NASA Technical Reports Server (NTRS) 19980213237: Polarized Nuclei in a Simple Mirror Fusion Reactor

18 Pages·0.71 MB·English

by  NASA Technical Reports Server (NTRS)

NASA-TN-112506
POLARIZED NUCLEI IN A SIMPLE
FUSION REACTORS
MIRROR FUSION REACTOR
KEYWORD8: polarized nucleL
mirror reactor, alpha particles
DAVID A. NOEVER National Aeronautics and Space Administration
Marshall Space Flight Center, ES76, Huntsville, Alabama 35812
r
Received November 29, 1993
Accepted for Publication July 14, 1994
ises 50% higher reactivity and 50% more nuclear power
for the same conditions (or the same power at lower op-
The possibifity of enhancing the ratio of output to erating pressures if beta is seriously constrained). Like-
input power Q in a simple mirror machine by polariz- wise, the sin 20 angular distribution of generated alpha
ing deuterium-tritium (D-T) nuclei is evaluated. Tak- particles should reduce alpha-particle end losses from
ing the Livermore mirror reference design mirror ratio the mirror. Consequently, more of the alpha-particle
of 6.54, the expected sin 20 angular distribution of fu- 3520 keV should be deposited in the reactant plasma
sion decay products reduces immediate losses of alpha leading to a net reduction in the required external heat-
particles to the loss cone by 7.6% and alpha-ion scat- ing power.
tering losses by -50%. Based on these findings, alpha- Enhanced alpha-particle trapping results from
particle confinement times for apolarized plasma shouM
1. the reduced number of alpha particles immedi-
therefore be 1.11 times greater than for isotropic nu-
ately "born" into the loss region
clei. Coupling this enhanced alpha-particle heating with
the expected >50% D-T reaction cross section, a cor- 2. the reduced alpha-ion scattering losses of the re-
responding power ratio for polarized nuclei, Qpotarized, maining alpha particles.
isfound to be 1.63 times greater than the classical un-
For these alpha-particle loss mechanisms, analytical
polarized value Qclassica/. The effects of this increase in
and numerical enhancement factors are derived, and
Q are assessed for the simple mirror.
the ratio of polarized to isotropic alpha-particle con-
finement time is found. These results are incorporated
into an equilibrium fusion energy balance to translate
the additional alpha-particle heating into a higher ratio
of output to input power (henceforth referred to as Q).
INTRODUCTION
The availability of multiampere beams of nuclear ALPHA-PARTICLEHEATING
polarized atoms I-3 and their resistance to depolariza-
tion under fusion reactor conditions 4-9 has sparked in-
As larger experimental fusion machines approach
terest in the use of fuel polarization to enhance fusion
reactor conditions, alpha-particle heating is an increas-
reactor performance. For a magnetic mirror device, po-
ing issue of concern, t° Alpha particles generated pri-
larization of deuterium (D) and tritium (T) nuclei par-
marily by the reaction D + T --*4He + n carry fully
allel to the B field offers two advantages:
one-fifth of the fusion reaction energy. Since this ki-
1. a 50°70increase in the effective nuclear cross sec- netic energy is "attached" to a charged species, some
tion portion is magnetically confined to be partitioned
among the fusion reactants. Alpha-particle studies prin-
2. fusion decay products that are emitted predom-
cipally seek to determine the dynamics of energy re-
inantly perpendicular to the magnetic field in a
laxation within the background plasma and the total
sin 20 angular distribution.
amount of energy deposited with each species (see
Because of these two effects, polarization of injected Ref. 11 for summary).
nuclei is expected to increase the ratio of output to input The principal alpha-particle energy loss terms are
power for the mirror. The higher cross section prom- assumed here to be
86 FUSION TECHNOLOGY VOL. 27 JAN. 1995
Noever POLARIZED NUCLEI
1. the immediate loss of alpha particles born into TABLE I
the loss cones
Polarized-to-lsotropic Ratio of Trapped Alpha Particles
2. slowing-down collisions with electrons and ions
Ratio of
3. pitch-angle scattering by collisions into the loss lsotropic Polarized Polarized
cones. Mirror Trapped Trapped to Isotropic
Ratio O Fraction Fraction Trapping
Anomalous energy losses such as Alfv6n wave coupling
will be ignored. Each loss term is considered in turn 14.9 15 0.97 0.999 1.03
with the ultimate objective being to develop a simula- 4.0 30 0.87 0.983 1.13
tion model for the energy coupling of the alpha par- 2.0 45 0.71 0.888 1.25
ticles to the plasma for both an isotropic and sin 20 1.3 60 0.50 0.690 1.38
alpha-particle distribution. 1.1 75 0.26 0.382 1.47
LOSS CONEALPHA PARTICLES
Ignoring ambipolar potentials, the probability of Fokker-Planck equation is written here as an ordinary
immediate alpha-particle loss for an isotropic distribu- diffusion equation describing the pitch-angle scattering
tion is Ptoss = 1- cos 00, where cos do = (i - R- i)I/2, and solved numerically by standard forward-difference
and R is the mirror ratio. Polarized D-T nuclei, on the techniques. Anomalous effects of an energy-dependent
other hand, have an alpha-particle distribution that is diffusion coefficient will not be considered.
proportional to sin 2do and can be written as The following assumptions are made:
da 1. There is a three-component plasma of Maxwell-
-- - sin 20 ,
dfl ian electrons and ions with an alpha-particle distribu-
tion f that is isotropic in physical space but arbitrary
valid to the first order in da/dfl, the fraction of unpo- in velocity space.
larized nuclei, or as a normalized distribution:
2. The alpha-electron scattering can be neglected
3 relative to ion-alpha scattering (a - v-4, which is small
P(12) = _ sin 20 . (1) for high-velocity electrons).
The probability that an alpha particle is born into the 3. There is a square-well uniform B field that runs
loss cone for polarized D-T nuclei is, therefore, parallel to the cylindrical mirror axis and rises discon-
tinuously at the ends (thus allowing the mirror field to
.fo0 foOO
appear only in the boundary conditions).
= -- sin 3dodCdOo = P(9) df] ,
87r .sO ,so
4. Only the high-energy end of the alpha-particle
where dfl is the differential solid angle dfl = sin 19dO dd_ distribution (where df/dt = 0 is a valid assumption) is
or considered.
Pro= = 1 - -_cos d0(sin 20o + 2) . (2) Using these assumptions, the Fokker-Planck equation
takes the form:
Table I summarizes the polarized-to-isotropic ratio of
trapped alpha particles:
1 a 1 a
-v-_ ig--V(V2/I) + vsin 0 00 (vl± sin 0) = S (3)
polarized trapped fraction = 1 (sin 200 + 2)
unpolarized trapped fraction 2
where h and I± are the parallel and perpendicular
which indicates the alpha-particle trapping enhance-
components of the flux density in velocity space, re-
ment by D-T polarization. As expected, the trapping
spectively, and S is a source function, i.e., the number
advantage of the distribution increases with loss cone
size. density of high-energy particles created per second by
fusion reactions. Neglecting parallel diffusion relative
to the perpendicular flux components (valid for vi ¢.
SLOWING-DOWN AND PITCH-ANGLESCATTERING v,_<<re), Eq. (3) can be written as
The dynamics of the slowing-down and scattering 1 0
[(v 3+ v3c)f]
process is governed by the Fokker-Planck equation, rsv20V
which describes each particle as a material point in
seven-dimensional phase and time space. Following
r, v_ Ox (1-x 2)axx =S, (4)
the analysis of Sivukhin _2and Anderson et al., _3the
FUSION TECHNOLOGY VOL. 27 JAN. 1995 _7
Noever POLARIZED NUCLEI
where ticles and the corresponding particle and energy losses
can be evaluated for isotropic and sin 20 alpha-particle
mle/2 e4ne
1 _ 16 (2r)l/2 lnA distributions. The unnormalized energy spectrum of
rs 3 ms (kTe) 3/2 trapped alpha particles is given by
Vc3 = (9-x) I/2 _m_e [Z]o e3 y?r ]
ms
f(v) = [ v_) +-v, 3. dx , (7)
rs = slowing-down time
and is shown for f in arbitrary units in Figs. 1and 2
v,. = critical velocity at which electrons and
for both isotropic (ISOT) and polarized (POL) dis-
ions contribute equally to the slowing-
down frictional force tributions at various mirror ratios. The peak in both
distributions around 0.1 contrasts with the standard
me, m,_ = electron and alpha masses, respectively Maxwellian distribution in that slow alpha particles are
preferentially scattered into the loss cones. This altered
he, Te, e = electron density, temperature, and charge,
distribution can be understood to arise directly from
respectively
the 1/v 4dependence of the Rutherford collision cross
In A = Coulomb logarithm. section and becomes more pronounced for smaller loss
cones.
Pitch-angle scattering is described by x = vu/v = cos 0,
where 0 is the angle between v and the B axis. The pa- The particle loss fraction P, normalized with re-
spect to the total number of generated particles (includ-
rameters r = Zeff / (2A _[Z]), Zef f = _,Z2ni /ne, [Z] =
ing those born into the loss cones), is given by
_(Z2ni/Ai)/ne, and Ai = mass of the ions (Ai) and
alpha particles (A,_) complete the left-hand side of
Eq. (4). The source function, S = S06(v - v,)H(x) fo _l [q,(x,0) - xI,(x,t)] dx
[where So is a constant related to the fusion rate and P(t) =
H(x) will be given explicitly later] closes the Fokker-
Planck description. oxL_1(x,O) dx
Integrating Eq. (4) from vff to v+ reduces the delta
source to the boundary condition: Particle loss fraction as a function of energy during
slowdown for both isotropic and sin: 0 initial distribu-
SoH(x)vZr5 tions in a 10-keV D-T plasma is shown in Fig. 3 for mir-
ftvff, x) - 3 3
ror ratios of 2 and 10.
Vc_+ Vc
The energy fraction Q was found in terms of this
Taking the other two boundary conditions asf(v,x = particle loss fraction by
+--XL) = 0 for the loss cone space, the problem is well
posed. Defining new variables xI,and t as
3 Q[P(t)] = fo P _V2(/) dP for0-<P-< 1
'11= (v3 + vc )f Uc_
and and is shown in Fig. 4 for mirror ratios of 2 and 10.
A total energy-scattering rate for the isotropic and
- 3 3 , (5) sin 20 alpha-particle distribution was found to be -6
t = 3 + Vclye,
and 2°70, respectively, for a mirror ratio of 10, and 12
Eq. (4) can be written as and 5070, respectively, for a mirror ratio of 2. The re-
. 0[ .] suits for the isotropic alpha-particle case are in good
agreement with the findings of Anderson et al., 13re-
Ot - Ox (l-x 2)_-x ' (6) ported as 6and 10°70for R = 10and R = 2, respectively.
Based on this analysis, injection of polarized D-T nu-
with boundary condition
clei has the potential to reduce alpha-particle scatter-
_(x,0) = Sov2rsH(x) , Ix[ < XL ing losses by 50070or more. However, given that losses
are responsible for negligible alpha-particle energy leak-
and
age, little improvement in overall Q can be expected
_(+--xL,t) = 0 , t >_0 , from this reduction.
where H(x) = 1for an isotropic alpha-particle distri-
bution and H(x) = 1 - x 2 for a sin 20 distribution. ALPHA-PARTICLECONFINEMENT TIMES
Numerical solution of Eq. (6) by forward-difference
methods [step-size ratio At/(Ax) 2< 0.5 for stability] To find an analytic expression for the ratio of po-
gives • as a function ofx and t. Using this steady dis- larized to isotropic alpha-particle confinement times,
tribution function, the energy spectrum of trapped par- a quasi-Maxwellian distribution function is assumed,
88 FUSION TECHNOLOGY VOL. 27 JAN. 1995
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12
0t.=0
LL 10
C "OL=15
O
8
C
C
0 6 2O
JC_
L-
C7 4
m
2
if)
01.=60
0
, , ,
0 2 4 10
Alpha Parlicle Energy E/Ea (x 0 1)
Fig. !. Effect of end loss on isotropic steady distribution function.
OL=15
6
t.L
.C_o
It.
c 4
.0I
.0
6
o
2
OL=45
0L = 60
0
o io
(x 0.t)
Alpha Particle Energy, E/E.
Fig. 2. Effect of end losson polarized steadydistribution function.
and Eq. (3) is solved to give the alpha-particle loss rates. of a full Fokker-Planck calculation, such a simplify-
While on first inspection, the assumption of Maxwell- ing assumption seems justified for the purposes of this
ian alpha-particle behavior may seem artificial, Gal- investigation.
braith and Kammash m4found a < 1% difference in the Equation (3) translated to (p, 0) space is
key parameter for this investigation, alpha-ion frac-
O O
tional heating, for a Maxwellian over a mirror alpha-
p _-_ (sin OIo) + sin 0 _p (p21p) = qp2 sin 0 , (8)
particle distribution. Therefore, given the complications
FUSION TECHNOLOGY VOL. 27 JAN. 1995 89
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(x 0.I)
8
ISOT
_POL
6
_ M:tror talio = 2
_ ///Mirror ratio = 10
\'%//
2 4 ;
(x o.1)
E/Ea
Fig. 3. Particle loss fraction versus alpha-particle energy.
where q is the source density of alpha particles and Ip
and I0 are the momentum and angular flux compo- F(p,O) = O(0)exp __p2) . (10)
nents, respectively. Following Sivukhin, 12
Equation (9) simply reduces to
1 OF OF
lo = - Doo -- -Doe + AoF
p 30 Op ' (11)
sin 0 ,
OF 1 OF
.... Dpo +mp F which, when coupled with the boundary conditions,
Ip --Dpp OP p 0-0 '
gives
and
F = qp____In_z sin O
Dop = Dpo • /91 sin 0o
AS before, the distribution function vanishes at the loss
Comparing this result with Eq. (10) gives
cone boundary, F = 0 for 0 = 0o, and by symmetry,
OF/O0 = 0 for 0 = 7r/2. sin d
O = Cln --
Since an exact solution requires knowledge of both sin 0o
the diffusion tensor D_a and the coefficient of dynam-
ical friction A to find the unknown distribution func- or
,in (_,2)
tion, the diffusion approximation is employed. By
replacing D,_ and A by known functions of p and 0, F=Cln si--no0exp _ ,
as if F were isotropic, Dw0 and Ao vanish. Further-
more, Dep = DN, Doo = D±, and Ap = A and are all where the normalization constant C is found from the
functions ofp only. Equation (8) can then be written as particle conservation requirement,
_0
D. 00 sin0 +sin0 p2 D, A n = (F(O,p) dp . (12)
d
= _qp2 sin 0 . (9)
Since an equilibrium alpha-particle loss term is de-
Since a quasi-Maxwellian distribution function is sired, the number of alpha particles lost should equal
sought, the number produced, or
90 FUSION TECHNOLOGY VOL. 27 JAN. 1995
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(x0.01)
18-
!5 "xNx /_ Mirror Ratio =2
",d/
Q
6
0 2 4 6 8 10
(x0.1)
EIEa
Fig. 4a. Energy loss fraction versus alpha-particle energy.
(x 0.01) 10
8 Mirror Ratio = 2
o .
t{-ll ISOT
4-
ISOT
.
Q POL
O-
o 2 4 6 8 10
(xO.1)
E/Ea
Fig. 4b. Particle and energy loss fractions versus alpha-particle energy.
N= £ qdp , (13) n=27rC ,_-oo In--sin-:0 sinOdO
O<O<x-0 'J_o $1n 0 0
where the vector dp = 2_rp 2sin 20 dO dp.
X fo=p 2exp( _p2
For an isotropic alpha-particle distribution, O(8) = l,
FUSION TECHNOLOGY VOL. 27 JAN. 1995 91
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both isotropic and polarized alpha-particle distributions
or
[Eq. (14)], and the difference in confinement times
C ,__ ,2 m[T,(n,c Jo2t-cos0o] r = n/N is found in the normalization [Eq. (12)]. In-
stead of
such that the loss rate (13) is (_/2 sin 0 In -s-in t9 dO
Joo sin ,90
(_)|/21nAe4n 2 3In(x/2+ l)-x/2 cos Oo • for the isotropic distribution, the polarized integral is
N= T3/2 In(cos_-O° ) -cos00 1
For a polarized alpha-particle source term, on the l -- (_)COS 2 Lg0
other hand, q = qo(p)9(O), where _(tg) is normal-
ized to give the same value for q as for the case of iso- × ('_/2 [2 In sinO 1 (cos 20 - cos 20o)]
WOo 3 sin Oo 6
tropic alpha particles [_,,(tg) = ll:
x sin 0 dO .
fo_/2 _(O)sin 0 dO = cos Oo ,
Based on this analysis, the ratio of alpha-particle con-
0
finement is
sin 0 1 (c°s 20 - cos 20o] sin 0 dO
1- (_)cos 20oWOo sin 0o 6 J
7.pol _
sin 0
7"iso sin 0 In --
fo x/2
sin 0o
o
l , 1,cos Oco°t][ln(
_O-ot -cosOo+_CO1S 30o]
(15)
00
In(cot _-) - cos 00
Figure 5summarizes these results for various mirror
where _t,(0) = Ksin 2Ogives ratios. As expected, Eq. (15) approaches 1in the limit
as the loss cone goes to _ and 7r/2. Figure 6 uses an
sin2O
approximation formula 15for alpha-particle slowing-
(14)
_I'(0) = 1 - (_)cos 2 t90 down behavior to demonstrate the effect of this en-
hanced alpha-particle confinement on the deposited
Solving Eq. (11) for F as before gives
plasma energy.
= -- xI'(0")sin 0" dO"
F q°Pz ff ' dtg' ( x/2
D± o sin 0' ,Jo' FUSIONENERGYBALANCE
and
To assess the impact of this enhanced alpha-particle
qop 2
trapping on calculated Q values, an equilibrium ion and
f_
Dill - (_)cos 20o] electron energy balance is performed. Following Kam-
mash, ]6the ion and electron injection power per unit
X [2_ In sin0 ! (cos 20- cos 20o) ] • volume is Wsi and W_e, respectively. The geometric
sin do 6
confinement time is 7"iand 7"efor ions and electrons,
Comparing this again with the quasi-Maxwellian form and when they escape, they carry Wz.i and Wz_.
gives a 0 dependence of The term W0 istaken here as the energy exchange
between each species (W,j = 0 for i = j). With these
C
definitions, the ion energy balance can be written as
O(0) = 1- (_)cos z 0o
d
dt(_,n_) =0= W,_+ Ws,+ WLi- We_
x [_ln ssiinn00o 61 (COS 2 t,q -- COS 2 bg0) ] •
where
Because of the previous normalization of the source
term, the alpha-particle loss rate N is the same for Wi_ = ln2(ov)Ui_ (keV/m3"s) •
92 FUSION TECHNOLOGY VOL. 27 JAN. 1995
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(x0.01)
118
115-
0
m
[E 112
(9
E
I--
109-
r-
(9
E /
(9
.m 106-
C
0
o
103
100
0 2 4 6 8 10 t2
Mirror Ralio BMAX/B
Fig. 5. Alpha-particle confinement times versus mirror ratio.
(;-,1000)
4
>
(9
v- ..v Unpolarized
....,,.Polarized
(9
i-
IU
m
t-
o.
<
c-
m
(9
0
3 6 9 12 15
(xO.1)
Confinement Time (Sec)
Fig. 6. Mean alpha-particle energy during slowdown.
FUSION TECHNOLOGY VOL. 27 JAN. 1995 93
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If Ui_,is that portion of the alpha-particle energy de- and analogously for including a C2 to account for re-
posited with the ions, then flectivity of synchrotron radiation from the wall with
a dimensional size parameter
W_i = S,Vi = Vi ni + n_2_(ov> ) (keV/m3.s) . D = B3/2(LB) I/2
\ 7i 4
The electron energy balance simplifies to
If the source Si = ion loss rate and re = thermalization
time, then (ov) (2Ui_, + 2Ve - 3Te) - (12Te - 8Ve)/nre
we, = _ n,7;./exchange - 2 z_ l - Tii + 9.6×1T•03-e'/s2 Ti ( l -- -T_ei)
and - 4.8 × IO-27C2TIel/4(Te + Ti) 3/2
w.= _-si_ =-r_ +-
/exchange 2 4 x 1+ 204]/ =0" (17)
(keV/m 3-s) . Combined, Eqs. (16) and 07) form a nonlinear set with
roots equal to the equilibrium ion and electron temper-
Combining these effects, the ion energy balance is
ature if the following quantities are specified:
1. ion and electron injection energies Vi, Ve
(ov)(2U_, + 21/,,- 3T,) (12T_: - 81/,,.)
t'/Ti 2. burnup fraction fL = (Si - LA/S = (1 + [4/
( )/ (ov)lnr)]) -I
(16)
T3/2 Ti 1 - = 0 . 3. ratio of ion to electron confinement times (ri/
e
re) : 'y
Similarly, the electron energy balance can be written 4. bremsstrahlung parameter C_
5. synchrotron radiation reflection adjustment C2
dt _neTe =0
6. size parameter D.
-=Wcte'+ Wse-{- Wei - WLe-- W x -- Wc , Equations (16) and (17) were solved using a two-
where dimensional Newton-Raphson iterative scheme in
(Ti,Te) space with the cross-section data called from an
n2
interpolating subroutine for the smoothed Glasstone-
Wc_e= _ (ov>Uote (keV/m 3-s) .
LovbergtV results. For comparison sake, relevant ma-
chine parameters were taken from the Livermore mirror
If U,_e is that portion of the alpha-particle energy
reference design 18(LMRD) and are summarized in Ta-
(keV/m 3-s) deposited with the electrons, then
ble II.
Figures 7 and 8 show equilibrium temperatures ver-
Wse = Ve( -rr-e/e "Jr- -n-2(a4v)) , sus burnup fraction for both polarized and unpolarized
fuel injection. As expected, polarized nuclei elevate
both the equilibrium ion and electron temperatures with
Wei= 1.2 X 10-18 T3e/2 1- , the more substantial energy gain found in electron tem-
peratures due to preferential alpha-electron energy ex-
Wte=:neT2ez=3: 3 Te(n_ +----_nZ(°v)) , change. The relevant ratio for Q calculations, however,
isthe ratio of ion energies, which is shown versus burnup
and fraction in Fig. 9.
3 dTe 10-21 )Ci n2Tie/2
Wx----CI _ r/e _- =(3 X QCALCULATIONS
where C_ is an adjustable parameter to account for
changing bremsstrahlung, A useful parameter for evaluating mirror reactor
machines is Q, the ratio of fusion power per unit vol-
ume (PF) tO the power per unit volume injected and
Wc=6X lO-28C2112Tlel/4(Te + Ti)3/2(l - Te )/D204 trapped in the plasma (P_,). In turn, P_. is related to
the beam energy Ein, and the equivalent injected cur-"
(keV/m3. s) , rent by the expression Pm = Em Ii,,. For steady-state
94. FUSION TECHNOLOGY VOL. 27 JAN. 1995
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(x 10)
10
Vi, Ve= 0 keY
C1=1 Eleclron Temperalures
C2= 0
7=ti/te=1
>
_D D=I
6
==
_cD_ 4
E
_D Ion Temperalufes
F-
• Polarized
0 x Isotropic
0 4 6 10 12
Fractional Burnup FB (x 0.01)
Fig. 7. Equilibrium temperature for isotropic and polarized alpha-particle distributions asafunction of fractional burnup.
(x IO)
10-
•.-..*.-"-..-.."•k....-._................,•,._..................'.7. 41---_-_- "'*
....._ .--_ .......... =___,=dmr/-- =====q
8-
>
6-
--1 v,.,_,oov ,.,,o,,op,c
Polarized
QE" 4-
c,=to:c2=o.o
O.}
I--
•lr'=ti/te=1.0
D-I
2-
O_ I I I I i I
0 2 4 6 8 10 12
Fractional Burnup FB (x 0.01)
Fig. 8. Equilibrium temperature for isotropic and polarized alpha-particle distributions as afunction of fractional burnup.
operation, the input current must equal the rate of par- where n is the ion density and r is the mean confine-
ticle losses given by ment time, or
f-;
Ein (rt 2dV
lo.t = dV = dV ,
P,, = Ei. --_TdV- (nr> J o
FUSION TECHNOLOGY VOL. 27 JAN. 1995 95