material

The dispersive power (ω) quantifies this effect: ω = (μᵥ - μᵣ)/(μy - 1), where μᵥ, μᵣ, and μy are the refractive indices for violet, red, and yellow light respectively.

This wavelength-dependent focusing forms the basis for both spectrum analysis and chromatic aberration in optical systems.

A spectrometer for optical measurements consists of three essential components:

1. Collimator (C):

   • Contains an adjustable slit (S) and an achromatic converging lens
   • Function: Produces a parallel beam of light from the source
   • The slit-to-lens distance is adjusted so the slit lies at the lens's focal point

2. Prism Table:

   • Horizontal circular platform that can rotate about a vertical axis
   • Function: Holds the dispersing element (prism, grating)
   • Includes graduated circular scale to measure angles of rotation

3. Telescope (T):

   • Contains objective lens and eyepiece in an astronomical arrangement
   • Function: Receives and focuses the light after reflection/refraction
   • Can rotate around the same vertical axis as the prism table
   • Has vernier scale to precisely measure angular positions

The entire system must be properly leveled using adjustment screws before measurements.

A spectrometer produces a pure spectrum through a precisely designed optical pathway with these critical components:

Essential Components:

1. Collimator Assembly:
   • Narrow entrance slit: Acts as a thin pencil of light source
   • Achromatic converging lens: Transforms divergent light into parallel rays
2. Dispersing Element (mounted on rotatable prism table):
   • Prism or diffraction grating: Separates light into component wavelengths
   • Must maintain ray parallelism within each wavelength group
3. Telescope Assembly:
   • Focusing lens (second achromatic lens): Collects dispersed parallel rays
   • Focal plane: Where separated wavelengths form distinct images

Critical Design Considerations:

• Each wavelength must occupy one specific spatial position in the spectrum
• Rays of identical wavelength must remain parallel after dispersion
• No overlapping of different wavelengths in the final image
• Proper alignment and achromatic lens quality to prevent chromatic aberration
• Rotatable mounting for the dispersing element to select specific wavelengths

Without this precise arrangement, an impure spectrum would form with diffused, overlapping color impressions that prevent accurate spectral analysis.

Minimum Deviation Principle

• Definition: The specific orientation of a prism where light deviation reaches its minimum value (δₘᵢₙ)
• Key characteristics at minimum deviation:
  • Light passes symmetrically through the prism
  • Refracted ray inside the prism runs parallel to the base
  • Refractive index calculation: μ = sin[(A + δₘᵢₙ)/2] / sin(A/2) where A is the prism angle

Critical Spectrometer Adjustments

1. Leveling: Ensure collimator axis, telescope axis, and prism table are all horizontal
2. Collimator adjustment: Position slit at collimator lens focal point to generate parallel light beam
3. Telescope adjustment: Set to infinity focus for parallel rays
4. Prism positioning: Place with refracting edge toward collimator

Measurement Procedure

1. Initially locate refracted image by rotating telescope
2. Iterative process:
   • Gradually move telescope toward direct position while adjusting prism orientation
   • Find precise position where further prism rotation in either direction makes image disappear or deviation increase
   • This position marks the minimum deviation
3. Calculate the angle between direct ray position and minimum deviation position

Application

This technique provides the most accurate method for determining refractive indices of prismatic materials across different wavelengths of light.