220 Week 6: Moving Charges and Magnetic Force
available to physicians practicing modern medicine. It is also not terribly easy to understand even
for physics majors because to completely understand it one has tounderstand alotabout both
quantum mechanics and spin relaxation to do a completely proper jobof it. This is especially true
given that there are multiple somewhat distinct methods (that provide some degree of choice in
contrast and resolution) that all come under the general headingof MRI and are all options on the
hardware that can accomplish different purposes.
However, at this point youshouldknowenoughto understand a sort of a “toy model” of just
how at least one or two of the MRI methods work, including the one that is arguably the most
important (conceptually) to understand. This section is devoted to presenting just such a toy model.
It deliberately omits most of the discussion of the quantum mechanics involved, while necessarily
introducing certain very general terms and describing in a qualitative manner the key processes
related to those terms. This section should very definitely be viewedas “optional” for most students
but may serve as an introductory reference for students who are interested or who are confused by
other descriptions.
Note well that this presentation ismy ownconception of the process, and while I do have some
research experience with the related quantum theory of photon resonance and photon echos, I am far
from being an expert on MRI and nuclear spin echos in particular in thecontext of MRI or otherwise.
Those who are more expert than I who read this and find errors areencouraged to contact me to
correct them, as long as the correction preserves the general semi-classical, functional presentation I
am attempting that is as appropriate for introductory physics non-major students interested in the
life sciences or medicine as it might be for future physics majors.
MRI is primarily used medically to map the density ofhydrogen nuclei– protons – in the human
body. Many of these protons are bound up inwater moleculesand are screened to some extent from
electromagnetic fields and radiation by the surrounding molecular electron clouds. One can then
treat them as “isolated” protons and add a phenomenological correction that describes the effect of
small variations in their local fields.
From our discussion above, weclassically expect the magnetic dipole moment of an isolated
proton to be given by an expression such as:
~mp=e
2 mpS~ (481)
There are several problems with this, however. The proton is not,in fact, a homogenous ball of
spinning mass and charge. It is a composite particle made up of threequarks, andtheyhave spin
angular momentumS~as shown (a purely quantum mechanical kind of angular momentum),not
the classical “orbital” angular momentum described by~L. It is therefore simpler to describe the
actual ratio of the magnetic dipole moment of a proton to its actualspin angular momentum as a
semi-empirical parameter and define:
~mp=γS~ (482)whereγis the so-calledgyromagnetic ratiofor the proton and where we will continue to useL~as the
symbol for its nuclear spin angular momentum. A concise discussion of this and accepted values for
γare available on Wikipedia: http://www.wikipedia.org/wiki/Proton magnetic moment, although
this article uses±~/2 as its spin angular momentum, which is technically the magnitude ofSz, not
its total angular momentum which would usually be given as
√
s(s+ 1)~^2 =√
3 / 2 ~for a spin-^12
particle.
This is where the subtleties of quantum mechanics start to kick in. Inthe absence of any strong
magnetic field, the total spin angular momentum of any collection of protons will be approximately
zero. The spins willnotnecessarily be aligned with any givenz-axis direction (which is, after all,
arbitrary). We can pick az-axis and try to describe the spin state of any given proton in termsof
eigenstates ofSzrelative to that direction, but if we do all we will get is that spins are incompletely