As I’ve been delving quite deeply into numerical solutions of quantum mechanics lately, I thought I would take a step back and write about something a little less… well… less. One thing about quantum mechanics that is sometimes a bit mind-boggling is that classical interpretations of certain systems can be helpful to understand things about the quantum mechanics of the same system.
Nuclear magnetic resonance (NMR) dovetails quite nicely with my on-going series about how magnetism works. You may be familiar with NMR from a common medical technique that makes heavy use of it: Magnetic Resonance Imaging (MRI). The imaging technique of MRI makes use of NMR to build density maps of the human anatomy. The imaging technique accomplishes this feat by using magnetism to excite a radio signal from NMR active atomic nuclei and then create a map in space from the 3D distribution of NMR signal intensity. NMR itself is due specifically to the quantum mechanics of spin, particularly spin flipping, but it also has a classical interpretation which can aid in understanding what these more difficult physics mean. The system is very quantum mechanical, don’t get me wrong, but the classical version is actually a pretty good analog for once.
I touched very briefly on the entry gate to classical NMR in this post. The classical physics describing the behavior of a compass needle depicts a magnetized needle which rotates in order to follow the lines of an external magnetic field. For a magnetic dipole, compass needle-like behavior will tend to dominate how that dipole interacts with a magnetic field unless the rotational moments of inertia of that dipole are very small. In this case, the compass needle no longer swings back and forth. So, What does it do?
Let’s consider again the model of a compass needle awash in a uniform magnetic field…
This model springs completely from part 3 of my magnetism series. The only difference I’ve added is that dipole points in some direction while the field is along the z-axis. The definition of the dipole is cribbed straight from part 4 of my magnetism series and is expressing quantum mechanical spin as ‘S.’ We can back off from this a little bit and recognize that spin is simply angular momentum, where I transit to calling it ‘L’ instead so that I can slip away from the quantum. In this particular post, I’m not delving into quantum!
In this formula, ‘q’ is electric charge, ‘m’ is the mass of the object and ‘g’ is the gyromagnetic ratio which regularizes spin angular momentum to a classical rotational moment.
I will crib one more of the usual suspects from my magnetism series.
I originally derived this torque expression to show how compass needles swing back and forth in a magnetic field. In this case, it helps to stop and think about the relationship between torque and angular momentum. It turns out that these two quantities are related in much the same manner as plain old linear momentum and force. You acquire torque by finding out how angular momentum changes with time. Given that magnetic moment can be expressed from angular momentum, as can torque, I rewrite the equation above in terms of angular momentum.
This differential equation has the time derivative of angular momentum (signified in physicist shorthand as the ‘dot’ over the quantity of ‘L’) equal to a cross product involving angular momentum and the magnetic field. If you decompress the cross product, you can get to a fairly interesting little coupled differential equation system.
This simplifies the cross product to the two relevant surviving terms after considering that the B-field only lies along one axis. This gives a vector equation…
I’ve expressed the vector equation in component form so that you can see how it breaks apart. In this, you get three equations, one for each hatted vector which connect to each dimension of a three dimensional angular momentum. These can all be written separately.
I’ve grouped the B-field into the coefficient because it’s a constant and I’ve tried to take control of my reckless highlighting problem so that you can see how these differential equations are coupled. The z-axis of the angular momentum is easy since it must solve as a constant and since it’s decoupled from ‘x’ and ‘y’. The other two are not so easy. The coefficient is a special quantity which is called the Larmor frequency.
This gives us a fairly tidy package.
I’ve always loved the solution of this little differential equation. There’s a neat trick here from wrapping the ‘x’ and ‘y’ components up as the two parts of a complex number.
You then just take a derivative of the complex number with respect to time and work your way through the definitions of the differential equation.
After working through this substitution, the differential equation is reduced to maybe the simplest first order differential equation you could possibly solve. The answer is but a guess.
Which can be broken up into the original ‘x’ and ‘y’ components of angular momentum using the Euler relation.
There’s an argument here that ‘A’ is determined by the initial conditions of the system and might contain a complex phase, but I’m going to just say that we don’t really care. You can more or less just say that all angular momentum is distributed between the x, y and z components of the angular momentum, part of it a linear combination that lies in the x-y plane and the rest pointed along the z-axis.
And, as the original casting of the problem is in terms of the magnetic dipole moment, I can switch angular momentum back to the dipole moment. Specifically, I can use the pseudo-quantum argument that the individual dipoles possess half-integer spin magnitude angular momentum as hbar over 2.
This gives an expression for how the classical atom sized spin dipole will move in a uniform magnetic field. The absolute value on the charge in the coefficient constrains the situation to reflect only the size of the magnetic moment given that the angular momentum was considered to be only a magnitude. Charge appears a second time inside the sine and cosine terms concerning the Larmor frequency: for example, if the charge is negative, the negative sign on the frequency will cause the sine to switch from negative to positive while the cosine is unaffected.
A classical magnetic dipole trapped in a uniform magnetic field pointed along the Z-axis will undergo a special motion called gyroscopic “precession.” In this picture, the ellipses are drawn to show the surfaces of a cylinder in order to follow the positions of the dipole moment vectors with time. Here, the ellipses are _not_ an electrical current loop as depicted in the first image above. The dipole moment vector traces out the surface of a cone as it moves; when viewed from above, the tip of the dipole moment with a +q charge sweeps clockwise while the -q charge sweeps CCW. This motion is very similar to a child’s top or a gyroscope…
This image taken from Real world physics, hopefully, you’ve had the opportunity to play with one of these. A mentioned, the direction of the gyroscopic motion is determined by the charge of the dipole moment. As also mentioned, this is a classical model of the motion and it breaks down when you start getting to the quantum mechanics, but it is remarkably accurate in explaining how a tiny, atom sized dipole “moves” when under the influence of a magnetic field.
Dipolar precession appears in NMR during the process of free induction decay. As illustrated in my earlier blog post on NMR, you can see the precession:
In the sense of classical magnetization, you can see the signal from the dipolar gyroscopes in the plot above. One period of oscillation in this signal is one full sweep of the dipole moment around the z-axis. As the signal here is nested in the “magnetization” of the NMR sample, the energy bleeding out of the magnetic dipoles into the radiowave signal that is actually observed saps energy from the system and causes the precession in the magnetization to die down until it lies fully along the z-axis (again, classical view!) In its relaxed state, the magnetization points along the axis of the external field, much as a compass needle does. The compass needle, of course, can’t precess the way an atomic dipole moment can. And, as I continue repeatedly to point out, this is a classical interpretation of NMR… where the quantum mechanics are appreciably similar, though decidedly not the same.
Because such a rotating dipole moment cannot exhibit this kind of oscillation indefinitely without radiating its energy away into electromagnetism, some action must be undertaken in order to set the dipole into precession. You must somehow tip it away from pointing along the external magnetic field, at which time it will begin to precess.
In my previous post on the topic, I gave several different explanations for how the dipoles are tipped in order to excite free induction decay. Verily, I said that you blast them with a radio frequency pulse in order to tip them. That is true, but very heavy handed. Classical NMR offers a very elegant interpretation for how dipoles are tipped.
To start, I will dial back to the picture we started with for the precession oscillation. In this set up, the dipole starts in a relaxed position pointing along the z-axis B-field and is subjected to a radio frequency pulse that is polarized so that the B-field of the radio wave lies in the x-y plane. The Poynting vector is somewhere along the x-axis and the radiowave magnetic field is along the y-axis.
In this, the radiowave magnetic field is understood to be much weaker than the powerful static magnetic field.
You can intuitively anticipate what must happen for a naive choice of frequency ‘ω.’ The direction of the magnetic dipole will bobble a tiny bit in an attempt to precess around the superposition of the B0 and B2 magnetic fields. But, because the B0 field is much stronger than the B2 field, the dipole will remain pointing nearly entirely along the z-axis. We could write it out in the torque equation in order to be explicit.
Without thinking about the tiny time dependence on the B2 field, we know the solution to this equation from above for atomic scale dipoles. The Larmor frequency would just depend on the vector sum of the two fields. This is of course a very naive response and the expected precession would be very small and hard to detect since the dipole is not displaced very far from the average direction of the field at any given time, again expecting B2 to be very small. And, if B2 is oscillatory, there is no point where the time average of the total field lies off the z-axis. The static field tends to dominate and precession would be weak at best.
Now, there is a condition where an arbitrarily weak B2 field can actually have a major impact on the direction of the magnetic dipole moment.
This series of algebraic manipulations takes a cosinusoidal radiowave B-field and splits it into two parts. If you squint closely at the math, the time dependent B-fields present in the last line will spring out to you as counter-rotating magnetic fields. I got away with doing this by basically adding zero.
Why in the world did I do this? This seems like a horrible complexification of an already hard-to-visualize system.
To understand the reason for doing this, I need to make a digression. In physics, one of the most useful and sometimes overlooked tools you can run across is the idea of frame of reference. Frame of reference is simply the circumstance by which you define your units of measurement. You can think about this as being synonymous with where you have decided to park your lawn chair in order to take stock of the events occurring around you. In watching a train trundle past on its tracks, clearly I’ve decided my frame of reference is sitting someplace near the tracks where I can measure that the train is moving with respect to me. I can also examine the same situation from inside the train car looking out the window, watching the scenery scroll past. Both situations will yield the same mathematical observations from two different ways of looking at the same thing.
In this case, the frame of reference that is useful to step into is a rotating frame. If you’re on the playground, when you sit down on a moving merry-go-round, you have shifted to a rotating frame of reference where the world will appear as if it rotates around you. Sitting on this moving merry-go-round, if you watch someone toss a baseball across over your head, you would need to add some sort of fictitious force into your calculation to properly capture the path the ball will follow from your point of view. This means reinventing your derivative with respect to time.
This description of the rotating frame time derivative is simply a matter of tabulating all the different vectors that contribute to the final derivative. (The vectors here are misdrawn slightly because I initially had the rotating vector backward.) The vector as seen in the frame of reference moves through the rotation according to displacements that are due both to the internal (in) rotation and whatever external (ext) displacements contribute to its final state. The portion due to the rotation (rot) is a position vector that is simply shifted by the rotation at an angle I called ‘α’ where the rotation is defined with positive being in the right-handed sense –literally backward (lefthanded) when seen from within the rotating frame. The angular displacement ‘α’ is equal to the angular speed ‘Ω’ times time as Ωt and it can be represented by a vector that is defined to point along the z-axis. The little bit of trig here shows that the rotating frame derivative requires an extra term that is a cross product between the vector being differentiated and the rotational velocity vector.
How does this help me?
I’ve once again converted torque and magnetic moment into angular momentum in order to reveal the time derivative. It is noteworthy here that the term involving the Larmor frequency directly, the first term on the right, looks very similar to the form of the rotating frame if the Larmor frequency is taken to be angular velocity of the rotating frame. Moreover, I have already defined two other magnetic field terms that are both rotating in opposition to each other where I have not selected their frequencies of rotation.
A rotating frame could be chosen where the term involving the static magnetic field will be canceled by the rotation. This will be a clockwise rotation at the speed of the Larmor frequency. If the frequency of rotation of B2 is chosen to be the Larmor frequency, the clockwise rotating B2 field term will enter into the rotating frame without time dependence while the frequency of the other term will double. As such, one version of the B2 field can be chosen to rotate with the rotating frame.
In the final line, the primed unit vectors are taken to be with the rotating frame of reference. So, two things have happened here: the effect of the powerful static field is canceled out purely by the rotation of the rotating frame and the effect of the counter rotating field, spinning around at twice the Larmor frequency in the opposite direction, is on average in no direction. The only remaining significant term is the field that is stationary with respect to the rotating frame, which I’ve taken to be along the y’-axis.
The differential equation that I’ve ended up with here is exactly like the differential equation solved for the powerful static field by itself far above, but with a precession that will now occur at a frequency of ω2 around the y’-axis.
If I take the starting state of the magnetic dipole moment to be relaxed along the z-axis, no component will ever exist along the y’-axis… the magnetic dipole moment will purely inhabit the z-x’ plane in the rotating frame.
As long as the oscillating radiowave magnetic field is present, the magnetic dipole moment will continue to revolve around the y’-axis in the rotating frame. In the stationary frame, the dipole moment will tend to follow a complicated helical path both going around the static B-field and around the rotating y’-axis.
If you irradiate the sample with radiowaves for 1/4 of the period associated with the ω2 frequency, the magnetic dipole moment will rotate around until it lies in the x-y plane. You then shut off the radiowave source and watch as the NMR sample undergoes a free induction decay until the magnetization lies back along the static B-field.
This is a classical view of what’s going on: a polarized radiowave applied at the Larmor frequency will cause atomic magnetic dipoles to torque around in the sample until they are able to undergo oscillation. Once the radiowave is shut off, the magnetization performs a free induction decay. Applying the radiowave at the Larmor frequency is said to be driving the system at resonance since the static B-field will always be strong enough to overwhelm the comparably weak radiowave magnetic field.
I’ve completely avoided the quantum mechanics of this system. The rotating frame of Larmor precession is fairly accurate for describing what’s happening here until you need to consider other quantum mechanical effects present in the signal, such as spin-spin coupling of neighboring NMR active nuclei. Quantum mechanics are ultimately what’s going on, but you want to avoid the headaches associated with that wherever possible.
I do have it in mind to rewrite a long defunct post that described the quantum mechanics of the two-state system specifically in how it describes NMR. It will happen someday, honestly!