1AMQ, Part IV Many Electron Atoms 4 Lectures

1AMQ P.H. Regan & W.N.Catford

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1AMQ, Part IV Many Electron Atoms4 Lectures

Spectroscopic Notation, Pauli Exculsion Principle

Electron Screening, Shell and Sub-shell Structure

Characteristic X-rays and Selection Rules. Optical Spectra of atoms and selection

rules. Addition of Angular Momentum for Two

electrons.

•K.Krane, Modern Physics, Chapter 8• Eisberg and Resnick, Quantum Physics, Chapters 9 and 10.


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Pauli Exclusion Principle and Spectroscopic Notation.

A complete description of the state of an electron in an atom requires 4 quantum numbers, n, l, ml and ms.

For each value of n, there are 2n2 different combinations of the other quantum numbers which are allowed.

The values of ml and ms have, at most, a very small effect of the energy of the states, so often only n and l are of interest for chemistry.

Spectroscopic Notation, uses letters to specify the l value, ie. l= 0, 1, 2, 3, 4, 5….

has the designation s, p, d, f, g, h,…...


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Summary of the Allowed Quantum Numbers Specifying the Allowed Quantum Numbers of an Atoms.

Symbol Name

n principal quantum number

l orbital quantum number

ml magnetic quantum number

ms spin quantum number

Symbol Allowed Values Physical Property

n n=1,2,3,4,… size of orbit, rn=a0n2

l l=0,1,2,3,…,(n-1) AM & orbit shape

ml -l, -l+1,…..,(l-1),+l projection of L

ms +1/2 and -1/2 projection of s.


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Atoms with Many Electrons

Electrons do not all collect in the lowest energy orbit (evident from chemistry).

This experimental fact can be accounted for using the Pauli Exclusion Principle which states that `no two electrons in a single atom can have the same set of quantum numbers (n,l,ml,ms).’ (Wolfgang Pauli, 1929).

For example the n=1 orbit (K-shell) can hold at most 2 electrons,

n l ml ms

1 0 0 +1/2

1 0 0 -1/2

Electrons in an atom fill the allowed states

(a) beginning at the lowest energy

(b) obeying the Exclusion Principle


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Energies of Orbitals in Multielectron Atoms.

The energies of the subshells are affected by the presence of other electrons, particularly by the screening of the nuclear charge.

In high-Z atoms, the inner subshells are also affected by the electrons in the higher subshells.

Shell Structure of Atoms.The n values dominates the determination of the radius of each subshell (as shown in the solutions to the Schrodinger equation).

For the penetrating orbitals (s and p), the probability of being found at a small radius is balanced by some probability of also being found at a larger radius.

Hence, subshells with the same n are grouped into `shells’ with about the same average radius from the nucleus.


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Conventionally, the shells are designated by letter, eg, K shell, n=1

L shell, n=2

M shell, n=3

Subshells correspond to different l values within each shell.

According to the Pauli Principle, each subshell has a maximum occupancy (number of electrons) which is given by,

(2l+1) x 2 = number of possible ml values x x no. of ms values for each ml .

Example: s subshells (2.0+1).2 =2 electrons

p subshells (2.1+1).2=6 electrons

Periodic Table of Elements.Inspection of the table of electronic structure show that this determines the chemical properties of that element (in particular, the number of valence electrons, ie. number in outermost shell is very important)


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Characteristic X-rays and Selection Rules.

Characteristic X-rays: are emitted by atoms when electrons make transitions between inner shells (note a vacancy must be created before this can happen).

An incident photon, electron or alpha-particle can knock out an e- from the atom. Electrons at higher excitation energies cascade down to fill the vacancy. A vacancy in the K shell can be filled with an e- from the L shell (a K transition) or the M shell (K) etc.

n=4, N n=3, M n=2, L

n=1, K

K series

L series

M series

K

K series

L

L series


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We can calculate the K

energy for each element approximately. Consider an L-shell electron, about to fill a K-shell vacancy.

K

L+Ze

Approximately, the L electron orbits an `effective’ charge of +(Z-1)e. To allow for penetrating orbits, say +(Z-b)e, where we expect b to be approximately equal to 1.

We can use Bohr theory, for an electron orbiting a charge +(Z-b)e, to estimate the X-ray energies.

2/172

222

1

))(105( )(75.0

2

1

1

1)( and ,

12 between transition

)( chargenuclear effective

sbZxbZcR

bZcRhE

nn

ebZ

H

H

i

Moseley’s Law

This compares to experiment to within 0.5% witha value for the intercept, b of very close to 1.


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1,0j and 1 l are observed to occur.

Fine Structure of X-ray SpectraThe subshells are split in energy by the spin-orbit interaction, into (l+1/2) and (l-1/2) levels. Note that not all the energetically allowed transitions are observed. The notation LI, LII, LIII etc. is used for these levels

Only transitions which obey the selection rules

The selection rules are related to the underlying physics of (a) the spatial properties (symmetry) of the charge oscillations that produce transitions and (b) the angular momentum (spin) carried away by the photon itself (at least 1h).


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Optical Spectra of Atoms.

Example: Sodium, Z=11, thus the electron configuration in the atomic ground state is 1s22s22p63s1

The first excited state has 1s22s22p63p1

The lowest energy levels of the atom are due to excitations of electrons between outer levels (where the smallest energy gaps and the first vacancies exist).

For electrons in the n=3 electron orbitals, the nuclear charge (Z=+11e) is screened by the inner 10 electrons from the n=1 and n=2 shell (Q=-10e). Thus the energy is similar to that of the n=3 Bohr orbit for hydrogen.

Screening effects give an energy shift of the levels, which depends on the l of the orbital.


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Optical photons are emitted when valence (outer) electrons make transitions between energy levels. Eg. 3p->3s , E=2.10 eV (3.37 x10-19 J), thus hc/E) = 589 nm (ie. yellow street lamps).


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1,0 and 1,0 lj mm

Not all of the energetically possible transitions are allowed. The selection rules are (as in x-rays)

10,j and 1 lThe transition 3p->3s shows fine structure giving the sodium D lines, ie. D2, 589.0 nm and D1, =589.6 nm.

If an external B field is applied, the Zeeman effect gives a further splitting of the levels. The Zeeman splitting is generally smaller (for typical laboratory size eternal fields) than the fine structure (which increases as approx. Z4).

A further selection rule is observed, namely,

Summary:• Optical spectra arise from transitions of valence e-s.• Optical photon energies are approx. several eV.• Inner e-s are left undisturbed by optical transitions.• When more energy is injected, and inner electrons are removed, x-rays (with energies ~keV) are emitted when these vacancies are filled


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The Helium Atom: 2 Active Electrons

While Sodium has, for low excitations, only one active electron for optical transitions, helium and other elements can have more.

L2

L1


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Ground State of Helium: two electrons, both with l=0 (same (1s) orbital), gives configuration of 1s2

n l ml ms

1 0 0 +1/2

1 0 0 -1/2

Using the coupling rules for l and s: i) Lmax = 0+0 = 0, Smax=1/2+1/2=1ii) Lmin = |0-0| = 0, Smin=|1/2-1/2|=0iii) L=0 (and thus ML=0), S=1 or 0iv) ML=0, MS=-1,0,+1 are the only possibilities NB. Use capital `L’ and `S’ for > 1 electron.

The Pauli Principle means that S=1 is not allowed, since would need ms=+1/2 for both e-s for S=1.

ie. for 1s2, can only have (L=0, S=0), ie. groundstate of helium has L=0, S=0, hence, J=0


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Excited State of the Helium Atom.

The first excited state (above the ground state) has configuration 1s12s1.

n l ml ms

1 0 0 +1/2, -1/2

2 0 0 +1/2,-1/2

As for the ground state, L=0, S=0 or 1 from coupling of L1 and L2 vectors etc.

Allowed states are (L=0, S=0) and (L=0, S=1).

For S=0, only one state (singlet state)

For S=1, three states (ie. MS=-1,0,+1) (triplet state)

Hund’s rules lead to Etriplet<Esinglet.

(This is related to the PEP, aligned electrons tend to `repel’ each other).


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Transitions in Helium

Experimentally, the selection rules He are L=0,+-1 and S=0 (ie. no `spin-flips’).

Singlet (S=0) and triplet (S=1) states CAN NOT BE CONNECTED by transitions. 1s2 (singlet)

1s12s1, singletand triplet

The triplet 1s12s1 state is rather long lived or metastable (=`isomeric’). This is due to the fact that no decay can occur to the ground state via single photon emission (S=1 is forbidden).

It subsequently decays by transferring kinetic energy in collisions.

The singlet 1s12s1 state is also isomeric since a transition between 2s and 1s states would violate the l=+-1 selection rule for individual l values.

1AMQ, Part IV Many Electron Atoms 4 Lectures

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