# Understanding NMR

Under ‘the term NMR’, people may correctly understand two different subjects:

1. Why do subatomic particles, in this case nuclei, precess?
2. How do Engineers exploit this precession, in order to form 2D and 3D images, in ‘NMRI’?

In this posting, I am only going to address subject (1).

Precession and spin are easier to understand, when we can simply apply the Newtonian concepts. Quantum Mechanics today tends to obscure the subject of precession. And so for most of this post, I am going to make the somewhat daft assumption that the precession of subatomic particles, is Newtonian.

If a gyroscope is spinning along an arbitrary axis, and if we apply torque to its axis, this torque integrates into the spin vector – at an angle to the existing spin vector. Unless we are accelerating or slowing down its spin. This results in the spin vector rotating – and thus in precession.

But, if we have seen the demonstration in which an off-axis gyroscope is precessing on a passive pedestal, we also observe that eventually the phenomenon weakens, and that the practical axis seems to shift further and further in the direction gravity is pulling on it.

The reason this weakening takes place, is the fact that some additional torque is being applied to the gyroscope, against the direction in which it is precessing. Otherwise, it would just precess forever. This additional torque could be due to friction with the pedestal, due to air resistance, due to magnetism, or whatever.

An artillery shell is aerodynamically designed, so that as long as it has excess spin, interaction with the air will always push it in the direction of any existing precession, and so this type of object will tend to straighten its axis of spin, into the direction with which it is flying. This would be the equivalent to the gyro from before, straightening up and standing up against gravity again.

Atomic nuclei that have an odd mass number, also have a non-zero spin quantum number, thus having spin, and also have a magnetic dipole moment. The wanton assumption could be made that its magnetic dipole moment is always parallel to its axis of spin. But then if we visualize matter as consisting of nuclei that are separated by vast, less-dense clouds of electrons, it would seem to follow that each nucleus is always precessing in response to local magnetic fields.

And even if we were to apply an external magnetic field to such a system, it would follow that precession could not yet be detected externally, because the nuclei are all out-of-phase. Ostensibly, they would also continue to precess, and to stay out of phase, simply due to an applied magnetic field. The only big difference with the practical gyro should then be, that the magnitude of their spin-vector should never change, since this should be intrinsic.

But if we were to insist on this very Newtonian description, then something else should also happen that is not as obvious. Those thin wisps of electrons should not only react to the applied field, but also locally, to the field of each nucleus precessing. So if we assume conservation of energy, there would also be reactive torque acting on each nucleus, in response to its own precession, because the density of the electron clouds is not zero.

After a certain settling period which is measurable, the nuclei end up aligning themselves with the applied field, resulting in the state that has its lowest-possible potential energy. This takes milliseconds instead of the nanoseconds that some of these behaviors should take on the subatomic scale. Precession has still not been detected.

Likewise, the fact that subatomic decay can take years instead of nanoseconds, refutes certain mundane explanations, of what might be causing that.

Well, one thing that Scientists can do is compute what the dipole moment of such a nucleus is, as well as the magnitude of its angular momentum – spin – and to compute as a function of the applied field-intensity, with what frequency all the nuclei should be precessing… This frequency is called the “Larmor Frequency”.

And then, with he applied field turned on, radio-waves can be applied to the sample at 90 degrees to the applied field, that have a frequency equal to this Larmor Frequency. What happens is that the sample exhibits Resonance. This means that the sample absorbs the radio waves while they are being applied – thus storing energy – and that when they are shut off, the sample emits radio waves, again at right-angles to the applied magnetic field, until the stored energy has been expended.

This is NMR.

Most obviously the radio-waves have caused the precession of the nuclei to synchronize in phase. Less obviously, each nucleus has precessed along a path, while the external energy source was on, which was not a circle, but a kind of dense spiral, which started out facing with the applied magnetic field, and which ended up pointing against the applied magnetic field. This has caused each nucleus to go from its lowest potential energy, to its highest potential energy.

With the external energy source off, this exaggeratedly Newtonian precession went from the high-potential-energy position, through a process of precession, all the way to the low-potential-energy position again.

Now I should add, a constraint which comes directly from Quantum Mechanics: Here, it is assumed that a fundamental particle is allowed to have two, 3-dimensional properties, that are linked in the complex ways we all know: The position vector, and the velocity vector. From the velocity vector and the particle-mass, a momentum vector follows, which must be parallel to the velocity vector.

According to QM, All the other properties of the particle need to be real values, including possibly integers, in order to count as observable properties. If any of the other Eigenvalues have complex – i.e. 2D – values, they are taken not to be observable properties.

There is no room in this depiction of subatomic particles, for spin to be a vector, with an orientation anything other than the momentum-vector. There is a Spin Quantum Number, and where a particle could have +1, it could also hypothetically have -1. And thus results a QM depiction of spin, which needs to be 1-dimensional.

I, personally, do not know whether this is accurate. I just know that this is how the subject is depicted in QM. I also do not know how precession results. We can only observe that it does.

But then what remains conform to both schools of knowledge, is that each QM particle can have a Spin State which is either Up or Down. Up is the higher-potential-energy state, and Down has the lower potential energy. This state is as 1-dimensional as the Quantum Number.

“Spin-Flip” is a phenomenon which has been recorded by QM to exist, not only for nuclei, but also for the electrons bound to them, in cases where the electrons are not paired. An atomic clock has a time-base of Caesium-137, and defined by the spin-flip of its single valence electron, against the magnetic dipole of the nucleus it is bound to. This time-base is the exact Larmor Frequency that results.

This subject becomes a lot more odd, when Neutron Optics is studied, of Cold Neutrons, which have been polarized with an external magnetic field. Even though neutrons are known to have magnetic dipole moments as well, these dipoles cannot be measured in free space. But if a beam of neutrons is passed through a steel plate, with an external magnetic field applied to the steel, at right angles to the path, then a beam emerges whose properties have changed, as if the original beam had excess potential energy with neutrons in a linear orientation, at odds with the applied field, and as if passing through the steel dissipated this energy, resulting in an emerging beam which is polarized.

If our thinking was entirely Newtonian, we might respond by concluding that we have caused the neutrons to assume an axis of spin, not parallel to their axis of motion. But because the world is not quite so cooperative, what we have done instead, is we have caused the quantum spin, which only has a (constant) Real Eigenvalue, to become Uncertain, as if it had a Complex Number.

Corresponding to that, whether the neutrons are Up or Down, becomes a Probability Function of their DeBroglie Wavelength, and a function of how far they have traveled.

Dirk

(Edit : ) There is a detail which I did not read, contrarily to what I wrote above, but which seems logical. To achieve a left-handed or a right-handed (cold) neutron beam should be easy.

We would orient the magnetic field of the steel plate along the axis of travel, instead of at right angles to it.

(Edit 09/16/2016 : ) The problem with my own proposition above would be, that although it is possible to define whether a neutron is left-handed or right-handed, because the neutron is a neutral particle, it is not possible based on that alone, to define whether, say, a left-handed neutron corresponds to the North Pole facing forward, as opposed to the South Pole facing forward. So in my naive statement above, I made a mistake. If there was an electric charge of Positive or Negative associated with the neutron, this would become possible. Therefore, it may not be so easy, to produce a decidedly left-handed beam for example, even though the spin quantum number can be negated, where not zero. And the spin quantum number of the neutron is +/-(1/2) .

This spin quantum number also suggests, that a single inversion of whether its magnetism is Up or Down – along the original, sideways axis – should take place within one exact DeBroglie Wavelength, and after the second DeBroglie Wavelength, its Up / Down state should be restored to what it started as.

Further, it is imaginable that this neutron could simply lose energy, for whatever reasons. This would certainly translate into decreasing momentum, and therefore longer DeBroglie Wavelengths. Whatever intrinsic properties it has linked to such a combined property – such as spin which is confirmed, and magnetic states which have yet to be confirmed –  would simply remain so linked.

(Edit 09/14/2016 : ) The reason for which certain experiments in the past were done with Cold Neutrons, is the fact that their DeBroglie Wavelengths needed to be long enough, to be measurable. If the beam has a hypothetical property which is changing according to the DeBroglie Wavelength, and which is therefore alternating within ?a Picometer? , then this property will seem as if random, and will not be observed as having any coherency.

This limitation also prevents the observation of DeBroglie Wavelengths under mundane, non-laboratory conditions.