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The mobile electrons move very easily in the metal lattice, which proves that they are very weakly bound to the positively charged ions at the lattice. But why is that?

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    $\begingroup$ The full answer requires some quantum physics. The quick answer is that the effect of the positive nuclei is reduced/shielded by the inner electrons, and it becomes energetically more favorable for the outer conduction electrons to be "shared" by the nuclei, thus becoming mobile. $\endgroup$
    – G. Paily
    Jun 23, 2023 at 9:40
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    $\begingroup$ useful link hyperphysics.phy-astr.gsu.edu/hbase/Solids/band.html $\endgroup$
    – anna v
    Jun 23, 2023 at 10:20

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Electrons reside in different energy levels around the atomic nucleus. These different energy levels with continuous energy variation form what are called "Energy bands"[imagine them as bands around the atomic nucleus, each of which has fixed energy level]. The energy band which includes the energy level of the valence(outermost) electrons is called "Valence band". The energy band above the valence band is called "Conduction band". These are some terms you need to be familiar with to understand this.

Now coming to your question, with no external energy, all the valence electrons will reside in the valence band. But if the lowest level in the conduction band happens to be lower than the highest level of valence band, the electrons from valence band can easily move into the condcution band. Normally the conduction band is empty. But when it overlaps on the valence band, electrons can move freely. This is exactly the case with metallic conductors.

In short, only electrons in the conduction band can move freely, and in metal, the conduction band overlaps with the valence band, allowing electrons to move freely. All these electrons are present around the metal kernel.

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