Atomic Physics: Bohr Theory, Spectral Series, X-rays, and Lasers

Definition of Wave Number: Wave number ( $1/\lambda$ ) represents the number of waves present in a length of 1 meter.
Mathematical Equation: The equation for the wave number is derived from Bohr's theory as: $\frac{1}{\lambda} = R_H \times \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
Rydberg Constant ( $R_H$ ):
- Exact value: $1.097 \times 10^7 \, \text{m}^{-1}$ .
- Simplified calculation value (for non-calculator exams): $1 \times 10^7 \, \text{m}^{-1}$ .
State Definitions:
- $n_f$ : Final state or final shell where the electron transitions to (destination).
- $n_i$ : Initial state or initial shell where the electron originates from.
Electron Transitions:
- Excitation: Electron moves from lower to higher energy levels, absorbing energy.
- De-excitation: Electron moves from higher to lower levels, emitting energy in the form of a photon with a specific wavelength and energy.

Concept of Spectrum: A spectrum is essentially a graph-like representation used to plot the wavelengths of different emitted photons or electromagnetic radiations.
Line Spectrum vs. Continuous Spectrum:
- Rutherford's model predicted a continuous spectrum.
- The observed reality is a Line Spectrum, consisting of discrete spectral lines corresponding to specific wavelengths.
Emission Spectrum: This occurs when an electron releases energy (emits radiation) as it jumps from a high energy level to a low energy level.
Absorption Spectrum: This occurs when an electron acquires energy to move from a ground state to a higher excitation state. Note: This is not the primary focus of the current study session.

The series are classified based on the final state ( $n_f$ ) to which the electron falls:

Lyman Series:
- Final state: $n_f = 1$
- Initial states: $n_i = 2, 3, 4, \dots, \infty$
- Region: Ultraviolet (UV) region.
Balmer Series:
- Final state: $n_f = 2$
- Initial states: $n_i = 3, 4, 5, \dots, \infty$
- Region: Visible region (can be seen with the naked eye).
- Wavelength range: Approximately $400 \, \text{nm}$ to $720 \, \text{nm}$ ( $4000 \, \text{Å}$ to $7000 \, \text{Å}$ ).
Paschen Series:
- Final state: $n_f = 3$
- Initial states: $n_i = 4, 5, 6, \dots, \infty$
- Region: Infrared (IR) region (specifically Lower Infrared).
Brackett Series:
- Final state: $n_f = 4$
- Initial states: $n_i = 5, 6, 7, \dots, \infty$
- Region: Infrared (IR) region.
Pfund Series:
- Final state: $n_f = 5$
- Initial states: $n_i = 6, 7, 8, \dots, \infty$
- Region: Infrared (IR) region (specifically Far Infrared).

Core Principle: Energy and wavelength have an inverse relationship: $E = \frac{hc}{\lambda}$
Shortest Wavelength (Maximum Energy):
- Occurs when the electron transitions from the furthest possible distance ( $n_i = \infty$ ) to the destination shell ( $n_f$ ).
- Energy is maximum because the transition spans the widest gap.
Longest Wavelength (Minimum Energy):
- Occurs when the electron transitions from the shell immediately above the destination shell.
- Condition: $n_i = n_f + 1$ .

Balmer Shortest Wavelength:
- $n_f = 2$ , $n_i = \infty$
- $\frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right)$
- $\frac{1}{\lambda} = R \times \frac{1}{4} \Rightarrow \lambda = \frac{4}{R}$
- Using $R = 1 \times 10^7$ , $\lambda = 4 \times 10^{-7} \, \text{m} = 400 \, \text{nm}$ .
Balmer Longest Wavelength:
- $n_f = 2$ , $n_i = 3$
- $\frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right)$
- $\frac{1}{\lambda} = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{5}{36} \right)$
- $\lambda = \frac{36}{5R}$
- Result: Approximately $720 \, \text{nm}$ .

General Properties: X-rays are highly energetic electromagnetic radiations with very short wavelengths.
Wavelength Order: Typically on the order of $10^{-10} \, \text{m}$ or $1 \, \text{Angstrom}$ .
Order Comparison: Visible light has an order of $10^{-7} \, \text{m}$ , whereas X-rays are $10^{-10} \, \text{m}$ .

Apparatus Components:
- Cathode: Fires fast-moving electrons.
- High Voltage Source ( $V$ ): Used to accelerate electrons from the cathode to the target.
- Target Metal: Usually a Tungsten disc or similar high-density metal.
Electron Speed Formula: Potential energy provides the kinetic energy for the electron: $q \Delta V = \frac{1}{2}mv^2$ $v = \sqrt{\frac{2q \Delta V}{m}}$

Characteristic X-Rays:
- Process: A fast-moving incoming electron strikes an inner-shell electron (e.g., in the K-shell) and knocks it out of the atom (Knockout Phenomenon).
- Vacancy Fulfillment: A vacancy or 'hole' is created. Electrons from higher shells (L, M, N) jump down to fill this gap, emitting a photon.
- Labeling ( $K_\alpha, K_\beta$ ):
  - Letter (K, L, M) identifies where the vacancy occurred.
  - Greek letter indicates the jump distance: $\alpha$ is from the adjacent shell, $\beta$ is from one shell further away.
- Characteristic Property: These values are constant for a specific element (e.g., Tungsten) because the shell energy levels are fixed for that element.
Continuous X-Rays (Bremsstrahlung):
- Process: A fast-moving electron passes near the nucleus. The nucleus's positive charge exerts an attractive force, slowing the electron down.
- Bremsstrahlung Meaning: German for "Braking Radiation."
- Energy Conversion: The kinetic energy lost by the electron during deceleration is emitted as a photon.
- Energy Range: Energies can vary continuously because the closeness of the electron to the nucleus (and thus the deceleration) is not quantized.

Definition: Light Amplification by Stimulated Emission of Radiation.
Energy States:
1. Ground State: Most stable, lowest energy.
2. Excited State: Highly unstable, very short life ( $10^{-8} \, \text{s}$ ).
3. Metastable State: Partial stability, longer life ( $10^{-3} \, \text{s}$ ).

Induced Absorption (Step 1):
- Ground state electrons are given energy via Optical Pumping to jump into the Excited State.
Spontaneous Emission (Step 2):
- Electrons naturally and quickly fall from the Excited State to the Metastable State. This happens without external intervention.
Induced/Stimulated Emission (Step 3):
- Population Inversion: The process is repeated until more electrons reside in the Metastable state than in the Ground state.
- Stimulation: An external photon triggers the electrons in the metastable state to fall to the ground state simultaneously.
- Output: The result is a highly directional, intense, and phase-coherent beam of light known as a Laser.

Homework Assignments:
- Calculate the Longest and Shortest wavelengths for Lyman, Paschen, Brackett, and Pfund series.
- Determine the first three spectral lines for Balmer, Paschen, Brackett, and Pfund series.
Clarification on Balmer Series Constraints: A student asked if $n_i = 1$ is possible for the Balmer series. The response: No, because that would require energy absorption to move from shell 1 to shell 2, creating an absorption spectrum. We are studying the emission spectrum, where electrons must move from higher shells to shell 2.
M-Alpha Photon Scenario: If an $M_\alpha$ photon is emitted, where was the vacancy? The vacancy was in the M-shell, and the electron jumped from the N-shell to fill it.