Post on 18-Mar-2018
transcript
Brian Wecht, the TA is back!
Pl. give all regrade requests to him
Quiz 4 is This Friday
Physics 2D Lecture Slides Lecture 14: Feb 3rd 2004
Vivek Sharma
UCSD Physics
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(atom is neutral)
• 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 +
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
Probing Within an Atom with α Particles
• Most α particles pass thru gold foil with nary a deflection
• SOME (≅10-4) scatter at LARGE angles Φ• Even fewer scatter almost backwards Why
“Rutherford Scattering” discovered by his PhD Student (Marsden)
Rutherford Discovers Nucleus (Nobel Prize)
Force on α-particle due to heavy Nucleus
•Outside radius r =R, F ∝ Q/r2
•Inside radius r < R, F ∝ q/r2 = Qr/R2
•Maximum force at radius r = R2
particle trajectory is hyperbolic
Scattering angle is related to impact par.
Impact Parameter cot2
kq Qb
m vα
α α
α
θ⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
Rutherford Scattering: Prediction and Experimental Result
2 2 4
22 2 41
4 ( / 2)2
k Z e NnAn
R m v Sinα α ϕ∆ =
⎛ ⎞⎜ ⎟⎝ ⎠
•# scattered Vs φ depends on :
•n = # of incident alpha particles
•N = # of nuclei/area of foil
•Ze = Nuclear charge
• Kα of incident alpha beam
•A= detector area
Rutherford Scattering & Size of Nucleus
2
distance of closest appoach r size of nucleus
1Kinetic 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
∝
( )( )
-
Al
215
2
15
-10
2
For K =7.7.MeV, Z 13
2
ial of
the
Size of Nucleus = 10
Siz
Nucleus:
21 K = 8
2
4.9
e of Ato
m = 1
10
0
2
kZ
Ze em v MeV k
er m
K
kZer
K
m
m
r
α
α
α β
α
α
−
=
⇒ =
=
×
=
=⇒
=
nucleus
nucleus
Dimension Matters !
-15
-10
Size of Nucleus = 10
Size of Atom = 10
m
m
•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
Rutherford Atom & Classical Physics
Continuous & Discrete spectra of Elements
Visible Spectrum of Sun Through a Prism
Emission & Absorption Line Spectra of Elements
Kirchhoff’ Experiment : “D” Lines in Na
D lines darken noticeably when Sodium vapor introduced
Between slit and prism
Emission & Absorption Line Spectrum of Elements
•Emission line appear dark because of photographic exposure
Absorption spectrum of Na
While light passed thru Na vapor
is absorbed at specific λ
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 ?
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
( )
1
2 e
eU r k
r
KE m v
= −
=
Reduced Mass of 2-body system
• 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) separation
µ= (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)
me
F V
me
+e
r
-e
General Two body Motion under a central force
reduces to
Allowed Energy Levels & Orbit Radii in Bohr Model
2
2 2
2
22
2
2
E=KE+U =
Force Equality for Stable Orbit
Coulomb attraction = CP Force
Total En
1
2
2 2
Negative E Bound sy
erg
stem
Thi
y
s
E = KE+U= - 2
e
e
e
m v
m v e
ek
r
m vek
r
Kr
r
E k
ek
r
=
−
=
⇒
⇒
=
⇒ much energy must be added to
the system to break up the bound atom
22
0
20
2 210
2
2
2
2
0
, 1 , 2
Radius of Electron Orbit :
,
1substitute in KE=
2 2
1 B
10.529 10
Quantized orbits of rotat
ohr Radius
In ge
,....
; 1 , 2,...neral .
ion
n
e
nr
mvr n
a mmk
nv
mr
r
kem v
r
n
n
n a n
e
e
mka
−
=
⇒ =
=
⇒
= ⇒
= = ∞
= =
= ×
∞
=
n
Energy Level Diagram and Atomic Transitions
2
2 20
2
2
2
20
2
2 20
2
2 20
2
i
0
2
since , n =quantum num ber
Interstate transition:
1 1
2
1 1 1
2
13.6, 1, 2, 3..
2
1 1
2
n
n
n
f
i
n
f
f
f
i
i
i
f
keE K U
r
kef
ha n n
f ke
c hca
keE eV n
a n n
ke
n
r a n
a
n
E h
n
E
n
f E
n
λ
−= = − = ∞
⎛ ⎞−= −⎜
⎛ ⎞= −⎜ ⎟⎜ ⎟
⎝ ⎠⎛ ⎞
= = −
−= + =
=
→
∆
⎜
= = −
⎟⎜ ⎟
⎜⎝
⎠
⎟⎟⎠
⎝
2 2
1 1 = R
f in n
⎛ ⎞−⎜ ⎟⎜ ⎟
⎝ ⎠
Hydrogen Spectrum: as explained by Bohr
Bohr’s “R” same as
the Rydberg Constant
R
derived emperically
from photographs of the
spectral series
2 2
202n
ke ZE
a n
⎛ ⎞= −⎜ ⎟
⎝ ⎠
Another Look at the Energy levels
2 2
202n
ke ZE
a n
⎛ ⎞= −⎜ ⎟
⎝ ⎠
Bohr’s Atom: Emission & Absorption Spectra
photon
photon
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 G
r r= →