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Ordering of the Elements

Byungsung O

Department of Physics, CNU

byung@cnu.ac.kr / x6544

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.

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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

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Continuous X-Ray Spectrum

Bremsstrahlung (braking radiation)

:: cutoff wavelength,

independent of target materials

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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

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Stimulated Emission

LASER = Light Amplification by Stimulated Emission of Radiation

• 3 ways that radiation can interact with matter

• Thermal distribution

(Boltzmann) :

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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