use a correct arrow for logical causality

docs/lecture_8.md

@@ -33,9 +33,9 @@ $$
 
 Depending on the sign, you can get different forms of magnetism:
 
-$\chi>0\rightarrow$ _paramagnetism_: the material tends to magnetise along the local field.
+$\chi>0\Rightarrow$ _paramagnetism_: the material tends to magnetise along the local field.
 
-$\chi<0\rightarrow$ _diamagnetism_: the material tends to magnetise opposite to the local field.
+$\chi<0\Rightarrow$ _diamagnetism_: the material tends to magnetise opposite to the local field.
 
 Further on, we will discuss two forms of spontaneous magnetisation: _ferromagnetism_ and _anti-ferromagnetism_. Unlike the ones mentioned above, these forms persist even in the absense of a magnetic field.
 
@@ -50,11 +50,11 @@ Examples:
 
 - Fe = [Ar]4s$^2$3d$^6$
 
-$S=2$, $L=2$, $J=4$ $\rightarrow$ $^{2S+1}L_J =\ ^5{\rm D}_4$
+$S=2$, $L=2$, $J=4$ $\Rightarrow$ $^{2S+1}L_J =\ ^5{\rm D}_4$
 
 - V = [Ar]4s$^2$3d$^3$
 
-$S=\frac{3}{2}$, $L=3$, $J=\frac{3}{2}$ $\rightarrow$ $^{2S+1}L_J =\ ^4{\rm F}_{3/2}$
+$S=\frac{3}{2}$, $L=3$, $J=\frac{3}{2}$ $\Rightarrow$ $^{2S+1}L_J =\ ^4{\rm F}_{3/2}$
 
 ![](figures/hund.svg)
 
@@ -117,8 +117,8 @@ $$
 ### Atoms in solids
 Until now we have considered magnetic atoms in free space. When embedded inside a solid, many things change. Below, we will discuss two effects:
 
-- Interaction of magnetic atoms with other magnetic atoms $\rightarrow$ _Heisenberg model_
-- Interaction of spins with non-magnetic atoms $\rightarrow$ _Crystal field_
+- Interaction of magnetic atoms with other magnetic atoms $\Rightarrow$ _Heisenberg model_
+- Interaction of spins with non-magnetic atoms $\Rightarrow$ _Crystal field_
 
 ### Heisenberg model
 Consider a one-dimensional chain of spins $\frac{1}{2}$ that are nearest-neighbor coupled with equal coupling strength $J$:
@@ -164,9 +164,9 @@ $$
 \psi({\bf r}_1,{\bf r}_2,{\bf s}_1,{\bf s}_2)=-\psi({\bf r}_2,{\bf r}_1,{\bf s}_2,{\bf s}_1)
 $$
 
-Coulomb interaction favors symmetric spatial wavefunction, resulting in a preferred antisymmetric spin wavefunction $\rightarrow J>0$.
+Coulomb interaction favors symmetric spatial wavefunction, resulting in a preferred antisymmetric spin wavefunction $\Rightarrow J>0$.
 
-- _Superexchange interaction_ – When magnetic atoms are connected via one non-magnetic mutual neighbor, simultaneous exchange of electrons with the neighbor can favor anti-alignment $\rightarrow J<0$.
+- _Superexchange interaction_ – When magnetic atoms are connected via one non-magnetic mutual neighbor, simultaneous exchange of electrons with the neighbor can favor anti-alignment $\Rightarrow J<0$.
 
 ![](figures/superexchange.svg)
 
@@ -175,7 +175,9 @@ Coulomb interaction favors symmetric spatial wavefunction, resulting in a prefer
 ![](figures/RKKY.svg)
 
 ### Crystal field
-For a free atom, the orbitals are spherically symmetric (_spherical harmonics_). Inside a crystal, it can happen that, due to the Coulomb interaction with neighboring non-magnetic atoms, the degeneracy between orbitals is broken. As a result, electrons can no longer complete a full circular orbit around an atom, causing the orbital angular momentum to be _quenched_: ${\bf L}\rightarrow 0$.
+
+For a free atom, the orbitals are spherically symmetric (_spherical harmonics_). Inside a crystal, it can happen that, due to the Coulomb interaction with neighboring non-magnetic atoms, the degeneracy between orbitals is broken.
+As a result, electrons can no longer complete a full circular orbit around an atom, causing the orbital angular momentum to be _quenched_: ${\bf L}\Rightarrow 0$.
 
 ![](figures/crystalfield.svg)