Nuclear Reactors

Transcription

1 Nucler Rectors Nucler Physics t BAU This course Prerequisites Nucler nd Rdition Physics 74 / l di ti Advnced Sttisticl Mechnics edu 1

2 Grding Review Test 05% Mid-term Exm 0% Projects, quizzes nd HWs 5% Finl Exm 50% Homeworks re due fter one week unless otherwise nnounced. Remrks or questions mrked in red without being nnounced s homeworks should be lso seriously considered! Some tsks cn (or should) be sent by emil: sed@dbbneh.com b

3 Review Test Review relevnt mteril in 74. Red Lmrsh Chs 1, nd 3. Will do the test fterwrds. 3

4 Projects Rdition Protection nd Shielding (Due December 6 th ). Nucler fuel cycles with emphsis on front ends (Due November 9 th ). Work s tem. Divide nd orgnize the job mong you. Try to explore locl pplicbility. Presenttion: Will be scheduled lter. Other smll projects will be nnounced in clss. 4

5 Nucler Rection Energetics (revisited) Conservtion Lws Chrge, Bryon number, totl energy, liner momentum, ngulr momentum, prity, (isospin??). p b b θ p X φ py Y m c m c = T T = i f f i +ve Q-vlue exoergic rection. -ve Q-vlue endoergic rection. T + T = Q + b +ve Q-vlue rection possible if T 0. -ve Q-vlue rection not possible if T 0. (Is T > Q sufficient?). Conservtion of momentum Y T Q 5

6 Nucler Rection Energetics (revisited) Conservtion of momentum. We usully do not detect Y. Show tht: T b = m m T b cosθ ± m m T b cos θ + ( m m + m Y m b Y + m b HW 1 )[ m Y Q + ( m Y m The threshold energy (for T ): (the condition occurs for θ = 0º). my + mb T Th = Q my + mb m +ve Q-vlue rection possible if T 0. Coulomb brriers.!!! Neutrons vs. chrged prticles. -ve Q-vlue rection possible if T > T Th. ) T ] 6

7 Nucler Rection Energetics (revisited) HW 1 (continued) The double vlued sitution occurs between T Th nd the upper limit T \. m T \ = Q Y m m Y Double-vlued l in forwrd cone. cos ( + m m Q + m Y b )[ Y ( θ Y mx = mmbt m Q m ) T ] 7

8 Nucler Rection Energetics (revisited) HW 1 (continued) Discuss thoroughly the 7 Li(p,n) rection. During the discussion emphsize on the cse when the incident proton bem is 30 kev bove the threshold. Use your computing skills. Discuss the elstic nd inelstic scttering of neutrons reltions. neutrons using these 8

9 Nucler Rection Energetics (revisited) 9

10 Nucler Rection Energetics (revisited) Wht bout neutron induced rections? 10

11 Nucler Rection Energetics (revisited) Wht bout neutron induced rections? 11

12 Nucler Rection Energetics (revisited) 1

13 Q Nucler Rection Energetics (revisited) If the rection reches excited sttes of Y ex = m X c + m c ( m Y c + E ex ) m bc = Q 0 58 Ni(α,p) 61 Cu E ex even less. less proton energy Highest proton energy See Figures 11.4 in Krne 13

14 Nucler Rection Energetics (revisited) 14

15 Neutron Interctions (revisited) Chdwick s discovery. Neutrons interct with nuclei, not with toms. (Exceptions). Recll from Nucler Physics 74: o Inelstic scttering (n,n \ ). Q = -E* Inelstic gmms. Threshold? o Elstic scttering (n,n). Q =?? (Potentil nd CN). Neutron modertion? o Rditive cpture (n,γ). Q=?? CptureQ = gmms. o (n,α), (n,p). Q =?? Absorption Rections. o (n,n),, (n,3n), Q =?? Energetic neutrons on hevy wter cn esily eject the loosely bound neutron. o Fission. (n,f). HW Exmples of such exo- nd endo-thermic rections with Q clcultions. 15

16 Neutron Scttering (revisited) Elstic or inelstic. Anlogous to diffrction. Alternting mxim nd minim. First mximum t λ θ h R λ = R = p R o 1 3 A Minimum not t zero (shrp edge of the nucleus??) Cler for neutrons. Protons? High energy, lrge ngles. Why? Inelstic Excited sttes, energy, X-section nd spin-prity. it 16

17 Rection Cross Section (revisited) Probbility. Projectile will more probbly hit trget X if re is lrger. Clssiclly: σ = π(r +R X ). Clssicl σ =??? (in b) n + 1 H, n + 38 U, 38 U + 38 U Quntum mechniclly: σ = π D. m + mx h h D = = m CM X me µ X EX Coulomb nd centrifugl brriers energy dependence of σ. Wht bout neutrons? Nture of force: Strong: 15 N(p,α) 1 C σ ~ 0.5 b t E p = MeV. Electromgnetic: 3 He(α,γ) 7 Be σ ~ 10-6 b t E α = MeV. Wek: p(p,e + ν)d σ ~ 10-0 bte p = MeV. Experimentl chllenges to mesure low X-sections.. 17

18 Rection Cross Section (Simple terms) A (Are of wht??!!) v X Monoenergetic neutrons of speed v (cm.s -1 ) nd density n (cm -3 ) Position of neutron 1 s before rriving t trget Trget with N toms.cm -3 or NAX toms. Volume = va contining nva neutrons tht hit the whole!! trget in 1 s. Bem Intensity I nva/a = nv (cm - s -1 ) Number of neutrons intercting with trget per second I, A, X nd N = σ t INAX 18

19 Rection Cross Section (Simple terms) Number of neutrons intercting with trget per second = σ t I N A X Totl cross section Totl number of nuclei in the trget Number of interctions with single nucleus per second = σ t I Interprettion nd units of σ. t nva = IA neutrons strike the trget per second, of these σ t I neutrons interct with ny single nucleus. Thus, σ ti = AI σ t A mesures the probbility for neutron to hit nucleus (per unit re of trget). t) Effective cross-sectionl re of the nucleus. 19

20 Rection Cross Section (Simple terms) Number of neutrons intercting with trget per second = σ t I N A X Totl cross section Volume of the trget Number of interctions per cm 3 per second (Collision Density) Mcroscopic totl cross section. Probbility per unit pth length. F t = σ t I N = I Σ t Σ t = N σ t I ( X ) = I e 0 λ = 1 Men free pth length t Σ Σ t 0 t X

21 Rection Cross Section (Simple terms) Homogeneous Mixture Σ = Σ x + Σ y = N x σ + x N y σ y Molecule l x m y n N x =mn, NNN y =nn σ = mσ + x n σ y given tht events t x nd y re independent. 1

22 Rection Cross Section (revisited) I \ θ,φ Detector for prticle b dω cm dσ = b prticles / s I dr b \ N \ Typicl nucleus (R=6 fm): geometricl πr 1 b. Typicl σ: <µb bt to >10 6 b.

23 Rection Cross Section (revisited) Mny different quntities re clled cross section. Krne Tble 11.1 dr = r (, ) Units! d Ω = sinθθ d θ d φ σ = dσ dω = Differentil cross section σ(θ,φ) or σ(θ ) or cross section!! π sinθdθ π dσ dφ d Ω d Ω d 0 0 σ t for ll b prticles. t dr b Angulr distribution d σ = dω dσ de dω θ φ 4π r ( θ, φ ) \ \ 4πI N Doubly differentil d σ dedω 3

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25 n-tof CERN 5

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27 Different Fetures (revisited) 1/v Fst neutrons should be moderted. 35 U therml cross sections σ fission 584 b. σ scttering 9 b. σ rditive cpture 97 b. Fission Brriers 7

28 Neutron Induced Rections (revisited) X(n,b)Y σ n D Y + b H II C C H I X + n For therml neutrons Q >> E n 1 1 E v Γ b (Q+E n ) Γ n (E n ) v n P l n ( E ) n Probbility to penetrte the potentil brrier Γ b (Q) constnt P o (E therml ) = 1 Q E n P >o (E therml ) = 0 Non-resonnt σ 1 ( E ) n n v 8

29 Neutron Induced Rections (revisited) 9

30 Sttisticl Fctor (revisited) L = lh = bp = b = ld h b D b 1) l, mx l 1 l σ = πb π b = l + + HW 3 π D ( b ) = µ ( u) E ( 1) π D CM ( kev J + 1 σ mx = π D X (1 + δ X ) (J + 1)(J + 1) X ω ) 30

31 Resonnce Rections (revisited) J π Excited E x + X Y + b Q > 0 Stte Entrnce b + Y X + Q < 0 Chnnel Exit + X Chnnel Compound b+y Nucleus C* Inverse Rection σ σ J + 1 X = πd X (1 + δx ) + II I + ( J + 1)( J X + 1) QM J Sttisticl Fctor (ω) + 1 Identicl prticles Y b H C C H X Nture of force(s). Time-reversl invrince. by = πd by (1 + δby ) + X H I C C H II b + (J b + 1)(JY + 1) HW 4 σ σ X by =?? 31 Y

32 Resonnce Rections (revisited) Projectile Projectile Trget Q-vlue Q-vlue Trget Q + E R = E r E γ = E + Q - E ex Direct Cpture σ γ Y H + γ (ll energies) X σ γ Resonnt Cpture (selected energies with lrge X-section) f H γ Er Er H CN + E + X σ γ Γ Γ b 3

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34 Resonnce Rections (revisited) Dmped Oscilltor Oscilltor strength response f ( ω δ ω ) + ( ) o δ = 1 Dmping t0 fctor eigenfrequency σ ( E) ( E Γ Γ b Γ E ) + ( ) R Γt o = h 34

35 σ ( E) = πd X Resonnce Rections (revisited) J + 1 Γ Γb (1 ) ( + δ X J + 1)( J + 1) ( E E ) + + X R Breit-Wigner formul All quntities in CM system Only for isolted resonnces. σ σ R Γ Γ Γ Γ b e σ R Γb HW 5 σ e = Γ Rection Elstic scttering ( ) Γ Γ = Γ + Γ b Usully Γ >> Γ b. HW 5 When does σ R tke its mximum vlue? 35

36 Resonnce Rections (revisited) J + J X + l = J (-1) l π(j ) π(j X ) = π(j) (-1) l = π(j) ( ) Nturl prity. J π Entrnce Chnnel +X Excited Stte Compound Nucleus C* E x Exit Chnnel b + Y 36

37 Resonnce Rections (revisited) Wht is the Resonnce Strength? Wht is its significnce? In wht units is it mesured? J + 1 Γ Γb ωγ = (1 + δ X ) ( J + 1)( J + 1) Γ + X oss secti ion Cr E C Chrged prticle rditive cpture (,γ) (Wht bout neutrons?) ωγ Γ ωγ Γ γ Energy 37

38 Neutron Resonnce Rections (revisited) 38

39 Neutron Activtion Anlysis (revisited) (Z,A) + n (Z, A+1) ) β - γ (β-delyed γ-ry) (Z+1, A+1) Project 1 NAA nd U 39