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Modern Physics (PHY 251)
Lecture 19
Joanna Kiryluk
Fall Semester Lectures
Department of Physics and Astronomy, Stony Brook University
• Models of an Atom
Reminder
Models of the Atom
Ernest Rutherford – “the father of nuclear physics”
§ 1911: Rutherford proposed the existence of
a massive nucleus as a small central part of
an atom
J.J. Thompson’s
Plum Pudding Model (1904)
Rutherford’s
Planetary Model (1911)
1
Reminder
Rutherford experiment
§ Rutherford designed an experiment to use a particles as atomic
bullets (then new technology).
§ The “gold-foil experiment” was performed in 1909 by Hans Geiger
and Ernest Marsden (under Rutherford’s direction)
2
Reminder
Rutherford scattering - Differential Cross Section
Here a nucleus = point like particle (i.e. no structure)
dN dσ
=
Nα ⋅ n,
dΩ dΩ
2
2 &2
#
&
#
dσ
1
zZe
1
=%
×
( ×%
(
dΩ $ 4πε 0 ' $ 2Mv 2 ' sin 4 (θ 2 )
Coulomb interaction
dA ( r sin θ ) ×
dΩ = 2 =
r
dΩ = 2π sin θ dθ
In the spherical
coordinate system
3
Reminder
Result: Rutherford model of Atom
Rutherford type of experiment has been applied with higher
energy beams to study structure of nuclei and nucleons
4
Probing Nuclei Structure
with High Energy Electron Beams
Elastic scattering: e− + A
p → e− + pA
R. Hofstadter, Department of Physics, Stanford University, Stanford, CA,
"Electron Scattering and Nuclear Structure", Rev. Mod. Phys. 28, 214–254 (1956)
Robert Hofstadter
Nobel Prize 1961
Electrons are point like
(elementary), sensitive to
charge only, can be accelerated
to high energies.
http://www.nobelprize.org/nobel_prizes/physics/laureates/1961/hofstadter-lecture.pdf
5
Form Factors and Charge Density
Robert Hofstadter
Results of electron scattering off various
nuclei charge density distributions.
*
" dσ %
" dσ %
2 2
$
' =$
' ⋅ F (q )
# dΩ &exp # dΩ &Mott
Charge distribution (nuclei or nucleon)


ρ ( x ) = Ze ⋅ f ( x )
and its Fourier transform

F (q ) =
∫e

iq⋅ x/
http://www.nobelprize.org/nobel_prizes/physics/laureates/1961/hofstadter-lecture.pdf

f ( x) d 3x
6
Even higher electron beams in scattering experiments revealed
that nucleons (protons, neutrons) are not point-like:
they consist of point like particles (quarks, and gluons)!
Nucleon Structure
nucleon
Small distances ….
9
How? Scattering experiment (Rutherford, again …)
Use high energy beam of particles to look inside the nucleon
Extra
Even higher electron beams in scattering experiments revealed
that nucleons are not point-like:
they consist of point like particles (quarks, and gluons)!
Parameterization of the measured lepton-nucleon cross section by ”structure functions” / “form
factors” and learning about nucleon structure from these structure functions.
Nobel Prize 1990
Extra
Even higher electron beams in scattering experiments revealed
that nucleons are not point-like:
they consist of point like particles (quarks, and gluons)!
Form factors,
Extra
Extra
Number of quarks: e+e- scattering
http://hyperphysics.phy-astr.gsu.edu/hbase/particles/qevid.html
Extra
Standard Model of Fundamental Interactions
(Elementary Particle Physics)
electron
u u
d quarks
proton
Hydrogen atom
Extra
Next step: Electron-Ion Collider (EIC)
Research done at Stony Brook U.
EIC R&D prof. A. Deshpande, prof. T. Hemmick (Exp. Nuclear Physics)
https://www.bnl.gov/rhic/eic.asp
16
Models of the Atom
Ernest Rutherford – “the father of nuclear physics”
§ 1911: Rutherford proposed the existence of
a massive nucleus as a small central part of
an atom
J.J. Thompson’s
Plum Pudding Model (1904)
Rutherford’s
Planetary Model (1911)
13
The Problem of Stability of Atoms
Size of the atoms
Continuous spectrum of radiation
(will come back to this later)
14
De’Broglie to rescue:
Source: C. Rogers
15
De’Broglie to rescue:
Source: C. Rogers
16
De’Broglie to rescue:
A.
B.
C.
D.
E.
1
5
10
20
Cannot determine from picture
Source: C. Rogers
17
De’Broglie to rescue:
Source: C. Rogers
18
Next
Source: C. Rogers
19
The Problem of Stability of Atoms
Size of the atoms
Continuous spectrum of radiation
Atomic Spectra
Atomic Spectra
A single element makes emission and absorption lines at the same wavelength
Spectral lines are fingerprints of atoms
(astronomy students: Kirchhoff’s laws)
Extra material
A single element makes emission and absorption lines at the same wavelength
Spectral lines are fingerprints of atoms
28
Atomic Spectra
Source:
Electric discharge passing through a monoatomic gas, for example hydrogen
Atomic Spectra
Emission spectrum
l
(wavelength)
e.g. hydrogen spectrum
Electromagnetic radiation emitted by free atoms is concentrated
at a number of discrete wavelengths.
Application: spectroscopy
Atomic Spectra
e.g. hydrogen spectrum
Electromagnetic radiation emitted by free atoms is concentrated
at a number of discrete wavelengths.
Balmer (1885) Hydrogen:
o
series limit: λ = 3646 A
κ = 1 λ = RH (1 2 2 −1 n 2 )
reciprocal wavelength
RH =
Rydberg constant
for hydrogen (H)
Hα : n = 3
Hβ : n = 4
Hγ : n = 5
etc
n = 3, 4, 5,....
n2 " o %
λ = 3646 2
A'
$
n − 4# &
http://en.wikipedia.org/wiki/Physical_constant
Atomic Spectra
RH =
κ = 1 λ = RH (1 m 2 −1 n 2 )
n = m +1, m + 2,....
Atomic Spectra
For alkali element atoms (e.g. Li, Na, K)
" 1
1 %
'
κ = 1 λ = R$
−
2
2
$# ( m − a ) ( n − b) '&
For the particular series:
m − fixed, n − variable
a, b − constants
33
Rydberg constant for the particular element
Next we will show how Bohr’s model of the atom
explains discrete emmission/absorption lines
§ Frequency of the electromagnetic radiation emitted when the electron
makes a transition from the quantum state ni to the quantum state nf
Bohr’s Postulates
Bohr
§ postulated that classical radiation theory did not hold for
atomic systems
§ applied Planck’s ideas of quantized energy levels to orbiting
electrons and postulated that:
electrons in atoms are confined to stable, non-radiating energy
levels and orbits (stationary states)
§ applied Einstein’s concept of the photon to arrive at an
expressions for the frequency of the light emitted when
electron jumps from one stationary state (i) to another (f)
ΔE = Ei − E f = hν
§ postulated that the electron orbital momentum is quantized
L = n
Bohr’s Model of the Atom
§ Justification of Bohr’s postulates: comparison with
experimental observations
§ Assumptions: Single electron atom of charge Ze ,
me << M
§ Stability of the electron (classical)
1
Ze 2
v2
F=
× 2 = me = ma
4πε 0 r
r
§ Orbital angular momentum
Centripetal acceleration
keeps the electron in its
circular orbit
  
"π %
L = L = r × p = mervsin $ ' = merv
#2&
L = merv=n
e
#2&
e
L = merv=n
1
Ze 2
v2
F=
× 2 = me = ma
4πε 0 r
r centripetal acceleration
Optional (classical mechanics)


F = ma
a) planar motion
φ
dθφ •
ω=
=φ
θ
dt
φ
φ
d 2φ
θ ••
α = 2 = θφ
dt
Bohr’s Model of the Atom
§ Bohr orbits
2
4
πε

2
0
r ≡ rn = n 2 ×
≡
n
a0
2
me Ze
Ze 2
1
v ≡ vn =
×
4πε 0  n
where n =1,2,3, …. – quantum number
Z = 1, n = 1:
c ≈ 197eV ⋅ nm = 197MeV ⋅ fm
e2
1
α em ≡
≈
4πε 0 c 137
o
r1 ≡ a0 ~ 0.053nm = 0.53 A
v1 ~ 2.2 ×10 6 m / s ~ 0.01c non-relativistic
Bohr’s Model of the Atom
§ Total energy of an electron moving in one of the orbits
E = Ekin + E pot
∞
Ze 2
Ze 2
E pot = − ∫
dr = −
2
4πε 0 r
r 4πε 0 r
(E
pot
#r→∞
##
→ 0)
me Z 2 e 4
1
E ≡ En = −
× 2
2
2
( 4πε0 ) 2 n
Quantization of the orbital angular momentum of the electron
leads to a quantization of its total energy
Hydrogen atom
13.6
Z = 1 ⇒ En = − 2 eV
n
Bohr’s Model of the Atom
§ Electron energy levels
13.6
Z = 1 ⇒ En = − 2 eV
n
Excited states: n >1
http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html
Math
Reminder
Spherical coordinate system
r ∈ [0, ∞)
φ ∈ [0, 2π], θ ∈ [0, π]
Transformations between the cartesian and spherical coordinates:
Math
Spherical coordinates, solid angle
Reminder
A
Ω≡ 2
r
W is the same for these
three surfaces:
The differential solid angle:
dA ( r sin θ ) × ( rdθ ) × dφ
dΩ = 2 =
r
r2
dΩ = sin θ dθ dφ
rsinq
r
rdq
(rsinq)df
Math
Spherical coordinates, solid angle
Reminder
A
Ω≡ 2
r
dΩ
dΩ
r
dΩ
The differential solid angle:
dA ( r sin θ ) × ( rdθ ) × 2π azimuthally symmetric case
dΩ = 2 =
integrated over the azimuthal angle f
r
r2
dΩ = 2π sin θ dθ