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Chapter 4: Atomic Structure

The Nuclear Atom

The Atom as the smallest division of an element

quantization of electric charge

oil drop experiments q = ne

e/m => mass of electrons

neutral atoms as “natural” state

“Plum Pudding” model

BUT.....

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Rutherford scattering (alpha particles from heavy nuclei)

= test of “plum pudding” model

alpha particles emitted in some radioactive decays

speeds ~ 2E7 m/s

q = +2e, m ~ 8000 x me ( is a He4 nucleus)

alpha source

lead collimator

thin foil

light flash

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Expected (from plum pudding): small scattering angles, no back scattering

Results: some larger scattering angles, including some back scattering

The Nuclear Atom:

small heavy nucleus (99.8% of atom’s mass) with positive electric charge

~ 1/100,000 radius of atom

electron “cloud” => electrons orbit nucleus

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N( )

N i

ntZ2e4

(8 0 )2 r 2 KE2 sin4( /2)

N( )

N i

fraction of incident particles scattered at

n number of atoms per volume in foil

Z Atomic number (number of protons in nucleus)

r distance from foil to screen

KE initial KE of alpha particles

t = foil thickness

45 90 135 180

Rutherford Scattering (theoretical results):

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Rutherford’s ingredients:

Newtonian Mechanics (F = ma)

Coulomb Interaction

=> Distance of closest approach F

1

4 0

q1q2

r 2;PE

1

4 0

q1q2

r

Conservation of Energy

Ei( at large distance ) E f ( at turn around )

KE0 0PE

KE 1

4 0

Ze2e

R R

1

4 0

2Ze2

KE

Example: The maximum KE of alpha particles from natural sources is 7.7 MeV. What is the distance of closest approach for a gold nucleus? (ZAu = 79)

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Electron Orbits: planetary models of the atom

for the purposes of this discussion, take electron orbits to be circular

Hydrogen: single electron atom

FE 1

4 0

e2

r 2; PE

1

4 0

e2

r

Fc mv 2

rFE 1

2mv 2 1

2

1

4 0

e2

r KE 1

2PE

also v e

4 0 mr

E KE PE 12PE PE 1

2PE

1

8 0

e 2

r

Example 4.1: The ionization energy of Hydrogen is 13.6 eV (the energy required to liberate the electron from the atom). Find the orbital radius and speed of the electron in a hydrogen atom.

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Problems with the nuclear atom:

accelerating charges radiate

orbits cannot be stable!!

considerable problems with atomic spectra

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Atomic Spectra

emission line spectra (from thin, hot gas or vapor)

spectrum tube contains rarified gas or vapor through which a high voltage is discharged

screen or film

prism

collimating slitgas

disc

harg

e tu

be

Hydrogen

Helium

Mercury

typical emission spectra

emission spectra

vs.

absorption spectra

700nm 400nm

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Hydrogen spectral series: patterns in the spectra

10 100 1000 10000

Balmer1

n

R1

22

1

n2

n 3,4,5, (visible light)

R 0.01097nm 1

Lyman1

n

R1

12

1

n2

n 2,3,4, (UV)

Paschen1

n

R1

32

1

n2

n 4,5,6, (IR)

Brackett1

n

R1

42

1

n2

n 5,6,7, (IR)

Pfund1

n

R1

52

1

n2

n 6,7,8, (IR)

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Bohr Atom

electron in orbit about nucleus

atomic size ~ electron orbit radius (or see example 4.1)

= 0.053 nm

compare de Broglie wavelength with radius

rnmnmrm

r

e

h

mr

ev

mv

h

233.0,053.0with

4

)orbit"planetary " (from4

0

0

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Bohr’s original hypothesis: quantize angular momentum of circular orbits

nnn p

hnrn 2

. . .

22

2

nhrmvr

mv

hnn

nh

nLmvrL

nnnn

n

n

Bohr’s hypothesis justified by de Broglie wave theory

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Energy in the Bohr Atom

eVJEn

EE

nh

me

r

eE

me

hra

anme

hnr

n

r

m

r

e

h

rn

n

nn

n

nnn

nn

6.131018.2,

1

88

nm05292.0

24

2

1812

1

2220

4

0

2

20

2

10

02

20

22

0

E = 0eV n =

E = -13.6eV n = 1

E = -3.40eV n = 2

n = 3

. . .

. . .

E > 0eV free electron

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Ei E f h E f hc

1

Ei E f

hc(any atom )

Ei E1

ni

2, E f

E1

n f

2(hydrogen )

1

E1

hc

1

n f

2

1

ni

2

R

E1

hc!

1

E1

hc

1

n f

2series limit (ni )

Origin of Line Spectra

Discrete Energy levels + conservation of energy + photons

n f =1 -> Lyman, n f =2 -> Balmer,

n f =3 -> Paschen, etc.

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Example 4.2: An electron collides with a hydrogen atom in its ground state(lowest energy) and excites it to a state of n = 3. How much energy was given to the hydrogen atom in this inelastic collision?

Example 4.3: Hydrogen atoms in state of high quantum number have been created in the laboratory. (a) Find the quantum number of the Bohr orbit in a hydrogen atom whose radius is 0.0100mm. (b) What is the energy of a hydrogen atom in this state?

Example 4.4: Find the longest wavelength present in the Balmer series of hydrogen

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The Correspondence Principle

A new theory should encompass an old theory where the old theory was successful.

Quantum theory approximates the results of classical mechanics when:

quantum numbers are large

h 0

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Classical treatment of radiation from “planetary” hydrogen: frequency of emitted light = frequency of orbits (+ harmonics)

31

3320

4

20

22

30

0

22

8with

424

2

nh

E

nh

mef

me

hnr

mr

ef

mr

ev

r

vf

n

Quantum transition from n np with p << n

fpn

p

h

E

npn

pnp

h

E

npnEh

31

22

21

221

2

)(

21

)(

1

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Refining the Bohr Atom

nuclear motion: electron and nucleus orbit each other (each orbit center of mass).

Two body problem =>

center of mass motion +

relative motion (with reduced mass)

99945.0'

:

'1

8

''

'

21

2220

4

m

mhydrogen

n

E

m

m

nh

emE

Mm

mMm

n

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Example 4.6: A “positronium” atom consists of an electron and a positron. Compare the spectrum of positronium to that of hydrogen

Example 4.7: Muons are elementary particles with mass 207me and +-e of charge. A muonic atom is formed by a negative muon with a proton. Find the radius of the first Bohr orbit and the ionization energy of the atom.

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Atomic spectra

Atoms have discrete set of allowed energies

ALL changes in atom’s energy have to be to an allowed state

Absorption and emission spectra from conservation of energy

Franck-Hertz Experiment: inelastic scattering of electrons by atoms ->atom only absorbs energy to E = e V

Eh

A

VV

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The Laser: bright, monochromatic, coherent light source

Excited State: state above ground state

decays to lower states, with emission of photon (or other mechanism for energy transfer).

Metastable State: “sort of stable” state

state with a longer life time than ordinary excited states

lifetime ~ 1E-3 s vs. 1E-8 s for ordinary states

Three kinds of transitions

E hh

h

h

h

Induced Absorption

Spontaneous Emission

Induced Emission(Stimulated)

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pumping process

fast emission to metastable state

laser transition: stimulated emission

hh

h

h’

Energy levels for 4-level laser

LightAmplification byStimulatedEmission ofRadiation

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Other considerations:

“recycling” inducing photons and selecting lasing transition: the laser cavity

Fabret-Perot Interferometer = standing waves

“Tunable” Dye Lasers

Semiconductor Lasers

Chemical Lasers

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Chapter 4 exercises:3,4,5,6,7,8,11,12,13,14,15,16,18,19,21,22,29,30,31,32,33,35