R. J. WilkesEmail: ph116@u.washington.edu

Physics 116

Session 32Models of atoms

Nov 22, 2011

Thomson Rutherford

Announcements • Exam 3 next week (Tuesday, 11/29)

• Usual format and procedures• I’ll post example questions on the website tomorrow

afternoon, as usual• We’ll go over the examples in class Monday 11/28

Enjoy your holiday weekend!

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Lecture Schedule (up to exam 3)

Today

4

Let’s back up a bit: Subatomic discoveries ~100 years ago

• J. J. Thomson (1897) identifies electron: very light, negative charge• E. Rutherford (1911) bounces “alpha rays” off gold atoms

• We now know: α = nucleus of helium: 2 protons + 2 neutrons• “Scattering experiment” = model for modern particle physics

– Size of atoms was approximately known from chemistry– He finds: scattering is off a much smaller very dense core (nucleus )

• Rutherford’s nuclear model of atom: dense, positively charged nucleus surrounded by negatively charged lightweight electrons

• Niels Bohr (1913): applies Planck/Einstein quanta to atomic spectra– Atoms have fixed energy states: they cannot “soak up” arbitrary energy– Quanta are emitted when atom “jumps” from high to low E state– Assumed photon’s energy E=hf, as Planck and Einstein suggested– Simple model of electrons orbiting nucleus, and “classical” physics (except

for quantized E) gives predictions that match results well (at least, for hydrogen spectrum)

Next topics: atoms, nuclei, radioactivity, subatomic particles

5

Back to the puzzles of 1900

Excite a low-pressure sample of noble gas (like neon) with an electric discharge: Pass this light through a slit and prism and you see sharp, separated lines, NOT a continuous rainbow:

Boltzmann’s thermodynamics + Maxwell’s electrodynamics explain only continuous spectra: • physical quantities are described by real numbers (decimals) • Electric charges in atoms can oscillate at any frequency...emit any wavelength of light

wavelength (in Angstroms = 10-10 m) "Holes" in the rainbow?

What causes these sharp lines, in both emission and absorption spectra?

look closely at spectrum of sunlight and you see dark lines in it

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Atomic spectra

• Nice illustration of progress of a science:1. Masses of data collected (“bug collections”)2. Empirical rules discovered suggesting underlying regularities3. Rules lead to models of atomic structure4. Models lead to a refined theory that (eventually) can explain everything –

and make predictions of as yet unseen phenomena, to provide a test• Theory has to be testable and refutable! (otherwise: speculation)

• Example of item 2: Hydrogen’s line specrum (1885-)– Heat hydrogen in a tube and run through a diffraction grating and you see

lines with wavelengths that satisfy the rule (Balmer, 1885)

– Outside the visible range, similar series of lines are found, in different EM wavelength regions, named after the rule-finders:

1λ= R 1

′n 2 −1n2

⎛⎝⎜

⎞⎠⎟

, ′n = 1,2, 3K n = ′n +1( ), ′n + 2( ), ′n + 3( )K

R = Rydberg constant

1λ= R 1

22 −1n2

⎛⎝⎜

⎞⎠⎟

, n = 3, 4,5K R = 1.097 ×107 m−1( )

n’ Series name (range)

1 Lyman (UV)

2 Balmer (visible)

3 Paschen (IR)

Early ideas about atoms

• Atom = concept since Democritus; physical evidence circa 1900• “Plum pudding model” (J. J. Thomson): electrons are very small

negative (q=-e) particles; atoms are larger, and neutral (q=0)– perhaps positive charge occupies a blob the size of the atom, and the

electrons are like plums in a pudding?

• Nuclear model (Rutherford, 1911)– Alpha-rays (q=+2e) scatter off atoms as if there were a tiny hard core, like

a billiard ball: large scattering angles, sometimes even knocked backwards– Perhaps positive charge occupies only a small volume in the atom, and

most of the mass is in this nucleus?

Thomson Rutherford Rutherford experiment

Radioactive mineral in a lead box with a pinhole

Phosphorescent screen

Gold foil

Beam of “alpha-rays”

Bohr’s model of the atom

• Semi-classical synthesis, combines Planck/Einstein quanta with Maxwell/Newton physics

• Assume (N. Bohr, 1911)1. Electrons are negative particles, occupying circular orbits around a

positively charged nucleus (Rutherford model + classical physics)2. Only certain orbits are allowed: ones where electron’s angular momentum

L = integer multiple of hbar (quantized) 3. Electrons do not radiate while in stable circular orbits (contrary to

Maxwell!)4. Radiation occurs only when electrons move between allowed orbits,

absorbing or releasing energy (quantum jumps)

• Bohr found this model explained the hydrogen series relationships – Assumption 1 means electron speed/momentum depends on radius

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Ln = nh h = h / 2π( )

mv2

r=

ke2

r2 ⇒ v2 =ke2

rm

L = mv( )r ⇒ vn =Ln

mr=

nh2πrnm

Bohr’s model of the atom

– Assumption 2 defines allowed radii: equate v from assumption 1 with v derived from quantization condition:

– All the constants above were known fairly well in 1911: r1=5.3 x 10-11 m– Assumption 4 means allowed radii correspond to energy levels (quantized)

– Put in the value of r from above:

– Energy released when electron jumps from one n to another:

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v2 =ke2

rm=

Ln

mrn

⎛⎝⎜

⎞⎠⎟

2

=nh

2πrnm⎛⎝⎜

⎞⎠⎟

2

⇒ rn = n2 h2

4π 2mke2

⎛

⎝⎜⎞

⎠⎟, n = 1,2, 3K

E = K +U =mv2

2−

kZe2

r=

kZe2

2r−

kZe2

r= −

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kZe2

r

• Z=number of + charges in nucleus (Z=1 for hydrogen)

• Negative means we must supply this much energy to extract the electron from the atom

En = −2π 2mk2e4h2

h2

⎛

⎝⎜⎞

⎠⎟Z 2

n2 = − 13.6eV( )Z 2

n2 , n = 1,2, 3K

∆E ni → nf( )= 2π 2mk2e4

h2

⎛

⎝⎜⎞

⎠⎟1nf

2 −1ni

2

⎛

⎝⎜

⎞

⎠⎟

∆E = hf = hcλ⇒

1λ=∆Ehc

=2π 2mk2e4

h3c⎛

⎝⎜⎞

⎠⎟1nf

2 −1ni

2

⎛

⎝⎜

⎞

⎠⎟ = 1.097 ×107 m−1( ) 1

nf2 −

1ni

2

⎛

⎝⎜

⎞

⎠⎟

Bohr explains hydrogen spectra: Lyman series has nf=1, Balmer has nf=2, etc

Rydberg constant !

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Familiar misleading picture of an atom

• We’ve all seen this– Electrons like tiny planets orbiting popcorn-ball nucleus at center

• You know better– Nucleus is tiny (would be invisible on this picture’s scale)– Particles (protons and electrons) are not really at any point in

space – probability distribution describes their locationYou can observe an electron’s path,but to do so you must knock it out of the atom!

Electron tracks in a cloud chamber (1937)sciencemuseum.org.uk

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deBroglie revisited (this time in context)

• Einstein says photons simultaneously have wave and particle character…• Bohr can explain hydrogen spectra with orbiting electrons that have quantized

angular momentum and energy• De Broglie (1923): if we

– Assume e’s have a wave character on the same basis as photons have particle character:

– Calculate the wavelengths corresponding to Bohr’s allowed e orbits

p =hλ

for photons ⇒ λ =hp

for electrons

Ln = rnmv =

p = mv = hλ

⇒ = for electrons

DeBroglie found that Bohr’s orbit rules corresponded to having circumference of orbit exactly fit m (integer number) wavelengths!Other radii not allowed because overlapping waves “interfere destructively”. Semi-classical picture: related quantum facts to well-known classical phenomena

BohrdeBroglie

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“Wave mechanics”

• E. Schrödinger (1927): particles obey a wave equation which can be used to understand subatomic phenomena– Wave equation defines behavior of a wave function

• Example: particle’s motion can be described by giving its position and momentum at any time:

wave function = Ψ(x, p , t) …this means Ψ depends on x, p and time– Mathematical form ensures proper wavelike behavior of particles– Interference effects (constructive and destructive) are possible!

– Wave function contains all information about quantum system (particle, or atom, or nucleus, or whatever)

• Deep consequence: any question you may ask that cannot be answered by solving the wave equation for a completely-specified wave function has no physical meaning !

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Analogy to E-M waves (this is a cultural supplement)

• E-M wave = moving, time-varying electric and magnetic fields– We can measure E field amplitude (volts per meter) with special hardware– More commonly, we measure intensity of light (energy/sec)

• Intensity = (amplitude) 2 (this gave Schrödinger a hint! )

• For your cultural benefit: look at and compare some wave equations– Here is the equation describing waves on a string:

– Here is the wave equation governing E-M waves:

– Here is Schrödinger’s wave equation

Notice a difference: Schrödinger’s is “first order” equation (no squares)

2 2

2 2 2

( , )1 at position xf x t vertical displacementf along stringspeed on s

fv wavex t gt r nv i

=⎧∂ ∂= ⎨ =∂ ∂ ⎩

2 2

2 2 2

( , )1 at position xE x t EE Ec ligh spetx c t ed

=⎧∂ ∂= ⎨ =∂ ∂ ⎩

2 ( , )' ( )2 / 2

x t Schrodinger wave functionPlanck s consta "h - barnt "x m t π

Ψ =⎧∂Ψ − ∂Ψ= ⎨ =∂ ∂ ⎩

h

h

These are called differential equations:they involve partial derivatives(concept from calculus: derivative = rate of change)

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Interpreting Schrodinger’s wave function

• What is the wave function “made of”?– Wave function Ψ is not a physical quantity like momentum or E

• Has no units, cannot be directly measured or detected– Wave function squared gives probability of finding particle at

position x (or with momentum p)

-0.2-0.1

00.10.20.30.40.5

-32 -22 -12 -2 8 18 28

X

Psi

-0.050

0.050.1

0.150.2

-32 -22 -12 -2 8 18 28

X

Psi2

Wavefunction Ψ(x) (has no units!) of a particle, vs position x

Probability of finding particle described by Ψ(x) at position x:

P(x) = Ψ2