2006-7 Quantum Theory Slides Lecture 4

Quantum theory and atomic spectroscopy Lecture 4 Spin

The story so far…. • Atomic subshells are made up of sublevels, which we label with the quantum number ml • The number of sublevels is given by 2l + 1 • ml is a manifestation of the space quantization of angular momentum • The number of electrons in each sublevel is limited to just 2 • Now read on …

Today’s question is…… • Why can a sublevel only hold a maximum of two electrons? • Why did Stern and Gerlach see just two spots in their space quantization experiment? • By the end of this lecture, we will understand atoms like helium a little better…..

Bohr’s model of the atom • Bohr based his famous paper on the idea that discrete energy levels existed in atoms (proved it for hydrogen) • Spectral lines are transitions between different energy levels- important! He calculated the energies! • Franck and Hertz quantified the relationship between mercury lines and “excitation” electrons in cathode ray tube

The Bohr Festival: Discovery of hafnium • Bohr claimed chemists believed unknown element 72 must be rare – earth • He predicted it wouldn’t be • Coster and Hevesy find it is a transition metal! Named hafnium in honour of Bohr! • But… was he really so brave? Ask Bury! • Bohr was wrong though about the subshells- no proper understanding of the subshell ordering then

The crisis of 1922 • (1) Bohr’s model actually predicted the wrong levels, but since no one understood the ordering of the levels noone knew how to fix it • (2) The old theory was hopeless at explaining the spectrum of helium, which is a bit worrying as it only has two electrons • (3) The spectrum of sodium, with its doublet of lines and anomalous Zeeman effect could not be explained • All this was solved by a single new type of angular momentum • Now read on …

Spectra of many-electron atoms • 1919- KosselSommerfeld laws of displacement and interchange • Odd electronsdoublet spectra • Even electronssinglet and triplet spectra • Explanation next year!

The Aufbau principle • This is a set of empirical rules (1925) for determining the ground electronic state of an atom • 1) We fill an atom lowest energy levels 1st 2) For each n, there are n subshells (l) 3) In general, the higher n, l the higher the energy 4) For each l, there are 2l + 1 sublevels 5) Only a maximum of 2 electrons share n, l and ml • The latter is the result of the Pauli exclusion principle “No two electrons can have the same 4 quantum numbers” – yeap, Pauli made up a new number!

Order of orbitals • The order of the relevant sublevels does depend on the atomic number • At high Z, the subshells do group according to the quantum number n • This is due to electron- electron interactions

Explaining the periodic table

• We can explain the arrangement of elements in terms of filling subshells • Add Pauli exclusion principle! But what was the 4th quantum number?

That crazy Stern-Gerlach result • In silver there is just one valence electron – an s electron • Therefore, its orbital angular momentum is zero! • We should get one spot, not two! • Is the fact the electron is unpaired significant?

Goudsmit and Uhlenbeck

• They weren’t the first, but their timing was good! • They suggested an electron has an intrinsic angular momentum- called spin • It is a new type of angular momentum!

Electron spin • Spin can take halfinteger quantum numbers • Thus this is the missing number Pauli was looking for! • An electron can either spin up or down! So can state Pauli exclusion principle as “No two electrons can have the four same numbers n, l, ml and ms”

Hund’s rule • We need this when dealing with partially filled subshells with l > 0 • Only applies to ground electronic term of an atom • Maximise the spin of unpaired electrons to minimise the energy • There is no simple explanation- leave it till next year!

Why do electron’s have spin? • Young physicist Dirac decides to solve spin problem • He tries to combine quantum theory with special relatively • Derives Dirac equation and shows all fundamental particles MUST have spin! And it must be ½! • Equation also predicts the existence of antiparticles • 1932- anti-electron discovered!

Total atom spin • Just like orbital angular momentum, add all the electron spins to find the total, S n

S = ∑ si i

• For a closed shell, the total spin is 0

Term Symbols • You need to know the configuration as you need (a) total L and (b) total S • Electronic term expressed as 2S+1L

2S 2S++11==multiplicity multiplicity

e.g gnd state silver is 2S It also explains the Stern- Gerlach result with silver (Fraser)!

Term Symbols in spectroscopy • •

We represent transitions in terms of Term symbols not individual electrons The energy states of atoms are determined by all the electrons in the atom and labelled 2S+1L

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States with same configurations are labelled numerically, from lowest in energy to highest e.g. 12S is lower in energy than 22S

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A transition is often represented in this manner 32P → 22S with the highest energy state on the left

Term Symbol test • • • • •

Write down the Term symbols for the following electron configurations? (a) d1p1 (b) s2p1 (c) f14 s1 Write down (a) the value of L and (b) the value of S for the following term symbols 2S 5D 2P 6G

Answers will be posted on the web!

Spin and selection rules • In light atoms, there is a spin selection rule in optical transitions ∆S = 0 • We can analyse the calcium atom as a two electron system • Poor transitions are good for atomic clocks!

Many – electron atoms

• The degeneracy of hydrogen levels is lost • This is the result of electron – electron interactions we now call electron correlation Great! Let’s consider helium….

The helium atom • Significant differences between helium and hydrogen spectra • Firstly, we seem to have two sets of energy levels! • They correspond to different values of total spin • At one time, thought to be two forms of helium!!!! • Secondly, orthohelium energy levels much lower! This last point couldn’t be fully explained until the discovery of quantum mechanics!

Answers to the earlier questions.. • There are two spots in the Stern- Gerlach experiment because the valence electron in silver carries only spin • The spin of an electron is ½ • Pauli exclusion principle limits 2 electron to an orbital • Stern – Gerlach is always a test of space quantization (sublevels)

Hang on……

• Where do these quantum numbers come from? • Why is angular momentum and energy in atoms quantised? • Next time, we will see that one completely new concept can explain all!