Effective Atomic Number Rule in Coordination Chemistry

The total electron number surrounding the nucleus of a metal atom in a metal complex is defined by the Effective Atomic Number rule (EAN) or simply the Effective Atomic Number. This number is composed of the metal electrons and the bonding electrons of the atom from the surrounding electron-donating molecules and atoms.

EAN for an Atom

The effective atomic number theory Zeff (also called the effective nuclear charge, sometimes) of an atom is the proton number that an electron in the element 'sees' effectively because of screening by inner-shell electrons. It is an electrostatic interaction measure between the positively charged protons and negatively charged electrons in the atom. One may view the electrons in an atom as being 'stacked' by the energy, which is outside the nucleus; the lowest energy electrons (like the 1s electrons and 2s electrons) occupy the space, which is closest to the nucleus, and electrons of higher energy are further located from the nucleus.

The electron's binding energy, or the energy required to remove the electron from the atom, is given as the electrostatic interaction function between the positively charged nucleus and negatively charged electrons. For instance, in iron, the atomic number is 26, the nucleus contains 26 protons. The electrons, which are closest to the nucleus, will nearly 'see' all of them. However, the electrons that are further away can be screened from the nucleus by other electrons in between and also feel less electrostatic interaction resultantly.

The 1s electron of the iron (which is the closest one to the nucleus) sees an effective atomic number theory (which is the number of protons) of 25. The reason behind why it is not 26 is due to some of the electrons present in the atom end up repelling the others by giving a nucleus' net lower electrostatic interaction. A way of envisioning this particular effect is to imagine the 1s electron sitting on one side of the nucleus's 26 protons, with the other electron sitting on the other side; every electron will feel less than the attractive force of 26 protons due to the other electron contributes a repelling force. In iron, the 4s electrons that are furthest from the nucleus feel an effective atomic number of only 5.43 due to the 25 electrons in between it, including the nucleus screening the charge.

Effective atomic numbers are more useful not only in understanding why the electrons further from the nucleus are much more weakly bound to that of closer to the nucleus but also due to the reason they may tell us when to use the simplified methods of calculating other interactions and properties. For instance, lithium with the atomic number 3 contains two electrons in the 1s shell and one in the 2s shell. Since the two 1s electrons screen the protons to provide an effective atomic number for the 2s electron that is close to 1, we may treat this 2s valence electron with a hydrogenic model.

In a mathematical way, the effective atomic number theory Zeff may be calculated using the methods called "self-consistent field" calculations, whereas, in the simplified situations, it is just taken as the atomic number subtracting with the number of electrons between the electron being considered and the nucleus.

For a Mixture or Compound

The alternative definition of the effective atomic number is entirely different from that, which is given above. The material's atomic number exhibits a fundamental and strong relationship with the nature of radiation interactions within that respective medium. There exist many mathematical descriptions of various interaction processes, which are dependent on the atomic number, Z.

When dealing with the composite media (it means a bulk material that is composed of more than one element), one, thus, encounters the difficulty of defining the value Z. In this context, an effective atomic number is equivalent to the atomic number, but it is used for compounds (for example, water) and the mixtures of various materials (such as bone and tissue).

This is of the most interest concerning radiation interaction with the composite materials. For the bulk interaction properties, it may be useful to describe an effective atomic number for a composite medium, and, based on the context, this can be done in various methods. Such methods are

1. A simple mass-weighted average,

2. A power-law type method with a few (very approximate) relationships to the radiation interaction properties or

3. The methods involve calculation depending on the interaction cross-sections. The latter is given as the most accurate approach (it means, Taylor 2012), and often, the other more simplified approaches are inaccurate even when used within a relative fashion for comparing the materials.

In several scientific publications and textbooks, the simplistic and often dubious given below - sort of method is employed. One of the proposed formula for the effective atomic number, Zeff, is given below (Murty 1965):

Z\[_{eff}\] = \[\sqrt[2.94]{f_{1} \times (Z_{1})^{2.94} + f_{2} \times (Z_{2})^{2.94} + f_{3} \times (Z_{3})^{2.94} + ...}\]

where,

f\[_{n}\] = fraction of the total electron number associated with every element, 

Z\[_{n}\] = atomic number of every element.