Optical spectra comprise discrete emission or absorption lines and bands in the ultraviolet, visible, and infrared regions arising from electronic transitions between atomic or molecular energy levels. These spectra provide fundamental insights into electronic configurations, coupling schemes, and fine structures of atoms.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic understanding of atomic excitation, emission, Bohr's frequency condition, hydrogen spectrum series (Lyman, Balmer, Paschen, Brackett, Pfund), and simple selection rules.
PGT extension
Detailed treatment of fine structure, spin-orbit interaction, L-S and j-j coupling schemes, Zeeman effect, Stark effect, Pauli's exclusion principle, and term symbols for multi-electron atoms.
Prerequisites
Bohr's theory of hydrogen atom
Electron spin and magnetic moments
Pauli's exclusion principle
Origin of Optical Spectra
Optical spectra are produced when valence electrons undergo transitions between discrete electronic energy states accompanied by the emission or absorption of photons.
The frequency of the emitted radiation is given by Bohr's frequency condition: h$\nu = E_{2} - E_{1}$.
Spectra are classified into line spectra (atomic gases), band spectra (molecules), and continuous spectra (incandescent solids).
Hydrogen Spectrum and Spectral Series
The atomic spectrum of hydrogen consists of distinct series of lines corresponding to transitions terminating at a specific lower principal quantum number n1.
Lyman series (n1 = 1, UV region), Balmer series (n1 = 2, Visible region), Paschen series (n1 = 3, IR region), Brackett (n1 = 4), and Pfund (n1 = 5).
Wave numbers of these spectral lines are accurately predicted by the Rydberg formula using the reduced mass of the electron-proton system.
Fine Structure and Coupling Schemes
High-resolution optical spectra reveal that single spectral lines split into closely spaced multiple components, known as fine structure, caused by spin-orbit interaction.
For lighter atoms, individual orbital angular momenta (Li) couple to form total orbital angular momentum L, and spin angular momenta (Si) couple to form total spin S (Russell-Saunders or L-S coupling).
For heavier atoms where spin-orbit interaction dominates over electrostatic repulsion between electrons, individual ji = li + si couple to form total J (j-j coupling).
Selection Rules for Optical Transitions
Transitions between states are governed by quantum mechanical selection rules derived from parity and angular momentum conservation.
For single-electron atoms, the primary selection rules are Δ$l = \pm 1$ and Δn = any integer.
For multi-electron systems under L-S coupling, the selection rules are ΔL = 0, ±1; Δ$S = 0; \DeltaJ = 0$, ±1 (with J = 0 to J = 0 forbidden).
Derivations
Rydberg Formula for Hydrogen-like Atoms
Conditions: Bohr's model with circular orbits and Coulomb electrostatic force providing centripetal acceleration, incorporating reduced mass μ.
Write down the expression for total energy of an electron in the nth orbit: En = -\frac{m e^4 Z^2}{8 \epsilon_0^2 h^2 n^2}.
Apply Bohr's frequency condition for a transition from upper state n2 to lower state n1: $\nu = \frac{E_{n_2} - E_{n_1}}{h}$.
Substitute energy values and convert frequency to wave number \bar{\nu} = \frac{\nu}{c} = \frac{E_{n_2} - E_{n_1}}{hc}.
Express wave number in terms of Rydberg constant R∞ = \frac{m e^4}{8 \epsilon_0^2 c h^3}.
Conditions: Perturbation theory approach to evaluate magnetic energy of electron spin in the magnetic field generated by orbital motion of the nucleus.
Calculate the effective magnetic field B experienced by the electron in its rest frame due to the nuclear motion: $B = \frac{1}{2m_0^2 c^2 r} \left( \frac{dV}{dr} \right) \vec{L}$.
Determine the potential energy perturbation due to interaction between magnetic dipole moment μs and magnetic field B: Δ$E = -\vec{\mu}_s \cdot \vec{B}$.
Substitute μs = -\frac{e}{m_0}\vec{S} and take the expectation value over quantum states characterized by L, S, and J.
Express the resulting energy shift in terms of total angular momentum quantum numbers J, L, and S: $E_{SO} = \frac{A}{2} [J(J+1) - L(L+1) - S(S+1)]$.
$\nu = frequency of characteristic X-ray line$, Z = atomic number, a and b = constants Hz1/2
Inner-shell atomic transitions
Maximum Spectral Lines from Level n
$$N = \frac{n(n-1)}{2}$$
n = principal quantum number of excited state dropping to ground state Integer number of lines
Single excited atom
Graphs and experiments
Grotrian Diagram (Energy Level Diagram)
Draw: Horizontal energy levels grouped by principal quantum number n and orbital angular momentum l, with vertical arrows indicating allowed optical transitions.
Interpret: Illustrates permitted transitions, selection rules, and the branching ratios of atomic emission lines.
Exam use: Directly tested in PGT exams to identify allowed vs forbidden transitions and term structures.
Hydrogen Emission Spectrum Series
Draw: A logarithmic or linear wavelength scale showing converging spectral lines grouped into Lyman, Balmer, and Paschen series converging to distinct series limits.
Interpret: Demonstrates the 1/n2 dependence of energy levels and convergence to ionization potential at the series limit.
Exam use: TGT questions on identifying spectral regions (UV, Visible, IR) corresponding to specific series.
Normal Zeeman Effect Splitting
Draw: Splitting of a single spectral line into three components (ν0, ν0 ± Δν) when subjected to an external transverse or longitudinal magnetic field.
Interpret: Direct proof of space quantization of angular momentum and magnetic moment associated with orbital motion.
Exam use: Conceptual and numerical questions regarding frequency separation and polarization states.
Solved numericals
TGT core
Find the shortest wavelength present in the Paschen series of hydrogen spectrum. (Given Rydberg constant $R = 1.097 x 10^7 m^-1)$
Identify the Paschen series lower state principal quantum number: n1 = 3.
For the shortest wavelength (maximum energy transition), the upper state is n2 = ∞.
Apply the Rydberg formula: 1/$\lambda = R (1/n_{1}^{2} - 1/n_{2}^{2}) => 1/\lambda = 1.097 x 10^7 (1/3^2 - 1/\infty ^2)$.
Calculate $\lambda = 9 / (1.097 x 10^7) = 8.20 x 10^-7 m$.
Answer: 8.20 x 10^-7 m (or 820 nm)
PGT extension
Calculate the Landé g-factor for a 2P3/2 state in L-S coupling.
Identify quantum numbers for 2P3/2 state: multiplicit$y = 2S+1 = 2$, so $S = 1/2$.
Letter P corresponds to orbital quantum number L = 1.
A spectral line of wavelength 589 nm undergoes Normal Zeeman effect in a magnetic field of 0.5 T. Calculate the wavelength shift (Δλ).
Frequency shift in Normal Zeeman effect is given by Δ$\nu = (e B) / (4 \pi m)$.
Substitute values: $e = 1.6 x 10^-19 C$, B = 0.5 T, $m = 9.1 x 10^-31 kg$.
Δ$\nu = (1.6 x 10^-19 * 0.5) / (4 * 3.1416 * 9.1 x 10^-31) = 8.0 x 10^-20 / 1.143 x 10^-29 = 6.99 x 10^9 Hz$.
Relate wavelength shift to frequency shift: Δ$\lambda = (\lambda^2 / c) * \Delta\nu$.
Substitute $\lambda = 589 x 10^-9 m$, $c = 3 x 10^8 m/s$, Δ$\nu = 6.99 x 10^9 Hz$.
Δ$\lambda = ((589 x 10^-9)^2 * 6.99 x 10^9) / (3 x 10^8) = (3.469 x 10^-13 * 6.99 x 10^9) / (3 x 10^8) = 8.08 x 10^-12 m$.
Answer: 8.08 x 10^-12 m (or 0.00808 nm)
Assumptions and limitations
Bohr-Sommerfeld model assumes non-relativistic electron velocities, which fails for inner-shell electrons in heavy elements where relativistic mass correction is mandatory.
Russell-Saunders (L-S) coupling assumes electrostatic interactions dominate over spin-orbit interactions, which breaks down for heavy atoms with high atomic number Z (where j-j coupling must be used).
Common misconceptions
Common error: All spectral series in hydrogen atom lie strictly within the visible region of the electromagnetic spectrum.
Correct: Only the Balmer series lies partially in the visible region; Lyman is in the ultraviolet, and Paschen, Brackett, Pfund are in the infrared region.
The energy differences defining each series place them in entirely different electromagnetic wavelength bands.
Common error: In L-S coupling, individual electrons maintain independent spin and orbital angular momenta that do not interact with each other.
Correct: Individual spin and orbital momenta couple vectorially to form resultant vectors L and S, which subsequently interact via spin-orbit coupling.
Electrostatic and magnetic interactions between electrons enforce collective coupling rather than isolated behavior.
Quick revision
Bohr's frequency condition equates photon energy to energy level differences: h$\nu = \DeltaE$.
The Rydberg formula calculates wave numbers of hydrogen spectral lines using principal quantum numbers.
Russell-Saunders coupling combines individual li and si into L and S before forming $J = L + S$.
Landé g-factor determines the magnetic splitting behavior of multi-electron atomic states in weak magnetic fields.
Must remember
Optical spectra originate from transitions of valence electrons between discrete energy levels, satisfying Bohr's frequency condition $h\nu = E_2 - E_1$.
The Rydberg wave number formula is given by $\bar{\nu} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$.
Russell-Saunders (L-S) coupling is applicable for lighter atoms where individual $L_i$ couple to $L$ and $S_i$ couple to $S$.
j-j coupling is predominant in heavier atoms where individual spin-orbit interactions exceed electrostatic repulsion between electrons.
Primary selection rules for single-electron atoms are $\Delta l = \pm 1$ and $\Delta n =$ any integer.
Selection rules for multi-electron L-S coupling: $\Delta L = 0, \pm 1$; $\Delta S = 0$; $\Delta J = 0, \pm 1$ (with $J=0 \to J=0$ forbidden).
Term symbol notation is represented as $^{2S+1}L_J$, where $S$ is total spin, $L$ is total orbital letter, and $J$ is total angular momentum.
Glossary
Bohr Frequency ConditionThe relation $h\nu = E_2 - E_1$, equating the energy of an emitted or absorbed photon to the difference between two discrete electronic energy levels.
Rydberg Constant ($R_H$)A fundamental physical constant characteristic of the spectrum of hydrogen, determining the wave numbers of spectral lines.
L-S Coupling (Russell-Saunders)Coupling scheme where individual orbital angular momenta combine to form total $L$ and individual spins combine to form total $S$, which then couple to form total $J$.
j-j CouplingCoupling scheme where each electron's orbital and spin angular momenta couple first ($j_i = l_i + s_i$), and these individual $j_i$ vectorially sum to total $J$.
Landé g-factorA dimensionless factor $g$ used in the analysis of Zeeman splitting to evaluate the contribution of orbital and spin angular momenta to magnetic moment.
L-S Coupling vs j-j Coupling
Feature
L-S Coupling
j-j Coupling
Atomic Weight Range
Lighter atoms
Heavier atoms
Interaction Dominance
Electrostatic repulsion > spin-orbit interaction
Spin-orbit interaction > electrostatic repulsion
Coupling Sequence
$\sum l_i \to L$, $\sum s_i \to S$, then $L + S \to J$
$l_i + s_i \to j_i$, then $\sum j_i \to J$
Hydrogen Spectral Series Regions
Series
Lower State ($n_1$)
Spectral Region
Lyman
$1$
Ultraviolet (UV)
Balmer
$2$
Visible and Near-UV
Paschen
$3$
Infrared (IR)
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following spectral series of hydrogen lies entirely in the ultraviolet region?
Explanation: The Lyman series corresponds to transitions ending at $n_1 = 1$, placing all its spectral lines in the ultraviolet (UV) region.
2. According to Bohr's frequency condition, the frequency $\nu$ of emitted radiation during a transition between states of energy $E_2$ and $E_1$ ($E_2 > E_1$) is given by:
Explanation: Bohr's frequency condition is $h\nu = E_2 - E_1$, which rearranges to $\nu = \frac{E_2 - E_1}{h}$.
3. What is the lower principal quantum number $n_1$ for the Balmer series of hydrogen?
Explanation: The Balmer series corresponds to electron transitions terminating at the principal quantum number $n_1 = 2$.
4. Which spectral series of hydrogen falls partially within the visible region of the electromagnetic spectrum?
Explanation: Only the Balmer series has lines falling within the visible region (specifically $H_\alpha, H_\beta, H_\gamma, H_\delta$). Lyman is UV, while Paschen and Brackett are in the infrared.
5. In Russell-Saunders (L-S) coupling, individual orbital angular momenta $l_i$ couple to form total orbital angular momentum $L$, and individual spins $s_i$ couple to form total spin $S$. This coupling scheme is primarily valid for:
Explanation: L-S coupling applies mainly to lighter atoms where electrostatic interactions between electrons are much stronger than individual spin-orbit interactions.
6. What is the primary selection rule for the orbital quantum number $l$ in single-electron atomic transitions?
Explanation: For single-electron atomic optical transitions, the parity change imposed by dipole radiation dictates the selection rule $\Delta l = \pm 1$.
7. What is the maximum number of spectral lines emitted when an excited hydrogen atom in the principal quantum number state $n$ returns to the ground state?
Explanation: The total number of unique transitions from level $n$ down to the ground state (and intermediate states) is given by combination formula $\frac{n(n-1)}{2}$.
8. What is the term symbol notation used in Russell-Saunders coupling?
Explanation: The standard term symbol is $^{2S+1}L_J$, where $2S+1$ is the multiplicity, $L$ is the letter denoting total orbital angular momentum, and $J$ is total angular momentum.
9. Which of the following quantum mechanical transitions is strictly forbidden under L-S coupling selection rules?
Explanation: The selection rule for spin in L-S coupling is $\Delta S = 0$. Therefore, transitions with $\Delta S = 1$ are forbidden.
10. In j-j coupling, the coupling order for an individual electron's quantum numbers is:
Explanation: In j-j coupling, each electron's spin and orbital angular momenta couple first to form individual $j_i = l_i + s_i$, and then these $j_i$ couple to form total $J$.
11. Fine structure in optical spectra (such as the doublet nature of sodium D-lines) is primarily caused by:
Explanation: Fine structure arises from the interaction between the spin magnetic moment of the electron and the magnetic field experienced due to its orbital motion around the nucleus (spin-orbit coupling).
12. What is the unit of the Rydberg wave number formula result ($\bar{\nu}$)?
Explanation: Wave number $\bar{\nu}$ is the inverse of wavelength, having SI units of reciprocal meters ($\text{m}^{-1}$).
13. Under the normal Zeeman effect in a weak magnetic field, a single spectral line splits into how many components for a transition between singlet states ($S=0$)?
Explanation: For singlet transitions ($S=0$), a single line splits into three components ($m_J = -1, 0, +1$), known as the normal triplet.
14. Which of the following correctly expresses the correction to the Rydberg constant due to the finite mass $M$ of the nucleus compared to electron mass $m$?
Explanation: Using reduced mass $\mu = \frac{mM}{m+M}$, the Rydberg constant is corrected as $R = R_\infty (1 + m/M)^{-1}$.
15. The transition $J = 0 \to J = 0$ is:
Explanation: For multi-electron systems under L-S coupling, the selection rules explicitly forbid $J = 0 \to J = 0$ transitions.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The wavelength of the first line of the Balmer series of hydrogen is proportional to which combination of Rydberg constant $R$?
Explanation: For the first line of the Balmer series, $n_1 = 2$ and $n_2 = 3$. The wave number is $\bar{\nu} = R \left(\frac{1}{2^2} - \frac{1}{3^2}\right) = R \left(\frac{1}{4} - \frac{1}{9}\right) = \frac{5}{36}R$. The wavelength $\lambda = \frac{1}{\bar{\nu}} = \frac{36}{5R}$.
2. In the term symbol $^2P_{3/2}$, the values of total spin $S$, total orbital angular momentum quantum number $L$, and total angular momentum $J$ are respectively:
Explanation: From multiplicity $2S+1 = 2 \implies S = 1/2$. The letter $P$ corresponds to $L = 1$. The subscript gives $J = 3/2$.
3. What is the region of the electromagnetic spectrum where the Paschen series of hydrogen lines is located?
Explanation: The Paschen series corresponds to transitions ending at $n_1 = 3$, which places all its spectral lines in the infrared region.
4. According to selection rules for multi-electron atoms under L-S coupling, which change in total spin $S$ is allowed?
Explanation: The selection rule for total spin in L-S coupling is $\Delta S = 0$, meaning intersystem combinations are forbidden.
5. Which of the following correctly describes line spectra?
Explanation: Line spectra are characteristic of atomic gases where valence electrons transition between discrete quantized energy levels.
6. In a hydrogen atom, if an electron transitions from $n = 4$ to $n = 2$, the color or region of the emitted photon belongs to:
Explanation: A transition terminating at $n_1 = 2$ from $n_2 = 4$ is the second line ($H_\beta$) of the Balmer series, which lies in the visible spectrum.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the ratio of the longest wavelength of the Lyman series to the longest wavelength of the Balmer series in hydrogen atom spectra?
2. Which coupling scheme is most appropriate for heavy atoms like mercury ($Z = 80$) where spin-orbit interactions are exceptionally large?
Explanation: For heavy atoms, spin-orbit interaction of individual electrons dominates over electrostatic repulsion, making j-j coupling appropriate.
3. What is the degeneracy of a state characterized by orbital quantum number $L = 2$ and spin quantum number $S = 1/2$ (ignoring nuclear spin)?
Explanation: The total spatial and spin degeneracy for a given $L$ and $S$ state without external magnetic fields is given by $(2L+1)(2S+1) = (5)(2) = 10$ counting $J$ components, but space-spin product degeneracy is $(2L+1)(2S+1) = 6$ independent orbital-spin orientations before $J$ splitting, and $\sum (2J+1) = 6$ total sublevels.
4. In the Zeeman effect, the frequency shift $\Delta \nu$ of a normal Zeeman triplet component in a magnetic field $B$ is given by:
Explanation: The normal Zeeman splitting frequency shift is $\Delta \nu = \frac{eB}{4\pi m}$.
5. Which transition satisfies the selection rule $\Delta l = \pm 1$?
Explanation: An $s$ state has $l = 0$ and a $p$ state has $l = 1$, giving $\Delta l = 1$ ($s \to p$), which satisfies the electric dipole selection rule.
6. What is the term symbol for an electron configuration with $L = 0$ and $S = 1/2$?
Explanation: With $S=1/2$, multiplicity $2S+1 = 2$. With $L=0$, the letter is $S$. $J = L + S = 1/2$. Thus, the term symbol is $^2S_{1/2}$.
7. The spectral lines of atomic hydrogen are best explained by considering:
Explanation: Atomic spectra are strictly quantized, arising from transitions between discrete electronic energy states as described by quantum theory and Bohr's model.
8. For a hydrogen-like single-electron ion (such as $\text{He}^+$), how does the Rydberg wave number scale with atomic number $Z$?
Explanation: The modified Rydberg formula for hydrogen-like ions includes $Z^2$ factor: $\bar{\nu} = R_H Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$, making wave numbers proportional to $Z^2$.
9. What does Landé g-factor evaluate in atomic spectroscopy?
Explanation: The Landé g-factor gives the correction for the effective magnetic moment of an atom in L-S coupling relative to its total angular momentum $J$.
10. Which of the following statements regarding continuous spectra is correct?
Explanation: Incandescent solids and dense glowing bodies emit continuous spectra spanning a broad, unbroken range of thermal radiation wavelengths.