A Unified Numerically Solvable Framework for Complicated Kinetic Plasma Dispersion Relations
Huasheng Xie
TL;DR
This paper introduces a comprehensive, numerically solvable framework for kinetic plasma dispersion relations accommodating multiple species, arbitrary drifts, and collisions, applicable to various plasma environments including space, astrophysics, and fusion.
Contribution
It develops the most general matrix-based numerical method for solving kinetic plasma dispersion relations with multiple species, drifts, and collision effects, extending previous models significantly.
Findings
Provides a unified dispersion relation framework for diverse plasma conditions.
Enables solving complex plasma wave problems without initial root guesses.
Applicable to space, astrophysics, fusion, and laser plasma studies.
Abstract
A unified numerically solvable framework for dispersion relations with arbitrary number of species drifting at arbitrary directions and with Krook collision is derived for linear uniform/homogenous kinetic plasma, which largely extended the standard one [say, T. Stix, {\em Waves in Plasmas}, AIP Press, 1992]. The purpose of this work is to provide a kinetic plasma dispersion relation tool not only the physical model but also the numerical approach be as general/powerful as possible. As a very general application example, we give the final dispersion relations which assume further the equilibrium distribution function be bi-Maxwellian and including parallel drift, two directions of perpendicular drift (i.e., drift across magnetic field), ring beam and loss-cone. Both electromagnetic and electrostatic versions are provided, with also the Darwin (a.k.a., magnetoinductive or magnetostatic)…
Click any figure to enlarge with its caption.
Figure 1
Figure 2
Figure 3| =- 1.0467968598346571444 - 2.1018525680357924343i | =0.37861161238699661757 - 1.3509435854325440801i | ||
| =0.54679685983465714437 + 0.037196505239893094435i | =1.2358876534356917085 - 1.2149821325576149719i | ||
| = | = | ||
| =- 5.5833741816150427087 - 11.208550459628098648i | =0.27393621805538084727 - 1.9417870375760945628i | ||
| =- 0.7399178112200519477 - 0.83951828462027428396i | =- 1.4652340919391423883 - 1.7896202996033145873i | ||
| =- 0.017340112270400811857 - 0.04630643962629377424i | =2.2376877251342932158 - 1.6259410241203623422i | ||
| =5.8406321051054954683 - 0.95360275132203964347i | =- 0.83925396636792203153 - 1.8919952115314256943i | ||
| = | = | ||
| =- 47.913598578418315281 - 106.98699311451399461i | =0.22536708628380726987 - 2.4862558428460328565i | ||
| =- 20.148858425809293248 + 12.874749056250453631i | =1.1590491549279069691 - 2.4061921257040740764i | ||
| =- 4.5311004339957471789E-3 + 6.3311756354943215316E-4i | =2.9785703941315209704 - 2.0490809954949754985i | ||
| =0.2150040123642351701 + 0.20042340981056393122i | =2.2568587892309227294 - 2.2080229126485700572i | ||
| =0.43131038679231352184 - 4.1505366661190555077i | =1.6738373878120108271 - 2.3235155478934783777i | ||
| =66.920673705505055584 + 20.747375125403268524i | =0.68229440981712468 - 2.4598334422617114946i | ||
| = | = | ||
| =- 10.020983259474214017 - 14.728932929429874883i | =0.22660012611958088508 - 2.0716877594897791206i | ||
| =- 0.58878169153449514493 + 0.19067303610080007359i | =- 1.7002921516300350075 - 1.882247422161272446i | ||
| =- 0.27475707659732384029 + 3.617920717493884482i | =1.1713932508560117853 - 1.9772503319208541098i | ||
| =4.5713742777499515344E-4 + 2.7155393843737098852E-4i | =3.0666201126826972102 - 1.5900208259325997176i | ||
| =0.017940627032508378515 - 0.036436053276701248142i | =2.3073274904105782764 - 1.7546732543728200654i | ||
| =10.366124263145749629 - 2.5069048649816145967i | =0.68720052490601906567 - 2.0402885259758440187i | ||
| = | = | ||
| =- 86.416592794839804566 - 147.57960545984972964i | =0.19664397441136646085 - 2.5854046363167904821i | ||
| =- 22.962540986214500398 + 46.211318219085729914i | =1.0004276870893045112 - 2.5277610669350594581i | ||
| =- 8.8757833558787660662 - 11.561957978688249474i | =1.4263380087098663429 - 2.4694803409658086505i | ||
| =- 0.025134802434111256483 + 0.19730442150379382482i | =2.382753075769737514 - 2.2903917960623787648i | ||
| =- 5.6462830661756538039E-3 - 2.7884991898011769583E-3i | =2.9566517643704010427 - 2.1658992556376956217i | ||
| =2.8262945845046458372E-5 + 2.6335348714810255537E-5i | =- 3.6699741330155866185 - 2.0087276133120462601i | ||
| =2.3290098166119338312 - 0.57238325918028725167i | =1.8818356204685089975 - 2.3907395820644127768i | ||
| =115.45666014287557906 - 2.8617578808752183449i | =0.59330036294742852232 - 2.5662607006180515205i | ||
| = | = | ||
| =- 579.77656932346560644 - 844.01436313629880827i | =0.16167711630587375808 - 2.9424665391729649011i | ||
| =- 179.52530851977905732 - 86.660002027244731382i | =1.1509135876493567245 - 2.874554296549015316i | ||
| =- 52.107235029274485215 + 453.3246806707749413i | =0.81513635269214329287 - 2.9085569383176322447i | ||
| =- 2.1607927691932962178 + 0.63681255371973499384i | =2.2362950589041724111 - 2.7033607074680388479i | ||
| =- 0.018283386874895507814 - 0.21941582055233427677i | =2.6403561313404041541 - 2.6228400297078984517i | ||
| =- 6.819511737162705016E-5 + 3.2026091897256872621E-4i | =3.5620497451197056658 - 2.4245607245823420556i | ||
| =- 2.8986123310445793648E-6 - 9.9510625011385493369E-7i | =4.1169251257106753931 - 2.3036541720854573609i | ||
| =2.3382228949223867744E-9 - 4.0404517369565098657E-9i | =4.8034117493360317933 - 2.1592490859689535413i | ||
| =0.01221466589423530596 + 0.00097890737323377354166i | =3.0778922349246567316 - 2.5301774598854448463i | ||
| =7.3718296773233126912 - 12.575687057120635407i | =- 1.8572088635240765004 - 2.7720571884094886584i | ||
| =44.078424019374375065 - 46.322124026599601416i | =1.496988132246689338 - 2.8290855580900544693i | ||
| =761.62579175738689742 + 185.11797721443392707i | =- 0.48636891219330428093 - 2.9311741817223824196i | ||
| = | = |
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A Unified Numerically Solvable Framework for Complicated Kinetic Plasma Dispersion Relations
Hua-sheng XIE
[email protected], [email protected]
Hebei Key Laboratory of Compact Fusion, Langfang 065001, China
ENN Science and Technology Development Co., Ltd., Langfang 065001, China
Abstract
A unified numerically solvable framework for dispersion relations with arbitrary number of species drifting at arbitrary directions and with Krook collision is derived for linear uniform/homogenous kinetic plasma, which largely extended the standard one [say, T. Stix, Waves in Plasmas, AIP Press, 1992]. The purpose of this work is to provide a kinetic plasma dispersion relation tool not only the physical model but also the numerical approach be as general/powerful as possible. As a very general application example, we give the final dispersion relations which assume further the equilibrium distribution function be bi-Maxwellian and including parallel drift, two directions of perpendicular drift (i.e., drift across magnetic field), ring beam and loss-cone. Both electromagnetic and electrostatic versions are provided, with also the Darwin (a.k.a., magnetoinductive or magnetostatic) version. The species can be treated either magnetized or unmagnetized. Later, the equations are transformed to the matrix form be solvable by using the powerful matrix algorithm [H. S. Xie and Y. Xiao, Plasma Science and Technology, 18, 2, 97, 2016], which is the first approach can give all the important solutions of a linear kinetic plasma system without requiring initial guess for root finding and thus can be extremely useful to the community. To our knowledge, the present model would be the most comprehensive one in literature for the distribution function constructed bases on Maxwellian, which thus can be applied widely for study waves and instabilities in space, astrophysics, fusion and laser plasma. We limit the present work to non-relativistic case.
keywords:
Plasma physics , Kinetic dispersion relation , Waves and instabilities , Matrix eigenvalue
††journal: XXX
1 Introduction
Due to the complicated evolution of charged particles and electromagnetic field, one of the most important feature of plasma is the numerous waves and instabilities. The fundamental features of linear waves and instabilities in uniform/homogenous plasma can be described by dispersion relation and are discussed by many authors in monographes [3, 5, 4] and textbooks (cf. [6]). In standard treatment of kinetic plasma dispersion relation, the velocity space equilibrium distribution function is assumed to be , thus can not treat the cases with drift across magnetic field. To including the arbitrary directions of drift and collision would largely extend the application range of the dispersion relation, with the instabilities in the non-uniform shock [14, 11] be one of numerous of them.
In this work, we try to provide a most comprehensive linear kinetic plasma dispersion relation tool to the community, which includes lots of new capabilities in both the physical model and algorithm. The tool is largely benefit from the powerful matrix approach of PDRK solver [1], which is the first algorithm can yield all the important kinetic solutions without initial guess for root finding. The new tool is named as PASS (Plasma wAve and inStability analySis), which includes the present work be kinetic version PASS-K succeed from PDRK [1] (hereafter ’PASS-K’ is refered equivalent to ’PDRK’), the fluid version PASS-F succeed from PDRF [16], and possibly more.
In the following sections, we firstly derive the most general kinetic dispersion relation equation with drift across the magnetic field and Krook collision in section 2. In section 3, we assume a very general extended Maxwellian equilibrium distribution function for all species and derive the corresponding final dispersion relation. In section 4, the corresponding equations suitable be solved by PASS-K matrix algorithm are derived. In sections 5, we give some benchmark results. In section 6, we give a summary and some discussions.
2 The General Non-relativistic Dispersion Relation
Considering that the widely used magnetized kinetic dispersion relation in literature such as in Ref.[3] does not including the drift across magnetic field, we firstly derive our new models. A similar magnetized model is derived only recently in Ref.[11], but which is still not as general as the present one. We limit our study to non-relativistic model. To help the reader, we will give detailed steps of our derivations.
2.1 Starting equations
We consider only the collisionless case or with a Krook collision, the kinetic equation for each species is the Vlasov equation with Krook collision at the right hand side
[TABLE]
where the distribution function , and , and are the charge, mass and the number density of species , respectively. When the collision frequency , the equation reduces to a collisionless case. The Maxwell equations for fields can be either
[TABLE]
for electromagnetic case, or
[TABLE]
for electrostatic case, where
[TABLE]
and is the speed of light. The accelerate term can be caused by other external forces such as the gravity and other (magneto-)hydro-dynamic forces due to spatial inhomogeneity, which then would cause drift motions across the magnetic field. A typical example is the low hybrid drift instability (LHDI) in space (e.g., in current sheet and shock cases) and fusion [9] plasma (e.g., in mirror and field-reversed configuration).
To extend the application range, we will also given the Darwin model [16] version. In Darwin mode, all field variables can be divided into two parts: the transverse (T, divergence free) part and the longitudinal (L, curl free) part, i.e.,
[TABLE]
The corresponding field equations are
[TABLE]
where for our usage we need only the last two of them. The only difference from full electromagnetic model is that the term is dropped in the Darwin model. The Darwin model which eliminates the high frequency electromagnetic wave , is particular interesting in theoretical study and kinetic particle-in-cell and Vlasov simulations, i.e., which can use large time steps in simulation and saves the computation resource a lot [10].
We assume the zero-order term be a homogenous system with , , , and . The conventional derivation is further assume and , whereas we treat also and in this work.
Without loss of generality, we assume the background magnetic field in direction, i.e., . The zero-order equations
[TABLE]
where . We introduce a cylindrical velocity coordinates111When , we have assumed the system to be Galilean invariant. It is not sure how much influence to the result yet. This issue is subtle, because the Vlasov equation is Galilean invariant and the electromagnetic field equations are Lorentz invariant and the total system is neither Galilean nor Lorentz invariant. with and , where and . We have , and . We have also , , and . Eq.(15) is
[TABLE]
Thus we have if , i.e., . And, we have assume the force balance in the z-direction, i.e., .
The first order kinetic equation is (we have dropped the ’1’ subscript)
[TABLE]
and Fourier transform the equation using , and also , we obtain
[TABLE]
where is the cyclone frequency. Without loss of generality, we can assume222Ref.[11] assumed the across magnetic field drift at direction but , which will limit that all species can only drift at direction. Later, they limit their discussion also to only . And thus their result is our result at and . the wave vector , which gives and . We study four cases:
Electromagnetic or Darwin, magnetized: , . 2. 2.
Electromagnetic or Darwin, unmagnetized: (), . 3. 3.
Electrostatic, magnetized: , (). 4. 4.
Electrostatic, unmagnetized: (), ().
For convenient to theoretical study, we let the user to choose whether a species is magnetized or unmagnetized333A typical case is in shock study, say Ref.[14], where electron are magnetized and drift ions are treated unmagnetized., i.e., say for a electromagnetic case, the different species can be either magnetized (labeled as ’m’) or unmagnetized (labeled as ’u’).
For unmagnetized case, Eq.(20) reduces to
[TABLE]
where or \bm{k}\cdot\bm{v}=k_{z}v^{\prime}_{\parallel}+k_{x}v^{\prime}_{\perp}\cos\phi^{\prime}+{\color[rgb]{1,0,0}k_{x}v_{dsx}}, which gives the solution for is
[TABLE]
For magnetized case, Eq.(20) reduces to
[TABLE]
where x_{s}=-\frac{\omega-k_{\parallel}v^{\prime}_{\parallel}{-\color[rgb]{1,0,0}k_{x}v_{dsx}+i\nu_{s}}}{\omega_{cs}} and . The solution for magnetized is much complicated. Instead of the approaches using the method of characteristics in Refs.[3] and [11], we using the approach similar to Ref.[6], which solves the differential equation directly. That is to say, Eq.(23) is a first order linear differential equation of the form
[TABLE]
which has a general solution
[TABLE]
where the integration factor term in our case is
[TABLE]
and thus
[TABLE]
with
[TABLE]
We notice that Eq.(27) can not reduce to (22) by set , i.e., the magnetized version can not reduce to unmagnetized version by set . Thus we should treat them separately.
2.2 Electrostatic case
For electrostatic case, , , and only Poisson equation for field is required. We have
[TABLE]
The unmagnetized species
[TABLE]
where or . The corresponding charge density is
[TABLE]
The velocity space integral may not be easy for complicated , and probably better calculate at coordinates instead of .
For magnetized species
[TABLE]
and
[TABLE]
which is not yet in a useful form. Let us do some further calculations step by step. We have used cylindrical coordinates with , and note . Use , we have
[TABLE]
Now, we use the following expansion
[TABLE]
where it the th order Bessel function. And we have
[TABLE]
and
[TABLE]
We further use
[TABLE]
and thus in Eq.(34)
[TABLE]
where we have used . Further integral out in Eq.(34) gives
[TABLE]
where because , and is Kronecker delta. Thus we find the final form is very similar to the standard Harris dispersion relation form, except the term .
Combine the magnetized and unmagnetized species and substitute them to the Poisson equation (29), we obtain the final electrostatic dispersion relation
[TABLE]
where .
2.3 Electromagnetic case
For electromagnetic case, , the field equations we needed are
[TABLE]
If we assume the relation between current and electric field be
[TABLE]
which will be calculated later via the kinetic equation, we obtain from Eq.(44)
[TABLE]
where can be expressed in terms of the dielectric tensor and gives the dispersion relation
[TABLE]
where is the unit tensor, and we have used . And the relation to conductivity tensor is
[TABLE]
with .
Now, we calculate .
For unmagnetized species
[TABLE]
where we have used , and , are dyadic product tensors. The result is similar to the one in Ref.[14], except our new term. Note also that the tensor .
For magnetized species
[TABLE]
We calculate firstly
[TABLE]
with
[TABLE]
where we have used , , .
terms: (\omega-k_{z}v^{\prime}_{\parallel}-k_{x}v^{\prime}_{\perp}\cos\phi^{\prime}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})(E_{x}\cos\phi^{\prime}+E_{y}\sin\phi^{\prime})+[v^{\prime}_{\perp}(E_{x}\cos\phi^{\prime}+E_{y}\sin\phi^{\prime})+v^{\prime}_{\parallel}E_{z}+v_{dsx}E_{x}+v_{dsy}E_{y}]k_{x}\cos\phi^{\prime}=(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})(E_{x}\cos\phi^{\prime}+E_{y}\sin\phi^{\prime})+(v^{\prime}_{\parallel}E_{z}+v_{dsx}E_{x}+v_{dsy}E_{y})k_{x}\cos\phi^{\prime}=(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})E_{y}\sin\phi^{\prime}+[(\omega-k_{z}v^{\prime}_{\parallel})E_{x}+k_{x}v_{dsy}E_{y}+k_{x}v^{\prime}_{\parallel}E_{z}]\cos\phi^{\prime}.
terms: (\omega-k_{z}v^{\prime}_{\parallel}-k_{x}v^{\prime}_{\perp}\cos\phi^{\prime}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})E_{z}+[v^{\prime}_{\perp}(E_{x}\cos\phi^{\prime}+E_{y}\sin\phi^{\prime})+v^{\prime}_{\parallel}E_{z}+v_{dsx}E_{x}+v_{dsy}E_{y}]k_{z}=(\omega-{\color[rgb]{1,0,0}k_{x}v_{dsx}})E_{z}+k_{z}v_{dsx}E_{x}+k_{z}v_{dsy}E_{y}+(k_{z}v^{\prime}_{\perp}E_{x}-k_{x}v^{\prime}_{\perp}E_{z})\cos\phi^{\prime}+k_{z}v^{\prime}_{\perp}E_{y}\sin\phi^{\prime}.
Our Eq.(52) is the same as in Ref.[11] when set our and , and their .
Thus, using and ,
[TABLE]
And thus we have
[TABLE]
with
[TABLE]
Write out each terms:
. [agree with Umeda18, except that his four terms can cancel]
- 2.
\Pi_{12}=(\frac{n}{y_{s}}J_{n}U_{s2}+iJ^{\prime}_{n}U_{s4}+J_{n}U_{s6})(v^{\prime}_{\perp}\frac{n}{y_{s}}+v_{dsx})J_{n}=(\frac{n}{y_{s}}J_{n}k_{x}v_{dsy}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+iJ^{\prime}_{n}[(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]+J_{n}k_{z}v_{dsy}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}})(v^{\prime}_{\perp}\frac{n}{y_{s}}+v_{dsx})J_{n}\\ =J_{n}^{2}v_{dsy}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})(\frac{n\omega_{cs}}{v^{\prime}_{\perp}}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}})+iJ_{n}J^{\prime}_{n}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]. [agree with Umeda18]
- 3.
. [agree with Umeda18, except that his two terms can cancel]
- 4.
\Pi_{22}=(\frac{n}{y_{s}}J_{n}U_{s2}+iJ^{\prime}_{n}U_{s4}+J_{n}U_{s6})(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})=(\frac{n}{y_{s}}J_{n}k_{x}v_{dsy}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+iJ^{\prime}_{n}[(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]+J_{n}k_{z}v_{dsy}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}})(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})\\ =J_{n}v_{dsy}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})[\frac{n\omega_{cs}}{v^{\prime}_{\perp}}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]+iJ^{\prime}_{n}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})[(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]. [agree with Umeda18]
- 5.
\Pi_{13}=(\frac{n}{y_{s}}U_{s3}+U_{s7})(v^{\prime}_{\perp}\frac{n}{y_{s}}+v_{dsx})J_{n}^{2}=(\frac{n}{y_{s}}[k_{x}v^{\prime}_{\parallel}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}-k_{x}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]+(\omega-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}})(v^{\prime}_{\perp}\frac{n}{y_{s}}+v_{dsx})J_{n}^{2}\\ =J_{n}^{2}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[n\omega_{cs}\frac{v^{\prime}_{\parallel}}{v^{\prime}_{\perp}}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+(\omega-n\omega_{cs}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]. [agree with Umeda18]
- 6.
. [agree with Umeda18, except that his two terms can cancel]
- 7.
\Pi_{23}=(\frac{n}{y_{s}}U_{s3}+U_{s7})J_{n}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})=(\frac{n}{y_{s}}[k_{x}v^{\prime}_{\parallel}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}-k_{x}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]+(\omega-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}})J_{n}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})\\ =(v_{dsy}J_{n}^{2}-iv^{\prime}_{\perp}J_{n}J^{\prime}_{n})[n\omega_{cs}\frac{v^{\prime}_{\parallel}}{v^{\prime}_{\perp}}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+(\omega-n\omega_{cs}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]. [agree with Umeda18]
- 8.
\Pi_{32}=(\frac{n}{y_{s}}J_{n}U_{s2}+iJ^{\prime}_{n}U_{s4}+J_{n}U_{s6})v^{\prime}_{\parallel}J_{n}=(\frac{n}{y_{s}}J_{n}k_{x}v_{dsy}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+iJ^{\prime}_{n}[(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]+J_{n}k_{z}v_{dsy}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}})v^{\prime}_{\parallel}J_{n}\\ =v^{\prime}_{\parallel}J_{n}^{2}(\frac{n\omega_{cs}}{v^{\prime}_{\perp}}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}})v_{dsy}+iv^{\prime}_{\parallel}J_{n}J^{\prime}_{n}[(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]. [agree with Umeda18]
- 9.
\Pi_{33}=(\frac{n}{y_{s}}U_{s3}+U_{s7})v^{\prime}_{\parallel}J_{n}^{2}=(\frac{n}{y_{s}}[k_{x}v^{\prime}_{\parallel}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}-k_{x}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]+(\omega-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}})v^{\prime}_{\parallel}J_{n}^{2}\\ =v^{\prime}_{\parallel}J_{n}^{2}[n\omega_{cs}\frac{v^{\prime}_{\parallel}}{v^{\prime}_{\perp}}\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+(\omega-n\omega_{cs}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]. [agree with Umeda18]
Note: x_{s}=-\frac{\omega-k_{\parallel}v^{\prime}_{\parallel}{-\color[rgb]{1,0,0}k_{x}v_{dsx}+i\nu_{s}}}{\omega_{cs}} and .
At and limit, our result reduces to exactly the same result as the Eq.(25) of Ref.[11]. By further set , our result reduces to the standard without across magnetic field drifts one in Ref.[6] and the non-relativistic case in Ref.[3]. We should also note that when , the matrix elements are not symmetric or antisymmetric any more.
The in electromagnetic dispersion relation is
[TABLE]
2.4 Darwin model case
For Darwin model, we still have , and thus the calculation of current is exactly the same as in Eq.(45), i.e., the kinetic solutions to the distribution function and current do not need change as in the above electromagnetic model, which simplify our derivation a lot.
We discuss the change of the field equation here. In Fourier space, we have
[TABLE]
which can satisfied , and . We have used the tensor relation and , where , and are vectors. The field equations we needed are
[TABLE]
i.e.,
[TABLE]
or
[TABLE]
where the only change from the electromagnetic model is that the term is changed to be , or term is changed to be . And thus the Darwin dispersion relation is
[TABLE]
Eqs.(43), (48) and (73) with in (68) are our starting electrostatic, electromagnetic and Darwin dispersion relations with drift across magnetic field. The above dispersion relations are valid for arbitrary non-relativistic distribution functions. Later, we will limit our study to treat a special case of the distribution function .
3 The Dispersion Relation for Extend Maxwellian Distribution
The extend Maxwellian distribution here means bi-Maxwellian distribution with loss cone, parallel and perpendicular drifts and ring beam. We model it use the following equilibrium distribution function.
3.1 Equilibrium distribution function
We assume equilibrium distribution function , with , , and
[TABLE]
where r_{sa}=\Big{(}\frac{1-\alpha_{s}\Delta_{s}}{1-\alpha_{s}}\Big{)} and r_{sb}=\Big{(}\frac{-\alpha_{s}+\alpha_{s}\Delta_{s}}{1-\alpha_{s}}\Big{)}, and
[TABLE]
with and .
That is, the equilibrium distribution function is separated to two sub-distributions: , , ,
[TABLE]
and
[TABLE]
where , , are the drift velocities in (perpendicular 1), (perpendicular 2) and (parallel) directions, respectively. And, is the perpendicular ring beam velocity, and is the complementary error function, with and . The and are the parallel and perpendicular thermal velocities and corresponding temperatures are and . We define the temperature anisotropic . The parameters and determine the depth and size of the loss-cone444Actually, the user can also use two different specieses to represent the loss cone distribution, with one of them has negative density. Then, they can also have different , and . Many complicated distribution function can be constructed based on our model, which we leave it to the user. For example, Ref.[17] constructed the shell distribution based on ring beam model.. Here, , for max loss cone and no loss cone. If or , i.e., and , the above equation reduced to no loss cone case. Note .
We will use the same distribution in magnetized and unmagnetized versions to simplified the notations. For unmagnetized version, to remove the trouble of integral, we study only . How to justify the assumed distribution to the realistic physics problem is leave to the user. For example, the linearized equation (i.e., the dispersion relations) can also run when the zero order current and charge density are not zero. Thus, the user should justify whether it is reasonable.
Several terms are particular useful for further discussions:
[TABLE]
where we have used that .
3.2 Notations
Note the definition of , i.e., , not as in Ref.[12]. Other notations: , , , , a_{s\sigma}={\color[rgb]{1,0,0}\sqrt{2}}k_{\perp}\rho_{cs\sigma}, , . Note: for electron (), and thus also and . [Note the definition in previous bi-Maxwellian version: , , , , is the modified Bessel function.]
We use ’ES3D’ or ’ES’, ’EM3D’ or ’EM’, ’Darwin’ to represent the electrostatic, electromagnetic and Darwin version, respectively. And with suffix ’-U’ and ’-M’ to represent the corresponding unmagnetized and magnetized species.
3.3 Some integrals and functions
Here, we also clarify the correctly treat of the plasma dispersion function for . The standard definition of plasma dispersion function is
[TABLE]
where is the Landau contour to analytic continuation from to . However, for our usage the above definition to plasma dispersion function with is only correct when , where for example . To correctly capture the physics, one should be careful of the analytic continuation for both and . From standard derivation of the dispersion relations based on Laplacian transformation instead of Fourier transformation which considered the causality, say Ref.[6], we obtain the correct analytic continuation one should be
[TABLE]
where refers to the principal value integral. The corresponding -pole expansion should also be modified for , which will be discussed later. We can find easily that for , one can use , which will simplify our later usage.
To simplify the notation, we define also the function , and have
[TABLE]
where we have used . Later, we will find by using will simplify the notations a lot, and also will make the matrix linear transformation quite straightforward, say,
[TABLE]
For Bessel function, we have (p256 of Ref.[3]): , , , , , , , , , and .
We define
[TABLE]
For our usage , and or [math]. Here, we calculate , and using numerical integral. When , the above integrals reduce to the conventional Maxwellian form with modified Bessel function and , i.e.,
[TABLE]
where . And , , . Thus, for species with , we will still use the Bessel function form and .
Note, we have
[TABLE]
We have checked in Matlab, the speed of numerical integral is also fast, compared to using the Bessel function. To short the notation, we would also use such as and .
For short the notations using , , , , and .
Note also: and .
Expansion and at would be useful. For and , , with be the Euler function. Thus, we have and . To , we have
[TABLE]
where we have used , , , , , , , , , and defined , .
3.4 Electrostatic dispersion relation
We derive the electrostatic dispersion relation in this subsection base on Eq.(43) and the distribution function (3.1). The term
[TABLE]
and thus
[TABLE]
and thus the magnetized term in Eq.(43) is
[TABLE]
We find the above result is the same as in Ref.[13] ring-beam case, except that our new (1) \omega_{sn}=\omega-k_{z}v_{dsz}-n\omega_{cs}{\color[rgb]{1,0,0}-k_{x}v_{dsx}+i\nu_{s}}, and (2) the summation for loss cone.
For the unmagnetized species
[TABLE]
where we have chosen the transformation
[TABLE]
Further consider
[TABLE]
we need a new transformation to let the two variables change to a single variable , which is
[TABLE]
i.e.,
[TABLE]
where kv_{ts\sigma}=\sqrt{k_{x}^{2}v_{\perp ts\sigma}^{2}+k_{z}^{2}v_{zts}^{2}}=v_{zts}\sqrt{k_{x}^{2}{\color[rgb]{1,0,0}/\lambda_{Ts\sigma}}+k_{z}^{2}}. To transform the integral to be able to use the function, we need term be separate. It is not easy to do so, except only when we set , i.e., and . At case, we can have . Thus, to make life easy, we set for unmagnetized study555It is also rare to meet ring beam in unmagnetized case. Thus, using for unmagnetized species will not limit too much to the application of present model. Actually, our present model is much general than Ref.[14] and the non-relativistic case in Ref.[15]..
We define , and thus . The unmagnetized term in Eq.(43) is
[TABLE]
where we have used . We note that the term is integralled to be zero which makes the result simplified a lot.
Thus we obtain the final electrostatic dispersion relation
[TABLE]
The above dispersion relation Eq.(99) is very general, except that required for unmagnetized species.
3.5 Electromagnetic dispersion relation
The electromagnetic case is much complicated than the electrostatic case.
3.5.1 Unmagnetized terms
We start from the unmagnetized term firstly. We use the same transformation for as in the unmagnetized electrostatic case. Again, we assume for unmagnetized species, and calculate term by term:
[TABLE]
[TABLE]
Define (note: ), we have
[TABLE]
Note also:
[TABLE]
The odd function term from and can be integralled to vanish, and thus we have omitted them in the above final expressions. Note
[TABLE]
and we have used:
,
- 2.
\Big{[}\frac{k_{x}}{kv_{ts\sigma}}x+\sqrt{\lambda_{Ts\sigma}}\frac{k_{z}}{kv_{ts\sigma}}z\Big{]}(v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}x-v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}z+v_{dsz})\to\frac{k_{x}}{kv_{ts\sigma}}x(v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}x+v_{dsz})-\frac{k_{z}}{kv_{ts\sigma}}\frac{k_{x}v_{zts}^{2}}{kv_{ts\sigma}}z^{2}=\frac{k_{x}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}x^{2}+\frac{k_{x}v_{dsz}}{kv_{ts\sigma}}x-\frac{k_{z}}{kv_{ts\sigma}}\frac{k_{x}v_{zts}^{2}}{kv_{ts\sigma}}z^{2},
- 3.
\Big{[}\frac{k_{z}}{kv_{ts\sigma}}x-\frac{k_{x}}{kv_{ts\sigma}}\frac{1}{\sqrt{\lambda_{Ts\sigma}}}z\Big{]}(v_{\perp ts\sigma}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}x+v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}z+v_{dsx})\to\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}x^{2}+\frac{k_{z}v_{dsx}}{kv_{ts\sigma}}x-\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}z^{2},
- 4.
\Big{[}\frac{k_{z}}{kv_{ts\sigma}}x-\frac{k_{x}}{kv_{ts\sigma}}\frac{1}{\sqrt{\lambda_{Ts\sigma}}}z\Big{]}(v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}x-v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}z+v_{dsz})\to\frac{k_{z}^{2}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}x^{2}+\frac{k_{z}v_{dsz}}{kv_{ts\sigma}}x+\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}z^{2}.
And
[TABLE]
with terms in
.
- 2.
.
- 3.
.
- 4.
.
- 5.
.
- 6.
.
Define , with \bm{P}_{s}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}, and . We calculate term by term, similar to
P_{s11}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}_{11}=-1{\color[rgb]{1,0,0}+}\\ 2\int dxdydzg\frac{-i\nu_{s}(\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}x^{2}+\frac{k_{x}v_{dsx}}{kv_{ts\sigma}}x+\frac{k_{z}^{2}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}z^{2})+\Big{[}\frac{k^{2}}{kv_{ts\sigma}}x+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)z\Big{]}[v_{\perp ts\sigma}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}x+v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}z+v_{dsx}]^{2}}{kv_{ts\sigma}(\zeta_{s\sigma}-x)}=-1+\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{x}v_{dsx}}{kv_{ts\sigma}}Z_{1}+\frac{k_{z}^{2}v_{zts}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}}{kv_{ts\sigma}}(v_{\perp ts\sigma}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}})^{2}Z_{3}+2\frac{k^{2}}{kv_{ts\sigma}}(v_{\perp ts\sigma}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}})v_{dsx}Z_{2}+[\frac{1}{2}\frac{k^{2}}{kv_{ts\sigma}}(v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}})^{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}^{2}+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)(v_{\perp ts\sigma}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}})v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}]Z_{1}+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{dsx}Z_{0}\Big{\}}\\ =-1+\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{x}v_{dsx}}{kv_{ts\sigma}}Z_{1}+\frac{k_{z}^{2}v_{zts}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}k_{x}^{2}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}Z_{3}+2\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsx}Z_{2}+[\frac{1}{2}v_{\perp ts\sigma}^{2}\frac{k^{2}k_{z}^{2}v_{zts}^{2}}{k^{3}v_{ts\sigma}^{3}}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}^{2}+(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}^{2}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}]Z_{1}+\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}v_{dsx}Z_{0}\Big{\}},
we have others
P_{s12}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}_{12}=0{\color[rgb]{1,0,0}+}\\ 2\int dxdydzg\frac{-i\nu_{s}0+\Big{[}\frac{k^{2}}{kv_{ts\sigma}}x+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)z\Big{]}[v_{\perp ts\sigma}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}x+v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}z+v_{dsx}]v_{dsy}}{kv_{ts\sigma}(\zeta_{s\sigma}-x)}=\frac{2}{kv_{ts\sigma}}\Big{\{}\frac{k^{2}}{kv_{ts\sigma}}\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsy}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsy}Z_{1}+\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{dsy}Z_{0}\Big{\}}\\ =\frac{2v_{dsy}}{kv_{ts\sigma}}\Big{\{}\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}Z_{1}+\frac{1}{2}\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}Z_{0}\Big{\}},
- 2.
P_{s21}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}_{21}=0{\color[rgb]{1,0,0}+}\\ 2\int dxdydzg\frac{-i\nu_{s}\frac{k_{x}v_{dsy}}{kv_{ts\sigma}}x+\Big{[}\frac{k^{2}}{kv_{ts\sigma}}x+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)z\Big{]}[v_{\perp ts\sigma}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}x+v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}z+v_{dsx}]v_{dsy}}{kv_{ts\sigma}(\zeta_{s\sigma}-x)}\\ =\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}\frac{k_{x}v_{dsy}}{kv_{ts\sigma}}Z_{1}+\frac{k^{2}}{kv_{ts\sigma}}\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsy}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsy}Z_{1}+\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{dsy}Z_{0}\Big{\}}\\ =\frac{2v_{dsy}}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}\frac{k_{x}}{kv_{ts\sigma}}Z_{1}+\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}Z_{1}+\frac{1}{2}\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}Z_{0}\Big{\}},
- 3.
P_{s22}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}_{22}=-1{\color[rgb]{1,0,0}+}\\ 2\int dxdydzg\frac{-i\nu_{s}y^{2}+\Big{[}\frac{k^{2}}{kv_{ts\sigma}}x+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)z\Big{]}[v_{\perp ts\sigma}^{2}y^{2}+v_{dsy}^{2}]}{kv_{ts\sigma}(\zeta_{s\sigma}-x)}\\ =-1{\color[rgb]{1,0,0}+}\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}\frac{1}{2}Z_{0}+(\frac{1}{2}v_{\perp ts\sigma}^{2}+v_{dsy}^{2})\frac{k^{2}}{kv_{ts\sigma}}Z_{1}\Big{\}},
- 4.
P_{s13}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}_{13}=0{\color[rgb]{1,0,0}+}2\int dxdydzg\\ \frac{-i\nu_{s}(\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}x^{2}+\frac{k_{z}v_{dsx}}{kv_{ts\sigma}}x-\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}z^{2})+\Big{[}\frac{k^{2}}{kv_{ts\sigma}}x+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)z\Big{]}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}x+v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}z+v_{dsx}][\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}x-v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}z+v_{dsz}]}{kv_{ts\sigma}(\zeta_{s\sigma}-x)}\\ =\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{z}v_{dsx}}{kv_{ts\sigma}}Z_{1}-\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}}{kv_{ts\sigma}}\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}Z_{3}+\frac{k^{2}}{kv_{ts\sigma}}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsz}+v_{dsx}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]Z_{2}+[\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsz}-\frac{1}{2}\frac{k^{2}}{kv_{ts\sigma}}v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}-\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}+\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]Z_{1}+[\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{dsz}-\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{dsx}v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}]Z_{0}\Big{\}}\\ =\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{z}v_{dsx}}{kv_{ts\sigma}}Z_{1}-\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}Z_{3}+\frac{k^{2}}{kv_{ts\sigma}}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsz}+v_{dsx}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]Z_{2}+[\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsz}-\frac{1}{2}\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{3}k_{z}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}+\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{3}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}]Z_{1}+[\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}-\frac{1}{2}(\lambda_{Ts\sigma}-1)v_{dsx}\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}]Z_{0}\Big{\}},
- 5.
P_{s31}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}_{31}=0{\color[rgb]{1,0,0}+}2\int dxdydzg\\ \frac{-i\nu_{s}(\frac{k_{x}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}x^{2}+\frac{k_{x}v_{dsz}}{kv_{ts\sigma}}x-\frac{k_{x}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}z^{2})+\Big{[}\frac{k^{2}}{kv_{ts\sigma}}x+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)z\Big{]}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}x+v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}z+v_{dsx}][\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}x-v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}z+v_{dsz}]}{kv_{ts\sigma}(\zeta_{s\sigma}-x)}\\ =\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{x}v_{dsz}}{kv_{ts\sigma}}Z_{1}-\frac{k_{x}k_{z}v_{zts}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}}{kv_{ts\sigma}}\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}Z_{3}+\frac{k^{2}}{kv_{ts\sigma}}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsz}+v_{dsx}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]Z_{2}+[\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsz}-\frac{1}{2}\frac{k^{2}}{kv_{ts\sigma}}v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}-\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}+\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]Z_{1}+[\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{dsz}-\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{dsx}v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}]Z_{0}\Big{\}}\\ =\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{x}v_{dsz}}{kv_{ts\sigma}}Z_{1}-\frac{k_{x}k_{z}v_{zts}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}Z_{3}+\frac{k^{2}}{kv_{ts\sigma}}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsz}+v_{dsx}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]Z_{2}+[\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsz}-\frac{1}{2}\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{3}k_{z}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}+\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{3}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}]Z_{1}+[\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}-\frac{1}{2}(\lambda_{Ts\sigma}-1)v_{dsx}\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}]Z_{0}\Big{\}},
- 6.
P_{s23}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}_{23}=0{\color[rgb]{1,0,0}+}\\ 2\int dxdydzg\frac{-i\nu_{s}\frac{k_{z}v_{dsy}}{kv_{ts\sigma}}x+\Big{[}\frac{k^{2}}{kv_{ts\sigma}}x+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)z\Big{]}[v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}x-v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}z+v_{dsz}]v_{dsy}}{kv_{ts\sigma}(\zeta_{s\sigma}-x)}\\ =\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}\frac{k_{z}v_{dsy}}{kv_{ts\sigma}}Z_{1}+\frac{k^{2}}{kv_{ts\sigma}}v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{dsy}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}v_{dsy}Z_{1}-\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}v_{dsy}Z_{0}\Big{\}}\\ =\frac{2v_{dsy}}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}\frac{k_{z}}{kv_{ts\sigma}}Z_{1}+\frac{k^{2}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}Z_{1}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{0}\Big{\}},
- 7.
P_{s32}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}_{32}=0{\color[rgb]{1,0,0}+}\\ 2\int dxdydzg\frac{-i\nu_{s}0+\Big{[}\frac{k^{2}}{kv_{ts\sigma}}x+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)z\Big{]}[v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}x-v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}z+v_{dsz}]v_{dsy}}{kv_{ts\sigma}(\zeta_{s\sigma}-x)}\\ =\frac{2}{kv_{ts\sigma}}\Big{\{}\frac{k^{2}}{kv_{ts\sigma}}v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{dsy}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}v_{dsy}Z_{1}-\frac{1}{2}\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}v_{dsy}Z_{0}\Big{\}}\\ =\frac{2v_{dsy}}{kv_{ts\sigma}}\Big{\{}\frac{k^{2}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}Z_{1}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{0}\Big{\}},
- 8.
P_{s33}^{u}=\Big{\{}-\bm{I}+\int dv^{3}\frac{\Big{[}-i\nu_{s}(\bm{v}\frac{\partial f_{s0\sigma}}{\partial\bm{v}})+(\bm{k}\cdot\frac{\partial f_{s0\sigma}}{\partial\bm{v}})(\bm{v}\bm{v})\Big{]}}{(\omega-\bm{k}\cdot\bm{v}+i\nu_{s})}\Big{\}}_{33}=-1{\color[rgb]{1,0,0}+}\\ 2\int dxdydzg\frac{-i\nu_{s}(\frac{k_{z}^{2}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}x^{2}+\frac{k_{z}v_{dsz}}{kv_{ts\sigma}}x+\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}z^{2})+\Big{[}\frac{k^{2}}{kv_{ts\sigma}}x+\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)z\Big{]}[v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}x-v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}z+v_{dsz}]^{2}}{kv_{ts\sigma}(\zeta_{s\sigma}-x)}=-1+\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{z}^{2}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{z}v_{dsz}}{kv_{ts\sigma}}Z_{1}+\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}}{kv_{ts\sigma}}(v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}})^{2}Z_{3}+2\frac{k^{2}}{kv_{ts\sigma}}v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{dsz}Z_{2}+[\frac{1}{2}\frac{k^{2}}{kv_{ts\sigma}}(v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}})^{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}^{2}-\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{zts}\frac{k_{z}v_{zts}}{kv_{ts\sigma}}v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}]Z_{1}-\frac{k_{x}k_{z}}{kv_{ts\sigma}}\frac{v_{\perp ts\sigma}}{v_{zts}}(\lambda_{Ts\sigma}-1)v_{zts}\frac{k_{x}v_{\perp ts\sigma}}{kv_{ts\sigma}}v_{dsz}Z_{0}\Big{\}}\\ =-1+\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{z}^{2}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{z}v_{dsz}}{kv_{ts\sigma}}Z_{1}+\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}k_{z}^{2}v_{zts}^{4}}{k^{3}v_{ts\sigma}^{3}}Z_{3}+2\frac{k^{2}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}Z_{2}+[\frac{1}{2}\frac{k^{2}k_{x}^{2}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}^{2}-(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}^{2}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}]Z_{1}-(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}Z_{0}\Big{\}},
where the argument for is .
Note, with the ’’ means that to integral out the , and terms, say, , , , , ), we have used
.
- 2.
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- 3.
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- 4.
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- 5.
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- 6.
.
Although the above expressions are much complicated and need be carefully, they can be solved easily by PASS-K matrix approach. We have also checked that, when , and , the result can reduce to exactly the same form as in Ref.[14] .
3.5.2 Magnetized terms
Next, we calculate the magnetized terms.
Define , with \bm{P}_{s}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime}_{\parallel}}{(\omega-k_{\parallel}v^{\prime}_{\parallel}{-\color[rgb]{1,0,0}k_{x}v_{dsx}+i\nu_{s}}-n\omega_{cs})}\bm{\Pi}_{s\sigma}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\bm{\Pi}_{s\sigma} and \omega_{sn}=\omega-k_{z}v_{dsz}-n\omega_{cs}{-\color[rgb]{1,0,0}k_{x}v_{dsx}+i\nu_{s}}, i.e., . We have also
[TABLE]
with , and f_{s0\sigma}=\frac{1}{\pi^{3/2}v_{zts}v_{\perp ts\sigma}^{2}A_{s\sigma}}\exp\Big{[}-\frac{v^{\prime\prime 2}_{\parallel}}{v_{zts}^{2}}-\frac{(v^{\prime}-v_{dsr})^{2}_{\perp}}{v_{\perp ts}^{2}}\Big{]}, i.e., . Since we have defined , we do not need transform .
Thus,
P_{s11}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\Pi_{s\sigma 11}=\sum_{n=-\infty}^{\infty}\Big{\{}[\frac{n\omega_{cs}}{k_{x}}(n\omega_{cs}+k_{x}v_{dsx}-i\nu_{s})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}\frac{Z_{0}}{k_{z}v_{zts}}+A_{n0\sigma}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})\frac{Z_{1}}{v_{zts}^{2}}](\frac{n\omega_{cs}}{k_{x}}+v_{dsx})\Big{\}}-{\color[rgb]{1,0,0}\sum_{n}A_{nbs\sigma}\frac{n\omega_{cs}}{k_{x}v_{\perp ts\sigma}^{2}}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})}.
- 2.
P_{s12}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\Pi_{s\sigma 12}=\sum_{n=-\infty}^{\infty}\Big{\{}v_{dsy}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})(n\omega_{cs}\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}\frac{Z_{0}}{k_{z}v_{zts}}+A_{n0\sigma}\frac{Z_{1}}{v_{zts}^{2}})+i(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[(n\omega_{cs}-i\nu_{s})\frac{B_{nbs\sigma}}{v_{\perp ts\sigma}}\frac{Z_{0}}{k_{z}v_{zts}}+\frac{B_{n0\sigma}v_{\perp ts\sigma}Z_{1}}{v_{zts}^{2}}]\Big{\}}.
- 3.
P_{s21}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\Pi_{s\sigma 21}=\sum_{n=-\infty}^{\infty}\Big{\{}\frac{n\omega_{cs}}{k_{x}}(n\omega_{cs}+k_{x}v_{dsx}-i\nu_{s})[-iv_{\perp ts\sigma}B_{nbs\sigma}+v_{dsy}A_{nbs\sigma}]\frac{1}{v_{\perp ts\sigma}^{2}}\frac{Z_{0}}{k_{z}v_{zts}}+[-iv_{\perp ts\sigma}B_{n0\sigma}+v_{dsy}A_{n0\sigma}](\frac{n\omega_{cs}}{k_{x}}+v_{dsx})\frac{Z_{1}}{v_{zts}^{2}}\Big{\}}.
- 4.
P_{s22}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\Pi_{s\sigma 22}=\sum_{n=-\infty}^{\infty}\Big{\{}v_{dsy}[-iv_{\perp ts\sigma}B_{nbs\sigma}+v_{dsy}A_{nbs\sigma}]\frac{n\omega_{cs}}{v_{\perp ts\sigma}^{2}}\frac{Z_{0}}{k_{z}v_{zts}}+v_{dsy}[-iv_{\perp ts\sigma}B_{n0\sigma}+v_{dsy}A_{n0\sigma}]\frac{Z_{1}}{v_{zts}^{2}}+v_{\perp ts\sigma}[v_{\perp ts\sigma}C_{nbs\sigma}+iv_{dsy}B_{nbs\sigma}](n\omega_{cs}-i\nu_{s})\frac{1}{v_{\perp ts\sigma}^{2}}\frac{Z_{0}}{k_{z}v_{zts}}+v_{\perp ts\sigma}[v_{\perp ts\sigma}C_{n0\sigma}+iv_{dsy}B_{n0\sigma}]\frac{Z_{1}}{v_{zts}^{2}}\Big{\}}-{\color[rgb]{1,0,0}\sum_{n}(C_{nbs\sigma}+i\frac{v_{dsy}}{v_{\perp ts\sigma}}B_{nbs\sigma})}.
- 5.
P_{s13}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\Pi_{s\sigma 13}=\sum_{n=-\infty}^{\infty}\Big{\{}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[n\omega_{cs}(\frac{Z_{1}}{k_{z}}+v_{dsz}\frac{Z_{0}}{k_{z}v_{zts}})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}+A_{n0\sigma}(v_{zts}Z_{2}+(k_{z}v_{dsz}-i\nu_{s})\frac{Z_{1}}{k_{z}})\frac{1}{v_{zts}^{2}}]\Big{\}}.
- 6.
P_{s31}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\Pi_{s\sigma 31}=\sum_{n=-\infty}^{\infty}\Big{\{}\frac{n\omega_{cs}}{k_{x}}(n\omega_{cs}+k_{x}v_{dsx}-i\nu_{s})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}(\frac{Z_{1}}{k_{z}}+v_{dsz}\frac{Z_{0}}{k_{z}v_{zts}})+A_{n0\sigma}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})(\frac{Z_{2}}{v_{zts}}+v_{dsz}\frac{Z_{1}}{v_{zts}^{2}})\Big{\}}.
- 7.
P_{s23}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\Pi_{s\sigma 23}=\sum_{n=-\infty}^{\infty}\Big{\{}[v_{dsy}A_{nbs\sigma}-iv_{\perp ts\sigma}B_{nbs\sigma}][(\frac{Z_{1}}{k_{z}}+\frac{Z_{0}v_{dsz}}{k_{z}v_{tsz}})\frac{n\omega_{cs}}{v_{\perp ts\sigma}^{2}}]+[v_{dsy}A_{n0\sigma}-iv_{\perp ts\sigma}B_{n0\sigma}][v_{zts}Z_{2}+(k_{z}v_{dsz}-i\nu_{s})\frac{Z_{1}}{k_{z}}]\frac{1}{v_{zts}^{2}}\Big{\}}.
- 8.
P_{s32}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\Pi_{s\sigma 32}=\sum_{n=-\infty}^{\infty}\Big{\{}[n\omega_{cs}\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}\frac{Z_{1}}{k_{z}}+A_{n0\sigma}\frac{k_{z}v_{zts}Z_{2}}{v_{zts}^{2}}]v_{dsy}+[n\omega_{cs}\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}\frac{Z_{0}}{k_{z}v_{zts}}+A_{n0\sigma}\frac{Z_{1}}{v_{zts}^{2}}]v_{dsy}v_{dsz}+i[(n\omega_{cs}-i\nu_{s})\frac{B_{nbs\sigma}}{v_{\perp ts\sigma}}\frac{Z_{1}}{k_{z}}+B_{n0\sigma}\frac{v_{\perp ts\sigma}Z_{2}}{v_{zts}}]+iv_{dsz}[(n\omega_{cs}-i\nu_{s})\frac{B_{nbs\sigma}}{v_{\perp ts\sigma}}\frac{Z_{0}}{k_{z}v_{zts}}+B_{n0\sigma}\frac{v_{\perp ts\sigma}Z_{1}}{v_{zts}^{2}}]\Big{\}}.
- 9.
P_{s33}^{m}=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{2\pi v^{\prime}_{\perp}dv^{\prime}_{\perp}dv^{\prime\prime}_{\parallel}}{(\omega_{sn}-k_{z}v^{\prime\prime}_{\parallel})}\Pi_{s\sigma 33}=\sum_{n=-\infty}^{\infty}\Big{\{}[n\omega_{cs}(\frac{v_{zts}Z_{2}}{k_{z}}+v_{dsz}\frac{Z_{1}}{k_{z}})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}+A_{n0\sigma}(Z_{3}+(k_{z}v_{dsz}-i\nu_{s})\frac{Z_{2}}{k_{z}v_{zts}})]+v_{dsz}[n\omega_{cs}(\frac{Z_{1}}{k_{z}}+v_{dsz}\frac{Z_{0}}{k_{z}v_{zts}})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}+A_{n0\sigma}(\frac{Z_{2}}{v_{zts}}+(k_{z}v_{dsz}-i\nu_{s})\frac{Z_{1}}{k_{z}v_{zts}^{2}})]\Big{\}}-{\color[rgb]{1,0,0}\sum_{n}\frac{1}{2}A_{n0\sigma}}.
where the argument for is , and we have used , , . If we further use , , , , the last terms in , and yields to . We derive the above equation base on (define , i.e., )
.
- 2.
v^{\prime}_{\perp}\Pi_{s\sigma 12}=J_{n}^{2}v_{dsy}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})(n\omega_{cs}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\perp}}+v^{\prime}_{\perp}k_{z}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\parallel}})+iv^{\prime}_{\perp}J_{n}J^{\prime}_{n}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\perp}}+k_{z}v^{\prime}_{\perp}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\parallel}}]\to J_{n}^{2}v_{dsy}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})(n\omega_{cs}\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}k_{z}\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}})+iv^{\prime}_{\perp}J_{n}J^{\prime}_{n}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[(\omega^{\prime}_{sn}+n\omega_{cs}-i\nu_{s})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+k_{z}v^{\prime}_{\perp}\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]\to J_{n}^{2}v_{dsy}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})(n\omega_{cs}\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}\frac{k_{z}v^{\prime\prime}_{\parallel}}{v_{zts}^{2}})+iv^{\prime}_{\perp}J_{n}J^{\prime}_{n}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[(n\omega_{cs}-i\nu_{s})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}\frac{k_{z}v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]+iv^{\prime}_{\perp}J_{n}J^{\prime}_{n}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}(\omega^{\prime}_{sn}).
- 3.
.
- 4.
v^{\prime}_{\perp}\Pi_{s\sigma 22}=J_{n}v_{dsy}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})(n\omega_{cs}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\perp}}+v^{\prime}_{\perp}k_{z}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\parallel}})+iv^{\prime}_{\perp}J^{\prime}_{n}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})[(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\perp}}+k_{z}v^{\prime}_{\perp}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}}]\to J_{n}v_{dsy}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})(n\omega_{cs}\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}k_{z}\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}})+iv^{\prime}_{\perp}J^{\prime}_{n}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})[(\omega^{\prime}_{sn}+n\omega_{cs}-i\nu_{s})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+k_{z}v^{\prime}_{\perp}\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]\to J_{n}v_{dsy}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})(n\omega_{cs}\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}\frac{k_{z}v^{\prime\prime}_{\parallel}}{v_{zts}^{2}})+iv^{\prime}_{\perp}J^{\prime}_{n}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})[(n\omega_{cs}-i\nu_{s})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}\frac{k_{z}v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]+iv^{\prime}_{\perp}J^{\prime}_{n}(-iv^{\prime}_{\perp}J^{\prime}_{n}+v_{dsy}J_{n})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}(\omega^{\prime}_{sn}).
- 5.
v^{\prime}_{\perp}\Pi_{s\sigma 13}=J_{n}^{2}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[n\omega_{cs}v^{\prime}_{\parallel}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\perp}}+v^{\prime}_{\perp}(\omega-n\omega_{cs}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\parallel}}]\to J_{n}^{2}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[n\omega_{cs}(v^{\prime\prime}_{\parallel}+v_{dsz})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}(\omega^{\prime}_{sn}+k_{z}v^{\prime\prime}_{\parallel}+k_{z}v_{dsz}-i\nu_{s})\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]\to J_{n}^{2}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[n\omega_{cs}(v^{\prime\prime}_{\parallel}+v_{dsz})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}(k_{z}v^{\prime\prime}_{\parallel}+k_{z}v_{dsz}-i\nu_{s})\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]+J_{n}^{2}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})v^{\prime}_{\perp}\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}(\omega^{\prime}_{sn}).
- 6.
.
- 7.
v^{\prime}_{\perp}\Pi_{s\sigma 23}=(v_{dsy}J_{n}^{2}-iv^{\prime}_{\perp}J_{n}J^{\prime}_{n})[n\omega_{cs}v^{\prime}_{\parallel}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\perp}}+v^{\prime}_{\perp}(\omega-n\omega_{cs}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\parallel}}]\to(v_{dsy}J_{n}^{2}-iv^{\prime}_{\perp}J_{n}J^{\prime}_{n})[n\omega_{cs}(v^{\prime\prime}_{\parallel}+v_{dsz})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}(\omega^{\prime}_{sn}+k_{z}v^{\prime\prime}_{\parallel}+k_{z}v_{dsz}-i\nu_{s})\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]\to(v_{dsy}J_{n}^{2}-iv^{\prime}_{\perp}J_{n}J^{\prime}_{n})[n\omega_{cs}(v^{\prime\prime}_{\parallel}+v_{dsz})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}(k_{z}v^{\prime\prime}_{\parallel}+k_{z}v_{dsz}-i\nu_{s})\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]+(v_{dsy}J_{n}^{2}-iv^{\prime}_{\perp}J_{n}J^{\prime}_{n})v^{\prime}_{\perp}\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}(\omega^{\prime}_{sn}).
- 8.
v^{\prime}_{\perp}\Pi_{s\sigma 32}=v^{\prime}_{\parallel}J_{n}^{2}(n\omega_{cs}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\perp}}+v^{\prime}_{\perp}k_{z}\frac{\partial f_{s0}}{\partial v^{\prime}_{\parallel}})v_{dsy}+iv^{\prime}_{\parallel}v^{\prime}_{\perp}J_{n}J^{\prime}_{n}[(\omega-k_{z}v^{\prime}_{\parallel}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0}}{\partial v^{\prime}_{\perp}}+k_{z}v^{\prime}_{\perp}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\parallel}}]\to(v^{\prime\prime}_{\parallel}+v_{dsz})J_{n}^{2}(n\omega_{cs}\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}k_{z}\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}})v_{dsy}+i(v^{\prime\prime}_{\parallel}+v_{dsz})v^{\prime}_{\perp}J_{n}J^{\prime}_{n}[(\omega^{\prime}_{sn}+n\omega_{cs}-i\nu_{s})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+k_{z}v^{\prime}_{\perp}\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]\to(v^{\prime\prime}_{\parallel}+v_{dsz})J_{n}^{2}(n\omega_{cs}\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}\frac{k_{z}v^{\prime\prime}_{\parallel}}{v_{zts}^{2}})v_{dsy}+i(v^{\prime\prime}_{\parallel}+v_{dsz})v^{\prime}_{\perp}J_{n}J^{\prime}_{n}[(n\omega_{cs}-i\nu_{s})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}\frac{k_{z}v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]+i(v^{\prime\prime}_{\parallel}+v_{dsz})v^{\prime}_{\perp}J_{n}J^{\prime}_{n}\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}(\omega^{\prime}_{sn}).
- 9.
v^{\prime}_{\perp}\Pi_{s\sigma 33}=v^{\prime}_{\parallel}J_{n}^{2}[n\omega_{cs}v^{\prime}_{\parallel}\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\perp}}+v^{\prime}_{\perp}(\omega-n\omega_{cs}-{\color[rgb]{1,0,0}k_{x}v_{dsx}})\frac{\partial f_{s0\sigma}}{\partial v^{\prime}_{\parallel}}]\to(v^{\prime\prime}_{\parallel}+v_{dsz})J_{n}^{2}[n\omega_{cs}(v^{\prime\prime}_{\parallel}+v_{dsz})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}(\omega^{\prime}_{sn}+k_{z}v^{\prime\prime}_{\parallel}+k_{z}v_{dsz}-i\nu_{s})\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]\to(v^{\prime\prime}_{\parallel}+v_{dsz})J_{n}^{2}[n\omega_{cs}(v^{\prime\prime}_{\parallel}+v_{dsz})\frac{(v^{\prime}_{\perp}-v_{dsr})}{v_{\perp ts\sigma}^{2}}+v^{\prime}_{\perp}(k_{z}v^{\prime\prime}_{\parallel}+k_{z}v_{dsz}-i\nu_{s})\frac{v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}]+{\color[rgb]{1,0,0}v^{\prime}_{\perp}J_{n}^{2}\frac{(v^{\prime\prime}_{\parallel}+v_{dsz})v^{\prime\prime}_{\parallel}}{v_{zts}^{2}}(\omega^{\prime}_{sn})}.
where ’’ means we have omitted the coefficient ’’.
We find by set , , these reduce exactly the same one as in Ref.[12] ring beam case. By set , , we have also checked that they can reduce to the one in Ref.[11] for drift across magnetic field case.
3.5.3 Final Form
The final form of the electromagnetic dispersion relation is the combine of the above unmagnetized and magnetized terms to in Eq.(68), and then to Eq.(48).
4 Transform to PASS-K matrix Equation
The conventional root finding approach to solve the above dispersion relations can only give one solution at one time and heavily depends on initial guess. The Cauchy contour integral approach [22] can locate all the solutions in a selected complex domain, however which still can not give all the important solutions and is also difficult for complicated dispersion relation. To solve the dispersion relation using PASS-K matrix approach [1], which can give all the important solution at one time, we need two further steps:
(1) Do -pole expansion of function, i.e.,
[TABLE]
where and are constants for given , as given in Ref.[1] for ;
- 2.
(2) Do linear transformation to a equivalent matrix eigenvalue problem. The standard eigenvalue library can solve all the eigenvalues of a matrix.
The first step with has been used well for more than thirty years in WHAMP [2] code; The second step is firstly developed in the first version of PASS-K/PDRK code [1]. We derive the corresponding equations step by step.
We note: , , and [8]. And also
[TABLE]
For , we have
[TABLE]
which is written to one compact form for both and , with . And
[TABLE]
where we have used . And
[TABLE]
where we have used . And
[TABLE]
where we have used . In the above and , we find for it could be simply by doing the variables change with: and . Thus, in the later usage, to simplify the notation, all is changed to be , and the meaning of for is the in the default . Thus for both and , we have a single compact form
[TABLE]
with . And the only change is that for and for , with be the for .
Typically, for magnetized species \zeta_{sn}=\frac{\omega-k_{z}v_{dsz}-n\omega_{cs}{-\color[rgb]{1,0,0}k_{x}v_{dsx}+i\nu_{s}}}{k_{z}v_{zts}} and unmagnetized species , with (which always ), we have corresponding and , respectively. For : , and .
4.1 The Electrostatic case
We solve Eq.(99) for electrostatic case. The corresponding linear transformation is straightforward and simple.
[TABLE]
Notation: \zeta_{sn}=\frac{\omega-k_{z}v_{dsz}-n\omega_{cs}{-\color[rgb]{1,0,0}k_{x}v_{dsx}+i\nu_{s}}}{k_{z}v_{zts}}, , kv_{ts\sigma}=\sqrt{k_{x}^{2}v_{\perp ts\sigma}^{2}+k_{z}^{2}v_{zts}^{2}}=v_{zts}\sqrt{k_{x}^{2}{\color[rgb]{1,0,0}/\lambda_{Ts\sigma}}+k_{z}^{2}}.
Define: , , , and .
The equivalent linear system can be
[TABLE]
or sparse matrix one
[TABLE]
where and is short notation for both unmagnetized and magnetized species and . We find the only singularity in the above final form occurs at , which requires . And thus the final form can be applied for arbitrary . Some other solvers in literature may meet singularity for or , and may be incorrect for .
It is also obvious that the magnetized species can not reduce to unmagnetized species by set . The major difference is in magnetized species and in unmagnetized species.
4.2 The Electromagnetic case
The electromagnetic case is much complicated. However, the linear transformation for is still similar to the original PASS-K/PDRK derivation.
To seek an equivalent linear system, the Maxwell’s equations
[TABLE]
do not need to be changed. We only need to seek a new linear system for .
4.2.1 The unmagnetized terms
Considering the defination , after -pole expansion, we have
P_{s\sigma 11}^{u}=-1+\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{x}v_{dsx}}{kv_{ts\sigma}}Z_{1}+\frac{k_{z}^{2}v_{zts}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}k_{x}^{2}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}Z_{3}+2\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsx}Z_{2}+[\frac{1}{2}v_{\perp ts\sigma}^{2}\frac{k^{2}k_{z}^{2}v_{zts}^{2}}{k^{3}v_{ts\sigma}^{3}}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}^{2}+(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}^{2}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}]Z_{1}+\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}v_{dsx}Z_{0}\Big{\}}\simeq-1+2\sum_{j=1}^{J}\frac{b_{j}}{\omega-c_{sj\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}c_{j}^{2}+\frac{k_{x}v_{dsx}}{kv_{ts\sigma}}c_{j}+\frac{k_{z}^{2}v_{zts}^{2}}{2k^{2}v_{ts\sigma}^{2}})+\frac{k^{2}k_{x}^{2}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}c_{j}^{3}+2\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsx}c_{j}^{2}+[\frac{1}{2}v_{\perp ts\sigma}^{2}\frac{k^{2}k_{z}^{2}v_{zts}^{2}}{k^{3}v_{ts\sigma}^{3}}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}^{2}+(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}^{2}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}]c_{j}+\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}v_{dsx}\Big{\}}=-1+\sum_{j=1}^{J}\frac{p_{11sj\sigma}}{\omega-c_{sj\sigma}},
- 2.
P_{s\sigma 12}^{u}=\frac{2v_{dsy}}{kv_{ts\sigma}}\Big{\{}\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}Z_{1}+\frac{1}{2}\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}Z_{0}\Big{\}}\simeq 2v_{dsy}\sum_{j=1}^{J}\frac{b_{j}}{\omega-c_{sj\sigma}}\Big{\{}\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}c_{j}^{2}+\frac{k^{2}v_{dsx}}{kv_{ts\sigma}}c_{j}+\frac{1}{2}\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}\Big{\}}=\sum_{j=1}^{J}\frac{p_{12sj\sigma}}{\omega-c_{sj\sigma}},
- 3.
P_{s\sigma 21}^{u}=\frac{2v_{dsy}}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}\frac{k_{x}}{kv_{ts\sigma}}Z_{1}+\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}Z_{1}+\frac{1}{2}\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}Z_{0}\Big{\}}\simeq 2v_{dsy}\sum_{j=1}^{J}\frac{b_{j}}{\omega-c_{sj\sigma}}\Big{\{}-i\nu_{s}\frac{k_{x}}{kv_{ts\sigma}}c_{j}+\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}c_{j}^{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}c_{j}+\frac{1}{2}\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}\Big{\}}=\sum_{j=1}^{J}\frac{p_{21sj\sigma}}{\omega-c_{sj\sigma}},
- 4.
P_{s\sigma 22}^{u}=-1{\color[rgb]{1,0,0}+}\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}\frac{1}{2}Z_{0}+(\frac{1}{2}v_{\perp ts\sigma}^{2}+v_{dsy}^{2})\frac{k^{2}}{kv_{ts\sigma}}Z_{1}\Big{\}}\simeq-1{\color[rgb]{1,0,0}+}2\sum_{j=1}^{J}\frac{b_{j}}{\omega-c_{sj\sigma}}\Big{\{}-i\nu_{s}\frac{1}{2}+(\frac{1}{2}v_{\perp ts\sigma}^{2}+v_{dsy}^{2})\frac{k^{2}}{kv_{ts\sigma}}c_{j}\Big{\}}=-1+\sum_{j=1}^{J}\frac{p_{22sj\sigma}}{\omega-c_{sj\sigma}},
- 5.
P_{s\sigma 13}^{u}=\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{z}v_{dsx}}{kv_{ts\sigma}}Z_{1}-\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}Z_{3}+\frac{k^{2}}{kv_{ts\sigma}}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsz}+v_{dsx}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]Z_{2}+[\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsz}-\frac{1}{2}\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{3}k_{z}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}+\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{3}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}]Z_{1}+[\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}-\frac{1}{2}(\lambda_{Ts\sigma}-1)v_{dsx}\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}]Z_{0}\Big{\}}\simeq 2\sum_{j=1}^{J}\frac{b_{j}}{\omega-c_{sj\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}c_{j}^{2}+\frac{k_{z}v_{dsx}}{kv_{ts\sigma}}c_{j}-\frac{k_{x}k_{z}v_{\perp ts\sigma}^{2}}{2k^{2}v_{ts\sigma}^{2}})+\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}c_{j}^{3}+\frac{k^{2}}{kv_{ts\sigma}}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsz}+v_{dsx}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]c_{j}^{2}+[\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsz}-\frac{1}{2}\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{3}k_{z}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}+\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{3}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}]c_{j}+[\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}-\frac{1}{2}(\lambda_{Ts\sigma}-1)v_{dsx}\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}]\Big{\}}=\sum_{j=1}^{J}\frac{p_{13sj\sigma}}{\omega-c_{sj\sigma}},
- 6.
P_{s\sigma 31}^{u}=\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{x}v_{dsz}}{kv_{ts\sigma}}Z_{1}-\frac{k_{x}k_{z}v_{zts}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}Z_{3}+\frac{k^{2}}{kv_{ts\sigma}}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsz}+v_{dsx}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]Z_{2}+[\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsz}-\frac{1}{2}\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{3}k_{z}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}+\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{3}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}]Z_{1}+[\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}-\frac{1}{2}(\lambda_{Ts\sigma}-1)v_{dsx}\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}]Z_{0}\Big{\}}\simeq 2\sum_{j=1}^{J}\frac{b_{j}}{\omega-c_{sj\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{x}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}c_{j}^{2}+\frac{k_{x}v_{dsz}}{kv_{ts\sigma}}c_{j}-\frac{k_{x}k_{z}v_{zts}^{2}}{2k^{2}v_{ts\sigma}^{2}})+\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}c_{j}^{3}+\frac{k^{2}}{kv_{ts\sigma}}[\frac{k_{x}v_{\perp ts\sigma}^{2}}{kv_{ts\sigma}}v_{dsz}+v_{dsx}\frac{k_{z}v_{zts}^{2}}{kv_{ts\sigma}}]c_{j}^{2}+[\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}v_{dsz}-\frac{1}{2}\frac{k^{2}k_{x}k_{z}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{3}k_{z}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}+\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{3}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}]c_{j}+[\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}k_{z}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}-\frac{1}{2}(\lambda_{Ts\sigma}-1)v_{dsx}\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}]\Big{\}}=\sum_{j=1}^{J}\frac{p_{31sj\sigma}}{\omega-c_{sj\sigma}},
- 7.
P_{s\sigma 23}^{u}=\frac{2v_{dsy}}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}\frac{k_{z}}{kv_{ts\sigma}}Z_{1}+\frac{k^{2}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}Z_{1}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{0}\Big{\}}\simeq 2v_{dsy}\sum_{j=1}^{J}\frac{b_{j}}{\omega-c_{sj\sigma}}\Big{\{}-i\nu_{s}\frac{k_{z}}{kv_{ts\sigma}}c_{j}+\frac{k^{2}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}c_{j}^{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}c_{j}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}\Big{\}}=\sum_{j=1}^{J}\frac{p_{23sj\sigma}}{\omega-c_{sj\sigma}},
- 8.
P_{s\sigma 32}^{u}=\frac{2v_{dsy}}{kv_{ts\sigma}}\Big{\{}\frac{k^{2}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}Z_{1}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{0}\Big{\}}\simeq 2v_{dsy}\sum_{j=1}^{J}\frac{b_{j}}{\omega-c_{sj\sigma}}\Big{\{}\frac{k^{2}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}c_{j}^{2}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}c_{j}-\frac{1}{2}(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}\Big{\}}=\sum_{j=1}^{J}\frac{p_{32sj\sigma}}{\omega-c_{sj\sigma}},
- 9.
P_{s\sigma 33}^{u}=-1+\frac{2}{kv_{ts\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{z}^{2}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}Z_{2}+\frac{k_{z}v_{dsz}}{kv_{ts\sigma}}Z_{1}+\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{2k^{2}v_{ts\sigma}^{2}}Z_{0})+\frac{k^{2}k_{z}^{2}v_{zts}^{4}}{k^{3}v_{ts\sigma}^{3}}Z_{3}+2\frac{k^{2}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}Z_{2}+[\frac{1}{2}\frac{k^{2}k_{x}^{2}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}^{2}-(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}^{2}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}]Z_{1}-(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}Z_{0}\Big{\}}\simeq-1+2\sum_{j=1}^{J}\frac{b_{j}}{\omega-c_{sj\sigma}}\Big{\{}-i\nu_{s}(\frac{k_{z}^{2}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}c_{j}^{2}+\frac{k_{z}v_{dsz}}{kv_{ts\sigma}}c_{j}+\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{2k^{2}v_{ts\sigma}^{2}})+\frac{k^{2}k_{z}^{2}v_{zts}^{4}}{k^{3}v_{ts\sigma}^{3}}c_{j}^{3}+2\frac{k^{2}k_{z}v_{zts}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}c_{j}^{2}+[\frac{1}{2}\frac{k^{2}k_{x}^{2}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsz}^{2}-(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}^{2}v_{zts}^{2}v_{\perp ts\sigma}^{2}}{k^{3}v_{ts\sigma}^{3}}]c_{j}-(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsz}\Big{\}}=-1+\sum_{j=1}^{J}\frac{p_{33sj\sigma}}{\omega-c_{sj\sigma}}.
In the above, for example, p_{11sj\sigma}=2b_{j}\Big{\{}-i\nu_{s}(\frac{k_{x}^{2}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}c_{j}^{2}+\frac{k_{x}v_{dsx}}{kv_{ts\sigma}}c_{j}+\frac{k_{z}^{2}v_{zts}^{2}}{2k^{2}v_{ts\sigma}^{2}})+\frac{k^{2}k_{x}^{2}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}c_{j}^{3}+2\frac{k^{2}k_{x}v_{\perp ts\sigma}^{2}}{k^{2}v_{ts\sigma}^{2}}v_{dsx}c_{j}^{2}+[\frac{1}{2}v_{\perp ts\sigma}^{2}\frac{k^{2}k_{z}^{2}v_{zts}^{2}}{k^{3}v_{ts\sigma}^{3}}+\frac{k^{2}}{kv_{ts\sigma}}v_{dsx}^{2}+(\lambda_{Ts\sigma}-1)\frac{k_{x}^{2}k_{z}^{2}v_{\perp ts\sigma}^{4}}{k^{3}v_{ts\sigma}^{3}}]c_{j}+\frac{k_{x}k_{z}^{2}}{k^{2}v_{ts\sigma}^{2}}(\lambda_{Ts\sigma}-1)v_{\perp ts\sigma}^{2}v_{dsx}\Big{\}}, and others are similar and thus we have not written them out explicitly.
We find the result is very simple by use , which would also make the EM3D-M case be much simpler than the previous PASS-K [1] derivation. This is also why we use in the ES3D case, which gives a more compact form.
Thus, we obtain the relations between and , which has the following form (with )
[TABLE]
with the coefficients
[TABLE]
Considering that are complex number and , and are real number, the singularity of the above form is also only .
4.2.2 The magnetized terms
Similarly to the unmagnetized case, considering the defination , after -pole expansion, we have
P_{s\sigma 11}^{m}\simeq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{J}\frac{k_{z}v_{zts}b_{j}}{\omega-c_{snj}}\Big{\{}[\frac{n\omega_{cs}}{k_{x}}(n\omega_{cs}+k_{x}v_{dsx}-i\nu_{s})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}\frac{1}{k_{z}v_{zts}}+A_{n0\sigma}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})\frac{c_{j}}{v_{zts}^{2}}](\frac{n\omega_{cs}}{k_{x}}+v_{dsx})\Big{\}}-{\color[rgb]{1,0,0}1}=-1+\sum_{n}\sum_{j}\frac{p_{11snj}}{\omega-c_{snj}}.
- 2.
P_{s\sigma 12}^{m}\simeq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{J}\frac{k_{z}v_{zts}b_{j}}{\omega-c_{snj}}\Big{\{}v_{dsy}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})(n\omega_{cs}\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}\frac{1}{k_{z}v_{zts}}+A_{n0\sigma}\frac{c_{j}}{v_{zts}^{2}})+i(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[(n\omega_{cs}-i\nu_{s})\frac{B_{nbs\sigma}}{v_{\perp ts\sigma}}\frac{1}{k_{z}v_{zts}}+\frac{B_{n0\sigma}v_{\perp ts\sigma}c_{j}}{v_{zts}^{2}}]\Big{\}}=\sum_{n}\sum_{j}\frac{p_{12snj}}{\omega-c_{snj}}.
- 3.
P_{s\sigma 21}^{m}\simeq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{J}\frac{k_{z}v_{zts}b_{j}}{\omega-c_{snj}}\Big{\{}\frac{n\omega_{cs}}{k_{x}}(n\omega_{cs}+k_{x}v_{dsx}-i\nu_{s})[-iv_{\perp ts\sigma}B_{nbs\sigma}+v_{dsy}A_{nbs\sigma}]\frac{1}{v_{\perp ts\sigma}^{2}}\frac{1}{k_{z}v_{zts}}+[-iv_{\perp ts\sigma}B_{n0\sigma}+v_{dsy}A_{n0\sigma}](\frac{n\omega_{cs}}{k_{x}}+v_{dsx})\frac{c_{j}}{v_{zts}^{2}}\Big{\}}=\sum_{n}\sum_{j}\frac{p_{21snj}}{\omega-c_{snj}}.
- 4.
P_{s\sigma 22}^{m}\simeq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{J}\frac{k_{z}v_{zts}b_{j}}{\omega-c_{snj}}\Big{\{}v_{dsy}[-iv_{\perp ts\sigma}B_{nbs\sigma}+v_{dsy}A_{nbs\sigma}]\frac{n\omega_{cs}}{v_{\perp ts\sigma}^{2}}\frac{1}{k_{z}v_{zts}}+v_{dsy}[-iv_{\perp ts\sigma}B_{n0\sigma}+v_{dsy}A_{n0\sigma}]\frac{c_{j}}{v_{zts}^{2}}+v_{\perp ts\sigma}[v_{\perp ts\sigma}C_{nbs\sigma}+iv_{dsy}B_{nbs\sigma}](n\omega_{cs}-i\nu_{s})\frac{1}{v_{\perp ts\sigma}^{2}}\frac{1}{k_{z}v_{zts}}+v_{\perp ts\sigma}[v_{\perp ts\sigma}C_{n0\sigma}+iv_{dsy}B_{n0\sigma}]\frac{c_{j}}{v_{zts}^{2}}\Big{\}}-{\color[rgb]{1,0,0}1}=-1+\sum_{n}\sum_{j}\frac{p_{22snj}}{\omega-c_{snj}}.
- 5.
P_{s\sigma 13}^{m}\simeq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{J}\frac{k_{z}v_{zts}b_{j}}{\omega-c_{snj}}\Big{\{}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})[n\omega_{cs}(\frac{c_{j}}{k_{z}}+v_{dsz}\frac{1}{k_{z}v_{zts}})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}+A_{n0\sigma}(v_{zts}c_{j}^{2}+(k_{z}v_{dsz}-i\nu_{s})\frac{c_{j}}{k_{z}})\frac{1}{v_{zts}^{2}}]\Big{\}}=\sum_{n}\sum_{j}\frac{p_{13snj}}{\omega-c_{snj}}.
- 6.
P_{s\sigma 31}^{m}\simeq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{J}\frac{k_{z}v_{zts}b_{j}}{\omega-c_{snj}}\Big{\{}\frac{n\omega_{cs}}{k_{x}}(n\omega_{cs}+k_{x}v_{dsx}-i\nu_{s})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}(\frac{c_{j}}{k_{z}}+v_{dsz}\frac{1}{k_{z}v_{zts}})+A_{n0\sigma}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})(\frac{c_{j}^{2}}{v_{zts}}+v_{dsz}\frac{c_{j}}{v_{zts}^{2}})\Big{\}}=\sum_{n}\sum_{j}\frac{p_{31snj}}{\omega-c_{snj}}.
- 7.
P_{s\sigma 23}^{m}\simeq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{J}\frac{k_{z}v_{zts}b_{j}}{\omega-c_{snj}}\Big{\{}[v_{dsy}A_{nbs\sigma}-iv_{\perp ts\sigma}B_{nbs\sigma}][(\frac{c_{j}}{k_{z}}+\frac{v_{dsz}}{k_{z}v_{tsz}})\frac{n\omega_{cs}}{v_{\perp ts\sigma}^{2}}]+[v_{dsy}A_{n0\sigma}-iv_{\perp ts\sigma}B_{n0\sigma}][v_{zts}c_{j}^{2}+(k_{z}v_{dsz}-i\nu_{s})\frac{c_{j}}{k_{z}}]\frac{1}{v_{zts}^{2}}\Big{\}}=\sum_{n}\sum_{j}\frac{p_{23snj}}{\omega-c_{snj}}.
- 8.
P_{s\sigma 32}^{m}\simeq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{J}\frac{k_{z}v_{zts}b_{j}}{\omega-c_{snj}}\Big{\{}[n\omega_{cs}\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}\frac{c_{j}}{k_{z}}+A_{n0\sigma}\frac{k_{z}v_{zts}c_{j}^{2}}{v_{zts}^{2}}]v_{dsy}+[n\omega_{cs}\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}\frac{1}{k_{z}v_{zts}}+A_{n0\sigma}\frac{c_{j}}{v_{zts}^{2}}]v_{dsy}v_{dsz}+i[(n\omega_{cs}-i\nu_{s})\frac{B_{nbs\sigma}}{v_{\perp ts\sigma}}\frac{c_{j}}{k_{z}}+B_{n0\sigma}\frac{v_{\perp ts\sigma}c_{j}^{2}}{v_{zts}}]+iv_{dsz}[(n\omega_{cs}-i\nu_{s})\frac{B_{nbs\sigma}}{v_{\perp ts\sigma}}\frac{1}{k_{z}v_{zts}}+B_{n0\sigma}\frac{v_{\perp ts\sigma}c_{j}}{v_{zts}^{2}}]\Big{\}}=\sum_{n}\sum_{j}\frac{p_{32snj}}{\omega-c_{snj}}.
- 9.
P_{s\sigma 33}^{m}\simeq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{J}\frac{k_{z}v_{zts}b_{j}}{\omega-c_{snj}}\Big{\{}[n\omega_{cs}(\frac{v_{zts}c_{j}^{2}}{k_{z}}+v_{dsz}\frac{c_{j}}{k_{z}})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}+A_{n0\sigma}(c_{j}^{3}+(k_{z}v_{dsz}-i\nu_{s})\frac{c_{j}^{2}}{k_{z}v_{zts}})]+v_{dsz}[n\omega_{cs}(\frac{c_{j}}{k_{z}}+v_{dsz}\frac{1}{k_{z}v_{zts}})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}+A_{n0\sigma}(\frac{c_{j}^{2}}{v_{zts}}+(k_{z}v_{dsz}-i\nu_{s})\frac{c_{j}}{k_{z}v_{zts}^{2}})]\Big{\}}-{\color[rgb]{1,0,0}1}=-1+\sum_{n}\sum_{j}\frac{p_{33snj}}{\omega-c_{snj}}.
In the above, say, p_{11snj}=k_{z}v_{zts}b_{j}\Big{\{}[\frac{n\omega_{cs}}{k_{x}}(n\omega_{cs}+k_{x}v_{dsx}-i\nu_{s})\frac{A_{nbs\sigma}}{v_{\perp ts\sigma}^{2}}\frac{1}{k_{z}v_{zts}}+A_{n0\sigma}(\frac{n\omega_{cs}}{k_{x}}+v_{dsx})\frac{c_{j}}{v_{zts}^{2}}](\frac{n\omega_{cs}}{k_{x}}+v_{dsx})\Big{\}}, and other terms are similar.
It is thus easy to find that after -pole expansion, the relations between and has the following form (with )
[TABLE]
with the coefficients
[TABLE]
In numerical test, considering the cut of summation , i.e., with , we find the above original form of , and , are better than the below form [to undetstand]
[TABLE]
It is readily to see that all the singularities from in are removable. The singularities at in , , , , are also removable. Thus, the overall equations have no singularity and will not meet numerical difficulty. In the solver, to short the code, if we set for magnetized species in EM version. For example, we can set .
Combining Eqs. (111), (112) and (114), the equivalent linear system for electromagnetic dispersion relation can be obtained as
[TABLE]
which yields a sparse matrix eigenvalue problem, where and so on. The symbols , and used here do not have direct physical meanings but are analogy to the perturbed velocity and current density in the fluid derivations of plasma waves. The elements of the eigenvector still represent the original electric and magnetic fields. Thus, the polarization of the solutions can also be obtained in a straightforward manner. The dimension of the matrix is . And another good aspect of the final PASS-K matrix equation is that it is valid for arbitrary real number of and , i.e., and the only requirement is .
4.3 The Darwin case
Based on the electromagnetic result, the Darwin model case is straightforward, where the linear system for is the same to electromagnetic case. We only need to modify the the linear system of the Maxwell’s equations, which are also straightforward
[TABLE]
and the matrix eigenvalue problem becomes , where is still the same as the electromagnetic one from Eq.(117) and changes from unit matrix to
[TABLE]
Though may not be full rank matrix, the standard eigenvalue library, such as ’eig()’ in Matlab, can solve the eigenvalue problem well.
4.4 The polarizations
The matrix solver can obtain directly666In principle, PASS-K matrix can also obtain as in standard matrix . To obtain group velocity or do ray tracing, we may also need and . from the matrix eigenvalue problem. Considered that the magnitude of the wave has no meaning for a linear system, we should do normalizations. We set and .
Some other useful: electric field energy , magnetic field energy , energy flux Poynting vector . .
5 Benchmark
There exists numerous applications of this newly developed updated version PASS-K tool, we only show some typical benchmarks to the reader to get a flavor of it.
The first benchmark is the ring beam case in Ref.[12], which is to make sure the function , and are treated correctly in our model. The results are shown in Fig.1, with very good agreement with Min’s [17] code. The second benchmark shown in Fig.2 is the mixed of magnetized and unmagnetized species in Ref.[14] for the instabilities driven by perpendicular beam in shock, which also show good agreement. And, the treatment of ion to be magnetized species also shows close result to the unmagnetized ion model, which implies that the unmagnetized ion assumption is valid in that case. This also gives us confidence of the validity of our magnetized model, since that the equations are totally different but yield similar solution as should be. The third benchmark shown in Fig.3 is for the Darwin model in Ref.[16], which is the same as the one solved using accurate function with conventional iterative root finding in Fig.5 of Ref.[16]. Here, we have also shown other branches and branches. The symmetry between and solutions implies that the function is treated correctly for in this new solver.
The benchmark parameters for above cases are listed below.
Defaultly we use SI unit: , , , , .
- 2.
Parameters for Ref.[12] benchmark: . , , , , , .
- 3.
Parameters for Ref.[14] benchmark: , , , , . , , , , , .
The purpose of the present work is to provide the foundation of this new tool. And thus, the applications to new examples would be discussed in other works.
6 Summary and Discussion
In summary, a powerful new kinetic dispersion relation tool is developed, which largely extend both the physical models and numerical capacilities of other works in literature. The advantages of this new version of PASS-K tool is that it contents many new features (anisotropic temperature/loss cone/drift in arbitrary direction/ring beam/collision, unmagnetized/magnetized, electrostatic/electromagnetic/Darwin, etc) and can be widely applied. And compared to some other solvers, the () and () cases are not singular in PASS-K. What is more, the modes are also correctly treated. The most attractive feature is that it does not require initial guess for root finding and thus will not miss solutions. Thus, we think that this is a unified tool what exactly dreamed of by the plasma community.
The limitation of the present PASS-K approach is that it can only used for cases when the 2D velocity integral are decoupled, i.e., usually required . Typically, the present PASS-K approach can not be used to oblique propagation -distribution function [19, 20] and relativistic [3] or other arbitrary distribution[18] cases. However, extensions are possible. For example, for -distribution, we can use the similar -pole for corresponding function777The -distribution , which is slightly different from the standard one and thus the corresponding transformation is also slightly different. of parallel velocity integral and use Gaussian quadrature for perpendicular velocity integral, which will yield a 2D -pole expansion and then can be transformed to a equivalent linear matrix eigenvalue problem, which is solvable though the corresponding matrix dimension could be much larger than the Maxwellian one, depends on how many nodes are used for perpendicular velocity integral. And thus, with 2D -pole expansion, the PASS-K approach can also be used for other distributions and non-uniform magnetic field drift wave case such as in gyrokinetic model [21]. The discussion of these extensions are out of the range of the present work. Besides the conventional iterative root finding approaches, it is still required to develop a new powerful algorithm, say, similar to PASS-K approach, to solve the more challenged relativistic kinetic plasma dispersion relation generally.
7 Acknowledgments
The author thanks the many suggestions, discussions and benchmarks from Richard Denton, Xin Tao, Jin-song Zhao, Zhong-wei Yang (helped the benchmark of Fig.2), Chao-jie Zhang (helped the benchmark of unmagnetized version), Can Huang, Wen-ya Li, Liang Wang, Kyungguk Min (provided the benchmark data in Fig.1), Yang Li, and many others. This work, which is particular useful, say for examples to study the loss-cone and LHDI instabilities in magnetic confinement fusion and beam instabilities in inertial fusion, is supported by the compact fusion project in ENN group.
Appendix A The PASS-K Solver
The update version of PASS-K/PDRK code includes more features, but still consists of two parts: the main program and the input data file. The input file “passk.in” has the follow structure (the case in Fig.2)
qs(e) ms(mp) ns(m^-3) Tzs(eV) Tps(eV) alphas Deltas vdsz/c vdsx/c vdsy/c vdsr/c nu_s m_or_u(1/0) 1 1.0 0.8e0 0.111 0.111 1.0 1.0 0.0 -8.333e-3 0.0 0.0 0.0 0 1 1.0 0.2e0 0.111 0.111 1.0 1.0 0.0 3.333e-2 0.0 0.0 0.0 0 -1 1.1111e-3 1.0e0 0.010 0.010 1.0 1.0 0.0 0.0 0.0 0.0 0.0 1
One can add the corresponding parameters for additional species. Here, ’qs(e)’, ’ ms(mp) ’, ’ns(m-̂3)’, ’Tzs(eV) ’, ’Tps(eV) ’, ’alphas’, ’Deltas’, ’vdsz/c’, ’vdsx/c ’, ’ vdsy/c’, ’ vdsr/c ’, ’nu_s ’, ’m_or_u(1/0) ’, are the charge in electron charge unit , mass in proton mass unit , density , parallel temperature , perpendicular temperature , loss cone parameters and , drift velocities , , , ring beam velocity , collision frequency , and whether the species is magnetized, respectively. Normalizations and other parameters can be set in PASS-K main program (more information can be found at: http://hsxie.me/codes/pdrk/ or https://github.com/hsxie/pdrk/).
Appendix B Relation to the dispersion relation of drift instabilities in inhomogeneous plasma
In conventional derivation [3, 5, 4] of drift instabilities in inhomogeneous magnetized plasma, the distribution function is assumed to depend on space inhomogeneous. We take the electrostatic density inhomogeneous case as in Chap. 4 of Ref.[5] for example to compare with our model. The conventional derivation, i.e., Eqs.(4.2.1) and (4.2.2) in Ref.[5], which ignores all terms of order and valids only for and , gives
[TABLE]
with , and , and where we have ignored the term. Here, is parameter for space inhomogeneous and is for force such as gravity. The corresponding dispersion relation in our model is
[TABLE]
with , and we have used . If we consider the perpendicular drift in our model is due to space inhomogeneous and gravity , with , and , by using force balance and , we have . We noticed that Eqs.(121) and (123) are exactly the same, except that in the conventional derivation (121) the density inhomogeneous drift velocity in not included in the argument of function .
That is to say, our model can be used to study the drift modes in inhomogeneous plasma although we assume the derivation is under homogeneous plasma assumption. It is also not easy to say which of the the two derivations, i.e., the conventional one and the present one, is more accurate to describe the drift modes in inhomogeneous plasma, because that in our derivation the inhomogeneity contributes to the drift velocity and all other steps are rigorous, whereas the conventional derivation ignores high order terms in more than one steps to obtain the final dispersion relation.
Appendix C More -pole coefficients of function
Ref.[1] has provided the typical -pole coefficients of function with . Here, following Ref.[1], we update them to more accurate data and even beyond the double precision (16 digit number) number with the help of symbolic computation. In the Table 1, means we keep equations for , and means we keep equations for , as described in Ref.[1].
Appendix D The Possible Numerical Inaccuracy
Since PASS-K solves the dispersion relation use matrix eigenvalue approach, the computational time to obtain all solutions for a dimensions matrix is , with usually . Thus, the numerical with cut off to is crucial. For larger , the matrix dimension could be large and the computation will slower. Usually is sufficient for many cases and the computation is very fast, which can yield all solutions for a typical single run in less than 1 second. For larger , say , the sparse matrix approach to search solutions around some initial guesses are still fast and thus the PASS-K tool has not limitation on this.
However, there exists several possible numerical inaccuracies, which should be careful of:
(1) The -pole expansion of may give some artificial growing mode with growth rate for , which however can be distinguished and reduced by using , etc.
- 2.
(2) The cut off in , which should be checked by using larger to make sure the results are convergent.
- 3.
(3) Rand off error of eigen solver library, due to that the default numerical data is double precision (16 digit number) in the solver. This in principle can be solved by using high precision digit data.
- 4.
(4) Others, such as singularity in Darwin matrix, degree of accuracy of functions , , and , etc.
The above inaccuracies mainly occur at extremely small and large . The third inaccuracy is not easy to distinguish at this moment, and however, we have not met this problem for most benchmarks.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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