Download Physics 116 Models of atoms

Physics 116
Thomson
Session 32
Models of atoms
Nov 22, 2011
R. J. Wilkes
Email: [email protected]
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!
Lecture Schedule
(up to exam 3)
Today
3
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
4
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:
look closely at spectrum of sunlight and you see dark lines in it
wavelength (in Angstroms = 10-10 m)
"Holes" in the rainbow?
What causes these sharp lines, in both emission and absorption spectra?
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
5
Atomic spectra
•
Nice illustration of progress of a science:
1.
2.
3.
4.
•
Masses of data collected (“bug collections”)
Empirical rules discovered suggesting underlying regularities
Rules lead to models of atomic structure
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)
1⎞
⎛ 1
= R ⎜ 2 − 2 ⎟ , n = 3, 4, 5K
⎝2
λ
n ⎠
1
–
7
m −1
)
R = Rydberg constant
Outside the visible range, similar series of lines are found, in different EM
wavelength regions, named after the rule-finders:
n’
Series name (range)
1⎞
⎛ 1
= R ⎜ 2 − 2 ⎟ , n ′ = 1, 2, 3K
⎝ n′
λ
n ⎠
1
(R = 1.097 × 10
n = (n ′ + 1), (n ′ + 2 ), (n ′ + 3)K
1
Lyman (UV)
2
Balmer (visible)
3
Paschen (IR)
6
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?
Phosphorescent screen
Radioactive mineral
in a lead box with a
pinhole
Gold foil
Beam of
“alpha-rays”
Thomson
Rutherford
Rutherford experiment
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) Ln = nh (h = h / 2π )
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
mv 2 ke2
ke2
2
= 2 ⇒v =
r
rm
r
L
nh
L = (mv )r ⇒ vn = n =
mr 2π rn m
8
Bohr’s model of the atom
– Assumption 2 defines allowed radii: equate v from assumption 1 with v
derived from quantization condition:
2
2
2
⎞
⎛
⎞
⎛
⎞
L
ke
nh
h2
2
2⎛
n
v =
=⎜
=
⇒
r
=
n
n
⎜⎝ 2π r m ⎟⎠
⎜⎝ 4π 2 mke2 ⎟⎠ , n = 1, 2, 3K
rm ⎝ mr ⎟⎠
n
n
– 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)
mv 2 kZe2 kZe2 kZe2
1 kZe2 • Z=number of + charges in nucleus
E = K +U =
−
=
−
=−
(Z=1 for hydrogen)
2
r
2r
r
2 r • Negative means we must supply this
much energy to extract the electron
– Put in the value of r from above:
from the atom
⎛ 2π 2 mk 2 e4 h 2 ⎞ Z 2
Z2
En = − ⎜
⎟⎠ n 2 = − (13.6eV ) n 2 , n = 1, 2, 3K
h2
⎝
– Energy released when electron jumps from one n to another:
(
∆E ni → n f
⎛ 2π 2 mk 2 e 4 ⎞ ⎛ 1
1⎞
=⎜
⎟⎠ ⎜ n 2 − n 2 ⎟
h2
⎝
⎝ f
i ⎠
)
Rydberg constant !
⎛ 1
1 ∆E ⎛ 2π 2 mk 2 e4 ⎞ ⎛ 1
1⎞
1⎞
hc
7 −1
=
1.097
×
10
⇒ =
=⎜
−
m
−
∆E = hf =
⎜ 2
2⎟
⎟⎠ ⎜ n 2 n 2 ⎟
λ
λ hc ⎝
h 3c
⎝ f
⎝ n f ni ⎠
i ⎠
Bohr explains hydrogen spectra: Lyman series has nf=1, Balmer has nf=2, etc
(
)
9
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 location
You 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
10
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:
h
h
p=
for photons ⇒ λ =
for electrons
p
λ
– Calculate the wavelengths corresponding to Bohr’s allowed e orbits
Bohr
Ln = rn mv =
p = mv =
h
λ
deBroglie
⇒ = 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
11
“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 !
12
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:
⎧ f ( x, t ) = vertical displacement at position x along string
∂2 f
1 ∂2 f
=
⎨
∂x 2 v 2 ∂t 2
⎩ v = wave speed on string
These are called
– Here is the wave equation governing E-M waves:
⎧ E ( x, t ) = E at position x
∂2 E 1 ∂2 E
=
⎨
∂x 2 c 2 ∂t 2
⎩ c = light speed
– Here is Schrödinger’s wave equation
∂Ψ − h 2 ∂Ψ
=
2m ∂t
∂x
differential equations:
they involve partial
derivatives
(concept from calculus:
derivative = rate of
change)
⎧ Ψ ( x, t ) = Schrodinger wave function
⎨
⎩h = Planck ' s constant / 2π ("h - bar" )
Notice a difference: Schrödinger’s is “first order” equation (no squares)
13
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)
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
0.5
0.4
0.2
Psi2
Psi
0.3
0.1
0
-0.1
-0.2
-32
-22
-12
-2
X
8
18
28
0.2
0.15
0.1
0.05
0
-0.05
-32 -22 -12 -2
8
18 28
X
14