Chemical potential and Fermi level

In summary, Chemical potential is a measure of the energy required to add one particle to a system while keeping the temperature, volume, and number of particles constant. It is closely related to the Fermi level, which describes the energy level at which the probability of finding an electron is 50%. The chemical potential is affected by factors such as temperature, pressure, and the number of particles present, as well as the properties of the material. In most cases, the chemical potential increases with increasing temperature, and these concepts are essential in understanding the behavior of electrons in materials and their applications in materials science.
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My question is aimed to know where fermi level depends on Density of states or not
Does fermi level (in metals) depend on the density of states? I am asking this because from fermi-dirac distribution it seems like that fermi level is non-dependent of DOS, but there is chemical potential in fermi-dirac distribution, which is said to be dependent of DOS.
 
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Say there are 2N electrons in the system, I think N-th number of states counting from the ground state would become Fermi energy level. In that sense not density but integrated total number of states and energy levels structure would matter.
 
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