Spin Quantum Number: Definition, How To Calculate & Significance

In quantum mechanics, as you try to make analogies between classical quantities and their quantum counterparts, it isn't uncommon for those analogies to fail. Spin is a perfect example of this.

In order to understand spin and the subsequent distinction between orbital and intrinsic angular momentum, it is important to understand the structure of an atom and how electrons are arranged within it.

The simplified Bohr model of the atom treats electrons as though they are planets orbiting a central mass, the nucleus. In reality, however, electrons act as diffuse clouds that can take on a number of different orbital patterns. Because the energy states they can occupy are quantized, or discrete, there are distinct orbitals or regions that different electron clouds exist in at different energy values.

Note the word ​orbital​ instead of ​orbit​. These electrons do not orbit in nice circular patterns. Some electrons might occupy a diffuse spherical shell, but others occupy states that create different patterns than might look like a barbell or a torus. These different levels or orbitals are often referred to as shells as well.

Because electrons have spin, but are also occupying a state in an orbital of an atom, they have two different angular momenta associated with them. The orbital angular momentum is a result of the shape of the cloud the electron occupies. It can be thought of as analogous to the orbital angular momentum of a planet about the sun in that it refers to the electrons motion with respect to the central mass.

Its intrinsic angular momentum is its spin. While this can be thought of as analogous to the rotational angular momentum of an orbiting planet (that is, the angular momentum resulting from a planet rotating about its own axis), this isn't a perfect analogy since electrons are considered point masses. While it makes sense for a mass that takes up space to have an axis of rotation, it doesn't really make sense for a point to have an axis. Regardless, there is a property, called spin, which acts in this way. Spin is also often referred to as intrinsic angular momentum.

Within an atom, each electron is described by four quantum numbers which tell you what state that electron is in and what it's doing. These quantum numbers are the principal quantum number ​n​, the azimuthal quantum number ​l​, the magnetic quantum number ​m​ and the spin quantum number ​s​. These quantum numbers are related to each other in different ways.

The principal quantum number takes on integer values of 1, 2, 3 and so on. The value of ​n​ indicates which electron shell or orbital the particular electron is occupying. The highest value of ​n​ for a particular atom is the number associated with the outermost shell.

The azimuthal quantum number ​l​, which is sometimes referred to as the angular quantum number or the orbital quantum number, describes the associated subshell. It can take on integer values from 0 to ​n​-1 where ​n​ is the principal quantum number for the shell that it is in. From ​l​, the magnitude of the orbital angular momentum can be determined via the relationship:

\(L^2=\hbar^2l(l+1)\)

Where ​L​ is the orbital angular momentum of the electron and ℏ is the reduced Planck constant.

The magnetic quantum number ​m​, often labeled ​_m_l​ to make it clear that it is associated with a particular azimuthal quantum number, gives the projection of the angular momentum. Within a subshell, the angular momentum vectors can have certain allowed orientations, and ​m_l​ labels which of those a particular electron has. ​m_l​ can take on integer values between -​l​ and +​l_​.

In general, the spin quantum number is denoted with an ​s​. For all electrons, however, ​s​ = ½. An associated number ​_m_s​ gives the possible orientations of ​s​ in the same way ​m_l​ gave the possible orientations of ​l​. The possible values of ​m_s​ are integer increments between ​-s​ and ​s​. Hence for an electron in an atom, ​m_s_​ can be either -½ or +½.

Spin is quantized via the relationship:

\(S^2=\hbar^2s(s+1)\)

where ​S​ is the intrinsic angular momentum. Hence knowing ​s​ can give you the intrinsic angular momentum just as knowing ​l​ can give you the orbital angular momentum. But again, within atoms all electrons have the same value of ​s​, which makes it less exciting.

Particle physics aims to understand the workings of all fundamental particles. The standard model classifies particles into ​fermions​ and ​bosons​, and then further classifies fermions into ​quarks​ and ​leptons​, and bosons into ​gauge​ and ​scalar bosons​.

Leptons include ​electrons​, ​neutrinos​ and other more exotic particles like the ​muon​, the ​tau​ and associated ​antiparticles​. Quarks include the ​up and down quarks​ that combine to form ​neutrons​ and ​protons​, as well as quarks named ​top​, ​bottom​, ​strange​ and ​charm​ and their associated antiparticles.

Bosons include the ​photon​, which mediates electromagnetic interactions; the ​gluon​, the ​_Z⁰ boson​, the ​W⁺​ and ​W^-​ bosons and the ​Higgs_​ boson.

The fundamental fermions all have spin 1/2, though some exotic combinations can have spin 3/2 and theoretically higher, but always an integer multiple of 1/2. Most bosons have spin 1 except the Higgs boson, which has spin 0. The hypothetical graviton (not yet discovered) is predicted to have spin 2. Again, theoretically higher spins are possible.

Bosons do not obey number conservation laws while fermions do. There is also a "law of conservation of lepton" number and "of quark" number, in addition to other conserved quantities. Interactions of the fundamental particles is mediated by the energy-carrying bosons.

The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state at the same time. On a macroscopic scale, this is like saying two people can't occupy the same place at the same time (though fighting siblings have been known to try).

What this means for the electrons in an atom is that there are only so many "seats" at each energy level. If an atom has a lot of electrons, then many of them must end up in higher energy states once all of the lower states are full. An electron's quantum state is completely described by its four quantum numbers ​n​, ​l​, ​_m_l​ and ​m_s_​. No two electrons within a single atom can have the same set of values for those numbers.

For example, consider allowed electron states in an atom. The lowest shell is associated with quantum number ​n​ = 1. The possible values of ​l​ are then 0 and 1. For ​l​ = 0, the only possible value of ​_m_l​ is 0. For ​l​ = 1, ​m_l​ can be -1, 0 or 1. Then ​m_s​ = + 1/2 or -1/2. This makes the following combinations possible for the ​n_​ = 1 shell:

•​l​ = 0, ​_m_l​ = 0,

​m_s_​ = 1/2
•​l​ = 0,

​_m_l​ = 0,

​m_s_​ = -1/2
•​l​ = 1,

​_m_l​ = -1,

​m_s_​ = 1/2
•​l​ = 1,

​_m_l​ = -1,

​m_s_​ = -1/2
•​l​ = 1,

​_m_l​ = 0,

​m_s_​ = 1/2
•​l​ = 1,

​_m_l​ = 0,

​m_s_​ = -1/2

•​l​ = 1,

​_m_l​ = 1,

​m_s_​ = 1/2
•​l​ = 1,

​_m_l​ = 1,

​m_s_​ = -1/2

Therefore, if an atom has more than eight electrons, the rest of them must occupy higher shells such as ​n​ = 2 and so on.

Boson particles do not obey the Pauli exclusion principle.

The most famous experiment for demonstrating that electrons must have intrinsic angular momentum, or spin, was the Stern-Gerlach experiment. To understand how this experiment worked, consider that a charged object with angular momentum should have an associated magnetic moment. This is because magnetic fields are created by moving charge. If you send current through a coil of wire, for example, a magnetic field will be created as if there were a bar magnet sitting inside of, and aligned with, the axis of the coil.

Outside of an atom, an electron will not have orbital angular momentum. (That is, unless it is moved in a circular path by some other means.) If such an electron were to travel in a straight line in the positive ​x​-direction, it would create a magnetic field that wraps around the axis of its motion in a circle. If such an electron were passed through a magnetic field aligned with the ​z​-axis, its path should deviate in the ​y​-direction slightly as a result.

However, when passed through this magnetic field, an electron beam splits in two in the ​z​-direction. This could only happen if electrons possess an intrinsic angular momentum. Intrinsic angular momentum will cause the electrons to have a magnetic moment that can interact with the applied magnetic field. The fact that the beam splits into two indicates two possible orientations for this intrinsic angular momentum.

A similar experiment was first performed by German physicists Otto Stern and Walter Gerlach in 1922. In their experiment, they passed a beam of silver atoms (which do not have a net magnetic moment due to orbital effects) through a magnetic field and saw the beam split in two.

Since this experiment made it clear that there were exactly two possible spin orientations, one that was deflected upward and one that was deflected downward, the two possible spin orientations of most fermions are often referred to as "spin up" and "spin down."

Fine structure splitting of energy levels or spectral lines in a hydrogen atom was further evidence of electrons having spin, and that spin having two possible orientations. Within the electron orbitals of an atom, every possible combination of ​n​, ​l​ and ​_m_l​ comes with two possible ​m_s_​ values.

Recall that within a given atom, only very specific wavelengths of photons can be absorbed or emitted, depending on the allowed, quantized energy levels within that atom. Absorption or emission spectra from a given atom reads like a bar code that is specific to that atom.

The energy levels associated with the different spin ​_m_s​ values for fixed ​n​, ​l​ and ​m_l​ are very closely spaced. In the hydrogen atom, when spectral emission lines were closely examined at high resolution, this so-called ​doublet​ was observed. What looked like a single emission line associated with just the ​n​, ​l​ and ​m_l_​ quantum numbers was actually two emission lines, indicating a fourth quantum number with two possible values.