[PDF] - Physics 2D Lecture Slides Lecture 14: Feb 3 rd 2004 Vivek Sharma PDF Document

SLIDE 1

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

SLIDE 2

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

r

+

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

SLIDE 3

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)

SLIDE 4

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 = R

2

particle trajectory is hyperbolic Scattering angle is related to impact par. Impact Parameter cot 2 kq Q b m v

α α α

α θ ⎛ ⎞⎛ ⎞ = ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠

SLIDE 5

Rutherford Scattering: Prediction and Experimental Result

2 2 4 2 2 2 4

1 4 ( / 2) 2 k Z e NnA n R m v Sin

α 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 1 Kinetic 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 v

α α β

α α ∝

( )( )

Al

2 15 2 15

10

2

For K =7.7.MeV, Z 13 2 ial of the Size of Nucleus = 10 Siz Nucleus: 2 1 K = 8 2 4.9 e of Ato m = 1 10 2 kZ Ze e m v MeV k e r m K kZe r K m m r

α α α β α α −

= ⇒ = = × = = ⇒ = nucleus nucleus

SLIDE 6

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

SLIDE 7

Continuous & Discrete spectra of Elements Visible Spectrum of Sun Through a Prism

SLIDE 8

Emission & Absorption Line Spectra of Elements Kirchhoff’ Experiment : “D” Lines in Na

D lines darken noticeably when Sodium vapor introduced Between slit and prism

SLIDE 9

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 2 2 1 2 1

M M F= G k r r Q Q ⇒

Laws of E&M destroy this equivalent picture : Why ?

SLIDE 10

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

e U 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

SLIDE 11

Allowed Energy Levels & Orbit Radii in Bohr Model

2 2 2 2 2 2 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 e k r m v e k r K r r E k

e k r

= − = ⇒ ⇒ = ⇒ much energy must be added to the system to break up the bound atom

2 2 2 2 2 10 2 2 2 2

, 1 ,2 Radius of Electron Orbit : , 1 substitute in KE= 2 2 1 B 1 0.529 10 Quantized orbits of rotat

hr Radius

In ge ,.... ; 1 ,2,... neral . io

n n e

n r mvr n a m mk n v mr r ke m v r n n n a n e e mk a

−

= ⇒ = = ⇒ = ⇒ = = ∞ = = = × ∞ =

n

Energy Level Diagram and Atomic Transitions

2 2 2 2 2 2 2 2 2 2 2 2 2 2 i

2 since , n =quantum number 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

ke E K U r ke f ha n n f ke c hca ke E 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 i

n n ⎛ ⎞ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

SLIDE 12

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 2

2

n

ke Z E a n ⎛ ⎞ = −⎜ ⎟ ⎝ ⎠ Another Look at the Energy levels