IONIZATION ENERGY and WORK FUNCTION

IONIZATION ENERGY and WORK FUNCTION 

This is a revised material of a ionization energy variation presented in Atomic structure book. 

Background and actual interpretation

The energy required to remove one electron from an isolated, gas-phase atom, when this atom is not hooked up to others like in a solid or a liquid, is called ionization energy (IE).
M(g) ----> M+(g) + e-
Generally speaking, ionization energies decrease down a group of the periodic table, and increase left-to-right across a period. Ionization energy exhibits a strong negative correlation with atomic radius. There is a dependence of ionization energy from nuclear charge, number of energy levels, and shielding.
As the nuclear charge increases, the attraction between the nucleus and the electrons increases and it requires more energy to remove the outermost electron and that means there is higher ionization energy. Along periodic table it can be observed an increase of ionization energy with increasing nuclear charge.
In the same column of periodic table, the effect of increased nuclear charge is balanced by the effect of increased shielding, and the number of energy levels becomes the predominant factor. With more energy levels, the outermost electrons (the valence electrons) are further from the nucleus and are not so strongly attracted to the nucleus. Thus the ionization energy of the elements decreases as you go down the periodic table because it is easier to remove the electrons.
Ionization energies differ significantly, depending on the shell from which the electron is taken. For instance, it takes less energy to remove a p electron than an s electron, even less energy to extract a d electron, and the least energy to extract an f electron. It is considered that s electrons are held closer to the nucleus, while f electrons are far from the nucleus and less tightly held.
The periodic nature of ionization energy for the last electron of first 20 elements is presented in fig. 1. With each new period the ionization energy starts with a low value. Within each period there is an increasing energy value with some saw teeth. The variation inside a period corresponds to the sublevels in the energy levels.

Figure 1. Ionization energy variation
For H which has only a single electron moving around nucleus there will be a single value for the ionization potential.
For other elements, the removal of each subsequent electron requires even more energy, so a distinct and increasing value of ionization potential is measured for closer electrons of nucleus; it becomes more difficult to remove additional electrons because they are closer to the nucleus and thus held more strongly by net positive charge of nucleus.
More generally, the nth ionization energy of an atom is the energy required to strip it of an nth electron after the first n - 1 have already been removed.
In order to explain the ionization energy, quantum theory does not have a ,,special formula” so, mathematically, the old Bohr treatment is accepted.

Besides ionization energy a new physical unit was necessary to be accepted – work function - based, mainly, on photoelectric experiments. Work function is the amount of energy needed to remove an electron from a bulk material (solid or liquid).
In some scientific texts, the ionization potential and work function of any metal is considered the same, but their values are different for semiconductors or insulators. In fact, quantum mechanics define work function as the energy required to remove an electron from Fermi level to vacuum. The work function is a characteristic property for any solid face of a substance with a conduction band (empty or partially filled). For a metal, the Fermi level is inside the conduction band. For an insulator, the Fermi level lies within the band gap, indicating an empty conduction band; in this case, the minimum energy to remove an electron is about the sum of half the band gap, and the work function.
Other texts present an empirically known correlation between the atomic ionization potential (IP) and the metal work function (WF) :

                                                IP/WF ≈2
Experimental facts show that work function depends on the orientation of the crystal and on crystallization type. For example Ag:4.26, Ag(110):4.64, Ag(111):4.74.

Why the actual explanation is erroneous…

In actual theory of atomic structure, ionization potential plays a secondary importance. The variation of ionization potential of last outer electron is used only as support for chemical periodicity. The variation of ionization potentials of different electrons from the same element or the variation of ionization potentials of the same inner electron from different elements does not present any significance in actual quantum mechanic.
In fact quantum mechanic is able to solve the Schrödinger equation only for hydrogen or hydrogenous atoms types. Therefore it is an absurd idea, its pretention to explain the variation of ionization potentials in periodic system. Firstly, in case of simple hydrogen atom, quantum mechanic must explain why for different atoms the ionization potential is the same. If electron does not follow a certain trajectory around nucleus, and its description is like a cloud, the ionization potential for different atoms must have a statistical distribution.
It is important to emphasize that ionization potential must play a fundamental role in the atomic structure. This because electrons are arranged in shells and in every shell again a difference in ionization energy is observed. In quantum mechanic the difference of ionization energy for the electrons on the same shell is given by electron spin energy and eventually interaction between electrons.
In proposed theory, different values of ionization potential are given due to the different orbits of electrons movements around nucleus.

In our calculus a database with ionization potentials found at following address .... was used.
Without making any supposition about arrangement of electron around nucleus let’s analyze the ionization potential for isoelectronic series. By isoelectronic series we mean the same number of electron but an increasing number of protons respectively neutrons in nucleus. Due to the limited space for display in tab 1 are presented ionization potentials for first 15 elements, but the facts presented for these elements are valid for all elements in periodic system.

Analyzing the ionization potential of first isoelectronic series (one electron around nucleus) we observe a quadratic dependency related to the atomic number Z. The quadratic dependence is easy to be observed for first isoelectronic series but for other series is hidden by a constant factor addition in the energy expression. In order to arrive to a linear dependency we will work with square root of ionization potential and we will make also some simple mathematical tricks.

We define relative ionization potential of kth electron of an element as kth ionization potential divided by ionization potential of hydrogen electron. For example in case of hydrogen the relative potential is 1, and for helium we have two relative ionization potentials; 1,8 for one electron and 3.99 for the second electron. For other elements the modality of relative ionization potential calculus are the same. In tab 2, are presented square root of relative ionization potentials for the first 15 elements.
With this simple modification, the distribution of square root of relative ionization potential for first electron (first isoelectronic series) in different atoms is linearly related to the atomic number Z and this is observed from tab. 2, even without a graphical representation.

The variation of square root of relative ionization potentials for first 36 isoelectronic series related to the atomic numbers Z is presented in fig 1.17 and 1.18; fig. 1.17 is a detailed part of 1.18 and is presented for a better visualization of ionization potential variations.

The same linear dependence is observed also for higher isoelectronic series, but a picture with such amount of information doesn’t give any supplementary information. In pictures the isoelectronic series are positioned from left to right. For first two isoelectronic series, two parallel lines with the same slope are obtained when the number atomic is increased from helium to lead. The second line representing the energy of second electron in different atoms is a little bit shifted related to the first isoelectronic series due to a factor which represent a new appearing interaction. We can observe also a coupling between energy of first electron and the energy of second electron when we change to different elements (different atomic numbers); the slope of energy variation for first two isoelectronic series is constant from Helium to last element (the checking was made up Z= 90).
As it is observed the linear dependency is respected for every electron from these isoelectronic series and also for higher isoelectronic series.

From the graphical representation of ionization energy we can observe that there is a coupling of electrons in pairs of minimum 2 electrons with the same slope of energy variation and for higher Z, a coupling in more pairs of two electrons having the same slope for energy variation. For example after first pair of electrons, a number of four pairs (eight electrons orbit) present the same slope in energy variation. These distributions of ionization potentials contradict quantum mechanic theory and also wave–corpuscular hypothesis. It is impossible for an electron having a complicated movement, given as a probability, to present a linear dependency of ionization potentials. 
As consequences we can suppose, for the moment, that ,,adding” one or more electrons to a hydrogenoid atom will have as consequences modification of energy interaction with a simple additional term; the variation is related to the atomic number.

In the same time for the Moseley low new explanation can be formulated. The jump of electron from a superior level to another inferior level, with both energy levels linear dependent on atomic number will produce a photon with an energy proportional with this difference. In conclusion the ecranation factor due to other electrons from atoms, actual accepted by quantum mechanic is wrong and the formulation of Moseley low must be corrected.
Therefore, the proposed theory, presents a simple and easy to follow variation of potential ionization in periodic system which suppose a precise trajectory of electron around nucleus. Quantum theory explanation based on density of electron probability around nucleus and wave-corpuscle duality are ruled out.
The linear dependency of relative ionization potential energy is accurate also for so called d layer and f layer in actual quantum mechanic. The only difference observed in distribution of relative ionization potential for different layer is represented by different slope.
A detailed discussion of these aspects will be made at electron arrangements on multielectron atoms.

Quantum mechanic collapses completely when, further, the concept of work function is analyzed. 
The work function value for metals must be equal with ionization energy, because in this case, based on actual model of metallic bond, electrons are free to move inside crystal but find a confining potential step U at the boundary of the metal.
In a comparative way, for some elements, in tab. 1 the values for their first ionization values and work functions are presented.

Table 1. Work functions and ionization potential values

 As is observed, as a general rule, the ionization energies for all metals are greater then work function. Sodium, a alkaline metal has a work function of 2,36 eV and a ionization potential value of 5,13 eV and on the other side Aluminium with a ionization potential of 5,98 eV has a work function of 4,20 eV. For mercury the work function is 4,47 eV and the ionization potential is 10,43 eV. It is very strange how the actual theoreticians didn’t observe these differences and didn’t make a quantum interpretation of this difference for metallic structure. 

But this is not the nightmare of quantum mechanics. It is well known that metallic oxides present a lower work function then metals and even lower then alkaline metals. For example, a tungsten cathode has a work function equal with 4,54eV and if the tungsten is covered by thorium oxide the work function is 2,5 eV.
How can explain quantum mechanic this modification?
There is no explanation and in fact quantum mechanic predict that tungsten covered by thorium oxide must have work function greater then pure tungsten. In metal electrons are free to move, but in metal oxide, according to actual quantum mechanic electrons are part of ionic or polar covalent bounds and they are not free. Therefore it is not necessary a deep math treatment to observe the inconsistency of quantum mechanics.

Further improvements are foreseen to the present text.