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8/10/2019 Einstein Models of Solids
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Einstein Model ofSolids
July 26, 2010
Chapter 12, Section 1-2
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Statistics Why
Does heat flow from hot cold
Does a ball lose energy as it bounces Why doesnt
An ice cube heat up a room temperature drink
A ball go higher with each bounce
These are physically possible
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Reversibility Some things look silly when played
backwards
A puddle freezing into an ice cube A person flying out of the pool onto a diving board
Other things dont
Elastic collisions
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Einstein Model of Solids Each atom in a solid is
attached to its
neighbors by springs
Seems valid, atoms do
vibrate
There is 1 pair of
springs per dimension
A 3-D oscillator Fairly complicated to
solve for motion in 3-D
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Energy and 3-D Oscillator Recall
So 1 3-D oscillator is actually 3 1-D oscillators
Much simpler
Energy is quantized
1 quanta =
2222
2222
zyx
zyx
ssss
pppp
22
2
2
22
22
2
1
22
1
22
1
2
2
1
2
zsz
ys
y
xsx
sspringvibvib
skm
psk
m
psk
m
pE
skm
pUKE
a
s
m
k
0
0
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Effective Spring Constant Recall from earlier:
Youngs Modulus allows the calculation of
the interatomic spring constant
How does this constant compare to
the constant from the Einstein Model?
2 springs per direction *2
Each spring is length *2
d
kY is,
ises kk ,, *4 a
is
quantam
kE
,4
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Clicker Question #1 How many oscillators are in a 3-D system
with 100 atoms?
A) 100 B) 150
C) 200
D) 300
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Distributing 4 Quanta Look at 1 atom (3 oscillators)
How can energy be split among these oscillators?
Similar to splitting 4 pieces of candy among 3 children
There are 3 ways to give 1 child all the candy
400Set 3
040Set 2
004Set 1
Child 3Child 2Child 1
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More 4 quanta You dont need to give everything to 1 child
Give 3-1-0:
Or 2-2-0, 2-1-1:
There are 15 different ways to split up 4 things
among 3 people
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Microstates and Macrostates Microstate
A particular arrangement on q-quanta among
n-oscillators Ex: Child 1 gets 4, all others get none
Macrostate
The collection of microstates that all have
q-quanta All microstates with the same total energy
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Fundamental Assumption Al l microstates are equally l ikely
Given enough time and random trials each of the
possible arrangements of quanta would occur
1/15th of the time
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4 Quanta and 2 Atoms 6 oscillators (3 per
atom)
Split the quanta between
each atom
Find the number of ways
to distribute them
There are 126 different
microstates A pain to find them all
A formula is needed
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A Formula is Needed With numbered balls
the order is important
12345 54321
The number ofsequences is 5!
In general for m
numbered balls
The number ofsequences is m!
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Still Looking Colored balls the order
is less important
The particular red ball
doesnt matter Ex:
There will be fewer
sequences
!!
!
gr
grr1 r2 r3
g2g1
312123211232121 rggrrrggrrrggrr
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Almost There Quanta and Oscillators
are like the red and
green balls
Quanta (q) dot
Oscillator (N) wall
The final formula is
!1!!1
Nq
Nq
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Large Numbers Factorials are very large
This is only 100 atoms!
Look at it once/sec. How long before you see all 100
quanta in 1 oscillator?
9610*7.1
300,100
Nq
!1!
!1
Nq
Nq
yearst
yearst
st
universe
avg
avg
10
88
96
10
10*17.3
10*12