Einstein Coefficients and Discharge Lamps in Spectroscopy

Posted on Oct 31, 2024 in Physics

Einstein Coefficients for Radiation

Induced Absorption

dP₁₂/dt = B₁₂R_v: The probability per unit time of a molecule or element transitioning from the ground state to an excited state.

Induced Emission

ⁱ₂₁dP/dt = B₂₁R_v: The probability per unit time of a molecule transitioning from an excited state to the ground state.

Spontaneous Emission

^e₂₁dP/dt = A₂₁: The transition probability per unit time for a molecule to spontaneously transition from an excited state to the ground state.

Where B₁₂, B₂₁, and A₂₁ are the Einstein coefficients for each process, and R_v is the density of incident radiation.

Relationship Between Coefficients

Consider a system with two energy levels and populations N₁ (ground state) and N₂ (excited state). The total probability is:

dP₁₂/dt = B₁₂R_vN₁

dP₂₁/dt = B₂₁R_vN₂ + A₂₁N₂

At equilibrium, these probabilities are equal:

B₁₂R_vN₁ = B₂₁R_vN₂ + A₂₁N₂

The ratio of populations is:

N₂/N₁ = (B₁₂R_v) / (A₂₁ + B₂₁R_v) (*)

Consider N molecules in thermal equilibrium. According to Boltzmann statistics, the population of each level i is:

N_i = N(g_ie^-E/KT) / Σg_je^-E/KT

Where g_i is the degeneracy of level E_i.

For a two-level system:

N₂/N₁ = (g₂/g₁)e^{-(E2 – E1)/KT}

As temperature increases, the population of E₂ increases and E₁ decreases. Equating this with equation (*):

R_v = (A₂₁/B₂₁) / ((g₁B₁₂/g₂B₂₁)e^(E2-E1)/KT – 1)

Comparing this with Planck’s law:

R_v = (8πv²/c³)(hv / (e^hv/KT – 1))

We can obtain expressions for the Einstein coefficients by noting that E₂ – E₁ = hv:

B₁₂ = (g₂/g₁)B₂₁

A₂₁ = (8πhv³/c³)B₂₁

Discharge Lamps or High-Pressure Arc Lamps

These lamps have high irradiance in the UV and visible regions. They consist of a protective quartz ampoule filled with gas, a tungsten anode, and a tungsten cathode. The cathode tip has a high electric field, promoting gas ionization and initiating the discharge. When the gas ionizes, a plasma forms, and intense incandescent light appears, superimposed on the gas emissions.

Xenon Lamps

Xenon lamps exhibit emission lines of Xe (750-1000 nm). They are used to mimic the solar spectrum, as their spectrum approximates a black body at 5000-6000 K.

Xenon-Mercury Lamps

These lamps combine the emission spectra of Xe and Hg. The spectrum is relatively smooth due to Xe’s continuous emission, with peaks from Hg emissions. The Hg peaks are broadened due to high-pressure interactions.

Population Inversion in a Two-Level System

In a two-level system under optical pumping, population inversion (N₂ > N₁) cannot be achieved. As radiation intensity increases, the transition probabilities due to external radiation dominate over spontaneous emission. Assuming equal degeneracy (g₁ = g₂), the Einstein coefficients for absorption and induced emission are equal (B₂₁ = B₁₂), leading to equal populations (N₁ = N₂). The excited level population will never exceed the ground level population.

Characteristics of Lamp Spectra

The graph shows spectra of two Xe discharge lamps (6251 75W and 6263 75W), a Xenon-Mercury lamp (6281 100W), a deuterium lamp (63162 30W), and an incandescent lamp (6332 50W QTH).

Incandescent Lamp (QTH)

Incandescent lamps consist of a tungsten filament heated by an electric current to 2500-3500 K. The emitted spectral density resembles Planck’s law. At 3000 K, the emission is primarily in the visible range, producing white light. Higher or lower temperatures shift the spectrum towards UV or IR, respectively. Quartz is commonly used for the lamp’s coating due to its transparency and thermal/mechanical resistance. Halogen gas inside the lamp prevents tungsten particles from adhering to the quartz surface.

Low-Pressure Discharge Lamps

These lamps contain a gas (e.g., mercury, deuterium, sodium) that is ionized by an electric current. The emitted light is primarily due to the intrinsic emission of the gas molecules.

• Mercury Lamps: Excitation of mercury occurs at specific energy levels, resulting in a spectrum of monochromatic lines.
• Deuterium Lamps: These lamps have a smooth continuum between 200 and 400 nm. Above 400 nm, the spectrum is irregular with high-irradiance peaks. They have a relatively short lifespan.