In compounds which Varma characterized by homogeneous mixed-valence, each ion is assigned (at least by physicists) the same, non-integer, valence which is a result of a quantum mechanical superposition of two integral valences occuring on each ion.
Compounds exhibiting this type of mixed-valence include, CePd3, TmSe, SmB6 where the valences of the ions are 3.45, 2.72 and 3.7 for Ce, Tm, and Sm, respectively. In TmSe, the valence of 2.72 for the Tm ion is a result of valence fluctuations of this ion between the Tm2+ and Tm3+ states.
A distinctive experimental signature of homogeneous mixed valence is that the ground state is a spin singlet (and so has not net magnetic moment) even if one or both of the two oxidation states of the metal ion have a non-zero spin and magnetic moment. This is seen in the temperature dependence of the magnetic susceptibility. At high temperatures it has a Curie form characteristic of a magnetic moment. At low temperatures it saturates to a finite value. It also means there can be an absence of an Electron Spin Resonance (ESR) signal normally associated with metal ions with integer valence.
Other signatures were reviewed by Varma, including inter-ionic distances in the crystal structure that are intermediate between those normally associated with metal ions with integer valence. There can also be large Debye-Waller factors associated with large fluctuations in these distances.
After the discussion following Elvis' dissertation, I am thinking that there is not enough rigor on either side of the aisle here.
ReplyDeleteYou state that homogeneous mixed valence results from a superposition. Of course, it makes sense to label both limits in the same state space (a Hilbert space, here). If homogeneous mixed valence (HMV) is a superposition and inhomogeneous MV (IMV) is not, then this means you are choosing a preferred basis for the Hilbert space. Otherwise, one could just choose the HMV eigenstates as a basis, and the IMV states would then be in a superposition.
Let's say we have a two state system, with states of integer charge |1> and |2>. What you are saying is that the ground state of an IMV is either |1> or |2> but the ground state of an HMV is |1+2> or |1-2>. There is a factor of 1/sqrt(2) in the latter case, for normalization.
We would then conclude that if |1> and |2> are orthogonal (presumed so, since they are distinguishable). Then so are |1+2> and |1-2>, but each of the HMV states is non-orthogonal to both IMV states. This implies that if the charge distribution is all that is being measured, then it the IMV and HMV cases are only distinguishable if a _complete_ set of _consistent_ measurements are performed (in the sense of a partition of the identity i.e. a POVM).
It is interesting, in this context, to point out that your criteria for distinguishing IMV's and HMV's all involve measurements of the magnetism. In contrast, the most memorable example that Jeff brought after the dissertation was the UV-Vis absorbance. The UV-Vis absorbance intensity is a measure of the coupling through the electric field, not the magnetic field.
Doesn't this seem at all strange to anyone?
Whoops. I guess what I meant to say was that you need more than a complete set, because projective measurements wouldn't distinguish the nonorthogonal states.
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