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AtomsMatti Hotokka

Department of Physical ChemistryÅbo Akademi University

Jyväskylä 2008

Dark lines in the sun’s spectrum

� First observed by Wollaston 1802

� Rediscovered by Fraunhofer 1814Fraunhofer’s notation is used still today

What is seen?

Hα

Hβ

Hγ

Hδ

FC f h

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Fraunhofer’s notationA few examples of lines

Symbol Atom Wavelength [nm]C H 656.3D Na 589.6 & 589.0F H 486.1

A (band) O2 760B (band) O2 687E Fe 527.0b Mg 518.4 & 517.3c Fe 495.8f H 434.0g Ca 422.7h H 410.2H Ca 396.8

See http://www.harmsy.freeuk.com/fraunhofer.html

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λ =−

=nn

G n2

2 43 4 5, , , �

1 4 12

1 12

1 3 4 52 2 2 2λ= −

��

�

��

�= −

��

�

��

�

=G n

Rn

nH , , , �Discovered by Wollaston/FraunhoferMathematical formulation by JohannBalmer in 1885

Balmer series

Generalised by Johannes Rydberg 1888

Rydberg’s constant RH = 109 668 cm-1.

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Rutherford (1911)

Classical physicsBohr (1913)

Non-classical particlesDeBroglie (1925)

Electrons as standing wavesSchrödinger and Heisenberg (1926, 1925)

Modern quantum theory

Atom modelsReasonably accurate models

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Electrons follow only certain paths (orbits)Energy is absorbed/emitted at transitionsfrom one orbit to another. Transitionenergy is EEmitted frequency is E/hCoulomb attraction and centrifugal forceThe electron’s energy is a multiple of ½h ,

= circulation frequency

Bohr

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E m en hn

e= −4

2 2028 ε

∆E m eh n

e= −

��

��

�4

202 2 28

12

1ε

Energy of the electron

Bohr

(Or �)

Transition energy from level 2 to level nModern quantum theory gives the same answer.

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Modern quantum theory

�

1s

3s

2p2s

3p 3d

Hydrogen atom only: The orbitalenergy depends only on theprincipal quantum number n.

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What is seen?

Lyman 121.58 nm102.58 nm97.26 nm

Balmer 656.53 nm486.32 nm434.21 nm

Paschen 1.8758 �m1.2822 �m1.0942 �m

Brackett 2467.5 cm-1

3807.9 cm-1

4616.1 cm-1

Pfund 1304.4 cm-1

2148.6 cm-1

2673.1 cm-1

In the visible

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Almost like hydrogen: one valence nselectronDifference: 2s and 2p have differentorbital energies; likewise 3s, 3p and 3d etc

Alkali metals

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0

-1

-2

-3

-4

-5

-6

0

-10000

-20000

-30000

-40000

-50000

2 S 2 P 2 D 2 F

2s

3s

4s

5s

6s

2p

3p

4p

5p

6p

3d

4d

5d6d7d

4f

5f6f7f

671

323

274256

813

497

427399

611

460413

1870

Principal

Diffuse

Sharp

FundamentalEnergy levelsand transitionsof lithium

Selection rule:qn � goes to

�+1 or �-1

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M L

MJL

Spectral terms

M = 2S+1 = spin multiplicity; S = total electronic spin

L = total orbital angular quantum number

Can be calculated in several ways. However, this may betrickier than for molecules because atomic orbitals mayincorporate more degenerate electrons.

Sum of spin and orbital angular momenta, J, may be added. It is important if relativistic effects need to be considered.

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The totally filled orbitals may be ignored.

Spectral termsFor a few simple cases

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Previously: � = ±1Quite strict rule: S = 0

More selection rules

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Relativity splits levels that should have thesame energy

Sodium energy levels

E

3s 2 S1/2

0.0

3p 2 P1/216956.183

3p 2 P3/216973.379

589.7554 nm589.1579 nm

E

3p 2 P1/216956.183

3p 2 P3/216973.379

4d 2 D5/234548.754

4d 2 D3/234548.789

568.4206 nm568,9768 nm568.9779 nm

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Emission spectrum of sodium

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The orbital angular momenta of the twoelectrons are combined as a vector sumThe spin angular momenta likewise

� Two electrons give a singlet or a triplet state

Many-electron atom: helium

S=0

� � � �S=1 S=1

� � � �

S=1

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0

-1

-2

-3

-4

-5

0

-10000

-20000

-30000

-40000

1 S 1 P 0 1 D 1 F 0 3 S 3 P 0 3 D 3 F 0

1s 2

2s; metastable state

2s2p

2p

3s3s 3p 3d3d3p

4s4s 4p 4d4d 4f4f4p5s

58.4 nm

2058

728

505

668

492

502

1870

1083

389

319707

471588

447

1869

Helium: Two completelyseparate sets of levels, onefor singlet and one for triplet.

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Electron configuration

� E.g., N: (1s)2 (2s)2 (2p)3

Atoms generally

� [J]

-6.81x10-17

-2.47x10-18

-4.12x10-18

1s

2px, 2py, 2pz

2s

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Electronic statesNitrogen atom

E

(1s)2(2s)2(2p)2(3p)

(1s)2(2s)2(2p)3

(1s)2(2s)2(2p)2(3s)

4S

2D

4S

2P

4P2P

2D

4P

4D2S 2P

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The transitions shown here are forbidden.Relativity makes them occur anyway, butthey are quite weak.

Transitions

E

Oxygen3 P 0.0

1 D 15867.7

1 S 33792.4

630 nm

558 nm

E

Nitrogen4 S 0.0

2 D 19223

2 P 28840

520 nm

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Ψ Ψm el m mr s r s( , ) ( ) ( ),

� �= χ

< >=< >< >Ψ Ψ Ψ Ψmel

n el mel

el m m nr s r s r r s s( , )| | ( , ) ( )| | ( ) ( )| ( ), ,

� � � �µ µ χ χ

In non-relativistic approximation the wf ofstate m can be written as

Transition moment

The transition moment is

Thus there is the selection rule S=0.The remaining selection rules can bederived from the spatial transition moment.

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L=0,±1; however not L=0 � L=0i

�

i even � odd or odd � even (Laporte)J=0,±1; however not J=0 � J=0S=0

Selection rulesNon-relativistic theory

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AASLaser ablation (LIBS)ICP-MS (or other detection)

Plasma