page_title: Nuclear magnetic resonance
section_title: Spin energy in a magnetic field
hierarchical_section_title: Nuclear magnetic resonance / Theory of nuclear magnetic resonance / Values of spin angular momentum / Spin energy in a magnetic field
caption_attribution_description: English: This is an adaptation of EPR splitting.gif which is located on the EPR page in order to suit the NMR page.
context_page_description: Nuclear magnetic resonance is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field and respond by producing an electromagnetic signal with a frequency characteristic of the magnetic field at the nucleus. This process occurs near resonance, when the oscillation frequency matches the intrinsic frequency of the nuclei, which depends on the strength of the static magnetic field, the chemical environment, and the magnetic properties of the isotope involved; in practical applications with static magnetic fields up to ca. 20 tesla, the frequency is similar to VHF and UHF television broadcasts. NMR results from specific magnetic properties of certain atomic nuclei. Nuclear magnetic resonance spectroscopy is widely used to determine the structure of organic molecules in solution and study molecular physics, crystals as well as non-crystalline materials. NMR is also routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging.
context_section_description: Consider nuclei with a spin of one-half, like ¹
  H
  , ¹³
  C
   or ¹⁹
  F
  . Each nucleus has two linearly independent spin states, with m =  or m = − (also referred to as spin-up and spin-down, or sometimes α and β spin states, respectively) for the z-component of spin. In the absence of a magnetic field, these states are degenerate; that is, they have the same energy. Hence the number of nuclei in these two states will be essentially equal at thermal equilibrium.
  If a nucleus is placed in a magnetic field, however, the two states no longer have the same energy as a result of the interaction between the nuclear magnetic dipole moment and the external magnetic field. The energy of a magnetic dipole moment  in a magnetic field B₀ is given by:
  Usually the z-axis is chosen to be along B₀, and the above expression reduces to:
  or alternatively:
  As a result, the different nuclear spin states have different energies in a non-zero magnetic field. In less formal language, we can talk about the two spin states of a spin  as being aligned either with or against the magnetic field. If γ is positive (true for most isotopes used in NMR) then m =  is the lower energy state.
  The energy difference between the two states is:
  and this results in a small population bias favoring the lower energy state in thermal equilibrium. With more spins pointing up than down, a net spin magnetization along the magnetic field B₀ results.
caption: Splitting of nuclei spin energies in an external magnetic field