phys 342 - lecture 19 notes - f12
TRANSCRIPT
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Lecture 19
Hydrogen-Like Atoms
Atoms with all but one electrons stripped
Energy level:
En
13.6eVn
2 Z
2
Bohr radius:
rn
n2a0
Z
Bohrs theory can be adapted to other hydrogen-like systems.
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Lecture 19
Many-Electron Systems
Bohrs shell hypothesis:
Electrons in an atom are arranged in shells.
Each shell is associated with each of the principle quantum
numbers n, with its radius increasing as n2 and its energy
decreasing as n-2 .
There can be no more than 2n2 electrons in each shell.
Electrons fill the innermost shell first and then outer shells.
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Lecture 19
Shells have letter names:
K shellfor n= 1
L shellfor n= 2
Energy Diagram and X-rays
The atom is most stable in its ground state.
When it occurs in a heavy atom, the
radiation emitted is an x-ray. It has the
energyE (x-ray) =Eu!E".
If there is a vacancy in an inner shell, an
electron from higher shells will fill thevacancy at lower energy.
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Lecture 19
X-ray Lines
K
K
Vacancies may be created by collisional
or photo ionization.
The x-rays have names:
L shell to K shell: K!x-ray
M shell to K shell: K"x-ray
Moseley found this relation
holds for the K#x-ray: f1/2 An(Z bn )
Kseries: due to transitions from n>1 to n=1, b1=1
Lseries: due to transitions from n>2 to n=2, b2=7.4
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Lecture 19
Moseley Plot
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Lecture 19
Wave and Particle Duality
de Broglies hypothesis:
A photon, quantum of light, has all the properties of a particle
of normal matter. This is the wave-particle duality of light.
The wave properties and particle properties of light are relatedin the following way:
p
h
h
Ef == !,
The relation above is a perfectly general one, applying
to radiation and matter alike.
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Lecture 19
Plane Waves
For a wave that propagates alongx-axis, we have
Wave equation:2
2
22
21
tvx !
!=
!
! ""
!"#
$%&
+')
*,
= -.
/0T
txAtx 2sin),(Solutions:
Definitions:
period:T
initial phase:!
wavelength:"=vT
frequency:f=1/T
wave number: k=2#/"
angular frequency: $=2#/T
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Lecture 19
Phase Velocity
Alternative expression for the plane wave:
[ ]!"# ++= tkxAtx sin),(in the negativexdirection:
[ ]!"# +$= tkxAtx sin),(in the positivexdirection:
more definitions:
phase of the wave: !" +=# tkx
phase velocity: the velocity of a point that moves with awave at constant phase.
kTvvphase
!"==#
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Lecture 19
Wave Description of a Particle
Consider a particle of rest mass m0that moves with a velocity v.
according to de Broglieshypothesis, there is a characteristic
frequency (or wavelength)f0(or "0) associated with the particle,
and, in the reference frame of the particle, we have
2
00 cmEhf ==
Then, in the same reference frame, it is possible to write theequation of a stationary vibration associated with the particle
(internal vibration), e.g.,
002sin tf!" =
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Lecture 19
Wave in the Laboratory Frame
Using Lorentz transformation to convert the internal vibration
to a reference frame at rest (with respect to which the particle
travels), i.e.,
$& '= 20 cvx
tt (
we have
!#
$& '(
$%& '==
w
xtf
cvxtftf
)
*))+
2sin
2sin2sin2000
where
v
cw
ff
2
0
=
= !
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Lecture 19
De Broglies Hypothesis Recovered
vm
h
cmv
hc
hfv
hc
fv
c
f
w
02
0
22
0
22
0
22
1
11
!
"
""#
=
$
=
$
=
$
==
p
h=!"
h
mv
v
c
h
p
v
cwf
22
===
! h
Ef =!
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Lecture 19
An Alternative Approach
From the point of view of an observer (at rest), however, we have
the following scenarios:
The mass of the particle mis greater than its rest mass, so
the characteristic frequency is now given by
2
2
0
1 !"=
cmhf 0
2
0
1
ff
f !"
=
#
=
Decrease of the frequency of the internal vibration, due totime dilation:
0
0
1
Tf =
2
02
0
1 1
1/
1!
!"=
"
= fT
f
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Lecture 19
de Broglies Solution
In the observers frame of reference, a wave is described by
tf
w
v
ft
w
vttf
w
x
tf
12sin
12sin
2sin
2sin
!
!
!
!"
#$%&
'()
*=
+,-
./0
$%&
'() *=
$%&
'()
*=
( )
2
22
2
1 11
c
v
w
v
ffw
v
f
==
!==
"#$
%&'
!
(
(
2cvw=!