Measurement of Screening Enhancement to Nuclear Reaction
Rates using a Strongly-Magnetized, Strongly-Correlated Nonneutral Plasma
Dan Dubin, UCSD
Experimental collaborators:
John Bollinger, Marie Jensen NIST Boulder
Supported by the NSF/DOE partnership
An example of
High Energy Density Physics
at
Low Energy Densities
How can a nonneutral plasma have anything to do with nuclear reaction rates??
Nonneutral plasma: collection of charges of like sign :
eg. pure ion plasma (Be+)
pure ion plasmas can be confined for days in the static electric and magnetic fields of a Penning trap
B ~ 4 TeslaE ~ 10Volt/cm ~ 30 kHzn ~ 108 cm-3
T ~ 0.001K - 104 K
Nuclear reactions are NOT happening.
But something analogous to nuclear reactions IS happening!
Nuclear reactions in the sun
Reaction rate Required distance of closest approach b ~ a few Fermi, ~10-12 cm (nuclei tunnel the rest of the way through the Coulomb barrier)
Relative Energy E required for close encounter:
€
E = e2 /b ~ 105 eV >> Tsun ~ 100 eV
€
= dE1
Te−E /T σ (E)∫
E
e-E/T (E) ~ e-c/E
EGamow
1/2
[ c 2 ~ Nuclear Rydberg ~ 105 eV]
Gamow peak:
€
d
dEe−E /T −c /E1/2
= 0
EGamow = (c T /2)2 /3
€
∝e−3EGamow / 2T
is dominated by superthermal nuclei with E >> T
Bethe (1939), Gamow and Crutchfield (1949), …
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Ion-Ion Collisions in a strong magnetic field
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2 4 6 8 10
10
20
30
40
50
E
E||
E +E||
€
⊥€
⊥
time
No exchange of parallel andcyclotron energy
Cyclotron freq. c >> all other dynamical frequencies
Energy E of cyclotron motion is an adiabatic invariant
€
⊥
Low parallel energy (strongly-magnetized collision):
0.5 1 1.5 2 2.5 3 3.5 4
10
20
30
40
50
60
E
E||
E +E||
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⊥€
⊥
Higher parallel energy:
time
Adiabatic invariant is broken in close collisions
B
B
Release of cyclotron energy requires close collisionsto break the adiabatic invariant :
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b /v|| <~1/Ωc
b
Collision timescale
B
Higher parallel energy collision:
So K is internal energy, like nuclear energy.
€
⊥
or
Close collisions release this energy€
bΩc /v|| <~1
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bΩc /v|| >>1In cold, strongly-magnetized plasma, most collisions have Only superthermal ions release the cyclotron energy
Adiabaticity parameter
Equipartition rate of cyclotron temperature T and parallel temperature T is analogous to nuclear reaction rate:
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dT⊥dt
= ν (T − T⊥)
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= dE||1
Te−E|| /T σ (E||)∫
E||
e-E /T (E||) ~ e-c/E
EGamow
3/2
|| ||
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~ e−(π / 2) bΩc / v|| = e−π Ωce2 /mv||
3
€
d
dEe−E /T −c /E3/ 2
= 0
EGamow = (3 c T /2)2 /5
= T(3π κ / 32)2 /5, κ ≡ b Ωc / v = Ωce2 /mv 3
€
c = π Ωce2 m /2 /4
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∝e−5EGamow / 3T = e−2.044κ 2/5
O’Neil + Hjorth ‘85
mean adiabaticity parameter
Theory and experiment for equipartition rate(measured on pure electron plasma)
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~ e−2.044κ 2/5
Beck, Fajans and Malmberg Phys. Plasmas ‘96, Glinsky, O’Neil and Rosenbluth Phys. Fluids B ‘93
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=b Ωc / v
= e2Ωc /mv 3
€
B = 6.31 Tesla,
n = 8 ×108 cm−3
What effect does Debye screening have on the rate (nuclear or equipartition)?
Debye screening decreases energy required for a given distance of closest approach b
Debye screening:
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E|| = e2e−b /λ D /b
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E|| = e2 /bNo screening:
less energy needed to get the same differential rate
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= dE||∫ e−(E||−e2 /λD ) /T
Tσ (E||)
= ee2 /(λ DT||)ν o
enhancementfactor f
rate for no shielding
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= dE||1
Te−E|| /T σ (E||)∫
≡ ν o
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≅e2(1− b /λ D ) /b, b << λ D
= e2 /b − e2 /λ D
Salpeter ‘55
Screening Enhancement factor f for equipartition is identical to enhancement factor for nuclear reactions
Release of cyclotron energy in a close collision of guiding centers is analogous to release of nuclear energy in close collision of nuclei
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f = f (Γ) = ee2 /λ DT|| = e 3Γ3/2, assuming Γ = e2 /(aT) <<1
Both nuclear and equipartition rates are enhanced by screening: because close collisions are more probable when they are screened
Eg. in solar interior: n~1023 cm3 T ~106 K ~0.1, f ~1.05
>>1 in a white dwarf, a giant planet interior, or a nonneutral plasma:
f is very large (Salpeter and van Horn, 1969) and has never been verified experimentally
Ichimaru and Iyetomi:
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ln f (Γ) =1.148Γ − 0.00944Γ lnΓ − 0.000168Γ(lnΓ)2
DeWitt and Slattery:
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ln f (Γ) =1.056299Γ +1.039957 Γ0.323064 − 0.545823lnΓ −1.13232
I. Strong shielding regime: close collisions still dominate:
€
⇒e2
T κ 2 /5<< a⇒ << 2/5
Rate is still given by = f() o (Proof: seeDubin, PRL in press)
€
e2
EGamow<< a interparticle spacing
II. Pycnonuclear regime: > 2/5 : theory TBD
1
10
100
1000
104
105
0 2 4 6 8 10
strong shielding enhancement factor
IchimaruDeWitt
f
Rate enhancement due to screening is huge at large ,Predictions for it differ (dynamical screening controversy: J. Bahcall 2002)f has never been tested experimentally in the strong shielding or pycnonuclear regime.
MD Simulations of equipartitioncan measure the rate enhancement factor f()
N=200 ions, c/p = 12.4. Parameters chosen so that =1.25/T
Start with T >> T. Increase T instantaneously, twice.
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⊥
Rapid equipartitionwhen T ~ 0.2
10-7
10-6
10-5
0.0001
0.001
0.1 1 10
/
c
T
€
=dT⊥/dt
T − T⊥
10-7
10-6
10-5
0.0001
0.001
0.1 1 10
/
c
T
o
10-7
10-6
10-5
0.0001
0.001
0.1 1 10
/
c
T
= f() o
Simulation with T < T
0.001
0.01
0.1
1
0 1 104 2 104 3 104
time
ex80-86
parallel temperature T
cyclotron temperature
=.25/T=42.4/T3/2
€
⊥
As T decreases, decreases and equilibration is suppressed
Measured equipartition ratefor several simulations:
o : theory for 2-body equipartition rate
=foc/p=12.4
= 1.25/T, = 42.4/T3/2
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=dT⊥/dt
T − T⊥
1
10
100
1000
0 1 2 3 4 5 6
measured enhancement factors for different simulations
Ichimaru
DeWitt
f ()
f = /o
Dubin, PRL in press
Experimental evidence of enhanced equipartition
Laser-cooled Be+ ion cloud,initial T~ 0.001 K.
Ion-neutral collisionsCauses slow heating
At time t=0 turn off laser cooling.
Pure Be+
Dirty cloud, dark ions(BeH+ etc)
Rapid heating in a dirty cloud
Parallel Temperature jumpdue to coupling to hot cyclotron motion
of dark ions
~ 1-10 hertz ~ 1010 0
Marie Jensen et al. PRL in press
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Proof that heating step is due to to dark ion cyclotron motion
Add rf noise to trap electrode at dark ion cyclotron freq.
Parallel energy is heated resonantly but only when T is sufficiently large
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T at 1 sec T(t)