Transcript 1AMQ, Part II Quantum Mechanics

1AMQ, Part IV Many Electron Atoms
4 Lectures
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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.
1AMQ P.H. Regan &
W.N.Catford
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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
1AMQ P.H. Regan &
W.N.Catford
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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 alphaparticle 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 efrom the L shell (a
Ka transition) or the
M shell (Kb) etc.
n=4, N
n=3, M
n=2, L
n=1, K
L series
K series
M series
La
L series
Ka
K series
1AMQ P.H. Regan &
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Moseley’s Law
We can calculate the Ka
K
energy for each element
approximately. Consider +Ze
L
an L-shell electron,
about to fill a K-shell
vacancy.
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.
effect ivenuclear charge   ( Z  b)e
t ransit ionbet ween
ni  2  n1  1
1 1 
E  h , and  cRH ( Z  b)  2  2 
1 2 
2
  0.75cRH ( Z  b) 2    (5 x107 )(Z  b) s 1/ 2
This compares to experiment to within 0.5% with
a value for the intercept, b of very close to 1.
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Fine Structure of X-ray Spectra
The subshells are split in energy by the spinorbit 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
l  1 and j  0,1
are observed to occur.
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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1AMQ P.H. Regan &
W.N.Catford
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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
l(hc/E) = 589 nm (ie. yellow street lamps).
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Not all of the energetically possible transitions
are allowed. The selection rules are (as in x-rays)
l  1 and j  0,1
The transition 3p->3s shows fine
structure giving the sodium D lines, ie.
D2, l589.0 nm and D1, l=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,
m j  0,1 and ml  0,1
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=1
ii) Lmin = |0-0| = 0, Smin=|1/2-1/2|=0
iii) L=0 (and thus ML=0), S=1 or 0
iv) 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. ground
state of helium has L=0, S=0, hence, J=0
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W.N.Catford
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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 1s12s1, singlet
L=0,+-1 and S=0 (ie. no `spin-flips’). and triplet
Singlet (S=0) and triplet (S=1) states CAN
NOT BE CONNECTED by transitions.
1s2 (singlet)
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.
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