# [Pw_forum] the unit of DOS

Iurii TIMROV iurii.timrov at polytechnique.edu
Fri May 6 15:39:07 CEST 2011

Thank you, Giovanni!

Best regards,
Iurii

Iurii TIMROV
Doctorant (PhD student)
Laboratoire des Solides Irradies
Ecole Polytechnique
F-91128 Palaiseau
+33 1 69 33 45 08
timrov at theory.polytechnique.fr

>
> 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,
>> the DOS reads:
>>
>> 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
>
> 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
>
> --
>
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>
> Giovanni Cantele, PhD
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