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Atomic Structure
The mystery of ...
... the quantum atom
The Quantum Atom

Rutherford's picture of an atom, while a good one and a big step forward in our understanding of atomic structure, had this one - fatal - flaw; electrons could not "orbit" the atomic center, slowly and constantly give up some of their energy, and then spiral down into the center of the atom. A new idea was needed.

That's when Neils Bohr, a Danish physicist started thinking, and by 1913 he had what he thought was the answer.

Bohr realized that if Plank (and Einstein) were right, and energy came in "lumps" or quanta, this changed the internal picture of the atom quite a bit.

In the Rutherford model of an atom, with electrons in orbits, the smooth loss of kinetic energy by the electrons (in the form of radiation) resulted in their decaying spiral to destruction. But if Plank was right, energy could not be lost "smoothly" or "steadily" like water draining out of a leaky tub, it had to be gained or lost in "chunks" or whole quanta.

Calculations showed that the amount of kinetic energy in an electron is so small that it is hardly larger than that found in a photon of light. So, any time an electron lost (or gained) a whole quantum of energy it would be a very big deal, with serious consequences, not just a slow spiral to lower orbits. What would happen?

Ground State (and above)

The way Bohr saw it, electrons would respond to the gaining or losing a whole quanta of energy by dramatically changing their position relative to the atomic center. Bohr suggested that an electron with absolutely the lowest amount of energy it could contain would be positioned in the ground state . Although it was not completely accurate, he thought of this as being a fixed orbit quite close to the atomic center.


Such an electron could not "spiral down" to destruction as predicted by the classical model because it could not lose energy in a steady stream. It was trapped by the fact that it had to lose a whole quantum of energy all at one time, and that was not easy, so it orbited continuously in a track around the center waiting for something to happen.

Adding a photon

What could change this picture was the arrival of a whole quantum of energy from some other source, and the best source was a photon of light.

If a photon of light made a direct hit on an electron, the quantum of energy the light was carrying would be transferred to the electron and the photon would cease to exist, it would vanish.

The newly energized electron would now have too much energy to remain in the ground state orbit, and would suddenly and spectacularly jump up to an orbit that was further away from the atomic center. These more distant orbits were called the excited states , and could only be occupied by more energetic electrons.

Bohr then gave considerable thought to the correct position of these "orbits". He assumed that they would have to be circular (like orbiting planets), and the position of the orbit would have to account for the fact that electrons have mass and are moving (so they would have angular momentum), and the energy they were carrying was "quantized" into chunks.

He put all these considerations into a formula:

n
=
principal quantum number

This formula gives us the first important piece of information about the electron and what it is doing. The symbol p is the angular momentum - the basic position of the electron - h is the Plank constant (all to do with quanta) and n is a symbol often called the principal quantum number , and can only have whole number values such as 1, 2, 3, 4 etc.

Under normal circumstances any electron in any atom must have the lowest possible quantum number. So in the hydrogen atom, for example, there is only one electron and that electron must normally have a quantum number of "1", and so be found in the lowest possible orbit, closest to the atomic center and in the "ground state".

If this electron were hit by a photon of light, then the quantum of energy the light was carrying would be transferred to the electron, it would double in energy content and move to a higher orbit, one of the "excited states". It would also now have a quantum number of "2" instead of its original quantum number of 1.

The quantum number, therefore, gives us the first definite piece of information about the location and properties of an electron. The "quantization" of our picture of the atom had begun.

Three more
Quantum Numbers


Bohr had pictured the electron orbits around the atomic center as being perfectly circular, but this was too simple. There are very few perfect circles in nature, and orbits in atoms are no exception.

Later, in 1916, the German physicist Arnold Sommerfeld refined Bohr's "easy" picture with one a bit more complex. In this modified view the electron obits were not circular, but elliptical. But there are many kinds of ellipses possible (certainly more than one), and this changed the calculations in subtle ways, as each ellipse has a slightly different angular momentum. To take account of the possibility of elliptical orbits, Sommerfeld introduced another number; the orbital quantum number (sometimes called the "angular momentum quantum number"), which usually has the symbol "L.

[Note: in most books the lower-case letter "l" ("el") is used as the symbol for the orbital quantum number, but since each user can change computer fonts to a wide variety of styles, a lower-case "l" is often confused with the numeral "1" ("one") on typical web-pages. So, for clarity in these discussions the upper-case letter "L" will be used instead as the symbol for the orbital quantum number.]

L
=
orbital quantum number

Like the principal quantum number, the orbital quantum number can have values of 0, 1, 2, 3, 4, etc., but only up to a whole number value of one less that the electrons principal quantum number (i.e. up to a value of n - 1).

So, if an electron had a principal quantum number of 2
(i.e. n = 2), then L can only be "0" or "1".

If n = 3, then L can equal 0, 1, or 2, but not 3.

m
=
magnetic quantum number


There are two more quantum numbers associated with each electron; the magnetic quantum number written as m, and the spin quantum number, written as s.

Our earth and all the planets orbit the sun in various elliptical paths of different sizes, but all in the same plane. Basically they travel in two dimensions and can be drawn as if they were all on the same flat piece of paper. It is not that easy with electron elliptical orbits. Some of their orbits could be at different angles, and thus move in three dimensions - something that is very hard to represent on a two dimensional piece of paper. (Technically this property is called the "angular momentum vector").

To make it easy to picture what is going on, the magnetic quantum number can be thought of as defining the amount of "tilt" there is to the orbit.

The possible values for m follow the same rules as for L, except that negative numbers are now allowed (the "tilt" of the orbit can be either "up" or "down"). So, for n = 2, the possible values for m would be 0, 1, or -1.

s
=
spin quantum number


Our earth spins on its axis. Electrons also "spin" (or at least that is how it is envisioned!), and can spin either to the right (clockwise) or to the left (anticlockwise). The spin quantum number is used to define which way the electron is spinning.

There are only two possible values for s for any value of n. These values are usually written as +1/2 and -1/2, meaning either a clockwise spin or an anticlockwise spin.

But what do these numbers tell us about the electrons?

Exclusion Principle

Austrian physicist Wolfgang Pauli worked out the significance of these numbers in 1925. He suggested that no two electrons in any given atom could have exactly the same values for all four quantum numbers.

This became known as the Pauli exclusion principle
- "No two electrons in any atom may have the same sets of quantum numbers".

Thus, for any orbit where the principal quantum number was the value of "1" (n = 1) there could only be a maximum of 2 electrons.

It works like this. If n = 1, then L = 0, and m = 0 (neither number can be greater than n), but the electron could have either of the two possible spin quantum number values, +1/2 or -1/2.

This means that in every atom of every element, there can only be a maximum of two electrons in the lowest orbit, closest to the atomic center at the basic, minimum ground state!

With higher principal quantum numbers it gets a bit more complicated, and is best illustrated using a table, thus:

Number of possible orbits
n L m
2 0 0 1st
2 1 0 2nd
2 1 1 3rd
2 1 -1 4th

This means that there are a total of four possible orbits when the principal quantum number is a value of 2. Each one of these four possible orbits can have electrons of opposite spin quantum numbers (+1/2 and -1/2) there are two such electrons in each case, so there is a grand total of eight (8) electrons possible altogether in this zone or region of the atom.

Repeating the logic and the calculations shows that when n = 3, the atom would have orbits enough to accommodate eighteen (18) electrons, and so on up and up to higher and higher orbits and number of possible electrons.

Shells
and Sub-Shells

Keeping track of where the orbits and electrons are within any atom can quickly get complicated. Although the use of the quantum numbers, and the Pauli exclusion principle, defines each electron unambiguously, it is not easy to keep writing out n = 1, L = 0, m = 0, s = +1/2, etc.

For convenience (and for historical reasons) therefore the electrons found in all the various orbits of each of the principal quantum number locations (n = 1, 2, 3, etc.) are said to be in the same "shell".

A shell is a group of electrons in a given atom that all have the same principal quantum number.

When the principal quantum number is greater than "1" (i.e. n = 2, or 3, or 4, etc.) the orbital quantum number (L) can be 0, 1, 2, ... n-1. This forms a series of sub-groups within the larger "shell", and these sub-groups are called sub-shells.

A sub-shell is a group of electrons in a given atom that all have the same principal quantum number and the same orbital quantum number (L).

All the electrons in a particular sub-shell are called by a lowercase letter, thus;

Value of L letter
0 s
1 p
2 d
3 f

Within each orbit defined by the principal, orbital and magnetic quantum numbers the electrons can only vary in one more property; the spin quantum number (do they spin to the "right" or to the "left"). The Pauli exclusion principle says that all four quantum numbers must be different - thus, only two electrons can occupy any one of these positions in space.

When physicists began to find out that electrons do not really behave like little planets circulating around the sun (the way Neils Bohr pictured them), they stopped using the term "orbit" (which implied a circular, or elliptical path around the atomic center), and started using the term "orbital" in its place.

An orbital is a group of up to 2 electrons that have the same principal, orbital and magnetic quantum numbers, but different spin quantum numbers.

Thus;

Value of n Value of L number of
electrons
name of
sub-shell
1 0 2 1s
2 0 2 2s
2 1 6 2p

Using this system it is possible to write out the position and "location" of all the electrons in a particular atom using just a few numbers and letters - like this;


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© 2003, Professor John Blamire