Nuclear Binding Energy

Nuclear Binding Energy...increases almost linearly with A; average B/A about 8 MeV per nucleon nuclei most tightly bound around A=60 below A=60, we can release energy by nuclear fusion above A=60, we can release energy by nuclear fission

Liquid Drop Model...semiempirical mass formula A Z /3 1 / 3 B = a V A as A ac Z Z 1 A a sym A av A volume term a V = 15.56 MeV a S A / 3 surface term a S = 17.3 MeV ac Z Z 1 A 1 / 3 Coulomb term A Z a sym A { a p A a C = 0.697 MeV 1 / asymmetry term a sym = 3.385 MeV N, Z even pairing term a p = 1.00 MeV A odd 0 a p A 1 / N, Z odd

Liquid Drop: Bethe-Weizsäcker Formula A Z /3 1 / 3 B = a V A as A ac Z Z 1 A a sym A

Liquid Drop: Bethe-Weizsäcker Formula A Z /3 1 / 3 B = a V A as A ac Z Z 1 A a sym A

Spontaneous Fission in Liquid Drop Model Q=B A1, Z 1 B A, Z B A, Z Z 1 A1 assuming = = y1 Z A / 3 /3 1 Z A = =y Z A Q=a S A A A /3 y 1 y =1 Z 1 Z Z a C 1 /3 1 /3 1 /3 A A1 A

Spontaneous Fission in Liquid Drop Model maximum energy release: Q Q = =0 y 1 = y y1 y symmetric fission Z /3 Q=0.37 a C 1/ 3 0.6 a S A A e.g. 38 9 U Q 00 MeV ~106 > energy release in chemical reactions!

Fission and Deformation volume term: 4 4 3 a b = R 3 3 constant a S A a S A 1 surface term: 5 Z Z a C 1/ 3 1 Coulomb term: a C 1/ 3 5 A A as /3 Z Binding energy: B=B B 0 =a C A A a 5 C (decrease) /3 /3

Fission and Deformation as Z A ac Nucleus unstable under deformation Z 47 A...which is only the case for nuclei with Z > 114 and A > 70

Potential Barrier remember α decay... instability condition from deformation not sufficient potential barrier Ef (fission activation energy) has to be climbed (maximum at scission point)

Potential Barrier a / 3 Z S E f =ac A A ac 5 Tunneling Spontaneous Fission External energy Induced fission

Spontaneous Fission: Tunneling remember α decay... 1 P exp k b a m 1 m 1 k= B Q = Ef ℏ ℏ tunneling probability depends on Z/A

Spontaneous Fission: Tunneling remember α decay... 1 P exp k b a m 1 m 1 k= B Q = Ef ℏ ℏ k m tunneling probability depends on fragment mass large fragment mass low tunneling probability 6 less probable than α decay) ( 38 : spontaneous fission 10 U 9

Spontaneous Fission: Lifetimes

Induced Fission...occurs when a nucleus captures a low energy neutron receiving enough energy to climb the fission barrier prompt neutrons If excitation energy > fission activation energy, fission occurs for ''zero'' energy neutrons (thermal neutrons)

Fission Activation Energy

Induced Fission If excitation energy < fission activation energy, fission occurs only if neutrons have enough kinetic energy e.g. for n U * U 39 U 9 E f 6 MeV E n 0 E 5 MeV E n =1.4 MeV E=6.4 MeV fission 35 9 N even U no thermal fission N odd energy energy 38 9 38 9 deformation deformation

Uranium Fission Cross Section

Fission Products: Energy Distributions Heavy fragments have approximately equal and opposite momenta: 1 M 1 v 1 M = 1 M1 M v mean neutron energy MeV 5 MeV per fission to neutrons 165 MeV to heavy fragments rest to prompt s, and decays from radioactive fragments ( 1 MeV to )

Reactors Natural heavy nuclei: 3 90 Th 14 34 9 U14 0.055 % 35 9 U 143 0.07 % 38 9 U 146 99.745 % fission induced by thermal neutrons possible for nuclei with odd N nuclei with even N can be bred in a reactor by thermal neutrons; neutron capture leads then to a nucleus with odd N 35 9 U is the only natural fuel for conventional reactors

Fission Fragments masses of fragments unequal tend towards magic numbers for Z, N

Prompt and delayed neutrons fragments tend to have the same Z/N ratio as parent neutron rich nuclei which emit prompt neutrons (10-16 s) β decay of fragments X, Y more slowly delayed neutrons from decay of excited states with E > En (1 delayed n / 100 fissions)

Delayed neutrons Example: highly excited state with E > En (neutron 93 separation energy) in 38Sr Delayed neutrons essential for control of nuclear reactors

Fission timeline

Chain reaction -3 prompt (fast) neutrons for every fission neutron reproduction factor: N n 1 k= Nn Nn+1, Nn,... number of thermal neutrons of generation n+1, n... k =1 pile critical k 1 pile subcritical k 1 pile supercritical

Critical Mass = mass, at which the chain reaction starts by itself Note: not a natural constant, but given by the ratio of surface to volume for an arbitrarily shaped nuclear fuel material Natural heavy nuclei: 3 90 34 9 Th 14 35 9 U14 0.055 % U 143 38 9 U 146 0.07 % 99.745 % Critical masses (for spherical shape using n reflector): 33 9 U 141 7.5 kg 3 Th breeding 35 9 U 143.8 kg 39 94 Pu 145 5.6 kg > 90 % 38 enrichment breeding U

Critical Reactor Condition Goal: steady energy release reactor has to operate at k = 1 moderator to slow down fast neutrons via elastic collisions (large energy transfer); requires light nucleus, e.g. 1C problem: natural U contains 99.3 % 38U, with a large cross section for n capture need to thermalise n away from 38U to avoid capture, e.g. in rods of 1C

Critical Reactor Condition Goal: steady energy release reactor has to operate at k = 1 control of reaction rate (= number of n) by absorption, e.g. rods of 113Cd problem: time between generations (fission and daughter n inducing another fission): 10-3 s but: mechanical control of rods at times << s not possible

Critical Reactor Condition What happens if the number of neutrons cannot be controlled? N(t) number of neutrons at time t (k-1) change in number of n in one cycle τ cycle (fission fission) mean time ( 10-3 s) dt N t dt = N t k 1 N t N t N 0 t dn dt = k 1 N 0 dt dn = k 1 N t t N t = N 0 exp k 1 Example: k = 1.01, τ = 0.001 s, t = 1 s N 1s = N 0 e 10 000 N 0

Critical Reactor Condition Note: reactor will not explode, when it becomes supercritical Reactor heats up kinetic energy of neutrons increases fission cross section drops reactor stabilizes at very high temp meltdown Solution: - design reactor to be subcritical to prompt n - use delayed n (after few seconds) to get to critical condition

A natural reactor What happened 1.7 billion years ago? - original 35U content 3% - water filtering through crevices to ore layer moderator to maintain chain reaction water heats up and evaporates chain reaction stops, until water cools and condensates or new water comes in

Radiocarbon Dating CO absorbed by organic matter: 98.89% 1C 1.11% 13C 10-1 14C radioactive 14C (t1/ = 5730 y): formed in upper atmosphere by cosmic ray bombardment: n + 14N 14C + p 14C + O 14CO 14C production rate relatively constant over for thousands of years living organic material reaches equilibrium of its carbon content with atmospheric carbon organism dies means it goes out of equilibrium initially 15 decays per minute for 1g of carbon major assumption: relatively constant production of 14C over past 50000 years......and fluctuations come from cosmic radiation intensity, solar activity, earth magnetic field...

Radiocarbon Dating 800 Something drastic happened in the 1900 s! 14 C ( ) 600 400 00 0-5000 -0000-15000 -10000-5000 0 5000 Calendar year courtesy of Kristina Stenström, Lund U

14C in atmospheric CO (1900-007) 14C-specific activity 800 14 C ( ) 600 400 00 0 INTCAL98 Vermunt, CO Schauinsland, CO Måryd, Juncus Absorption in biosphere and oceans 400 Bq/kg Bomb-pulse dating Bomb-effect 300 Bq/kg Combustion of fossil fuels 6 Bq/kg 1900 1910 190 1930 1940 1950 1960 1970 1980 1990 000 010 Calendar year courtesy of Kristina Stenström, Lund U

Bomb-pulse dating Study carbon cycle turnover times in biological tissue Dynamics of fat cell turnover History of atherosclerotic placks Checking vintages of wine Burchuladze et al, 1989 Zoppi et al, 004 Spalding et al, 008 causing heart attacks or strokes Age of senile plaques in Alzheimer s disease Lovell et al, 00 Retrospecive cell dating in the brain Spalding et al, 005 courtesy of Kristina Stenström, Lund U