Bremsstrahlung Emission From Ion–Dust Grain Coulomb Scatterings in Dusty Plasmas

PHYSICS OF PLASMAS VOLUME 7, NUMBER 2 FEBRUARY 2000

Bremsstrahlung emission from ion–dust grain Coulomb scatteringsin dusty plasmas

Young-Dae Junga)

Department of Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Koreaand National Institute for Fusion Science, Toki 509-5292, Japan

Hiro TawaraNational Institute for Fusion Science, Toki 509-5292, Japan

~Received 21 July 1999; accepted 21 October 1999!

Bremsstrahlung processes in ion–dust grain Coulomb scatterings in dusty plasmas are investigatedusing the classical trajectory method. Attractive interaction between the projectile ion and its ownimage charge inside the positively charged dust grain is included to obtain the total interactionpotential between the projectile ion and the dust grain. The classical straight-line trajectory methodis applied to represent the differential radiation cross section as a function of the scaled impactparameter, projectile energy, and photon energy. The result shows that the induced image chargeenlarges the bremsstrahung cross section near the dust surface. ©2000 American Institute ofPhysics.@S1070-664X~00!01902-9#

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I. INTRODUCTION

Bremsstrahlung emission1–8 in plasmas has receivemuch attention, since these processes have been widelyin plasma diagnostics in laboratory and astrophysical pmas. Recently, bremsstrahlung processes in weakly4,7 andstrongly coupled plasmas5,6 have been extensively investgated using the classical trajectory method. In recent yethere has been a considerable interest in the dynamicgases and plasmas containing dust particles or higcharged aerosol, including collective effects and strong etrostatic interaction between the charged components. Tdust–plasma interactions occur not only in space, but alsthe laboratory. Various atomic processes in dusty plashave been investigated in order to get information on plasparameters in dusty plasmas.9 Dust particles are charged bcollecting electrons and ions, as they do in the low-pressdischarges, but also by emitting electrons. The later proccan lead to a positive electric charge in the dust grain.cently, the formation of regular structures of dust grains wobserved in a laminar flow of a thermal plasma, in whicase the grain charge was positive due to thermal elecemission.10 To the best of our knowledge, the bremsstralung in collisions of ions with positively charged dust graiin dusty plasmas has not been investigated as yet. Thuthis paper we investigate bremsstrahlung processes inion–dust grain Coulomb scatterings in dusty plasmas.interaction potential between the projectile ion and the potively charged dust grain is obtained by the repulsive pdue to the dust charge and the attractive part due toimage charge inside the dust grain. The classical straight-~SL! trajectory method1,11–14 is applied to obtain the differ-

a!Permanent address: Department of Physics, Hanyang University, AnKyunggi-Do 425-791, South Korea. Electronic [email protected]

7151070-664X/2000/7(2)/715/4/$17.00

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ential bremsstrahlung radiation cross section as a functiothe scaled impact parameter, projectile energy, and radiaphoton energy.

In Sec. II, we derive the ion-dust grain Coulomb inteaction potential using the image charge method. In Sec.we discuss the classical expression of the bremsstrahcross section in scattering of ions with positively chargdust grains in dusty plasmas. In Sec. IV we obtain the alytic form of the differential radiation cross section using tcomponents of the force parallel (F i) and perpendicular(F') to the projectile velocity. We also investigate the imacharge effects on the radiation cross section for soft and hphoton cases. The result shows that the induced imcharge significantly enlarges the differential bremsstrahcross section near the surface of the dust grain. FinallySec. V, discussions are given.

II. ION–DUST GRAIN INTERACTION

Introducing polar spherical coordinates with their cenat the dust grain, we can evaluate the electrostatic potef(r ) produced by the dust chargeQ with the surroundingplasma:9

f~r !5Qexp@2~r 2a!/lD#

r, ~1!

wherea is the radius of the dust grain andlD is the Debyelength15 of the background plasma. Here, for the sakesimplicity, the dust particles are assumed spherical. For tcal laboratory and astrophysical dusty plasmas, it has bknown [email protected] When a very small~sizeaq!a! particleof electric chargeq is placed near the dust grain, the inteaction potential energy could be evaluated by a sum offields. The first would be produced by the dust grain andsecond by the image chargeq852(a/r )q located at pointr 85a2/r inside the dust sphere. Hence, when the projecion is placed within the Debye radius (a,r ,lD), the inter-

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716 Phys. Plasmas, Vol. 7, No. 2, February 2000 J. D. Jung and H. Tawara

action potential energyV(r ) between the ion (q5ze) andthe positively charged dust grain (Q5Ze) can be written as

V~r !5qQ

r1

qq8

ur2r 8 r u, ~2!

5zZe2

r2

az2e2

r 22a2 , ~3!

wherer is the position vector of the projectile ion from thcenter of the dust grain andr ([r /r ) denotes the unit vectorHere, the first term in Eq.~3! represents the repulsive inteaction between the ion and the positive dust charge andsecond term represents the attractive interaction betweenion and its own negative image charge inside the dust grThen, the total forceF acting on the projectile ion by the dusgrain is given by

F~r !52,V~r !5FzZe2

r 3 22az2e2

~r 22a2!2G r . ~4!

In the following section, we shall discuss the bremsstrahlucross section due to the ion–dust grain Coulomb scatteri

III. BREMSSTRAHLUNG CROSS SECTION

The classical expression of the bremsstrahlung crsection11 is given by

dsb52pE dbbdwv~b!, ~5!

whereb is the impact parameter anddwv is the differentialprobability of emitting a photon of frequencyv within dvwhen a projectile particle changes its velocity in collisiowith a static target system. For all impact parameters,dwv isgiven by the Larmor formula for the emission spectrum ononrelativistic accelerated charge:

dwv58p

3\c3 US d2d~ t !

dt2 DvU2 dv

v, ~6!

wheredv is the Fourier coefficient of the dipole momentd ofthe system. For the ion–dust particle system, (d2d/dt2)v canbe presented as

S d2d~ t !

dt2 Dv

5S Ze

M2

ze

miDFv , ~7!

whereM andmi are the dust mass and the ion mass, resptively, andFv is the Fourier coefficient of the forceF(t):

Fv51

2p E2`

`

dteivtF~ t !. ~8!

In the SL trajectory method, the position of the projectile ican be represented asr (t)5b1vt with b•v50, wherev isthe velocity of the projectile ion. The use of the ion–dugrain Coulomb interaction@Eq. ~4!# and the SL trajectoryimpact-parameter approach allows us to derive analyticmulas for the Fourier coefficients of the force:

Fmv5zZe2

pvaFmv ~m5i ,' !. ~9!


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Here, the Fourier coefficientsF iv and F'v are the compo-nents of force parallel and perpendicular to the projecvelocity, respectively, and given as follows:

F iv5 i E0

`

dtt sinjtF 1

~t21b2!3/222~z/Z!

~t21b221!2G ,~10!

5 i F jK0~jb!2z

Z

p

2

je2jAb221

Ab221G , ~11!

F'v5E0

`

dtb cosjtF 1

~t21b2!3/222~z/Z!

~t21b221!2G ,~12!

5jK1~jb!2z

Z

p

2b~jAb22111!

e2jAb21

~ b221!3/2, ~13!

whereb([b/a) is the scaled impact parameter,t([vt/a) isthe scaled time,j[va/v, andK0(jb) and K1(jb) are thezeroth- and first-order modified Bessel functions16 of the sec-ond kind, respectively. These expressions, Eqs.~11! and~13!, are quite reliable for the small impact-parameter regiIf we neglect the image charge effects on the interactpotential, the Fourier coefficientsF iv8 andF'v8 are just foundto be

F iv8 5 i jK0~jb!, ~14!

F'v8 5jK1~jb!. ~15!

The differential bremsstrahlung cross section due toion–dust grain Coulomb scatterings is then found to be

dsb516z2Z2a3a0

2Ry

3E

m

m S Zm

M2

zm

miD 2 dv

v E dbb~ uF ivu2

1uF'vu2!, ~16!

where a(5e2/\c>1/137) is the fine structure constana0(5\2/me2) is the Bohr radius, Ry(5me4/2\2>13.6 eV)is the Rydberg constant,m is the electron rest massm@5Mmi /(M1mi)# is the reduced mass, andE([mv2/2)is the kinetic energy of the projectile ion.

IV. RADIATION CROSS SECTION

The differential radiation cross section17 is defined by

dxb

de[

dsb

\dv\v, ~17!

516z2a3a0

2

3E

m

m S Zm

M2

zm

miD 2E dbb~ uF ivu21uF'vu2!,

~18!

where e~[\v! is the photon energy. In the nonrelativisilimit, the parameterj can be rewritten asj5ZAm/m3(a/a0)( e/2AE) where e([\v/Z2 Ry) is the scaled pho-ton energy andE([E/Z2 Ry) is the scaled kinetic energy othe projectile ion. Then, the scaled doubly differential rad


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717Phys. Plasmas, Vol. 7, No. 2, February 2000 Bremsstrahlung emission from ion-dust grain Coulomb . . .

FIG. 1. The total scaled doubly differential bremsstralung cross section (d2xb /dedb) in units of pa0

2 as afunction of the scaled impact parameter (b5b/a) whenE55.4431023 eV ande51.6331023 eV, i.e., the softphoton radiation case (e/E50.3). The dashed line isthe bremsstrahlung cross section including the imacharge effect. The solid line is the bremsstrahlung crosection neglecting the image charge effect.

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tion cross section, i.e., the scaled differential radiation crsection per scaled impact parameter, can be presented

d2xb

dedb/pa0

25S d2xb

dedbDi

Y pa021S d2xb

dedbD'

Y pa02,

~19!

516z2Z2a3

3pE

m

m S m

M2

z

Z

m

miD 2

3b@ uF iv~ b,e,E!u21uF'v~ b,e,E!u2#. ~20!

In order to investigate the contribution of the imacharge effects on the radiation cross section, we sea50.01mm, Z5200,z51 ~proton projectile!, and the densityof the dust grain isr>2 g cm23. The result@Eq. ~20!# is quitereliable for a low-energy domain (E,Z2 Ry). Figure 1shows the total scaled doubly differential bremsstrahlucross section@(d2xb /dedb)/pa0

2# as a function of the scaleimpact parameter for the soft radiation photon case (e/E50.3), i.e.,E55.4431023 eV ande51.6331023 eV sincethe classical expression of the bremsstrahlung cross se@Eq. ~5!# is known to be reliable for low-energy projectiles11

(E,Z2 Ry). Here, the dashed line is the bremsstrahlucross section including the image charge effect and the sline is the bremsstrahlung cross section neglecting the imcharge effects. As we can see in this figure, the induimage charge enlarges the differential bremsstrahlung ration cross section forb,2 since the induced image chargeffect is important near the surface of the dust grain. Fig


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2 shows the scaled doubly differential bremsstrahlung crsection for the hard radiation photon case (e/E50.9), i.e.,E55.4431023 eV ande54.9031023 eV. As we see in Fig.2, the image charge effect is found to be stronger than thathe soft photon case, since the image charge effectj is thekey parameter of the image charge effect in Eq.~13!. Theimage charge effects are especially significant for hard pton cases. Hence, it is found that the image charge effplay an important role in the bresstrahlung in ion–dust grcollisions in dusty plasmas. Hence, the image charge effplay an important role in the bresstrahlung in ion–dust grcollisions in dusty plasmas. Because of hugely enhanbremsstrahlung cross sections near the dust surface, theeasily lose their energy due to the ion–dust grain Coulobremsstrahlung process in dusty plasmas. Recently, seexperiments on dust–plasma interactions were performea dusty plasma device~DPD!.18,19Also, a recent paper20 sug-gested the use of UV radiation to form a new type of plascrystal consisting of positively charged dust and electroHence, in the future, we may detect and resolve the ctinuum spectra from dust plasmas due to the Coulobremsstrahlung processes, and, then, we can determinphysical information of the dust plasmas.

V. DISCUSSIONS

We investigate the ion–dust grain Coulomb bremsstrlung processes in dusty plasmas. The interaction potenbetween the projectile ion and the dust grain is obtained

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FIG. 2. The total scaled doubly differential bremsstralung cross section (d2xb /dedb) in units of pa0

2 as afunction of the scaled impact parameter (b5b/a) whenE55.4431023 eV and e54.9031023 eV, i.e., thehard photon radiation case (e/E50.9). The dashed lineis the bremsstrahlung cross section including the imacharge effect. The solid line is the bremsstrahlung crosection neglecting the image charge effect.


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718 Phys. Plasmas, Vol. 7, No. 2, February 2000 J. D. Jung and H. Tawara

the repulsive part due to the positive dust charge andattractive part due to the negative image charge insidedust grain. The classical straight-line trajectory methodapplied to the motion of the projectile ion in order to invetigate the variation of the scaled doubly differential radiaticross section as a function of the scaled impact paramprojectile energy, and photon energy. The bremsstrahlradiation cross section is obtained by the parallel componof the force to the projectile velocity, i.e., (d2xb /dedb) i ,and the perpendicular component of the force to the protile velocity, i.e., (d2xb /dedb)' . The result shows that thinduced image charge inside the dust grain enlargesbremsstrahung cross section near the surface of thegrain. Hence, it should be noted that the image charge effplay an important role on the bresstrahlung in ion–dust grcollisions in dusty plasmas. Since the bremsstrahlung csection including the image charge effect is greater thanneglecting the image charge effect, the image charge ewould be measured using a comparison of the bremsstlung spectra from the ion Coulomb collisions with the coducting dust grains and with the nonconducting dust graThese results provide useful information on the bremsstlung processes in ion–dust grain scatterings in dusty pmas.

ACKNOWLEDGMENTS

One of the authors~Y.-D. Jung! thanks financial supporby the Japan Society for the Promotion of Science~JSPS!while visiting the National Institute for Fusion Science, ToJapan. The authors would like to thank the anonymouseree for suggesting improvements to this text.

This work was supported by the Korean Ministry


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Education through the Brain Korea~BK21! Project, by theResearch Fund of Hanyang University~Project No. HYU-99-040!, and by the interdisciplinary research program of tKorea Science and Engineering Foundation through GNo. 1999-1-111-001-5.

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Documents

Bremsstrahlung Emission From Ion–Dust Grain Coulomb Scatterings in Dusty Plasmas