Ch1-Nuclear Reaction PDF

Chapter 1 Nuclear Reaction Nuclear Physics 2 PHY3602  when Tow parts Introduction to Nuclear Reactions - interact to Nuclear reactions invo...

Chapter 1 Nuclear Reaction Nuclear Physics 2 PHY3602  when Tow parts Introduction to Nuclear Reactions - interact to Nuclear reactions involve changes within the nucleus of an atom, often - eachother Produce New to elements transforming elements. If energetic particles from a reactor or accelerator (or even from a radioactive source) are allowed to fall upon bulk matter, there is the possibility of a nuclear reaction taking place. The first such nuclear reactions were done in Rutherford’s laboratory, using a particle from a radioactive source that produced a phenomenon, : 5I 1591969) known ever since as Rutherford scattering, gave us the first evidence - for the existence of atomic nuclei. X + Y Y  Nuclear Reaction I I 14 A typical nuclear reaction is written as following O Outgoing y x+X Y+ y   Interacts Products Projectile x Target X je gein - Residual Y X (x , y) Y >  Nuclear Reaction The nuclear reaction consists of: Outgoing y 1. The interacts: X , X - a. The projectile x which is a particle has  definite small mass, kinetic energy and  2 direction such as, alpha particle , Projectile x Target X -deuteron d, proton p and neutron n. Sometimes be nucleus of light elements Residual Y b. The target which is a stationary nucleus X. Heavy nucleus A > 150 ,Medium 40 < A < 150, Light nucleus A < 40 2. The Products (yields): Y L , In most the nuclear reactions the products & - consist of two parts, residual nucleus Y and outgoing particle y , so the linear momentum and kinetic energy laws can easy be applied Sometimes the products are three or more parts. -  Ojf Nuclear Reaction Conditions The nuclear reaction must satisfy four Conservation Laws: - 1. Energy conservation: Total energy is conserved. 2. Momentum conservation: Linear and angular momentum are conserved - - - - 3. Charge conservation: Total charge remains constant. & - 4. Atomic mass conservation: Protons and neutrons are conserved - - & &  i COULOMB SCATTERING: Because the nucleus has a distribution of electric charge, it can be studied by the electric (Coulomb) scattering of a beam of charged particles. This scattering may be - - either elastic or inelastic. Elastic Coulomb scattering is- & called Rutherford scattering because early (1911-1913) experiments on the scattering of a particles in Rutherford’s laboratory by Geiger and Marsden led originally to the discovery of the existence of the nucleus.  Types of nuclear reactions: - 1- Scattering: & - & in this reaction, the projectile particle x and the outgoing particle y are the same and the target nucleus is the residual nucleus - There are two cases: a- Elastic scattering: - &i Kinetic energy Aconserved. Example: Rutherford scattering. The target nucleus in the ground state 𝑿 𝒙, 𝒙 𝑿 b- Inelastic - scattering: dig g Energy excites the nucleus. Example: Neutron scattering. The target nucleus in the excited state Origi 𝑿 𝒙, 𝒙 𝑿∗ 4- Compound reaction In this reaction, the projectile particle and target nucleus form a compound nucleus for short time interval (about 10 -16sec). The compound nucleus decays by more than one way and no rule controls these ways, such as:- -N 𝟔𝟑 𝟏z 𝟑𝟎𝒁𝒏 + 𝟎𝒏 & + & -N O ∗ 𝒑 + 𝟔𝟑 𝟐𝟗𝑪𝒖 ⟶ 𝟔𝟒 𝟑𝟎𝒁𝒏 𝟔𝟐# 𝟑𝟎𝒁𝒏 + 𝟏𝟎𝒏 + 𝟏𝟎𝒏 - xX - Y 𝟔𝟐 𝟐𝟗𝑪𝒖 + 𝟏𝟎𝒏 + 𝒑 - & -- 5- Others reactions such as Fusion when Light nuclei combine and Fission when Heavy & & nucleus splits. - Types of nuclear reactions Epis M X X X - assis ↳ O - - X  = ENERGETICS OF NUCLEAR REACTIONS Q=Reaction Energy x+X Y+ y T= kinetic energy Interacts Products Outgoing y 𝑻𝒕𝒉=threshold energy  M= mass  Projectile x Target X m Residual Y V= velocity 𝒗  ENERGETICS OF NUCLEAR REACTIONS The Reaction Energy Q - The - energy is liberated or required for nuclear reaction can be calculated from mass- energy relation: -𝑸 = 𝑴𝑿 + 𝒎𝒙 − 𝑴𝒀 + 𝒎𝒚 𝒖 From conservation energy: (let TX =0 for target) - 𝑸+𝑻 =𝑻 +𝑻 𝒙 𝒀 𝒚 Where Tx , TY and Ty are the kinetic energy of projectile, residual and outgoing 5 - The reaction is exothermal If Q > 0 and the energy is divided between the - outgoing particle and residual nucleus depending on the outgoing particle angle  & · Div z - The reaction is endothermal If Q < 0 and the energy is required (threshold energy) for & the target nucleus to done the reaction from the projectile particle kinetic energy Tx. - Threshold Energy is the Minimum energy required for a reaction.  Exothermal Reaction - 𝑻𝒚 = 𝑸 𝑴𝒀 𝑴𝒀 + 𝒎𝒚 exothermal If Q > 0 𝑨𝒀 ≈ 𝑸 𝑨𝒀 + 𝑨𝒚 Endothermal Reaction wo - 𝑴𝑿 + 𝒎𝒙 𝑻𝒕𝒉 = 𝑸 𝑴𝑿 endothermal If Q < 0 𝑨𝑿 + 𝑨𝒙 ≈ 𝑸 𝑻𝒕𝒉=threshold energy 𝑨𝑿  Ex: Calculate Q for the reaction 𝟏𝟔𝟖𝑶 𝟒𝟐𝜶, 𝟐𝟏𝒅 𝟏𝟖 𝟗𝑭 where MO = 15.994915 u MF = 18.000938 u m = 4.002603 u md = 2.014102 Soln 𝑸 = 𝑴𝑿 + 𝒎𝒙 − 𝑴𝒀 + 𝒎𝒚 𝒖 𝑸 = 𝑴𝑶 + 𝒎 − 𝑴𝑭 + 𝒎𝒅 𝒖 𝑸 = 𝟏𝟓. 𝟗𝟗𝟒𝟗𝟏𝟓 + 𝟒. 𝟎𝟎𝟐𝟔𝟎𝟑 − 𝟏𝟖. 𝟎𝟎𝟎𝟗𝟑𝟖 + 𝟐. 𝟎𝟏𝟒𝟏𝟎𝟐 × 𝟗𝟑𝟏. 𝟓 = −𝟏𝟔. 𝟑𝟐 𝑴𝒆𝑽 This reaction is endothermal, and the threshold energy is: endothermal If Q < 0 𝑨𝑶 +𝑨 𝟏𝟔+𝟒 𝑻𝒕𝒉 =  𝑸= × 𝟏𝟔. 𝟑𝟐 = 𝟐𝟎. 𝟒 𝑴𝒆𝑽 𝑨𝑶 𝟏𝟔  Ex: Calculate Q for the reaction 𝟏𝟎𝟓𝑩 𝜶, 𝒑 𝟏𝟑 𝟔𝑪 where MB = 10.012939 u MC = 13.003354 u m = 4.002603 u mp = 1.007825 u Soln 𝑸 = 𝑴𝑿 + 𝒎𝒙 − 𝑴𝒀 + 𝒎𝒚 𝒖 𝑸 = 𝑴𝑩 + 𝒎𝜶 − 𝑴𝑪 + 𝒎𝒑 𝒖 = 𝟏𝟎. 𝟎𝟏𝟐𝟗𝟑𝟗 + 𝟒. 𝟎𝟎𝟐𝟔𝟎𝟑 − 𝟏𝟑. 𝟎𝟎𝟑𝟑𝟓𝟒 + 𝟏. 𝟎𝟎𝟕𝟖𝟐𝟓 × 𝟗𝟑𝟏. 𝟓 𝑸 = 𝟒. 𝟎𝟔 𝑴𝒆𝑽 𝑨𝑪 𝟏𝟑 𝑻𝒑 = 𝑸 = × 𝟒. 𝟎𝟔 = 𝟑. 𝟕𝟕 𝑴𝒆𝑽 𝑨𝒑 + 𝑨𝑪 𝟏𝟑 + 𝟏 Ex: Calculate Q for the reaction 𝟐𝟎𝟗𝑩𝒊 𝒑, 𝒅 𝟐𝟎𝟖 𝑩𝒊 where MBi209 = 208.980394 u MBi208 = 207.979731 u mp = 1.007825 u md = 2.014102 u Soln 𝑸 = 𝑴𝑩𝒊𝟐𝟎𝟗 + 𝒎𝒑 − 𝑴𝑩𝒊𝟐𝟎𝟖 + 𝒎𝒅 𝒖 = 𝟐𝟎𝟖. 𝟗𝟖𝟎𝟑𝟗𝟒 + 𝟏. 𝟎𝟎𝟕𝟖𝟐𝟓 − 𝟐𝟎𝟕. 𝟗𝟕𝟗𝟕𝟑𝟏 + 𝟐. 𝟎𝟏𝟒𝟏𝟎𝟐 × 𝟗𝟑𝟏. 𝟓 𝑸 = −𝟓. 𝟐𝟑 𝑴𝒆𝑽 This reaction is endothermal, and the threshold energy is: 𝑨𝑩𝒊𝟐𝟎𝟗 + 𝑨𝒑 𝟐𝟎𝟗 + 𝟏 𝑻𝒕𝒉 = 𝑸= × 𝟓. 𝟐𝟑 = 𝟓. 𝟐𝟔 𝑴𝒆𝑽 𝑨𝑩𝒊𝟐𝟎𝟗 𝟐𝟎𝟗  Ex: Calculate Q for the reaction 27Al ( ,  )31P where MAl = 26.981539 u MP = 30.973765 u m = 4.002603 u Soln 𝑸 = 𝑴𝑿 + 𝒎𝒙 − 𝑴𝒀 + 𝒎𝒚 𝒖 𝑸 = 𝑴𝑨𝒍 + 𝒎 − 𝑴𝑷 𝒖 = 𝟐𝟔. 𝟗𝟖𝟏𝟓𝟑𝟗 + 𝟒. 𝟎𝟎𝟐𝟔𝟎𝟑 − 𝟑𝟎. 𝟗𝟕𝟑𝟕𝟔𝟓 × 𝟗𝟑𝟏. 𝟓 𝑸 = 𝟗. 𝟔𝟕 𝑴𝒆𝑽 Ex: Calculate Q for the reaction 15N (d , n )16O where MN = 15.000108 u MO = 15.994915 u md = 2.014102 u mn 1.008665 u Soln 𝑸 = 𝑴𝑿 + 𝒎𝒙 − 𝑴𝒀 + 𝒎𝒚 𝒖 𝑸 = 𝑴𝑵 + 𝒎𝒅 − 𝑴𝑶 − 𝒎𝒏 𝒖 = 𝟏𝟓. 𝟎𝟎𝟎𝟏𝟎𝟖 + 𝟐. 𝟎𝟏𝟒𝟏𝟎𝟐 − 𝟏𝟓. 𝟗𝟗𝟒𝟗𝟏𝟓 − 𝟏. 𝟎𝟎𝟖𝟔𝟔𝟓 × 𝟗𝟑𝟏. 𝟓 𝑸 = 𝟗. 𝟗 𝑴𝒆𝑽 𝑨𝑶 𝟏𝟔 𝑻𝒏 = 𝑸= × 𝟗. 𝟗 = 𝟗. 𝟑𝟐 𝑴𝒆𝑽 𝑨𝒏 + 𝑨𝑶 𝟏𝟔 + 𝟏  Reaction Cross Sections Measures the probability of a nuclear reaction. A beam of incident projectiles may not all of them interact with the target but some of them pass without any reaction. Units: Expressed in barns (1 barn = 10-24 cm2). Dependence: 1. Energy of the incident particle. 2. Properties of the target nucleus. 3. Quantum effects (e.g., resonance).  The Reaction Rate The reaction rate is the No of reactions take place per second and can be calculated from the following: The substance consists of atoms arranged in order. But most the atom is empty, so for projectile particle the distance among nuclei is so much and probability of reaction takes place is slightly. To study probability of reaction suppose a specimen of L substance has surface area AT and thickness L. AT The No. of incident particles on the surface is N0. N0 dx x The study takes a slide at distance x and has thickness dx. N(x) N(x)-dN The No. of non-interacting particles reaches the slide N(x) The No. of interacting particles inside the slide dN The No. of non-interacting particle leaves the slide N(x)-dN.  The probability of reaction takes place in slide 𝒅𝑵Τ𝑵(𝒙) 𝒅𝑵 𝑵𝒐 𝒐𝒇 𝒊𝒏𝒕𝒆𝒓𝒂𝒄𝒕 𝒑𝒂𝒓𝒕𝒊𝒄𝒍𝒆𝒔 𝒂𝒍𝒍 𝒏𝒖𝒄𝒍𝒆𝒊 𝒄𝒓𝒐𝒔𝒔 𝒔𝒆𝒄𝒕𝒊𝒐𝒏 𝒂𝒓𝒆𝒂 = = 𝑵(𝒙) 𝑵𝒐 𝒐𝒇 𝒊𝒏𝒄𝒊𝒅𝒆𝒏𝒕 𝒑𝒂𝒓𝒕𝒊𝒄𝒍𝒆𝒔 𝒕𝒐𝒕𝒂𝒍 𝒂𝒓𝒆𝒂 𝒇𝒂𝒄𝒊𝒏𝒈 𝒕𝒉𝒆 𝒊𝒏𝒄𝒊𝒅𝒆𝒏𝒕 𝒑𝒂𝒓𝒕𝒊𝒄𝒍𝒆𝒔 𝒅𝑵 𝒏𝝈𝑨𝑻 𝒅𝒙 = = (+)𝒏𝝈𝒅𝒙 𝑵(𝒙) 𝑨𝑻 where, n is No of nuclei per unit volume 𝝆𝑵𝑨 𝒏= 𝑨 ( is the substance density) and  is the interaction cross-section 𝝈 = 𝝅𝒓𝟐 Nuclear cross section (or interaction cross section) is that area around the nucleus facing the incident r r particles, within which the incident particles will lead to an interaction with the nucleus By integration from 0 to x to calculate the non-interacting particles in x: 𝑵(𝒙) 𝒙 𝒅𝑵 න = − න 𝒏𝝈𝒅𝒙 𝑵(𝒙) 𝑵𝟎 𝟎 𝑵(𝒙) 𝑵(𝒙) 𝒍𝒏𝑵 𝑵𝟎 = −𝒏𝝈𝒙 𝒍𝒏 = −𝒏𝝈𝒙 𝑵𝟎 𝑵(𝒙) = 𝒆−𝒏𝝈𝒙 𝑵𝟎 𝑵(𝒙) = 𝑵𝟎 𝒆−𝒏𝝈𝒙 The No. of interacting particles from 0 to x ∆𝑵 = 𝑵𝟎 − 𝑵(𝒙) = 𝑵𝟎 − 𝑵𝟎 𝒆−𝒏𝝈𝒙 = 𝑵𝟎 𝟏 − 𝒆−𝒏𝝈𝒙 The reaction rate R: 𝑵𝟎 𝟏 − 𝒆−𝒏𝝈𝒙 𝑹 = 𝚫𝑵Τ𝚫𝒕 = 𝚫𝒕 The No of incident particles N0 per unit time is defined by the flux : 𝑵𝟎 𝜱= Thus 𝚫𝒕 𝑹 = 𝜱 𝟏 − 𝒆−𝒏𝝈𝒙 For all the specimen 𝑹 = 𝜱 𝟏 − 𝒆−𝒏𝝈𝑳 From the exponential expansion 𝒆−𝒚 = 𝟏 − 𝒚 + 𝒚𝟐 − 𝒚𝟑 + ⋯ Thus 𝒆−𝒏𝝈𝑳 ≈ 𝟏 − 𝒏𝝈𝑳 𝑹 = 𝜱 𝟏 − 𝟏 − 𝒏𝝈𝑳 = 𝜱𝒏𝝈𝑳  Ex: Determine the 113Cd absorption ratio to neutrons with thickness 0.1 mm and density 8650kg/m3 and  =20000 b Soln 𝑳 = 𝟏 × 𝟏𝟎−𝟒 𝒎  = 𝟖𝟔𝟓𝟎 𝒌𝒈/𝒎𝟑 𝝈 = 𝟐 × 𝟏𝟎−𝟐𝟒 𝒎𝟐 𝝆𝑵𝑨 𝟖𝟔𝟓𝟎 × 𝟔. 𝟎𝟐𝟐 × 𝟏𝟎𝟐𝟔 𝒏= = = 𝟒. 𝟔𝟏 × 𝟏𝟎𝟐𝟖 𝒏𝒖𝒄𝒍𝒆𝒖𝒔Τ𝒎𝟑 𝑨 𝟏𝟏𝟑 𝒏𝝈𝑳 = 𝟒. 𝟔𝟏 × 𝟏𝟎𝟐𝟖 × 𝟐 × 𝟏𝟎−𝟐𝟒 × 𝟏 × 𝟏𝟎−𝟒 = 𝟗. 𝟐𝟐 𝑹Τ𝜱 = 𝟏 − 𝒆−𝒏𝝈𝑳 = 𝟏 − 𝒆−𝟗.𝟐𝟐 = 𝟎. 𝟗𝟗𝟗𝟗  Experimental Techniques A typical nuclear reaction study requires a beam of particles, a target, and a detection system such as Geiger counters. The Optical Model The optical model is useful only in discussing average behavior in reactions such as scattering. The calculation using the optical model does not deal with where the absorbed particles go; they simply disappear from the elastic channel. The optical model is successful in accounting for elastic and inelastic scattering and leads us to an understanding of the interactions of nuclei. It is similar to cloudy crystal balls, when struck by a beam of particles, they partially absorb the beam, partially scatter it, and partially transmit it in a way analogous to the behaviour of light.  Problems 1. Calculate energy “Q” for the reaction 168𝑂 21𝑑, 42𝛼 14 7𝑁 where; MO = 15.994915 u MN = 14.003074 u m = 4.002603 u and md = 2.014102 u. 12 12 2 22 2. Calculate energy “Q” for the reaction 6𝐶 6𝐶, 1𝑑 11𝑁𝑎 where; MC = 12.000 u, MNa = 21.99443742 u and md = 2.014102 u. 23 1 12 12 3. Calculate energy “Q” for the reaction 11𝑁𝑎 1𝑝, 6𝐶 6𝐶 where; MC = 12.00000 u, MNa = 21.99443742 u and mp = 1.00727647 u. 4. Find the missing numbers “letters A, B, C, D, E, F, G and I”. 𝟏𝟔 𝟒 𝟏 𝐁 𝐀𝐍 𝟐𝛂, 𝟏𝐩 𝟖𝐎 𝟑 𝟏 𝟏 𝐃 𝟏𝐇 𝟏𝐇, 𝐂𝐇 𝟏𝐇 𝐄 𝟏𝟐 𝟒 𝟐𝟎 𝟔𝐂 𝐅𝐂, 𝟐𝛂 𝟏𝟎𝐍𝐞 𝟐 𝐆 𝟏 𝟑 𝟏𝐇 𝟏𝐇, 𝐈𝐧 𝟐𝐇𝐞