Overview

• Wavelength: 1 m - 5 Km
• Measures: Magnetic spin transitions
• Typical Uses: Organic molecules, Structure
• Key Formulae:
  • Larmor Frequency: ${\nu }_L=|\gamma |\frac{B}{2\pi }$
  • Chemical Shift: $\delta =\frac{{\nu }_s-{\nu }_{ref}}{{\nu }_{ref}}\times {10}^6$
• Selection Rules:
  • $\mathrm{\Delta }M=\pm 1$
  • $\mathrm{\Delta }E=h\nu$

Important Concepts

Spin Quantum Numbers

• Total Spin Quantum Number (l):

  • l = 0, 1/2, 1, 3/2, ...
  • Proton/Neutron Combinations:

    Protons	Neutrons	Spin Quantum Number (l)
    Even	Even	0
    Odd	Odd	1, 2, 3, ...
    Even	Odd	1/2, 3/2, 5/2, ...
    Odd	Even	1/2, 3/2, 5/2, ...

• Spin Angular Momentum Vector Magnitude: $\sqrt{l(l+1)}\hslash$

• Magnetic Quantum Number (m): $m=-l$ to $+l$

• Nuclear Magnetic Moment ($\mu$):

  • $\mu =\gamma l$
  • ${\mu }_z=\gamma l_z=m\hslash \gamma$

Energy and Polarization

• Energy Difference: $\mathrm{\Delta }E=\hslash |\gamma |B$
• Polarization: $p=\frac{n_a-n_b}{n_a+n_b}\approx \frac{\mathrm{\Delta }E}{2K_{B}T}$
• Boltzmann Distribution: $\frac{N_a}{N_b}=e^{\frac{\mathrm{\Delta }E}{k_{B}T}}$
• Energy of Levels: $E(m_l)=-m_I\gamma B_0\hslash$
• Allowed Magnetic Dipole Transitions: $\mathrm{\Delta }E=\gamma B_0\hslash$

Larmor Frequency and Chemical Shift

• Larmor Frequency: ${\nu }_L=\frac{|\gamma |B}{2\pi }$
• Shifted Larmor Frequency: ${\nu }_L=\gamma B_0(1-\sigma )/2\pi$
• Chemical Shift (ppm): $\delta ={10}^6\times \frac{{\nu }_s-{\nu }_{ref}}{{\nu }_{ref}}$
• Frequency Separation (Hz): $\frac{({\delta }_1-{\delta }_2)\times {\nu }_{ref}}{{10}^6}$
• Receiver Reference Frequency (Rx): ${\nu }_{offset}=\nu -{\nu }_{rx}$

Applications and Calculations

Calculating Impurity Concentration

• 13C Satellite Peaks:
  • Integration of 13C peaks gives 0.5% relative to the proton peak. If impurity peaks integrate to less than a 13C peak, the impurity is less than 0.5% of the compound concentration.

NMR Limit of Detection (LOD)

• LOD Formula:
  • $LOD=3\times \frac{H_{signal}}{H_{Noise}}$

Sections to Explore

• NMR Theory
• NMR Exchange
• NMR Coupling
• NMR Relaxation
• Nuclear Overhauser Effect (NOE)
• Evan's Method

Visual Aids

• Element Nuclear Spin Quantum Numbers: