Summary of Nuclear Magnetic Resonance Spectroscopy
Nuclear Magnetic Resonance Spectroscopy: A Student Guide
Introduction
Nuclear Magnetic Resonance (NMR) theory explains how atomic nuclei with spin interact with magnetic fields and electromagnetic radiation. This material focuses on the quantum description of those interactions, the energy levels that arise in a magnetic field, and practical calculations used to determine resonance frequencies. It is aimed at a Not attending student and emphasizes clear concepts, worked examples, and useful comparisons.
Basic quantum concepts
Nuclear spin and spin quantum number
- Atomic nuclei can have an intrinsic angular momentum called spin. The spin of a nucleus is characterized by the spin quantum number $I$.
- A nucleus with spin quantum number $I$ has $2I+1$ discrete spin states.
Definition: The spin quantum number $I$ is the intrinsic angular momentum of a nucleus and determines how many magnetic states the nucleus can occupy.
- Common useful nuclei and their spins: $^{1}\mathrm{H}$, $^{13}\mathrm{C}$, $^{19}\mathrm{F}$, $^{31}\mathrm{P}$ all have $I=\tfrac{1}{2}$, so each has two spin states: $m = +\tfrac{1}{2}$ and $m = -\tfrac{1}{2}$.
Magnetic moment and magnetogyric ratio
- A spinning charged nucleus produces a magnetic moment $\mu$ oriented along its spin axis.
- The magnetic moment is proportional to the nuclear angular momentum $P$:
$$\mu = \gamma P$$
where $\gamma$ is the magnetogyric ratio (also called gyromagnetic ratio). Each nuclide has a characteristic $\gamma$.
Definition: The magnetogyric ratio $\gamma$ is the constant of proportionality between a nucleus's magnetic moment and its angular momentum; it determines how strongly the nucleus interacts with a magnetic field.
Energy levels in an external magnetic field
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When a nucleus with spin $I=\tfrac{1}{2}$ is placed in an external magnetic field $B_0$, its magnetic moment aligns either with or against $B_0$, producing two distinct energy levels corresponding to $m=+\tfrac{1}{2}$ and $m=-\tfrac{1}{2}$.
-
The energy of a magnetic moment in a magnetic field is given by:
$$E = -\mu \cdot B_0$$
Replacing $\mu$ with $\gamma P$ and choosing the projection of angular momentum that corresponds to magnetic quantum number $m$, the quantized energy for a spin-$\tfrac{1}{2}$ nucleus can be written as
$$E = -\frac{\gamma m h}{2\pi} B_0$$
where $h$ is Planck's constant and $m=\pm\tfrac{1}{2}$.
- The energy difference between the two spin states determines the frequency of radiation that can be absorbed or emitted when the nucleus transitions between states.
Resonance frequency (Larmor frequency)
- The resonance (Larmor) frequency $\nu_0$ at which a nucleus absorbs electromagnetic radiation is proportional to the magnetic field $B_0$ and the magnetogyric ratio $\gamma$:
$$\nu_0 = \frac{\gamma B_0}{2\pi}$$
- This simple linear relation allows calculation of the frequency for any nucleus in a known magnetic field.
Table: Magnetic properties of common spin-1/2 nuclei
| Nucleus | Magnetogyric ratio $\gamma$ (rad,T^{-1},s^{-1}) | Isotopic abundance (%) | Relative sensitivity |
|---|---|---|---|
| $^{1}\mathrm{H}$ | $2.6752 \times 10^{8}$ | 99.98 | 1.00 |
| $^{13}\mathrm{C}$ | $6.7283 \times 10^{7}$ | 1.11 | 0.01 |
| $^{19}\mathrm{F}$ | $2.5181 \times 10^{8}$ | 100.00 | 0.83 |
| $^{31}\mathrm{P}$ | $1.0841 \times 10^{8}$ | 100.00 | 0.06 |
Worked examples
Example 1: Proton resonance frequency in a 4.69 T magnet
- Given $B_0 = 4.69\ \mathrm{T}$ and $\gamma$ for $^{1}\mathrm{H}$ is $2.6752 \times 10^{8}\ \mathrm{rad,T^{-1},s^{-1}}$.
$$\nu_0 = \frac{\gamma B_0}{2\pi}$$
Substitute values:
$$\nu_0 = \frac{(2.6752 \times 10^{8}\ \mathrm{rad,T^{-1},s^{-1}})(4.69\ \mathrm{T})}{2\pi}$$
Evaluating gives
$$\nu_0 \approx 2.00 \times 10^{8}\ \mathrm{s^{-1}} = 200\ \mathrm{MHz}$$
Example 2: Carbon-13 resonance frequency in a 4.69 T magnet
- Use $\gamma$ for $^{13}\mathrm{C}$: $6.7283 \times 10^{7}\ \mathrm{rad,T^{-1},s^{-1}}$.
$$\nu_0 = \frac{(6.7283 \times 10^{7})(4.69)}{
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NMR Theory Basics
Klíčová slova: Nuclear Magnetic Resonance (NMR) Theory, Nuclear Magnetic Resonance (NMR) Spectroscopy, NMR Spectroscopy
Klíčové pojmy: Nuclei spin is quantified by $I$ and yields $2I+1$ states, Spin-1/2 nuclei have two states: $m=+\tfrac{1}{2}$ and $m=-\tfrac{1}{2}$, Magnetic moment: $\mu=\gamma P$ where $\gamma$ is nucleus-specific, Energy of a spin state: $E = -\dfrac{\gamma m h}{2\pi} B_0$, Larmor (resonance) frequency: $\nu_0 = \dfrac{\gamma B_0}{2\pi}$, $^{1}\mathrm{H}$ has high $\gamma$ and high natural abundance, $^{13}\mathrm{C}$ has much lower sensitivity because of lower $\gamma$ and abundance, Higher $B_0$ increases resonance frequency and signal-to-noise ratio, Instrument RF must match $\nu_0$ for excitation/detection, Use the table of $\gamma$ values to compute frequencies for different nuclei