1. Proton precession in magnetic fields
Amanda J. Fagan
Wittenberg University
March 19, 2014

2. Outline Problem statement Precession Frequency Sample Problem Sources
Questions?

3. Problem statement
a) Show that a spinning magnetic moment in the presence of a magnetic field leads to a precession of the magnetic moment.
b) Derive an expression for the frequency of precession.
c) Evaluate using the given parameters

4. Precession
Precession occurs when magnetic moment is placed in external magnetic field
Because elementary particles have intrinsic angular momentum and an external magnetic field is exerting a torque
Related to torque, then, by the equation
where µ is the magnetic moment, B is the external magnetic field, and θ is the angle between them.
-magnetic moment bar magnert, spin aligns with external B field

5. Precession θ L ∆L 𝛕
We also know that the torque is related to angular momentum by the equation
where ɸ is the angular position
Can use this to find frequency!
Because the magnetic field is constant, the torque will be constant, and therefore we can use delta L/delta t instead of having to take the full derivative

6. Larmor Frequency where is known as the gyromagnetic ratioθ L ∆L 𝛕
Since the angular momentum is only changing position (ɸ), we can say that
Larmor frequency is characteristic frequency
gyromagnetic ratio is specific to each type of particle being looked at
We can rearrange this in order to get that
where
is known as the gyromagnetic ratio
and is constant depending on the type of particle.

7. Larmor Frequency where is a constant known as the nuclear magneton and
is known as the Larmor frequency
and represents the rate of precession
So in this case, we’re going to be looking at the proton’s gyromagnetic ratio. If we were looking at a different particle, the formula will be similar but the g-factor and the magneton will be different (Bohr magneton for the electron)
We then can find the gyromagnetic ratio of the proton by
where
is a constant known as the nuclear magneton
and
is a dimensionless constant
known as the proton g-factor.
Therefore,

8. which can convert to Hz by dividing by 2π to give
Sample Problem
Find the precession frequency of a magnetic moment of nuclear magnetons in a 2.00 T magnetic field, with the angle between the field and spin axis of 90 degrees.
We’ve already done most of the work!
which can convert to Hz by dividing by 2π to give

9. Sample Problem YES! Does our answer of make sense?
85.5 MHz is in the radio range (RF pulses used in MRI)
the known reduced gyromagnetic ratio is which is half of the 85.5 MHz we obtained, and therefore agrees with the accepted value

10. Sources http://sites.fas.harvard.edu/~phys191r/pdf/ap12.pdf

11. Questions?