Practical Calculation of Nuclear Fusion Power for a Toroidal Plasma Device with Magnetic Confinement

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1 Procdings of IC18, 8 Practical Calculation of Nuclar Fusion Powr for a oroidal Plasma ic with Magntic Confinmnt P.R. Goncharo St.Ptrsburg Polytchnical Unirsity, 19551, Russia An algorithm has bn dlopd and ralizd as a FORRAN cod to calculat th olum intgral powr of a magntic confinmnt nuclar fusion ractor and th fusion rat function in a gnral cas taking as input data th nucli nrgy distributions, fusion cross-sctions and th magntic surfac gomtry. wo fast and simpl analytic modls of practical magntic flux surfac shaps ha bn introducd and th corrsponding Jacobian dtrminants ha bn found. h dlopd mthod has bn applid to obtain th radial profils of th nuclar fusion raction rat and th olum intgral powr for both Maxwllian and suprathrmal and particl distributions. Gaussian krnl mpirical probability dnsity stimation has bn proposd to rconstruct th ion nrgy probability dnsity function from xprimntally obtaind random sampls of scaping nutral atom nrgy. Nutral particl diagnostic databas may sr as a basis for an xprimntally confirmd calculation tchniqu for ractor powr and ignition critrion. Kywords: fusion ractor powr, nuclar fusion rat, ion distribution function, high nrgy particls, non-maxwllian rat cofficints, plasma olum intgration 1. Introduction h powr of a magntic confinmnt fusion ractor P [W] and th fulfillmnt of th ignition critrion ar quantitatily dtrmind by th nuclar fusion raction rat [m -3 s -1 ] intgratd or th plasma olum using th known magntic surfac gomtry: whr 3 P ( r) d r = ( ρ, ϑ, ϕ) R( ρ, ϑ, ϕ) J d ρdϑdϕ, (1) R ρ R ϑ Z Z J = () ρ ϑ 1 is th Jacobian dtrminant for th transformation from cylindrical coordinats ( R, Z, ϕ ) to flux coordinats ( ρ, ϑ, ϕ ) with ρ, ϑ and ϕ dsignating th magntic surfac labl, th poloidal angl and th toroidal angl rspctily. Azimuthally symmtric magntic surfacs ar assumd in (). h rat of th nuclar fusion raction btwn th spcis and, n n =, (3) 1 δ in turn, is proportional to th rat cofficint = σ ( ) aragd or th locity distribution functions of th racting spcis by intgrating or th six-dimnsional locity spac: 3 3 = σ ( ) f ( ) f ( ) d d, (4) whr = is th rlati locity and is th Kronckr symbol rflcting th fact that whn th racting spcis ar idntical as in and ractions, and thir dnsity is n, th rat is proportional to 1 n Cn = n ( n 1). h FORRAN cod for nuclar fusion rat and powr calculation is basd on (1) - (4) and nuclar cross-sction approximations from [1-3]. Particl dnsity profils and th magntic surfac gomtry ar usd as input data. Eithr analytic ion distribution functions basd on thortical modls, or xprimntally obtaind ion distributions may b usd for th calculations.. Analytic Modls of Magntic Surfacs A rigorous tratmnt rquirs that Grad-Shafrano quation solutions ar usd. In ordr to incras th computation spd and simplify th cod, two analytic modls of magntic surfacs ha bn usd in th form of nstd shiftd -shapd curs and nstd shiftd llipss in th poloidal cross-sction. h -shapd last closd flux surfac () quation in cylindrical coordinats is δ author s -mail: goncharo@phtf.stu.na.ru 89

Procdings of IC18, 8 ( κ κ ) ± Z ( R) = ± R 1 κ R κ R κ κ R κ, (5) 3 4 5 6 7 whr κ1 = γ, κ = 1 γ, κ3 = 1 4γ, R 4 ( 3R ( 1) ) ( γ ) R δ, 6 1 κ = κ δ γ, 1 5 = 3R κ κ δ κ = and κ7 = δ R ar dtrmind by

On mor dimnsional input paramtr [m] is to simulat th Shafrano shift. h magntic axis position is thn Rin R R =, (6) whr th innr plasma radius R in [m] is calculatd from th quation (5).

2 Procdings of IC18, 8 ( κ κ ) ± Z ( R) = ± R 1 κ R κ R κ κ R κ, (5) whr κ1 = γ, κ = 1 γ, κ3 = 1 4γ, R 4 ( 3R ( 1) ) ( γ ) R δ, 6 1 κ = κ δ γ, 1 5 = 3R κ κ δ κ = and κ7 = δ R ar dtrmind by thr input paramtrs: r plasma radius R [m], poloidal cross-sction width δ [m] and dimnsionlss γ to control th poloidal cross-sction width to hight ratio. On mor dimnsional input paramtr [m] is to simulat th Shafrano shift. h magntic axis position is thn Rin R R =, (6) whr th innr plasma radius R in [m] is calculatd from th quation (5). h poloidal angl ϑ,π is dfind in th sam way as th polar [ ) angl assuming R = R, Z = to b th cntr point and Z to b th polar axis. As for th transformation btwn th cylindrical coordinats ( R, Z, ϕ ) and flux coordinats ( ρ, ϑ, ϕ ), th azimuthal (toroidal) angl ϕ [,π ) prsrs. For a gin point ( ρ, ϑ ) in th poloidal plan th coordinat transformation to cylindrical ( R, Z ) is prformd as follows. First, ϑ = π π ϑ sign π ϑ (7) is calculatd, whr sign( y ) = 1 for y < and sign( y ) = 1 f o r y. I f ϑ =, t h n Z = ; R = R ρ ( R R ). I f ϑ = π, t h n Z = ; R = R ρ ( Rin R ). I f ϑ = π, t h n R = R; Z = ρz ( R ) sign( π ϑ). Othrwis, th nonlinar quation Z R ( R ) R tanϑ = is rsold numrically with rspct to R, R. hn, within [ ] in (8) ρ ( ) R = R R R, R sought Z = ρz R sign π ϑ. (9) h llips-shapd quation in cylindrical coordinats is whr ± Z R = ± κ R κ R κ, (1) 1 b a 1 3 κ =, 3 κ = b R a a and κ = b b R a a ar dtrmind by thr Fig. 1. a) -shapd and b) llips-shapd isolins ρ = const for R in = 1.5 m, R = 3 m, =.1 m. Fig.. Jacobian dtrminant for a) -shapd and b) llips-shapd magntic surfac modls. 9

3 Procdings of IC18, 8 dimnsional input paramtrs: r plasma radius R [m], major and minor llips smiaxs a [m] and b [m]. Again, on mor dimnsional input paramtr [m] is to simulat th Shafrano shift. h magntic axis position is thn R = R a. (11) h coordinat transformation from ( ρ, ϑ ) to cylindrical ( R, Z ) is prformd using formulas (8) and (9) analogously to th cas of -shapd curs Z R gin by (1) substituting th propr function instad of (5). Fig. 1 shows ρ = const isolins obtaind by using th coordinat transformation procdurs for th two modl magntic surfac shaps. h paramtrs ar δ = 1.5 m, γ =.3646 for Fig. 1 a); a =.75 m, b = 1.5 m for Fig. 1 b), R in = 1.5 m, R = 3 m, =.1 m for both. Sinc R and Z ar implicit functions of ρ and ϑ, cntral diffrnc driati formulas ar usd to calculat th lmnts of th Jacobian dtrminant (). Lft and right diffrnc driatis ar usd at th rang,1 ϑ,π xtrmitis. h rsulting J ρ [ ], [ ) shown in Fig. nabls on to calculat th olum intgral (1) for ithr of th two magntic surfac typs. hs fast simpl analytic modls or thir combination may b usd as a satisfactory practical approximation of a tokamak magntohydrodynamic quilibrium whnr chord or olum intgration, or mapping of plasma paramtrs as functions of magntic surfacs to ral spac coordinats is rquird in physical and nginring tasks. For stllarator configurations mor complicatd full 3 modls ar ndd. 3. Nuclar Fusion Rat Cofficints 3.1. Mononrgtic bam and Maxwllian targt For th mononrgtic and monodirctional distribution of spcis intracting with a Maxwllian targt, 3/ m m f ( ) = δ ( V), f = π, (1) th rat cofficint (4) is rducd to 1/ mv ( BM ) 1 m (, ) V m = V π m m V σ ( )sinh d. (13) noting th atomic locity unit as cm / s and introducing a dimnsionlss ariabl m constants A = dimnsionlss function hn, y = and dimnsionlss and B = V, considr a AB y Ay AB AB y Ay AB ( y) = y. (14) ( BM ) A σ (y) A = 5, B = 3 A = 1, B = Fig. 3. Function ( y) in th intgrand in (15). V, m = ( y) ( y) dy π B. (15) o aoid arithmtic orflows, th constant should not b takn sid th intgral. ( y) is a AB nonngati, slightly asymmtrically bll-shapd xponntially dcaying function as shown in Fig. 3. h maximum position dpnds on th paramtrs. Qualitatily, function ( y) can ry roughly b thought of as a Gaussian cur corrsponding to th Maxwllian distribution of targt particls, shiftd along th abscissa axis by a alu corrsponding to th bam particl locity. h xact maximum condition is xprssd by a nonlinar quation AB z tanh z = z AB, (16) whr z = AB y. Equation (16) can b sold numrically, taking as an initial lft-hand approximation to its root th positi root of th quadratic quation corrsponding to th unit right hand sid. Assuming th cross-sction σ ( y) to b a smoothr function compard to ( y), th intgral in y 91

4 Procdings of IC18, 8 (15) can first b aluatd or a finit intral around th maximum of ( y). hn, at ry subsqunt stp on should broadn th initially chosn arbitrary intgration limits until th intgral alu bcoms clos nough to th on obtaind at th prious stp. hus, th rquird rlati prcision may b achid. 3.. Isotropically distributd projctils and Maxwllian targt. In th cas whn th locity distribution of spcis is xprssd by an isotropic function F( ) and th distribution of spcis is Maxwllian as in (1), th rat cofficint is calculatd as ( FM ) ( BM 4 ) = π F (, m ) d. (17) h tchniqu to comput ( BM ) (, m ) is dscribd abo. h practical ralization of formula (17) dpnds on th spcific function F( ). In particular, for suprathrmal ion distributions occurring du to fast nutral bam injction hating th uppr intgration limit in practic appars to b finit. Anothr charactristic cas of a bll-shapd xponntially dcaying intgrand implis that on should stp-by-stp broadn th intgration rang around th maximum position comparing th intgral alus until th rquird rlati prcision is achid Isothrmal Maxwllian cas. his is an important particular cas of 3. abo. For two Maxwllian spcis and at thrmal quilibrium = = th rat cofficint is ( MM ) 3/ ( ) = π µ γ σ ( y) f ( y) dy, (18) y whr th dimnsionlss function f ( y) = y γ, th dimnsionlss ariabl y = and th µ dimnsionlss constant γ =. h alu, as abo, dnots th atomic locity unit and µ = m m ( m m ) is th rducd mass. h maximum of f ( y ) is attaind at y = 1/ γ. Assuming th cross-sction σ ( y) to b a smoothr function compard to f ( y ), th intgral in (18) can first b aluatd or a finit intral around th maximum of f ( y ). hn, at ry subsqunt stp on should broadn th initially chosn arbitrary intgration limits until th intgral alu bcoms clos nough to th on obtaind at th prious stp. hus, th rquird rlati prcision may b achid. 4. Fast Nutral Bam Injction-Inducd Suprathrmal Ion istribution In ordr to dscrib th nutral bam injction hating-inducd fast ion distribution, on can us th classical nonstationary slowing down distribution function for a dlta-lik fast ion sourc S = 4π ( inj ) S F ( ) inj π * Κ, t inj = rf 3 3 c inj rf (19), () whr Κ is a normalization constant, th slowing down tim 3m τ s = 4 π n Z cub of th critical locity 3 c Λm 3 π 1, (1) = 3 m m, () Λ is th Coulomb logarithm, and, as shown in [4], 3 t / τ 1/3 c ( ) ( c ) * 3 3 s 3, t =. (3) h normalization constant Κ is dtrmind by numrical intgration. h ion locity = E / m, inj is th injction locity corrsponding to th injction nrgy E inj, th alus S and in (19) dtrmin th sourc rat and pak width, rspctily, and t is th tim sinc th commncmnt of th fast particl sourc action. 5. Exprimntally Obtaind Suprathrmal Ion istributions Radial and angl dpndnc of th ion distribution function is studid xprimntally by mans of passi lin-intgral and also acti localizd charg xchang 9

5 Procdings of IC18, 8 nutral particl diagnostics [5, 6]. An xtnsi diagnostic databas of this kind should nabl on to prdict th local ion distribution function olution for a gin plasma discharg rgim in a crtain dic, for a spcific hating mthod and tim diagram. Using th diagnostic data in th form of an array of nrgis (E 1,, E N ) of scapd nutral particls masurd along a crtain obsration dirction, whr N is th total numbr of particls collctd during a crtain tim intral, on can construct an mpirical probability dnsity function a) n (ρ), cm -3 1x1 14 9x1 13 8x1 13 7x N ( K ) j f E K h 1 E E =, > Nh j= 1 h (4) b) 6.x1 3 ρ z with Gaussian krnl function K( z) = π. h optimal krnl bandwidth h slction algorithm and th ion distribution function rconstruction ar discussd in [7]. hus, a possibility xists to prform a corrct xprimntally confirmd calculation of th tim olution of th local fusion rat cofficint (4) and th fusion ractor powr (1). Particular MH quilibrium data can thn b usd rathr than analytic approximations.,, (ρ), V 4.x1 3.x ρ 6. Calculation Exampls Assuming th radial dnsity profils to b a ( 1 ρ ) n ( ρ) = n () n (1) n (1) (5),,,,,,,, and radial tmpratur profils q ( 1 ρ ) ( ρ) = () (1) (1), (6),,,,,,,, lt us introduc an additi non-maxwllian distortion of th form () to th dutrium distribution so that 3/ m m = f A (1 A) F( ), (7) π whr A 1. A = 1 corrsponds to th pur undistortd Maxwllian cas. h rat cofficint for th intraction of dutrium particls distributd according to (7) ( µ ) = A A(1 A) ( MM ) ( FM ) (1 A) ( FF ) r b (8) is thn calculatd using (17) and (18). h last trm in (8) accounts for th tail-tail particl intraction rat. It is considrd ngligibl bcaus th non-maxwllian distortion is assumd to b small, i.. (1 A) 1. In th cas whn dutrium componnt distributd according to (7) intracts with Maxwllian tritons th rat cofficint is 3/ m m f ( ) =, (9) π ( µ ) = A (1 A). (3) ( MM ) ( FM ) h fusion rat radial profil and intgral powr calculation gin blow is for (,n)h 4 raction. Fig. 4 a) shows th lctron dnsity profil with a = 4, b =, 14 n () = 1. 1 cm -3 14, n (1) =.7 1 cm -3. h Fig. 4. a). Radial profil of dutrium and tritium dnsity; b). Radial profil of dutrium and tritium tmpratur. nucli dnsitis ar n = n = n /. Fig. 4 b) shows th = = with lctron and ion tmpratur profil q = 1.5, r =,,, (1) = 5 V. 3,, () = 5. 1 V, Calculations ha bn prformd for thr ariants of th dutron locity distribution function shown in Fig. 5, namly, for undistortd Maxwllian distribution and for th cass whn thr xists.5% or 5% population of suprathrmal particls dscribd by th classical slowing down modl of bam particls with th injction nrgy E inj = 15 kv. h non-locality, i.., th radial dpndnc 93

6 Procdings of IC18, 8 a) b) f (E) f (E) 1 A = ρ =.1 ρ =.45 ρ = E, kv 1 A = ρ =.1 ρ =.45 ρ =.75 R (ρ), cm -3 s -1.x x x x1 1 A =.95 P =.9x1 18 s -1 (-shap) P =.88x1 18 s -1 (E-shap) A =.975 P = 1.5x1 18 s -1 (-shap) P = 1.5x1 18 s -1 (E-shap) A = 1. P = 1.16x1 17 s -1 (-shap) P = 1.14x1 17 s -1 (E-shap) Fig. 6. Radial profils of (,n)h 4 raction rat for thrmal tritons and thr ariants of dutron nrgy distribution and th corrsponding plasma olum intgral powr alus for two typs of magntic surfac gomtry. ρ E, kv population lads to approximatly 5 tim incras in th fusion powr compard to th pur Maxwllian cas. c) 1 A = Summary f (E) Fig ρ =.1 ρ =.45 ρ = E, kv utron nrgy distribution function a) in th undistortd Maxwllian cas for A = 1 and in th prsnc of suprathrmal tail with b) A=.975 and c) A =.95. of th ion distribution function in this modl is du to th radial profils of n and, dtrmining th slowing down tim and th critical locity alus. For ach of th thr ariants of th distribution th radial profil of (,n)h 4 raction rat has bn calculatd as wll as th plasma olum intgral powr for two typs of magntic surfacs with -shapd and lliptical poloidal cross-sctions as shown in Fig. 1. Slight diffrncs in th magntic surfac shap, as xpctd, ha no significant dirct gomtrical influnc on th powr. Calculation rsults ar shown in Fig. 6. In this xampl th prsnc of 5% suprathrmal dutron A gnral practical algorithm has bn ralizd to calculat th nuclar fusion rat and powr in a toroidal magntic plasma confinmnt dic. Volum intgration is prformd using analytical approximations of magntic surfacs. A dtaild dscription of locity spac intgration tchniqu for bam-maxwllian, bi-maxwllian and isotropic function - Maxwllian cass has bn gin. Fusion rat and powr calculations can b don using ithr thortical or xprimntally obtaind nucli nrgy distribution functions. A significant contribution to th nuclar fusion raction rats coms from suprathrmal ions from high-nrgy distribution tails. hrfor, th production and good confinmnt of fast ions play th ssntial rol. Rliabl xprimntal data and thortical undrstanding of th formation of fast ion distribution tails ar rquird. h ion distribution function rflcts th kintic ffcts, th singl particl confinmnt proprtis dpnding on th particular magntic configuration, th finit ffcts such as MH inducd fast ion losss, radial lctric fild ffcts, tc. As a mthod to instigat th ion componnt distribution function and its olution du to th application of hating, masurmnts of kintic nrgy distributions of nutral atoms scaping from th plasma may b usd, which ar oftn rfrrd to as nutral particl analysis (NPA) diagnostics. Multidirctional passi masurmnts proid 94

7 information on th angular anisotropy, fast ion confinmnt, and distribution tail shaps. Lin-intgral nrgy-rsold nutral fluxs ar obtaind at diffrnt obsration angls. Spcial mathmatical tchniqus ar rquird for th corrct data analysis [5]. Anothr approach is to crat a localizd dns targt for charg-xchang in th plasma. A diagnostic pllt ablation cloud can b usd for this purpos (pllt charg xchang, or PCX mthod). im-rsold masurmnts of th nutral flux from th cloud as it mos across th plasma column rsult in radially-rsold information on th fast particl nrgy distribution [6]. Smooth normalizd probability dnsity functions for th nucli nrgis can thn b calculatd from NPA data using th mthod gin in [7]. hus, xprimntally confirmd calculations of nuclar fusion rat and powr ar possibl on th basis of diagnostic data. Procdings of IC18, 8 Rfrncs [1] G.H. Mily, H. ownr, N. Iich, Rpt. COO-18-17, Unirsity of Illinois (1974). [] B.H. uan, Rpt. BNWL-1685, Brookhan National Laboratory (197). [3] H.-S. Bosch, G.M. Hal, Nucl. Fusion, 3, 611 (199). [4] J.G. Cordy and M.J. Houghton, Nucl. Fusion, 13, 15 (1973). [5] P.R. Goncharo,. Ozaki t al., R. Sci. Instrum., 79, 1F311 (8). [6] P.R. Goncharo,. Ozaki t al., R. Sci. Instrum., 79, 1F31 (8). [7] P.R. Goncharo,. Ozaki t al., Plasma and Fusion Rsarch, 3, S183 (8). 95

Collisions between electrons and ions

Collisions between electrons and ions DRAFT 1 Collisions btwn lctrons and ions Flix I. Parra Rudolf Pirls Cntr for Thortical Physics, Unirsity of Oxford, Oxford OX1 NP, UK This rsion is of 8 May 217 1. Introduction Th Fokkr-Planck collision