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PLASMA

REVISTA MEXICANA DE F´ISICA 49 SUPLEMENTO 3, 153–155

NOVIEMBRE 2003

Analysis of the dust crystal vibrational frequency modes in the presence of a supersonic ion flow C. Cereceda, J. Puerta, and P. Martin Departamento de F´ısica, Universidad Sim´on Bol´ıvar, Apartado Postal 89000, Caracas 1080-A, Venezuela Recibido el 11 de enero de 2002; aceptado el 18 de noviembre de 2002

In previous works we analyzed in a simple way the vibration modes of Coulomb quasi crystals. In this work, we propose a similar study but taking into account the presence of a supersonic ion flow in the calculation of the microfield around dust grains, due to the fact that the suspended dust particles are under the influence of a potential that induces such flow. This strongly coupled Coulomb system supports a vertical mode of oscillation with frequency depending on the grain size, mass, charge and interparticle distances. Comparisons with frequencies calculated by other authors using standard microfield distributions are presented. Keywords: Plasma; dust; crystal oscillation. En trabajos anteriores analizamos de una manera simple los modos de vibraci´on de cuasi cristales coulombianos. En este trabajo, proponemos un estudio similar pero tomando en cuenta la presencia de un flujo i´onico en el c´alculo del microcampo que rodea los granos de polvo, debido al hecho de que las part´ıculas de polvo suspendidas est´an bajo la influencia de un potencial que induce dicho flujo. Este sistema fuertemente acoplado permite un modo vertical de oscilaci´on con frecuencia dependiente del tama˜no del grano, masa, carga y distancia interpart´ıcula. Se presentan comparaciones con las frecuencias calculadas por otros autores usando distribuciones de microcampo t´ıpicas. Descriptores: Plasma; polvo; oscilaci´on de cristales. PACS: 52.27.Lw

1. Introduction In previous works [1-3], we have used an improved screening potential [4,5], in order to describe the oscillations of dust particle crystals, instead of the usual Debye screening potential (or Yukawa type), which is a good approximation for r ¿ λD . We used the screening potential Φ calculated from an approximated solution of the Poisson-Boltzmann equation [6] where the dust particle, with charge Q = −Zd e, has spherical shape with radius Ro . Ions were supposed to have thermal equilibrium distribution function and electrons were considered as a uniform background. In this work we study the effect on the screening potential due to the supersonic ion flow in the sheath where the dust crystal particles levitates over a negative electrode. The observed density of the ions in the sheath is a function of the sheath potential and equivalently of their flow speed [7,8]. Also, because of the flow of ions to the negative electrode, there is a cylindrical symmetry instead of the usual spherical one.

2. The model In the sheath region, the density of the ions ni(V ) is a function of their flow speed V which is higher than the ion acoustic speed of the plasma in dust crystal experiments [9,10]. According to the discussion given in the introduction we can write the Poisson-Boltzmann equation for the screening po-

tential Φ around a dust particle of charge Q as ∇2 Φ(r) = −4πeni(V ) e−ZeΦ(r)/Ti ³ ´ + 4π ene(V ) eeΦ(r)/Te + Qδ(r)

(1)

where the equilibrium ion and electron densities in the sheath are modified by the self consistent screening potential around dust particles. This is taken into account by the Boltzmann factors. The screening potential can be cast in the form Q Φ(r) = + φ(r), (2) r because ϕ(r) = Q/r is the potential due to a central charge Q in vacuum for Poisson equation solved with spherical symmetry : 4π ∇2 ϕ(r) = −4πQδ(r) = − 2 Qδ(r), (3) r and where φ(r) stands for the potential due to the screening by ions and electrons. By this way the equation for the screening potential is simplified to be ´ ³ (4) ∇2 φ(r)=−4πe ni(V ) e−ZeΦ(r)/Ti −ne(V ) eeΦ(r)/Te As a first approach to the problem, in the linear approximation, we consider a small screening potential Φ(r) ¿ Ti , Te and the Boltzmann factor can be linearized: · ¸¶ µ ZeΦ(r) 2 ∇ φ(r) = − 4πe Zni(V ) 1 − Ti · µ ¸¶ eΦ(r) + 4πe ne(V ) 1 + . (5) Te

154

C. CERECEDA, J. PUERTA, AND P. MARTIN

Although electron density is a little bit smaller than ion density in the sheath, we neglect [11] this small difference: Zni(V ) ' ne(V ) , leading to µ ¶ Z 1 Φ(r) ∇2 φ(r) = 4πe2 Zni(V ) + Ti Te · ¸ 1 Q = 2 + φ(r) . (6) λ r Here λ is the dynamic screening length. The ion density in the sheath is determined [7,8] equating the ion energy inside the sheath and the initial ion energy at its entrance to the sheath: 1 1 mi vi2 = mi V 2 − ZeΨ(z) , (7) 2 2 with Ψ(z) the sheath potential and V the velocity of the ions in the sheath, this velocity is of the order of the ion acoustic speed cs = (Te /mi )1/2 . In dust crystal experiments [9,10] this entrance velocity is found to be a little bit higher than cs . Since the ion flux ni vi is constant, Ã !1/2 1 2 m V i 2 ni(V ) = n0 1 (8) 2 2 mi V − ZeΨ(z) n0 is the equilibrium plasma density, i.e. far from the sheath region. As we are interested in the thin region of the sheath around a horizontal chain formed by the dust grains, we use the value of the sheath potential at the dusty chain equilibrium position z = 0 (Ψ(z=0) = Ψo ) and corresponding velocity V(z=0) = Vo : µ ¶1/2 M2 nio(V ) = n0 (9) M 2 − 2ZeΨo /Te where M = Vo /cs is the Mach number. It is worth to notice that in the absence of sheath (Ψo = 0), ni(0) = n0 , and −2 −1/2 λ = λD = (λ−2 . For r 6= 0, by writing the Di + λDe ) potential as φ(r) = (Q/r)(W (r) − 1) with the appropriate boundary condition W (0) = 1, Poisson equation is simplified as follows: Q 2 Q Q ∇ W (r) + (W (r) − 1) ∇2 = 2 W (r) (10) r r rλ 1 ∇2 W (r) = 2 W (r). (11) λ The flow of ions in the negative z direction defines azimuthal symmetry around z-axis and the calculation of the screening potential can be performed in cylindrical coordinates ρ and z by the separation of variables method with W (r) = W (ρ, z) = R(ρ)S(z): 1 d Rρ dρ

µ ¶ dR 1 ∂2S 1 ρ + − 2 = 0, dρ S ∂z 2 λ µ 00

S −

1 1 + 2 λ2 Λ

³ ρ ´2 Λ

R = 0,

(14)

where 1/Λ2 is the separation constant. The solutions are S(z) = Ae−z

√

1/λ2 +1/Λ2

R(ρ) = BJo

³ρ´ Λ

(15)

(16)

which satisfy the boundary condition W (ρ = z = 0) = 1, with A = B = 1 in addition to W (ρ, z → ∞) = 0. By this way, the dynamic ellipsoidal screening potential has been calculated to be: Φ(ρ, z)

+Q r (W (r) − 1) ¡ ρ ¢ −z√1/λ2 +1/Λ2 , =Q J o r Λ e =

Q r

(17)

p where r = ρ2 + z 2 . The value of Λ must be of the order of 3λ in order to recover linear Debye potential in z direction in the case of no ion flow (Ψo = 0).

3. Applications Our dynamic screening potential is compared in Figs. 1 and 2 with Debye potential for the parameters in dust crystal experiments [9,10]. It is seen that the dynamic screening length (solid line) is shorter than that of Debye (dashed line) in z direction (Fig. 2) and three times larger in ρ direction (Fig. 1). This result is in agreement with recent simulations [11] which take into account the ion flow and show enhancement in the horizonal spacing between particles in dust crystals. Dust charge p is Q = −3x104 e, Mach number M 2 = 1.1 and λDe = Ti /(4πe2 ne ) = 2 × 10−2 cm, w = z/λ, x = ρ/λ, Ti = Te /10.

(12)

¶ S = 0,

ρ2 R00 + ρR0 +

(13)

F IGURE 1. Comparison between dynamic screening potential (solid line) and Debye potential (dashed line) as a function of normalized radial distance ρ (x).

Rev. Mex. F´ıs. 49 S3 (2003) 153–155

ANALYSIS OF THE DUST CRYSTAL VIBRATIONAL FREQUENCY MODES IN THE PRESENCE OF. . .

F IGURE 2. Comparison between dynamic screening potential (solid line) and Debye potential (dashed line) as a function of normalized z distance (w).

The oscillation frequency of linear vertical oscillations of a one dimensional chain of dust grains is [9] ¯ ¯ γ 4Q ¯¯ dΦ ¯¯ ω2 = + sin2 (kro /2) , (18) m mro ¯ dρ ¯ρ=ro with estimated width of the sheath Ã ¸1/2 ! · M2 eΨo /Te γ=4πe |Q| no e − , (19) M 2 −2ZeΨo /Te where double bar | | stands for absolute value, and Ψo is determined numerically from µ ¶2 1 mg = eeΨo /Te − 1 + M 2 8πno Te Q Ãr ! 2ZeΨo × 1− −1 . (20) Te M 2 In Fig. 3 we show the frequency calculated from the dynamic screening potential dΦ dρ |ro (solid line) and that calculated from Debye potential. The dust grain mass is m = 0.6 × 10−9 g, M 2 = 1.1, Ti = Te /10, x0 = r0 /λ is the normalized interparticle distance in the chain. For heavier (which is usual)

1. C. Cereceda, J. Puerta, and P. Martin, International Congress on Plasma Physics, Proceedings I (2000) 328. 2. C. Cereceda, J. Puerta, and P. Martin, IEEE Conference-Record Abstracts 00CH37087 (2000) 145. 3. C. Cereceda, J. Puerta, and P. Martin, Bull. American Phys. Soc. 45 (2000) 306. 4. C. Cereceda, J. Puerta, and P. Martin, Physica Scripta T 84 (2000) 206. 5. J. Puerta and C. Cereceda, Astr. and Space Sci. 256 (1998) 349.

155

F IGURE 3. Dust crystal oscillation frequency versus dust particle distance calculated from the dynamic screening potential (solid line) and from Debye potential (dashed line).

or less charged dust particles and for higher Mach numbers, the difference is reduced. This result is due to the stronger effect of the sheath potential on the oscillation frequency (first term γo /m, simple numerical comparison) compared with the screening potential.

4. Conclusions Our results show that the effect of the supersonic ion flow on this mode of oscillation of dust crystal chains is enough taken into account via the sheath potential [first term of the right member of Eq. (18)], which is calculated from the flow velocity V of ions. The smallness of the correction here found indicates that our previous use of the static screening potential neglecting the ion flow is well suited to describe the oscillations for the typical parameters in dust crystal experiments: inter particle distance, coupling parameter Γ = ZeQ/(λDi kTi ) and dust radius. Here we have also found that the screening in the z-direction is shorter than Debye length, however this screening length is about three times the Debye length in the ρ-direction. Hence, we found that the ion flow determines a spatial anisotropy of the dust crystal, however its effect on the frequency of oscillation of the crystal can be neglected.

6. R. Perez and P. Martin, Astr. and Space Sci. 256 (1998) 263. 7. J. Wesson and Contrib., Tokamaks. (2nd. Ed., Chap. 9, section 9.2, Clarendon Press, Oxford, 1997). 8. R.D. Hazeltine, F.L. Waelbroeck, The framework of plasma physics (Perseus, Reading, 1998) Chap. 3. 9. S. Vladimirov, Plasma Phys. Contr. Fusion 41 (1999) 467. 10. Y. Nakamura, Plasma Phys. Contr. Fusion 41 (1999) 469. 11. G. Joyce, M. Lampe, and G. Ganguli, IEEE Trans. Plasma Sc. 29 (2001) 238.

Rev. Mex. F´ıs. 49 S3 (2003) 153–155

REVISTA MEXICANA DE F´ISICA 49 SUPLEMENTO 3, 153–155

NOVIEMBRE 2003

Analysis of the dust crystal vibrational frequency modes in the presence of a supersonic ion flow C. Cereceda, J. Puerta, and P. Martin Departamento de F´ısica, Universidad Sim´on Bol´ıvar, Apartado Postal 89000, Caracas 1080-A, Venezuela Recibido el 11 de enero de 2002; aceptado el 18 de noviembre de 2002

In previous works we analyzed in a simple way the vibration modes of Coulomb quasi crystals. In this work, we propose a similar study but taking into account the presence of a supersonic ion flow in the calculation of the microfield around dust grains, due to the fact that the suspended dust particles are under the influence of a potential that induces such flow. This strongly coupled Coulomb system supports a vertical mode of oscillation with frequency depending on the grain size, mass, charge and interparticle distances. Comparisons with frequencies calculated by other authors using standard microfield distributions are presented. Keywords: Plasma; dust; crystal oscillation. En trabajos anteriores analizamos de una manera simple los modos de vibraci´on de cuasi cristales coulombianos. En este trabajo, proponemos un estudio similar pero tomando en cuenta la presencia de un flujo i´onico en el c´alculo del microcampo que rodea los granos de polvo, debido al hecho de que las part´ıculas de polvo suspendidas est´an bajo la influencia de un potencial que induce dicho flujo. Este sistema fuertemente acoplado permite un modo vertical de oscilaci´on con frecuencia dependiente del tama˜no del grano, masa, carga y distancia interpart´ıcula. Se presentan comparaciones con las frecuencias calculadas por otros autores usando distribuciones de microcampo t´ıpicas. Descriptores: Plasma; polvo; oscilaci´on de cristales. PACS: 52.27.Lw

1. Introduction In previous works [1-3], we have used an improved screening potential [4,5], in order to describe the oscillations of dust particle crystals, instead of the usual Debye screening potential (or Yukawa type), which is a good approximation for r ¿ λD . We used the screening potential Φ calculated from an approximated solution of the Poisson-Boltzmann equation [6] where the dust particle, with charge Q = −Zd e, has spherical shape with radius Ro . Ions were supposed to have thermal equilibrium distribution function and electrons were considered as a uniform background. In this work we study the effect on the screening potential due to the supersonic ion flow in the sheath where the dust crystal particles levitates over a negative electrode. The observed density of the ions in the sheath is a function of the sheath potential and equivalently of their flow speed [7,8]. Also, because of the flow of ions to the negative electrode, there is a cylindrical symmetry instead of the usual spherical one.

2. The model In the sheath region, the density of the ions ni(V ) is a function of their flow speed V which is higher than the ion acoustic speed of the plasma in dust crystal experiments [9,10]. According to the discussion given in the introduction we can write the Poisson-Boltzmann equation for the screening po-

tential Φ around a dust particle of charge Q as ∇2 Φ(r) = −4πeni(V ) e−ZeΦ(r)/Ti ³ ´ + 4π ene(V ) eeΦ(r)/Te + Qδ(r)

(1)

where the equilibrium ion and electron densities in the sheath are modified by the self consistent screening potential around dust particles. This is taken into account by the Boltzmann factors. The screening potential can be cast in the form Q Φ(r) = + φ(r), (2) r because ϕ(r) = Q/r is the potential due to a central charge Q in vacuum for Poisson equation solved with spherical symmetry : 4π ∇2 ϕ(r) = −4πQδ(r) = − 2 Qδ(r), (3) r and where φ(r) stands for the potential due to the screening by ions and electrons. By this way the equation for the screening potential is simplified to be ´ ³ (4) ∇2 φ(r)=−4πe ni(V ) e−ZeΦ(r)/Ti −ne(V ) eeΦ(r)/Te As a first approach to the problem, in the linear approximation, we consider a small screening potential Φ(r) ¿ Ti , Te and the Boltzmann factor can be linearized: · ¸¶ µ ZeΦ(r) 2 ∇ φ(r) = − 4πe Zni(V ) 1 − Ti · µ ¸¶ eΦ(r) + 4πe ne(V ) 1 + . (5) Te

154

C. CERECEDA, J. PUERTA, AND P. MARTIN

Although electron density is a little bit smaller than ion density in the sheath, we neglect [11] this small difference: Zni(V ) ' ne(V ) , leading to µ ¶ Z 1 Φ(r) ∇2 φ(r) = 4πe2 Zni(V ) + Ti Te · ¸ 1 Q = 2 + φ(r) . (6) λ r Here λ is the dynamic screening length. The ion density in the sheath is determined [7,8] equating the ion energy inside the sheath and the initial ion energy at its entrance to the sheath: 1 1 mi vi2 = mi V 2 − ZeΨ(z) , (7) 2 2 with Ψ(z) the sheath potential and V the velocity of the ions in the sheath, this velocity is of the order of the ion acoustic speed cs = (Te /mi )1/2 . In dust crystal experiments [9,10] this entrance velocity is found to be a little bit higher than cs . Since the ion flux ni vi is constant, Ã !1/2 1 2 m V i 2 ni(V ) = n0 1 (8) 2 2 mi V − ZeΨ(z) n0 is the equilibrium plasma density, i.e. far from the sheath region. As we are interested in the thin region of the sheath around a horizontal chain formed by the dust grains, we use the value of the sheath potential at the dusty chain equilibrium position z = 0 (Ψ(z=0) = Ψo ) and corresponding velocity V(z=0) = Vo : µ ¶1/2 M2 nio(V ) = n0 (9) M 2 − 2ZeΨo /Te where M = Vo /cs is the Mach number. It is worth to notice that in the absence of sheath (Ψo = 0), ni(0) = n0 , and −2 −1/2 λ = λD = (λ−2 . For r 6= 0, by writing the Di + λDe ) potential as φ(r) = (Q/r)(W (r) − 1) with the appropriate boundary condition W (0) = 1, Poisson equation is simplified as follows: Q 2 Q Q ∇ W (r) + (W (r) − 1) ∇2 = 2 W (r) (10) r r rλ 1 ∇2 W (r) = 2 W (r). (11) λ The flow of ions in the negative z direction defines azimuthal symmetry around z-axis and the calculation of the screening potential can be performed in cylindrical coordinates ρ and z by the separation of variables method with W (r) = W (ρ, z) = R(ρ)S(z): 1 d Rρ dρ

µ ¶ dR 1 ∂2S 1 ρ + − 2 = 0, dρ S ∂z 2 λ µ 00

S −

1 1 + 2 λ2 Λ

³ ρ ´2 Λ

R = 0,

(14)

where 1/Λ2 is the separation constant. The solutions are S(z) = Ae−z

√

1/λ2 +1/Λ2

R(ρ) = BJo

³ρ´ Λ

(15)

(16)

which satisfy the boundary condition W (ρ = z = 0) = 1, with A = B = 1 in addition to W (ρ, z → ∞) = 0. By this way, the dynamic ellipsoidal screening potential has been calculated to be: Φ(ρ, z)

+Q r (W (r) − 1) ¡ ρ ¢ −z√1/λ2 +1/Λ2 , =Q J o r Λ e =

Q r

(17)

p where r = ρ2 + z 2 . The value of Λ must be of the order of 3λ in order to recover linear Debye potential in z direction in the case of no ion flow (Ψo = 0).

3. Applications Our dynamic screening potential is compared in Figs. 1 and 2 with Debye potential for the parameters in dust crystal experiments [9,10]. It is seen that the dynamic screening length (solid line) is shorter than that of Debye (dashed line) in z direction (Fig. 2) and three times larger in ρ direction (Fig. 1). This result is in agreement with recent simulations [11] which take into account the ion flow and show enhancement in the horizonal spacing between particles in dust crystals. Dust charge p is Q = −3x104 e, Mach number M 2 = 1.1 and λDe = Ti /(4πe2 ne ) = 2 × 10−2 cm, w = z/λ, x = ρ/λ, Ti = Te /10.

(12)

¶ S = 0,

ρ2 R00 + ρR0 +

(13)

F IGURE 1. Comparison between dynamic screening potential (solid line) and Debye potential (dashed line) as a function of normalized radial distance ρ (x).

Rev. Mex. F´ıs. 49 S3 (2003) 153–155

ANALYSIS OF THE DUST CRYSTAL VIBRATIONAL FREQUENCY MODES IN THE PRESENCE OF. . .

F IGURE 2. Comparison between dynamic screening potential (solid line) and Debye potential (dashed line) as a function of normalized z distance (w).

The oscillation frequency of linear vertical oscillations of a one dimensional chain of dust grains is [9] ¯ ¯ γ 4Q ¯¯ dΦ ¯¯ ω2 = + sin2 (kro /2) , (18) m mro ¯ dρ ¯ρ=ro with estimated width of the sheath Ã ¸1/2 ! · M2 eΨo /Te γ=4πe |Q| no e − , (19) M 2 −2ZeΨo /Te where double bar | | stands for absolute value, and Ψo is determined numerically from µ ¶2 1 mg = eeΨo /Te − 1 + M 2 8πno Te Q Ãr ! 2ZeΨo × 1− −1 . (20) Te M 2 In Fig. 3 we show the frequency calculated from the dynamic screening potential dΦ dρ |ro (solid line) and that calculated from Debye potential. The dust grain mass is m = 0.6 × 10−9 g, M 2 = 1.1, Ti = Te /10, x0 = r0 /λ is the normalized interparticle distance in the chain. For heavier (which is usual)

1. C. Cereceda, J. Puerta, and P. Martin, International Congress on Plasma Physics, Proceedings I (2000) 328. 2. C. Cereceda, J. Puerta, and P. Martin, IEEE Conference-Record Abstracts 00CH37087 (2000) 145. 3. C. Cereceda, J. Puerta, and P. Martin, Bull. American Phys. Soc. 45 (2000) 306. 4. C. Cereceda, J. Puerta, and P. Martin, Physica Scripta T 84 (2000) 206. 5. J. Puerta and C. Cereceda, Astr. and Space Sci. 256 (1998) 349.

155

F IGURE 3. Dust crystal oscillation frequency versus dust particle distance calculated from the dynamic screening potential (solid line) and from Debye potential (dashed line).

or less charged dust particles and for higher Mach numbers, the difference is reduced. This result is due to the stronger effect of the sheath potential on the oscillation frequency (first term γo /m, simple numerical comparison) compared with the screening potential.

4. Conclusions Our results show that the effect of the supersonic ion flow on this mode of oscillation of dust crystal chains is enough taken into account via the sheath potential [first term of the right member of Eq. (18)], which is calculated from the flow velocity V of ions. The smallness of the correction here found indicates that our previous use of the static screening potential neglecting the ion flow is well suited to describe the oscillations for the typical parameters in dust crystal experiments: inter particle distance, coupling parameter Γ = ZeQ/(λDi kTi ) and dust radius. Here we have also found that the screening in the z-direction is shorter than Debye length, however this screening length is about three times the Debye length in the ρ-direction. Hence, we found that the ion flow determines a spatial anisotropy of the dust crystal, however its effect on the frequency of oscillation of the crystal can be neglected.

6. R. Perez and P. Martin, Astr. and Space Sci. 256 (1998) 263. 7. J. Wesson and Contrib., Tokamaks. (2nd. Ed., Chap. 9, section 9.2, Clarendon Press, Oxford, 1997). 8. R.D. Hazeltine, F.L. Waelbroeck, The framework of plasma physics (Perseus, Reading, 1998) Chap. 3. 9. S. Vladimirov, Plasma Phys. Contr. Fusion 41 (1999) 467. 10. Y. Nakamura, Plasma Phys. Contr. Fusion 41 (1999) 469. 11. G. Joyce, M. Lampe, and G. Ganguli, IEEE Trans. Plasma Sc. 29 (2001) 238.

Rev. Mex. F´ıs. 49 S3 (2003) 153–155