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Solid State Question: A Confusing Integral in Ashcroft

  1. Sep 25, 2015 #1
    I am trying to undersand Ashcroft's derivation of energy density on page 38. I am following most of the arguments save EQ 2.30. Where did the vector go? I did follow my professors argument for converting to polar/spherical when the argument of a function is a modulus of a vector — exploiting symmetry to make a nice easy integral. However, it is a whole other matter two have an actual vector in the integrand. I would expect it to be dotted into something in order to have it equal a scalar (other than zero). I am scratching my head here. The steps leading up to this are a little long, so I am just leaving it as a reference and hoping you guys have access to this textbook.

  2. jcsd
  3. Sep 25, 2015 #2
    I think it's not a vector in the integrand.
    $$d\mathbf{k} = dk_x \, dk_y\,dk_z $$
    This may be confusing since the total derivative of ##\mathbf{k}## would also be denoted ##d\mathbf{k}##, but I think represents something different. However I believe in this case it just means an integral over k space. So just a somewhat confusing notation.
  4. Sep 25, 2015 #3
    Perfect, thank you.

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