## Abstract

This work proposes new assumptions in place of some basic erroneous assumptions of *free electron theory* to determine *n*, the number of highest energy electrons, and Fermi energies are recalculated accordingly. The bulk modulus of metals are calculated using both the number of highest energy electrons in atoms proposed in this work and original *free electron theory* assumptions. When compared to measured values there is a much better agreement with values determined by this work than those using the original *free electron theory* assumptions. There is fairly good agreement between Fermi energies calculated by the standard theoretical equation and estimates using an approximation proposed by this work.

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There are serious problems [13] with the assumption that each atom contributes one electron (or more) for conduction. For example, it is assumed that in a cm^{3} of copper there are 8.49 × 10^{22} conduction electrons [5,15,16] and with each conduction, the electron travels at a velocity of about 1.6 × 10^{6} m s^{–1}. Hence, the total amount of kinetic energy generated by the conduction electrons is approximately 1 × 10^{7} J. The electrons are slowed down by collisions and the mean time between collisions is of the order of 10^{–14} s^{4}, which means there are 10^{36} collisions in a cm^{3} of copper. The calculated electron drift velocity [25] in a copper wire carrying a current of 10 Amps is less than 10^{–3} m s^{–1}. Most of the 10^{7} J of kinetic energy resulting from the collisions have to be dissipated as heat and is enough to cause the wire to melt quickly.

Secondly, the calculated values of *R*_{s}, the radius of the sphere occupied by a conduction electron, for some metals are larger than the size of the metal atoms which cannot be correct. Take the examples of rubidium and caesium, *R*_{s} are 275 and 298 pm, respectively, and *n* for the two metals [4,5,15] are 1.15 × 10^{28} and 0.91 × 10^{28} m^{–3}, respectively. Therefore, in a cubic metre of space, the total calculated volume of conduction electrons equals 4/3*π* (2.75 × 10^{–10})^{3} × 1.15 × 10^{28} m^{3}and 4/3*π*(2.98 × 10^{-–10})^{3} × 0.91 × 10^{28} m^{3}, which are 1.002 and 1.009 m^{3}, respectively, and physically impossible. Even if the total volume of the conduction electrons occupies slightly less than one cubic metre, this still violates the rules of closest packing of spheres. More details are available in the original work [14].

Except for the metals in Groups 1 and 2, most metals have more than one valency. The number of conduction electrons per atom and the Fermi energy have been calculated for a sample of different metals [5,15,16]. Some of the particular metals and their specified valencies, including the following elements with their valencies used for the original calculations, are written in brackets: Nb (1), Mn (2), Fe (2), Zn (2), Al (3), Ga (3), Sn (4), Pb (4), Bi (5) and Sb (5). Without exception, all the above-mentioned metals have more than one valency. There is no theoretical justification to use these respective valencies over other valid valencies.

For example, Advanced Inorganic Chemistry [16] reported that niobium exhibits valencies from 1 to 5; manganese with the greatest range of valencies from 1 to 7; iron has valencies from 1 to 6. Chemistry of the Elements [18] describes similar compounds and valencies for the above-listed metals.

Bonding was not very well understood until the 1930s when *The Nature of the Chemical Bond* was published. It was one of the first publications which described bonding in detail [6]. When the *free electron theory* was formulated, less was understood of valencies and bonding. The majority of metals have more than one valency and there is no experimental or theoretical proof which valency is the correct valency to use to determine the value of *n* for calculating the correct Fermi energy and valency should not be a factor for producing the value of the Fermi energy. If valency is a variable in determining *n* (the number of conduction electrons or the Fermi energy), using valency can produce a different set of values of *n* and very different values of Fermi energies which can be many times bigger or smaller than the correct value.

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Lang, P.F. Calculation of the Fermi energy and bulk modulus of metals.
*Bull Mater Sci* **45**, 112 (2022). https://doi.org/10.1007/s12034-022-02692-7

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DOI: https://doi.org/10.1007/s12034-022-02692-7