Resonance self-shielding in the epi-thermal region
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1 Resoace self-shieldig i the epi-thermal regio Workshop o Nuclear Data for Activatio Aalysis Trieste, Italy, 7 18 March 005 P. Schillebeeckx, K. Volev ad M. Moxo peter.schillebeeckx@cec.eu.it EC JRC IRMM, B Geel, (B) Istitute for Referece Materials ad Measuremets (IRMM) Geel, Belgium
2 Neutro resoaces ad NAA Basic priciples of NAA Nuclear reactio theory Neutro resoaces ad NAA Resoace self-shieldig for a parallel beam Resoace self-shieldig for a isotropic beam Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
3 1. Basic priciples of NAA Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
4 Neutro capture process Neutro Eergy Resoaces Cross sectio NRCA E * = S A + E A + 1 PGAA S E γ A+1 X T 1/ Eγ (I)NAA + A A + 1 * X X A+ 1 X + γ Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
5 Capture cross sectio σ(,γ) / bar Co(,γ) Neutro Eergy / ev The probability that a eutro iteracts with a ucleus is expressed as a cross sectio σ, which has the dimesio of a area The uit of a cross sectio is take as : 1 bar, 1 b = 10-4 cm To calculate reactio probabilities we express the target thickess i atoms per bar : m A = 0.60 m : atomic mass ρ : desity i g/cm 3 t : thickess i cm : target thickess i at/b A ρ t Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
6 Total reactio rate 10 4 Neutro Flux 59 Co : σ(,γ) The total reactio rate per atom: σ(,γ) / bar Neutro flux per ev R = 0 ϕ(e depeds o: ) σ γ (E ϕ(e ) the eutro flux ad ) de σ γ (E ) the capture cross sectio Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
7 Neutro flux i a thermal reactor Neutro flux per ev 10-1 Total spectrum Maxw. (kt = 0.05 ev) 1 / E 1+α Fissio The eutro flux i a thermal reactor is a sum of three compoets Maxwellia distributio with maximum at E = kt 1/E 1+α distributio due to moderatio process of the fast eutros (epithermal spectrum) Watt spectrum of fissio eutros Neutro Eergy / ev At a eutro guide, the eutro flux ca be described by a Maxwellia distributio (cfr. PGAA at Budapest, i.e o resoace shieldig!) Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
8 Reactio rate i a thermal reactor The total reactio rate per atom is: R = ϕ(e ) σ(e 0 ) de To solve the itegral oe separates betwee the thermal ad the epi-thermal regio: R E Cd = ϕ(e ) σ(e ) de + ϕ(e 0 E E 3 Cd ) σ(e ) de + fast with E Cd = 0.55 ev Covetios : Høgdahl & Westcott covetio Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
9 Reactio rate Importace of resoace regio σ o : capture cross sectio at thermal g w : Westcott g-factor (expresses the deviatio of σ γ from 1/v) ϕ t : thermal eutro flux G t : thermal self-shieldig I R : resoace itegral ϕ e : epi-thermal eutro flux G R : resoace self-shieldig R G t ϕ σ t o g w + G R ϕ e I R Cd-ratio measuremets F Cd = R G ϕ R Cd e I R σ o, g w, I R, G R, F Cd are iflueced by the resoace structure Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
10 Trasmissio through cadmium 10 4 at Cd e - Cd σ t Ideal σ(,t) / bar 10 Trasmissio factor 0.5 E Cd = 0.55 ev Cd = at /b (1 mm) Neutro Eergy / ev Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
11 Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05. Nuclear reactio theory
12 Eergy differetial cross sectios 10 6 σ t U(,tot) = 38 U(,) + 38 U(,γ) Cross-sectio / bar 10 σ γ σ Γ E o D σ tot = σ + σ γ Thermal Resoace Regio : D > Γ Resolved Resoace Regio : R < D Uresolved Resoace Regio : R > D High Eergy Regio : D < Γ Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
13 Eergy Differetial Cross Sectios 10 4 σ t 10 5 σ t 56 Fe Au + σ σ Cross sectio / bar σ γ Cross sectio / bar σ γ Neutro Eergy / ev No capture without scatterig - Relative cotributio of σ ad σ γ to σ t may be differet - Boudaries of the RRR differ Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
14 Importace of resoace structure σ tot / bar at Fe + IRMM 1993 ENDF/ B-VI 0 Trasmissio Trasmissio (d Fe = 40 cm) Neutro Eergy / MeV Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
15 Resoace structure 8000 σ R σ t A cross sectio as a fuctio of E shows a resoat structure, which ca be described by a Breit-Wiger shape : σ t / bar 4000 Γ σ t ~ ( E E ) + ( Γ ) R 1 0 E R Neutro Eergy / ev with Γ E R atural lie width (FWHM) resoace eergy Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
16 Bohr s hypothesis : Compoud ucleus reactio Two step process (1) Formatio of compoud ucleus σ C* E* E σ C* ( E ) π = g k Γ = r Γ ( E E ) + ( Γ ) r R Γ Γ (r =, γ,f,...) A X A X + σ c* () Decay of compoud ucleus P r P r = Γr Γ (r =, γ,f,...) γ Partial cross sectio σ r = σ C* P r -S E * = S + A+1 X A E A + 1 Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
17 Breit Wiger formula Resoace part of the cross sectio Total Cross Sectio (,tot) σ t ( E ) = π g k ( E E ) + ( Γ ) R Γ Γ J + 1 g = ; k = waveumber (I + 1) Elastic Cross Sectio (,) σ Γ ( E ) = σ ( E ) Γ t Capture Cross Sectio (,γ) σ γ Γ ( E ) = σ ( E ) Γ t γ Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
18 e.g. 109 Ag s-wave at E o = 134 ev (E o, Γ, Γ γ, J π ) σ R (bar) = Γ = Γ + Γ γ.608x10 E (ev) R 6 m A + 1 ma gγ Γ σ t / bar σ R Γ σ t E o = 134 ev Γ = ev Γ γ = ev J π = 1 - g = 3/4 σ / bar σ R Γ Γ Γ E R σ Neutro Eergy / ev 0 E R Neutro Eergy / ev σ γ / bar Γ σ γ R Γ Γ E R σ γ Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
19 Eergy depedece of eutro width The eergy depedece of the eutro width is due to the cetrifugal-barrier peetrability, which depeds o the agular mometum of the icomig eutro l ad E The eutro width Γ depeds o E s-wave (l = 0) 0 E Γ ( E ) = Γ 1eV σ γ σ(,γ) / bar ( E ) π = g k 10 6 s - wave p - wave ( E E ) + ( Γ ) E R = 1 ev Γ γ Γ R = 100 mev Γ = 10 mev Γ γ p-wave (l = 1) Γ (E ) = Γ 1 E 1eV ka 1+ k a Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
20 3. Neutro Resoaces ad NAA Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
21 Ifluece of resoace structure o: Thermal capture cross sectio 1/v behaviour of the capture cross sectio Westcott g w - factor Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
22 σ o ad cotributio of s-wave resoaces # Ag(,γ) σ o, l= x10 6 m A + 1 ma N j= 1 gγ E Γ o j γj Rj σ γ / bar 10 σ 0 = 91 b with σ o i bar E R, Γ o, Γ γ i ev 10 0 E R = 5. ev Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
23 σ o ad cotributio of s-wave resoaces # σ(,γ) / bar 10 6 σ γ, total σ γ for E r = 1 ev σ γ for E r = - 1 ev σ o If σ o > σ o, l= m A + 1 ma j= 1 gγ Additioal cotributio from boud states (egative resoaces) E* 6 E N E Γ o j γj Rj 10 0 A X Neutro Eergy / ev A X σ c* γ -S A+1 X Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
24 1/v behaviour of capture cross sectio Capture cross sectio σ γ ( E ) = π g k ( E E ) + ( Γ ) Γ Γ R γ Neutro width for a s-wave eutro 0 E Γ ( E ) = Γ 1eV For E << E R witk k E σ γ 1 E = E ( ) 1 v σ(,γ) / bar 10 6 s-wave 1/v E R = 1 ev Γ γ = 100 mev Γ = 10 mev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_ Neutro Eergy / ev
25 Westcott g w - factor: deviatio of 1/v behaviour # Co(,γ) E R = 13 ev E R 176 Lu(,γ) = 0.14 ev σ(,γ) E 1/ / bar ev 1/ g w = Neutro flux at: 300 K σ(,γ) E 1/ / bar ev 1/ 10 4 g w = Neutro flux at: 300 K Neutro Eergy / ev Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
26 Westcott g w - factor: deviatio of 1/v behaviour # The Westcott g w factor is temperature depedet 176 Lu(,γ) Lu(,γ) σ(,γ) E 1/ / bar ev 1/ Neutro flux at: 10 K 300 K 1000 K Neutro Eergy / ev Westcott Factor, g w (T) Temperature / K 176 Lu(,γ) temperature moitor Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
27 Data base of resoace parameters Evaluated data libraries (see IAEA INDC website) JEF ENDF-B JENDL CENDL Compilatio by S.F.Mughabghab Neutro Resoace Parameters ad Thermal Cross Sectios Part A & B NNDC, BNL, 1984 Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
28 Data base of σ o, g w -factor ad I R Compilatio by S.F.Mughabghab, BNL, USA Thermal eutro capture cross sectios, resoace itegrals ad g w -factors INDC(NDS) 440 February 003 Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
29 4. Resoace self-shieldig for a parallel beam Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
30 Self-shieldig ad multiple scatterig Parallel eutro beam o a foil or a disc Study the basic effects (parallel beam is ot directly applicable to NAA, but experimetal verificatio of procedures is possible) Ifluece of resoace structure o the self-shieldig factor Doppler effect Total correctio factors (due to self-shieldig & scatterig) Iterferece effects Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
31 Parallel beam o thi sample For a relatively thi sample, o scatterig oly self-shieldig ϕ(x) dx = ϕ x= 0 e ρσ t x dx t R(E ) ρ σ (E ) ϕ(x) 0 γ dx ϕ (x) R(E ) ϕ x= 0 σ γ (E ) (1 e σ σ t t (E (E ) ) ) x self-shieldig factor Ifiitely thi sample γ R thi σ γ rσ t ( 1 e ) = σ for σ << 1 σ t γ t Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
32 Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05 γ γ γ = j R j R R o R 1 R Self-shieldig ad multiple scatterig θ θ + + = A A ' si m m cos m m m E E t ) e (1 R t σ σ σ γ
33 Self-shieldig ad multiple scatterig factor Calculatio Aalytical expressios REFIT (parallel beam+ disc) SAMMY (parallel beam+ disc) Mote Carlo simulatios E1 0 SAMSMC (parallel beam + disc) MCNP (o geometry limitatios, use probability tables!!) G R = E E E 1 de de dω dω t t 0 dx dx Σ γ Σ γ (E) Φ(x, Ω,E) (E) Φ(x = 0, Ω,E) Defiitios Self-shieldig without scatterig : G R,0 Self-shieldig + 1 scatterig : G R,1 Self-shieldig + 1,,... scatterig : G R, Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
34 Examples : 56 Fe(,γ), 59 Co(,γ) ad 197 Au(,γ) Reactio E R / ev Γ / ev Γ γ / ev Γ / ev D / ev 56 Fe Co Au Au(,γ) 59 Co(,γ) 56 Fe(,γ) σ(,γ) / bar Neutro Eergy / ev Neutro Eergy / ev Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
35 Self-shieldig + multiple scatterig for 56 Fe(,γ) #1 56 Fe(,γ) Fe = 1x10-3 at/b (t = 0.1 mm) Yield per atom / bar 10 R θ = 90 o θ = 180 o R 0 R 1 R Neutro Eergy / ev E R = 1.15 kev i 56 Fe θ E / kev E / kev 90 o 1.19 kev 1.15 kev 180 o 1.35 kev 1.15 kev 90 o & 180 o 1.80 kev 1.15 kev θ = 90 o θ = 180 o E E ' ' = E = E m m A A m m A A Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
36 Self-shieldig + multiple scatterig for 56 Fe(,γ) # Fe = at/b, t = 1.3 mm Reactio yield [arb. uits] R Reactio yield x 10 3 R R 1 R 0 Neutro Eergy / ev Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
37 Self-shieldig + multiple scatterig for 197 Au(,γ) #1 197 Au(,γ) Au = 6x10-4 at/b (t = 100µm) Yield per atom / bar 10 4 R R 0 R 1 R Reactio yield [arb. Uits] o Exp. - REFIT Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
38 Self-shieldig + multiple scatterig for 197 Au(,γ) # Reactio yield [arb. uits] Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
39 1. Ifluece of the resoace structure o G R #1 59 Co(,γ) 59 Co foil (1-e - σ t) / σ t G R,0 G R,0 (E ) thickess <σ t > σ t,max at /b µm 47 b b 5x x x x10-6 at/b 5x10-5 at/b 5x10-4 at/b (55.0 µm) Neutro Eergy / ev σt < σt > < e > e < e σ t > e < σ > t (1+ var σ t...) Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
40 Ifluece of the resoace structure o G R # σ(,tot) / b 0 15 This evaluatio Gregoriev et al. Uttly et al. Phoeitz et al. Iwasaki et al. G R, Th(,γ) = 1.588x10-3 at/b t = 0.5 mm HARFOR (aalytical model) SAMSMC (Mote Carlo) MCNP (probability tables) MCNP (average cross sectios) Neutro Eergy / kev Neutro Eergy / kev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
41 Ifluece of the Doppler effect #1 Yield per atom / kbar = 6x10-6 at/b t = 1 µm 197 Au(,γ) σ γ 300 K 1000 K Yield per atom / kbar = 6x10-5 at/b t = 10 µm 197 Au(,γ) σ γ 10 K 300 K 1000 K Neutro Eergy / ev Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
42 Ifluece of the Doppler effect # Au(,γ) Temperature Correctio factor G R, t = 1 µm t = 10 µm G R (E ) Temperature 10 K 300 K 1000 K 10 K K K Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
43 Reactio yields ad correctio factors for 56 Fe(,γ) 56 Fe(,γ) Yield per atom / bar σ γ 1x10-3 at/b (0.1 mm) 1x10 - at/b (1. mm) 1x10-1 at/b (1 mm) Neutro Eergy / ev Foil thickess Correctio factor at / b mm G R,0 G R,1 G R, 1x x x E R = ev Γ = ev Γ γ = ev Γ = ev D = 1.45 ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
44 Reactio yields ad correctio factors for 59 Co(,γ) Yield per atom / bar 10 3 σ γ x10-6 at/b 5x10-5 at/b 5x10-4 at/b 59 Co(,γ) Neutro Eergy / ev Foil thickess E R = 13.0 ev Γ = 5.15 ev Γ γ = 0.47 ev Γ = 5.6 ev D = 0.47 ev Correctio factor at / b µm G R,0 G R,1 G R, 5x x x Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
45 Reactio yields ad correctio factors for 197 Au(,γ) Yield per atom / bar 10 5 σ γ x10-6 at/b 6x10-5 at/b 6x10-4 at/b 197 Au(,γ) Neutro Eergy / ev Foil thickess Correctio factor at / b µm G R,0 G R,1 G R, 6x x x E R = 4.9 ev Γ = ev Γ γ = 0.14 ev Γ = ev D = ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
46 Iterferece effects, Uraium #1 Yield per U atom / bar U(,γ) 35 U(,γ) + 35 U(,f) U(,γ) LEU Powder Neutro Eergy / ev LEU Powder with at/b U 38 U wt% 35 U 3.00 wt% 34 U 0.03 wt% Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
47 Iterferece effects, Cd-ratio measuremets # Cross sectio / bar σ γ σ t 59 Co at Cd F Cd = R G ϕ R Cd e I R 10 - e -σ t for 1mm Neutro Eergy / ev Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
48 Iterferece effects, Cd-ratio measuremets #3 R Co i Co + Cd / R Co Co(,γ) i a mixture of 59 Co ad at Cd Co = at/b F Cd = R G ϕ R Cd e I R Neutro Eergy / ev F Cd is iflueced by the resoace structure of the cross sectios Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
49 5. Resoace self-shieldig for a isotropic beam Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
50 Isotropic beam (NAA applicatios) Experimetal data Oly self-shieldig, o scatterig & Doppler effect Self-shieldig + Doppler Self-shieldig + scatterig Uiversal method Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
51 Experimetal G R factors, F. De Corte et al Foil 94 Zr(,γ) 96 Zr(,γ) 99 Mo(,γ) 99 Mo(,γ) Wire (diameter) Foil (thickess) 0.95 G R Thickess / µm Thickess / µm Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
52 Theoretical calculatio, discussio #1 Reactio E R / ev Γ / ev Γ γ / ev Γ / ev D / ev 56 Fe Co Zr Zr Au D = with k 4 kt E m A R the Boltzma cos tat Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
53 Theoretical calculatio, discussio 96 Zr(,γ) # Foil Thickess (µm) G R for 96 Zr(,γ) G R,0 (theory, o scatterig) No Doppler Trubey* With Doppler Roe* Experimet De Corte *Take from P. De Neve, Thesis, Mol, 199 De Corte 87, Aggregraatsproefschrift Hoger Oderwijs, Uiversiteit Get, 1987 Γ = 0.1 ev Γ γ = 0.6 ev Γ = 0.47 ev D = 0.56 ev Importace of Doppler effect icreases with thickess especially for D Γ Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
54 Theoretical calculatio, discussio 94 Zr(,γ) #3 Foil G R for 94 Zr(,γ) thickess G R,0 with Doppler Experimet (µm) Roe* De Corte *Take from P. De Neve, Thesis, Mol, 199 De Corte 87, Aggregraatsproefschrift Hoger Oderwijs, Uiversiteit Get, 1987 Γ Γ γ = 1.3 ev = 0.10 ev Γ = 1.33 ev D = 1.55 ev Importace of multiple scatterig for Γ > Γ γ Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
55 Theoretical calculatio, discussio 59 Co(,γ) #4 Foil Thickess (µm) Theory (R o + R 1 ) No Doppler Lopes ad Avila G R for 59 Co(,γ) Experimet Eastwood ad Werer M.C. Lopes ad J.M. Avila, Nucl. Sci. & Eg., 104 (1990) 40 T.E. Eastwood ad R.D. Werer, Nucl. Sci. & Eg., 13 (196) 385 Γ Γ γ = 5.15 ev = 0.47 ev Γ = 5.6 ev D = 0.47 ev Importace of multiple scatterig for Γ > Γ γ Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
56 Uiversal curve #1 E. Martiho et al., Uiversal curve of epithermal eutro self-shieldig factors i foils, wires ad spheres, J. Appl. Rad. Ad Isot., 58 (003) 371 Express G R as a fuctio of the variable z = Σ t (E r ) Γ γ Γ y y = R for wires y = R for spheres y = t for foils Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
57 Uiversal curve # A1 A G res = + 1+ (z / z p 0) A z = Σ t (E ) r Γ γ Γ y y = R for wires y = R for spheres y = 1.5t for foils Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
58 Summary The importace of the resoace structure for NAA Thermal capture cross sectio 1/v behaviour of cross sectios Westcott g w - factor, g w (T) Self-shieldig ad multiple scatterig correctios, G R A first estimate of the correctio factor for self-shieldig ad scatterig is give by E. Martiho et al. More accurate correctio factors ca be calculated by aalytical expressios ad Mote Carlo simulatios if all relevat effects are accouted for Resoace structure (e.g. MCNP use probability tables) Neutro scatterig Doppler effect Solutio: F. De Corte et al., J. Radioaal. ad Nucl. Chem., 179 (1994) 93 I geeral, the best way to solve the problem of self-shieldig is to avoid it Resoace self-shieldig i the epithermal regio, P. Schillebeeckx 15_03_05
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