2006-7 Module 113 - Lecture 4

Module 113- Quantum theory and atomic spectroscopy LECTURE 4 – Atomic structure 9. Electronic Configurations and the Pauli Principle The electronic configuration of an atom is the complete set of n, l and ml quantum numbers that the electrons possess. Of particular importance is the ground state- the lowest energy configuration. Can we predict what electronic configuration will have the lowest energy? Our current understanding of the so-called ‘building up’ or AUFBAU principle began with Niels Bohr and his identification of the quantised energy levels of atoms. But when Bohr extended this to other elements, he found that the ordering of the energy levels was not trivial, despite his apparent “prediction” concerning the nature of the then unknown element hafnium. The Aufbau principle came about more from experimental evidence and by 1925 J.D. MAIN-SMITH and Edmund Clifton STONER had unravelled the main points and in its modern form is as follows 1) In the ground state, the electrons take those quantum numbers that have the lowest energy first. 2) For a given value of n, there are n different values of the orbital quantum number l . The energy separation for different n with the same l is much greater than for different l with the same n. 3) For a given value of l , the higher the principal quantum number, the higher the energy of that electron. 4) For a given n, the energy of the electron increases for higher values of l e.g. 3p > 3s in energy 5) There is a maximum number of electrons with a given n and l , determined by the total number of ml sublevels, which is given by 2l + 1. 6) Only a maximum of two electrons can share the same n, l and ml quantum numbers. The aufbau principal in the above form is correct, but it can still leave to uncertainties in determining the ground state configurations because very often the high l states of a lower n quantum number can have similar energies to the low l levels of the next n in the sequence. A well known example of this are the 4s and 3d electrons, which is why potassium with a single 4s electron comes before scandium which has a single 3d electron in the Periodic Table. Which numbers are lower is reflected in the periodic table, which can be used to determine the sequence of electron energies.

A great webpage to read about electron configurations is http://chem.stthomas.edu/pages/genchem/data/ch05/ch05_12.htm where you can also practice working out the electronic configurations of elements. Returning to the question at the beginning of the lecture: we now understand why there are six p-electrons: there are three p-sublevels for each principal quantum number n. Ok, but why are there two electrons in each sublevel? Well, that is because there is a missing quantum number!

10. Electron Spin Silver atoms were, on reflection, a very unfortunate choice for a demonstration of space quantisation in orbital angular momentum. Today, we know that silver has an odd number of electrons. It also has fully closed shells except for a single 5s level, by which we mean that all the n, l and ml states with n < 5 have two electrons in each ml level. Now, if a subshell is fully closed i.e. all the available ml levels have two electrons, then the net angular momentum of all those electrons is zero. This is a great help in finding the total angular momentum of an atom because we can forget all about the closed subshells and just concentrate on those l levels that have less than their maximum 2l + 1 electrons. In plain English, we just need to concentrate on the one row in the periodic table that contains the atom we’re interested in. Therefore, in silver, the 5s electron is the culprit for the observed space quantisation. But hang on, a 5s electron has no orbital angular momentum! There should have been no splitting of the atomic silver beam in the magnetic field! Why were there two spots? This was because that single electron possessed its own intrinsic angular momentum called SPIN. For an electron, we represent spin by the quantum number s, which is a bit confusing since s is the label for l = 0, but there you go, it wasn’t up to me. What is the value of this electron spin i.e. just what is the value of s? Well, if it is truly an angular momentum, argued its “discoverers” George UHLENBECK and Samuel GOUDSMIT in 1925, then it must have 2s + 1 sublevels, and since there are just two electrons in an orbital, the value of s must be ½. Many scientists were absolutely horrified by this idea. They had grown accustomed to quantum theory being associated with integer values; in fact, even today many people will erroneously say that quantum mechanics is all about integers. Electron spin is associated with the electron magnetic moment, meaning that electrons will attempt to align themselves in a magnetic field in order to minimise their energy.

To summarise: The quantum number s represents the electron spin. It takes a value ½. For each electron, there are two possible spin states, given by the ms quantum number: ms = ½ and ms = -½. The spin magnetic quantum number ms is the missing fourth number from Pauli’s exclusion principle. If an (n, l , ml ) sublevel contain two electrons, they assume the values ms = +½ and ms = –½. The electrons are said to be paired. They cannot both take the same ms quantum number because all four quantum numbers for two electrons in the same atom must not be identical. This final result stems from the work of one of the true scientific geniuses of the last one hundred years, Wolfgang PAULI, who proposed his famous exclusion principle, which states: “any electron in an atom can be described by four quantum numbers, but no two electrons can have the same 4 numbers” So to our first three numbers n, l and ml Pauli has added a fourth: but he had no idea what it was at the time because he proposed his principle before the discovery of electron spin. All he knew was that it only take two values because only two electrons can have the same n, l and ml quantum numbers (rule 6). The discovery of electron spin revealed the final quantum number. The idea of electron spin had been raised a number of times prior to 1925: in 1920, Arthur COMPTON had suggested it but there were a number of arguments against it, such like a spinning electron with this much angular momentum would be rotating at faster than the speed of light! But no one could argue that electron spin seemed to explain the Pauli exclusion principle and in 1927 Ronald FRASER used it to explain the Stern-Gerlach measurements in silver. Pauli grudgingly accepted it and everyone had to abandon any classical description of spin and accept that this intrinsic angular momentum of electrons was a purely quantum phenomenon. It was left to Paul DIRAC, in 1928, to demonstrate that by combining quantum physics and relativity theory, you find that the electron had to have an intrinsic angular momentum! An electron with a particular set of n, l ml and ms values is often represented as an arrow within a sublevel (of a subshell) box.

Each box represents an n, l , ml level. The arrows inside represent the ms value of the electron: up represents ms = +½ and down ms = –½. So paired electrons are presented as

Hence, this is the ground state configuration of helium.

By analogy with our introduction of L earlier, we can introduce a total spin S, defined as

n

S = ∑ si

(4.1)

i

The electron spins in the ground state of helium are opposed, so the vector sum of the spins gives S=0

11. Term Symbols We can define the total spin quantum number, S, as the magnitude of S. For light atoms (1st and 2nd row), we can label an electronic configuration by using its multiplicity, which is defined by the number Multiplicity = 2S + 1

(4.2)

The multiplicity of an atom is added as a superscript to the total orbital quantum number. For example, the ground state of helium above is given by 2S+1L

=

2(0)+1L(

= 0) = 1S

Furthermore, we can now state that the normal Zeeman effect that we may observe on a 2P → 1S transition in helium takes place, more accurately, in the transition 21P → 11S. Can you think of a 2P → 1S transition where you wouldn’t see a normal Zeeman effect? Now consider oxygen. It has four 2p electrons to distribute in three sublevels. What is the lowest energy configuration? Well, from experiments it is found that it is

This is a reflection of Hund’s rule which states that there will be a maximum number of unpaired electrons consistent with the requirement that lowest energy n, l ml and ms states are fully occupied first. Hund’s rule only applies to the ground state configuration of an atom and is almost useless in other circumstances so you must always use it with care. Actually, these little boxes are not very realistic and to see why on the next page are the lowest sublevel energies of the helium atom, with the possible electronic transitions marked. Note that only transitions with ∆ l = ±1 are marked because these are the most important in atomic spectroscopy. There are some important things to note from this diagram. Firstly, there seem to be two sets of sublevels: one for S =0 and one for S = 1. How can this be? Well, remember, an subshell is just another way of describing the three quantum numbers n, l and ml of an electron. When two or more electrons are present, however, the energy of the electron will by influenced by its neighbours. Now, we know from Hund’s rules that electrons prefer to be parallel rather than have opposite spins: in practise this means that the relative orbital energies of any electron pair will be reduced if their spins are parallel to one another. Try to get away from the notion that an orbital exists: it is really a short-hand description for a particular electron and remember the little orbital boxes are a simplification.

Secondly, the 1s subshell in helium is much lower in energy than in hydrogen. This is simply a result of the increased nuclear charge. So the energy of 1s electrons will vary from element to element.

Thirdly note that subshells with the same n but different l are no longer degenerate (unlike the case for the hydrogen sublevels). We will explain why at the end of this course.

Spin and selection rules

Electronic transitions in light atoms tend to obey the selection rule ∆S = 0, so you observe two completely different electronic spectra in helium depending one whether you begin with helium in a singlet (S = 0) or triplet (S = 1) electronic state. In fact, for many years parahelium and orthohelium were thought to be two separate atomic forms of helium!

Below is an energy level diagram for the calcium atom,

concentrating on the lowest energy states of the atom that are used in a type of atomic clock. We can picture calcium as a two electron system, since 18 of its electrons are in closed atomic shells. Note that the clock transition involves a singlet ↔ triplet transition which ought to be forbidden. However, calcium is not a light atom so we find that although the transition is very weak it is not strictly forbidden. This is an example of an intercombination line, meaning a transition (line) between two states with different spins. Note how all the transitions obey the ∆ l = ± 1 selection rule: this is a stricter rule than the spin one and tends to be obeyed more often than not. Also notice how, unlike hydrogen, transitions where ∆n = 0 are now allowed, since unlike in hydrogen electron energy levels with same n but different l are no longer degenerate. You will have noticed a subscript attached to each Term Symbol: this number refers to the total angular momentum J and is the sum of the spin and the orbital angular momentum of the electrons. This is a very important property of atoms, with a powerful selection rule to boot, but we won’t deal with it this year! It is also related to the question of the split sodium D-line. This leads us to today’s closing questions • •

But why is the electronic structure of an atom described in terms of 4 quantum numbers? Why are they integers (or half-integers)?

Over the next three lectures I will attempt to answer this, and will introduce a whole new theory of matter. Now this has got to be more exciting than carbohydrates and copper sulphate solutions. Key points in lecture 4 •

• •

•

The subshells of atoms are affected by a magnetic field. This can be seen as a splitting of spectral lines. The splitting of the lines reveals that atomic energy levels can be degenerate: this degeneracy depends on the orbital and spin quantum numbers. The electron spin is intrinsic to the electron and does not depend on its orbital angular momentum. The spin of an electron is always ½. The arrangement of electrons in the energy levels of an atom are dominated by the Pauli exclusion principle, which states for electrons in an atom that no two electrons can have the same four quantum numbers. This limits the number of electrons sharing the same n, l and ml quantum numbers (a subshell) to just two. Using the aufbau principle the electronic configuration of the ground state of an atom can be determined if the number of electrons present is known. The energy of an electron depends on its interaction with other electrons in the atom. This is the origin of Hund’s rule, which states that electrons in atoms prefer to adopt parallel rather than opposing spins.