🔬 Electron Paramagnetic Resonance (EPR) Spectroscopy - Splitting
EPR (or ESR) is used to study species with unpaired electrons ($S \neq 0$). The structure of the EPR spectrum is governed by two main types of splitting:
- Fine Structure: Splitting due to interactions between the unpaired electron spins ($S$) themselves, especially in complexes with $S > 1/2$.
- Hyperfine Structure (HFS) / Superhyperfine Structure (SHFS): Splitting due to the interaction between the unpaired electron spin ($S$) and the magnetic moment of a nearby nucleus ($I$).
I. Fine Structure (Zero-Field Splitting - ZFS)
Fine structure arises in transition metal complexes where the total spin quantum number $S$ is greater than $1/2$ (i.e., multiple unpaired electrons). The interaction between the electron spins and the local electric field of the ligands causes the spin states ($m_s$) to split even when no external magnetic field is applied (Zero-Field Splitting).
🔑 Key Formula: Fine Structure Lines
The number of observable transitions ($\Delta m_s = \pm 1$) in a solid-state or frozen solution spectrum is given by:
Where $S$ is the total spin quantum number ($S = n_{\text{unpaired}} \times 1/2$).
Example: High-Spin $\text{Mn}^{2+}$ ($d^5$)
- Ion: $\text{Mn}^{2+}$
- Configuration: $d^5$ (High Spin)
- Unpaired Electrons: 5
- Total Spin ($S$): $5/2$
- Calculation: $2S = 2 \times (5/2) = \mathbf{5 \text{ lines}}$
The five lines correspond to the transitions: $m_s = \pm 5/2 \rightarrow \pm 3/2$, $\pm 3/2 \rightarrow \pm 1/2$, and $-1/2 \rightarrow +1/2$.
II. Hyperfine Structure (HFS/SHFS)
Hyperfine splitting occurs when the unpaired electron interacts with the magnetic moment of an atomic nucleus ($I \neq 0$). This nucleus can be the central metal ion (HFS) or a coordinating ligand atom (SHFS).
🔑 Key Formula: Hyperfine Lines
The number of lines due to a single set of $\mathbf{n}$ equivalent nuclei with nuclear spin $\mathbf{I}$ is given by:
A. HFS from Central Metal Nucleus
Each fine structure line will be further split by the central metal nucleus, provided its nuclear spin ($I$) is non-zero.
Example 1: $\text{Mn}^{2+}$ HFS
- System: $\text{Mn}^{2+}$ (High Spin, $d^5$)
- Interacting Nucleus: $\text{Mn}^{55}$
- Nuclear Spin ($I$): $\mathbf{5/2}$ (Crucial value for $\text{Mn}$)
- Number of Nuclei ($n$): 1
- Calculation: $2nI + 1 = 2(1)(5/2) + 1 = \mathbf{6 \text{ lines}}$
In a high-resolution $\text{Mn}^{2+}$ spectrum, the central transition line is split into 6 lines due to HFS.
Example 2: $\text{Cu}^{2+}$ HFS
- System: $\text{Cu}^{2+}$ ($d^9$)
- Interacting Nucleus: $\text{Cu}^{63}$ / $\text{Cu}^{65}$
- Nuclear Spin ($I$): $\mathbf{3/2}$ (Crucial value for $\text{Cu}$)
- Number of Nuclei ($n$): 1
- Calculation: $2nI + 1 = 2(1)(3/2) + 1 = \mathbf{4 \text{ lines}}$
B. SHFS from Ligand Nuclei
The electron also interacts with nuclei of the coordinating ligands. This is called Superhyperfine Splitting (SHFS). The total number of lines is the product of all splittings.
Example: $[\text{Cu}(\text{ethylenediamine})_2]^{2+}$ (A CSIR/GATE/SET favorite)
- Metal HFS: $\text{Cu}^{2+}$ ($I=3/2$, $n=1$) $\rightarrow$ 4 lines
- Ligand SHFS: The complex coordinates via four equivalent $\text{N}$ atoms.
- Interacting Nucleus: $\text{N}^{14}$
- Nuclear Spin ($I$): $\mathbf{1}$ (Crucial value for $\text{N}^{14}$)
- Number of Nuclei ($n$): 4
- SHFS Calculation: $2nI + 1 = 2(4)(1) + 1 = \mathbf{9 \text{ lines}}$
Total Number of Lines (Product Rule)
The total number of observable lines is the product of the individual splittings:
Total Lines for $[\text{Cu}(\text{en})_2]^{2+}$:
III. Relative Intensity of Lines
When the splitting is caused by $n$ equivalent nuclei with $I=1/2$ (e.g., protons), the intensity ratio follows Pascal's Triangle (binomial coefficients).
| $n$ (Equivalent $I=1/2$ Nuclei) | Number of Lines ($2nI+1$) | Intensity Ratio |
|---|---|---|
| 1 | 2 | 1:1 |
| 2 | 3 | 1:2:1 |
| 3 | 4 | 1:3:3:1 |
| 4 | 5 | 1:4:6:4:1 |
| 6 (Benzene Radical) | 7 | 1:6:15:20:15:6:1 |
IV. Exam Focus Summary
| Structure Type | Cause | Formula | $\text{Mn}^{2+}$ ($d^5, S=5/2, I=5/2$) Result |
|---|---|---|---|
| Fine Structure | Electron Spin-Spin Interaction ($S > 1/2$) | $2S$ | 5 lines |
| Hyperfine Structure (HFS) | Electron Spin-Metal Nucleus Interaction ($I_{\text{Metal}} \neq 0$) | $2I_{\text{Metal}} + 1$ | 6 lines |
| Superhyperfine (SHFS) | Electron Spin-Ligand Nucleus Interaction ($I_{\text{Ligand}} \neq 0$) | $2nI_{\text{Ligand}} + 1$ | Depends on Ligands |
- $\mathbf{^{1}\text{H}: I = 1/2}$
- $\mathbf{^{14}\text{N}: I = 1}$
- $\mathbf{^{55}\text{Mn}: I = 5/2}$
- $\mathbf{^{63/65}\text{Cu}: I = 3/2}$