22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion


 Peregrine Dalton
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1 .54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References  J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (AddisnWesley, Reading, 966) T study neutrn diffusin we g back t the neutrn transprt equatin and btain an equatin nly in the spatial variable. We first eliminate the Ω dependence by integrating the transprt equatin ver Ω, getting an equatin with tw unknwns, φ(, ret,) = dωφ(, re, Ω,) t (.) JrEt (,,) = dωωφ (, re, Ω,) t (.) Then we invke Fick's law t eliminate J, thus btaining φ( ret,, ) = [ DE ( ) Σ t ( E)](, φ ret,) + SrEt (,,) + v t f s (.3) ν f ( E) de' Σ ( E') φ( re, ', t) + de' Σ ( E') φ( re, ', tfe ) ( ' E) T reduce further, we cnsider nly steadystate slutins, and integrate ver all energy t arrive at [ D ( ν f a)] φ( r) S( r) + Σ Σ = (.4) where φ() r = deφ(, r E) (.5) D ded( E) φ( r, E) deφ (, r E) (.6) and a similar expressin like (.6) fr the macrscpic crss sectin Σ. The verhead dentes energy average weighed by the flux as indicated in (.6) (recall als (9.7)). In writing (.4) we have made use f the statement f neutrn cnservatin,
2 def( E ' E) = (.7) We need t keep in mind that in (.4) we are als assuming that the external surce is timeindependent, and mre significantly that D is independent f psitin, which wuld be the case if φ (, re) were separable in r and E (this is nt true in general). Eq.(.4) is a secndrder differential equatin with cnstant cefficients. Since the Schrdinger equatin, in the case f cnstant ptential, is als f this frm, it is wrthwhile t make nte f the analgy between the prblem f neutrn diffusin and the prblem f a particle cnfined in a ptential well, particularly in the rle f the bundary cnditins. T keep the ntatins simple we will drp the verhead bar n the material cnstants with the understanding that they are t be regarded as energy averaged quantities. Bundary Cnditins The bundary cnditins t be impsed n φ () r are quite similar t thse impsed n the wave functin in slving the Schrdinger equatin. Because we are dealing with a physical quantity, the neutrn distributin in space, φ () r must be psitive and finite everywhere r zer. Als, the distributin must reflect the symmetry f the prblem, such as φ( x) = φ( x) in a slab system with x = 0 being at the center f the slab. Then there are the usual bundary cnditins at a material interface; flux and currents must be cntinuus since there are n surces r sinks at such interfaces. All these cnditins have cunterparts in slving the wave equatin. The ne bundary cnditin which requires sme discussin is the statement f n reentrant current acrss the bundary between a medium and vacuum. Let this surface be lcated at the psitin x = x in a slab gemetry. The physical cnditin is J ( x ) = 0. While the definitin f J  given in Eq. (9.0), Chap9, is perfectly crrect, we can evaluate J accrding t its physical meaning and by making the assumptins that the scattering is istrpic in LCS, the medium in nn absrbing, and the flux is slwly varying. Then J  is apprximately given by the fllwing integral (see Fig. ), Fig.. The gemetrical set up fr estimating the current f neutrns crssing a unit area A at the rigin (with nrmal ˆn ) after scattering istrpically in the vlume element d 3 r abut r. (Adapted frm Lamarsh, p. 6.) cs ˆ A θ () (.8) Σt 3 z J r = Σsφ r e d r 4π r uhs
3 where the integral extends ver the upper half space (uhs) f the medium because we are interested in all thse neutrns which can crss the unit area A frm abve (in the directin ppsite t the unit nrmal). The part f the integrand in the parenthesis is fractin f neutrn ging thrugh the unit area A if there were n cllisins alng the way, with A being subtended at an angle θ frm the elemental vlume d 3 r at r. The fact that neutrns can cllide n the way t the unit area is taken int accunt by the factr exp( Σ r ). T carry ut the indicated integral we need t knw φ () r Since we have t assumed the flux is slwly varying we can expand abut the rigin (the lcatin f the unit area) and keep nly the first term in the expansin, Then, φ() r φ(0) + r φ (.9) π π / Σ z (0) s Σt cs sin r = ϕ θ θ θ φ ( φ/ ) ( φ/ ) ( φ/ ) 4π ϕ= 0 θ= 0 0 [ ] J d d dre x x y y z z (.0) where we have taken A =. Writing ut the Cartesian cmpnent x, y and z in terms f the spherical crdinates (, r θ, ϕ ), x = rsinθ csϕ, y = rsinθ sinϕ and z = rcsθ, we find the ϕ integratin renders the terms cntaining x r y equal t zer. The term cntaining z can be easily integrated t give (after shifting the unit area frm being at the rigin as in Fig. t a slab gemetry with the unit area n the surface at x = x ) r φ( x ) D dφ J ( x ) = + 4 dx x dφ dx x = φ( x ) D (.) (.) Eq.(0.) is nt really a bnafide cnditin n φ ( x ) because the gradient dφ / dx is nt knwn. T find anther relatin between the flux and gradient, we interpret the latter as a finite difference, dφ φ( x ) φ( x ') = dx x ' x x x' > x (.3) where we use the negative sign because we knw the gradient must be negative. Nw we chse x' such that we knw the value f the flux at this psitin. Hw is this pssible? Suppse we chse x' t be the distance where the flux linearly extraplates frm x = x t zer. Calling this distance x' = x + δ (see Fig. ), we then have frm (.3) 3
4 Fig.. Schematic f the extraplated bundary cnditin, with δ being the distance beynd the actual bundary where a linear extraplatin f the flux at x = x wuld vanish. The dtted curve shws the variatin f the flux that wuld be btained frm transprt thery. (Adapted frm Lamarsh, p. 35.) dφ dx x = φ ( x ) δ (.4) Cmbining this with (.9) we btain fr the extraplated distance δ = D. Cnventinally instead f the physical cnditin f n reentrant current, ne ften applies the simpler mathematical (and apprximate) cnditin f φ( x + D) = φ( x ) = 0 (.5) where x = x + D. Eq.(.5) is called the extraplated bundary cnditin; it is cmmnly adpted because f its simplicity. One can use transprt thery t d a better calculatin f the extraplated distance δ. We have seen that in simple diffusin thery this turns ut t be D, r / 3Σ tr. The transprt thery result, when there is n absrptin, is 0.7/ Σ tr. The rather small difference between diffusin thery and transprt thery shuld nt be taken t mean that the flux near the surface is always accurately given by diffusin they. As can be seen in Fig., diffusin thery typically verestimates the flux at the surface relative t transprt thery. Diffusin Kernels (Green's Functins) One can slve the neutrn diffusin equatin fr the flux shape crrespnding t varius lcalized surces. This is tantamunt t the standard prblem f finding the Green's functin fr a pint surce and then integrating the result t btain slutins fr ther simple surce distributins. Since this kind f calculatins is well described in the standard references, we will give nly sme f the results here. Cnsider a plane surce at x = 0 in an infinite medium which emits istrpically s neutrns/cm /sec. The diffusin equatin reads 4
5 d dx κ φ ( ) 0 pl x = x 0 (.6) with κ =Σ / D > 0 (κ is real). The slutin fr the case f a plane surce is a s κ x φ pl ( x) = e (.7) Dκ where we have applied the surce cnditin, [ AJ ( x)] = D[ dφ( x)/ dx] = s / (.8) x x x x with A being a unit area. Eq.(.8) is simply the statement that if s is the number f neutrns emitted per unit area per sec by the plane, then half f the neutrns wuld cme ut in the +x directin. Suppse nw instead f a plane surce we have a pint surce at the rigin emitting s neutrns/sec. The equatin becmes = r 0 (.9) ( κ ) φ pt ( r) 0 with slutin s κ r φ pt () r = e (.0) 4π rd Cmparisn f (.7) and (.0) suggests that the tw kernels are related, and that ne can be btained frm the ther. This cnnectin is actually quite general and fllws directly frm the prperty f the Green's functin. Since the diffusin equatin is linear, ne can superpse the cntributins frm different pint surces t make the slutin t any distributed surce, φ (.) 3 () r = d r'(') s r φpt ( r r') s Applying this t the plane surce distributin, ne btains π pl ( x) = dz ( z) d d pt ( x + ) 0 0 (.) φ δ ρ ρ ϕφ ρ where the integral is written ut in cylindrical crdinates with x being the perpendicular distance frm the surce plane. Carrying ut the integratins, ne finds 5
6 φ ( x) = π dγγφ ( γ ) (.3) pl x pt which ne can verify is cnsistent with (.7) and (.0). One can invert (.3) by differentiating t give dφ pl ( x) φ pt () r = dx π x= r (.4) The relatin (.3) als helps us t understand why the pint surce kernel is singular at the rigin and yet the plane surce kernel is regular everywhere. The Cncept f Buckling in Criticality (We recmmend the student t first study Chap befre reading this sectin, since the cncept f criticality is discussed in detail in Chap.) We nw turn t the prblem f criticality and shw hw diffusin thery can be used t estimate the nnescape prbabilities that appear in the multiplicatin cnstant. T d this it is instructive t ask what culd be a measure f the reactr size besides the bare system dimensins. Recall that the extraplated bundary cnditin, such as (.5), expresses the idea f an extraplated distance as an incremental length beynd the actual system bundary. Thus it is nt surprising that a useful gemetric measure f system size shuld invlve the extraplated distance. Hw des this cme abut naturally in the cntext f bundary cnditins fr slving the diffusin equatin? We will examine this cnnectin thrugh the example f a critical spherical reactr, a system in which the materials prperties and the gemetric size are in balance such that its multiplicatin cnstant is unity. Cnsider a spherical reactr f radius R cmpsed f materials fr which all the crss sectins, scattering, absrptin, and fissin, are nnzer, and there is n external surce. The diffusin equatin fr this system is + = r R (.5) ( α ) φ( r) 0 with α = ( νσ f Σ a)/ D >0. (Nte α 0 means at best k =, and any finite system must therefre be subcritical, i.e., the system cannt maintain a nnzer steady state flux in the absence f a surce.). The physical slutin t (.5), after applying the cnditin f finite flux at the rigin, is just sinαr φ () r = A (.6) r 6
7 We als can cnclude that A must be psitive and that α R must be π. We can apply ne mre bundary cnditin, that at the reactr surface r = R. Since (.6) is a hmgeneus equatin, we knw that we will nt be able t determine A. Thus the cnditin at R has t impse a cnstraint n α, the nly ther cnstant left in the descriptin. (The analgy with energy quantizatin in quantum mechanics when slving the wave equatin fr a certain shape f the ptential shuld be quite apparent at this pint.) We have already seen that the prper bundary cnditin fr a materialvacuum interface is n reentrant current, J ( R) = 0. In diffusin thery this is apprximately φ( R) D + n ˆ φ r R 4 = = 0, r Applying this t (.6) gives R dφ φ dr r= R R = (.7) D R αrctαr= (.8) D The slutin t (.8), α R, is seen t depend n the magnitude f the rati R/D, clse t zer if R << D and clse t π if R >> D. The latter is the mre physically cmmn situatin fr any interesting value f D, i.e., reactr material. S we write αr = π ε, with ε being small. The left hand side f (.8) then becmes Nting that π ε π π αrctαr + = = (.9) ε ε π α R α = ( π ε)/ R= ( π / R)( ε/ π) ( π / R)( + ε/ π) Frm (.9) and (.8) we als have (.30) π R ε = D, r Rε / π = D (.3) Cmbining (.30) and (.3) we thus btain with R being the 'extraplated' radius, π α B g (.3) R R R+ D (.33) 7
8 Thus we arrive at the same result as in the case f the slab reactr befre. One can apply the surface bundary cnditin as φ ( R ) = 0, rather than J ( R) = 0. Eq. (.3) als serves t intrduce the quantity B g, called 'gemetric buckling' in reactr physics, presumably because it has t d with the shape ("buckling") f the flux and it depends nly n the size (gemetry) f the system. The implicatin f (.3) is that in rder fr the critical spherical reactr t have a physical slutin satisfying the bundary cnditins in diffusin thery, the cnstant α has t have the value specified by (.3). Hwever, recall that in writing the diffusin equatin (.5), the cnstant α already was defined by the materials prperties. We will rewrite this definitin as νσ f Σa α = B m (.34) D thereby intrducing the quantity B m, 'materials buckling', in analgy with the gemetric buckling. Therefre, the nly way t satisfy bth the materials cnstraint, represented by (.5) and (.3), and the system size cnstraint, represented by (.9) and (.3), is t require B B (.35) m g which we can regard as the cnditin fr system criticality, the balance between materials prperties and system size. T see what this relatin can lead t, we rewrite it as νσ f Σa = DB Σa g (.36) r η f = + LBg (.37) with νσ / Σ η and f a f L D/ a Σ, L being called the diffusin length. We purpsely write (.37) in the frm f a critical cnditin, explicitly shwing the multiplicatin cnstant k = k PNL having the value f unity. With this identificatin we can pick ff an expressin fr the nnleakage prbability, P NL (.38) + LB g Eq.(.38) is useful because it prvides a quick estimate, in the cntext f simple diffusin thery, f the nnescape prbability that appears in the multiplicatin cnstant. 8
9 Ging back t (.36), we see that anther way t interpret the balance cnditin is the requirement νσ Σ = DB (.39) f a g The left hand side represents the effective crss sectin fr 'neutrn gain', whereas the right hand side represents the 'neutrn lss', with DB playing the rle f a 'leakage crss sectin'. This bservatin makes it pssible t cmpare the effects f neutrn interactins, in the sense f scattering and reactins measured in the frm f macrscpic crss sectins (r the mean free path), with thse f neutrn diffusin, in the sense f diffusin and surface bundary cnditin in terms f D and the gemetric buckling, n the same basis. An appreciatin f this simple equivalence is a primary reasn that we can give fr studying neutrn diffusin thery. g 9
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