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Chapter 30

The Atom

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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

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c) Nuclear atom

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- positive nucleus ~ 10-15 m- electrons orbits: ~ 10-10 m- mostly open space- originally a planetary model

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d) Problem with the planetary model

An accelerating electron radiates. Why doesn’t it loseenergy, and spiral inward, collapsing the atom?

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2. Atomic Spectra

a) Light emission from gas discharge

Atoms or molecules are excited electrically, and emit light at characteristic wavelengths.

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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.

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(visible)UV

IR

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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

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1

λ= R

1

9−

1

n2

⎛ ⎝

⎞ ⎠; n = 4,5,6,... Paschen series

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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

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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.

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€

E i − E f = hf = hc

λWhat are the rules that restrict energy levels?

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b) Classical energies and radii

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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

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rn =h2

4π 2mke2

⎛

⎝ ⎜ ⎞

⎠ ⎟n2

Z; n = 1,2,3,...

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€

rn =h2

4π 2mke2

⎛

⎝ ⎜ ⎞

⎠ ⎟n2

Z; n = 1,2,3,...

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rn = 5.29 ×10−11m( )n2

Z; n = 1,2,3,...

Bohr radius: n = 1, Z = 1

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r1 = 5.29 ×10−11 m

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Energy with quantized L (and r)

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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

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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.

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e) Energy level diagram

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En = − 13.6eV( )1

n2 ; n = 1,2,3,...

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• 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+?

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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

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5. X-rays

a) The phenomenon

- discovered in 1896 by Roentgen

- collisions of energetic electrons with metals produce short wavelength radiation

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- 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.

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b) Spectrum

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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

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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.

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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

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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)

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X-ray shoe fitter

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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

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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

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c) Amplification

If the probability for causing stimulated emission > .5, amplification occurs

LightAmplification byStimulatedEmission ofRadiation

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d) Population Inversion

To promote emission over absorption, more upper statesare required, than lower states.

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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

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f) Resonator

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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

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7. Holography

a) Parallax and 3dAlignment depends on position of observer

2 eyes give slightly different image, interpreted as 3d

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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

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d) Holography concept- record intensity and phase of wave, so original wave can

be reproduced

Film records wave information

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Film reproduces wave

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e) Recording a hologram

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f) Viewing a hologram

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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

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Reference

L

€

=Δy sinθ =Lλ

dsinθ

y ≅ Lsinθ = d

y

Developed film is grating