Etoos Tips and Formulas

1. NUCLEAR COLLISIONS We can represent a nuclear collision or reaction by the following notation, which means X (a, b) Y a + X  Y+b (bombarding particle) (at rest) We can apply : (a) Conservation of momentum (b) Conservation of charge (c) Conservation of mass- energy For any nuclear reaction a + X  Y+b K1 K2 K3 K4 By mass energy conservation (a) K1 + K2 + (m + mx) c2 = K3 + K4 + (mY + mb)c2 (b) Energy released in any nuclear reaction or collision is called Q value of the reaction 2 (c) Q = (K3 + K4) – (K1 + K2) =  K p   K R    m R   m p  c (d) If Q is positive, energy is released and products are more stable in comparison to reactants. (e) If Q is negative, energy is absorbed and products are less stable in comparison to reactants. Q    B.E  products    B.E  reactants 2. Nuclear Fission 3. Nuclear Fusion In 1938 by Hahn and Strassmann. By attack of a particle splitting of a heavy nucleus (A > 230) into two or more lighter nuclei. In this process certain mass disappears which is obtained in the form of energy (enormous amount) A + p  excited nucleus  B + C + Q It is the phenomenon of fusing two or more lighter nuclei to form a single heavy nucleus. A + B  C + Q (Energy) The product (C) is more stable then reactants (A and B) and mc < (ma + mb) and mass defect m   m a  m b   m c  amu Energy released is E = m 931 MeV The total binding energy and binding energy per nucleon C both are more than of A and B. E  E c   E a  E b  4. Radioactivity Radioactive Decays : Generally, there are three types of radioactive decays (a)  decay (b)  – and  + dacay (c)  decay (a)  decay : By emitting  particle, the nucleus decreases it’s mass number and move towards stability. Nucleus having A > 210 shows a decay. (b)  decay : In beta decay, either a neutron is converted into proton or proton is converted into neutron. (c)  decay : When an  or  decay takes place, the daughter nucleus is usually in higher energy state, such a nucleus comes to ground state by emitting a photon or photons. Order of energy of photon is 100 keV 120 NUCLEAR PHYSICS 5. Laws of Radioactive Decay : The rate of disintegration isdirectly proportional to the number of radioactive atoms present at that time i.e., rate of decay  number of nuclei. Rate of decay =  (number of nuclei) i.e., dN   N dt where  is called the decay constant. This equation may be expressed in the form dN  dt . N N t NN  dN dN   t t    nl n    dt N  NN 0 dt NN 0  0 t where N0 is the number of parent nuclei at t = 0. The number that survives at time t is therefore N = N 0 e and t= N  2.303 log10  0    Nt  t N = N0 e where  = decay constant (a) Half life t1 2  l n2  (b) Average life t av  1  Within duration t1 2  50 % of N0 decayed and 50 % of N0 remains active Within duration t av  63 % of N0 decayed and 37 % of N0 remains active t (c) Activity R = N = R0 e (d) 1 Bq = 1 decay/s. (e) 1 curie = 3.7 × 1010 Bq. (f) 1 rutherford = 106 Bq (g) After n half lives Number of nuclei left = N0 2n Probability of a nucleous for survival of time t  6. N N 0 e t   e t N0 N0 Parallel radioactive disintergration Let initial number of nuclei of A is N0 then at any time number of nuclei of A, B & C are given by N0 = NA + NB + NC  dN A d   NB  NC  dt dt A disintegrates into B and C by emitting , particle. Now, dN C dNB d  1 N A and   2 N A   N B  N C     1   2  N A dt dt dt dNA A t tt t dN eff effeff11 122 2 tteffeff t eff t11 122 2 NN       A    1 22 2 N  1 1 AA A dt t dt tt 1 tt t 2 120 7. Radioactive Disintegration with Successive    A  B Production  (  rate of production) dN A    N A ......(i) dt When NA is maximum N A max   rate of production    By equation (i) 8. dN A  0    N  0  0     N A  0 , dt t t dN A   0   N A 0 dt , Number of nuclei is N A   1  et  Equivalence of mass and energy E = mc2 Note : - 1u = 1.66 × 10-27 kg = 931 MeV or c2 = 931.5 MeV/u 9. Binding energy of ZXA BE = mc2 = [Zmp + (A – Z)mn – mx]c2 = [ZmH + (A – Z)mn – mx)c2 10. Q-value of a nuclear reaction For a + X  Y + b or X (a, b) Y ; Q = (Ma + Mx – MY – Mb)c2 11. Radius of the nucleus R = R0A1/3 where R0 = 1.3 fm = 1.3 × 10–15 m From Bohr Model n1 = 1, n2 = 2, 3, 4......K series n1 = 2, n2 = 3, 4, 5......L series n1 = 3, n2 = 4, 5, 6......M series 122