Nuclear Magnetic Resonance

By Bodo Kaiser


Usage, Examples, Content




  1. Introduction
  2. Overview
  3. Measurement
  4. Classic Description
  5. Quantum Description
  6. Refinements
  7. Prospects
  8. Closing


Apparatus, Setup

MRI Apparatus

MRI apparatus

From USRE32619 (1988)

NMR Apparatus

NMR apparatus

From Agilent


  • What is the setup of a NMR apparatus?
  • What is the measurement procedure?
  • What magnetisations do we observe?


RF Pulse, Free Induction Decay, Spectra, Bloch Equations, Time Constants

RF Pulse

Free Induction Decay


Bloch Equations

$$\begin{align} M_L(t)=M_0\left(1-e^{-t/T_1}\right) && M_T(t)=M_0e^{-t/T_2} \end{align}$$

Time Constants

Tissue Type $T_1$ in ms $T_2$ in ms
Fat 240-250 60-80
Blood (deoxygenated) 1350 50
Blood (oxygenated) 1350 200
Gray Matter 920 100
White Matter 780 90
Liver 490 40
Kidney 650 60-75
Muscles 860-900 50

From Wikipedia for magnetic field $B=1.5T$

Image Weighting


  • What is the FID and how is it measured?
  • What time constants are there and what do they describe?

Classic Description

Magnetic Moment, Nuclei, Precession, Rotating Frame, Magnetisation

Magnetic Moment

The magnetic moment $\mu$ is defined by $$\boldsymbol{\tau}=\boldsymbol{\mu}\times\boldsymbol{B}.$$

Moving charges possess a magnetic moment $$\boldsymbol{\mu}=\frac{q}{2m}\boldsymbol{l}.$$

Gyromagnetic constant $$\gamma=\frac{q}{2m}.$$

Relevant Nuclei

Nucleus $\gamma$ $\left[10^6\frac{rad}{s T}\right]$ $f$ $[10^6Hz]$ $m/n$
$^1H$ 267.5 42.6 99.98
$^2H$ 40.8 6.5 0.02
$^{10}B$ 28.3 4.6 18.8
$^{11}B$ 86.1 13.7 81.2
$^{13}C$ 67.2 10.7 1.11
$^{19}F$ 252.0 40.1 100
$^{23}Na$ 70.8 11.3 100
$^{31}P$ 108.1 17.2 100

From Wikipedia1, Wikipedia2 and Bergmann und Schaefer (1992) for magnetic field $B=1T$

Larmor Precession

Torque changes angular momentum $$\dot{\boldsymbol{l}}=\boldsymbol{\tau}.$$

Magnetic moment precesses in magnetic field $$\dot{\boldsymbol{\mu}}=\gamma\boldsymbol{\mu}\times\boldsymbol{B}_0.$$

Magnetic Pulse

Equation of motion in rotatin frame $$\dot{\boldsymbol{\mu}}_\varphi =\boldsymbol{\mu}_\varphi\times\left(\gamma\boldsymbol{B}_0-\boldsymbol{\omega}_0\right).$$

Magnetic field with pulse in rotating frame $$\boldsymbol{B}_\varphi=B_0\boldsymbol{n}_\parallel +R(-\omega_0t)R(\omega_1t)B_1\boldsymbol{n}_\perp.$$

New equation of motion in rotating frame $$\dot{\boldsymbol{\mu}}_\varphi =\boldsymbol{\mu}_\varphi\times\boldsymbol{B}_1(t).$$

Energy Distribution

Potential energy stored in magnetic field $$V(\theta)=-\mu B_0\cos\theta.$$

Boltzmann distribution of energy in equilibrium $$P(\theta)=\frac{\exp(-V(\theta)/k_B T)} {\int_0^\pi d\theta\sin\theta\exp(-V(\theta)/k_B T)}.$$

Boltzmann factor of antiparallel magnetic state $$P_{\uparrow\downarrow}=\exp\left(-\frac{\Delta V}{k_B T}\right) \approx 1+\frac{2\mu B_0}{k_B T} \overset{\text{H}}{\underset{B_0=3T\\T=38°C}{\approx}} 1+2{\cdot}10^{-5}.$$

Magnetisation I

Net magnetisation for $I=\frac{1}{2}$:

$$\begin{align} \boldsymbol{M}_0 &=\mu(N_\uparrow-N_\downarrow)\boldsymbol{n}\\ &=\mu N\frac{1-\exp(-\mu B_0/kT)}{1+\exp(-\mu B_0/kT)}\boldsymbol{n}\\ &=\mu N\tanh\left(\mu B_0/2kT\right)\boldsymbol{n}\\ &\approx\frac{N\mu^2}{2kT}\boldsymbol{B}_0 =\frac{N\gamma^2l^2}{2kT}\boldsymbol{B}_0 =\chi_0\boldsymbol{B_0} \end{align}$$

Magnetisation II


  • What did we do?
  • What assumptions did we make?
  • What orders of magnitude do we have?

Quantum Description

Angular Momentum, Magnetic Moment, Hamiltonian, Precession, Magnetisation

Nuclear Angular Momentum

The nucleus possesses total angular momentum $$\hat{\boldsymbol{I}} =\sum_{i=1}(\hat{\boldsymbol{l}}_i+\hat{\boldsymbol{s}}_i)$$

with eigenvalues $$\begin{align} \hat{\boldsymbol{I}}^2\vert I,m\rangle&=I(I+1)\hbar^2\vert I,m\rangle\\ \hat{\boldsymbol{I}}_z\vert I,m\rangle&=m\hbar\vert I,m\rangle \end{align}$$

Nuclear Magnetic Moment

Magnetic moment $$\hat{\boldsymbol{\mu}}= \frac{e_0}{2m_p}\sum_{i=1}(g_{li}\hat{\boldsymbol{l}}_i+g_{si}\hat{\boldsymbol{s}}_i).$$

Nucleon $g_l$ $g_s$
Proton 1 5.58
Neutron 0 -3.82

Expectation value $$\langle\hat{\boldsymbol{\mu}}\rangle= g_I\frac{e_0}{2m_p}\langle\hat{\boldsymbol{I}}\rangle=\gamma\langle\hat{\boldsymbol{I}}\rangle.$$

Nucleus $I$
$^1H$ $1/2$
$^2H$ $1$
$^{10}B$ $3$
$^{11}B$ $3/2$
$^{13}C$ $1/2$
$^{19}F$ $1/2$
$^{23}Na$ $3/2$
$^{31}P$ $1/2$

From Wikipedia

Energy Eigenvalues

Hamiltonian of nucleus $$\hat{H}= -\gamma\hat{\boldsymbol{\mu}}{\cdot}\boldsymbol{B}_0= -\gamma\hat{\boldsymbol{I}}{\cdot}\boldsymbol{B}_0.$$

has energy eigenstates $$\begin{align} \hat{H}\vert I,m\rangle &=-\gamma\hat{\boldsymbol{I}}{\cdot}\boldsymbol{B}_0\vert I,m\rangle\\ &=-\gamma B_0\hat{I}_z\vert I,m\rangle\\ &=-\gamma B_0\hbar m\vert I,m\rangle \end{align}$$

therefore enery levels of $$\Delta E=\hbar\gamma B_0=\hbar\omega_0.$$

Larmor Precession

Total angular momentum changes $$\begin{align} i\hbar\frac{d}{dt}\hat{I}_i &=\left[\hat{I}_i, -\gamma B_j \hat{I}_j\right]\\ &=-\gamma\left[\hat{I}_i, \hat{I}_j\right]B_j\\ &=-i\hbar\gamma\epsilon_{ijk}\hat{I}_kB_j \end{align}$$

equals classical movement $$\frac{d}{dt}\langle\hat{\boldsymbol{\mu}}\rangle =\gamma\langle\hat{\boldsymbol{\mu}}\rangle\times\boldsymbol{B}(t).$$


Energy stored in magnetic field now discrete $$E_m=-\gamma\hbar m B_0.$$

Boltzmann distribution of energy in equilibrium $$P_m=\frac{\exp(-E_m/k_B T)} {\sum^I_{m=-I} \exp(-E_m/k_B T)}.$$

Net magnetisation ($I=\frac{1}{2}$) $$\boldsymbol{M}_0 =\mu N\tanh\left(\hbar\gamma B_0/2kT\right)\boldsymbol{n} \approx\frac{N\gamma^2\hbar^2}{4kT}\boldsymbol{B}_0.$$


  • How does quantum differ from classical description?
  • Where do we need to be careful with the quantum description?
  • What effects do we still neglect?


Relaxation, Spin Sequences

Revision Relaxation

Spin-Lattice Relaxation

Caused by thermal energy dissipation.

Responsible for intensity loss in FID (determines recycle time).

Strongly depends on magnetic field and frequency.

Spin-Spin Relaxation


$T_2$ caused by tissue structure.

$T_2^\prime$ caused by local inhomogeneities.

$$2T_1\geq T_2^\ast\geq T_2^\prime\geq T_2$$


From O. Dietrich (2001), see Literature

Stimulated Spin-Echo

From O. Dietrich (2001), see Literature

Inversed Spin-Echo

From O. Dietrich (2001), see Literature


  • What are marcoscopic effects behind $T_1,T_2,T_2^\prime$?
  • What is the basic idea behind spin echos?
  • What pulse sequences can we use for specific $T_1,T_2,T_2^\prime$?
  • What tissue type are $T_1,T_2,T_2^\prime$ sensitive to?
  • What is the general time inequation that connects $T_1,T_2,T_2^\prime$?


In Vivo MR Spectroscopy, Transformations

In Vivo MR Spectroscopy

From M. Kim et al, 2001., see Literature

CT Transformation

From D. Nie et al, 2001., see Literature

US Transformation


  • How does In Vivo NMR differ from MRI?
  • What is the best medical imaging technique?


Literature, Questions


    Thank You