# [Pw_forum] the unit of DOS

Giovanni Cantele giovanni.cantele at spin.cnr.it
Fri May 6 15:19:31 CEST 2011

On May 6, 2011, at 2:12 PM, Iurii TIMROV wrote:

> I would also like to clarify for myself the units of DOS. By definition,
>
> DOS(E) = \sum_n \int delta(E - E_n(k_x,k_y,k_z)) dk_x dk_y dk_z / (4 \pi^3),
>
> where \delta is the Dirac delta function, k_x,k_y,k_z are the 3 components
> of the wave vector, n is the band index, E is the energy (Ref. Ashcroft
> and Mermin). According to this equation the unit of DOS is
> 1/(Energy*length^3).
>
> Is the following statement correct?:
> "The Density of States of a system is the number of states per interval of
> energy -in the unit cell volume-".
>
> Could somebody comment on this? Is there a mistake in the above thinking?

I would say that the correct statement is (of course it is just  a matter of definition!):

The Density of States per unit volume of a system is the number of states per interval of
energy -in the unit cell volume-

There is just a volume factor difference between the two definitions.

Usually:
DOS(E) dE = number of energy levels in the energy range from E and E+dE

and according to this definition
\int_E0^E1 DOS(E) dE = total number of states between E0 and E1 (adimensional).

This is what the dos.x executable included in Quantum-ESPRESSO computes.

According to the above definition:

DOS(E) = \sum_n \int delta(E - E_n(k_x,k_y,k_z)) dk_x dk_y dk_z *V / (4 \pi^3)

If you carefully read the chapter 8 of Ashcroft-Mermin, it says:
"....one can define a density of levels per unit volume (or "density of levels" for short)....."
and Eq. (8.57) (provided we're looking to the same edition!) is exactly the definition you gave
(so, "per-unit-of-volume" definition).

Giovanni

--

Giovanni Cantele, PhD
CNR-SPIN and Dipartimento di Scienze Fisiche
Universita' di Napoli "Federico II"
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