PROTON PRECESSION IN MAGNETIC FIELDS

Amanda J. FaganWittenberg University

March 19, 2014

OUTLINE• Problem statement

• Precession

• Frequency

• Sample Problem

• Sources

• Questions?2

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 parameters3

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. 4

PRECESSION

• 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!

5

θL

∆L𝛕

LARMOR FREQUENCY

where is known as the gyromagnetic ratio

6

Since the angular momentum is only changing position (ɸ), we can

say that

We can rearrange this in order to get that

θL

∆L𝛕

and is constant depending on the type of particle.

LARMOR FREQUENCY

is known as the Larmor frequency

where

is a constant known as the nuclear magnetonand is a dimensionless constant

known as the proton g-factor.

Therefore, 7

and represents the rate of precession

We then can find the gyromagnetic ratio of the proton by

SAMPLE PROBLEM• Find the precession frequency of a magnetic

moment of 1.00 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!

8

which can convert to Hz by dividing by 2π to give

SAMPLE PROBLEM

• 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

Does our answer of make sense?

YES!

SOURCES• http://sites.fas.harvard.edu/~phys191r/pdf/ap12.pdf

• http://www.columbia.edu/itc/chemistry/photochem/spin/06.pdf

• http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/larmor.html

• http://hdl.handle.net/1721.1/10503

• http://physics.nist.gov/cgi-bin/cuu/Value?gammap

• http://physics.nist.gov/cgi-bin/cuu/Value?mun

• http://physics.nist.gov/cgi-bin/cuu/Value?gp

• http://physics.nist.gov/cgi-bin/cuu/Value?gammapbar

• http://eradiology.bidmc.harvard.edu/LearningLab/central/minja.pdf

QUESTIONS?