Tài liệu Compton scattering of photons from electrons in thermal (maxwellian) motion electron heating

s, h Affiitive Actios@@@porturdty Empbyes ... ..-, .— . . -.. ,... .- ..—. -- . .. . . . ..-— —-- — -- . . . . s ~ - - . -.. ... ... .-—. —---—. - ----—.- .- .. . . . ---- . - . . . DKXAIMER .- -. -. ~ reportwaspreparedasanaccmartof worksponsoredbyanagencyof the UnitedStatesGovernment. Neitherthe UrdtedStatcaCovernrmntnor~y ~ncy t+epflnor anyof Sheir,employees, makesany wiriwntj,exprewor implied,or assumesanylegalliabilityor responsibility forthe accuracy,completeness, or usefulnessof anyinfo~tion, apparatus,product,or processdisclosed,or represe.nta that its w would not Infringeprivz~ely ownedrights. Referencehereinto anyspcciflccommercialproduct,proccas,or serviceby tradename,trademark,manufacturer,or otherwise,doesnot neosaarilymrrstituteor implyits endorsement,recommendation, or favori!rg bythe Uniteds.ta~sGovernment or anyagencythereof, The viewsandoptions of authorsexpressedhereindonot necessarilystateor reflectthoseof the United StatssCovernmcntor anyagencythereof. . . ,..— .. . . -—.—. . - LA-1OO65-MS UC-34 Issued: April 1984 ComptonScattering of Photonsfrom Electrons in Thermal (Maxwellian) Motion:Electron Heating JosephJ.Devaney . —@ L&----&..: ...)= xm.;”; !m?g’-+, = —=a m ! ‘–“.. ,. . ..Ty ;.. - ,.. —-... :’=l_., .. ,. ,., . - . ,, TABLE OF CONTENTS PAGE EXECUTIVE SUMMARY ........ ..00.0....0.0.0.. .......0................... vii ● ● ABSTRACT ..................................**.*....*..**............ ... 1 I. INTRODUCTION.................................................... 1 II. THE EXACT COMPTON SCATTERINGOF A PHOTON FROM A RELATIVISTIC MAXWELL ELECTRON DISTRIBUTION ................................... 3 NUMERICALVERIFICATIONOF THE EXACT COMPTON SCATTERINGFROM A MAXWELL DISTRIBUTION .0.0.00................... ..0.0..0...0.0000 10 THE WIENKE-LATHROPISOTROPICAPPROXIMATIONFOR THE COMPTON SCATTERINGOF A PHOTON FROM A RELATIVISTICMAXWELL ELECTRON DISTRIBUTION .000...0......... ..00..0. .0.......00..0.0......... 11 NUMERICAL VERIFICATIONOF THE WIENKE-LATHROPISOTROPIC APPROXIMATIONOF COMPTON SCATTERING ............................. 14 COMPARISONOF THE WIENKE-LATHROPAPPROXIMATIONTO THE EXACT COMPTON SCATTERINGOF PHOTONS FROM A MAXWELLIAN ELECTRON GAS .... 14 WIENKE-LATHROPFITTED APPROXIMATIONTO COMPTON SCATTERING ....... 14 VIII. THE TOTAL COMPTON SCATTERINGCROSS SECTION AT TEMPERATURET ..... 16 ● III. ● Iv. ● v. m. VII. Ix. ● APPLICATIONTO MONTE CARLO (OR OTHER) CODES: THE EXACT EQUATIONS 00...0......0000......... ....0..0 ..0.0......0..0.0 . 17 APPLICATIONTO MONTE CARLO (OR OTHER) CODES: THE WIENKE-LATHROP ISOTROPICAPPROXIMATION ...0...0 .......................... 0.... 18 APPLICATION TO MONTE CARLO (OR OTHER) CODES: THE WIENKE-LATHROP FITTED APPROXIMATION .0..0.. .0...00. .00000.0 0.000..0.0000.0.. 26 THE MEAN SCATTERED PHOTON ENERGY, HEATING: THE EXACT THEORY ... 26 XIII. THE MEAN SCATTERED PHOTON ENERGY, HEATING: THE WIENKE-LATHROP ISOTROPICAPPROXIMATION ........ ....00.. .0...0........000...... 29 ● x. ● ● ● ● XI. ● XII. ● ● ● XIV. ● ● ● THE MEAN SCATTEREDPHOTON ENERGY, HEATING: THE WIENKE-LATHROP ONE-PARAMETERFITTED APPROXIMATION ............................0. 31 TABLE OF CONTENTs (coNT) PAGE xv. XVI. COMPARISONSOF THE MEAN SCATTEREDPHOTON ENERGY AND THE MEAN HEATING AS GIVEN BY THE EXACT, THE ISOTROPICAPPROXIMATION,AND THE FITTED APPROXIMATIONTHEORIES ............................... 31 A. ScatteredPhoton Energy ..................................... 31 B. Heating ..................................................... 35 RECOMMENDATIONS................................................. 35 ACKNOWLEDGMENTS....................................................... APPENDIX A. 39 $-INTEGRATIONOF THE EXACT COMPTON DIFFERENTIAL CROSS SECTION ............................................ 40 APPENDIX B. &INTEGRATION OF THE ISOTROPICAPPROXIMATIONCOMPTON DIFFERENTIALCROSS SECTION .0....00 0.....00..0...00. .0.. 43 REFERENCES ............................................................ 44 ● I , vi ● EXECUTIVE SUMMARY The Compton differentialscatteringof photons from a.relativisticMaxwell Distributionof electrons is reviewed and the theory and numerical values verified for applicationto particle transport codes. We checked the Wienke exact covariant theory, the Wienke-Lathropisotropic approximation,and the WienlceLathrop fitted approximation. Derivation of the approximationsfrom the exact theory are repeated. The IUein-Nishinalimiting form of the equationswas verified. Numerical calculations,primarily of limiting cases, were made as were comparisonsboth with Wienke’s calculationsand among the various theories. An approximate (Cooper and Cummings), simple, accurate, total cross section as a function of photon energy and electron temperatureis presented. Azimuthal integrationof the exact and isotropic cross sections is performed but rejected for practical use because the results are small differencesof large quantitiesand are algebraicallycumbersome. The isotropic approximationis good for photons below 1 MeV and temperatures below 100 keV. The fitted approximationversion discussed here is generally less accurate but does not require integration,replacing the same with a table or with graphs. We recommend that the ordinary Klein-Nishinaformula be used up to electron temperaturesof 10 keV (errors of — < 1.5% in the total cross section and of about 5% or less in the differentialcross section.) For greater accuracies,higher temperatures,or better specific detail and no temperature or photon energy limits, the exact theory is recommended. However, the exact theory effectivelyrequires four multiple integrationsso that within its accuracy and temperatureand energy limits the Wienke-Lathropisotropic approximation is a simpler solution and is thereby recommendedas such. The mean energy of a photon scattered from a Maxwell distributedelectron gas is calculatedby four methods: exact; the Wienke-Lathropisotropicand one-parameterfitted approximations;and the standard (temperatureT = O) Compton energy equation. To about 4% error the simple Compton (T = O) equation is adequate up to 10-keV temperature. Above that temperaturethe exact calculation is preferred if it can be efficientlycoded for practical use. The isotropic approximationis a suitable compromisebetween simplicityand accuracy, but at the extreme end of the parameter range (T = 100 kev incident vii photon energy v’ = 1 keV, scatteringangle 0 = 180°) the error is as high as -28%. For mid-range values like 10 to 25 keV, the errors are generallya percent or so but range up to about 8X (25 keV, 1800). The fitted approximation is generally found to have large errors and is consequentlynot recommended. The energy deposited in the electron gas by the Compton scatteringof the photon, i.e., the heating, is only adequatelygiven by the exact expression for all parametersin the ranges 1 —— < T < 100 keV and 1 —— < v’ < 1000 keV. For low depositionsthe heating is the differencebetween two large quantities. Thus if one quantity is approximate,orders of magnitude errors can occur. However, for scattered photon energy v >> T the isotropicapproximationdoes well (0.13% error for v’ = 1000 keV, 0 = 180°, T = 10 keV, v = 790.7 keV; and 2.7% error for v’ = 1000 keV, Cl= 90°, v = 583.5 keV, T = 100 keV). The regular T= O Compton also does well for T —< 10 keV and v >> T (0.7% for v’ = 1000 keV, e = 180°, T = 10 keV, v = 790.7 keV; and 0.08% for v’ = 1000 keV, 0 = 180°, T = 1 keV, v = 795.9 keV). The fitted approximationis without merit for heating. viii COMPTON SCATTERINGOF PHOTONS FROM ELECTRONS IN THERMAL (MAXWELLIAN)MOTION: ELECTRON HEATING by Joseph J. Devaney ABSTRACT The Compton differentialscatteringof photons from a relativisticMaxwell distributionof electrons is reviewed. The exact theory and the approximatetheories due to Wienke and Lathrop were verified for applicationto particle transport codes. We find that the ordinary (zero temperature)Klein-Nishinaformula can be used up to electron temperaturesof 10 keV if errors of less than 1.6% in the total cross section and of about 5% or less in the differential cross section can be tolerated. Otherwise, for photons below 1 MeV and temperaturesbelow 100 keV the Wienke-Lathropisotropic approximationis recommended. Were it not for the four integrationseffectivelyrequired to use the exact theory, it would be recommended. An approximate (Cooper and Cummings), simple, accurate, total cross section as a function of photon energy and electron temperatureis presented. I. INTRODUCTION This report criticallyreviews the exact Compton differentialscattering of a photon from an electron distributedaccording to a relativisticMaxwell velocity distribution. We base our study on the form derived by Wienke using field theoreticmethods.1-7 (ParticularlyEq. (1) of Ref. 1, whose derivation is presented in Ref. 2.) Wienke was the first known to this writer to point out the simplicityand power of deriving the Compton effect for moving targets by the coordinatecovariant (i.e., invariant in form) techniques of modern field theory. His derivation is equivalent to,’4but replaced, earlier methods8 which involved the tedious and obscure making of a Lorentz transformationto the rest frame of the target electron, applying the KleinNishina Formula, and making a Lorentz transformationback to the laboratory frame. We also criticallyreview the Wienke-Lathropisotropicapproximation to the exact formula which selectivelysubstituteselectron averages into the exact formula so obviating integrationover the electron momenta and colatitude. The electron directions in a Maxwell distributionare, of course, isotropic,hence the name chosen by Wienke and Lathrop. We verify the theory for the exact expressionand the plausibilityarguments for the isotropicexpression. We verify in detail numerical comparisons between the two theories at selected electron temperaturesand initial photon energies. We rewrite the formulas in a form suitable for application,especially for the Los Alamos National LaboratoryMonte Carlo neutron-photoncode, MCNP.1O As a further approximation,Wienke and Lathrop have reduced the iso9 which tropic approximationto a one- or two-parameterfitted approximation, we also review. As always, the choice between the methods is complexityversus accuracy and limitationsof parameter ranges. We include a simple, accurate estimate of the total Compton cross section. We give the mean scattered photon energy and the mean heating of the electron gas by the photon scattering. Both quantitiesare given as a function of the photon scatteringangle, (3;the electron temperature;and the incident photon energy, v’. We compare these means, v> and I-D: as calculatedexactly, as calculatedwith the Wienke-Lathropisotropicand one-parameterfitted approximations,as well as with the unmodified,regular, T = O, Compton energy equation results. Recommendationsare offered. Because much of this report is devoted to derivationand verification, we recommend that a user-orientedreader turn first to the recommendationsof Section XVI, then for differentialcross sections, Sections IX to XI as desired, which give applicationstogether with referenceto Figs. 2 through 8, which show the accuracy of the cross-sectionapproximations. For scattered photon energies and heating, refer first to Section xv for comparisonsand errors, particularlyFig. 10 and Tables 11 and III, then as desired Sections XIII to XIV. Refer to the Table of Contents for further guidance. 2 II. THE EKACZ (XMPTON SCATTERINGOF A PNOTON FROM A REMTIVISTICMAXWELL ELECTRONDISTRIBUTION We will follow the notation and largely the method of Wienkel>9 and first choose the “natural”system of units in which h = c = 1 and kT ~ T in keV. Let the incoming photon and electron energies be v’ and s’, and the outgoing be v and c, with correspondingelectron momenta ~’ and ~, and photon momenta ~’ and ~, respectively. The angle between $’ and ~ shall be (3;q is oriented relative to some fixed laboratory direction by azimuthal angle Oq. The angle between ~’ and ~’ shall be a’ and that between ~ and $’ shall be a. The azimuthal angle between the spherical triangle sides ~’ and <’$’ shall be 4. Thus on the surface of a sphere with origin of the vectors ~,~’, and $’ at the center of the sphere and intersectionslabelled on the surface thereof, we have the spherical triangle shown in Fig. 1. Fig. 1. Angular relations. The law of cosines applied to the triangle of Fig. 1 gives a = cos a’ cos e+sinaf cos sine COS$ . (1) In terms of energy, e, and (vector)momentum,~, the four-vectormomentum is (2) Its square is 2=2 -$2 (3) ● Now all four vectors are required to transform (Lorentz transformation)alike, in particularas the line element ds = [dt,d~] (4) so as to keep (ds)2 invariant. Thus P2 is invariant. In the rest frame ~ = o, s ~ m, so that generally for any four-momentum P of mass m, 2 =m 2 . (5) In particular,for photons,m = o, so that their four-momenta,Q, satisfy Q2= o (photons) . (6) Considernow the Compton scatteringof four-momentaP’ + Q’ into P + Q. Four-momentumis conserved,so p=p~ +Q? -Q . (7) The square of Eq. (7) yields P’Q’ = P’Q + Q’Q 4 , (8) where we have used Eqs. (6) and (5): 222 P2 = P’ = m , Q 2 = Q’ = O . I Expanding the four-momentumproducts into energy and three-momentumproducts by, for example, PtQt = E*V’ _;h$f , (9) and then using Fig. 1 to determine that, for example, +9 p .;1 = plq? Cos a’, (lo) where p’ and q’ are, of course the magnitude of the 3-momenta~’ and ~f, we get finally for Eq. (8) &’v’ - p’v’ cos a’ = c’v - p’v cos a +vv’ - vv’ cos 9 , (11) where we have used q’2 = v’2, q2 = V2 for the zero rest mass photons. The Compton collision,of course, also conserves 3-momentumso that (12) If now we square Eq. (12) and use the relation +2 p +m2=E2 (13) plus scalar products determined from Fig. 1 as we did for Eqs. (10) and (11), we get & ?-l L = . . & ‘L+VL+V’L . + 2p’v’ cos a’ - 2p’v cos a - 2vv’ cos 0 . (14) The Compton cross section for scattering a photon into a direction solid angle do, and into an energy interval dv, from a relativisticMaxwell electron distribution,f(~’), is given by Wienkel to be I r’ 0 dcs d~dv = _ [ 2 d3&f(p’) s fi” () 6 (c’ +v’ ‘E ‘V) “ K , (15) where K= [( 2 2 2 m m Ctv’-ptv’ cos a’ - c’v-p’v cos a ) +2 ( + m’ m’ c’v’-p’v’ cos a’ - E’v-p’v cos a 1 Etv’-ptv’cos a’ + c’v-p’v cos a c’v’-p’v’ cos a’ c’v-p’v cos a m is the electron rest energy (or mass, c = 1), ) (16) ‘ and r. E e2/mc2 is the classicalelectron radius. We integrateover final photon energy in order to remove the ~-function. However, the 6-function is not in the form 6(v - Vo), where V. is constant because Eq. (14) shows that e = e(v). We must first use the identity u -1 (17) 6(X-XO) , ti(f(x))= g x o from which, using Eq. (14), then differentiating,then substituting C+v = e’+v’, and then Eq. (11), E(+) 6(E+V-E’ ‘V’) ‘E, -p, ~osa,6(V-vo) (18) , for some constant outgoing photon energy, Vo. Substitutingin Eq. (15), integratingover v, and replacing V. by v (i.e., v is now the outgoing photon energy), we get do —.— dQ r’ O 2 f d3;’ f(p’) c E’(s’ m’ - pt cos a,) ‘>)’ K “ (19) Formulas (15) and (16) have been checked by independentcalculationby C. Zemach, TheoreticalDivision, Los Alamos National Laboratory.* In the form of Eqs. (19) and (16) the formulas are the same as those of Pauli8 and Ginzburg and Syrovat-Skiillprovided one corrects for the electron motion. The number of events per unit time are equal to the flux times the density of electrons times the cross section times the volume. For an electron of velocity ~, the number of events per unit time is increased by the factor ++ () V.c 1 -— c’ +, +, v c’ - P “~ = s’ . ~? -p’ cosa’ E’ (20) s where again c ~ 1. The relativisticMaxwell electron distributionis f(~’) = (4TCy)-1e- ,2 2 7/ p % T (21) or for E’ =m+K’ (22) . K’ being the electron kinetic energy, then -1 e-m/T . e-K’/T f = (41rf) (23) 9 where the normalizationconstant y is given by 1/2 y = m2TK2(m/T) . m2T(~) ea’T 1 + ~15 ~, 8m + *&)2 . [ (24) where K2 is the modified Bessel function of the second kind and order. We use e-K’/T f= /2% 9 “ 2xm2T 1 + 15(;) + ~+)’ [ - ~~)3 + ~(&14] which is good for T << 4 MeV, more accurately, for T < 400 keV. *Informationfrom C. Zemach, November 1982. (25) We generally prefer to describe electrons by their kinetic energy, K’, rather than momentum ~’, so using &’2 = p’2 + m2 and c’ = m + K’ we have 2 P’ dp’ = (K’ +m) ● d~-v dK~ (26) , also d3$’ = p’2 dp’ dcos a’ d$ (27) . We put (28) thus dp = -sin 8 de. For convenience,define K = &’/m ‘1 =l+(K’/m) (29) , Em ‘1 (E’ - p’ Cos a’) = K - A- Cos (x, , (30) and K 2Sm ‘1 (&’ - p’ cos a) = K - Gcos a (31) ● Substitutinginto the energy equation ((11)j we get simply K mv‘ = Vv‘(1 - p) + K2mv 1 (32) . Substitutinginto Eq. (16) and with a little rearranging, K = ‘1 ; ;)2 - 2(’ -“) +5+5 VK 2 ‘1K2 [ ‘1K2 V ‘K 1 (33) , 1 and the cross section, Eq. (19), becomes r2 do - 0 d$2 T J x K dK’ (K’ + m) -. ● d$ ● d cos a’ . f(Kt) ● ~ KK ● IV (~~ (34) The expressions (32), (33), (34), (25) (or (23) and (24))> and (1) constitute the exact Compton scattering cross section of a photon of energy v’ from a relativisticMaxwell distributionof electrons, f(Kt), into the solid angle d!2= sin f3dO d$q. The o-integrationcan be carried out analytically, see Appendix A, but we found the result both too cumbersomeand too inaccurate for practical use. The latter was caused by small differencesof very large, but exact, quantities that our calculatorcould not handle. In the limit T + O, Eq. (34) or Eq. (19) should reduce to the KleinNishina Formula (i.e., Compton for zero velocity electrons). So it does, as we now show. Observe that the relativisticMaxwellian is normalized such that ! fd~’ = 1 (35) and from Eq. (25) for T + O that for K’ # O,f = O and for K’ = O, f = -, so that we may write f d3;’ + T+O 6(P’) dp’ (36) because the tkfunction,6(x), is defined as 6(X # o) = o ~ 6(X) dx = 1 . (37) SubstitutingEq. (36) in Eq. (19), using Eq. (33) and Eqs. (29), (30), and (31), (i.e., K = 1 = K~ = K2), we get the usual Compton energy equation from Eqo (32). mv’ = Vv’(1 - ~) + vm (initiallystationaryelectron) (38) and the Klein-NishinaFormula10~12for unpolarizedlight, do —= d$2 as we r2 2 v — ()[ ;+:, 2° ~ should. + p2-1 1 , (39) III- NUMER.ICAL VERIFICATIONOF THE EWX DISTRIBUTION COMPTONSCATTERINGFROM A MAXWELL Integrationover $ in Eq. (34) is possible analyticallybut is both complicatedand leads to small differencesin large quantities (see Appendix A). Accordingly,we have used Simpson’s rule to integrateover $, a’, and K’ to provide dcs/dOvs 0. By symmetry the cross section is independentof $q. We used angular intervalsof 22.50° and variable electron kinetic energy intervals appropriateto yielding an error of about 1% or less. Our numerical calculationsagree with Wienke and Lathrop9 to about 1% or better except at T = 1 keV, v’ = 1000 keV, and 13= 30°, where the agreementwas only 3.5% because of our using a coarse K’ interval. We found and checked with Wienke that his latest communication(Ref. 9, February 15, 1983) has an erroneous exact curve in Fig. 6. (T = 100 keV, v’ = 1 keV); however, we agree with ear- lier Wienke-Lathropexact calculationsfor these parameters. The parameter sets for which we have numericallychecked the Wienke-Lathropexact angular distributionsare given in Table I. TABLE I P~~R%TS~~I~~E~~~~-~OP~~ FOR du/deWERE VERIFIED TEMPERATURE T (keV) INCIDENT PHOTONENERGY V’(kev) 100 1 100 1000 1 1 1 1000 PHOTONSCATTERINGANGLES,e 45°, 90”, 00° 135” 45°, 90°, 35” 60°, 140° 30° (As noted, agreement is within the error of our Simpson’srule approximationerror, i.e., = 1% except, 1, 1OO(),30°: = 3.5%.) 10 I Iv. POR THE COMPTONSCATTERINGOF THE wlENKE-LATHROPTSOTROPICAPPROXIMATION A PHOTONFROM A REUTIVISTIC MAXWELLELECTRONDISTRIBUTION The exact Compton formula (34) requires three integrationsto provide the differentialscattering cross section dci/dQ. It requires four integrations to provide the total scatteringcross section (the fifth integrationover @q is trivial, yielding 2X because of symmetry). Accordingly,an approximateform of Eq. (34) without such integrationscould be quite useful when the errors of the approximationcan be tolerated. We derive the Wienke-Lathropisotropic approximationby a plausibilityargument. This approximationremoves the integration over p’ and a’. In place of averaging the covariant Compton expression over a Maxwell spectrum,key parameters are averaged in that expression,an inexact but reasonableapproach leading to a simpler expression. Because the relativisticMaxwell distributionis isotropic,it is clear that =o (40) , where the average, < >, is the Maxwell average over K’ and at. The first approximationthen is to replace cos a’ by its average cos ar + = 0 (41) so that (42) and Eq. (1) becomes cosa=sinf3 cos~ . (43) Consequently,by Eq. (30) ‘1 ‘K “ (44) Since (Eq. (24)) K ~ c’/m, one might, for example, choose K = /m> but Wienke chooses rather to set E’ =/ +m2 (45) , which he shows and we verify and expand to be SubstitutingEq. (42) into Eq. (31) yields ‘2 = K - = ● (sin ~ .0s $) . (47) Eq. (32) reduces by Eq. (44) to v —. v’ K : . (48) (1-~) + K2 Rememberingthat the Maxwell averaging operator, J d3$‘ — f (p’) d4 (i.e., except for $) now applies solely to cos a’, where it gives zero, and to 2, where it yields Eqs. (45) and (46), we perform that averaging in Eq. (19) with K taken to be Eq. (33) to obtain the isotropic approximationfor the differentialcross section 12 2 dd ‘O —=%” dQ 2n 2 1 ~ J[ (;) (1 - p)2 _ 2(1 - V) + V’K + ‘:2 ~KK2)2 KK2 VK2 v K o where r. ~ e2/mc2 is the 1 d+ , (49) electron radius, K2 is given by Eq. (47)s classical K by Eq. (46), v = cos i3,and v/v’ is given by Eq. (48). dcs do do —d$l- sineded$ = 2nsinedEI q (50) because of @q symmetry of the problem. ~US x do w= r2 o 2 sin 8 J o “T d$ (~) (1 - p)2 - 2(1 - P) + v’ K + ‘;2 v KK 2 ‘2 v K [ (KK2)2 . 1 (51) The @-integrationof Eqs. (49) or (51) can be performed analytically,but we found the result to be small differencesof very large quantities leading to inaccuraciesin small computers as well as to be algebraicallycumbersome. Accordingly,we found it simpler to use direct numerical integrationby Simpson’s rule. For angles, only 8 intervals gave sufficientaccuracy for our purposes. The approximation(49) also reduces in the limit T + o to the KleinNishina formula12>10for unpolarized light, as it should. In Eq. (46) setting T = O gives K = 1, which in Eq. (47) then gives K2 = 1, i.e., K2 is now no longer a function of $, and thus yields from Eq. (48) 1 )1 —= v’ (T + O) (1 - p) + 1’ + (52) which is identical to Eq. (38), and from Eq. (49) (with ~ d$ = 2x) yields the unpolarizedKlein-NishinaFormula (39) da —=— d$2 r’ O 2 ● V (-) V’ 2 “[ : +++p2 -1 1 ● (T + O) (53) 13