Lecture notes on Nuclear Reactions

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Introduction to nuclear reactions Introduction to Nuclear Science Simon Fraser University Spring 2011 NUCS 342 March 14, 2011 NUCS 342 (Lecture 21) March 14, 2011 1 / 322 The energetics 3 Cross section 4 Neutron-induced reactions 5 Charged-particle induced reactions 6 Tunneling Outline 1 Notation NUCS 342 (Lecture 21) March 14, 2011 2 / 323 Cross section 4 Neutron-induced reactions 5 Charged-particle induced reactions 6 Tunneling Outline 1 Notation 2 The energetics NUCS 342 (Lecture 21) March 14, 2011 2 / 324 Neutron-induced reactions 5 Charged-particle induced reactions 6 Tunneling Outline 1 Notation 2 The energetics 3 Cross section NUCS 342 (Lecture 21) March 14, 2011 2 / 325 Charged-particle induced reactions 6 Tunneling Outline 1 Notation 2 The energetics 3 Cross section 4 Neutron-induced reactions NUCS 342 (Lecture 21) March 14, 2011 2 / 326 Tunneling Outline 1 Notation 2 The energetics 3 Cross section 4 Neutron-induced reactions 5 Charged-particle induced reactions NUCS 342 (Lecture 21) March 14, 2011 2 / 32Outline 1 Notation 2 The energetics 3 Cross section 4 Neutron-induced reactions 5 Charged-particle induced reactions 6 Tunneling NUCS 342 (Lecture 21) March 14, 2011 2 / 32Notation Binary nuclear reactions The binary reaction between the projectile a and target nucleus A producing a nucleus B and outgoing particle b a +A B +b (1) is abbreviated as A(a;b)B. The entrance channel is a +A. The exit channel is B +b. NUCS 342 (Lecture 21) March 14, 2011 3 / 32The energetics TheQ-value The heat or Q-value for the reaction is the energy released in the reaction, which can be calculated from the mass di erence between the entrance and exit channels: 2 Q = (M +M M M )c : (2) n n;a n;A n;B n;b Exothermic reactions release heat and have Q 0. Endothermic reactions have Q 0. Endothermic reaction require heat above the threshold value of Q to proceed. This heat comes from the kinetic energy of the projectile. NUCS 342 (Lecture 21) March 14, 2011 4 / 32The energetics TheQ-value The Q value can be extracted from the atomic masses: 2 Q = Q +(Z +Z Z Z )m c +B (Z )+B (Z )B (Z )B (Z ): a n a A B b e e a e A e B e b (3) If all the exit channel charge is in the produced nuclei charge conservation requires Z +Z Z Z = 0. a A B b Therefore Q = Q + B : (4) a n e The di erence between electron binding in the entrance and exit channel is usually small compared to the nuclear Q value: n B = +B (Z ) +B (Z )B (Z )B (Z ) Q : (5) e e a e A e B e b n Therefore, usually Q  Q . a n NUCS 342 (Lecture 21) March 14, 2011 5 / 32The energetics Atomic and nuclearQ-values For reactions involving charged lepton production the atomic Q value is di ers from the nuclear Q value since the condition Z +Z Z Z = 0 is not ful lled. a A B b Let us consider the Hydrogen burning reaction which proceeds via the weak interactions + p +p d +e + : (6) The atomic Q value is 2 Q = (2M M 2 )c = 1:44 MeV: (7) a a;H a; H The nuclear Q value is 2 Q = (2M M )c = 0:93 MeV: (8) n n;p n;d Atomic Q value includes the energy released in annihilation of the positron, while nuclear Q value does not include this energy. NUCS 342 (Lecture 21) March 14, 2011 6 / 32The energetics TheQ-value Atomic masses are usually tabulated in terms of atomic mass excess: 2 M = (MAM )c (9) a u with M being the atomic mass unit: u 1 2 12 2 M c = M( C )c = 931:50 MeV: (10) u 12 Note that M 6= M but M ' M . a n a n M a The ratio  0:1% 1% represents the conversion factor from M a mass to energy in nuclear reactions. NUCS 342 (Lecture 21) March 14, 2011 7 / 32Cross section Q value Q value measures the energy liberated or consumed in a nuclear reaction. Q values are relatively easy to measure and are usually known with high precision from mass measurements. The Q value carry very limited information on reaction rates: for endothermic reactions the rate is zero if the kinetic energy of the projectile is not sucient to overcome the Q value threshold. Other than that, the Q value alone does not allow to predict reaction rates. NUCS 342 (Lecture 21) March 14, 2011 8 / 32Cross section Beer-Lambert law in optics Transmission intensity (number of particles per second) I for a beam 1 of particles traversing a thin target with thickness x depends on the target thickness and initial intensity:   x I = I exp : (11) 1 0 l This also applies to elementary particles passing through thin targets. NUCS 342 (Lecture 21) March 14, 2011 9 / 32Cross section Mean free path Parameter l referred to as the mean free path is the mean distance for particles to be removed from the beam. Intensity decreases due to the beam interactions with atoms or nuclei in the target. Mean free path has to be related to the number of atoms per unit volume, referred to as the number density N. The relation is the inverse proportionality: larger N smaller l. The proportionality constant multiplying N is called the cross section . With this de nition the relation between l, n and  is 1 l = (12) N NUCS 342 (Lecture 21) March 14, 2011 10 / 32Cross section The reaction rate Inserting the cross section to the Beer-Lambert law results in: I = I exp (Nx): (13) 1 0 Number of particles removed from the beam is equal to the number of reactions per second or the reaction rate: R = I I = I (1 exp (Nx)): (14) 0 0 A thin target is a target for which x = Nx 1: (15) l In this case exp (Nx) 1Nx and R = I Nx: (16) 0 NUCS 342 (Lecture 21) March 14, 2011 11 / 32Cross section The cross section The dimension of a cross section is that of an area: " 1 1 2  = ;  = = m : (17) 1 Nl m 3 m The cross section represents an e ective area for interaction of a single target atom. For a in nitesimally thin target Ndx is the number of atoms per target surface area. This arises from a fact that probability of an atom to be screened from the interaction with the beam by another atom is very small. In such a case Ndx is the fraction of atoms interacting with the beam. NUCS 342 (Lecture 21) March 14, 2011 12 / 32Cross section The unit of a cross section 28 2 The unit of a cross section is a barn 1 b= 10 m . From the nuclear interaction points of view the target is mostly 10 empty. Indeed a separation between atoms is 10 m while a 15 nuclear sizes are on the order of 10 m. Consequently nuclear cross sections are small. If a reaction has 1 barn cross section it implies that it is as easy to hit the target nucleus as to hit a barn. NUCS 342 (Lecture 21) March 14, 2011 13 / 32Cross section Geometric approximation If we knew the radii of the target R and the projectile R the t p geometric approximation for the cross section would be an area of a disc with the radius of R = R +R S t p 2 2  =R =(R +R ) : (18) t p s 1 3 With the nuclear radius de nition R = R A and R = 1:3 fm the N 0 0 geometric cross sections are: 1 1  = 0:2 b for H+ H, 1 238  = 2:8 b for H+ U, 238 238  = 4:8 b for U+ U, NUCS 342 (Lecture 21) March 14, 2011 14 / 32Cross section The proper treatment The proper derivation of a reaction cross section includes quantum mechanical treatment of interaction between the target and projectile. There are signi cant discrepancies between the realistic cross sections and geometrical estimates. For example, real cross sections are energy dependent, while the geometric approximation is not. Various fundamental interactions give very di erent scales for typical cross sections. Assuming 2 MeV for the projectile energy + 20 for the weak force and p(p;e )d reaction  10 b, 3 7 6 for the E-M force and He( , ) Be reaction  10 b, 15 12 for the strong force and N(p, ) C reaction  0:5 b, If possible, cross sections are measured rather than derived. NUCS 342 (Lecture 21) March 14, 2011 15 / 32

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