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Department of Physics, CNU Byungsung OByungsung O
Pauli Exclusion Principle
No two electrons confined to the same trap (or atom) can
have the same set of values for their quantum numbers
Multiple Electrons in Rectangular Traps
1. One-dimensional trap: Two quantum numbers = 1, 2, 3, …
(wavefunction state along ) and = +1/2 or -1/2.
2. Rectangular corral: Three quantum numbers = 1, 2, 3, …
(wavefunction state along ), = 1, 2, 3, (wavefunction…
state along ), and = +1/2 or -1/2.
3. Rectangular box: Four quantum numbers = 1, 2, 3, …
(wavefunction state along ), = 1, 2, 3, (wavefunction…
state along ), = 1, 2, 3, (wavefunction state along…
), and = +1/2 or -1/2.
Department of Physics, CNU Byungsung OByungsung O
Finding the total energy
Adding electrons to a rectangular trap:
1. Use energy level diagram.
2. Start at lowest energy level and move up as lower levels
become filled.
Department of Physics, CNU Byungsung OByungsung O
Energy levels of multiple electrons in a 2D
infinite potential well
• Seven electrons are confined to a 2D square infinite potential with
.
• What is the configuration for the ground state of the system?
For a 2D square infinite potential,•
1 1 ±1/2 21 2 ±1/2 52 1 ±1/2 52 2 ±1/2 81 3 ±1/2 103 1 ±1/2 10...
.
.
....
Department of Physics, CNU Byungsung OByungsung O
Energy levels of multiple electrons (cont'd)
# 1 1 1 -1/2 22 1 1 +1/2 23 1 2 -1/2 54 1 2 +1/2 55 2 1 -1/2 56 2 1 +1/2 57 2 2 -1/2 8
Total 32
• What is the total energy of the system in its ground state
configuration, as a multiple of (= )?
From the above table,•
Department of Physics, CNU Byungsung OByungsung O
Energy levels of multiple electrons (cont'd)
• How much energy must be transferred to the system to jump to its
first excited state?
The first excited state energy: out of the possible 3 transitions•
below
→
→
→
Since → is the smallest,
the first excited state total energy is
Department of Physics, CNU Byungsung OByungsung O
Building the Periodic Table
• Four quantum numbers , , , and identify the quantum
states of individual electrons in a multi-electron atom.
• Subshells are labeled by letters:
= 0 1 2 3 4 5 . . .s p d f g h . . .
Example: =3, =2 3d subshell→
• Sometimes, shells are also labeled by letters
= 1 2 3 4 . . .K L M N . . .
Example: =1 K-shell→
Department of Physics, CNU Byungsung OByungsung O
Building the Periodic Table
=0 (s) =1 (p) =2 (d)=0 -1 / 0 / +1 -2 / -1/ 0 / +1 / +2Energ
y
3
2
1 ↑ 1s1
=0 (s) =1 (p) =2 (d)=0 -1 / 0 / +1 -2 / -1/ 0 / +1 / +2E
nerg
y
3
2
1 ↑↓ 1s2
Department of Physics, CNU Byungsung OByungsung O
Building the Periodic Table
=0 (s) =1 (p) =2 (d)=0 -1 / 0 / +1 -2 / -1/ 0 / +1 / +2Energ
y
3
2 ↑↓
1 ↑↓ 1s2
2s2
=0 (s) =1 (p) =2 (d)=0 -1 / 0 / +1 -2 / -1/ 0 / +1 / +2E
nerg
y
3
2 ↑↓ ↑↓ ↑↓ ↑↓
1 ↑↓ 1s2
2s22p
6
Department of Physics, CNU Byungsung OByungsung O
Building the Periodic Table
=0 (s) =1 (p) =2 (d)=0 -1 / 0 / +1 -2 / -1/ 0 / +1 / +2E
nerg
y
3 ↑↓
2 ↑↓ ↑↓ ↑↓ ↑↓
1 ↑↓ 1s22s
22p
63s
2
=0 (s) =1 (p) =2 (d)=0 -1 / 0 / +1 -2 / -1/ 0 / +1 / +2E
nerg
y
3 ↑↓ ↑↓ ↑↓ ↑↓
2 ↑↓ ↑↓ ↑↓ ↑↓
1 ↑↓ 1s2
2s22p
63s
23p
6
Department of Physics, CNU Byungsung OByungsung O
Building the Periodic Table
=0 (s) =1 (p) =2 (d)=0 -1 / 0 / +1 -2 / -1/ 0 / +1 / +2Energ
y
4 ↑↓3 ↑↓ ↑↓ ↑↓ ↑↓
2 ↑↓ ↑↓ ↑↓ ↑↓
1 ↑↓ 1s22s
22p
63s
23p
64s
2
Department of Physics, CNU Byungsung OByungsung O
Building the Periodic Table
• For atoms with a larger number of electrons, the interactions
among the electrons causes shells with
the same but different to have
different energies (degeneracy lifted).
• Due to interactions it takes less energy to
start filling the 4s subshell before
completing the filling of the 3d subshell,
which can accommodate 10 electrons.
1s22s
22p
63s
23p
64s
23d
61s⇒ 2
2s22p
63s
23p
63d
64s
2
Department of Physics, CNU Byungsung OByungsung O
X Rays and Ordering of Elements
• X rays are short wavelength (10-10
m), high energy (~keV)
photons. Photons in the visible range: ~ 10-6
m; ~eV.
• Useful for probing atoms
Production of X-Ray
Department of Physics, CNU Byungsung OByungsung O
Continuous X-Ray Spectrum
Bremsstrahlung (braking radiation)
:: cutoff wavelength,
independent of target materials
Department of Physics, CNU Byungsung OByungsung O
Characteristic X-Ray Lines
• Energetic electron strikes atom in target, knocks out deep-lying
(low value). If deep-lying electron in =1 (K-shell), it
leaves a vacancy (hole) behind.
• Another electron from a higher energy shell in
the atom jumps down to the K-shell to fill
this hole, emitting an x-ray photon in the process.
Department of Physics, CNU Byungsung OByungsung O
Characteristic X-Ray Lines
• If the electron that jumps into the hole of K shell starts from
the = 2 (L-shell), the emitted radiation is the line.
• If it jumps from the = 3 (M-shell), the radiation is the
line. The hole left in the = 2 or = 3 shells is filled by
still higher lying electrons, which relax by emitting lower
energy photons (higher lying energy levels are more closely
spaced).
Department of Physics, CNU Byungsung OByungsung O
Ordering Elements
• Moseley (1913) measured the wavelengths of the emitted
x-rays from 38 elements.
• of vs. the position
of the element in the
periodic table : a straight
line
• Nuclear charge, not mass,
is the critical parameter
for ordering elements.
Department of Physics, CNU Byungsung OByungsung O
Accounting for the Moseley Plot
• Energy levels in hydrogen:
eV, for n=1,2,3, ...
• Approximate effective energy levels in multi-electron atom with
protons (replace × with ×)
eV
• energy :
(eV)
eV
• frequency :
×
× Hz
where × Hz1/2
Department of Physics, CNU Byungsung OByungsung O
Characteristic spectrum in x-ray production
• Co is bombarded with electrons. There is a second, fainted
characteristic spectrum due to an impurity in Co. The of the
lines are 178.9 pm (Co) and 143.5 pm (impurity). = 27.
Determine the impurity.
Since•
The impurity should be Zn
Department of Physics, CNU Byungsung OByungsung O
Stimulated Emission
LASER = Light Amplification by Stimulated Emission of Radiation
• 3 ways that radiation can interact with matter
• Thermal distribution
(Boltzmann) :
Department of Physics, CNU Byungsung OByungsung O
How laser works
• To get more stimulated emission than absorption, →
population inversion not in thermal equilibrium.→
The stimulated emitted photons have the same frequency•
(monochromatic) and the same phase (coherent)
• medium to get population inversion
• pumping (optically or electrically) electrons to the higher level
• resonator to amplify the beam
http://phet.colorado.edu/en/simulation/lasers
http://www.physics.uoguelph.ca/applets/Intro_physics/kisalev/java/laser/index.html
http://web.phys.ksu.edu/vqm/laserweb/Ch-3/F3s5p1.htm
Department of Physics, CNU Byungsung OByungsung O
He-Ne Gas Laser
• The first HeNe laser emitted at 1.15 μm (1960). The best
known and most widely used wavelength is 632.8 nm (1962).
Department of Physics, CNU Byungsung OByungsung O
Population inversion in a laser
• Consider a laser that emits at = 550 nm.
• Ratio of the population of atoms at Rm Temp.
In thermal equilibrium•
=(6.63×10
-34)(3.00×10
8)
= 3.616x10-18
J(550×10
-9)
= 2.26 eV
× ×≈1.3x10
-38.
• What temperature would the ratio be 1/2?
From•
=
(3.616×10-18
)= 38 000 K
(1.380×10-23
)(ln 2)
Department of Physics, CNU Byungsung OByungsung O
Lasers and Laser Light
1. Laser light is highly monochromatic: spread in wavelength as
small as 1 part in 1015.
2. Laser light is highly coherent: Single uninterrupted wave train
up to 100 km long. Can interfere one part of beam, with
another part that is very far away.
3. Laser light is highly directional: beam spreads very little.
Beam from earth to moon only spreads a few meters after
traveling 4x108
m.
4. Laser light can be sharply focused: can be focused into very
small spot so that all the power is concentrated into a tiny
area. Can reach intensities of 1017
W/cm2, compared to
103
W/cm2for oxyacetylene torch.
Department of Physics, CNU Byungsung OByungsung O
Laser have many uses:
• Small: voice/data transmission over optic fibers, CDs, DVDs,
scanners
• Medium: medical, cutting (from cloth to steel), welding
• Large: nuclear fusion research, astronomical measurements,
military applications