# Calculating Energy Levels in a Ni2+ Ion Crystal

• Nico045
In summary, the conversation discusses the energy levels and eigenstates of Ni2+ ions in a crystal, which are considered independent and subject to a potential. The Hamiltonian for this system is given by H0 = C(Sz^2 - S^2/3) and the eigenvalues are determined by solving the equation H0Ψ = KΨ. The quantum numbers for operators that commute with H0 can be used to find the eigenvalues. Adding a magnetic field in the z direction, given by the Hamiltonian H1 = 2uBBSz, changes the eigenvalues to include the term 2μBB. The eigenstate can be written as |S, M>, where S is the spin and M
Nico045
Thread moved from the technical forums, so no Homework Help Template is shown.
Hello, I am stuck at the beginning of an exercise because I have some trouble to understand how are the energy level in this problem :

In a crystal we have Ni2+ ions that we consider independent and they are submitted to an axial symmetry potential. Each ion acts as a free spin S=1. We have the hamiltonian :

H0=C(Sz2-S2/3)
C is a constant >0

- I need to know the energy level of each ion and the eigenstates.

After that, we add a magnetic field B oriented on the z axis (in interaction with the ion) which is given by the hamiltonian :
H1=2uBBSz

- Here again I have to find the levels of energy for H = H0+H1

Nico045 said:
H0=C(Sz2-S2/3)
C is a constant >0

- I need to know the energy level of each ion and the eigenstates.
So what are the eigenvalues and eigenstates of that Hamiltonian? (Hint: think in terms of the quantum numbers for operators that commute with H.)

I haven't studied much the eigenvalues of the spin but I can try. (I am not sure at all) :

Sz = +1/2 , -1/2
S=1

Then H0=C(1/4-1/3) = -C/12

eigenvalues K=-C/12

and I should look for something who satisfy :

H0Ψ=KΨ

Nico045 said:
I haven't studied much the eigenvalues of the spin but I can try. (I am not sure at all) :

Sz = +1/2 , -1/2
S=1
The eigenvalue of the ##\hat{S}^2## is ##S (S+1)##.

Nico045 said:
and I should look for something who satisfy :

H0Ψ=KΨ
Looking at it in terms of a wave function is not necessary. Do you know Dirac notation?

So the eigenvalue of S2 should be only 2 ? (since it is given that S=1)

Yes i know Dirac notation

Nico045 said:
So the eigenvalue of S2 should be only 2 ? (since it is given that S=1)
Correct.

Edit: assuming ħ = 1.

Nico045 said:
Yes i know Dirac notation
Then you should be able to write the eigenstate as a ket,

I have been a little busy, I'm sorry for not answering sooner.

It should be on this form then :

| eigenvalue of S2 , eigenvalue of Sz >
##\frac{C}{3}## for |2,1>

##\frac{C}{3}## for |2,-1>

##\frac{-2C}{3}## for|2,0>

And if we add a magnetic field B oriented on the z axis (in interaction with the ion), the hamiltonian is ##H=C(S_z^2 -\frac{1}{3}S^2) + 2 \mu_B B S_z## and the eigenvalues are :

##\frac{C}{3}+ 2 \mu_B B## for|2,1>
##\frac{C}{3} - 2 \mu_B B## for |2,-1>
##\frac{-2C}{3}## for |2,0>

Is that right ?

Last edited:
Looks fine, except
Nico045 said:
| eigenvalue of S2 , eigenvalue of Sz >
is not conventional. Normally, one would use ##| S, M \rangle## (in other words, use the value of the spin ##S##, not the eigenvalue of ##\hat{S}^2##).

## 1. What is a Ni2+ ion crystal and why is it important to calculate its energy levels?

A Ni2+ ion crystal is a type of crystal structure where nickel ions are surrounded by negatively charged ions. It is important to calculate its energy levels because understanding the energy levels of a crystal can provide insight into its physical and chemical properties, which can have various applications in fields such as materials science, chemistry, and physics.

## 2. How is the energy level of a Ni2+ ion in a crystal determined?

The energy level of a Ni2+ ion in a crystal is determined using quantum mechanical calculations. This involves solving the Schrödinger equation, which describes the behavior of electrons in a crystal, to determine the allowed energy levels for the ion within the crystal lattice.

## 3. What factors affect the energy levels of a Ni2+ ion in a crystal?

The energy levels of a Ni2+ ion in a crystal can be affected by several factors, including the strength of the crystal's bonding forces, the size and shape of the crystal lattice, and the presence of other ions or defects in the crystal structure.

## 4. How do energy levels in a Ni2+ ion crystal relate to its properties?

The energy levels in a Ni2+ ion crystal play a crucial role in determining its properties. For example, the energy levels can affect the absorption and emission of light, as well as the crystal's electrical conductivity and magnetic properties.

## 5. Can the energy levels of a Ni2+ ion crystal be experimentally measured?

Yes, the energy levels of a Ni2+ ion crystal can be experimentally measured using techniques such as X-ray diffraction, electron energy loss spectroscopy, and optical spectroscopy. These experimental methods can provide valuable information about the crystal's energy levels and properties.

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