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Units, Constants and Useful Formulas

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Nuclear Spin
------------

By analogy with the Bohr magneton we can define the NUCLEAR
MAGNETON as:

μN = eh/2mp

For individual electrons, protons and neutrons we can write:

μe = gμBS/h where g = -2.0023

= gμBms

μp = gμNS/h where g = 5.585691

= gμNms

μn = gμNS/h where g = -3.826084

= gμNms

Note the neutron has 0 charge but has a magnetic moment because
the component quarks are charged.

For combinations of neutrons and protons into nuclei, the
situation is more complex.  There is no simple formula to
predict the nuclear spin based on the number of protons
and neutrons within an atom. Nevertheless, there are some
general rules that apply to special cases:

# of protons   # of neutrons  Spin  Examples
------------   -------------  ----  --------
Even            Even         0    12C, 16O
Even            Odd         m/2   17O
Odd             Even        m/2   1H, 23Na, 31P
Odd             Odd          m    2H

Where m is an integer.

If these magnetic moments are placed in a static magnetic field
they will have a potential energy related to their orientation
with respect to that field.   In the case of an electron, the
lowest energy configuration is when the dipole moment is aligned
with the field.  The maximum energy is when the dipole is in
anti-alignment.  Due to the difference in charge, the opposite
is true for the proton.  In both cases, there will be precession
around the mgnetic field.  The Larmor frequency is given by:

ωp = μpB/h

Quantum mechanically the precession can be thought of as the
quantum energy of transition between the two possible spin
states for spin 1/2.  The process of transition can be expressed
as a photon energy according to the Planck relationship.  Thus,

E = hωp = μpB

------- +μpB
ΔE = 2μpB
------- -μpB

The energy difference between the 2 spins states is very small
in comparison to the thermal energy at room temp.  This means
that the degree of polarization that can be maintained at ordinary
temperatures is extremely small.

Nuclear Magnetic Resonance
--------------------------

The precession of the proton spin is used NMR.  Hydrogen is
placed in a strong magenetic field to polarize the proton spins
and then an RF field is applied to excite some of the nuclear
spins into a higher energy state.  The absorbtion of the RF
energy only occurs when the frequency of the radiation equals
the Larmor frequency - hence the term 'resonance'.  When the
RF field is removed, the spins tend to return to their lower
energy state (spin relaxation) in the process emitting a small
amount of radiation at the Larmor frequency.