OverviewElectron Magnetic Resonance (EMR), sometimes termed Electron Paramagnetic Resonance (EPR), is the name given to the magnetic behaviour of an electron immersed in an external magnetic field. This paper describes the behaviour and also covers some related topics. In so doing it uses the concept of a “bound” as opposed to a “free” electron. In EMR an electron has two key properties - its magnetic field and its gyroscopic behaviour. Its electric field plays no part. The electron's magnetic field The magnetic field of an electron is dipole in structure. This means it has two ends which we think of as north and south for historical reasons - magnets or lodestones were used for navigational purposes for millennia.
Figure 1: Electron’s magnet in external field Opposite poles attract, so when a magnet is in an external linear field the magnet’s north pole is attracted to the field's south pole and vice versa. The forces are equal and opposite so the magnet does not move in either direction. It does, however, rotate so that its north pole is as near as it can get to the field's south pole and vice versa. The stronger either the magnet or the field, the greater the rotational torque. The term "moment" is used in science to designate the strength of any rotational effect, and so we can use the expression "dipole moment" to indicate the strength of the magnet. This torque is an expression of the energy stored in the magnetic fields - when magnetic fields are in opposition they add, and when in alignment they subtract, so the energy difference gives rise to the torque. The “dipole moment” may be expressed as the peak torque Tpeak, which occurs when the magnet and the field are normal (orthogonal) to each other. We can also express the dipole moment as the amount of energy needed to rotating it from the fully aligned position (with north pole to south pole and vice-versa) to the orthogonal (normal) orientation . The energy is simply the integral of the torque over the rotation, and it just so happens that the energy and the peak torque are numerically identical. Both are linearly proportional the the strength of the external magnetic field, which is normally given in units of Tesla. The electron's dipole moment may then be given in vector torque as 9.28E-24 Newton-meters per Tesla of external field so that the torque T in Newton-meters is simply the vector cross-product of the dipole moment and the external field strength B...
...which integrates to give the dipole moment energy as 9.28E-24 joules per Tesla for a rotation from aligned to orthogonal orientation...
The amount of energy required to rotate the magnet from alignment (north-to-south) to opposition (north-to-north and south-to-south) to is simply double this, giving the full inversion energy...
Dipole moment and gyroscopic momentThe external field interacts with the magnet to produce an aligning torque. If this were all there was to it the magnet would simply swing into orientation. However, the gyroscopic moment of the electron comes into play here. The dipole moment and the gyroscopic moment lie on the same axis, so when a torque is applied to the magnet the gyroscope turns this into precession, just as with any gyroscope in the universe. The precession frequency v is the peak torque divided by the gyroscopic moment G...
...but the electron's gyroscopic moment is h/2, where "h" is Plank's constant, so...
...or alternatively...
- a well known equation! For the electron in a magnetic field of one Tesla, the frequency is...
Now if the precession did not release energy the electron would simply precess indefinitely. However the magnetic fields around an electron are extensive - the field around a bar magnet is simply the fields of many electrons acting together - and as the magnet rotates the magnetic fields sweep out over a large region.
Figure 2: Plan view of electron in electromagnetic waveWhether a rotating magnet radiates or absorbs energy, or neither, depends on its phase relationship with its radiant electromagnetic environment. If it is exactly in step (a phase of zero) energy is neither radiated nor absorbed. If it is phase-advanced the magnet will experience a drag from its surroundings and hence radiate energy. If it is phase-retarded, or lagging, the magnet will experience a driving torque from it surroundings and will hence absorb energy. The greater the phase difference, the greater the rate of energy loss or gain. Electric motors and radio aerials work in exactly the same way. The phase difference ‘a’ is measured by considering the phase of the magnetic component of the electromagnetic wave when the magnet is oriented normal to the direction of propagation, as shown in Figure 2. This is a plan view - a view from above. In Figure 2 the radiation lags behind the precession by a small phase angle, taking power from the whatever is driving the gyroscopic precession motion and radiating electromagnetic energy into space. The phase lag provides a damping torque on the precession motion which in turn allows the magnet to move into alignment with the field. The greater the phase lag (up to a maximum of -pi/2) the greater the rate of energy loss and hence the faster the electron moves into alignment. The energy flow is therefore: Energy is stored in the interaction between the electron’s magnetic field and the external magnetic field. This creates a torque applied to the gyroscope’s primary axis, causing precession. This precession causes the magnet to rotate and hence radiate electromagnetic energy. A reverse torque is applied to the precession axis as a result of this energy loss, which appears on the primary gyroscope axis as a precession into alignment with the original torque, so reducing the energy stored in the magnetic fields. Since there is rotation and torque on the primary axis, energy is injected on that axis, and this appears as the radiation energy lost from the secondary axis. During inversion of the magnet from opposition to alignment the energy radiated is...
...at a frequency v...
It is easy to reverse the process. We can move the magnet from alignment to opposition by pumping the system with an incident electromagnetic wave at the correct frequency v (this frequency must be exactly right - if it is off by a tiny fraction of a percent the precession rate will fall out of step with the electromagnetic wave and fail to absorb energy from it). In pumping the system the electromagnetic wave leads in advance of the gyroscope’s precession rather than (as above) lagging behind it. Again, the greater the phase lead (up to a maximum of +pi/2) the greater the rate of pumping and the faster the electron moves into opposition. The rate of pumping is proportional to the amplitude of the incident electromagnetic wave and the sine of the phase lead angle. It is also quite possible for the electron to simply resonate with the incident wave. The wave may simply be scattered. The incident and the radiated waves both have their magnetic components everywhere normal to the external magnetic field direction, so the electric field vector of the incident and scattered waves is parallel to the external magnetic field vector. If we take away the external field the electron will not have any aligning torque and so will not precess; it may simply rock back and forth to a trivial degree in the direction of propagation. The torque is zero at both alignment and opposition, so these two positions may be regarded as stable states. The energy is taken to be zero at the normal orientation between the magnet and field, so the energy of the aligned state is...
...and that of the opposition state is...
Then the electron may absorb a photon of frequency v, energy...
...in being pumped from the lower energy state (alignment) to the higher (opposition), or emit a photon in dropping from the higher to the lower. The protonIt is possible to do the same for the proton in Proton Magnetic Resonance (or PMR). This has the same gyroscopic moment ‘G’ as the electron, at h/2, but its magnetic dipole moment is only 1.41E-26 joules/Tesla. Hence in a field of one Tesla the precession frequency is... v = 2.(1.41E-26).|B| / h The precession frequency is so precisely a function of field strength and so readily measured to high precision, that PMR is used as a means of measuring magnetic fields to very high accuracy. Hydrogen nuclei in water are used to provide the protons. CaveatsI have described the simplest possible instance of EMR here. Nevertheless, a very basic concept is demonstrated by this experimental setup; an electron ”bound” or constrained by an external field can interact with a photon, while there is no mechanism for a “free” or unconstrained electron to do so. When dealing with EMR inside an atom many other factors come into play. Atoms do not like having their electron orbits disturbed by external magnetic fields, so that every atom is to some extent diamagnetic (repelled by magnetic fields). On the other hand electrons in an atom can be very sensitive to their magnetic orientation vis-à-vis an external field, leading to paramagnetism (attracted to magnetic fields), ferromagnetism, antiferromagnetism, ferrimagnetism, antiferrimagnetism and helical magnetism. In fact this sensitivity to external fields can be used to investigate the nature of atoms and molecules. |
