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Journal of Fusion Energy ISSN 0164-0313 J Fusion EnergDOI 10.1007/s10894-013-9594-0

Thermonuclear Fusion Reactivities forDrifting Tri-Maxwellian Ion VelocityDistributions

D. Nath, R. Majumdar & M. S. Kalra

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ORIGINAL RESEARCH

Thermonuclear Fusion Reactivities for Drifting Tri-MaxwellianIon Velocity Distributions

D. Nath • R. Majumdar • M. S. Kalra

Received: 28 November 2012 / Accepted: 4 January 2013

� Springer Science+Business Media New York 2013

Abstract The presence of strong magnetic fields in

magnetically confined thermonuclear fusion plasmas

introduces a considerable anisotropy in the ion velocity

distributions. Drifting or streaming of ionic species is

another source of anisotropy in fusion plasmas. To account

for these anisotropies, we consider in the present work a

generalization of the equilibrium Maxwellian distribution

in the form of a drifting tri-Maxwellian distribution func-

tion. It is shown that the calculation of thermonuclear

reactivities for these distributions is best carried out in a

transformed velocity space. Computational results for gy-

rotropic velocity distributions with drifts parallel and per-

pendicular to the magnetic field are presented.

Keywords Plasma � Fusion reactivity � Bi/tri-Maxwellian

distributions � Anisotropy � Drifts

List of symbols

a, A Dimensionless constants

b, B Dimensionless constants

c, C Dimensionless constants

f Distribution function

k, l As in Eq. 12

m Mass

r Vdrift/vrms

R T\/TkT Temperature (in energy units)

u Dimensionless velocity

v V - V0

V Velocity in laboratory frame

V0 Drift velocity

a, b, c Dimensionless constants

r Microscopic fusion cross section

hr vi Fusion reactivity

Subscripts

A, B Species

k Parallel

D Deuterons

\ Perpendicular

T Tritons

i x, y, z

Introduction

Fusion reactivities generally reported in the literature are for

two interacting Maxwellian species at the same temperature

[1–4]. In magnetically confined thermonuclear fusion plas-

mas under reacting conditions, the ion velocity distributions

can, however, deviate considerably from Maxwellian dis-

tributions. Especially the presence of strong magnetic fields

introduces anisotropy because it leads to different particle

velocities parallel and perpendicular to the magnetic field,

characterized by different kinetic temperatures. Another

source of anisotropy is drifting or streaming of ionic species

caused, for example, by neutral-beam-injection (NBI)

plasma heating. The interaction of the energetic charged

particles produced in exothermic nuclear reactions with the

plasma can also lead to deviations from the Maxwellian [5].

These enhance the tail part of the Maxwellian spectrum. On

D. Nath (&) � M. S. Kalra

Nuclear Engineering and Technology Program, Indian Institute

of Technology Kanpur, Kanpur 208 016, India

e-mail: dnath@iitk.ac.in

M. S. Kalra

e-mail: msk@iitk.ac.in

R. Majumdar

Department of Nuclear Engineering, North Carolina State

University, Raleigh, NC 27695-7909, USA

123

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DOI 10.1007/s10894-013-9594-0

Author's personal copy

the other hand, faster reactions or losses at the high energy

end of the spectrum can result in a tail-depleted Maxwellian

distribution. Impact of these deviations from the isotropic

Maxwellian distributions on fusion reactivities can be con-

siderable and worth investigating in connection with prac-

tical fusion reactors. Such information may play a significant

role in hot thermonuclear plasma diagnostics, characteriza-

tion, power balances, and the physics and engineering gain

factors [4].

The effect on fusion reactivities of Maxwellian tail

depletion and enhancement under isotropic conditions has

been studied earlier both analytically and numerically

[6–9]. A kinetic analysis of non-Maxwellian knock-on ions

in a self-sustained D-T, fusion plasma with an admixture of

other light elements has been used to show a reactivity

enhancement [5]. Some results related to the effect of

anisotropy as well as unequal ion temperatures on reac-

tivity have also been reported [8, 10].

In the present work, we consider a generalization of the

equilibrium Maxwellian distribution in the form of a

streaming tri-Maxwellian distribution function in which

each velocity component of a reacting species is charac-

terized by a different kinetic temperature with an associ-

ated drift. The formulation is very general and can

accommodate arbitrary drifts in the reacting species toge-

ther with the anisotropy resulting from unequal ion tem-

peratures parallel and perpendicular to the magnetic field as

a special case. It is shown that the calculation of fusion

reactivities for these distributions is best carried out in a

transformed velocity space resulting in a reduction of

computational time by two to four orders of magnitude

depending on the accuracy desired. These computations

would be prohibitively time-consuming if carried out in the

laboratory frame of reference. This formulation is then

applied to investigate the effect of bi-Maxwellian gyro-

tropic velocity distributions on fusion reactivities with or

without parallel or perpendicular drift or streaming. The

results of computations are presented for D-T, D-D, and

D-3He fusion reactions in the form of reactivity surfaces as

well as their projections on appropriate planes.

Tri-Maxwellian Distribution with Drifts

A simple generalization of the equilibrium Maxwellian

distribution allows us to represent a streaming anisotropic

plasma characterized by different component temperatures

and drifts. The unit-normalized shifted or drifting Max-

wellian distribution for one component of the velocity of a

species is given as below [11–13]:

fAðVAiÞ ¼mA

2pTAi

� �1=2

� exp �mA VAi � V0

Ai

� �2

2TAi

( ); ð1Þ

where TAi is the kinetic temperature of the component i, VAi0

is the corresponding drift, and other symbols are given in

the Nomenclature. The full streaming tri-Maxwellian

distribution for species A is obtained by multiplying the

three one-dimensional Maxwellians given in Eq. 1:

fAiðVAÞ ¼Y

i¼x;y;z

mA

2pTAi

� �1=2

� exp �mA VAi � V0

Ai

� �2

2TAi

( );

ð2Þ

and similarly for the species B:

fBiðVBÞ ¼Y

i¼x;y;z

mB

2pTBi

� �1=2

� exp �mB VBi � V0

Bi

� �2

2TBi

( ):

ð3Þ

In the present work, A will stand for D (deuterons) and

B will stand for D, T, or 3He, depending upon the fusion

reaction under consideration. In the formulation presented

below, in general, TAi = TBi; TAi = TAj; i = j. Thus the

formulation can accommodate different component

temperatures and drifts for each of reacting species.

In terms of the unit-normalized tri-Maxwellians in

Eqs. 2 and 3, the fusion reactivities or the reaction rate

parameters, rvh i, can be written as the following six-

dimensional integral in velocity space:

rvh i ¼ZZ

d3VAd3VBjVA � VBj

� rAB jVA � VBjð ÞfA VAð ÞfB VBð Þ:ð4Þ

where rAB is the microscopic fusion cross section between

species A and B and depends only on the relative speed

jVA � VBj.

Nondimensionalization and Velocity Space

Transformation

The analytical expressions for rAB for D-T, D-D, and

D-3He reactions are well established [14]. However, evalua-

tion of rvh iwith any reasonable accuracy by direct calculation

of the six-dimensional integral in Eq. 4 is computationally

prohibitive. In this section, we present a velocity space trans-

formation which reduces the six-dimensional integral in Eq. 4

to a three-dimensional integral which can be numerically

evaluated easily. The nondimensionalized velocities uA and

uB are obtained as

uA ¼vA

vN; uB ¼

vB

vN;

where vA ¼ VA � V0A

� �and vB ¼ VB � V0

B

� �, i.e., in

component form, vAi ¼ VAi � V0Ai

� �; i = x, y, z. Rest of

the parameters are given below:

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v2N

� �¼ v2

A

� �þ v2

B

� �

v2A

� �¼ 3TA

mA; v2

B

� �¼ 3TB

mB

TA ¼TAx þ TAy þ TAz

3; TB ¼

TBx þ TBy þ TBz

3:

In terms of the dimensionless velocities uA and uB, the

distribution functions in Eqs. 2 and 3 can be written as

fAiðuAÞ ¼1

2p

� �3=21ffiffiffiffiffiffiffiabcp � exp � u2

Ax

2a�

u2Ay

2b�

u2Az

2c

!;

ð5Þ

fBiðuBÞ ¼1

2p

� �3=21ffiffiffiffiffiffiffiffiffiffi

ABCp � exp � u2

Bx

2A�

u2By

2B�

u2Bz

2C

!;

ð6Þ

where the dimensionless parameters a, b, c and A, B, C are

given below

a ¼ TAx

3 TAþmAmB

TB

; A ¼ TBx

3mBmA

TAþTB

b ¼ TAy

3 TAþmAmB

TB

; B ¼ TBy

3mBmA

TAþTB

c ¼ TAz

3 TAþmAmB

TB

; C ¼ TBz

3mBmA

TAþTB

9>>>>>>>=>>>>>>>;: ð7Þ

We now transform the velocity space uA; uBð Þ to u;Uð Þgiven for each component as

ui ¼ uAi � uBi Ui ¼ kiuAi þ liuBi: ð8Þ

Note that u is simply the dimensionless relative velocity of

the interacting species whereas U is a linear combination of

uA and uB yet to be specified.

The fusion reactivity, in terms of the transformed

velocities u and U, can be written as

rvh i ¼ vN

Zd3uur uð ÞfAB uð Þ; ð9Þ

where

fAB uð Þ ¼ fAB uxð ÞfAB uy

� �fAB uzð Þ; ð10Þ

and

fAB uxð Þ ¼1

2pffiffiffiffiffiffiaAp

Z1

�1

dUx

� exp � Ux � kxuxð Þ2

2A

( )

� exp � Ux þ lxuxð Þ2

2a

( ):

ð11Þ

Similar expressions hold for y and z components. We

now choose kx and lx such that the Jacobian determinant of

the transformation in Eq. 8 has a magnitude of 1 and the

cross-terms containing the product uxUx in the exponent in

Eq. 11 become zero. So both these requirements are

satisfied if and only if

kx ¼A

aþ A; lx ¼

a

aþ A: ð12Þ

The integral in Eq. 11 can then be shown to yield

fAB uxð Þ ¼ffiffiffiap

rexp �au2

x

� �; ð13Þ

where a ¼ 1= 2aþ 2Að Þ. For y and z components, a and A

in Eq. 12 are replaced by b and B, c and C, respectively,

giving the following expressions for y and z components of

the relative velocity.

fAB uy

� �¼

ffiffiffibp

rexp �bu2

y

; ð14Þ

fAB uzð Þ ¼ffiffifficp

rexp �cu2

z

� �; ð15Þ

where b ¼ 1= 2bþ 2Bð Þ; c ¼ 1= 2cþ 2Cð Þ. Thus we finally

have,

fAB uð Þ ¼ffiffiffiffiffiffiffiffiabc

p3=2

rexp �au2

x � bu2y � cu2

z

: ð16Þ

Using this equation (Eq. 16) for the relative velocity

distribution, the reactivity in Eq. 9 can be easily evaluated

by a three-dimensional integral instead of a six-

dimensional integral implied in Eq. 4. We may mention

here that for axisymmetric situations investigated in the

present work, a further reduction is possible by

transforming to cylindrical coordinates. However, we do

not attempt it here to avoid more algebra. Further

reduction for the special case of axisymmetric

distributions loses the generality of Eq. 16, and, since

the computation is already greatly simplified, not much

further is gained by specializing Eq. 16 to specific

cases. The evaluation of the three-dimensional integral

in Eq. 16 is computationally quite acceptable and

straightforward.

Computational Results and Discussion

The general formulation developed in the preceding section

is now applied to fusion reactivity calculations for bi-

Maxwellian gyrotropic distributions with and without drifts

as well as for Maxwellian species with a relative drift.

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Reactivity Surface for Gyrotropic Distributions

Without Drifts

The anisotropic bi-Maxwellian two-temperature gyrotropic

velocity distribution (also referred as two-dimensional

pseudo-Maxwellian or pancake distribution) is given by

Stacey [3, 12]

fA Vð Þ ¼ m

2pT?

� �m

2pTk

� �1=2

� exp �mv2?

2T?�

mv2k

2Tk

( );

ð17Þ

where T\ and Tk denote the kinetic temperatures (in joules)

perpendicular and parallel to the magnetic field. Here we

consider two interacting species with same T\ and Tkwithout drifts. For this case the tri-Maxwellian formulation

presented in nondimensionalization section can be used by

simply substituting

TAx ¼ TAy ¼ TBx ¼ TBy � T?;

TAz ¼ TBz ¼ Tk;

V0Ai ¼ V0

Bi ¼ 0:

and for the above case we may define an effective kinetic

temperature, Tk, and a ratio R as follows:

Tk ¼2T? þ Tk

3; R ¼ T?

Tk:

Due to fast reactivity computations made possible by the

formulation presented in this work, it is possible to calculate

the reactivity surface over a wide range of Tk and R without

much computational effort. These are presented in Fig. 1 for

D-T, D-T, and D-3He, reactions using the microscopic cross

sections given in [14]. The projections of these surfaces for

selected temperatures are shown in Fig. 2. It can be seen

from these figures that for D-T reaction the gyrotropic

velocity distribution can enhance the reactivity considerably

at 5 keV as compared to the Maxwellain distribution

(R ¼ 1). At 15 keV, it has almost no effect on reactivity

while at 30 keV the reactivity for gyrotropic distribution is

less than the corresponding Maxwellian reactivity. For the

D-D and D-3He, reactions also, a similar behavior is

observed, i.e., at lower temperatures, the reactivity is

enhanced by the two-temperature distribution while at

higher temperatures, it is reduced as compared to the

corresponding Maxwellian reactivity.

Effect of Drift on Maxwellian Reactivities

To see the effect of relative drift on the fusion reactivity of

two Maxwellian species, we simply take all temperatures

(a)

(b)

(c)

Fig. 1 Reactivity surfaces for the gyrotropic velocity distributions

a D-T, b D-D, and c D-3He reactions

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same in Eqs. 2 and 3 and all drifts zero except one. Here

we consider deuterons as having drift velocity and define a

ratio

r ¼ Vdrifts

Vrms;

where Vrms ¼ 3T=mDð Þ1=2is the root mean square speed of

the deuterons. The computed reactivities at various tem-

peratures for D� T reaction are shown in Fig. 3 It can be

seen that drift enhances the reactivity considerably, par-

ticularly at lower temperatures.

(a)

(b)

(c)

Fig. 2 Projections of the reactivity surfaces for selected temperatures

a D-T, b D-D, and c D-3He reactions

(a)

(b)

Fig. 3 Effect of drift on Maxwellian reactivities for D� T reaction areactivity surface, b projection on rvh i � r plane at temperatures

5 keV; 15 keV, and 30 keV

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(a)

(b)

(c)

Fig. 4 Effect of parallel drift on bi-Maxwellian reactivities at

tempertures a 10 keV, b 15 keV, and c 20 keV

(a)

(b)

(c)

Fig. 5 Effect of perpendicular drift on bi-Maxwellian reactivities

a 10 keV, b 15 keV, and c 20 keV

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Effect of Parallel and Perpendicular Drifts on Bi-

Maxwellian Reactivities

As mentioned above, for the bi-Maxwellian gyrotropic

distribution, we take temperature x and y direction as T\

and z direction as Tk. The root mean square speed would

now be defined as 3Tk=mDð Þ1=2. The results are shown in

Figs. 4 and 5 for parallel and perpendicular drifts respec-

tively. It can be observed from theses figures that the rel-

ative drift, parallel or perpendicular, between the

interacting species enhances the reactivity. At 15 keV,

when Vdrift is comparable to Vrms r ¼ 1ð Þ, the fusion reac-

tivity almost doubles as compared to no drift ðr ¼ 0Þ.Similar effect is seen at other temperatures shown in these

figures.

Conclusion

A formulation for the calculation of thermonuclear fusion

reactivities for drifting/streaming tri-Maxwellian ion

velocity distributions is presented based on a transformed

velocity space. The velocity transformations proposed in

the present work reduce the six-dimensional reactivity

integral in the laboratory frame of reference into a three-

dimensional integral in the transformed velocity space. The

later can be evaluated very fast, resulting in a reduction of

computation time by two to four orders of magnitude,

depending on the accuracy desired. This formulation is

applied in evaluating the fusion reactivities for bi-Max-

wellian gyrotropic distributions with and without parallel

or perpendicular drift between two interacting species. The

effect of drift between two interacting Maxwellian species

is also investigated using the same formulation. It is found

that the relative drift increases the fusion reactivity at all

temperatures of interest in thermonuclear fusion. The

anisotropy introduced by the bi-Maxwellian distribution

increases the reactivity at lower temperatures and decreases

the same at higher temperatures as compared to the Max-

wellian reactivities.

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