1

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

2

X-ray Spectra

Recall the x-ray spectra shown a few lectures ago

3

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

4

X-ray Spectra

5

X-ray Spectra

6

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

7

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

8

Franck-Hertz Experiment

9

Franck-Hertz Experiment

10

Franck-Hertz Experiment

11

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

12

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

13

Frank-Hertz Experiment

14

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)

15

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

16

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

17

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

18

Wilson-Sommerfeld Quantization

∫ = nhPdq

cycle oneover integrate means and

coordinate ingcorrespond theis q andmomentum ofcomponent some is Pwhere

∫

19

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

20

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

==

==

−=

21

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