Property of quantum particles spin quantum number is the base of the quantum computer and MRI ( magnetic resonance imaging ) like machines.

Atomic or subatomic quantum particles like protons, neutrons, nuclei, and electrons possess four basic properties called quantum numbers.

1) Principal quantum number

2) Azimuth quantum number

3) Magnetic quantum number

4) Spin quantum number

All the above four quantum numbers define the quantum state of atomic or subatomic particles.

Spin Quantum Number Represents

Represent the angular movement of quantum particles and denoted by discrete values.


Symbols l and S are used to represent the value of quantum particles. Like electron spin value always denoted by S = +1/2 or -1/2.

In the classical spin, the quantum particle model denotes a gyroscopic rotating body but in reality, according to quantum mechanics is not actually spinning or rotating.

Spin quantum property interacts with magnetic or electromagnetic fields instead of the gravity field.

Only can be represented in the quantum values set of discrete numbers.

Nuclear spin quantum number

In the all elements of the periodic table have a nuclear spin value between 0 to 8 with 1/2 increment or decrement.

This property is more commonly used in NMR, spectroscopy, and MRI ( Magnetic Resonance Imaging ).


The number of nuclei is equivalent to the number of protons until the atomic number is less.

As the atomic number and mass increase, the number of neutrons also increases then neutron also needs to consider in a total of nuclear spin.

If for a given element the atomic mass is even and the number of protons, neutrons are also even then result in nuclear spin value will be zero.

And if the mass is even and the number of protons, neutrons are odd. The value of spin will be in integer.

Assume nuclear mass is odd and similar the number of neutrons also odd but the number of protons is even that the nuclear spin value will be half-integer.

Again nuclear mass is odd and the number of protons is odd value but neutrons are in even numbers.

For such a combination, the nuclear spin value will be half-integer.

Spin quantum number discovered by

Before the discovery of spin quantum number Zeeman observe in strong magnetic field non zero spin elements spectral lines are split into small parts called Zeeman Effect.

Zeeman Effect

Till this time there was no explanation for the effect also know as Paschen Back effect.

Later credit two physicists George Uhlenbeck and Samuel Goudsmit credited the discovery spin number.

The relation between Gyromagnetic Ratio or Larmor frequency (γ) and spin

μ = γ I 

Where μ = magnetic moment, I = spin for nucleus

μ = γ S 

Where μ = magnetic moment, S = spin for electron

How to Detect Spin

After two years of spin theoretical concept came into existence. There was no experimental evidence to support the theory.

Otto Stern and Walter Gerlach experiment to show such phenomena known as the Stern–Gerlach experiment.


In this experiment set up in a vacuum, it uses an electric furnace to produce silver atom vapors.

Silver atoms cross a wide slit and create a wide beam of silver atoms.

After cross slit passes through an inhomogeneous magnetic field then collect on the metal sheet.

As the result on the metal sheet silver atom form two wide lines instead of one.

This result explains by the spin quantum number which tells about up and down electron spin and acts like magnetic dipole deflect in different directions due to different magnetic poles.

Even More

Above try to explain about are not commonly available in textbooks. Still, a lot to explain like the Dirac equation for define energy levels and quantum mechanics.

The most common questions try to cover in the FAQ.

How do you find the spin quantum number?


For an atom of any element, there are two kinds of spin numbers.
First is nuclear spin quantum number and electron spin quantum number.
For nuclear numbers explain in detail in above in nuclear heading about how to evaluate nuclear spin value.
Rest for any electron there are only two spin values that exist either +1/2 or -1/2.
For example, Na (sodium) has an atomic mass is 23 and an atomic number is 11.
Thus we can find out the number of neutrons (23-11= 12). So the mass and number of protons are odd numbers but the number of neutrons is even.
Then the nuclear spin value will be in half-integer.
Now for 11 electron of Na atom spin value calculation.
atomic number = 11
the principal quantum number (main orbit ) will be n = 3 (2, 8, 1)
and the azimuth quantum number (sub orbital value) will be l(n-1)= 0,1 ( s, p)
Now the magnetic quantum number is m( -l,0,+l) = -1, 0, +1
Finally we know the electron distribution 1S2, 2S2, 2P6, 3S1 (Hund’s Rules ).
Thus the last electron is single in 3S1 orbital and the spin quantum number will be +1/2.

How do you know if the spin quantum number is positive or negative ?


According to Hund’s law can find out the electron position or can predict electron belongs to the witch suborbital.
Similarly, also define that the first electron in the suborbital shell will have a positive spin.
The next electron in the suborbital shell will be a negative spin. Thus the resultant spin of the suborbital shell becomes zero.

What is the importance of spin quantum number ?

Use to define the behavior of atomic or subatomic partials in an external magnetic field.
Also, explain the quantum state of the partials.
Though this can define and calculate the RF or electromagnetic energy absorption/ radiation.
Nuclear magnetic resonance, spectroscopy, and magnetic resonance imaging some of the examples are based on the spin quantum number.

What is spin formula?

\mu_{\text{eff}} = \sqrt{\vec{L}(\vec{L}+1)+ 4\vec{S}(\vec{S}+1)} \mu_B
μeff = effective magnetic moment for complex transition metal ions
{\vec {L}} & {\vec {S}} =angular momentum of the unpaired electrons
μB= Bohr magneton constant
\mu _{{\mathrm  {B}}}={\frac  {e\hbar }{2m_{{\mathrm  {e}}}}}
e = charge
ħ = Planck constant
me = electron rest mass 

Simple spin formula

\mu_{\text{eff}}= \sqrt{n(n+2)} \mu_B
n = number of unpaired electrons

Why do electron spin ?

Physicists found in their repeated experiments that unpaired electrons of elements deflect in the magnetic field.
And the concept is very well known that only moving charge can produce a magnetic field like electron and thus assume electron spin along its own axis.
As a result electron spin quantum number comes to define the phenomena.
The spin of an electron is different from an orbital rotation of the nucleolus.

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