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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