Nuclear Reactions. RELEVANCE: Basic Research reaction mechanisms structure exotic nuclei

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Bryan Wheeler
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1 Nuclear Reactions RELEVANCE: Basic Research reaction mechanisms structure exotic nuclei Applications: analytical tool neutron activation analysis astrophysics origin of elements space radiation effects satellites, Mars landing materials science T c (high) superconductors medical therapy tumor treatment nuclear power 25% of U.S. electricity DEFINITION: A collision between two nuclei that produces a change in the nuclear composition and/or energy state of the colliding species ; i.e., 2 nd order kinetics

2 1. Nomenclature A. Constituents 1. Projectile: nucleus that is accelerated (v > 0) ; ev a. neutrons reactors b. light ions A 4: AGS (NY), FNAL (IL) c. heavy ions A > 4 U: ANL (IL), MSU (MI), TAMU (TX), RHIC (NY) d. electrons (and photons) SLAC (CA), CEBAF (VA), MIT (MA) e. exotic beams: π, K, p : FNAL f. radioactive beams: MSU, ANL (IL) HHJRF (TN) 2. Target: fixed (v = 0), usually But now have colliding beams, RHIC, FNAL 3. Products ANYTHING PERMITTED BY CONSERVATION LAWS. Note: since projectile kinetic energy can be converted into mass, only limit on mass-energy is beam energy.

B. Notation: Target [projectile, light product(s)] heavy product 7 e.g., 3 Li + 208 82 Pb 212 At + 3n : 208 Pb( 7 Li, 3n) 212 At 197 Au + 18 C : 208 Pb( 7 Li, 18 C) 197 Au C.

3 B. Notation: Target [projectile, light product(s)] heavy product 7 e.g., 3 Li Pb 212 At + 3n : 208 Pb( 7 Li, 3n) 212 At 197 Au + 18 C : 208 Pb( 7 Li, 18 C) 197 Au C. Energetics: Q-values revisited 1. Q = (reactants) (products) = available energy 2. Example: 208 Pb( 7 Li, γ) reaction: Q = ( 208 Pb) + ( 7 Li) ( 215 At) (γ, ) Q = ( 1.263) 0 Q = MeV ENDOTHERMIC Accelerate 7 Li to MeV, no reaction. WHY?

4 II. Energetic Conditions for Reaction A. Energetic Threshold: E th & Excitation Energy: E* 1. DEFINITION: E th is the minimum projectile energy necessary to satisfy mass-energy and momentum conservation (i.e., compensate for Q) 2. Derivation: a. Apply conservation Laws projectile + target composite nucleus products 2 p E = ; p = 2m 2 2mE Mass-energy: E p + p + t = CN + E CN + E* Linear Momentum: E p = projectile E T = 0 kinetic energy E* = Excitation energy; When E* = 0 E p = E th pp + pt = pcn 2M p E p 0 2MCNE CN OR M p E p = M CN E CN A p E p = A CN E CN

5 b. Combining E p + p + t = CN + (A p /A CN )E p + E* E p (A p /A CN )E p = CN p T + E* E p (1 A p /A CN) = E* [ p + T CN ] A A E CN p A CN p = E* Q E p (A T /A CN ) = E* Q IF E* = 0 (minimum), then E p = E th and E th = (A CN /A T )( Q) IF E* > E th E* = (A T /A CN ) E p + Q c. Example: 208 Pb( 7 Li, γ) 215 At ; Q = MeV E th = Q A A CN T = ( MeV)(215/108) = MeV E CN = E th E p = = MeV i.e., of total of MeV, MeV goes into mass (M = E/c 2 ) and goes into kinetic energy of the composite nucleus.

6 d. Example: E p = 40.0 MeV, 208 E* = 40.0 MeV = 33.1 MeV 215 This energy is converted into heat Fermi Gas model: T E */A e. Accelerate to MeV Still no reaction. WHY? B. Coulomb Barrier Charged Projectiles Only 1. Nucleus-Nucleus Charge Repulsion 1 MeV K (kt) V coul = (Z e)(z e) d = Z Z e R + R p T p T p 2 T 2. Potential Energy Surface Nuclear Reactions begin to occur when tails of nuclear matter distributions overlap Bottom line: r 0 for reactions is greater than for potential well.

7 3. Net Result V 1.44 Z Z MeV fm 0.90 Z Z p T = r ( A + A ) ( A + A ) cm p T Coul = 0 p 1/3 T 1/3 p 1/3 T 1/3 MeV r 0 = 1.60 fm 4. Momentum Conservation CN must carry off some kinetic energy; same correction as for Mass-Energy Conservation (Q-value) V lab Coul = A A CN T V cm Coul 5. Example 208 Pb( 7 Li, γ) 215 At V lab Coul = (82)(3)(0.9) 1/3 ( / 3 ) = 29.1 MeV NOW THINGS HAPPEN 6. Diffuse nuclear surface fuzzes precision of Coulomb barrier E th is an EXACT condition V Coul is an APPROXIMATE condition

From the scattering angle and momentum, one can calculate the impact parameter and closest approach to the target

8 C. Centrifugal Barrier NOT A MINIMUM CONDITION Alpha Scattering Geometry The scattering of the alpha particle by the central repulsive Coulomb force leads to a hyperbolic trajectory. From the scattering angle and momentum, one can calculate the impact parameter and closest approach to the target nucleus. Rotational energy = ( + 1) 2I 2 where l = mvb and I = µr 2 where µ is the reduced mass of the system. The rotational energy is not available for reaction!

9 D. Summary of Energetic Factors 1. E p E th : Mass-Energy Conservation: ABSOLUTE CONDITION 1 st law lab 2. E p V Coul : Charge Repulsion Constraint BARRIER PENTRATION Probability low below this energy 3. E p E rot : No constraint since = 0 is always possible. 4. IN GENERAL V Coul > E th ; except for very light nuclei and neutral projectiles, e.g., neutrons

10 III. Reaction Probability: The Second-Order Rate Law Probability σ = cross section [SAME FOR CHEMICAL Rx] A. Schematic Picture 1. ON-OFF nature of nuclear force suggests a simple geometric model: Touching Spheres Model IF projectile and target touch, REACTION ; b R p + R T IF projectile and target don't touch, NO REACTION; b > R p + R T 2. Bottom Line: Probability is proportional to cross-sectional area Area = π(r P + R T ) 2 = σ b T σ R = π r 0 2 (A p 1/3 + A T 1/3 ) 2 r fm TOTAL REACTION CROSS SECTION 3. Unit: 1 barn = cm 2 = 1b

11 B. Definitions 1. Sequential Process COLLISION STAGE DECAY STAGE (entrance channel) (exit channel) τ ~ Mass-Energy s τ ~ s σ= REACTION CROSS SECTION Equilibration σ(a,b)= PRODUCTION CROSS SECTION Collision Probability Probability of forming a given product (many may be possible) 2. σ R = σ(a,b) i.e., Sum of all possible production σ's equals the total reaction σ 3. Example 40-MeV 4 He Th 236 U* products 234 Pa σ (α, pn) = b 91 σ (α, 2n) = b 234 U σ (α, 3n) = b 233 U σ (α, 4n) = b 232 U σ (α, f) = b fission σ R = b = σ (α, x)

12 C. Cross Section Measurements: Nuclear Reaction Rates 1. Review of Biomolecular Rate Law A + B C + D [Elementary Reaction only kind for nuclei] Rate = d [A] = k [A][B] dt i.e., a second-order rate process a. [A][B] factor: Collision Probability Defines collision geometry ; e.g., two gases or two liquids, molecular beam + gas, etc b. k = rate constant = probability of reaction IF collision occurs. k = f ( H a, T, structure, etc.) 2. Nuclear Case: Charged-Particle-Induced Reactions k = σ ; [A][B] = n p n T (projectile nuclei target nuclei); definition of n is geometry dependent R = Rate = Number of Product nuclei/time σ(a, b) OR Number of Reactions/time σ total σ = cross section must be given

13 3. Thick Target: General Case For thick targets, nuclear reactions remove a significant fraction of the beam: I 0 I I α n p = number of projectiles/time I 0 I = rate of reaction = di/dx x = target thickness 0 x R = ( di/dx) = σ n t I dx, where n t = ρ N 0 /(g at. wt.) i.e., number density (N/cm 3 ) I = I 0 e nσx Beer-Lambert Law 4. Thin Target Case GEOMETRY: Beam + infinitely thin target [thin = no shadowed nuclei] Beam Target = fixed (i.e., v = 0); Area of target > Area of beam N p N T Collision Probability: (N p /t)(n T /area) Reaction probability: σ = f(q, V c, Iπ, etc.) Thin target Result: Rate = Iσ(n t x)

14 a. Projectile Beam Target Faraday Cup To count projectiles, measure current i collected in Faraday cup I = i / particle charge q e.g., 10nA 12 C beam (stripped of electrons) 9 1nA = 6.28x10 e / s 10 10nA = 6.28x10 e / s However, each 12 C has a charge of +6. # 12 C/s = 6.28x10 6e / e C / s = 1.05x10 I is measured in Amps, since charged particles q = Z p e (ion charge) C/s = I 2 b. (n t x) = # t arg etatoms / cm = density x thickness c. TOTAL RATE R = Number of Reactions/unit time R = Iσ(n t x)

15 Problem: What is the production rate of 106 Sg if a 100 µg/cm 2 target of Cm is bombarded with a 1.0 µa beam of 22 Ne ions? σ ( 22 Ne, 4n) = 1.0 nb R = Iσ(n t x) σ= ( cm 2 )(10 9 ) = cm 2 (n t x) = ( )g ( atoms/mole) = atoms/cm g/mole cm 2 I = 1.0 µa e /µa / (10 e /Ne) = Ne/s R = ( cm 2 ) R = /s = 0.54/hr cm 17 ( /s). σ ( 22 Ne, f) = 2.5 b R = /s NOTE: 2 fragments/fission i.e., humongous fission background

Lecture 15: Nuclear Reactions. RELEVANCE: Basic Research reaction mechanisms structure exotic nuclei

Lecture 15: Nuclear Reactions. RELEVANCE: Basic Research reaction mechanisms structure exotic nuclei Lecture 15: Nuclear Reactions RELEVANCE: Basic Research reaction mechanisms structure exotic nuclei Applications: analytical tool neutron activation analysis astrophysics origin of elements space radiation