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Bohr Model
In addition to the atomic line spectra of single electron atoms, there were other successes of the Bohr Model
X-ray spectraFrank-Hertz experiment
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X-ray Spectra
Recall the x-ray spectra shown a few lectures ago
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X-ray SpectraMoseley found experimentally that the wavelengths of characteristic x-ray lines of elements followed a regular pattern
A similar formula described the L-series x-rays
( )214
3−= ZcRfKα
( )24.736
5−= ZcRfLα
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X-ray Spectra
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X-ray Spectra
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X-ray SpectraMoseley’s law can be easily understood in terms of the Bohr model
If a K (n=1) shell electron is ejected, an electron in the L (n=2) shell will feel an effective charge of Z-1 (Z from the nucleus – 1 electron remaining in the K shell)We have then for the n=2 to n=1 transition
Exactly agreeing with Moseley’s law
( )
( )2
222
14
321
1111
−==
⎟⎠⎞
⎜⎝⎛ −−=
ZcRcf
RZ
K λ
λ
α
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X-ray SpectraApplying the Bohr model to the L-series
Now there are two electrons in the K shell and several in the L shell thus we might expect a (Z-2-several)2 dependence
The data show Zeff = (Z - 7.4)Aside, based on the regular patterns in his data he showed
The periodic table should be ordered by Z not AElements with Z=43,61, and 75 were missing
222
2
365
31
21
effeffL ZcRcRZf =⎟⎠⎞
⎜⎝⎛ −=
α
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Franck-Hertz Experiment
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9
Franck-Hertz Experiment
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10
Franck-Hertz Experiment
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Franck-Hertz Experiment
The data showThe current increases with increasing voltage up to V=4.9V followed by a sudden drop in current
This is interpreted as a significant fraction of electrons with this energy exciting the Hg atoms and hence losing their kinetic energyWe would expect to see a spectral line associated with de-excitation of
nmeV
AeVeVhc 253
9.41024.1 4
=−×
==λ
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Franck-Hertz ExperimentAs the voltage is further increased there is again an increase in current up to V=9.8V followed by a sharp decrease
This is interpreted as the electron possessing enough kinetic energy to generate two successive excitations from the Hg ground state to the first excited stateExcitations from the ground state to the second excited state are possible but less probable
The observation of discrete energy levels was an important confirmation of the Bohr model
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Frank-Hertz Experiment
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Correspondence PrincipleThere were difficulties in reconciling the new physics in the Bohr model and classical physics
When does an accelerated charge radiate?
Bohr developed a principle to try to bridge the gap
The predictions of quantum theory must agree with the predictions of classical physics in the limit where the quantum numbers n become largeA selection rule holds true over the entire range of quantum numbers n (both small and large n)
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Correspondence PrincipleConsider the frequency of emitted radiation by atomic electrons
Classical
2320
4
2
22
02
2/1
3
2
14
find we
using
21
22
nhmef
kmenanr
mrke
rVf
classical
n
classical
ε
πππω
=
==
⎟⎟⎠
⎞⎜⎜⎝
⎛===
h
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Correspondence PrincipleQuantum
( )
( )
( )
classicalBohr
Bohr
Bohr
Bohr
fnh
mef
meE
hnE
hnnEf
nnn
hEf
nnhEf
==
=
=≈
⎟⎟⎠
⎞⎜⎜⎝
⎛
++
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
+−=
3320
4
20
2
4
0
30
40
220
220
14
42for ngsubstituti
22becomes largen for which 112
111
ε
πεh
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Wilson-Sommerfeld Quantization
Quantization appears to play an important role in this “new physics”
Planck, Einstein invoked energy quantizationBohr invoked angular momentum quantization
Wilson and Sommerfeld developed a general rule for the quantization of periodic systems
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Wilson-Sommerfeld Quantization
∫ = nhPdq
cycle oneover integrate means and
coordinate ingcorrespond theis q andmomentum ofcomponent some is Pwhere
∫
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Wilson-Sommerfeld QuantizationFor a particle moving in a central field (like the Coulomb field)
P = Lq = φ
Then
Which is just the Bohr condition
hnnhL
nhdL
nhLdPdq
==
=
==
∫∫∫
π
φ
φ
2
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Wilson-Sommerfeld QuantizationFor a particle undergoing simple harmonic motion
P = pX
q = x
tAmdtdxmp
tdtAdxtAx
kxdt
xdm
x ωω
ωωω
cos and
cos thensin solution has
law sNewton' 2
2
==
==
−=
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Wilson-Sommerfeld QuantizationContinuing on
Which is just the Planck condition
nhfnhE
nhEdE
nhtdtE
AmkAE
nhtdtAmdxpPdq x
==
==
=
==
===
∫
∫
∫∫ ∫
πω
πω
θθω
ω
ω
ωω
π
2
2cos2
cos221
21
oscillator harmonic simplea from Recalling
cos
2
0
2
2
222
222