Energy levels in alkali metals
Lectures 8-9: Multi-electron atoms
o
Alkali atoms: in ground state, contain a set of Z - 1 completely filled subshells with a single
valence electron in the next s subshell.
o
Electrons in p subshells are not excited in any low-energy processes. s electron is the single
optically active electron and core of filled subshells can be ignored.
o Alkali atom spectra
o Central field approximation
o Shell model
o Effective potentials and screening
o Experimental evidence for shell model
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Energy levels in alkali metals
Hartree theory
o
In alkali atoms, the l degeneracy is lifted: states with the same principal quantum number n
and different orbital quantum number l have different energies.
o
For multi-electron atom, must consider Coulomb interactions between its Z electrons and its
nucleus of charge +Ze. Largest effects due to large nuclear charge.
o
Relative to H atom, alkali terms lie at lower energies due to increased Coulomb attraction of
nucleus. This shift increases the smaller l is.
o
Must also consider Coulomb interactions between each electron and all other electrons in
atom. Effect is weak.
o
For larger values of n, i.e., greater orbital radii, the terms are only slightly different from
hydrogen.
o
Assume electrons are moving independently in a spherically symmetric net potential.
o
o
Also, electrons with small l are more strongly bound and their terms lie at lower energies.
The net potential is the sum of the spherically symmetric attractive Coulomb potential due to
the nucleus and a spherically symmetric repulsive Coulomb potential which represents the
average effect of the electrons and its Z - 1 colleagues.
o
These effects become stronger with increasing Z.
o
o
Non-Coulombic potential breaks degeneracy of levels with the same principal quantum
number.
Hartree (1928) attempted to solve the time-independent Schrödinger equation for Z electrons
in a net potential.
o
Total potential of the atom can be written as the sum of a set of Z identical net potentials V( r),
each depending on r of the electron only.
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Screening
o
Central field approximation
Hartree theory results in a shell model of atomic structure,
which includes the concept of screening.
o
For example, alkali atom can be modelled as having a valence
electron at a large distance from nucleus.
o
Moves in an electrostatic field of nucleus +Ze which is
screened by the (Z-1) inner electrons. This is described by the
effective potential Veff( r ).
o
At r small,
-e
r
The Hamiltonian for an N-electron atom with nuclear charge +Ze can be written:
+Ze
-(Z-1)e
where N = Z for a neutral atom. First summation accounts for kinetic energy of electrons ,
second their Coulomb interaction with the nuclues, third accounts for electron-electron
repulsion.
o Unscreened nuclear Coulomb potential.
o
o
o
Not possible to find exact solution to Schrodinger equation using this Hamiltonian.
o
Must use the central field approximation in which we write the Hamiltonian as:
At r large,
o Nuclear charge is screened to one unit of charge.
where Vcentral is the central field and Vresidual is the residual electrostatic interaction.
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Central field approximation
Central field approximation
o
o
The central field approximation work in the limit where
o
In this case, Vresidual can be treated as a perturbation and solved later.
o
By writing " = "1 ( rˆ1 )" 2 ( rˆ2 )K" N ( rˆN ) we end up with N separate Schrödinger equations:
where Ri(ri) are a set of radial wave functions and Yi(!i, "i ) are a set of spherical harmonic
functions.
o
Following the same procedure as Lectures 3-4, we end up with three equations, one for each
polar coordinate.
o
Each electron will therefore have four quantum numbers:
o l and ml: result from angular equations.
o n: arises from solving radial equation. n and l determine the radial wave function Rnl(r )
and the energy of the electron.
o ms: Electron can either have sip up (ms = +1/2) or down (ms = -1/2).
o
State of multi-electron atom is then found by working out the wave functions of the individual
electrons and then finding the total energy of the atom (E = E1 + E2 + … + EN).
!
with E = E1 + E2 + … + EN
o
As potentials only depend on radial coordinate, can use separation of variables:
Normally solved numerically, but analytic solutions can be found using the separation of
variables technique.
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Shell model
o
Shell model
Hartree theory predicts shell model structure, which only considers gross structure:
o
Periodic table can be built up using this shell-filling process. Electronic configuration of first
11 elements is listed below:
o
Must apply
1. Pauli exclusion principle: Only two electrons with opposite spin can occupy an atomic
orbital. i.e., no two electrons have the same four quantum numbers.
2. Hunds rule: Electrons fill each orbital in the subshell before pairing up with opposite spins.
1. States are specified by four quantum numbers, n, l, ml and ms.
2. Gross structure of spectrum is determined by n and l.
3. Each (n,l) term of the gross structure contains 2(2l + 1) degenerate levels.
o
Shell model assumes that we can order energies of gross terms in a multielectron atom
according to n and l. As electrons are added, electrons fill up the lowest available shell first.
o
Experimental evidence for shell model proves that central approximation is appropriate.
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Shell model
o
Shell model
Atomic shells listed in order of increasing energy. Nshell = 2(2l + 1) is the number of electrons
that can fill a shell due to the degeneracy of the ml and ms levels. Naccum is the accumulated
number of electrons that can be held by atom.
o
4s level has lower energy than 3d level due to penetration.
o
Electron in 3s orbital has a probability of being found close to nucleus. Therefore
experiences unshielded potential of nucleus and is more tightly bound.
3d
4s
o
Note, 19th electron occupies 4s shell rather than 3d shell. Same for 37th. Happens because
energy of shell with large l may be higher than shell with higher n and lower l.
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Shell model
Shell model
Radial probabilities for 1s 2s 3s
Radial probabilities for 3s 3p 3d
3s - red
3p - blue
3d - green
1s - red
2s - blue
3s - green
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Quantum defect
Quantum defect
o
Alkalis are approximately one-electron atoms: filled
inner shells and one valence electron.
o
#(l) depends mainly on l. Values for sodium are
shown at right.
o
Consider sodium atom: 1s2 2s2 2p6 3s1.
o
Can therefore estimate wavelength of a transition via
o
Optical spectra are determined by outermost 3s electron.
The energy of each (n, l) term of the valence electron is
o
For sodium the D lines are 3p ! 3s transitions. Using
values for #(l) from table,
where #(l) is the quantum defect - allows for penetration
of the inner shells by the valence electron.
o
Shaded region in figure near r = 0 represent the inner n
= 1 and n = 2 shells. 3s and 3p penetrate the inner shells.
o
Much larger penetration for 3s => electron sees large
nuclear potential => lower energy.
=> $ = 589 nm
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Shell model justification
Experimental evidence for shell model
o
Ionisation potentials and atomic radii:
o
Consider sodium, which has 11 electrons.
o
Nucleus has a charge of +11e with 11 electrons
orbiting about it.
o Ionisation potentials of noble gas elements are
highest within a particular period of periodic
table, while those of the alkali are lowest.
o
From Bohr model, radii and energies of the electrons
in their shells are
n2
13.6Z 2
rn = a0 and E n = "
Z
n2
o Ionisation potential gradually increases until
shell is filled and then drops.
o
First two electrons occupy n =1 shell. These electrons
see full charge of +11e. => r1 = 12/11 a0 = 0.05 Å and
!
!
E1 = -13.6 x 112/12 ~ -1650 eV.
o
o Filled shells are most stable and valence
electrons occupy larger, less tightly bound orbits.
o Noble gas atoms require large amount of energy
to liberate their outermost electrons, whereas
outer shell electrons of alkali metals can be
easily liberated.
Next two electrons experience screened potential by
two electrons in n = 1 shell. Zeff =+9e => r2 = 22/9 a0
= 0.24 Å and E2 = -13.6 x 92/22 = -275 eV.
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Experimental evidence
for shell model
o
Experimental evidence
for shell model
X-ray spectra:
o Enables energies
determined.
of
inner
shells
to
be
o Accelerated electrons used to eject core electrons
from inner shells. X-ray photon emitted by
electrons from higher shell filling lower shell.
o
Wavelength of various series of emission lines are found to obey Moseley’s law.
o
For example, the K-shell lines are given by
where % accounts for the screening effect of other electrons.
80 keV
40 keV
o
Similarly, the L-shell spectra obey:
o
Same wavelength as predicted by Bohr, but now have
and effective charge (Z - %) instead of Z.
o
%L ~ 10 and %K ~ 3.
o K-shell (n = 1), L-shell (n = 2), etc.
Wavelength (A)
o Emission lines are caused by radiative transitions
after the electron beam ejects an inner shell
electron. Continuum due to bremsstrahlung.
o Higher electron energies excite inner shell
transitions.
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Experimental evidence
for shell model
o
Assume net charge is ( Z - 1 )e.
o
Therefore, the potential energy is
o
Total energy of orbit is
o
Modified Bohr formula taking into account screening.
o
Can therefore easily show that
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