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CASIMIR FORCES FROM THE VACUUMENERGY, THE ENERGY OF NOTHING.
What is vacuum?
Assume that we manage to pump away all airmolecules in the vacuum chamber.
Do we have a vacuum?
Is the vacuum chamber completely empty?Do we have nothing inside?
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The answer to the first question is yes, we havea vacuum. We have no matter inside.
The answer to the second question is no!
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MATTER –– RADIATION
Sound waves need a medium to exist and topropagate through.
Radiation does not need a medium to exist andto propagate through.
We cannot send a sound wave through theinterior of the vacuum chamber.
We can send light through the vacuum chamber.
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Radiation is a solution to Maxwell’s equations:
∇⋅ =∇⋅ =
∇ × = −
∇ × = +
D
B
EB
H JD
4
0
1
4 1
πρ
∂∂
π ∂∂
c t
c c t
These are rated today as the most popularequations, more popular than Einstein’s
E = mc2.
These four equations describe all electro-magnetism. This is fantastic!!!
Almost all phenomena and interactions weexperience around us have electromagneticorigins.
Advice: When you come to courses inelectromagnetism put some extra effort intothese courses. (TFYY67, 4th year students).(www.ifm.liu.se/~boser/elma/)
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The solutions to Maxwell’s equations in vacuumare so-called electromagnetic-normal-vacuum-modes or photons.
They are so-called massless bosons.
They are in thermal equilibrium with thesurroundings.
The probability that a mode is occupied is
nk TB
εε
( ) ( ) += 1
1exp
So for T ≠ 0 we have photon-modes occupied!They are in thermal equilibrium with thesurroundings
Thus we have photons in the vacuum chamber!
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Now, If we lower the temperature to T = 0,will we still have something inside the vacuumchamber?
The answer is no!!! We have nothing inside.
Has this nothing no properties, can it not be usedfor anything?
It can!!!
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The vacuum is not “dead”! It has properties!
But this has nothing to do with ε0 and µ0 thatappear in SI-units.!!
These are just artefacts of the choice of unitsystem. They have nothing to do with physics.
The Maxwell equations I showed were in cgs-units. There all these obscuring artefacts areabsent.
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The mass-less bosons have the following energy:
E ni ii
= +( )∑ε 12
T = 0 → n = 0, but the energy is not zero!!
What remains is the zero-point energy or energyof vacuum fluctuations.
I am sure you will be told by some teacher thatthis half above will only lead to a constant thatyou can throw away.
If you do, send the teacher to me!
If the normal modes are changed, so is thisconstant!
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1948
is a very important year!
What happened?
Casimir published a famous paper on the forcebetween two neutral, perfectly smooth metalplates.
By assuming that the force was due to thechange in the total zero-point energy of allelectromagnetic modes as a function ofseparation between the plates he managed to getthe simple result:
F d
c
d( ) = − π 2
4240
Note: no material parameters like electrondensity, electron charge, electron mass..
Only the speed of light in vacuum!!
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In a real metal there are also modes that decayexponentially away from the interface, surfacemodes.
These give rise to the van-der-Waals force forsmall separations.
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The result for the attractive potential betweentwo gold plates:
10-19
10-17
10-15
10-13
10-11
10-9
10-7
10-5
0.001
0.1
0.0001 0.001 0.01 0.1 1 10 100 1000
T= 0KT = 300KvdW asymptoteCasimir asymptoteExperiments
|Pot
entia
l| (J
oule
/m2 )
d (µm)
Gold plates
12
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
0.1 1 10
T= 0KT = 300KvdW asymptoteCasimir asymptoteExperiments
|Pot
entia
l| (J
oule
/m2 )
d (µm)
Gold plates
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Two atoms
A dipole on atom 1 will give rise to a field at theposition of atom 2.
Atom 2 will be polarized by this field.
This induced dipole will give rise to a field at theposition of atom 1.
There will be self-sustained fields, normal modes
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6 normal modes
1 2
3 4
5 6
15
0
0.02
0.04
0.06
0.08
0.1
5 10 15 20 25 30 35
Ene
rgy
(Har
tree
)
r (a0)
Li-Li
1
2
3,5
4,6
The interaction energy for the coupled system is
E r n ri
ii
i
= ( ) +[ ] = ( )= =∑ ∑ω ω
1
612
12
1
6
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and the force
Fddr
E r= − ( )
Neglecting retardation we have
V rr
d
r
vdW ( ) = − ( ) ( )
= − ( ) ( )+
−∞
∞
∫32
1
32
0 0
6 1 2
1 26
1 2
1 2
πωα ω α ω
α α ω ωω ω
' '
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Let us now take retardation effects into account.
We arrive at
V rr
d i i e
r c
CPr c( ) = − ( ) ( )
× + (
∞−∫π
ωα ω α ω
ω
ω6 1
02
2
3 6 )) + ( ) + ( ) + ( )
5 22 3 4ω ω ωr c r c r c
which is the so-called Casimir-Polder potential
10-29
10-24
10-19
10-14
10-9
10-4
101 102 103 104 105
Pote
ntia
l (H
artr
ee)
r (a0)
Li-Li
ELi
= 0.06859 a.u.
αLi
(0) = 164.8 a.u.
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In summary so far:
The electromagnetic vacuum-normal-modes andelectromagnetic surface-normal modes give riseto a force between atoms, molecules, and allobjects.
The relative importance of these forces increaseswhen the objects become smaller.
The hot research field now is nano-science. Herethese effects are very important.
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More effects:
E ni ii
= +( )∑ε 12
1) One can manipulate the forces by changing the occupation numbers.
A couple of years ago there was a row in the media. I happened to calculate the force between two blood cells in presence of micro-wave radiation from cell phones.
I found it was enhanced by 10 orders of magnitude.
2) If we break a piece of solid in two we create two fresh surfaces
In doing so we create new electromagnetic surface-normal-modes. This costs energy.
This energy is the so-called surface energy.
3) If we split a liquid in two we also create new surface area. The surface modes give rise to thesurface tension.
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4) Capillary force.
5) Composites. The enhancement in strength comes from the surface modes.
6) Stability of colloids.
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Can mobile phones be dangerous?
Some years ago I calculated the van-der-Waals–force between two blood cells in thepresence of mobile phone radiation.
This radiation is in the micro wave region. Sametype as in microwave ovens.It makes the water molecules rotate.
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One excites the modes (increases n) I found that the force was very weak from thestart but was strongly enhanced by the radiation
I suggested that one should look for a tendency ofthe cells to stack like coin rolls
F1
F2 > F1
F1
F2 F2
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Now one has found just this!
German students have studied blood samplestaken from the ear and the finger before and afterthe person has been talking on a mobile phone.
Blood sample from the ear taken just beforemobile phone conversation:
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Blood sample from the ear taken after 20 secondsof mobile phone conversation:
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Blood sample from the ear taken 10 minutes aftermobile phone conversation:
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Blood sample from the finger taken just aftermobile phone conversation:
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Blood sample from the finger taken 10 minutesafter mobile phone conversation:
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Summary
All objects have electromagnetic fields inside andbound to the surface
Electromagnetic fields are important part ofnature
The bulkmodes affects the stability of objects
The surface modes give rise to surface energy,surface tension and van-der-Waals forcesbetween objects
The vacuum modes give rise to Casimir forcesbetween objects and are believed to affect theexpansion of the universe.
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If you become interested in the normal-modeapproach to interactions read more in my Book:
Bo E. Sernelius, Surface Modes in Physics,Wiley-VCH, Berlin 2001.
