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:

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E = e2 /b ~ 105 eV >> Tsun ~ 100 eV

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= 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:

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d

dEe−E /T −c /E1/2

= 0

EGamow = (c T /2)2 /3

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∝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

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E

E||

E +E||

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⊥€

⊥

time

No exchange of parallel andcyclotron energy

Cyclotron freq. c >> all other dynamical frequencies

Energy E of cyclotron motion is an adiabatic invariant

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⊥

Low parallel energy (strongly-magnetized collision):

0.5 1 1.5 2 2.5 3 3.5 4

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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.

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⊥

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

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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

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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

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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:

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⇒e2

T κ 2 /5<< a⇒ << 2/5

Rate is still given by = f() o (Proof: seeDubin, PRL in press)

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e2

EGamow<< a interparticle spacing

II. Pycnonuclear regime: > 2/5 : theory TBD

1

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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

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=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

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⊥

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)