Spin operators in the ZFS-PAS

[theflamer14]

Hi guys,

My question is regarding defining spin operators in the zero field splitting principal axis system. I am currently working on a S = 2 spin system, and know how to define the Sx, Sy, and Sz spin matrices. My question is, how do I rotate them to the zfs-PAS? Some papers I came across simply leave it at etc. [itex]\hat{S}[/itex][itex]_{x}[/itex]. I found a web page that uses:

R[itex]_{r}[/itex](θ) = e[itex]^{i σ_{r} θ/2 }[/itex]

r = x,y,z

(can't post the website because this is my first post! :P )

But it is for spin 1/2 systems. Also, it uses only one angle, [itex]\theta[/itex]. However, in my Zeeman Hamiltonian, I have my magnetic field, [itex]\vec{B}[/itex], specified by the polar angles ([itex]\theta[/itex] and [itex]\varphi[/itex]) of the vertices of a truncated icosahedron (buckeyball). So is there a need to incorporate both the angles into converting a normal spin operator, ex. [itex]{S}_{x}[/itex], into [itex]\hat{S}[/itex][itex]_{x}[/itex]? Any help in helping me understand/visualize is appreciated.

Regards,

Kiran

 

[eljose79]

Hi Kiran,

Thank you for your question. Defining spin operators in the zero field splitting principal axis system can be a bit tricky, especially for spin systems with higher spin values like S = 2. The key concept to understand here is the rotation of spin operators in the zfs-PAS. This involves using rotation operators, as you have mentioned in your post.

The rotation operator you have mentioned, R_{r}(θ) = e^{i σ_{r} θ/2}, is actually a general formula for spin operators in any spin system. The subscript r (x, y, or z) indicates the direction of the spin operator, and the angle θ represents the rotation angle. This formula can be used for spin 1/2 systems as well as higher spin systems like yours.

To answer your question about incorporating both angles (θ and φ) into the conversion of a normal spin operator to the zfs-PAS, the answer is yes. In your case, since your magnetic field is specified by the polar angles (θ and φ), you will need to incorporate both angles into your rotation operator. This will result in a slightly more complicated formula, but the concept remains the same.

I hope this helps in visualizing the rotation of spin operators in the zfs-PAS for your specific spin system. If you need further clarification or assistance, please don't hesitate to ask.