The Spin

 

The spin is an intrinsic form of angular momentum of elementary particles, composite particles and atomic nuclei. The spin of the electron was discovered initially from the Stern-Gerlach experiment which demonstrated the presence of two discrete angular momenta for silver atoms and their physical separation in a magnetic field. Additionally, the splitting of hydrogen spectral lines, termed fine structure, indicated the existence of energetic sublevels depending on an intrinsic angular momentum of the electron other than the orbital angular momentum associated to its movement around the nucleus. This could be explained in classical physics, if the electron was considered as a spinning ball of charge i.e. rotating around itself, thereby having a special angular momentum due to this rotation. For this reason, this intrinsic angular momentum was termed spin angular momentum or spin (S).

 

In classical physics, if representing the rotation of a sphere, e.g. an educational globe, we draw a vector for angular momentum which lies on the rotation axis of the sphere (and passes from its center), has a length which is proportional to the angular velocity of the rotation and a direction which depends on whether the rotation is clockwise or counter-clockwise. For example, it points up if the direction is clockwise and down if it is counter-clockwise.

 

In quantum physics, in the case of the electron for instance, the spin or angular momentum S is a quantized. We can determine its magnitude and although we cannot determine its direction in a coordinate system, we can calculate the projection on one axis, conventionally the z axis. This is represented by the the spin magnetic number ms which takes two values i.e. +1/2 for "spin up" and -1/2 for "spin down". The spin angular momentum is also associated to a spin magnetic (dipole) moment μs.

 

 

 

 

Effect of the magnetic field on the spin

 

A magnetic field exerts a torque on the spins due to which most will, similarly to a compass needle, tend to preferentially align themselves with the direction of the magnetic field, while some will tend to align themselves in the opposite direction. In is noted that in the first case, the configuration is termed "spin-up", "aligned" or "parallel" (aligned or parallel with respect to the magnetic field lines) and in the second case "spin-down", "anti-aligned" or "anti-parallel". The magnetic field induces spin polarization to a certain degree, or in other words, it polarizes the spins.

 

By examining the compass similarity, we can appreciate that in order to have the compass needle point to a direction other than that of the Earth's magnetic field, we need to rotate it and therefore do work and provide kinetic energy to the system which then transitions to a higher energy state (the stronger the magnetic field, the more energy we have to supply). It can therefore be inferred that the configuration that is in the opposite direction of the magnetic field, the anti-aligned state, is a higher energy state. Similarly, the aligned state is a low energy state.

 

It is mentioned that the magnetic field produces a small splitting of energy levels or that it creates magnetic sublevels, as due to the torque exerted, it attributes different magnetic potential energies to the spins. As most spins tend to preferentially align themselves with the direction of the magnetic field, a net alignment with it is produced. In the absence of a magnetic field all spins would have the same energy. This splitting, which is also termed Zeeman splitting is represented in Figure 1.

 

 

Figure 1: (From ref.) The application of a magnetic field causes a small splitting of energy levels producing a net alignment with the magnetic field.

 

 

Figure 2: From Wikipedia. Splitting of the energy levels of an electron by a magnetic field.


 

As the spin represents an intrinsic angular momentum which has been compared to the angular momentum of a spinning charged ball, the magnetic field exerts a torque on the spin similarly to the gravity field on a spinning top. Due to the gravity torque, the spinning top will wobble or precess, or in other words, it will perform a movement that draws a cone in space. Similarly, the spin will precess around the direction of the magnetic field with a frequency ω termed Larmor frequency. The Larmor frequency is given by the relationship:

 

ω = γ * Β

 

where γ is the gyromagnetic ratio (magnetogyric ratio) and Β the magnetic field strength.


 

In accordance with what was mentioned above, in the presence of the magnetic field, spins will tend to preferentially align themselves with the direction of the field, thereby adopting a low energy state while precessing around the direction of the field, or will align themselves opposite to the direction of the field, thereby adopting a high energy state, while precessing accordingly.


 

For a “spin up” electron i.e.  ms=+½ in a magnetic field:

 

E+½= -γ ħ B0 /2

P+½= 0.5000049

 

For a “spin down” electron i.e.  ms=-½ in a magnetic field:

 

E-½= +γ ħ B0 /2

P-½= 0.4999951

 

 

Max Planck had shown that the change in energy (ΔE) of an atomic system due to the emission of a photon of frequency (fo) is:

 

ΔE = h fo

 

where h is Planck's constant.


 

The energy difference between the two spin energy states is given by the relationship:

 

ΔE = γ ħ B0 = ħ ω

 

where γ is the gyromagnetic ratio, h is Plank’s constant and ħ = h / 2π 

 

 

It is noted that this is equivalent to ΔE = h fo.

 

The energy difference between the two states is proportional to the strength of the magnetic field (B).

 

 

 

 

Magnetic Resonance

 

If we provide to a spin system electromagnetic energy at the frequency of spin precession, the spin system will be able to absorb it and transition to a high energy state and subsequently emit it and transition to a lower energy state. This is termed Magnetic Resonance. It is used in nuclear magnetic resonance spectroscopy to determine material properties and identify substances and also in medical imaging techniques such as “Magnetic Resonance Imaging” (MRI). A brief description of the MRI technique will be provided further below.

 

For instance, in the case of the electron spin, there will be a transition from ms=+½ to ms=-½ and back in accordance with the relationship mentioned previously.

 

This can be accomplished by providing a radiofrequency as is the case in Magnetic Resonance Imaging (MRI). If we consider the magnetic component of the RF, i.e. the changing magnetic field (e.g. picture), then this will exert a torque/force on the spin system which will tip (flip) it or make it change direction for a specific angle depending on the duration of the pulse.

 

This angle is termed flip/tip angle (φ). If we apply this field perpendicular to the spin system and we wish to flip the spin by 90°(π/2) then we can use the following relationship to calculate the required duration of the pulse for φ=π/2:

 

φ=ω*T = γ*B*T.

 
 
 

Boltzmann Distribution of Spin Energy States - Thermal equilibrium

 

The influence of the magnetic field is counteracted by thermal collisions. Under the influence of the magnetic field and the thermal collisions, an equilibrium will be established for the population, termed thermal equilibrium which is described by the Boltzmann distribution:

 

N+/N− = exp [− ΔE / kT]

 

where N+ and N- represent the number of spins that are expected to be found in the spin up or spin down configuration.

 

The Boltzmann distribution predicts that at room/body temperature the amount of spins in the lower energy state is almost equal to that of spins in the higher energy state with a very small excess of spins ~3 x 10-6 in the spin up orientation.