4e : the quantum universemodphys.ucsd.edu/4es04/slides/4electure6-apr6.pdf4e : the quantum universe...
TRANSCRIPT
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Bragg Scattering photographic film
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Bragg Scattering: Probing Atoms With X-Rays
Incident X-ray detector
Constructive Interference when net phase difference is 0, 2π etcThis implied path difference traveled by two waves must be integral
multiple of wavelength : nλ=2dsinϑ
From X Ray (EM Wave) Scattering data, size of atoms was known to be about 10-10 m
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X-Ray Picture of a DNA Crystal and Discovery of DNA Structure !
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Where are the electrons inside the atom?
Early Thought: “Plum pudding” model Atom has a homogenous distribution of Positive charge with electrons embedded in them
• How to test these hypotheses? Shoot “bullets” at the atom and watch their trajectory. What Kind of bullets ? •Indestructible charged bullets Ionized He++ atom = α++ particles •Q = +2e , Mass Mα=4amu >> me , Vα= 2 x 10 7 m/s (non-relavistic) [charged to probe charge & mass distribution inside atom]
e-
e-
e-e-
e-
e-
e-
e- e-
e-
e-e-
e-e-
e-e-
e-
e-
e-
Positively charged matter
?
+ Core
or +
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Plum Pudding Model of Atom
• Non-relativistic mechanics (Vα/c = 0.1)• In Plum-pudding model, α-rays hardly scatter because
– Positive charge distributed over size of atom (10-10m)– Mα >> Me (like moving truck hits a bicycle)– predict α-rays will pass thru array of atoms with little scatter
(~1o)
Need to test this hypothesis Ernest Rutherford
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“Rutherford Scattering” discovered by his PhD Student (Marsden)
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Force on α-particle due to heavy Nucleus
2
particle trajectory is hyperbolicScattering angle is related to impact par.
Impact Parameter cot2
kq Qbm v
α
α α
α
θ⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
•Outside radius r =R, F ∝ Q/r2
•Inside radius r < R, F ∝ q/r2 = Qr/R2
•Maximum force at radius r = R
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Rutherford Scattering: Prediction and Experimental Result
2 2 4
22 2 414 ( / 2)
2
k Z e NnAnR m v Sinα α ϕ
∆ =⎛ ⎞⎜ ⎟⎝ ⎠
# scattered Vs φ depends on :n = # of incident alpha particlesN = # of nuclei/area of foilZe = Nuclear chargeKα of incident alpha beamA= detector area
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Rutherford Scattering & Size of Nucleus
2
distance of closest appoach r size of nucleus1Kinetic energy of = K = 2
particle will penetrate thru a radius r until all its kinetic energy is used up to do work AGAINST the Coulomb potent
m vα α βα
α
∝
( )( )
Al2
15
-15
-
2
2
10
For K =7.7.MeV, Z 13
ial of the
2
Nucleus:21 K = 8
2
Size of Nucleus = 1
0 Size of Atom =
4.9 10
2
10
kZe
Ze em v MeV k
kZer
r
rK
mm
m
K
α
α
α α β
α
−
=
⇒ =
=
⇒
=
= = ×
nucleus
nucleus
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Dimension Matters !
-15
-10
Size of Nucleus = 10Size of Atom = 10
mm
How are the electrons located inside an atom ?How are they held in a stable fashion ?
necessary condition for us to exist !All these discoveries will require new experiments and observations
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Where are the Electrons in an Atom ?
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Clues: Continuous & Discrete spectra of Elements
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Visible Spectrum of Sun Through a Prism
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Emission & Absorption Line Spectra of Elements
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Kirchhoff’ Experiment : “D” Lines in Na
D lines darken noticeably when Sodium vapor introduced Between slit and prism
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Emission & Absorption Line Spectrum of Elements
Emission line appear dark because of photographic exposure
Absorption spectrum of NaWhile light passed thru Na vaporis absorbed at specific λ
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Spectral Observations : series of lines with a pattern
• Empirical observation (by trial & error)• All these series can be summarized in a simple formula
2
7 1
2
1 1 1 , , 1, 2,3, 4..
Fitting to spectral line serie
s R=
data 1.09737 10
f i if i
R n n nn n
m
λ
−×
⎛ ⎞= − > =⎜ ⎟⎜ ⎟
⎝ ⎠
How does one explain this ?
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The Rapidly Vanishing Atom: A Classical Disaster ! Not too hard to draw analogy with dynamics under another Central Force
Think of the Gravitational Force between two objects and their circular orbits.
Perhaps the electron rotates around the Nucleus and is bound by their electrical charge
2 22
12
1M MF= G k r
r
Q Q⇒
Laws of E&M destroy this equivalent picture : Why ?
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Bohr’s Bold Model of Atom: Semi Quantum/Classical
1. Electron in circular orbit around proton with vel=v
2. Only stationary orbits allowed . Electron does not radiate when in these stable (stationary) orbits
3. Orbits quantized: – Mev r = n h/2π (n=1,2,3…)
4. Radiation emitted when electron “jumps” from a stable orbit of higher energy
stable orbit of lower energy Ef-Ei = hf =hc/λ
5. Energy change quantized• f = frequency of radiation
F V
me
+e
r
-e
2
2
( )
12 e
eU r kr
KE m v
= −
=
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Reduced Mass of 2-body system
me
F V
me
+e
r
-eGeneral two body motion under a central force
reduces to
• Both Nucleus & e- revolve around their common center of mass (CM)• Such a system is equivalent to single particle of “reduced mass” µ that
revolves around position of Nucleus at a distance of (e- -N) separationµ= (meM)/(me+M), when M>>m, µ=m (Hydrogen atom)Νot so when calculating Muon (mµ= 207 me) or equal mass charges rotating around each other (similar to what you saw in gravitation)
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Allowed Energy Levels & Orbit Radii in Bohr Model
22
0
20
2 210
2
2
2
2
0
, 1 , 2
Radius of Electron Orbit :
,
1substitute in KE=2 2
1 B
1 0.529 10
Quantized orbits of rotat
ohr Radius
In ge
,....
; 1 , 2,...neral .io
n
n
e
nr
mvr n
a mmk
nvmr
r
kem vr
n
n
n a n
e
e
mka
−
=
⇒ =
=
⇒
= ⇒
= = ∞
= =
= ×
∞
=
n
2
2 2
2
22
2
2
E=KE+U =
Force Equality for Stable OrbitCoulomb attraction = CP Force
Total En
12
2 2
Negative E Bound sy
erg
stemThi
y
s
E = KE+U= - 2
e
e
e
m v
m v e
ekr
m vekr
Kr
r
E k
ekr
=
−
=
⇒
⇒
=
⇒ much energy must be added to
the system to break up the bound atom
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Energy Level Diagram and Atomic Transitions
2
2 20
2
2
2
20
2
2 20
2
2 20
2
i
0
2since , n =quantum num ber
Interstate transition:
1 12
1 1 12
13.6 , 1, 2, 3..2
1 1
2
n
n
n
f
i
n
f
f
f
i
i
i
f
keE K Ur
kefha n n
f kec hca
keE eV na n n
ke
n
r a n
a
n
E h
n
E
n
f E
n
λ
−= = − = ∞
⎛ ⎞−= −⎜
⎛ ⎞= −⎜ ⎟⎜ ⎟
⎝ ⎠⎛ ⎞
= = −
−= + =
=
→
∆
⎜
= = −
⎟⎜ ⎟
⎜⎝
⎠
⎟⎟⎠
⎝
2 2
1 1 = Rf in n
⎛ ⎞−⎜ ⎟⎜ ⎟
⎝ ⎠
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Hydrogen Spectrum: as explained by Bohr
2 2
202n
ke ZEa n
⎛ ⎞= − ⎜ ⎟
⎝ ⎠
Bohr’s “R” same as Rydberg Constant Rderived empericallyfrom spectral series
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A Look Back at the Spectral Lines With Bohr’s Optic
2 2
202n
ke ZEa n
⎛ ⎞= − ⎜ ⎟
⎝ ⎠
Rydberg Constant
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Bohr’s Atom: Emission & Absorption Spectra
photon photon
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Some Notes About Bohr Like Atoms • Ground state of Hydrogen atom (n=1) E0= -13.6 eV• Method for calculating energy levels etc applies to all Hydrogen-
like atoms -1e around +Ze– Examples : He+, Li++
• Energy levels would be different if replace electron with Muons• Bohr’s method can be applied in general to all systems under a
central force (e.g. gravitational instead of Coulombic)
1 2 1 2If change ( )
Changes every thing: E, r , f etc"Importance of constants in your life"
Q Q M MU r k Gr r
= →