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


3/17

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


5/17

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


13/17


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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

Einstein Models of Solids

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