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SCHOOL OF STUDIES IN PHYSICS JIWAJI UNIVERSITY GWALIORTopic: Nuclear Fission: Mass and energy distribution of fission fragmentsNuclear FissionDecay process in which an unstable nucleus splits into two fragments of comparable mass. In1939, it was found by Hahn and Strassmann that heavy nuclei split into two or more lighternuclei spontaneously or after the absorption of neutron or ϒ-photon.1932: discovery of neutrons1939: official discovery by Otto Hahn and Fritz Strassmann fission of 235U Lise Meitner! (109Mn)1942: first “chain reacting pile” (E. Fermi)1945: first nuclear explosion in Alamogordo (New Mexico, USA)1972: discovery of Oklo (Gabon): unique natural nuclear reactor (1.8 10 6 y ago) very abnormal isotopic ratios of 235U/238U in uranium ores/The minimum energy required for fission to take place is known as activation energy. Foramount of energy required is 6.4 MeV. The fission reaction can be written as;AZX Fission ZA11 X ZA22 X EFWhere, EF represents energy released during the process and is given byE F M (Z , A)c 2 {M (Z1 , A1 ) M (Z 2 , A2 )}c 2Or, EF B(Z1 , A1 ) B(Z 2 , A2 ) B(Z , A)Fig. 1: Binding energy per nucleon vs number of nucleons235U
From the binding energy per nucleon (B/A) curve shown above it can be seen that B/A ismaximum for56Fe. In heavier nuclei B/A decreases and can be increased by splitting heavernuclei into smaller fragments. The fission fragments are usually unequal in size. Since heavynuclei have large N/Z ratio, the fragments contains excess of neutrons. These neutrons are calledPromt neutrons and are emitted as the fragments are formed. If fission is carried out bybombardment with incident particle, then it is known as induced fission.Symmetric FissionFission is known as symmetric fission if a nucleus breaks into two equal fragments. E.g., if238Usplits two equal fragments with A 119. The expected energy released in symmetric fission isE FS M (Z , A) 2M ( Z / 2, A / 2) B(Z / 2, A / 2) B(Z , A) . By binding energy formula, thebindingenergyofB( Z , A) aV A a S A 2 / 3 aCnucleus𝐴𝑍𝑋canbewrittenZ ( Z 1)( A 2Z ) 2 a 0 . And for two fragmentsAAA1 / 3B( Z / 2, A / 2) aV A 2a S ( A / 2) 2 / 3 2aCZ / 2( Z / 2 1)( A / 2 Z )2 a 2 0AA( A / 2)1 / 3𝐴/2𝑍/2𝑋′;as;
Neglectingthedifferenceofpairingenergy,wehaveaC 1 Z Z 1 1 / 3 (21 / 3 1) 1/ 3A 2 E FS a S A 2 / 3 (1 21 / 3 ) Since aC 0.585 MeV and aS 13.1 MeV. Therefore,Surface tension termE FS 3.42 A 2 / 3 0.22Z2A1 / 3Coulomb repulsion termThe surface tension term tries to retain nucleus as such. But the Coulomb force are destructiveforces and favour fission. When ‘Z’ increases Coulomb repulsion increases dominates oversurface tension. The nucleus does not remain stable and thus undergo spontaneous emission.Fig. 2: Fission Barrierand EF 0Mass and energy distribution of fission products
The mass distribution of fission products is represented via fission yield shown in the Fig. 2 inwhich percentage yield of different products is plotted against mass number. The yeild of agiven mass number can be found by measuring the long lived nuclei near the end of the chain orthat of stabble end products. The fission yeild represents the probability of forming that nuclide.The curve shows two peaks corresponding to light and heavy group of products. The distributionmust be symmertic about the centre that is for every heavy fragment there must correspond lightfragment. From the figure it is clear that fission induced by thermal neutrons is highlyassymetric. However, by increasing the energy of incidient neutron the probability of symmetricfisssion increases.Fig. 3: Mass distribution of fissionproductsThe kinetic energy of fission products can be determined by measuring the inonization producedin approprate ionization chamber. Considering the nucleus undergoing fission to be at rest and ifneutrons emitted can be neglected. Then by law of conservation of momentum we have
m1v1 m2 v 2m1 v 2 m2 v1E1 m2 E 2 m1By using E mv2/2Fig. 4: Energy diistribution of fission fragmentsThe masses are inversely proportional to kinetic energy. When K.E. distribution has beenmeasured the mass distribution can be obtained. The K.E. distribution has two peaks as well.when two fragments are formed they are in highly excited states, thereby de-excite by emissionof ϒ-rays or neutrons or by ꞵ -decay. The peaks near at 60 MeV and 95 MeV show the natureof assymetric fission. When energy of incidient neutrons is increassed the probability ofsymmetric fission increases. With 90 MeV of incidient neutron only one peak is observedcorresponding to division into two equal framents.
In heavier nuclei B/A decreases and can be increased by splitting heaver nuclei into smaller fragments. The fission fragments are usually unequal in size. Since heavy nuclei have large N/Z ratio, the fragments contains excess of neutrons. These neutrons are called Promt neutrons and are emitted as the fragments are formed.
A Course on Rough Paths With an introduction to regularity structures June 2014 Errata (last update: April 2015) Springer. To Waltraud and Rudolf Friz and To Xue-Mei. Preface Since its original development in the mid-nineties by Terry Lyons, culminating in the landmark paper [Lyo98], the theory of rough paths has grown into a mature and widely applicable mathematical theory, and there are by .
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CREATING THE BLUEPRINT FOR YOUR “HOUSE” Cynthia Grant, PhD University of Colorado-Denver Azadeh Osanloo, PhD New Mexico State University The theoretical framework is one of the most important aspects in the research process, yet is often misunderstood by doctoral candidates as they prepare their dissertation research study. The importance of theory-driven thinking and acting is emphasized .
In electric circuits this is expressed by Kirchoff’s law for currents: the total current that flows into a junction must also flow out of it. In the junction below I 1 flows in and I 2 and I 3 flow out, so Kirchoff’s law says: I 1 I 2 I 3 I 1 I 2 I 3 Quiz Use Kirchoff’s law to choose the correct answer for this diagram: I 3 I 4 I 1 I 2 (a) I 1 I 2 I 3 I 4 (b) I 1 I .
