BIBECHANA 18 (1) (2021) 134-139 134 BIBECHANA ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Department of Physics, Mahendra Morang A.M. Campus, TU, Biratnagar, Nepal Modulation frequency and velocity variation of ions in a magnetized plasma sheath for different obliqueness of the field Bhesha Raj Adhikari 1,2,*, Hari Prasad Lamichhane1, Raju Khanal1 1 Central Department of Physics, Tribhuvan University, Kirtipur, Kathmandu, Nepal 2 Department of Physics, Bhaktapur Multiple Campus, Tribhuvan University, Bhaktapur, Nepal * Email: b.r.adhikari@hotmail.com Article Information: Received: May 29, 2020 Accepted: August 1, 2020 Keywords: Bohm criterion Kinetic theory Sheath Presheath Modulation frequency Damping constant ABSTRACT The understanding of ion dynamics in magnetized plasma sheath is crucial for all applications of plasma. The velocity variation as well as modulation frequency of ions in a magnetized plasma sheath has been studied for different obliqueness of magnetic field. The governing Lorentz force equation has been solved numerically for the given boundary conditions as applicable in kinetic simulation of the sheath. For different obliqueness of the magnetic field, the average values, maximum amplitude, damping factor as well as frequency of oscillation are studied. The oscillating velocity components change at different rates depending on their orientation with respect to the field direction. Significant changes in the damping factor and modulation frequency has been observed for all components of velocity; however, the frequency of oscillation remains same. As the obliqueness increases, shoulder natures in the components of velocity are observed. DOI: https://doi.org/10.3126/bibechana.v18i1.29171 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ 1. Introduction The formation of non-neutral region in the vicinity of material wall is a ubiquitous feature of bounded plasma and it has wide range of growing applications in diverse fields (e.g. plasma confinement for fusion, sputtering, etching, surface treatment etc.) [1]. In plasma, the thermal velocity of electrons is higher as compared to that of the ions. Due to this, the wall is charged up negatively with respect to the core plasma. Therefore, negative potential attracts the ions and repels electrons forming a thin positive space charge region near to the wall. This positive space charge region, known as the ‘sheath’ separates the negatively charged material wall from the quasineutral ‘presheath’ plasma. The in-streaming ions have to satisfy the Bohm criterion to ensure the stability of the overall plasma [2, 3]. The problem related to sheath is one of the oldest problems in plasma physics [3, 4] and once the plasma-wall interaction is well understood it will be possible to control heat loading, energy transfer and particle flow towards the wall and overall bulk plasma behavior [5 - 7]. http://nepjol.info/index.php/BIBECHANA mailto:b.r.adhikari@hotmail.com https://doi.org/10.3126/bibechana.v18i1.29171 https://creativecommons.org/licenses/by-nc/4.0/ Bhesha Raj Adhikari et al. / BIBECHANA 18 (1) (2021) 134-139 135 The problem of sheath formation whenever it faces a material surface, its nature and evolution with time is significant for all plasma applications. This is a topic of interest and various works are reported recently as well; however, the study of modulation frequency as well as the time evolution of the ion velocity profile in a magnetized plasma sheath for varying obliqueness is still lacking [8-10]. This work focuses in the time variation of ion velocity, which shows oscillating nature, and its modulation frequency in a magnetized plasma sheath for different obliqueness of the field. The governing Lorentz force equation are formulated and solved numerically for the given boundary conditions as applicable in kinetic trajectory simulation (KTS) method of the sheath region [8, 11, 12]. In typical boundary layer problems, the sheath region is of several electron-Debye lengths which is much smaller than the characteristic extension of the plasma. Such a sheath can only be formed, if the Bohm criterion [2, 5] is satisfied which demands that the ions enter the sheath region with a high velocity, which cannot be generated by thermal ion motion alone [5]. In the presence of a magnetic field, in kinetic form, the Bohm criterion reads 22 || 11 sCv  (1) where, i e PS ei PS i s m TTk C )(  + = is the ion-acoustic velocity defined at the presheath side of the sheath edge, with k is Boltzmann constant, i and e the ion and electron polytropic constants, respectively, i psT and e psT the ion and electron temperatures at the presheath side of the sheath edge, respectively and mi is the mass of ion species. 2. Model and basic equations The 1d3v model of magnetized plasma sheath is shown schematically in Fig. 1. The region of interest is bounded by two parallel planes at 0=x and Lx = , and the plasma consists of only electrons and singly charged ions. The plane at Lx = represents the “sheath entrance” that faces the plasma side and 0=x is the material wall, assumed to be non-emitting in the present case. The uniform magnetic field B  acts on the xy-plane which makes an angle  with the normal to the wall. In the presence of oblique magnetic field, the presheath consist of two distinct regions: collisional presheath and magnetic presheath [3, 13]. The general schematic diagram of magnetized plasma-wall transition region is shown in Fig. 2 [11]. As the sheath region is characterized by sharp gradients in a small scale of the order of electron- Debye lengths, it is usually treated kinetically, where the particle distribution are governed by the Boltzmann equation [1] c v t f f m F fv t f         =++   ..   (2) where F  is the macroscopic force acting on the particles, and ct f         is the time rate of change of distribution function ),,( tvrf  due to collisions. The symbol  represents the usual space-gradient operator whereas the symbol v represents the gradient in velocity space. For collisionless cases, suitable for sheath region as their dimension are much less than the mean free path, the equation (2) changes to so called Vlasov equation: 0).(. =+++   →→→→ fBvE m q fv t f v (3) In the KTS method, the kinetic equations are solved along with other basic equations describing the plasma for given boundary and initial conditions. The distribution function at any point along the trajectory can be obtained if its value at one point (i.e., at the boundary) is known. Density of the species ‘s’ is then given by Bhesha Raj Adhikari et al. / BIBECHANA 18 (1) (2021) 134-139 136 Wall Plasma  Y O Z X L → E → B X ( ) ( ) + − = vxvfdxn ss  ,3 , (4) which then yields the space charge density as ( ) ( )= s ss xnqx   . (5) Thus obtained space charge density is then used in the Poisson’s equation to obtain the electrostatic potential ( )x ( ) ( ) , 0 2 2   x dx xd  −= (6) and the electric field is given by ( ) ( ) dx xd xE    −= (7) The kinetic equations (3)-(7) are solved along the collisionless trajectories for given boundary conditions. The ion velocity in the plasma sheath are computed using the Lorentz force equation ( ) BvEq dt vd m  += (8) where q is the charge of ion species whose mass is m. The components of equation (8) are m qBv m qE dt dv zx sin −= (9) m qBv dt dv zy cos = (10) m qBv m qBv dt dv yxz  cossin −= (11) Since the magnetic field is dominating near the sheath entrance over the electric field, which is almost zero, the ions gyrate as they move slowly forward towards the wall. As the ions move towards the wall from the sheath entrance the electric field starts dominating, and hence the gyration decreases, and the equation of the damped harmonic oscillator can be is written as [14] ( ) )(sin  ++= − tAevtv tk m (12) where k, A,  and  are damping constant, amplitude, frequency of oscillation and phase angle, respectively. The damping constant of oscillating velocity components defined by equation (9) can be expressed as [14] 12 2 1ln tt vv vv k m m −         − − = (13) where 1v and 2v are the resultant velocity at time 1t and 2t , respectively and mv is the mean value of oscillating parts of velocity. Fig. 1: Schematic geometry of the plasma-wall interaction model. Fig. 2: Schematic diagram of magnetized plasma- wall transition region. 3. Results and Discussion In order to study the temporal variation of velocity, the equations (9)-(11) are solved by using Runge- Kutta method for the given boundary conditions. Bhesha Raj Adhikari et al. / BIBECHANA 18 (1) (2021) 134-139 137 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -1 -0.5 0 0.5 1 1.5 x 10 4 Time (s) V el o ci ty ( m /s ) v x v y v z Furthermore, the amplitude of oscillation, damping factor, oscillation frequency and mean value are obtained by solving the equations (12) and (13) numerically. The temporal dependence of all three components of ion velocity have been calculated at constant magnetic field of 6 mT and for three different obliqueness of the field (30°, 60° and 75°), shown in Figs. 3, 4, and 5, respectively. For the obliqueness of 30°, the x, y and z-component of velocity varies about the mean value of 0.007 m/s, 6661 m/s and 11600 m/s respectively with same frequency of oscillation 100 Hz (Fig. 3). Also these components show damping nature of oscillation with damping factor 13.8 s-1, 13.1 s-1 and 13.0 s-1, respectively. Likewise, these component of velocity varies with maximum amplitude of 9794 m/s, 3138 m/s and 1550 m/s, respectively. It also shows that the modulation frequency of x, y and z-component of velocity is same which is equal to 15.4 Hz. Fig. 4 shows that at obliqueness 60°, the x, y and z- component of velocity varies about the average values of -0.008 m/s, 11570 m/s and 6718 m/s, respectively with same frequency of oscillation, 100 Hz. Also these components show damping nature of oscillation with damping factor of 15.2 s- 1, 13.7 s-1 and 12.3 s-1 respectively. Likewise, these components of velocity vary with maximum amplitude of 9794 m/s, 2650 m/s and 4632 m/s, respectively. The modulation frequency of x, y and z-component of velocity for this case is 16.2 Hz, 15.8 Hz and 16.2 Hz, respectively. At obliqueness of 75°, the x, y and z-component of velocity varies about the average values of 0.0058 m/s, 11580 m/s and 3137 m/s respectively, with same frequency of oscillation 100 Hz as shown in Fig. 5. Also these components show damping nature of oscillation with the damping factor as 12.9 s-1, 14.9 s-1, and 14.1 s-1, respectively. Likewise, these component of velocity varies with the maximum amplitude of 9794 m/s, 1690 m/s and 6657 m/s, respectively. The modulation frequency of x, y and z-components of velocity are 15.8 Hz, 16.0 Hz and 15.4 Hz, respectively. In this case, of obliqueness 75°, shoulder nature in the velocity components are observed around 0.05 second which was not the case when the angles were smaller. The obtained results are summarized in Table 1, where the average value, maximum amplitude, damping factor, frequency of oscillation and modulation frequency of all three component of ion velocity at obliqueness of 30°, 60° and 75°. The y- component of mean value (11580 m/s) is maximum at 75° compared to other angles. The amplitude is maximum (9793.99 m/s) of the x-component in two cases, when the angles are 30° and 75°, whereas amplitude is minimum (1550 m/s) for z-component at 30°. The damping factor of the z-component is minimum (13.0 s-1) when the angle is 30°, on the other hand it is maximum for y-component (14.9 s- 1) at 75°. Similarly, modulation frequency in the x- component and z-component is maximum (16.2 Hz) at angle 60°, whereas frequency of oscillation remained same for all components of velocity for each cases considered. Fig. 3: Temporal variation of velocity of ions for a magnetic field of 6 mT at angle 30°. Bhesha Raj Adhikari et al. / BIBECHANA 18 (1) (2021) 134-139 138 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -1 -0.5 0 0.5 1 1.5 x 10 4 Time (s) V el o ci ty ( m /s ) v x v y v z 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -1 -0.5 0 0.5 1 1.5 x 10 4 Time (s) V el o ci ty ( m /s ) v x v y v z Fig. 4: Temporal variation of velocity of ions for a magnetic field of 6 mT at angle 60°. Fig. 5: Temporal variation of velocity of ions for a magnetic field of 6 mT at angle 75°. Table 1: The average value, maximum amplitude, damping factor, frequency of oscillation and modulation frequency of ion velocity 4. Conclusions Velocity variation as well as modulation frequency of ions in a magnetized plasma sheath has been studied for different obliqueness of magnetic field. It has been observed that the ion velocity at presheath-sheath boundary is affected by varying the obliqueness of the field. Various parameters like average ion velocity, amplitude, modulation frequency, and the damping factor can be controlled by varying the obliqueness. However, the frequency of oscillation is independent of obliqueness of magnetic field. The changes in these parameters results due to the change in contribution of the magnetic field with obliqueness. Present work is useful in understanding the time evolution of the particles velocities, and hence, exact particle behavior in magnetized plasma sheath region which is important in diverse plasma applications: material processing, surface treatment, plasma etching, confinement of plasma in fusion devices, and many more. Acknowledgement B. R. Adhikari would like to acknowledge the University Grants Commission, Nepal for the Ph. D.  Average Value (m/s) Maximum Amplitude (m/s) Damping factor (s-1) Frequency of oscillation (Hz) Modulation frequency (Hz) vxm vym vzm vxa vya Vza vx vy vz vx vy vz vx vy vz 30° 0.007 6661 11600 9793.99 3138 1550 13.8 13.1 13.0 100 100 100 15.4 15.4 15.4 60° -0.008 11570 6718 9794 2650 4632 15.2 13.7 13.3 100 100 100 16.2 15.8 16.2 75° 0.0058 11580 3137 9793.99 1690 6657 12.9 14.9 14.1 100 100 100 15.8 16 15.4 Bhesha Raj Adhikari et al. / BIBECHANA 18 (1) (2021) 134-139 139 fellowship. The authors warmly thank Mr. Suresh Basnet for fruitful discussions. References [1] R. Chalise and R. Khanal, A kinetic trajectory simulation model for magnetized plasma sheath, Plasma Phys, Control. Fusion 54 (2012) 095006 (5pp). https://doi.org/10.1088/0741-3335/54/9/095006 [2] D. Bohm, The characteristics of electrical discharges in magnetic fields, Edited by A. Guthry and P.K. Wakerling, MC-Graw Hill, 1949. [3] R. Chodura, Plasma wall transition in an oblique magnetic field, Phys. 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