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The ion density distribution in a magnetized plasma sheath
Vacuum 83 (2009) 205–208
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
The ion density distribution in a magnetized plasma sheath Xiu Zou a, *, Minghui Qiu a, Huiping Liu a, Lijie Zhang a, Jinyuan Liu b, Ye Gong b a b
School of Science, Dalian Jiaotong University, Huanghe Road 794, Dalian 116028, China School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116023, China
a b s t r a c t Keywords: Ion density Plasma sheath Magnetic field
The ion density distribution of plasma sheath in an oblique magnetic field is investigated with a fluid model. We performed numerical simulations of the sheath. The results reveal that the magnetic field has significant effects on the plasma sheath, including ion density distribution and space charge density distribution. Two cases of ion incidence are considered here. Under suitable conditions, Lorentz force induces fluctuations in the ion density. And the magnetic field parallel to the board is responsible for these changes. The action states of ions are more complicated while the ions enter the sheath with an oblique incidence angle. Ions could gather in some regions, so that it leads to small peaks of the density curve. Also the space charge density in such regions is slightly higher. Ó 2008 Elsevier Ltd. All rights reserved.
ef ; Te
1. Introduction
ne ¼ ne0 exp
In recent years, the sheath formed between magnetic plasma and a particle-absorbing wall has received a considerable amount of attention [1–13]. In some sense, the effect of the magnetic field cannot be ignored, but introducing the magnetic field does make the problem more complicated. Many authors [1–4] investigated the structures of the plasma sheath in an oblique magnetic field by using the dynamic theory. But their work is mainly about the presheath. In this paper, we discussed the ion density distribution of the plasma sheath in an oblique magnetic field by using fluid method. As compared with our previous work [5] and Ref. [6], the case of ion entering the sheath with an oblique incidence angle is taken into account. A rough partition of magnetic field presented in Ref. [7] is consulted here.
where, f(V) is the spatial electrostatic potential and Te (eV) is the electron temperature. The ions are treated as a cold fluid governd by the number and momentum equations
2. Mathematical formulation and basic equations We follow the previous work [5,8,9] considering a plasma sheath with a horizontal wall below, which has one-dimensional coordinate space and three-dimensional speed space. The external oblique magnetic field is spatially uniform and constant in time, embedded in the (x, z) plane, and q is the angle of the magnetic field with respect to the x-axis direction (see Fig. 1). At the edge of the sheath, x ¼ 0, the electrostatic potential is taken to be zero, f ¼ 0. The sheath consists of isothermal electrons and fluid ions. The electrons are assumed to be in thermal equilibrium state, thus the density ne satisfies the Boltzmann relation [10–13] * Corresponding author. E-mail address: [email protected] (X. Zou). 0042-207X/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2008.03.032
(1)
V$ðni vi Þ ¼ 0;
(2)
v B ; xþe i mi ðvi $VÞvi ¼ eVfb c
(3)
where, ni, vi and mi denote ion density, ion velocity, and ion mass, respectively. Finally, the system is closed by Poisson’s equation v2 f ¼ 4peðni ne Þ: vx2
(4)
All the functions are assumed to vary only in the direction normal to the wall, V/ðv=vxÞb x . From Eqs. (2) and (3), we have vðni vix Þ ¼ 0; vx mi vix
vvi vf v B : ¼ e b xþe i c vx vx
(5) (6)
And we define cis ¼ ðTe =mi Þ1=2 is the ion sound speed, uic ¼ eB=mi c is the ion gyro frequency. For simplicity, we introduce dimensionless quantities: F ¼ ef=Te , x ¼ x=lD , ui ¼ vi =cis , Ne ¼ ne =ne0 , Ni ¼ ni =ni0 . And lD ¼ ðTe =4pne0 e2 Þ1=2 is electron Debye length.
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Fig. 1. Geometry of the magnetized plasma sheath model.
At the sheath edge x ¼ 0, from quasi-neutral condition, we have ne0 ¼ ni0 , so their ratio di ¼ ni0 =ne0 ¼ 1. From Eqs. (1), (4)–(6), we obtain Ne ¼ expðFÞ;
(7)
Mi ; uix
(8)
Ni ¼
4. Numerical results and discussions
vu vF 1=2 b0; b x þ di gi ui $ B uix i ¼ vx vx d2 F dx
2
Fig. 2. The dependence of the ion density distribution on the intensities of B for q ¼ 45 , uiy0 ¼ 0 and uiz0 ¼ 0.
¼ Ni Ne ;
(9)
(10)
where, in Eq. (8) Mi ¼ vix0 =cis is the ion Mach number, in Eq. (9) gi ¼ uic =upi is the ratio of ion gyro frequency to ion plasma frequency. b ¼ b Taking B x cos q þ b z sin q, then Eq. (9) can be written as 0 uix
vuix vF ¼ þ gi uiy sin q; vx vx
(11)
uix
vuiy ¼ gi ðuiz cos q uix sin qÞ; vx
(12)
uix
vuiz ¼ gi ðuiy cos qÞ: vx
(13)
In the following numerical simulations, some typical parameters are employed, such as n0 ¼ 5 108 ðcm3 Þ, Te ¼ 3ðeVÞ, and so on. The starting point is taken at x ¼ 0, where we set f ¼ 0, and vF=vxjx¼0 ¼ 0:01. 4.1. Ions enter the sheath only along the direction normal to the wall In the first place, we assume that the ions enter the sheath only along the x-axis direction. Their initial velocities in the y-axis and zaxis directions are zero (uiy0 ¼ 0 and uiz0 ¼ 0). Taking ion Mach number Mi ¼ 1, the numerical simulation results are shown in Figs. 2 and 3. As mentioned in Ref. [5], the magnetic field has gyral effect on the moving ions in the sheath. Lorentz force accelerates and decelerates the ion flow in the x-axis direction in each gyrate period, and induces fluctuations in the ion density distribution. However, ion flow usually cannot gyrate for one period, due to the limitation of some factors, such as sheath thickness, the intensity of the magnetic field, and so on. As a result, the density curves rise instantaneously, then drop, and do not have periodical fluctuations
From Eqs. (7)–(13), we obtain the numerical simulation results presented as follows.
3. Bohm’s criterion At the edge of sheath, x ¼ 0, F/0, Ni /1, and dF=dxs0. From the Eqs. (1), (4), and (11), we have
gi sin q Mi2 1 þ vF uiy0 : j
(14)
vx F¼0
If the ion enters the sheath with initial speed in the y-axis direction as zero, uiy0 ¼ 0, we obtain the Bohm’s criterion Mi2 1:
(15)
In this case, the Bohm’s criterion is the same as that in the absence of magnetic field. That is to say, only when the ion velocity in the x-axis direction is greater than or equal to the ion sound speed, can it enter the magnetized sheath. In another case, uiy0 s 0, the ion critical mach number is determined by Eq. (14).
Fig. 3. The dependence of the ion density distribution on the angles of B for B ¼ 2000 Gs, uiy0 ¼ 0 and uiz0 ¼ 0.
X. Zou et al. / Vacuum 83 (2009) 205–208
207
Fig. 6. The ion density distribution under various value of q for B ¼ 4000 Gs and ui0//B. Fig. 4. The ion density distribution under various value of uiy0 for B ¼ 200 Gs, q ¼ 20 and uiz0 ¼ 0.
(see Fig. 2). Fig. 2 indicates the dependence of the ion density distribution on the intensities of the magnetic field, while the angle of magnetic field is q ¼ 45 . As shown, ions are weakly magnetized in the weak magnetic field (B < 400 Gs), and the density distribution does not change obviously. While the ions are magnetized in the strong magnetic field, the density distribution curves change evidently. Also Fig. 3 shows the dependence of the ion density distribution on the angles of the magnetic field for B ¼ 2000 Gs. We can get an idea of the magnetic field component in the z-axis direction attribute to variation of ion density distribution. In one word when the ions enter the sheath only along the x-axis direction, the magnetic field holds up the ions moving to the wall. 4.2. Ions enter the sheath with an oblique incidence angle For the case of ions entering the sheath with an oblique incidence angle (uiy0 s 0 or uiz0 s 0), the action states of ions are more complicated. The ion critical mach number is calculated with Eq. (14). We get the numerical simulation results as followed. Fig. 4 shows the ion density distribution under various value of ion initial velocities in the y-axis direction for B ¼ 200 Gs, q ¼ 20 and uiz0 ¼ 0. Under the actions of the Lorentz force and the electron force, ions
Fig. 5. The ion density distribution under various value of q for B ¼ 4000 Gs, uiy0 ¼ 0 and uiz0 ¼ 0.3.
could gather in some regions, which brings on small peaks of the density curves. The fluctuations of the curve depend on both ion velocity and magnetic field. Fig. 5 indicates the ion density distribution under various value of angles of the magnetic field for B ¼ 4000 Gs, uiy0 ¼ 0 and uiz0 ¼ 0.3. Both the magnetic field components in the x-axis and z-axis directions vary with the angle. The reasons for the fluctuations of the curve are more complicated. From Figs. 4 and 5, we can see that when the ions have velocity component parallel to the board, Lorentz force induces fluctuations in the ion density under suitable conditions, including intensity and angle of the magnetic field. When the initial velocity of ion is parallel to the direction of magnetic field, at this moment, Lorentz force does not exist. However, after the accelerating action of electric field, the direction of velocity is no longer parallel to the magnetic field, and causes the density distribution slight changes (see Fig. 6). In Fig. 6, the ion density distribution under various value of angle of magnetic field is shown, while B ¼ 4000 Gs and ui0//B. 4.3. Space charge density distribution in the magnetized plasma sheath In the case of ions entering the sheath only along the direction normal to the wall, the dependence of the space charge density
Fig. 7. The dependence of the space charge density distribution on the intensities of B for q ¼ 30 , uiy0 ¼ 0 and uiz0 ¼ 0.
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X. Zou et al. / Vacuum 83 (2009) 205–208
(1) The magnetic field has significant effects on the ion density distribution and space charge density distribution, and the magnetic field component in the z-axis direction attribute to variation of ion density distribution. (2) The magnetic field holds up the ions moving to the wall, and Lorentz force induces fluctuations in the ion density under suitable conditions, including intensity and angle of the magnetic field. (3) The space charge density is slightly higher in some regions due to the fluctuations of the ion density, so the external magnetic field can make the shield distance of sheath longer. Acknowledgements This work has been supported by National Natural Science Foundation of China (Project 10605008). Fig. 8. The space charge density distribution under various value of uiy0 for B ¼ 200 Gs, q ¼ 20 and uiz0 ¼ 0.
distribution on the magnetic field is shown in Fig. 7. The solid line is the space charge density curve in the absence of magnetic field. Because of the characteristic of density distribution of ions and electrons, the space charge density distribution curve has a peak, which indicates that in this region, more positive particles gathering to shield the negative potential of the board. With the increase of intensity of magnetic field, the peak of the curve moves towards the edge of sheath. The reason is the effects of the Lorentz force on the density distribution of ions. So the external magnetic field can make the shield distance of sheath longer. While in another case of ions entering the sheath with an oblique incidence angle, ions could gather in some regions, so that the space charge density in such regions is slightly higher (see Fig. 8). 5. Conclusions The plasma sheath is easily influenced by the magnetic field. In the magnetized plasma sheath, two cases of ion incidence are discussed, and the following numerical conclusions are obtained.
References [1] Chodura R. Plasma-wall transition in an oblique magnetic field. Phys Fluid 1982;25:1628–33. [2] Kim G-H, Hershkowitz N, Diebold DA, Cho M-H. Magnetic and collisional effects on presheaths. Phys Plasmas 1995;2:3222–33. [3] Ahedo E. Structure of the plasma-wall interaction in an oblique magnetic field. Phys Plasmas 1997;4:4419–30. [4] Bornali Singha, Sarma A, Joyanti Chutia. Experimental observation of sheath and magnetic presheath over an oblique metallic plate in the presence of a magnetic field. Phys Plasmas 2002;9:683–90. [5] Zou X, Liu JY, Gong Y, Wang ZX, Liu Y, Wang XG. Plasma sheath in a magnetic field. Vacuum 2004;73:681–5. [6] Masoudi SF. Effect of pressure on plasma sheath in a magnetic field. Vacuum 2007;81:871–4. [7] Sato N. Magnetic effects in dusty plasma. Third International Conference On the Physics of Dusty Plasmas (New York) 2002:66–73. [8] Zou X, Liu JY, Wang ZX, Gong Y, Liu Y, Wang XG. Electronegative plasma sheath structure in a magnetic field. Chin Phys Lett 2004;21:1572–4. [9] Zou X. Characteristics of dust plasma sheath in an oblique magnetic field. Chin Phys Lett 2006;23:396–8. [10] Riemann KU. Theory of the collisional presheath in an oblique magnetic field. Phys Plasma 1994;1:552–8. [11] Stangeby PC. The Bohm-chodura plasma sheath criterion. Phys Plasma 1995;2: 702–6. [12] Valsaque F, Manfredi G. Numerical study of plasma-wall transition in an oblique magnetic field. J Nucl Materials 2001;290–293:763–7. [13] Baishya SK, Das GC. Dynamics of dust particles in a magnetized plasma sheath in a fully ionized space plasma. Phys Plasmas 2003;10:3733–45.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
The ion density distribution in a magnetized plasma sheath Xiu Zou a, *, Minghui Qiu a, Huiping Liu a, Lijie Zhang a, Jinyuan Liu b, Ye Gong b a b
School of Science, Dalian Jiaotong University, Huanghe Road 794, Dalian 116028, China School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116023, China
a b s t r a c t Keywords: Ion density Plasma sheath Magnetic field
The ion density distribution of plasma sheath in an oblique magnetic field is investigated with a fluid model. We performed numerical simulations of the sheath. The results reveal that the magnetic field has significant effects on the plasma sheath, including ion density distribution and space charge density distribution. Two cases of ion incidence are considered here. Under suitable conditions, Lorentz force induces fluctuations in the ion density. And the magnetic field parallel to the board is responsible for these changes. The action states of ions are more complicated while the ions enter the sheath with an oblique incidence angle. Ions could gather in some regions, so that it leads to small peaks of the density curve. Also the space charge density in such regions is slightly higher. Ó 2008 Elsevier Ltd. All rights reserved.
ef ; Te
1. Introduction
ne ¼ ne0 exp
In recent years, the sheath formed between magnetic plasma and a particle-absorbing wall has received a considerable amount of attention [1–13]. In some sense, the effect of the magnetic field cannot be ignored, but introducing the magnetic field does make the problem more complicated. Many authors [1–4] investigated the structures of the plasma sheath in an oblique magnetic field by using the dynamic theory. But their work is mainly about the presheath. In this paper, we discussed the ion density distribution of the plasma sheath in an oblique magnetic field by using fluid method. As compared with our previous work [5] and Ref. [6], the case of ion entering the sheath with an oblique incidence angle is taken into account. A rough partition of magnetic field presented in Ref. [7] is consulted here.
where, f(V) is the spatial electrostatic potential and Te (eV) is the electron temperature. The ions are treated as a cold fluid governd by the number and momentum equations
2. Mathematical formulation and basic equations We follow the previous work [5,8,9] considering a plasma sheath with a horizontal wall below, which has one-dimensional coordinate space and three-dimensional speed space. The external oblique magnetic field is spatially uniform and constant in time, embedded in the (x, z) plane, and q is the angle of the magnetic field with respect to the x-axis direction (see Fig. 1). At the edge of the sheath, x ¼ 0, the electrostatic potential is taken to be zero, f ¼ 0. The sheath consists of isothermal electrons and fluid ions. The electrons are assumed to be in thermal equilibrium state, thus the density ne satisfies the Boltzmann relation [10–13] * Corresponding author. E-mail address: [email protected] (X. Zou). 0042-207X/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2008.03.032
(1)
V$ðni vi Þ ¼ 0;
(2)
v B ; xþe i mi ðvi $VÞvi ¼ eVfb c
(3)
where, ni, vi and mi denote ion density, ion velocity, and ion mass, respectively. Finally, the system is closed by Poisson’s equation v2 f ¼ 4peðni ne Þ: vx2
(4)
All the functions are assumed to vary only in the direction normal to the wall, V/ðv=vxÞb x . From Eqs. (2) and (3), we have vðni vix Þ ¼ 0; vx mi vix
vvi vf v B : ¼ e b xþe i c vx vx
(5) (6)
And we define cis ¼ ðTe =mi Þ1=2 is the ion sound speed, uic ¼ eB=mi c is the ion gyro frequency. For simplicity, we introduce dimensionless quantities: F ¼ ef=Te , x ¼ x=lD , ui ¼ vi =cis , Ne ¼ ne =ne0 , Ni ¼ ni =ni0 . And lD ¼ ðTe =4pne0 e2 Þ1=2 is electron Debye length.
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X. Zou et al. / Vacuum 83 (2009) 205–208
Fig. 1. Geometry of the magnetized plasma sheath model.
At the sheath edge x ¼ 0, from quasi-neutral condition, we have ne0 ¼ ni0 , so their ratio di ¼ ni0 =ne0 ¼ 1. From Eqs. (1), (4)–(6), we obtain Ne ¼ expðFÞ;
(7)
Mi ; uix
(8)
Ni ¼
4. Numerical results and discussions
vu vF 1=2 b0; b x þ di gi ui $ B uix i ¼ vx vx d2 F dx
2
Fig. 2. The dependence of the ion density distribution on the intensities of B for q ¼ 45 , uiy0 ¼ 0 and uiz0 ¼ 0.
¼ Ni Ne ;
(9)
(10)
where, in Eq. (8) Mi ¼ vix0 =cis is the ion Mach number, in Eq. (9) gi ¼ uic =upi is the ratio of ion gyro frequency to ion plasma frequency. b ¼ b Taking B x cos q þ b z sin q, then Eq. (9) can be written as 0 uix
vuix vF ¼ þ gi uiy sin q; vx vx
(11)
uix
vuiy ¼ gi ðuiz cos q uix sin qÞ; vx
(12)
uix
vuiz ¼ gi ðuiy cos qÞ: vx
(13)
In the following numerical simulations, some typical parameters are employed, such as n0 ¼ 5 108 ðcm3 Þ, Te ¼ 3ðeVÞ, and so on. The starting point is taken at x ¼ 0, where we set f ¼ 0, and vF=vxjx¼0 ¼ 0:01. 4.1. Ions enter the sheath only along the direction normal to the wall In the first place, we assume that the ions enter the sheath only along the x-axis direction. Their initial velocities in the y-axis and zaxis directions are zero (uiy0 ¼ 0 and uiz0 ¼ 0). Taking ion Mach number Mi ¼ 1, the numerical simulation results are shown in Figs. 2 and 3. As mentioned in Ref. [5], the magnetic field has gyral effect on the moving ions in the sheath. Lorentz force accelerates and decelerates the ion flow in the x-axis direction in each gyrate period, and induces fluctuations in the ion density distribution. However, ion flow usually cannot gyrate for one period, due to the limitation of some factors, such as sheath thickness, the intensity of the magnetic field, and so on. As a result, the density curves rise instantaneously, then drop, and do not have periodical fluctuations
From Eqs. (7)–(13), we obtain the numerical simulation results presented as follows.
3. Bohm’s criterion At the edge of sheath, x ¼ 0, F/0, Ni /1, and dF=dxs0. From the Eqs. (1), (4), and (11), we have
gi sin q Mi2 1 þ vF uiy0 : j
(14)
vx F¼0
If the ion enters the sheath with initial speed in the y-axis direction as zero, uiy0 ¼ 0, we obtain the Bohm’s criterion Mi2 1:
(15)
In this case, the Bohm’s criterion is the same as that in the absence of magnetic field. That is to say, only when the ion velocity in the x-axis direction is greater than or equal to the ion sound speed, can it enter the magnetized sheath. In another case, uiy0 s 0, the ion critical mach number is determined by Eq. (14).
Fig. 3. The dependence of the ion density distribution on the angles of B for B ¼ 2000 Gs, uiy0 ¼ 0 and uiz0 ¼ 0.
X. Zou et al. / Vacuum 83 (2009) 205–208
207
Fig. 6. The ion density distribution under various value of q for B ¼ 4000 Gs and ui0//B. Fig. 4. The ion density distribution under various value of uiy0 for B ¼ 200 Gs, q ¼ 20 and uiz0 ¼ 0.
(see Fig. 2). Fig. 2 indicates the dependence of the ion density distribution on the intensities of the magnetic field, while the angle of magnetic field is q ¼ 45 . As shown, ions are weakly magnetized in the weak magnetic field (B < 400 Gs), and the density distribution does not change obviously. While the ions are magnetized in the strong magnetic field, the density distribution curves change evidently. Also Fig. 3 shows the dependence of the ion density distribution on the angles of the magnetic field for B ¼ 2000 Gs. We can get an idea of the magnetic field component in the z-axis direction attribute to variation of ion density distribution. In one word when the ions enter the sheath only along the x-axis direction, the magnetic field holds up the ions moving to the wall. 4.2. Ions enter the sheath with an oblique incidence angle For the case of ions entering the sheath with an oblique incidence angle (uiy0 s 0 or uiz0 s 0), the action states of ions are more complicated. The ion critical mach number is calculated with Eq. (14). We get the numerical simulation results as followed. Fig. 4 shows the ion density distribution under various value of ion initial velocities in the y-axis direction for B ¼ 200 Gs, q ¼ 20 and uiz0 ¼ 0. Under the actions of the Lorentz force and the electron force, ions
Fig. 5. The ion density distribution under various value of q for B ¼ 4000 Gs, uiy0 ¼ 0 and uiz0 ¼ 0.3.
could gather in some regions, which brings on small peaks of the density curves. The fluctuations of the curve depend on both ion velocity and magnetic field. Fig. 5 indicates the ion density distribution under various value of angles of the magnetic field for B ¼ 4000 Gs, uiy0 ¼ 0 and uiz0 ¼ 0.3. Both the magnetic field components in the x-axis and z-axis directions vary with the angle. The reasons for the fluctuations of the curve are more complicated. From Figs. 4 and 5, we can see that when the ions have velocity component parallel to the board, Lorentz force induces fluctuations in the ion density under suitable conditions, including intensity and angle of the magnetic field. When the initial velocity of ion is parallel to the direction of magnetic field, at this moment, Lorentz force does not exist. However, after the accelerating action of electric field, the direction of velocity is no longer parallel to the magnetic field, and causes the density distribution slight changes (see Fig. 6). In Fig. 6, the ion density distribution under various value of angle of magnetic field is shown, while B ¼ 4000 Gs and ui0//B. 4.3. Space charge density distribution in the magnetized plasma sheath In the case of ions entering the sheath only along the direction normal to the wall, the dependence of the space charge density
Fig. 7. The dependence of the space charge density distribution on the intensities of B for q ¼ 30 , uiy0 ¼ 0 and uiz0 ¼ 0.
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X. Zou et al. / Vacuum 83 (2009) 205–208
(1) The magnetic field has significant effects on the ion density distribution and space charge density distribution, and the magnetic field component in the z-axis direction attribute to variation of ion density distribution. (2) The magnetic field holds up the ions moving to the wall, and Lorentz force induces fluctuations in the ion density under suitable conditions, including intensity and angle of the magnetic field. (3) The space charge density is slightly higher in some regions due to the fluctuations of the ion density, so the external magnetic field can make the shield distance of sheath longer. Acknowledgements This work has been supported by National Natural Science Foundation of China (Project 10605008). Fig. 8. The space charge density distribution under various value of uiy0 for B ¼ 200 Gs, q ¼ 20 and uiz0 ¼ 0.
distribution on the magnetic field is shown in Fig. 7. The solid line is the space charge density curve in the absence of magnetic field. Because of the characteristic of density distribution of ions and electrons, the space charge density distribution curve has a peak, which indicates that in this region, more positive particles gathering to shield the negative potential of the board. With the increase of intensity of magnetic field, the peak of the curve moves towards the edge of sheath. The reason is the effects of the Lorentz force on the density distribution of ions. So the external magnetic field can make the shield distance of sheath longer. While in another case of ions entering the sheath with an oblique incidence angle, ions could gather in some regions, so that the space charge density in such regions is slightly higher (see Fig. 8). 5. Conclusions The plasma sheath is easily influenced by the magnetic field. In the magnetized plasma sheath, two cases of ion incidence are discussed, and the following numerical conclusions are obtained.
References [1] Chodura R. Plasma-wall transition in an oblique magnetic field. Phys Fluid 1982;25:1628–33. [2] Kim G-H, Hershkowitz N, Diebold DA, Cho M-H. Magnetic and collisional effects on presheaths. Phys Plasmas 1995;2:3222–33. [3] Ahedo E. Structure of the plasma-wall interaction in an oblique magnetic field. Phys Plasmas 1997;4:4419–30. [4] Bornali Singha, Sarma A, Joyanti Chutia. Experimental observation of sheath and magnetic presheath over an oblique metallic plate in the presence of a magnetic field. Phys Plasmas 2002;9:683–90. [5] Zou X, Liu JY, Gong Y, Wang ZX, Liu Y, Wang XG. Plasma sheath in a magnetic field. Vacuum 2004;73:681–5. [6] Masoudi SF. Effect of pressure on plasma sheath in a magnetic field. Vacuum 2007;81:871–4. [7] Sato N. Magnetic effects in dusty plasma. Third International Conference On the Physics of Dusty Plasmas (New York) 2002:66–73. [8] Zou X, Liu JY, Wang ZX, Gong Y, Liu Y, Wang XG. Electronegative plasma sheath structure in a magnetic field. Chin Phys Lett 2004;21:1572–4. [9] Zou X. Characteristics of dust plasma sheath in an oblique magnetic field. Chin Phys Lett 2006;23:396–8. [10] Riemann KU. Theory of the collisional presheath in an oblique magnetic field. Phys Plasma 1994;1:552–8. [11] Stangeby PC. The Bohm-chodura plasma sheath criterion. Phys Plasma 1995;2: 702–6. [12] Valsaque F, Manfredi G. Numerical study of plasma-wall transition in an oblique magnetic field. J Nucl Materials 2001;290–293:763–7. [13] Baishya SK, Das GC. Dynamics of dust particles in a magnetized plasma sheath in a fully ionized space plasma. Phys Plasmas 2003;10:3733–45.