The Lloyd formula and the muffin-tin CPA local density of states

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Abstract
The Lloyd formula (1967) for the density of states (DOS) of a finite cluster on nonoverlapping potentials in free space is derived in a direct way by local integration of the Green function of the system in the r representation. Similarly a new expression is derived from the DOS per atom in a disordered binary substitutional alloy, which is valid within the muffin-tin single-site coherent potential approximation. It is shown that this expression is equivalent to the one that is usually derived from the Lloyd formula, but that its numerical evaluation is simpler. Alternative expressions for the component DOS are also discussed.