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Applications of Nuclear Transformations

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1 Applications of Nuclear Transformations
Basic Concepts Reaction Thresholds Reaction Barriers Nuclear Reactors Fission Fusion Accelerators Van de Grafs Linacs Cyclotrons Cross Sections

2 3.1 Basic Concepts of Nuclear Reactions
Nuclear Reactions => Transformations or Transmutations Processes involving the reaction of one nucleus with another nucleus, or an elementary Particle, or a photon to give one or more nuclei and/or other elementary particle within 10-12 second or less. Many, many reactions possible and many, many products possible. Depends on type and/or energy of bombarding particle. Types of Bombarding Particle: p+ , n ,  , e- , mesons, nuclei (D, T, , etc) Sources of Bombarding Particle: Reactors, Accelerators.

3 3.1 (a) Energetic Comparison of Nuclear & Chemical Reactions
1919 – First nuclear reaction discovered by Rutherford: 14N(,p)17O 1933 – Curie’s: 27Al(,n)30P Others: 139La(12C,4n)147Eu In nuclear reactions, conservation of : A, charge, energy, momentum, ang. momentum, … But not for matter-antimatter annihilation (A not conserved !) Bombarding Particle notation: n , p , d , t ,  , e- ,  ,  ,

4 3.1 (a) Energetic Comparison of Nuclear & Chemical Reactions
Just like chem. Rxn., get E released or absorbed in Nuclear Reactions, but opposite convention to H. Where: Q = energy involved in nuclear reaction. positive => exoergic reaction (E released) negative => endoergic reaction (E required) Q is defined per nucleus transformed (single process) compare with H which is energy per mole of reaction (multi-atom process). For: 14N(,p)17O Q = MeV per 14M-atom transformed. = x10-6 erg per … = -4.57x10-14 cal per … = x10-13 J per … ( 1 eV = x10-19 J ; 1 mol atoms = 6.022x1023 atoms ) Q = -1.15x1011 J per mol of 14N-atoms ~ 105 times larger than the largest Ea (~103 kJ) of any chemical reactions known! Important Question? Does it only take a 1.2 MeV -particle to strike a 14N-atom at rest to produce the above nuclear reaction?

5 3.1 (b) Q-values and Reaction Thresholds
Answer: Q-value is not the only or correct estimate. Several other Factors: Threshold Energy Coulombic Repulsions Centrifugal Attractions Note: one can perform QM-tunnelling to get final product without reaching Q-value! (i) Q-values Can be obtained by doing a mass spectrographic expt. to get all the masses, e.g. 14N(,p)17O Reactants ( amu ) versus Products ( amu) Q = amu ( MeV/amu ) Q = MeV . Endoergic; MeV required for reaction of nucleus. Note: By measuring Q-values of reactions, one can calculate unknown masses of nuclides.

6 3.1 (b) Q-values and Reaction Thresholds
(ii) Reaction Thresholds Back to Question of whether Q-value of 1.19 MeV of -particle enough to produce reaction? Answer: NO ! Conservation of Momentum requires that 4/18 of Ek() be retained by product nuclides. Consider: p  momentum = mv Ek  kin. E of  = ½mv2 vN = 0 ( 14N at rest ) Ek’  kin. E of product nuclide ( 17O & 1H ) m’  mass of 17O & 1H Only remains available for reaction.

7 3.1 (b) Q-values and Reaction Thresholds
(ii) Reaction Thresholds Define: QTh = Threshold Energy  min E from momentum conservation Best situation is when a light particle strikes a heavy stationary target, since minimal E amount is retained in products, meaning more E left for nucl. rxn. Worst is when heavy particle strikes light stationary target.

8 3.1 (c) Nuclear Reaction Barriers
Besides momentum conservation leading to QTh > Q is the fact that Coulombic repulsion between (  and 14N ) nucleons creates so called “Barrier”. For example, Repulsion increases with decreasing distance until  reaches nuclear force field of 14N-nucleus, this fringe field is referred as a “Potential Barrier” Two General Types: Square-Well Potential versus Woods-Saxon Potential r = distance from center of nucleus R = distance at which V=Vo/2 Vo = potential at center of nucleus a = constant ~ 0.5 fm Other nuclides have similar Potential Barriers that can be calculated/simulated.

9 3.1 (c) Nuclear Reaction Barriers

10 3.1 (c) Nuclear Reaction Barriers
Define Potential Barrier: VC = coulombic repulsive potential (Barrier Height) Example: for a spherical nucleus approaching a square well potential, its VC is : R = effective nuclear reaction size ( R ~ roA1/3 , ro ~ 1.1 – 1.6 fm )

11 3.1 (c) Nuclear Reaction Barriers
Example: 14N(,p)17O reaction VC = 3.36 MeV Classically,  must have at least 18/14(3.36) = 4.4 MeV of kinetic energy to enter 14N nucleus and produce (,p) reaction even though QTh is only 1.53 MeV.

12 3.1 (c) Nuclear Reaction Barriers
For nuclear target of heaviest elements: VC for protons and deutrons ~ 12 MeV VC for -particle ~ 25 MeV For nuclear reactions involving heavy elements, must have machines that are capable of overcoming above VC ; so must be capable of generating/accelerating charged particles to energies of ~ 5 – 100 MeV Note that although VC is high in penetrating the nucleus; once inside, the particles emitted must also overcome and leak through VC . Therefore High Energy going in, but sill quite high energy coming out! (Charged particles emitted from nucleus is typically ~ > 1 MeV

13 3.1 (d) Barrier Penetrability & Estimation of Nuclear Size
Recall that -particles were used in Rutherford’s Gold-foil Scattering Experiment, but they are produced in nature by “Radioactivity” For example: (Kinetic Energy  T) 212Po   Pb T = 8.7 MeV ; t1/2 = 3x10-7 s 238U   Th T = 4.1 MeV ; t1/2 = 4.5x109 yr Q? Why is there such an enormous difference in t1/2 ? Barrier Penetrability Think of  having to penetrate a barrier before radioactivity emitted !

14 3.1 (d) Barrier Penetrability & Estimation of Nuclear Size
In fact, decay constant (  ) is proportional to the penetrability of this barrier. For example, for a square barrier: Sources of -particles with kinetic energies: T ~ 4 – 8 MeV

15 3.1 (d) Barrier Penetrability & Estimation of Nuclear Size
Au   Au (Elastic “Rutherford Scattering”) + 14N  17O p+ (1919 – discovery of proton) + 9Be  12C n (1932 – discovery of neutron) Fundamental difference among these reactions is the height of the barrier ( VB ). VB consists of two parts.

16 3.1 (d) Useful Units 1 f = 1 “fermi” = 10-13 cm = 10-15 m
1 MeV = 106 eV = 1.602x10-6 erg e2 = 1.44 MeV f ħ e = 197 MeV f

17 3.1 (d) Barrier Penetrability & Estimation of Nuclear Size
(ii) Estimation of Nuclear Size Classically: from E = T + V using “Classical Turning Point” and define “D” as “distance of closest approach” or “upper limit for size of nucleus. For gold foil scattering experiment: D ~ 38 f for Au.

18 3.1 (d) Barrier Penetrability & Estimation of Nuclear Size
(ii) Estimation of Nuclear Size More Modern estimates are performed using higher energy scattering experiments giving better and/or true nuclear sizes. Some can parametrize nuclear surface as: RN = roA1/3 where: ro => 1.1 to 1.3 f A = Z + N For example: 197Au

19 3.1 (e) Neutron Penetrability
Neutrons has no charge, therefore no coulombic repulsion. Found that even with neutrons of very low energy, one can easily react with the heaviest nuclei. 1934 (Fermi & co-workers): used neutrons to irradiate Ag (induced radiation) Neutron Activation Analysis Moderation easily using paraffin blocks Types of Neutrons for Different Targets: (i) Thermal neutrons ~ eV ; high reaction probability. (ii) Epithermal (resonance) neutrons ~ 1 keV . (iii) Fast neutrons ~ > 1 keV .

20 3.1 (f) Cross Sections  cross section = probability of a nuclear process ( cm2 ) Define: 1 barn ( b ) = cm2 Note: radii of heaviest nuclei ~ cm “big as a barn” For a beam of particles striking a thin target : Where: R = number of product nuclides produced per unit time I = number of incident (bombarding) particles per unit time n = number of target nuclei per cm2 of target x = target thickness  = cross section for process in cm2 often (nx) is given in weight per cm2

21 3.1 (f) Cross Sections Example: Consider a one hour bombardment of thin foil of 55Mn(s) having 10 mg/cm2 thickness using a 1A beam of 35 MeV -particles. Given (,2n) reaction = 200 mb, calculate (i) number of 57Co produced in 1 hour ; (ii) disintegration rate of 57Co ( t1/2 = 270 days.

22 3.1 (g) Kinetics of Nuclear Transformation & Radioactive Decay
Two cases: (i) bombarding of a stable target (ii) bombarding of a radioactive target (i) Stable Targets 𝑅= 𝜆∙𝑁 1− 𝑒 −𝜆∙𝑡

23 3.1 (g) Kinetics of Nuclear Transformation & Radioactive Decay
RA Targets om a high-flux neutron reactor Two types of decays: (a) Natural RA decay (b) Decay via Transmutation 𝑁= 𝑁 𝑜 ∙ 𝑒 −Λ∙𝑡 where: Λ=𝜆+𝑓∙𝜎 𝑑𝑁 𝑖+1 𝑑𝑡 = 𝜆 𝑖 ∙ 𝑁 𝑖 + Λ 𝑖+1 ∙ 𝑁 𝑖+1

24 3.2 Basic Concepts of Nuclear Fission
Fission  break up of a heavy nucleus into two approximately same size nuclides with release of tremendous energy. ~200 MeV per fission event (i.e./ 1 atom of 238U + 1n  )

25 3.2 Basic Concepts of Nuclear Fission

26 3.2 Basic Concepts of Nuclear Fission
Historical Late 1930’s: Uranium-235 fission discovered first by Enrico Fermi & co-workers in Rome and shortly after by Otto Hahn & co-workers in Berlin. August 1939: Albert Einstein wrote letter to President Franklin D. Roosvelt which outlined the possible military applications of nuclear fission and emphasized the danger that weapons based on fission would pose if they were developed by the Nazis. Late 1941: Decision made by the U.S. to build a bomb based on the fission reaction, which began the Manhattan Project. Dec. 2, 1942: Fermi & co-workers, in an abandoned squash court (Stagg Field West Stands) at the University of Chicago, achieved the first artificial self-sustaining nuclear fission chain reaction. [Pile of graphite blocks stacked layer by layer to form an appro. spherical assembly of aobut 40 tons of normal Uranium (metals and oxides) arranged cubically and imbedded in about 385 tons of concrete. Power level ~few kW (no cooling!).

27 3.2 Basic Concepts of Nuclear Fission
Historical July 1945: First Atomic Bomb, developed at Los Alamos National Laboratory in New Mexico; then tested July 16 on the Alamogordo test range. August 1945: the U.S. dropped atomic bombs on two Japanese cities, Hiroshima and Nagasaki.

28 3.2 Basic Concepts of Nuclear Fission
Historical 1972: The Oklo Phenomena. Team of French Scientist discovered Natural Fission Reactors (about 16 zones) had occurred in the Oklo Mines in Gabon, West Africa. Occurred about 2x109 years ago and lasted for about 7x105 years, with ~20 kW. An illustration of the reactor is shown in Figure 1 (from Cowan, 1976).

29 Avg # n’s per fission event
3.2 (a) Nuclear Fission Reactions Many, many possible reactions! Shown above are just two. 235U , a fissile nuclide, is ~0.72% of natural uranium. Q ~ 3.5x10-11 J per event ~ 2.1x1013 J per mole 235U ~ 218 MeV per event More than 200 isotopes of 35 elements have been found among the 235U fission products. Most of them radioactive. Isotope Thermal n’s Fast n’s Avg # n’s per fission event f (barns) 235U 580 1.44 2.42 239Pu 742 1.78 2.88 (232Th ->) 233U 531 2.20 2.49

30 3.2 (a) Nuclear Fission Reactions
Slow-moving (or thermal) neutrons are required in fission because the process requires initial absorption of neutron by nucleus, which excites the nucleus and deforms it. For 235U, about 85 percent of the time the deformed nucleus becomes unstable and splits into two fragments of unequal size (fission). Fast neutrons tend to bounce off nucleus and little fission occurs. Other particles ( d , p+ ,  ) can also induce fission, but their’s must overcome charged-particle Coulomb barrier. Special note: 238U not fissionable by thermal n’s; only by fast neutrons. Other important points: Thermal neutron cross-sections ( n ) Isotope n (barns) 1H 0.33 2H 0.0005 113Cd 27.00 114Cd 0.1 235U 580

31 3.2 (b) Chain Reaction System
Trick is that for each n used in neutron-induced reaction, gets one or more neutrons produced and these go on to induce more fissions => “self-sustaining” chain reaction.

32 3.2 (b) Chain Reaction System
Define: k  “multiplication factor” If k < 1 : “subcritical”, no sustaining reaction. If k = 1 : “critical”, steady state of n-fission, n from each fission on average cause another fission. If k >1 : “supercritical”, divergent chain process, “Bombs Away”!

33 3.2 (b) Chain Reaction System
(i) Equation for Neutron Generation Let: N = # of neutrons at time t No = # of neutrons at t=0  = average time between successive n-generations 𝑑𝑁 𝑑𝑡 = 𝑁(𝑘−1)  𝑁=𝑁𝑜∙ 𝑒 𝑘−1 ∙𝑡 𝜏 Example: The Atomic Bomb (Fission Bomb) 1945  is small & k >1 ; rapid heat, “Boom” ! Sudden combination of two subcritial masses of fissionable material to form a supercritical mass, giving enormous explosive intensity! (~20 kton) (ii) Delayed Neutrons Due to many excited states, get some called delayed neutrons. Their effect is to lenghthen  , so chain reaction can be controlled.

34 3.2 (c) General Reactor Design
Assembly of fissile material: 235U (~0.7% natural), enriched to 3-5%. 239Pu or 233U Arranged in some fashion that produces a controlled self-sustaining chain reaction environment. Normally, k =1 for steady state operation; but reaction is designed such that k ~ 1.01 or 1.02 so as to bring up neutron flux or power level.

35 3.2 (c) General Reactor Design
(i) Control-Rods Cd , B , Hf ( large n ) Moved in & out of reactor core to control reactor power level Control neutron flux level to prevent core from overheating and keep reaction chain self-sustaining Example:  ~ 10-3 s and k = 1.001 N = 2x104 in 10 s ; Too Fast! (“out of control”) => need moderator also! 𝑁=𝑁𝑜∙ 𝑒 𝑘−1 ∙𝑡 𝜏

36 3.2 (c) General Reactor Design
Moderators Graphite , H2O , D2O , Be , BeO , or organic compounds Used to slow down neutrons so they can be capture by 235U again before escaping. (iii) Coolant (Primary and Secondary) Air , He , CO2 , H2O , D2O , liquid metal (Na) (iv) Critical Size Reactor rod dimensions ~ 1.4 cm dia & 5.5 m long Either embedded in water or graphite or heterogeneous arrangement as rods

37 3.2 (c) General Reactor Design

38 3.2 (c) General Reactor Design

39 3.2 (d) Types of Reactors 1943: (i) Oak Ridge, Tennessee – “X-10 Reactor” air-cooled, 1000 kW graphite-U-reactor. (ii) Hanford, Washington water-cooled, graphite-moderated-Pu-production reactor. For ~10-20 years, only used reactors as research tool (wasted heat!). 1977: 224 operating, 21 countries. 68 in U.S. at 49,000 MW US ~16% electrical energy via nuclear France ~75% (49 operating +15 building)

40 3.2 (d) Types of Reactors Classification
Based on – speed or energy of neutrons fuel type moderators coolant (ii) Reactors for Electrical Power Generation PWR BWR PHWR

41 3.2 (d) Types of Reactors (iii) Convertors
Makes use of fertile nuclides to produce fissile products (iv) Breeders Produce more fissile products than consumed.

42 3.2 (d) Types of Reactors (v) Reactors for Propulson
Due to large shielding requirement: mainly ships and satellites Nuclear Power Submarines Extend submerge periods (limitless!) Faster speed, longer range (~600,000 km) PWR mainly (vi) Natural Reactors Oklo Mines: Gabon, W. Africa Underground PWR conditions ~ 2 billion years ago 233U ~ 3% (slightly enriched) ran for ~ (6-8)x105 yr ~ 20 kW Still has remnants; So what does this mean in terms of Waste !

43 3.2 (e) Problems of Fission Reactors
Reactor core cannot reach supercritical levels (k is too small) because concentration of fissile material ( 235U ) is too low (i.e. no fission bomb!) However, “Melt-Down” (overheat of reactor core) can lead to release of radioactive materials into the environment. Radiation Hazard: coolant contamination. Thermal Pollution: Cooling Towers. Hazard Control: Careful design, safety features, highly trained & qualified operators, interlocks, controls, etc. Containment Building: coolant loss & core melt-down. Nuclear Material Safeguard: theft of fissile material => terroristic threats! Radioactive Wastes Storage/Disposal: Underground storage => leaks?!

44 3.2 (f) Fission Fuel Generation
Uranium Refinement: on-site to reduce cost! ( UO2 , U3O8 ) pitch blend Ores (~3%) Refining Process Chemical Equation (ii) Isotope Enrichment: Gaseous diffusion Centrifugation Laser Separation (iii) Fuel Elements: export by France (iv) Fuel Reprocessing: expensive!

45 3.3 Nuclear Fusion Combination of two lighter elements.
In terms of grams, even more powerful than fission! Examples: Stars, Sun (73% H , 26% He, 1% others); H-bomb (1st test 1952) Difficulty in controlling fusion releasing power!

46 3.3 (a) Fusion Reactions

47 3.3 (a) Fusion Reactions

48 3.3 (b) CTR Requirments Major Problem: need great acceleration of these charged species up to energies to overcome Couloumbic barrier! 108 K -> high-density gases of charged particles ( “Plasmas” ) Goals of CTR: (i) Achieve Plasma T (ii) Keep Plasma long enough to produce useful energy from nuclear reaction (iii) Energy produced > Energy input

49 3.3 (c) Confinement Methods
Magnetic Confinement 108 K must be contained by magnetic field Linear Toroidal Superconducting Magnets Tokamak Inertial Confinement Fusion Laser Fusion

50 3.4 Particle Accelerators
Particle: electron, proton (realm of high E, Interm. E Physics) at lower energies: d , T , … 238U Define: Particle  “anything” that can be accelerated Note: (  , hv ) cannot be accelerated. Why need accelerators? search for “elementary particles”, of which of the most fundamental quantity is mass ! In order to “see” particles of smaller dimensions, must go to higher and higher energies. (Quarks ~ GeV) Basically 3 kinds of accelerators (i) Van de Graaff’s (ii) Linac’s (iii) Cyclotrons : only natural -emitters ( ~ 6-7 MeV )

51 3.4 (a) Electrostatic Generators
Application of direct voltage between two terminals 1932: Cockcroft & Walton, accelerated protons. Van de Graaff’s Static machine Accelerate ions (+ve and –ve) using electrostatic generators Builds up high potential, continuous transfer from belt to conducting surface and back. Limit: voltage breakdown. Accelerating Tubes - high voltage - system of accelerating electrodes - target at end of tube - well focused beams ( ~ 0.1 cm2 ) Tandem Van de Graaff’s ~ 1956 - 2 stage ~25 MeV protons - 3 stage ~45 MeV protons - Great stability, excellent E-resolution, cont. E-varaiability, wide choice of ion- beams (iv) Febetron: charged up (parallel) capacitor then discharge

52 3.4 (b) Linear Accelerators (Linac’s)
So far, with electrostatic accelerators, the full high potential corresponding to the final E of ions must be provided, limitation due to insulation problems. Therefore, to solve the insulation problems, use repeated small potential difference accelerations. Must choose proper synchronization at each phase at each electrode Use magnetic focusing lens. Each time the particle emerges from a drift section, it is accelerated

53 3.4 (b) Linear Accelerators (Linac’s)
Electron-Linac’s small mass, at low frequencies, can travel at ~ c. Largest SLAC (Stanford) 2 miles long 20 GeV Pulsed machine (360 pulses per sec, peak current ~ 50 mA) Typical 102 – 103 MeV, electron scattering research Improvements: increased intensity Higher duty cycle Better energy definition Applications: (low E linac’s) Photonuclear reactions Radiation therapy Industrial radiation processing

54 3.4 (b) Linear Accelerators (Linac’s)
(ii) Proton Linac’s Standing wave acceleration, mass increases markedly with increasing velocity Series of drift-tubes of increasing length Focus via Quadruple lenses  200 MeV Serves MeV p+ injector into p+ synchrotrons Largest: LAMPF (Los Alamos, New Mexico) 600 yards long, 800 MeV protons (iii) HILAC’s (heavy ion) Two sections: first section accel. Up to charged state ( ~ 1.5 MeV/amu ) second section accel. From charged state to final E. SUPERHILAC (LBL): C , N , O , Ne , Xe ( ~ 8.3 MeV/amu ) UNILAC (Germany): any up to U ( ~ 10 MeV/amu ) Electron-Linac’s are preferred over Cyclotrons since no less of electron energy due to Bremstrahlung. Proton-Linac’s are favored over proton-Linac’s mainly due to economic design problems.

55 3.4 (c) Cyclotrons Best known, most successful of all accelerating devices for positive ions First (1930, Lawrence) – 80 keV protons Use Magnetic field to bend beam in circles, get spiral path of ions. Electrodes  Dees Circular motion opposed by centrifugal force 𝑞𝑣𝐵 𝑐 = 𝑚 𝑣 2 𝑟 𝑇= 1 2 𝑚 𝑣 2 = 𝑞 2 𝐵 2 𝑟 2 2𝑚 𝑐 2 Cyclotron ( r = 44” ) Tp =~ 50 MeV TRIUMF ( r = 20 ft ) Tp ~ 500 MeV NAL ( r ~ 1 km ) Tp ~ 500 GeV

56 m= 𝑚 𝑜 1− 𝑣 2 𝑐 2 3.4 (c) Cyclotrons
Must deal with relativistic mass increase, recall: m= 𝑚 𝑜 1− 𝑣 2 𝑐 2 𝜈= 𝑞𝐵 2𝜋𝑚𝑐 Three possibilities: (i) Raise B (sector focused) -> cyclotron (ii) Lower frequency (FM) -> synchocyclotron (iii) Fix r , raise both B and freq -> synchrotron

57 3.4 (d) Photon & Neutron Sources
Photon Sources comes from secondary beams in electron accelerators i.e./ 3H ( p ,  ) 4He ~19.8 MeV Bremstrahlung Continuous X=rays when electrons are decelerated in Coulumbic fields of atomic nuclei (slowing-down radiation) Synchrotron Radiation Continuous spectrum of EM radiation emitted when relativistic electrons are bent in magnetic field Applications: solid state physics, radiation chemistry, photo e-spectroscopy, X-ray crystallography Neutron Sources Many produced from: Reactors; Accelerators, RA Sources

58 Applications of Nuclear Transformations
Basic Concepts Reaction Thresholds Reaction Barriers Nuclear Reactors Fission Fusion Accelerators Van de Grafs Linacs Cyclotrons Cross Sections

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