I Why are there not more ferromagnetic Materials?

Hellooo

I have another questions about that fact that we have only a few solids that are ferromagnetic. In ferromagnetics i also read a few time ago it is important to consider 2 things about the reason why there is no magnetizaion. And i dont really understand both reasons, so i hope you can help me. So the both things are:

1. "The hybridization breaks spherical spherical symmetry for the enviroment of each atom, which tends to quench any orbital component of the magnetic moment."
So i dont see really why the case of no spherical movement of an electron will give less orbital magnetic moment. The magnetic moment of the electron is defined by its motion around a core and this motion will be still a motion around the core.

2. "Bandstructure also inhibits spin polarization. If one starts with a system of unpolarized electrons and imagines flipping spins to create alignment, then there is a kinetic-energy cost associated with moving electrons from lower-energy filled band states to higher-energy unoccupied band states. As a result, most solids are not ferromagnetic."

So i read that 2. sentence a thousand times but i really dont understand it...or what the author meant...
So far i know the consider a band with an unpolarized spin and than he imagines spin flipping of some electron in this band (correct???). Then he says its cost kinetic energy to move an electron from the lower band to the higher bbut whats the connection to the spin flipping and why does it cost kinetic energy besides the potentiell energy?
 
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1. "The hybridization breaks spherical spherical symmetry for the enviroment of each atom, which tends to quench any orbital component of the magnetic moment."
So i dont see really why the case of no spherical movement of an electron will give less orbital magnetic moment. The magnetic moment of the electron is defined by its motion around a core and this motion will be still a motion around the core.
To begin, your quote from an unnamed source speaks of the "spherical symmetry of the environment" which you then took upon yourself to recast as "spherical movement of an electron" (a statement that is problematic in and of itself, but one that I am not going to address.) If we focus on what the author actually said, he is alluding to orbital quenching which takes place when the crystal field environment from the surrounding lattice (which is not spherically symmetric) is larger than the spin orbit interaction, thereby overriding Hund's 3rd rule. The simple hand-wavy explanation as to why this leads to orbital quenching is that the orbital components precess in the crystal field which result in them averaging to zero: ##<L_{i}> = 0## for ##i = x,y,z##. The non hand-wavy, mathematical reason why you get orbital quenching is entirely due to the fact that the crystal field Hamiltonian is real, ie: no imaginary component. That is quite a spectacular result when you stop to think about it. It is also straightforward to see why this leads to quenching. Namely, a real Hamiltonian guarantees that one can write out a full set of real eiganstates. However, since the crystal field Hamiltonian commutes with ##L_{i}## these eigenstates still carry a factor ##e^{im_{l}\phi}##. Thus, in order for the eigenstate to be real, linear combinations are required of ##\pm m_l## so that the complex exponentials take the form of cosines or sines, which in turn leads to our desired result of ##<L_{i}> = 0##.

consider a band with an unpolarized spin and than he imagines spin flipping of some electron in this band (correct???)
That is correct.

Then he says its cost kinetic energy to move an electron from the lower band to the higher bbut whats the connection to the spin flipping
If you have a band that is full up to the Fermi energy, then this implies that each filled energy state includes both a spin up and spin down electron. This means you cannot just flip an electrons spin while simultaneously leaving it in the same energy state, that would break the exclusion principle. Thus, if you want to flip its spin, then you need to move it to a free energy state. Since all the states are filled up to the Fermi energy, this guarantees that it will cost extra energy to move it above the Fermi energy.

why does it cost kinetic energy besides the potentiell energy?
The main take away is that the electron will need to move to a higher energy state, and that will cost energy. The bands themselves already encapsulate both the kinetic energy of the electrons and the potential energy from the lattice (at the very least). Indeed, depending on what new state the electron moves to, its own kinetic energy could increase or decrease, but its total energy will definitely go up. The question then is, who is supplying this energy and in what form? Thermal fluctuations are the typical culprit and they are kinetic at heart.
 

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