NUCLEAR MODELSNUCLEAR MODELS

ANALOGY BETWEEN A SMALL ANALOGY BETWEEN A SMALL LIQUID DROP AND A NUCLEUSLIQUID DROP AND A NUCLEUS

LIQUID DROP IS SPHERICAL IN SHAPE DUE TO LIQUID DROP IS SPHERICAL IN SHAPE DUE TO SURFACE TENSION FORCE ACTING TOWARDS THE SURFACE TENSION FORCE ACTING TOWARDS THE CENTRE.THE NUCLEUS IS ALSO ASSUMED TO BE CENTRE.THE NUCLEUS IS ALSO ASSUMED TO BE SPHERICAL IN SHAPE.SPHERICAL IN SHAPE.

DENSITY OF SPHERICAL DROP IS KNOWN TO BE DENSITY OF SPHERICAL DROP IS KNOWN TO BE INDEPENDENT OF ITS VOLUME.NUCLEAR DENSITY INDEPENDENT OF ITS VOLUME.NUCLEAR DENSITY IS ALSO FOUND TO BE INDEPENDENT OF IS ALSO FOUND TO BE INDEPENDENT OF NUCLEAR VOLUMENUCLEAR VOLUME

THE MOLECULES OF A LIQUID DROP ARE IN THE MOLECULES OF A LIQUID DROP ARE IN CONTINUAL RANDOM MOTION.EXTENDING THESE CONTINUAL RANDOM MOTION.EXTENDING THESE ANALOGY TO NUCLEUS,DUE TO RANDOM ANALOGY TO NUCLEUS,DUE TO RANDOM COLLISIONS BETWEEN VARIOUS COLLISIONS BETWEEN VARIOUS NUCLEONS,THESE MAY PICK UP SUFFICIENT NUCLEONS,THESE MAY PICK UP SUFFICIENT ENERGY AND ESCAPE.THE MODEL EXPLAINS ENERGY AND ESCAPE.THE MODEL EXPLAINS NUCLEAR REACTIONS AND NUCLEAR FISSION.NUCLEAR REACTIONS AND NUCLEAR FISSION.

LIQUID DROP MODELLIQUID DROP MODEL THIS MODEL WAS PROPOSED BY NIELS THIS MODEL WAS PROPOSED BY NIELS

BOHR.THIS MODEL IS BASED UPON THE FACT BOHR.THIS MODEL IS BASED UPON THE FACT THAT MOLECULES IN A LIQUID ARE HELD THAT MOLECULES IN A LIQUID ARE HELD TOGETHER BY SHORT RANGE INTERMOLECULAR TOGETHER BY SHORT RANGE INTERMOLECULAR FORCES KNOWN AS COHERENT FORCES.DENSITY FORCES KNOWN AS COHERENT FORCES.DENSITY OF LIQUID DROP IS VERY HIGH.OF LIQUID DROP IS VERY HIGH.

ASSUMPTIONSASSUMPTIONS

ALL NUCLEI ARE CONSIDERED TO BEHAVE LIKE ALL NUCLEI ARE CONSIDERED TO BEHAVE LIKE INCOMPRESSIBLE DROP OF LIQUID.INCOMPRESSIBLE DROP OF LIQUID.

NUCLEUS IS SPHERICAL IN SHAPE DUE TO NUCLEUS IS SPHERICAL IN SHAPE DUE TO SYMMETRICAL SURFACE TENSION FORCES.SYMMETRICAL SURFACE TENSION FORCES.

DENSITY OF LIQUID IS CONSTANT AND DENSITY OF LIQUID IS CONSTANT AND INDEPENDENT OF SIZE OR SHAPE OF LIQUID INDEPENDENT OF SIZE OR SHAPE OF LIQUID DROP.DROP.

BINDING ENERGY PER NUCLEON IS CONSTANT BINDING ENERGY PER NUCLEON IS CONSTANT JUST AS LATENT HEAT OF VAPOURISATION IS JUST AS LATENT HEAT OF VAPOURISATION IS CONSTANTCONSTANT

BINDING ENERGY PER NUCLRON IS PROPORTIONAL TO MASS NUMBER A JUST AS BINDING ENERGY PER NUCLRON IS PROPORTIONAL TO MASS NUMBER A JUST AS ENERGY REQUIRED TO EVAPORATE LIQUID IS PROPORTIONAL TO ITS MASS.ENERGY REQUIRED TO EVAPORATE LIQUID IS PROPORTIONAL TO ITS MASS.

SEMI –EMPIRICAL MASS FORMULASEMI –EMPIRICAL MASS FORMULATOTAL BINDING ENERGY OF A NUCLEUS IS GIVEN BYTOTAL BINDING ENERGY OF A NUCLEUS IS GIVEN BY

B.E=B.E=[[ZMZMPP+(A-Z)M+(A-Z)MNN-M-M]]NUCLEUSNUCLEUS - B/C - B/C22

NUCLEAR MASS IS GIVEN BYNUCLEAR MASS IS GIVEN BY

M M NUCLEUSNUCLEUS = =[ZM[ZMPP+(A-Z) M+(A-Z) MNN] – B/C] – B/C22

THIS IS CALLED SEMI-EMPIRICAL MASS FORMULA.THIS IS CALLED SEMI-EMPIRICAL MASS FORMULA.

TOTAL BINDING ENERGY TOTAL BINDING ENERGY EEBB=E=EVV+E+ESS+E+ECC+E+EAA+E+EPP

1)VOLUME ENERGY1)VOLUME ENERGY BINDING ENERGY PER NUCLEON HAS BINDING ENERGY PER NUCLEON HAS

CONSTT.VALUE OVER WIDE RANGE OF MASS CONSTT.VALUE OVER WIDE RANGE OF MASS NUMBERS.NUMBERS.

SO WE HAVE ESO WE HAVE EVV=a=aVVA where aA where aV V is a consttis a constt

.. 2) 2)SURFACE ENERGYSURFACE ENERGYSURFACE NUCELONS HAVE FEWER NEIGHBOURS SURFACE NUCELONS HAVE FEWER NEIGHBOURS

THAN NUCLEONS LYING DEEP INSIDE THE THAN NUCLEONS LYING DEEP INSIDE THE NUCLEUS.SO BINDING ENERGY REDUCEC.THIS NUCLEUS.SO BINDING ENERGY REDUCEC.THIS DECREASE IN BINDING ENERGY IS PROPORTIONAL DECREASE IN BINDING ENERGY IS PROPORTIONAL TO THE SURFACE AREA OF NUCLEUS.TO THE SURFACE AREA OF NUCLEUS.

EESS= -a= -assAA2/3 2/3

NEGATIVE SIGH INDICATES THAT THIS ENERGY NEGATIVE SIGH INDICATES THAT THIS ENERGY HAS TO BE DEDUCTED FROM THE TOTAL ENERGY.HAS TO BE DEDUCTED FROM THE TOTAL ENERGY.

3)COULOMB ENERGY3)COULOMB ENERGYCOLOUMB ENERGY OF A NUCLEUS IS COLOUMB ENERGY OF A NUCLEUS IS

PROPORTIONAL TO THEPOTENTIAL ENERGY OF Z PROPORTIONAL TO THEPOTENTIAL ENERGY OF Z PROTONS PACKED TOGETHER IN SYMMETRIC PROTONS PACKED TOGETHER IN SYMMETRIC ASSEMBLY OF RADIUS RASSEMBLY OF RADIUS R

EECC= -a= -accZ(Z-1)AZ(Z-1)A-1/3-1/3

4)ASYMMETRY ENERGY4)ASYMMETRY ENERGYFOR HEAVY NUCLEI,AS NO. OF NEUTRONS FOR HEAVY NUCLEI,AS NO. OF NEUTRONS

INCREASES,NUCLEUS ACQUIRES AN INCREASES,NUCLEUS ACQUIRES AN ASYMMETRICAL CHARACTER,DUE TO WHICH A ASYMMETRICAL CHARACTER,DUE TO WHICH A FORCE COMES INTO PLAY WHICH REDUCES THE FORCE COMES INTO PLAY WHICH REDUCES THE ASYMMETRY ENERGYASYMMETRY ENERGY

EEaa=-a=-aa a (A-2Z)(A-2Z)22/A/A

aaaa IS A CONSTANT IS A CONSTANT

PAIRING AND SHELL ENERGYPAIRING AND SHELL ENERGY NUCLEI WITH EVEN NUCLEI WITH EVEN ZZ AND EVEN AND EVEN NN ARE HIGHLY ARE HIGHLY

STABLE,EVEN Z- ODD N,OR ODD Z-EVEN N ARE STABLE,EVEN Z- ODD N,OR ODD Z-EVEN N ARE LESS STABLE.SO THIS EFFECT CONTRIBUTES LESS STABLE.SO THIS EFFECT CONTRIBUTES TOWARDS VOLUME ENERGY BYTOWARDS VOLUME ENERGY BY

EEPP=a=appAA-3/4-3/4

WHERE aWHERE ap p IS CONSTANTIS CONSTANT

COMBINING ALL THESE TERMS WE GET TOTAL COMBINING ALL THESE TERMS WE GET TOTAL BINDING ENERGY EBINDING ENERGY EBB=E=EVV+E+ESS+E+ECC+E+EAA+E+EPP

SO BINDING ENERGY PER NUCLEON IS GIVEN BYSO BINDING ENERGY PER NUCLEON IS GIVEN BY

EEBB/A/A= = aaV V -a-assAA2/3 2/3 -a-accZ(Z-1)AZ(Z-1)A-1/3-1/3 -a-aa a

(A-2Z)(A-2Z)22/A+a/A+appAA-3/4-3/4

ACHEIVEMENTS ACHEIVEMENTS

1) 1) STABLE NUCLEUSSTABLE NUCLEUS:-:- STABILITY OF LIQUID DROP IS DUE TO FORCE OF STABILITY OF LIQUID DROP IS DUE TO FORCE OF

COHESION BETWEENTHE MOLECULES.AND COHESION BETWEENTHE MOLECULES.AND STABILITY OF NUCLEUS IS DUE TO THE BINDING STABILITY OF NUCLEUS IS DUE TO THE BINDING ENERGY OF EACH NUCLEON.ENERGY OF EACH NUCLEON.

2)RADIOACTIVE NUCLEUS:-2)RADIOACTIVE NUCLEUS:- A LIQUIDA LIQUID EVAPORATES BY GAINING ENERGY EVAPORATES BY GAINING ENERGY

FROM ITS NEIGHBOURING MOLECULES DURING FROM ITS NEIGHBOURING MOLECULES DURING THE PROCESS OF COLLISIONS.SIMILARLY THE PROCESS OF COLLISIONS.SIMILARLY NUCLEON MAY LEAVE THE NUCLEUS BY GAINING NUCLEON MAY LEAVE THE NUCLEUS BY GAINING ENERGY FROM NEIGHBOURING NUCLEONS,THUS ENERGY FROM NEIGHBOURING NUCLEONS,THUS EXHIBITING THE PROCESS OF RADIOACTIVITYEXHIBITING THE PROCESS OF RADIOACTIVITY

ARTIFICIAL RADIOACTIVITY:-ARTIFICIAL RADIOACTIVITY:-LIQUID DROP MODEL ALSO EXPLAINS THE LIQUID DROP MODEL ALSO EXPLAINS THE

PHENOMENON OF ARTIFICIAL RADIOACTIVITYPHENOMENON OF ARTIFICIAL RADIOACTIVITY

FISSION:-FISSION:-LIQUID DROP MODEL ALSO EXPLAINS THE LIQUID DROP MODEL ALSO EXPLAINS THE

PHENOMENON OFNUCLEAR FISSIONPHENOMENON OFNUCLEAR FISSION

FAILURES:-FAILURES:-1)IT FAILS TO EXPLAIN THE HIGH STABILITY OF 1)IT FAILS TO EXPLAIN THE HIGH STABILITY OF

NUCLEI WITH MAGIC NO.SNUCLEI WITH MAGIC NO.S2)IT FAILS TO EXPLAIN THE MEASURED SPIN AND 2)IT FAILS TO EXPLAIN THE MEASURED SPIN AND

MAGNETIC MOMENTS OF THE NUCLEIMAGNETIC MOMENTS OF THE NUCLEI

DIFFERENCE BETWEEN SHEEL DIFFERENCE BETWEEN SHEEL

MODEL AND LIQUID DROP MODELMODEL AND LIQUID DROP MODEL IN LIQUID DROP MODEL,IT IS ASSUMED THAT IN LIQUID DROP MODEL,IT IS ASSUMED THAT

NUCLEONS INTERACT STRONGLY WITH NUCLEONS INTERACT STRONGLY WITH IMMEDIATE NEIGHBOURS.IMMEDIATE NEIGHBOURS.

BUT SHELL MODEL TREATS NUCLEONS BUT SHELL MODEL TREATS NUCLEONS INDIVIDUALLY AND IT IS ASSUMED THAT INDIVIDUALLY AND IT IS ASSUMED THAT NUCLEONS DO NOT INTERACT WITH EACH OTHERNUCLEONS DO NOT INTERACT WITH EACH OTHER

EVIDENCE IN FAVOUR OF SHELL MODELEVIDENCE IN FAVOUR OF SHELL MODEL

NUCLEONS HAVE A TENDENCY TO FORM PAIR NUCLEONS HAVE A TENDENCY TO FORM PAIR AND ITS DIFFICULT TO REMOVE A PAIRED AND ITS DIFFICULT TO REMOVE A PAIRED NUCLEON THAN UNPAIRED.NUCLEON THAN UNPAIRED.

NUCLEI WITH EVEN Z AND EVEN N ARE MOST NUCLEI WITH EVEN Z AND EVEN N ARE MOST ABUNDANT,ODD Z AND ODD N ARE LEAST ABUNDANT,ODD Z AND ODD N ARE LEAST ABUNDANT,EVEN N AND ODD Z OR EVEN Z AND ABUNDANT,EVEN N AND ODD Z OR EVEN Z AND ODD N COME IN BETWEENODD N COME IN BETWEEN

It has been found that the nuclei with It has been found that the nuclei with proton number or neutron number equal proton number or neutron number equal to certain numbers 2,8,20,28,50,82 and to certain numbers 2,8,20,28,50,82 and 126 behave in a different manner when 126 behave in a different manner when compared to other nuclei having compared to other nuclei having neighboring values of Z or N. Hence these neighboring values of Z or N. Hence these numbers are known as MAGIC NUMBERSnumbers are known as MAGIC NUMBERS

Nuclei with magic numbers of neutrons or Nuclei with magic numbers of neutrons or

protons have their first excited states at protons have their first excited states at higher energies than in cases of the higher energies than in cases of the neighboring nuclei neighboring nuclei

ASSUMPTIONSASSUMPTIONS

NUCLEONS FORM CLOSED SUB-SHELLS WITHIN NUCLEONS FORM CLOSED SUB-SHELLS WITHIN THE NUCLEUS JUST AS ELECTRONS IN CASE OF THE NUCLEUS JUST AS ELECTRONS IN CASE OF ATOMSATOMS

NUCLEONS IN NUCLEUS RE ARRANGED IN SHELL NUCLEONS IN NUCLEUS RE ARRANGED IN SHELL STRUCTURE .THE SHELLS GETBCLOSED WITH STRUCTURE .THE SHELLS GETBCLOSED WITH SUITABLE NO. OF PROTONS AND NEUTRONSSUITABLE NO. OF PROTONS AND NEUTRONS

EACH NUCLEON MOVES INDEPENDENTLY INSIDE A EACH NUCLEON MOVES INDEPENDENTLY INSIDE A NUCLEUS IN A FIXED ORBIT UNDER THE EFFECT NUCLEUS IN A FIXED ORBIT UNDER THE EFFECT OF CENTRAL OTENTIAL PRODUCED BY AVERAGE OF CENTRAL OTENTIAL PRODUCED BY AVERAGE INTERACTION BETWEEN REMAINING (A-1) INTERACTION BETWEEN REMAINING (A-1) NUCLEONS IN IT.NUCLEONS IN IT.

EACH NUCLEON IS ASSUMED TO POSSESS A SPIN EACH NUCLEON IS ASSUMED TO POSSESS A SPIN ANGULAR MOMENTUM AND ORBITAL ANGULAR ANGULAR MOMENTUM AND ORBITAL ANGULAR MOMENTUMMOMENTUM

Consider a nucleon moving Consider a nucleon moving independently in the harmonic independently in the harmonic oscillator potential which is oscillator potential which is spherically symmetric. The spherically symmetric. The Schrodinger equation given below Schrodinger equation given below can be solved in the Cartesian can be solved in the Cartesian coordinate system as well as in the coordinate system as well as in the spherical coordinate system. spherical coordinate system.

Success of shell modelSuccess of shell model

MAGIC NUMBERS:-shell model explains MAGIC NUMBERS:-shell model explains the existence of magic no.sthe existence of magic no.s

It has been found that the nuclei with It has been found that the nuclei with proton number or neutron number proton number or neutron number equal to certain numbers equal to certain numbers 2,8,20,28,50,82 and 126 behave in a 2,8,20,28,50,82 and 126 behave in a different manner when compared to different manner when compared to other nuclei having neighboring values other nuclei having neighboring values of Z or N. Hence these numbers are of Z or N. Hence these numbers are known as magic numbers. known as magic numbers.

SPIN :-SPIN :-Shell model successfully explains the ground Shell model successfully explains the ground state spins and magnetic moments of the nuclei.state spins and magnetic moments of the nuclei.

Even-even nuclides (both Z and A even) Even-even nuclides (both Z and A even) have zero intrinsic spin and even parity.have zero intrinsic spin and even parity.

Odd A nuclei have one unpaired Odd A nuclei have one unpaired nucleon. The spin of the nucleus is equal nucleon. The spin of the nucleus is equal to the j value of that unpaired nucleonto the j value of that unpaired nucleon

MAGNETIC MOMENT:- MAGNETIC MOMENT:- Shell model Shell model successfully explains the magnetic moments of successfully explains the magnetic moments of the nuclei. the nuclei.

For even-even nuclei magnetic moment is zeroFor even-even nuclei magnetic moment is zero For even-odd or odd-even magnetic moment For even-odd or odd-even magnetic moment

depends upon the last unpaired nucleon whether depends upon the last unpaired nucleon whether its proton or a neutron.its proton or a neutron.

STABILITY:-STABILITY:-VERY HIGH STABILTY AND HIGH VERY HIGH STABILTY AND HIGH BINDING ENERGY IS ALSO EXPLAINED ON THE BINDING ENERGY IS ALSO EXPLAINED ON THE BASIS OF CLOSED SHELLS.BASIS OF CLOSED SHELLS.

NUCLEAR ISOMERISM:-NUCLEAR ISOMERISM:-EXISTENCE OF EXISTENCE OF ISOBARIC,ISOTOPIC NUCLEI IN DIFFERENT ISOBARIC,ISOTOPIC NUCLEI IN DIFFERENT ENERGY STATE IS ALSO EXPLAINED ON THE BASIS ENERGY STATE IS ALSO EXPLAINED ON THE BASIS OF SHELL MODELOF SHELL MODEL

MAGIC NUMBERSMAGIC NUMBERS

It has been found that the nuclei with It has been found that the nuclei with proton number or neutron number proton number or neutron number equal to certain numbers equal to certain numbers 2,8,20,28,50,82 and 126 behave in a 2,8,20,28,50,82 and 126 behave in a different manner when compared to different manner when compared to other nuclei having neighboring other nuclei having neighboring values of Z or N. Hence these values of Z or N. Hence these numbers are known as magic numbers are known as magic numbers. numbers.

Experimental evidences for the existence of Experimental evidences for the existence of magic numbers; magic numbers;

1. The binding energy of magic numbered nuclei 1. The binding energy of magic numbered nuclei is much larger than the neighboring nuclei. Thus is much larger than the neighboring nuclei. Thus larger energy is required to separate a single larger energy is required to separate a single nucleon from such nuclei. nucleon from such nuclei.

2. Number of stable nuclei with a given value of Z 2. Number of stable nuclei with a given value of Z and N corresponding to the magic number are and N corresponding to the magic number are much larger than the number of stable nuclei with much larger than the number of stable nuclei with neighboring values of Z and N. For example, Sn neighboring values of Z and N. For example, Sn with Z=50 has 10 stable isotopes, Ca with Z=20 with Z=50 has 10 stable isotopes, Ca with Z=20 has six stable isotopes. has six stable isotopes.

3. Naturally occurring isotopes whose nuclei contain magic 3. Naturally occurring isotopes whose nuclei contain magic numbered Z or N have greater relative abundances. For numbered Z or N have greater relative abundances. For example, Sr-88 with N=50, Ba-138 with N=82 and Ce-140 example, Sr-88 with N=50, Ba-138 with N=82 and Ce-140 with N=82 have relative abundances of 82.56%, 71.66% with N=82 have relative abundances of 82.56%, 71.66% and 88.48% respectively. and 88.48% respectively.

4. Three naturally occurring radioactive series decay to the 4. Three naturally occurring radioactive series decay to the stable end product Pb with Z=82 in three isotopic forms stable end product Pb with Z=82 in three isotopic forms having N=126 for one of them. having N=126 for one of them.

5. Neutron absorbing cross section is very low for the nuclei 5. Neutron absorbing cross section is very low for the nuclei having magic numbered neutron number. having magic numbered neutron number.

6.Nuclei with the value of N just one more than the magic 6.Nuclei with the value of N just one more than the magic number spontaneously emit a neutron (when excited by number spontaneously emit a neutron (when excited by preceding beta-decay) E.g., O-17, K-87 and Xe-137. preceding beta-decay) E.g., O-17, K-87 and Xe-137.

7. Nuclei with magic numbers of neutrons or protons have 7. Nuclei with magic numbers of neutrons or protons have their first excited states at higher energies than in cases of their first excited states at higher energies than in cases of the neighboring nuclei. the neighboring nuclei.

8. Electric quadrupole moment of magic numbered nuclei is 8. Electric quadrupole moment of magic numbered nuclei is zero indicating the spherical symmetry of nucleus for zero indicating the spherical symmetry of nucleus for closed shells. closed shells.

9. Energy of alpha or beta particles emitted by magic 9. Energy of alpha or beta particles emitted by magic numbered radioactive nuclei is larger than that from other numbered radioactive nuclei is larger than that from other nuclei. nuclei.

SPECIAL FEATURES OF MAGIC NUCLEISPECIAL FEATURES OF MAGIC NUCLEI

The neutron (proton) separation The neutron (proton) separation energies (the energy required to energies (the energy required to remove the last neutron(proton)) remove the last neutron(proton)) peaks if N (Z) is equal to a magic peaks if N (Z) is equal to a magic number.number.

There are more stable isotopes if Z There are more stable isotopes if Z is a magic number, and more stable is a magic number, and more stable isotones if N is a magic numberisotones if N is a magic number

If N is magic number then the cross-If N is magic number then the cross-section for neutron absorption is section for neutron absorption is much lower than for other nuclides.much lower than for other nuclides.

The energies of the excited states The energies of the excited states are much higher than the ground are much higher than the ground state if either N or Z or both are state if either N or Z or both are magic numbers.magic numbers.