Effect of dust grains on the parametric

coupling of a lower hybrid wave driven ion

cyclotron wave in a tokamak plasma
Cite as: AIP Advances 12, 035026 (2022); https://doi.org/10.1063/5.0085062
Submitted: 12 January 2022 • Accepted: 21 February 2022 • Published Online: 14 March 2022

Amit Kumar, Ruby Gupta and Jyotsna Sharma

AIP Advances 12, 035026 (2022); https://doi.org/10.1063/5.0085062 12, 035026

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Effect of dust grains on the parametric coupling

of a lower hybrid wave driven ion cyclotron wave
in a tokamak plasma
Cite as: AIP Advances 12, 035026 (2022); doi: 10.1063/5.0085062
Submitted: 12 January 2022 • Accepted: 21 February 2022 •
Published Online: 14 March 2022

Amit Kumar,1 Ruby Gupta,2 and Jyotsna Sharma1,a)

AFFILIATIONS
1
Department of Physics, Amity School of Applied Sciences, Amity University Haryana, Manesar, Gurugram 122051, India
2
Department of Physics, Swami Shraddhanand College, University of Delhi, Alipur, Delhi 110036, India

a)
Author to whom correspondence should be addressed: plasmajyoti@gmail.com

ABSTRACT
In this article, the effect of dust charge fluctuations on the parametric upconversion of a lower hybrid wave into an ion cyclotron wave
and a side band wave in a two-ion species tokamak plasma is studied. When the oscillatory velocity of plasma electrons is a few percent
of the sound velocity, the lower hybrid wave becomes unstable and decays into two modes: an ion cyclotron wave mode and a low fre-
quency lower hybrid side band wave. Furthermore, a ponderomotive force by a lower hybrid pump and a side band wave is exerted on the
existing electrons, which drives the ion cyclotron decay mode. The presence of negatively charged dust grains and their shape, size, radius,
and density influence the instability. The growth rate of instability is calculated by considering typical existing D–T (Deuterium–Tritium)
dusty plasma parameters, and it is observed that the growth rate increases with the relative density of dust grains, number density of dust
grains, oscillatory velocity of electrons, and amplitude of pump waves. However, the normalized growth rate increases with the unsta-
ble wave frequency, and it also increases as we increase the ratio of deuterium to tritium density. Here, the growth rate decreases with
the increase in the size of dust grains and electron cyclotron frequency. The theoretical results summarized in the present study are able
to efficiently elaborate the complexity produced in plasma properties in a tokamak due to the dust–plasma interactions, which are briefly
discussed here.
© 2022 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0085062

I. INTRODUCTION fusion plasmas. Initially, high power lower hybrid waves having a
frequency range of 500 MHz to 1 GHz were the strong candidates
Complex plasmas, space plasmas, and plasmas in fusion devices for heating a magnetized plasma to thermonuclear temperatures,
contain many ions species, e.g., positive ions, negative ions, elec- but in recent years, the large amplitude lower hybrid waves are
trons, and negatively charged dust grains. The addition of dust useful at higher frequency (1–5 GHz) for driving non-inductive
grains modifies plasma properties and wave behaviors and leads currents in tokamak plasmas. Hence, it opened the possibility of
to the occurrence of additional wave modes. Dispersion rela- running tokamak in the steady state. Experimental observations2,3
tions for these wave modes become more complicated due to the show that the parametric instability occurs and a relationship exists
wave–particle interactions in the plasma.1–47 The interest in studying between ion heating and parametric instabilities. The propagation
the effect of dust grains on the parametric instabilities of large ampli- of lower hybrid wave (LHW) instability in a non-uniform medium
tude pump waves, lower hybrid current drive, and heating inside a was studied by Porkolab, and linear theory of lower hybrid paramet-
tokamak and in small plasma devices is growing very fast these days. ric instabilities via a non-uniform pump wave in a uniform plasma
The lower hybrid waves (LHWs) are highly electrostatic with a large was explained by Berger et al.5 Nishikawa and Liu6 found that a
Ð
→
wave propagation vector k along the magnetic field, which makes traveling pump wave produces instabilities with a finite frequency
them attractive for radio frequency heating of tokamak devices1 and shift from pump waves. Anomalous heating of a plasma near the

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Ð
→
lower hybrid frequency and the parametric instabilities were stud- hybrid wave having frequency ω1 and wave number k 1 transfers
ied by Kindel et al.7 The decay instability of lower hybrid waves in Ð
→
the oscillatory velocity v 1 to electrons. As a result, it decays into an
homogenous and inhomogeneous plasmas was explained by Ott;8 Ð
→
ion cyclotron wave having frequency ω and wave number k and a
here, the author carried out the expression of growth rates. Porko-
lower hybrid side band wave having frequency ω2 and wave num-
lab9 derived the dispersion relation for parametric instabilities near Ð
→ Ð
→ Ð → Ð →
the lower hybrid frequency. The author obtained the expression of ber k 2 in this complex plasma, where ω2 = ω − ω1 , k 2 = k − k 1 .
the growth rate and threshold for both the quasi-ion modes and The lower hybrid side band couples with the pump wave to exert a
purely growing mode. Later, Kuo and Chen10 explored the para- ponderomotive force on electrons that drives the low frequency ion
metric instabilities excited by localized pump waves near the lower cyclotron mode, and these emitted ion cyclotron waves propagate
hybrid frequency. Here, the expression of threshold growth rates, reverse to the direction of a lower hybrid wave.
threshold conditions, and spatial amplification factors are obtained In the present paper, the effect of dust charge fluctuations on
for the oscillating two stream instability. The threshold of paramet- the parametric upconversion of a large amplitude lower hybrid wave
ric instabilities near the lower hybrid frequency was observed by (LHW) into an ion cyclotron wave and a lower hybrid side band
Berger and Perkins,11 and they calculated the damping rate and low- wave is described. The growth rate of instability is analyzed in the
est threshold for decay into a lower hybrid wave plus a low frequency presence of negatively charged dust grains and presented in Sec. II.
sound wave or an ion Bernstein wave or a backward ion cyclotron Section III provides a detailed summary of results obtained, which
wave. Porkolab12,13 presented a good review of parametric instability are concluded in Sec. IV.
of lower hybrid waves. The nonlinear mechanism by which a lower
hybrid wave having a high phase velocity transfers its momentum II. INSTABILITY ANALYSIS
and energy to the low parallel velocity runways was explained by
Consider a large amplitude lower hybrid pump wave prop-
Liu et al.14 Later, the density threshold for parametric instability of
agating through the Deuterium–Tritium (D–T) complex plasma
lower hybrid waves inside a tokamak was explored by Liu et al.15
inside a tokamak having equilibrium electron density n0e , deuterium
Fisch18 studied the physics of confining a tokamak with radio fre-
quency currents. The author found that waves having a high phase ion density n0D , tritium ion density n0T , and dust grain density
velocity can produce a current in the reactor plasma so that steady n0d , and a static magnetic field is applied along the z direction,
state operation of a tokamak is possible with acceptable power dissi- Bŝ
z. The charge, mass, and temperature of electrons and deuterium
pation. The theory of parametric instabilities in a plasma containing and tritium ions are (−e, me , T e ), (ZD e, mD , T D ), and (ZT e, mT , T T ),
hot e− s and ions near the lower hybrid frequency was explained respectively, whereas Qd and md represent the charge and mass of
by Boldyrev.19 dust grains. The electrostatic potential of a large amplitude lower
The theory of electrostatic instabilities in a magnetized plasma hybrid pump wave propagating through the plasma is described as
excited by the help of a large amplitude pump wave oscillating at a
frequency close to the lower hybrid frequency was studied by Rambo ϕ1 = A1 exp−i(ω1 t−k1x x−k1z z) , (1)
and DeGroot.20 Chen and Birdsall21 exposed the problem of heating
caused by a large amplitude low frequency pump wave (ω0 ≃ ωpi )
in a magnetized plasma. Takase et al.22 presented the experimen- ⎛ ω2pe k21z ⎞
ω 21 = ω 2LH 1 + 2 ,
tal results of parametric instabilities obtained by radio frequency ⎝ ωpD + ωpT k21 ⎠
2
probes and CO2 laser scattering during the Alcator C lower hybrid
experiments. 1/2 1/2
ω2 +ω2 4 π n0D ZD2 e2
The current drive experiments on a number of tokamak where ωLH = ( 1+ω
pD
2 /ω2 )
pT
, ωpD = ( ) , and ω pT
mD
devices have been successfully performed, and the ion heating pe ce
1/2
has been observed to company the lower hybrid waves’ current 4 π n0T ZT2 e2
=( mT
) represent the frequency of the lower hybrid
drives. In the HL-1 M tokamak, ion heating was explained by
Liu,26 when the plasma density is greater than 3.5 × 1013 cm−3 , wave, deuterium ions, and tritium ions, respectively, and
1/2
and for low densities ∼1012 cm−3 , a high ion temperature was
0e 2
ωpe = ( 4 π mn e e ) and ωce = me Be sc represent the electron plasma
observed in the Triam-1 M tokamak, which was studied by frequency and cyclotron frequency of electrons, respectively.
Kuang et al.27 The heating of a lower hybrid wave in the HT-7 The lower hybrid pump wave transmits the oscillatory velocity
tokamak was investigated by Liu et al.31 The quasi-steady state Ð
→v 1 to electrons, given by the following equation of motion:
H-mode with large plasma density due to the injection of lower
∂Ð
→
hybrid current drive and lower hybrid heating with a threshold
v1 Ð
+→
v 1 ⋅ ∇Ð
→ e∇ ϕ1 Ð
power of 50 KW in the HT-6M tokamak was experimentally ( v 1) = −→
v1×Ð
→
ω ce . (2)
observed by Li et al.32 ∂t me
However, the physics of the parametric process of the cur-
rent study can be understood as follows: lower hybrid waves are After linearizing the equation of motion and replacing ∂
∂t
by −iω1
launched into a tokamak plasma by a phased array of waveguides Ð
→
and ∇ by i k 1 , we get
with the parallel phase velocity ranging from c/4 to c/2. As the
lower hybrid waves propagate toward the center in well-defined res- Ð→
− i ω1 Ð
→
v1 +Ð
→
v1×Ð
→
onance cones, they attain a large amplitude at higher densities and e i k 1 ϕ1
ω ce = . (3)
are prone to parametric instabilities. Now, a large amplitude lower me

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On simplifying the above equation, we obtain the parallel and and

axial components of the velocity, given as Ð→ Ð →
e ϕ1 ϕ2 ⎛ k x ⋅ k 1x × ωce ⎞
e k1z ϕ1 ϕp = − (ω1 kz − ωk1z ) ,
v1 z = − , (4) 2 m ωce ⎝ ikz ω1 ω2
e 2
⎠
me ω1
where ϕp represents the ponderomotive potential.
i e ϕ1 Ð
→ → Ð
→ The perturbed linear density of electrons is given as
Ð
→
v 1– =− e 2 (k1×Ðω ce + iω1 k 1 ). (5)
m ωce k2
NL = ( ) χe ϕ , (12)
Using a perturbed velocity in the continuity equation 4πe

and the nonlinear electron perturbed density is given as

+ ∇(n0e ⋅ Ð
→
∂n1
( v )) = 0,
∂t
k2
N NL = ( ) χe ϕp . (13)
we get the perturbed density of electrons as 4πe

n0e e ϕ1 k21z k21x Therefore, the total perturbed electron density is given as
n1 = − ( 2 − 2 ). (6)
me ω1 ωce
k2
n = N L + N NL = ( ) χe (ϕ + ϕp ) . (14)
Now, the large amplitude lower hybrid pump wave decays into 4πe
an electrostatic ion cyclotron wave mode of potential ϕ and a low
frequency lower hybrid side band of potential ϕ2 . These perturbed densities arise from the parallel ponderomotive
The potential of an ion cyclotron wave mode can be written as force and the self-consistent field of the low frequency mode.
Similarly, the linear perturbed density of deuterium and tritium
Ð
→→
(ωt− k ⋅Ð
r) ions is given as
ϕ = A exp−i . (7)
k2
In addition, the potential of a lower hybrid side band wave is n1D = −( ) χ1D ϕ (15)
4 π ZD e
given as
Ð
→ → and
(ω2 t− k 2 ⋅Ð
r)
ϕ2 = A2 exp−i , (8)
k2
Ð
→ Ð → Ð → n1T = −( ) χ1T ϕ , (16)
where ω2 = ω − ω1 and k 2 = k − k 1 . The perturbed velocity of 4 π ZT e
Ð→
electrons at the lower hybrid side band (ω2 , k 2 ) is written as where
e k2z ϕ2 ω2pe
v2 z = − , (9) χe ≈ 1 + 2
,
me ω2 k2 vth
2 ω2pj ω ω − nωcj (17)
Ð
→ i e ϕ2 Ð
→ → Ð
→ χ1j = [1 + ∑Z( )In (bj )e−bj ],
v 2– = − e 2 ( k 2 × Ð
ω ce + iω2 k 2 ). (10) 2
k2 vthj k vthj n kz vthj
m ωce
j = D, T,
The lower hybrid side band couples with the pump wave to
produce a low frequency ponderomotive force on electrons
Ð
→
F p = −me Ð→
v ⋅ ∇Ð → e 1/2 j 1/2
v . This ponderomotive force have two compo- where vth = ( 2Tme
) and vthj = ( 2T
mj
) are the thermal velocity of
Ð
→ Ð
→ Z eB
nents, F p– and F pz , perpendicular and parallel to the magnetic electron and deuterium and tritium ions and ωcj = mj j cs and I n (bD ),
Ð
→ I n (bT ) represent the ion cyclotron frequencies and the modified
field. The response of electrons to F p– is strongly suppressed by the
k2x vthD
2
applied magnetic field, and the response of electrons is significantly Bessel functions of order n of arguments bD and bT or bD =
Ð
→ 2ω2cD
considerable along F pz . Hence, the low frequency nonlinearity k2 v2
arises mainly through parallel ponderomotive force. and bT = 2ω 2 .
x thT
cT

The parallel component of ponderomotive force on electrons at The perturbed density of dust grains may be written as
Ð
→
ω, k can be written as k2
e
n1d = χ1d ϕ, (18)
4 π Q0d
Fpz = − ( Ð
m →
v 1 ⋅ ∇v2z + Ð
→
v 2 ⋅ ∇v1z ),
2 and the dust charge fluctuations can be written as
i.e.,
dQ1d n1i n1e
+ η Q1d = −∣I0e ∣( 0i − 0e ),
Fpz = iekz ϕp , (11) dt n n

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0e
where I 0e is the electron current and η = 0.01 ωpe a λDn n0i is the dust The ions’ perturbed density can be represented as
charging rate. a is the dust grain size. λD = ωvthpe is the Debye length.
vth is the thermal speed of electrons, and k22j
n2j = χ2j ϕ2 , (25)
4 π Zj e
i∣I0e ∣ n1i n1e
Q1d = [ 0i − 0e ], (19)
(ω + iη) n n
ω2 ω2 k2 ω2
where χ2 = ωpe2 − ωpe2 k2z2 and χ2j = − ω2pj , with j = D, T for deuterium
where n is the density of equilibrium ions: i = D represents the den-
0i ce 2 2 2
and tritium ions. In the present case, the nonlinearity arises as dust
sity of deuterium ions and i = T represents the density of tritium
grains and ions are ignored because they are suppressed by their
ions.
large masses. Using electron and ions perturbed density in Poisson’s
Now, putting all the values from Eqs. (14)–(16) into Eq. (19),
equation, we obtain
we get
2
i ( 4πn 0e e )
∣I0e ∣k
n0e n0e ∇2 ϕ2 = 4π(n2 e − n2D ZD e − n2T ZT e).
Q1d = − [χe (ϕ + ϕp ) + χ1D + χ 1T ]. (20)
(ω + iη) n0i n0i
After simplifying the above equation, we get
Substituting the values from Eqs. (14)–(16), (18), and (20) into
Poisson’s equation, we obtain
ω2pe ω2pe k22z ω2pD ω2pT 4πe
2 1D 1T
∇ ϕ = 4π(ne − n ZD e − n ZT e − n Q1d − Q0d n ), 0d 1d (1 + − 2 2 − 2 − 2 )ϕ2 = − 2 nNL
2 ,
ω2ce ω2 k2 ω2 ω2 k2

4πe NL
k2 k2 ε2 ϕ2 = − n2 , (26)
∇2 ϕ = 4π[ χe ( ϕ + ϕp ) e + χ1D ϕ ZD e k22
4πe 4 πZD e
Ð
→ Ð
+
k2
χ1T ϕ ZT e ] k2 k2⋅ →v ∗1
ε2 ϕ2 = (ε − χ e ) ϕ.
4 πZT e 2
k2 2 ω2
⎡ n0d i∣I0e ∣ k2 ⎤
⎢ 4π e n0e n0e n0e ⎥
− 4π⎢
⎢ (ω + iη) χe (ϕ + ϕp ) + χ1D n0i + χ1T n0i
⎥.
⎥ (21) After simplifying Eqs. (22) and (26), we obtain
⎢ ⎥
⎣ ⎦
After simplifying Eq. (21), we get ε ε2 = μ (27)

iβ iβ n0e and
[1 + χe (1 + ) + χ1D (1 + )
(ω + iη) (ω + iη) n0i
iβ n0e iβ iβ k2 U 2 ω k1z
+ χ1T (1 + ) + χ1d ] ϕ = − χe (1 + )ϕp , μ = −χe (1 + )(ε − χe ) 2 2 sin2 θ (1 − ), (28)
(ω + iη) n0i (ω + iη) ω + iη k2 4 ω2 ω1 kz

iβ where U = emk1xe ω∣ϕce1 ∣ represents the oscillatory velocity of plasma elec-

ε ϕ = − χe (1 + )ϕp . (22)
(ω + iη) trons, ∣ϕ1 ∣ is the amplitude of pump waves, and θ is the angle between
k1x and kx .
Ð
→
The density perturbation at ω, k couples with the oscillatory veloc- In the absence of the pump wave (μ = 0), ε = 0, the linear dis-
Ð
→
ity of electrons v 1 and produces a nonlinear density perturba- persion relation corresponding to an ion cyclotron mode and the
Ð
→ lower hybrid side band wave is termed resonant decay.
tion at the lower hybrid side band ω2 , k 2 . Solving the continuity
equation for the nonlinear density at the lower hybrid side band, We can obtain the growth rate as

+ 1 ∇ ⋅ (nÐ
→
∂nNL
2
∂t 2
v ∗ ) = 0, we obtain the value of nonlinear electron
1 μ
density as follows: Γ2 = − ∂ε ∂ε2 , (29)
∂ω ∂ω2
Ð
→ →∗
n k ⋅Ðv1
nNL
2 = . (23)
2 ω2 where
The perturbed linear density of electrons due to self-consistent
potential is given as ∂ε2 2 ω2pe
≈ (1 + 2 ) (30)
∂ω2 ω2 ωce
k22
nL2 = ( ) χ2 ϕ2 . (24)
4πe and

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∂ε 2ω2pd iβ 2ω2pD ωcD I1 (bD ) e−bD iβ 1

= − χ e ( ) + [ ][1 + ]
∂ω ω3 (ω + iη)2 2
k2 vthD (ω − ωcD )2 (ω + iη) d

2ω2pT ωcT I1 (bT ) e−bT iβ 1 2ω2pD ω I1 (bD ) e−bD iβ 1

+ [ ][1 + ] − 2 2 [1 − I0 (bD ) e−bD − ][ ]
2 2
k vthT (ω − ωcT ) 2 (ω + iη) d k vthD (ω − ωcD ) (ω + iη)2 d

2ω2pT ω I1 (bT ) e−bT iβ 1

− [1 − I 0 (b T ) e −bT
− ][ ]. (31)
2
k2 vthT (ω − ωcT ) (ω + iη)2 d

Using the values from Eqs. (28), (30), and (31) in (29), we where
obtain the value of the growth rate as
2 ω2pe
⎡ 2 2 ⎤1/2 A= (1 + 2 )
⎢ χe (1 + iβ
)(ε − χe ) 4kω2Uk2 sin2 θ (1 − ω k1z ⎥
) ω2 ωce
⎢ ω+i η ω1 kz ⎥
Γ=⎢ 2 2
⎥ , (32)
⎢ AC ⎥
⎢ ⎥
⎣ ⎦ and

2ω2pd iβ 2ω2pD ωcD I1 (bD ) e−bD iβ 1

C= − χe ( )+ 2 2 [ ][1 + ]
ω 3 (ω + iη)2 k vthD (ω − ωcD )2 (ω + iη) d

2ω2pT ωcT I1 (bT ) e−bT iβ 1 2ω2pD ω I1 (bD ) e−bD iβ 1

+ [ ][1 + ] − [1 − I0 (bD ) e−bD − ][ ]
2 2
k vthT (ω − ωcT ) 2 (ω + iη) d 2 2
k vthD (ω − ωcD ) (ω + iη)2 d

2ω2pT ω I1 (bT ) e−bT iβ 1

− [1 − I 0 (b T ) e −bT
− ][ ], (33)
2
k2 vthT (ω − ωcT ) (ω + iη)2 d

FIG. 1. The growth rate γ (s−1 ) as a

function of electron cyclotron frequency,
ωce .

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where β = ∣I0e ∣ n0d

represents the coupling parameter, which defines density n0e = 109 cm−3 , plasma ion density n0i = 109 –5 × 105 cm−3 ,
e n0e
the coupling between dust grains and plasma. Here, d = nn0e repre-
0i
density of dust grains n0d = 5 × 105 cm−3 , and dust grain size
sents the relative density of negatively charged dust grains. a = 1–5 μm. The growth rate γ (s−1 ) as a function of electron
cyclotron frequency ωce is plotted in Fig. 1 using the above-
mentioned parameters, and it is observed that the growth rate
III. RESULTS AND DISCUSSION decreases with the increase in electron cyclotron frequency due
to the fact that, as the electron cyclotron frequency increases, the
We solve Eq. (32) numerically to obtain the growth rate of the Ð
→ Ð →
unstable mode using typical parameters of the tokamak of Kumar applied magnetic field increases and the contribution of E × B drift
e
and Tripathi:35 ωωLH1 = 2, kω1z1c = 2, TTD = 1.5, T D = T T , ZD = 1, ZT = 1, decreases at higher values of magnetic field, which, in turn, declines
the growth rate. Using Eq. (32), we have plotted Fig. 2 where we
ω2pe mD mT
U
VthD
= 2, ω2ce
= 14 , me
= 3672, me
= 5508 (D–T plasma), electron observed that the growth rate increases linearly with the oscillatory

FIG. 2. The growth rate γ (s−1 ) with

the oscillatory velocity of plasma
electrons, U.

FIG. 3. The growth rate γ (s−1 ) vs

relative density d (= n0i /n0e ) of dust
grains.

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FIG. 4. The growth rate γ (s−1 ) with

dust grain size a (cm) for the same
parameters, as used in Fig. 1.

velocity of plasma electrons U. As the oscillatory velocity of plasma damping becomes more significant at higher values of d(= n0i /n0e ).
electrons is greater than the sound velocity, the lower hybrid wave The presence of dust particles of various sizes and shapes in the
becomes parametrically unstable, therefore decaying into an ion plasma affects the operation of a tokamak, and the dust particles hav-
cyclotron mode and a lower hybrid side band wave. Our theoretical ing high atomic number (Z) impurities such as tungsten can become
results of Fig. 2 are similar to the experimental observations of radioactive when came into contact with tritium ions. Hence, it
Porkolab12 [see Figs. 16(a) and 12(a)]. In Fig. 3, the growth rate modifies the growth rate and operation of the tokamak significantly.
γ (s−1 ) with relative density of dust grains d(= n0i /n0e ) is plotted In Fig. 4, the growth rate as a function of dust grain size is plotted
and we can see that the growth rate increases initially with the where we summarized that as the size of dust grains increases, the
increase in relative density of dust grains, but at higher values of growth rate decreases. This can be explained as follows: increasing
d(= n0i /n0e ), it increases slowly because the contribution of landau the dust grain size reduces the inter-grain spacing, so the charge

FIG. 5. The growth rate γ (s−1 ) as a

function of the number density of dust
grains n0d (cm−3 ).

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FIG. 6. The normalized growth rate

γ/ωcT as a function of normalized fre-
quency ω/ωcT for a different deuterium
0D
to tritium density ratio nn0T = 0.25 and
n0D
n0T
= 0.5.

present on the dust grains decreases. As a result, the potential of decreases the growth rate. The presence of dust in a plasma has
dust grains is reduced, which decreases the growth rate. It means a significant effect on the growth rate of the unstable mode; the
that the large-sized dust grains stabilize the instability. The varia- growth rate increases with the number density of dust grains and
tion of growth rate γ (s−1 ) with the number density of dust grains relative density of grains, whereas it decreases with the size of dust
n0d (cm−3 ) is displayed in Fig. 5. Here, we found that the growth grains. Therefore, the study about the causes and role of dust parti-
rate scales linearly with the number density of dust grains. This is cles and their sizes is very useful for fusion community in order to
due to the fact that increasing the density of dust grains changes the increase the efficiency of tokamaks. Our theoretical results may find
aggregate conduct of the plasma, which triggers the development applications in tokamaks.
of an electric field inside the plasma, altering the molecule direc-
tion in the plasma, which changes near-by plasma potential, and the
possibility of generating new waves inside a plasma is produced. It AUTHOR DECLARATIONS
means that adding dust into a plasma enhanced the instability, and
hence, the growth rate increased. In Fig. 6, the normalized growth Conflict of Interest
rate γ/ωcT as a function of normalized frequency ω/ωcT is plotted.
The authors have no conflict of interest.
Here, we observed that the normalized growth rate increases by
increasing the normalized frequency and ratio of deuterium to tri-
tium density. The growth rate is higher for n0D /n0T = 0.5 and smaller DATA AVAILABILITY
for n0D /n0T = 0.25. Our observed theoretical results of Fig. 6 are
The data that support the findings of this study are avail-
qualitatively similar to the experimental observations of Porkolab.12
able within the article and from the corresponding author upon
Therefore, our theoretical results are very significant in understand-
reasonable request.
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