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Chapter 30
The Atom
2
1. Rutherford & the Nuclear Atom
a) Thompson’s plum pudding atom
positive pudding; negative plums (electrons)
b) Rutherford scattering
Marsden: One in ~20,000 alpha particles scattered by > 90º
"It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you.” Rutherford
3
c) Nuclear atom
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
- positive nucleus ~ 10-15 m- electrons orbits: ~ 10-10 m- mostly open space- originally a planetary model
4
d) Problem with the planetary model
An accelerating electron radiates. Why doesn’t it loseenergy, and spiral inward, collapsing the atom?
5
2. Atomic Spectra
a) Light emission from gas discharge
Atoms or molecules are excited electrically, and emit light at characteristic wavelengths.
6
b) The emission spectrum
Solar
Neon
Hg
H
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c) The atomic hydrogen spectrum
- simplest atom- should be simplest to describe & understand
Discrete wavelengths observed for simple systems.
QuickTime™ and aTIFF (Uncompressed) decompressor
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(visible)UV
IR
8
Empirical formulas for observed series:
€
1
λ= R
1
4−
1
n2
⎛ ⎝
⎞ ⎠; n = 3,4,5,... Balmer series
€
1
λ= R 1−
1
n2
⎛ ⎝
⎞ ⎠; n = 2,3,4,... Lyman series
€
1
λ= R
1
9−
1
n2
⎛ ⎝
⎞ ⎠; n = 4,5,6,... Paschen series
9
Empirical formulas for observed series:
€
1
λ= R
1
22 −1
n2
⎛ ⎝
⎞ ⎠; n = 3,4,5,... Balmer series€
1
λ= R
1
12 −1
n2
⎛ ⎝
⎞ ⎠; n = 2,3,4,... Lyman series
€
1
λ= R
1
32 −1
n2
⎛ ⎝
⎞ ⎠; n = 4,5,6,... Paschen series
10
Empirical formulas for observed series:
€
1
λ= R
1
n12 −
1
n22
⎛
⎝ ⎜ ⎞
⎠ ⎟ Rydberg formula
R = 1.097 x 107 m-1 Rydberg constant€
n1 = 1,2,3,...
n2 = n1 +1,n1 + 2,n1 + 3,...
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3. The Bohr Atom
a) The concept - constrained classical physics
(i) Only certain orbits with well-defined energies allowed
(ii) “Stationary” orbits do not radiate
(iii) Emission or absorption occur only during transitions from one stationary orbit to another; energy related to wavelength by Planck’s relation.
12
€
E i − E f = hf = hc
λWhat are the rules that restrict energy levels?
13
b) Classical energies and radii
€
Total energy : E = KE + PE
€
E = 12 mv 2 −
kZe2
r
€
Circular motion : F =mv2
r
€
Coulomb force : F =kZe2
r2
}
€
mv 2 =kZe2
r
€
Combining, E =1
2
kZe2
r−
kZe2
r
€
E = −1
2
kZe2
r
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c) Quantization of energy levels
To get the Rydberg formula, energy should be proportional to 1/n2, or radius to n2
€
E = hc
λ= −
1
2
kZe2
r
€
1
λ= R
1
n12 −
1
n22
⎛
⎝ ⎜ ⎞
⎠ ⎟
Postulate: Quantize angular momentum according to
€
Ln = nh
2π
€
Then, using L = mvnrn, and mv 2 =kZe2
r, gives
€
rn =h2
4π 2mke2
⎛
⎝ ⎜ ⎞
⎠ ⎟n2
Z; n = 1,2,3,...
15
€
rn =h2
4π 2mke2
⎛
⎝ ⎜ ⎞
⎠ ⎟n2
Z; n = 1,2,3,...
€
rn = 5.29 ×10−11m( )n2
Z; n = 1,2,3,...
Bohr radius: n = 1, Z = 1
€
r1 = 5.29 ×10−11 m
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Energy with quantized L (and r)
€
En = −2π 2mk 2e4
h2
⎛
⎝ ⎜ ⎞
⎠ ⎟Z 2
n2 ; n = 1,2,3,...
€
En = − 13.6eV( )Z 2
n2
€
n = 1,2,3,...
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d) Line spectra for hydrogen (Z = 1)
€
ΔE = E i − E f
The energy of an emitted photon is given by
€
We had E i − E f = hf = hc
λ, so
€
=−2π 2mk 2e4
h2
⎛
⎝ ⎜ ⎞
⎠ ⎟1
ni2 −
1
n f2
⎛
⎝ ⎜ ⎞
⎠ ⎟
€
1
λ=
2π 2mk 2e4
h3c
⎛
⎝ ⎜ ⎞
⎠ ⎟1
ni2 −
1
n f2
⎛
⎝ ⎜ ⎞
⎠ ⎟
€
1
λ= R
1
n12 −
1
n22
⎛
⎝ ⎜ ⎞
⎠ ⎟
Same form as Rydberg formula
18
R = 1.097 x 107 m-1
€
1
λ= R
1
n12 −
1
n22
⎛
⎝ ⎜ ⎞
⎠ ⎟
€
1
λ=
2π 2mk 2e4
h3c
⎛
⎝ ⎜ ⎞
⎠ ⎟1
ni2 −
1
n f2
⎛
⎝ ⎜ ⎞
⎠ ⎟
€
2π 2mk 2e4
h3c= 1.097 ×107m−1
Constant in Bohr model is completely determined by known physical constants, and matches the Rydberg constant.
19
e) Energy level diagram
€
En = − 13.6eV( )1
n2 ; n = 1,2,3,...
20
21
• Examples
Problem 30.15What values of ni and nf give wavelength 410.2 nm in H tube?
Problem 30.17What values for ni give visible lines for nf = 4 in He+?
22
4. Bohr atom & de Broglie waves
Concept: Confined waves have discrete wavelengths:- guitar string- pipe organ Condition for standing wave in a
circular orbit:
€
nλ = 2πrUsing the de Broglie relation = h/p
€
nh
mv= 2πr
Rearranging gives Bohr’s condition:
€
mvr = nh
2π
A physical justification for Bohr’s ad hoc hypothesis
23
5. X-rays
a) The phenomenon
- discovered in 1896 by Roentgen
- collisions of energetic electrons with metals produce short wavelength radiation
24
- short wavelength => penetrating radiation
Mrs. Roentgen’s hand.
First published x-ray picture
"I have discovered something interesting, but I do not know whether or not my observations are correct.” Roentgen, during 7 weeks working in isolation to characterize the new rays.
25
b) Spectrum
26
c) Origins
- characteristic x-rays:- electrons remove inner shell electrons from metal atom- atom relaxes emitting x-ray photon
- continuous background- Bremsstrahlung (braking radiation)- decelerating electron produces radiation
27
Examples
Find minimum energy to eject K (n = 1) electrons from platinum (Z = 78) according to the Bohr model.
Find wavelength for K rays for Pt.
28
d) Cutoff wavelength
- independent of target
- Max energy of photon is energy of incident electron
€
Emax = eV
€
hc
λ 0
= eV
€
0 =hc
eV
€
For V = 45000V, λ 0 = 2.8 ×10−11m
29
e) Medical applications
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- x-ray images exploit varyiable absorption in tissues
Hand of Albert von Kolliker, made at the conclusion of Roentgen's lecture and demonstration at the Wurzburg Physical-Medical Society on 23 January 1896.
- CAT scans (computer assisted tomography)
30
X-ray shoe fitter
31
6. Laser
a) CoherenceCoherent waves: point source (spatial), fixed frequency (temporal)
- displacement simply predictable- e.g. water wave from single bobbing cork
Incoherent waves: multiple sources, mixture of frequencies- complex wave front- e.g. water wave from multiple corks at different frequencies (or sand thrown on water)
Ordinary light sources are highly incoherent- sources are widely spaced (compared to wavelength(- mixture of frequencies
32
b) Absorption, Spontaneous Emission, & Stimulated Emission
Absorption
Spontaneous emission
Stimulated emission- in phase- parallel- same wavelength- proposed by Einstein in 1916to preserve thermal eq’m
33
c) Amplification
If the probability for causing stimulated emission > .5, amplification occurs
LightAmplification byStimulatedEmission ofRadiation
34
d) Population Inversion
To promote emission over absorption, more upper statesare required, than lower states.
35
e) Pumping - produces population inversion
- ordinarily lower states more populated- more absorption than emission; no amplification
- metastable (longer lived) state (ms) and intermediate short-lived state (ns) allows population inversion- energy may be provided by (e.g.) light flashes or electrical pulses
36
f) Resonator
37
g) Properties
- low divergence (10-3 radians typical; 10-5 radians easy)- coherence length ~ 10 km (discharge lamp << 1 cm)- monochromatic: 106 narrower than discharge- 107 more photons per unit area of source- 1014 more photons per unit solid angle
38
7. Holography
a) Parallax and 3dAlignment depends on position of observer
2 eyes give slightly different image, interpreted as 3d
39
b) Photography- records intensity and color from one perspective- Original wave not reproduced; image is the same from everywhere --
eyes seem to follow observer
c) Stereo photography- 2 images recorded from different perspectives
- gives 3d image from one perspective- Original wave not reproduced; no parallax due to motion
40
d) Holography concept- record intensity and phase of wave, so original wave can
be reproduced
Film records wave information
41
Film reproduces wave
42
e) Recording a hologram
43
f) Viewing a hologram
44
g) Principle for a point object
ReferenceObject
€
mλ = d sinθ
sinθ ≅ Δy /L
Δy =Lλ
d
L
d
- consider two slits; one as object, and the other reference
film records pattern
45
Reference
L
€
=Δy sinθ =Lλ
dsinθ
y ≅ Lsinθ = d
y
Developed film is grating