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

Outline Problem statement Precession Frequency Sample Problem Sources Questions?

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

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

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

Larmor Frequency where is known as the gyromagnetic ratio θ 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.

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,

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

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

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?