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NMR Spectroscopy

Field: Spectroscopy

A spectroscopic technique to observe local magnetic fields around atomic nuclei.

Sequence of Expressions

Definition

Nuclear Spin

Let I\mathbf{I} be the nuclear spin angular momentum operator, defined in the context of quantum mechanics. The magnitude of the spin is quantized by the spin quantum number II, such that the allowed eigenvalues of the squared spin operator I2\mathbf{I}^2 are I(I+1)ℏ2I(I+1)\hbar^2. The projection of the spin onto an external magnetic field B0B_0 is quantized by the magnetic quantum number mIm_I, yielding the eigenvalues mIℏm_I\hbar, where mI∈{−I,−I+1,…,I}m_I \in \{-I, -I+1, \dots, I\}.
Definition

Chemical Shift

Define the chemical shift δ\delta (in ppm) as the normalized difference between the resonance frequency νsample\nu_{sample} of a nucleus in a specific chemical environment and the resonance frequency νref\nu_{ref} of a reference standard (e.g., TMS), relative to the spectrometer frequency νspec\nu_{spec}: \nδ=νsample−νrefνspec×106\delta = \frac{\nu_{sample} - \nu_{ref}}{\nu_{spec}} \times 10^6 \nWhere νsample\nu_{sample} and νref\nu_{ref} are measured in Hz.
Define the longitudinal magnetization Mz(t)M_z(t) and transverse magnetization Mxy(t)M_{xy}(t) as functions of time tt. The relaxation processes are governed by the following differential equations: \n\n1. Longitudinal Relaxation (T1T_1): The recovery of Mz(t)M_z(t) towards its equilibrium value M0M_0 is described by: Mz(t)=M0(1−e−t/T1)M_z(t) = M_0 \left(1 - e^{-t/T_1}\right) where T1T_1 is the longitudinal relaxation time constant.\n2. Transverse Relaxation (T2T_2): The decay of Mxy(t)M_{xy}(t) is described by: Mxy(t)=Mxy(0)e−t/T2M_{xy}(t) = M_{xy}(0) e^{-t/T_2} where T2T_2 is the transverse relaxation time constant. It is mathematically constrained that T2≤T1T_2 \le T_1.
Define the chemical shift δ\delta as the relative difference in the resonance frequency νmeas\nu_{meas} of a nucleus compared to a standard reference compound (e.g., TMS, νref\nu_{ref}), normalized by the spectrometer frequency νspec\nu_{spec} (or field strength B0B_0). The chemical shift in parts per million (ppm) is calculated by the ratio: δ (ppm)=νmeas−νrefνspec×106\delta \text{ (ppm)} = \frac{\nu_{meas} - \nu_{ref}}{\nu_{spec}} \times 10^6 where νmeas\nu_{meas} and νref\nu_{ref} are measured in Hertz (Hz), and νspec\nu_{spec} is the operating frequency in Hz. This dimensionless quantity quantifies the local electronic environment's influence on the shielding experienced by the nucleus.
Let NN be a nucleus with proton number ZZ and neutron number NeN_{e}. The nuclear spin angular momentum I\mathbf{I} is proportional to the total angular momentum of the constituent nucleons. For a nucleus to exhibit NMR activity, the net nuclear spin must be non-zero, i.e., I≠0\mathbf{I} \ne 0. This condition is met if the total number of nucleons A=Z+NeA = Z + N_{e} is odd, or if ZZ and NeN_{e} are both odd. Mathematically, the spin quantum number II must satisfy I=12∣Z−(Z mod 2)+Ne−(Ne mod 2)∣≥12I = \frac{1}{2} |Z - (Z \bmod 2) + N_{e} - (N_{e} \bmod 2)| \ge \frac{1}{2}. Isotopes with I=0I=0 (e.g., 12C^{12}\text{C}) are NMR inactive.
Define the angular resonance frequency ω\omega of a nucleus with gyromagnetic ratio γ\gamma subjected to a static magnetic field B0B_0 as: \nω=γB0\omega = \gamma B_0 \nAlternatively, relating frequency ν\nu (in Hz) to field strength B0B_0 (in Tesla): \nν=∣γ∣2πB0\nu = \frac{|\gamma|}{2\pi} B_0
Consider a nucleus with spin angular momentum I\mathbf{I} placed in a static external magnetic field B0=B0z^\mathbf{B}_0 = B_0 \hat{\mathbf{z}}. The system Hamiltonian H\mathcal{H} is given by: H=H0−γI⋅B(t)\mathcal{H} = \mathcal{H}_0 - \gamma \mathbf{I} \cdot \mathbf{B}(t) where γ\gamma is the gyromagnetic ratio and B(t)\mathbf{B}(t) is the time-dependent magnetic field. The resonance condition occurs when the applied oscillating field B1(t)=B1cos⁡(ωt)x^\mathbf{B}_1(t) = B_1 \cos(\omega t) \hat{\mathbf{x}} matches the Larmor frequency ω0\omega_0: ω0=γB0\omega_0 = \gamma B_0 The energy absorbed by the nucleus is quantized at this frequency, leading to transitions between spin energy levels.
Consider the applied magnetic field B0B_0 and the induced local magnetic field BindB_{ind} generated by the electron cloud surrounding a nucleus. The effective magnetic field BeffB_{eff} experienced by the nucleus is given by: \nBeff=B0−Bind=B0(1−σ)B_{eff} = B_0 - B_{ind} = B_0 (1 - \sigma) \nWhere σ\sigma is the shielding constant, a dimensionless quantity quantifying the degree of magnetic field opposition, and BeffB_{eff} determines the resonance frequency ωobs=γBeff\omega_{obs} = \gamma B_{eff}.
For two coupled neighboring nuclei, I1I_1 and I2I_2, the interaction Hamiltonian Hcoupling\mathcal{H}_{coupling} is defined by the scalar coupling constant JJ: \nHcoupling=2I1I2J\mathcal{H}_{coupling} = 2\text{I}_1 \text{I}_2 J \nThis interaction modifies the energy levels EE of the system, leading to an energy splitting proportional to JJ, which is measured in Hertz (Hz). The coupling constant JJ is independent of the applied magnetic field B0B_0.
Let Δνmin\Delta\nu_{min} be the minimum detectable frequency separation between two signals, ν1\nu_1 and ν2\nu_2. The spectroscopic resolution RR is defined by the ratio of the frequency separation to the linewidth (ΔνL\Delta\nu_{L}) at the signal peak: R=∣ν1−ν2∣ΔνLR = \frac{|\nu_1 - \nu_2|}{\Delta\nu_{L}} The practical resolution is limited by the acquisition time TacqT_{acq} and the Fourier Transform window function W(t)W(t), such that the minimum resolvable separation Δνres\Delta\nu_{res} satisfies Δνres∝1Tacq\Delta\nu_{res} \propto \frac{1}{T_{acq}}. For optimal resolution, the linewidth ΔνL\Delta\nu_{L} must be significantly smaller than the minimum required separation.