What Is the Effective Magnetic Moment in Quantum Mechanics?

[premagg]

1.I am confused about the topic effective magnetic moment.The equation is

U={L(L+1)+4S(S+1)}^(0.5)
Where u is effective magnetic moment,.

Please clear the various terms and units involved in this.
I will be greatly thankful if you could support it with example.(Ni+2)(Or at least clear the terms and units)

Homework Equations

The Attempt at a Solution

 

[Stevedye56]

http://wwwchem.uwimona.edu.jm:1104/spectra/MagMom.html

 

[Blueshift5]

The effective magnetic moment is a term used in quantum mechanics to describe the magnetic properties of an atom or molecule. It is a measure of the strength of the magnetic field produced by the spinning and orbital motion of the electrons in the system. The equation you have provided is known as the Landé g-factor, which is used to calculate the effective magnetic moment.

In this equation, L and S represent the orbital and spin angular momentum of the electrons, respectively. The terms (L+1) and (S+1) represent the possible orientations of the angular momentum vectors, while the square root term takes into account the quantum mechanical effects of spin-orbit coupling.

To understand this concept better, let's take an example of nickel (Ni) in its +2 oxidation state. This means that it has lost two electrons, leaving it with 28 electrons in total. In this state, the electronic configuration of Ni is 1s^2 2s^2 2p^6 3s^2 3p^6 3d^8. This configuration shows that there are 8 electrons in the 3d orbital, which contributes to the orbital angular momentum (L) of the system.

Now, let's calculate the effective magnetic moment of Ni+2 using the equation provided. Since there are 8 electrons in the 3d orbital, the value of L would be 2 (since 2+1=3 and 3d has a quantum number of 2). The value of S for Ni+2 is 1 (since there are 2 unpaired electrons in the 3d orbital). Plugging these values into the equation, we get:

U = {2(2+1)+4(1+1)}^(0.5)
U = {6+8}^(0.5)
U = {14}^(0.5)
U = 3.74

Therefore, the effective magnetic moment of Ni+2 is 3.74. This value tells us about the strength of the magnetic field produced by the electrons in the system.

It is important to note that the effective magnetic moment is a unitless quantity, as it is a ratio of magnetic field strength to the applied magnetic field. However, it is often expressed in units of Bohr magnetons (μB), which is a unit of magnetic moment commonly used in atomic and molecular physics. In this case, the effective magnetic moment of Ni+