chapter 1 correlations in trapped plasma

July 15, 2015 15:59 World Scientific Review Volume - 9in x 6in ”Anderegg Corr les Houches15” page 1

Chapter 1

Correlations in trapped Plasma

Francois Anderegg

University of California San Diego, Physics Dept. 0319,

La Jolla CA 92093, USA

[email protected]

This chapter describes the e↵ects of correlations on trapped plasma.When the potential energy between neighboring ions is larger than thethermal energy, the internal structure of trapped plasma changes, fromplasma to liquid to Coulomb crystal. These Coulomb crystals are sim-ilar to solid state one but the inner particle spacing is of the order of10µm, significantly larger than solid state material. The structure ofthese Coulomb crystal is obtained not from x-ray scattering but fromBragg scattering of UV light (⇠ 300nm). Correlations not only changeequilibrium properties of a plasma but also dynamical properties, in par-ticular this chapter will describe the dramatic increase of the collisionrate in the presence of correlations.

1. Introduction

The thermodynamic state of a one component plasma is determined by itsshape, size, and coupling parameter � :

� =1

4⇡✏0

q2

aws

kB

T(1)

where aws

is the Wigner-Seitz radius:

aws

⌘✓

3

4⇡n

◆ 1

3

(2)

aws

is the radius of a sphere whose volume equal the volume occupiedby one charged particle in a plasma of density n. The coupling parameter� represents the ratio of the potential energy between neighboring charged

1

Francois Anderegg


2 F. Anderegg

particle and the thermal energy. When � > 1 a one component plasma isstrongly coupled. Large value of coupling parameter can be achieved withhigh density n, or large charge per particle q, or low temperature T .

Strongly coupled one component plasmas are believed to exist indense astrophysical objects such as giant planets interior (T ⇠1eV, n ⇠1024cm�3, q = e, resulting in � ⇠ 10), white dwarf stars (T ⇠100eV,n ⇠ 1030cm�3, q = e, resulting in � ⇠ 10).1 Strongly coupled plasmasalso exist in less dense objects if the charge per particle is large, such asdust grain in a dusty plasma2 (T dust ⇠1eV, n ⇠ 1cm�3, q = 104 e, result-ing in � ⇠ 10), or if the temperature is low enough in laser cooled trappedion plasma (T ⇠ 10�5eV, n ⇠ 107cm�3, q = e, resulting in � ⇠ 10). Onesees that these vastly di↵erent physical systems, can all have a couplingparameter � ⇠ 10. Therefore astrophysical e↵ects depending on � can bestudied with laser cooled trapped ion, as will be described in section 3 ofthis chapter.

For coupling parameter � ⌧ 1 trapped ions are in a ”plasma state”, for� > 1 trapped ions are in a liquid state and for � & 172. trapped ions arein a solid state called a Coulomb crystal.

2. E↵ect of Correlations on Equilibrium properties

In a trapped ion plasma consisting of many particles (N � 1), even whenthe correlation parameter � is large, the mean electric potential is largecompared to the potential energy between neighboring ions.

e� � e2

4⇡✏0

aws

(3)

The mean electric potential determines the equilibrium shape of the plasmain the trap. The plasma shape remains essentially unchanged by the pres-ence of correlation, the correlations only change the internal structure ofthe plasma, not its shape.

In the absence of correlation, the ”one” particle distribution function isa Boltzman distribution.

f(r, v) = C exp

✓�1

kB

T[H + !P

✓

]

◆(4)

where the Hamiltonian H is:

H =mv2

2+ e(�

T

+ �p

) (5)


Correlations in Trapped Plasma 3

�T

is the trap potential and �p

is the plasma potential which is determinedby Poisson’s equation:

r2�p

=�en

✏0

(6)

and the canonical angular momentum P✓

in a uniform magnetic field in thez-direction is:

P✓

= mv✓

r +eB

2r2 (7)

In contrast, when correlations are present, the N particle distribution is aGibbs distribution:

f(r1

, v1

, r2

, v2

... rN

, vN

) = C exp

✓�1

kB

T[H(N) + !P

(N)

✓

]

◆(8)

where the N particle Hamiltonian is:

H(N) =NX

i=1

mv2i

2+ e

0

@�T

(ri

) +X

j>i

�ij

1

A (9)

where �T

is the trap potential and �ij

the interaction potential:

�ij

=e2

4⇡✏0

|ri

� rj

| + ”image charge” (10)

�ij

is a Green function when the image charges are included.3 In thischapter we assume that the image charge contribution is negligible, that isthe plasma is far from the wall of the trap.The N particle canonical angular momentum is:

P(N)

✓

=NX

i=1

✓mv

✓i

ri

+eB

2r2i

◆(11)

and the N particles distribution becomes:

f(r1

, v1

, r2

, v2

... rN

, vN

) = C exp

NX

i=1

m

2kB

T

hvi

+ !ri

✓i

i!

+ C exp

0

@�NX

i=1

1

kB

T

2

412

X

i 6=j

e2

4⇡✏0

|ri

� rj

| + e�T

3

5

1

A(12)

Useful quantities to be considered are the reduced distribution; for ex-ample the spatial distributions are obtained by integrating over velocities,and the N -particle spatial distribution is:

⇢(N)(r1

, r2

, ...rM

) =

Zd3r

M+1

...d3rN

f(r1

, ...rN

) (13)


4 F. Anderegg

The density is simply the first reduced distribution:

n(r) = N⇢(1)(r) (14)

The two particles distribution is:

⇢(2)(r1

, r2

) = ⇢(1)(r1

) ⇢(1)(r2

) [1 + g(r1

, r2

)] (15)

where g(r1

, r2

) represents the two body spatial correlations which measurethe extra probability beyond what would be expected of a completely ran-dom distribution of finding particle at r

1

and r2

.Since Coulomb interaction is a binary interaction, only n(r), T and

⇢(2)(r1

, r2

) are necessary to describe thermodynamic functions. No higherorder term are necessary. Figure 1 shows the correlation function g(r

1

, r2

)for a one component plasma for various value of correlation parameter.4

g(r

1,

r 2)!

Fig. 1. Correlation function for one component plasma.

In the absence of correlation, the correlation function is equal to zeroat large distance, and goes to minus one at short distance reflecting thee↵ect of Debye shielding. As the correlation parameter � increases, g(r

1

, r2

)increases at a distance of about 1.7 Wigner-Seitz radius; for distance smallerthan one Wigner-Seitz radius g(r

1

, r2

) decreases, indicating that short rangeorder is appearing in the system. These oscillations are evidence of thebeginning of a crystal lattice.



2.1. Correlation with Small Plasmas

Early numerical simulations5 solved the following equations for N = 400ions:

dxi

dt=

Ei

B⇥ z + U

i

z

dUi

dt=

mEi

ez

Ei

=�@�

@xi

e�(xi

, ..., xN

) =X

i>j

e2

4⇡✏0

|xi

� xj

| +X

i

1

2m!2

z

✓z2i

� ⇢2i

2

◆

(16)

where one can see the potential � is the same as the first term of equation10 and that the trap potential is the standard quadratic potential of aPenning trap. These simulations show that for small plasmas, boundarye↵ect dominates the geometry and inside the plasma ions are organized ina concentric shell structure.

N=400!

Fig. 2. a) Density as a function of spherical radius b) Observation of shell structure in

Penning trap.

The results of these simulations are shown on figure 2a, at low � thenormalized spatial distribution is uniform, in contrast for large � sharp”onion-like shell” are predicted. Figure 2b is a measurement of such struc-ture on a cloud of beryllium ions.6 The number of shells observed dependon the number of ions N , with N = 20, 1 shell is observed and withN = 15, 000, 16 shells are observed.


6 F. Anderegg

2.2. Correlation with Large Plasmas

The influence of the surface is limited for large plasma and their interioris comparable to an infinite size plasma. The lowest energy state for aCoulomb crystalline structure is a Body Center Cubic (bcc) structure fol-lowed closely by a face centered cubic (fcc) and a hexagonal close packed(hcp). These di↵erent structures have very small fractional di↵erence ofCoulomb energies per ion, typically < 10�4. Numerical simulations usinga planar model7 suggest that one need at least 60 inter-particle spacingsto observe bulk behavior; that is N > 105 ions are necessary in a sphericalplasma to exhibit bulk properties. More recently numerical simulations ofspherical plasma8 have suggested that a bcc lattice may be present in smallsections of a plasma with as little as 15,000 ions.

The inner particles spacing of Coulomb crystals is ⇠ 10µm, thereforeBragg scattering can be performed on them with ultraviolet light insteadof x-ray as for solid state crystals. Since Coulomb crystals contain onlya small number of ion compared to solid state crystals, the wavelength ischosen to be resonant with an atomic transition to increase the scatteringcross-section.

Figure 3 shows the NIST trap designed to study large coulomb crystals.

Fig. 3. NIST Cylindrical Penning trap for large crystal.



The inside diameter of the trap is 4.5cm and the magnetic field B =4.5T. In this trap Beryllium ions can be laser cooled down to T < 1mK. TheBragg scattering is done by an axial beam. The direct beam is deflectedto avoid saturating the CCD camera, and crossed polarizers are used tofurther minimize unwanted scattering from the entrance window.9

Fig. 4. Bragg scattering rings and frequency of ring location.

Scattering pictures are shown on figure 4. The location of scatteringrings indicates that the Coulomb crystal structure is bcc. Since the scat-tering picture are accumulated over several plasma rotations, it is not pos-sible to distinguish a single crystal from several smaller crystals in the trap.Similarly x-ray di↵raction on powdered crystal result in di↵raction rings.Locking the crystal rotation to the rotating wall and strobing the camera atthe rotating wall frequency determines unambiguously that a single crystalis in the trap. Figure 7 in the chapter describing the rotating wall techniqueshows a strobed di↵raction pattern corresponding to a bcc crystal.

bcc (111) plane predicted spacing: 14.4 µm measured: 14.6 ± 0.3 µm

bcc (100) plane predicted spacing: 12.5 µm measured: 12.8 ± 0.3 µm

Fig. 5. Real space images of bcc coulomb crystal.

Real space images of the plasma also confirm a bcc crystal. Dependingon the orientation of the probe laser beam the (100) plane or the (111)


8 F. Anderegg

plane can be illuminated as shown on figure 5. The spacing of the ions inthe (100) plane and the (111) plane are in agreement with a bcc crystal.

Summary of Correlation on Equilibrium Properties

For plasmas with N < 104 ions, the boundary e↵ects dominate and shellstructure is observed. For large plasmas, with N > 105 ions, bcc Coulombcrystals are observed in spheroidal plasma.

3. E↵ect of Correlations on a Dynamical property

In this section, we will briefly describe theory results and measurements thatuse non-neutral plasmas to model aspects of the physics of nuclear fusionreactions in dense, correlated plasmas, such as in giant planets, degeneratestars, and laser fusion plasmas. Nuclear reaction rates in dense correlatedplasmas are, according to theory, enhanced compared to rates predicted forreactions at lower density. In the astrophysics community, this theoreticale↵ect is known as the Salpeter enhancement.10 Here we use an analogybetween screened nuclear reactions in dense plasmas and perpendicular-to-parallel energy equipartition in strongly coupled and strongly magnetizednon-neutral plasmas.11,12

In a magnetized plasma (i.e. ⌦c

� !p

) with a perpendicular tempera-ture T? and a parallel temperature Tk, we consider the perpendicular-to-parallel collision rate ⌫?k defined as:

d

dtT? = ⌫?k (Tk � T?) (17)

with

⌫?k ⌘ nvb2 4p2I() g(�) (18)

Here the ”bare” collision rate 4p2 nvb2 is modified by a dynamical factor

I() depending only on the magnetization parameter

⌘p2

b

rc

(19)

and modified by an correlation factor g(�), depending only on the correla-tion parameter �. Where b is the distance of closest approach

b ⌘ e2

4⇡✏0

aws

kB

T(20)



I() suppress the perpendicular to parallel collision in the strongly mag-netized regime of > 1. In this regime only rare energetic collisions mixthe perpendicular energy E? and parallel energy Ek.

The correlation factor g(�) enhances these rare collisions due to particlecorrelations, in the cryogenic liquid regime of 1 < � < 10.

3.1. Strong Magnetization

I will briefly review the strong magnetization regime results, a rigorousderivation can be found in reference 11. For r

c

< b , the perpendicularenergy of two particles is an adiabatic invariant E? = E?1

+ E?2

, that isconserved by most collisions except by rare energetic (large Ek) collision.The cross-section for sharing perpendicular and parallel energy is a functionof Ek:

�(Ek) / exp

�⇡

✓b

rc

◆�⇡ exp

"�⇡

✓Cst

Ek

◆ 32

#(21)

E||!

e-E /T! σ(E||) !||!

∆E

EGamow!

Fig. 6. Graphic description of the integrand of equation 22 showing exponentially decay-

ing Maxwellian particle energy distribution and the exponentially growing cross-section�. The product of the particle distribution and the cross-section gives rise to the Gamow

peak. Shifting � by �E gives enhancement exp(�E/kBT ).

That is the cross-section for Ek � E? sharing increases exponentiallyfor large Ek. The collision rate is obtained by integrating the product ofthe maxwellian particles distribution and �(Ek).

⌫no corr

?k =

ZdEk

1

kB

Texp

✓�

Ek

kB

T

◆�(Ek) = nvb24

p2 I() (22)


10 F. Anderegg

where I() is:13

I() ⇡ 1.5 exp⇣�2.044

25

⌘(23)

The Maxwellian has almost no particles at large Ek and the cross-sectionis large at large Ek as shown on figure 6. Most of the collisions causingsharing of Ek and E? comes from the ”Gamow” peak shown on figure6. For example for = 20 the Gamow peak is located at 4v that is lessthan 2% of the particles participate in such ”rare” collision. The cyclotronenergy is released only by rare collision, analog to fusion reaction where theenergy stored in the nuclei is liberated only by rare energetic collision.

3.2. Correlations

Correlations enhance the perpendicular-to-parallel collision rate throughscreening e↵ects, by reducing the amount of parallel energy required bytwo ions to come within a distance ⇢. In the absence of shielding theenergy required is:

Ek =e2

4⇡✏0

⇢(24)

in contrast, with Debye screening the parallel energy required is smaller:

Ek =e2

4⇡✏0

⇢exp

✓�⇢

�D

◆' 1

4⇡✏0

e2

⇢� e2

�D

�(25)

In other words, screening reduces the parallel energy required to comewithin a distance ⇢. In the strong coupling regime of � > 1, the e↵ec-tive shielding distance is the inter-particle spacing, and the energy requiredfor a collision at distance ⇢ < a

ws

is:

Ek ' 1

4⇡✏0

e2

⇢� e2

aws

�(26)

This shift the cross-section �(Ek) by �Ek = e2/(4⇡✏0

aws

) as shown in fig-ure 6. Equivalently, the Maxwellian in equation 22 is shifted by�Ek/(kBT )= e2/(4⇡✏

0

aws

kB

T ) = �, therefore equation 22 becomes:

⌫corr?k 'Z

dEk1

kB

Texp

✓�

Ek

kB

T� e2

4⇡✏0

aws

kB

T

◆�(Ek)

= exp

✓e2

4⇡✏0

aws

kB

T

◆ZdEk

1

kB

Texp

✓�

Ek

kB

T

◆�(Ek)

= exp(�) ⌫no corr

?k

(27)



Correlation increases the perpendicular to parallel collision rate by a factorof roughly g(�) = exp(�). It is worth noting that the enhancement isindependent of �(E), as in the fusion case, where this e↵ect is known asthe Salpeter enhancement.

3.3. Experiments

To test the theoretically predicted correlation enhancement of the collisionrate, we use a magnesium ion plasma contained in a Penning-Malmbergtrap. The plasma density for this experiment is controlled with a rotatingwall (0.12⇥ 107 n 2⇥ 107cm�3), the temperature (2.5⇥ 10�6 T 1eV) is controlled by laser cooling, the uniform axial magnetic field waschanged in the range of 1.2T B 3T. The adiabaticity parameter isequal to one at a temperature T = 5 ⇥ 10�4eV and B=3T. The couplingparameter is � = 1 for n = 2⇥ 107cm�3. and T = 6⇥ 10�5eV.

The measured perpendicular-to-parallel collision rate is plotted in figure7 for two densities n = 2 ⇥ 107 and 0.12 ⇥ 107cm�3 labeled n

7

= 2 andn7

= 0.12 In both cases the collision rate is strongly suppressed when islarger than unity. The solid lines are theory curves obtained from equa-tion 22 using numerical values of I() from reference13 with no adjustableparameters.

Fig. 7. Measured collision rates for two densities versus temperature (lower axis) and (upper axis) compared to theory with and without correlations. The correlation pa-

rameter � for high and low density is also shown.


12 F. Anderegg

The dashed lines are from equation 27. At lower density the plasmais never strongly correlated, as shown by � which is also plotted for bothdensities. The n = 2 ⇥ 107 cm�3 density labeled n

7

= 2 has � > 1 atlow temperatures, and the measured collisionality is enhanced by severalorders of magnitude over theory neglecting correlations. At low density,the collision rate is measured down to a rate of 1 sec�1, setting an up-per limit on extraneous collisional e↵ects. Figure 8 shows the measuredcollision rate divided by the theoretical rate neglecting correlations, thatis, the Salpeter enhancement g(�). The enhancement depends on � butis independent of , as predicted by theory. The measured correlationenhancement g(�) is consistent with equilibrium shielding theories, and nosupport for alternative dynamic shielding e↵ects is seen.

10-1

101

103

105

107

109

1011

10-4 10-3 10-2 10-1 100 101

Γ

B = 3T n7 = 2

B = 3T n7 = 0.12

ν⊥

//

no c

orr

ν⊥

//

f (Γ

) =

B = 1.2T n7 = 1.8

!!

Fig. 8. Correlation enhancement g(�) of the collision rate versus the correlation pa-rameter �.

This work illustrates how laboratory non-neutral plasmas can be usedto study high energy density plasmas where the same enhancement appliesto rare energetic fusion collisions in hot, dense correlated plasmas such asstars.

Summary of Perpendicular to Parallel Collision Rate

Perpendicular to parallel collision rate are strongly suppressed in the”strong magnetization” regime of > 1. Only rare energetic collisions causeE? to Ek energy exchange. These rare energetic collisions are strongly en-hanced in the correlated liquid and crystal regime. Enhancement of up to109 over uncorrelated theory are observed. The same enhancement applies



to rare energetic fusion collisions in hot, dense, correlated plasmas such asstars.

4. Tutorial Problem

Devise a technique to measure the perpendicular to parallel collision rate⌫?k in a magnetized plasma. Hint: look at equation 17 to see which phys-ical quantities have to be perturbed to measure the collision rate ⌫?k.

A detailed description of two di↵erent techniques is given in reference 12.The first technique directly observes T? and Tk as they relax to a commontemperature. The parallel temperature is initially increased by small oscil-lating voltages applied at one end of the plasma. This direct measurement ispractical only for slow rates (⌫?k < 100sec�1) due to photon counting ratelimitations, and is not accurate for low temperatures (T < 10�4eV) sinceion-neutral collisions give a heating rate which dominates the temperatureevolution.

The second technique also uses small oscillating voltages applied at oneend of the plasma. This acts as a piston and alternatively compress andexpand the plasma in the axial direction. If the oscillation frequency f

osc

islarge compared to the collision rate, the cycle is a reversible process sinceheat is not transfer to the perpendicular degrees of freedom. Similarly, ifthe oscillation frequency is slow compare to the collision rate, the cycle isalso reversible, since heat is fully shared with the perpendicular degrees offreedom. In contrast if f

osc

⇠ ⌫?k, the process is not reversible: more workis done during the compression stroke and the excess appears as plasmaheat. The maximum heat per cycle occurs when 2⇡f

osc

C ' ⌫?k whereC is the specific heat at constant density of a magnetized plasma. In theabsence of correlations C = 1/3; and C increases slightly in the presence ofcorrelations.

Acknowledgements

This work was supported by National Science Foundation Grant PHY-1414570, Department of Energy Grant DE-SC0002451and Department ofEnergy High Energy Density Laboratory Plasma Grant DE-SC0008693.The author thanks the organizers of Les Houches Winter School for theopportunity to contribute to the workshop. The author also thanks Prof.C.F. Driscoll, Prof. D.H.E. Dubin and Prof. T.M.O’Neil for many years of


14 F. Anderegg

enlightening guidance and theoretical support.

References

1. Frontiers in High Energy Density Physics, NRC, The National AcademiesPress, Washington, DC (2003).

2. V.E. Fortov, A.G. Khrapak, S.A. Khrapak, V.I. Molotkov, O.F. Petrov,Dusty plasmas, Physics-Uspekhi, 47, 447-492, (2004). H. Thomas, G. Mor-fill, V. Demmel, J. Goree, B. Feuerbacher, and D. Mhlmann, Plasma Crystal:Coulomb Crystallization in a Dusty Plasma, Physical Review Letters 72, 652-656, (1994).

3. D.H.E. Dubin and T.M. O’Neil, Trapped Nonneutral Plasmas, Liquids, andCrystals (The Thermal Equilibrium States), Rev. Mod. Phys. 71, 87-172,(1999). T.M. O’Neil and D.H.E. Dubin, Thermal Equilibria and Thermody-namics of Trapped Plasmas with a Single Sign of Charge, Phys. Plasmas 5,2163-2193, (1998).

4. S.G. Kuzmin and T.M. O’Neil, Numerical Simulation of Ultracold Plasmas,Phys. Plasmas 9, 3743 (2002).

5. D.H.E. Dubin and T.M. O’Neil, Computer Simulation of Ion Clouds in aPenning Trap, Phys. Rev. Lett. 60, 511 (1988). D.H.E. Dubin, E↵ect of Cor-relations on the Thermal Equilibrium and Normal Modes of a NonneutralPlasma, Phys. Rev. E 53, 5268 (1996).

6. S.L. Gilbert, J.J. Bollinger, and D.J. Wineland, Shell-Structure Phase ofMagnetically Confined Strongly Coupled Plasmas, Phys. Rev. Lett. 60, 2022-2025 (1988).

7. D.H.E. Dubin, Correlation Energies of Simple Bounded Coulomb Lattices,Phys. Rev. A 40, 1140 (1989).

8. Hiroo Totsuji, Tokunari Kishimoto, Chieko Totsuji, and Kenji Tsuruta, Com-petition between Two Forms of Ordering in Finite Coulomb Clusters, Phys.Rev. Lett. 88, 125002, (2002).

9. W.M. Itano, J.J. Bollinger, J.N. Tan, B. Jelenkovic, X.-P. Huang, and D.J.Wineland, Bragg di↵raction from crystallized ion plasmas, Science 279, 686-689 (1998).

10. E. E. Salpeter, Electron Screening and Thermonuclear Reactions, Aust. J.

Phys. 7 , 373 (1954); E. E. Salpeter and H. Van Horn, Nuclear ReactionRates at High Densities, Astrophys. J., 155, 916 (1969).

11. D.H.E. Dubin, Modeling Nuclear Fusion in Dense Plasmas Using a CryogenicNon-neutral Plasma, Phys. Plasmas 15, 055705 (2008).

12. F. Anderegg, D. H. E. Dubin, T. M. O’Neil and C. F. Driscoll, Measure-ment of Correlation-Enhanced Collision Rates, Phys. Rev. Lett. 102, 185001(2009).

13. M.E. Glinsky, T.M. O’Neil, M.N. Rosenbluth, K. Tsuruta and S. Ichimaru,Collisional Equipartition Rate for a Magnetized Pure Electron Plasma, Phys.Fluids B 4, 1156 (1992).

chapter 1 correlations in trapped plasma

Documents