Nucl Phys ESH2012 Honours B

5/19/2018 Nucl Phys ESH2012 Honours B

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The second category is more complex and involves the transformation of

one type of nucleon into the other (via the so called Weak Interaction)by emission of beta particles ( ). These are electrons or their anti-

particles the positrons.

Electron capture and internal conversion are also part of the second

category.

Finally, during photon emission ( ) a particular nucleus passes from

one state to another - from an excited state to the ground state for

example.


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Effect of a magnetic and electric fields on the different types of radiation.

, and radiation interacting with matter.


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Radioactive Decay Law

Three years after the discovery of radioactivity in 18! it "as noted that

the decay rate of a pure radioactive substance decreases "ith timeaccording to an exponential la".

#adioactivity is a random process and is statistical in nature.

The probability that a particular parent nucleus "ill decay into daughternucleus at lo"er energy is$

1. The same for all nuclides of that species

%. &ndependent of the past history of the parent

'. lmost independent of external influence.


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&n all processes the decay is governed by a characteristic probability per

unit time. This is usually described by the "ell no"n radioactive decay

e*uations$

dN = - N dtand

N(t) = N0 e - t.

+articular radionuclides decay at different rates, each having its o"n decay

constant . The negative sign indicates thatNdecreases "ith each decay

event.

The mean lifetimeis ust the reciprocal of this decay probability = 1/ .

alf!life t1/2is the time taen for half of the nuclei in a sample to decay.


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1/2N0= N0 e -. or e +. = 2

t1/2 = ln(2) / = 0 .693 / , t1/2 = ln(2)

t1/2t1/2


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"ctivity

The radioactivity decay la" allo"s a prediction to be made of the number of

nuclei left after time t - difficult to measure

asier to count the number of decays bet"een times t1and t%.

et / be the change in the number of nuclei bet"een t and t,

N = N(t) N( t + t ) = N0e t( 1- e -

t)

&f t 00 1 or t 00 t1%, ignore higher order terms in the expansion of the

second exponential$

N = N0e t t

dN/dt = N0e t

2efine activityA as the rate of "hich decays occur in the sample,

A (t) = ) = A 0e t


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Total activityA$ /umber of decays a sample undergoes persecond.

#umber of particlesN$ the total number of particles in the sample.

$pecific activitySA$ number of decays per second per amount ofsubstance. ("here a3is the initial amount of

active substance )

d# % dt & ! #


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%'1Th is radioactive "ith t1%4 %5.! hours. t t43 a%'1Th sample

contains /3 4 5!333 nuclei. 6o" many%'1Th nuclei "ill remain after '

days7 hat is the total rate at "hich electrons are emitted at t43 andalso after ' days7

N(3 day) = No e-t(3day)

t1/2 = ln(2) / = 0 .693 / and = 0 .693 / t1/2 4 '.'* hours!*

N(3day) = !"000 e#$-(0.02%1 # %2 ) =&. '000

Actiity at t=o * dN / dt = - N0= (0.02%1 # !"000 ) = 1!20 e/o,r

= 2! e /in

Actiity ater t=%2 o,r = (0.02%1 # 000 ) =3." e /in

Note tat* A = A0e- t


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9easuring the number of decay N in a time interval t gives the activity ofthe sample only if t t1/2

N = t1t2=t1+ t

A dt =A dt

Determination of alf!Life

t1/2

= ln(2) / = 0.693 / , hence from find t1/2

T"o "ays to determine t1/2*

1. 9easure the activity as a function of time

A =A 0e tlnA = lnA 0 t y = c + # plot lnA versus t

slope 4

see figure


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%. &f t1/2 is large, thenA ' contant

measureA and mass of sample ( N ) hence t1/2 from e*uation belo"$

A (t) = ) =

N(t) ln2/t1/2

Daughter #uclei

#adioactive nucleus 1 decays into stable daughter nucleus % $

N1(t) = N0 e 1 t

N2(t) = N0 (1 - e - t) (eventually all of type 1 converts to type %)

:ometimes a nucleus can decay in t"o different "ays to a nucleus ;a< andnucleus ;b


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- (dN/dt) = - (dN/dt)a- (dN/dt) = N ( a + ) = N t

Total decay rate Total decay constant tis only one observed

N(t) = N0 e tt

The relative decay constants, a and , determine the probability for the

decay to proceed by mode a or b .

N1(t) = N0 e 1,tt

N2a(t) = ( a/ t)N0 (1 - e -1 ,tt)

N2(t) = ( / t)N0 (1 - e 1,tt)

:pecial case "ould be a mixture of t"o or more radionuclei "ith unrelated decay

schemes. xample$!=

>u (1%.? h) and!1

>u ( '.= h) see figure.


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Radiocarbon Dating

+hen neutron hits a nitrogen nucleus a

proton is released and #itrogen becomes

*!-arbon.

$pace is filled with cosmic rays

penetrating the upper layers of the earths

atmosphere.

rod,ction o 4oo5enic Ioto$e


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-arbon!* is absorbed by plants in the form of

carbon dio/ide. 0t enters the food chain throughthe herbivores to carnivores li1e humans


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+ith the death of the animal the

replacement of decayed -arbon

atoms by fresh ones out of the

environment stops.

They still contain normal and

radioactive -arbon atoms.

The amount of -arbon!* decreases

over the course of the years. This

amount halves every !.%30 year.


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W. LibbyNobel Price 1951


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Radiocarbon Dating

1=> is beta radioactive "ith half life t1%4 5?'3 years. '3 grams of carbon

from a "ooden sample from a prehistoric site eects %33 electronsmin. 6o"old is the sample7

The ratio of 1=> to 1%> in the "ooden samples "hen it "as part of a living tree

is 1.' x 13-1%. t that time (t43), the number of 1=> isotopes in the '3grams of

carbon "as $

/34 1.'x13-1% ('3grams 1%grams mole) x (!.3%' x 13%'atomsmole)

4 1.! x 131%isotopes

The decay constant of 1=> is$ = 3.!' t1%4 %.' x 13-13min-1

@riginal activity$ 34 /34 A4 =51 decays min

t present time the activity is measured to be 4 %33 decays min

4 3e-t and from this t 4 !?33 years


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"L2" DE-"3

For B C 8' ( C %13) the >oulomb repulsion of protons is strong enough that it

cannot be overcome by adding excess neutrons.

/ote, nuclear strong force (attractive)A>oulomb force (repulsive) 62

Therefore, all heavy nuclei are unstable and usually decay by emission of an -particle (=6e nucleus) to increase stability.

"45#6 "!4!57#! 8


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! Decay Energetics

>onservation of energy for the decay$

#c2 = #7 c2+ 8#7+ ac2+ 8a

9 = (#- #7 - a)c2= 8#7+ 8a

pply conservation of linear momentum to the system "ith the initial nucleus

:at rest$a = #7

nd note that -decays typically release 5 9eD. Thus, 8 c2use non-relativistic inematics, using 8 = 2/ 2and above e*uations$

8a= 9/( 1 + a/#7)

Q-value

931.502 MeV/u

Kinetic Energy


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:inceA ;;


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&t "as first notice by Geiger and /uttall that large disintegration energies

correspond to short half-lives and vice-versa.

The limiting cases are232! " 1.# $ 1010y% Q & #.0' MeV (

21'! " 1.0 $ 10-)*% Q & 9.'5 MeV ( /ote that a factor of% in energy means a

factor of 13%=in half-

lifeH

The theoreticalinterpretation of this

Geiger-/uttall rule in

1%8 "as one of the

first triumphs of

*uantum mechanics.

:ee figure 8.1 for the

inverse relationship

bet"een decayhalf-life and decay

energy.


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Theory of !emission

ssume that the -particle pre-exists inside the nucleus, see fig. 8.'. Themitted -particles moves in the >oulomb field of the daughter nucleus.

4laically* the -particle could not escape unless it has an energy toovercome the >oulomb barrier.

9,ant,?ecanically* there is a probability + that the -particle can penetrate the

barrier (tunnelling effect). + is very small but non-negligible. 2epends on height of the barrier relative to T 2epends on "idth of the barrier.

Estimate of probability 2

ssume that the a-particle is in

constant motion and is constrained

by the potential barrier.

2R


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8eory o a-decay

>onsider a system that consists of an -particle and a daughter nucleus.The relative potential energy of the system as a function of their

separation is sho"n belo". t the nuclear surface r 4 a the potential "ell

is a s*uare "ell. Ieyond the surface only the coulomb repulsion operates.

The alpha particle tunnels through the >oulomb barrier from a to b.

The >oulomb potential barrier

that the alpha particleexperiences is $

D(r) 4 %(B-%)e% =3r

D (9eD) 4 %.88(B-%) r(fm)

#(fm)J 1.% K=1'L ( -=)1'M


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Barrier Width


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The -particle collides "ith the barrier n times per second

= v /2B, v is the speed of the a-particle.

&f v& 2$10)+/*then ' 1021 -1

1$10-1#+ }2ecay probability per unit time =

&f = 1# 10-10y-1then ' 10 - 3

' 1021 -1

}Thus the -particle stries the barrier 13'8 times before tunnelling through./ote that fusion reactions in stars are responsible for energy release and are also

analysed using the barrier penetration approach.


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9uantum mechanical !decay

The >oulomb barrier in fig. 8.' has a height I at r 4 a, "here I is$

C = 1/oulomb potential energy.

C = 1/


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E/ample;

2 < /*'!= 2 < */*'!>'

< *'!> s a%c (/2 -2 )

#esults of the calculation are given in table 8.%, note that$

1. The trend in the change of half-lives is reproduced "ell

%. -values change only by about a factor of t"o "hile t1% changes over more

than t"enty orders of magnitudeH


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lso note that the spin and parity of an -particle are 3L and that in a decayprocess the total angular momentum carried by an a-particle is purely orbital in

character.

The intensity of distribution of a-particles emitted from a deformed nucleus

depends on the point of emission from the nucleus, see fig. 8. and 8.13.


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"l h d $ t


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"lpha!decay $pectrum

parent nucleus (in the ground state) can decay by a-emission to the ground state, a 3, of

the daughter nucleus, or to the excited states of the daughter nucleus, a1, a%, a', a=,A.etc.

Cy Day o e#a$le te decay o2!1

Eto a0 a13i oDn in i5 .13


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Figure 8.11 sho"s the measured discrete, -particle energy spectrum.

fter -particle decay, the excited %=? >f decays to the ground state by -decay,see fig.8.1%


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Ceta Fecay


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Ceta Fecay

lectrons "ere among the first to be observed in radioactive decay.

E/amples;

1=!> O1=

?/ L e- L O excess neutrons

1

13/e O 1

F L eLL O excess protons

/ote that the e-and edo not exist in an atomic nucleus but the decay does occur

inside the nucleus or for a free neutron$

n $ + e- + -= e -$ n + e++ += e+

The half-life$ t1%C 13 ms

&n comparison t1%C 13-15s (-decay)

t1%C 13-? s (-decay)


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Ieta emission is a very different process to alpha.

Transformation of a neutron into a proton or vice

versa through "hat is called

Fte Weak N,clear EorceGor

HWeak InteractionH.

The inverse process, capture of an orbital electron by the nucleus, "as not observed

until 1'8 "hen . lvareP detected the characteristic Q-ray emitted "hen filling a

vacancy.

&n 1'8 the Roliot->uries first observed the related process of positive electron emissionin radioactive decay (positron emission), t"o years after the discovery of the positron in

cosmic rays.

n $ + e- +

$ n + e++

_


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continuous range of energies is measured "ith an

end-point, Tmax, that is

characteristic of the particularnucleus (Tmax4 1-% 9eD)

Energy Release in !Decay

&f -decay "ere a t"o body process then all -particles "ould have a uni*ueenergy. For example, from mass differences %13Ii -decay "ould have -ineticenergy of T-(

%13 Ii) 4 1.1! 9eD, yet a continuous energy distribution is

observed.


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ithout a third particle in the process a number of conservation la"s are violated$

*. "ngular momentum

n $ + e - tree Gerion all ae $in 1/2J J J intrinic $in

0 or 1

}

Aomentum


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. Aomentum

&n a fe" cases the recoil of a daughter nucleus can be observed. 6o"ever, the

daughter does not move in the opposite direction to the eSas re*uired.

*D

?

e

>. Energy

$ F + e -

then a single value for Te- is expected. 6o"ever, a continuous spectrum is

observed.

The third decay product, , is massless (7) and moves "ith the speed of light

and its relativistic energyKris the same as its inetic energy.


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#eutrinos

/eutrinos "ere postulated to preserve conservation la"s in b-decay and must

have

1. Bero charge and rest mass

%. 6alf-integer spin

'. Tae a"ay ;missing momentum and energy

The neutrino interacts via the nuclear "ea interaction. :o very "ea that the

mean-free path of the neutrino in solid iron is J 13lyH

:tars produce e,g. the :un J13'8 sec or J 1315 m%s at the arth.

- decay


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y

&n general,

nuclear masses

>onvert nuclear masses to neutral atomic masses

hereCiis the binding energy of the ithelectron

6ere the electron masses cancel and

neglecting the difference in the electrons

binding energy$

F th


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Further,

nd it follo"s that each has a maximum "hen the other approaches Pero.

+- decay

&n general,


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Reines

#obel 2ri:e *CC@

The netrino was artificialy produced and


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The netrino was artificialy produced and

detected in a lab by Reines in *C=?. (Reines !

#obel 2ri:e *CC@)

#atural solar netrinos were detected for thefirst time ever by Reines and G.2.H. $ellschop

in *C@@ in $outh "frica at depths of about

>,'''m in the Rand mines.

>-l 8 6 >"r 8 e!

L..E. Sellco$ (F 4arid5e F@4Beearc Wit 1>0-1>>" Eo,nder o teSconland Beearc 4entre or N,clearScience- WI8S (noD i8ea MACSa,ten5) ).

-and + Energy $pectra


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gy p

#eutrino Aass


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The end-point and shape of the spectrum for decay can be an indicator asto the mass of the neutrino.

xample$> 6 >e 8 e! 8

Dalues range from !3 eD do"n to '3 eD for the latest and more precise

measurements.

>urrently the upper limit on the neutrino mass is 0 %3eD.

/ote that a value of as lo" as 5 eD "ould provide sufficient mass-energy

density to close the universe.

!


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$teven +einberg "bdus $alam

The 0nternational

Theoretical 2hysics

0nstitute in Trieste


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The Wand6bosons (the "ea bosons) are the elementary particles that mediate the"ea interactionU their symbols are W+ WO and6.

The

Wbosons have a positive and negative electric charge of 1 elementary charge

respectively and are each otherVs antiparticles.

The6boson is electrically neutral and is its o"n antiparticle.

ll three of these particles are very short-lived "ith a half-life of about 3P10O2! . Their

discovery "as a maor success for "hat is no" called theStandard ?odel of particlephysics.


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Ieta decay does not change the number A of nucleons in the nucleus but changes onlyits charge 6. Thus the set of all nuclides "ith the same A can be introducedU theseioaric nuclides may turn into each other via beta decay. mong them, severalnuclides are beta stable, because they present local minima of the mass excess$ if such anucleus has (A 6) numbers, the neighbour nuclei (A 6O1)and (A 6+1) have highermass excess and can beta decay into (A 6) but not vice versa. For all odd massnumbersAthe global minimum is also the uni*ue local minimum. For even A, there areup to three different beta-stable isobars experimentally no"nU for example, >""

"


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The rest mass energy (A) of nuclear

isobars (same value of ") along a

third axis perpendicular to the #%4

plane.

For odd "the isobars lie on a single

parabola as a function of 4. That it is

parabolic in B can be checed by

looing at the $emi!empirical Aass

formula. &n this case there is only asingle stable isobar to "hich the

other members on the parabola

decay by electron or positron

emission depending on "hether their

B value is lo"er or higher than thatof the stable isobar.

For even "the pairing term in the 9ass formula splits the parabola into t"o, one for

even 4and one for odd 4. The beta decay transition s"itches from one parabola to

the other and in this case there may be t"o or even three stable isobars. The odd

and even cases are illustrated in the figures above.

#EITR0#J J$-0LL"T0J#$


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#EITR0#J J$-0LL"T0J#$

&n five distinct measurements, :uper-Yamioande finds

neutrinos apparently XdisappearingX. :ince it is unliely that

momentum and energy are actually vanishing from the

universe, a more plausible explanation is that the types of

neutrinos "e can detect are changing into types "e cannot

detect. This phenomenon is no"n as ne,trino ocillation.

$ymmetries


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2arity (2) Jperation;hen a system obeys the

same la"s going from

r O -r , then the system isinvariant "rt parity.

-harge -onKugation (-)

Jperation ;

#eplace all particles "ith

antiparticles.

Time Reversal (T)

Jperation ;

t O -t $ #everse the direction

in time of all of the processes

in the system.

y

Three different G Belection

"ngular momentum


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g

does not change

direction upon

reflection.

ravity,

electromagnetism are

invariant with respect to

-, 2 and T

Testing 0nvariance of #uclear 0nteractions;

N l R ti


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+-operation$

turn the boo

page and stand

on your head

I is emitted in the same direction as the spin of . Iut in +-operation, I is emitted

opposite to the spin of . xperiment over a large ensemble of nuclei "ill sho" on

average isotropic behaviour (e*ual numbers along and opposite to the spin)

Nuclear Reactions :CPT Invariant

Beaction A + C 4 + F and Fecay* A C + 4


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Test +-operation directly$ align the spins of decaying nuclei and chec if the

decay products are emitted in a preferential direction.

Lee and 3ang *C=@$ ( puPPle)

lthough and seem identical (mass, spin, lifetime), they decay (in asimilar manner to -decay) to states of different parityHHH

They suggested that and are the same particle ( a Y meson) and in orderto decay to states of different parity$

the 2!operation should not be invariant for !decayMM


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-.$. +u ;

lign the -decaying!3

>ospins (T J 3.31 Y)

?3E of the -particles"ere emitted opposite to

the nuclear spin

!decay not invariant

with respect to 2


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2 operation is not a valid symmetry in !decay.

6o"ever, if "e consider a >-reflected experiment, then the -particles are no"

emitted preferentially along the magnetic field 6 Biolation of -!symmetry

xperiment "ith I@T6 + and > operations$ The original experiment is restored.

-2 invariant during !decay.

Electron -apture


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lectron capture competes "ith L- decay.

e- n (>)

n e L- decay )

The nucleus captures an orbital electron, usually from the Y or shells.

vacancy in the Y or shells is then filled by another orbital electron. Thisleads to emission of Q-ray "hich is characteristic of the daughter nucleusatom.

BQ/ e

- B-1QZ/L1

QE& + "

4( + "

4( 7 c2

- 8n, 8n& in:ing energy o; cature:n-*!ell electron "n&K,


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nother (electromagnetic) process "hich competes "ith gamma radiation,

especially if it is rather forbidden, is internal conversion. &n this process, the

nucleus transfers its excitation energy directly to an atomic electron (usually a Y-

shell electron) "hich is eected, carrying the excitation energy.

8e= K - C

"ith K = Ki K(nuclear levels) andC 4 binding energy of the electron in theatom

The increasing concentration of the Y-shell electrons near the nucleus as B

increases means that internal conversion competes increasingly "ell "ith photon

emission (-- decay ) as B increases. /ote that there is no photon involved hereU

the energy is transferred directly to the atomic electron.

9ost ! radioactive sources also emit internal conversion electrons. These stand

out as discrete electron conversion peas riding on a continuous bacground of

-decay electrons


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/ote that the &nternal >onversion occasionally dominates over the gamma-

decay "hen the gamma decay is unliely or absolutely forbidden by

conservation of angular momentum.

'8

'8

#o ray is observed as

the emission of photon

withM=0is impossible.I4


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amma raysare often produced$


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a) alongside other forms of radiation

such as alpha or beta. hen a nucleus

emits an \ or ] particle, the daughter

nucleus is sometimes left in an excitedstate. &t can then de-excite to a lo"er

level by emitting a gamma ray

b) Dia an induced nuclear reaction "here

nuclei are excited to excited states andde-excite bac to the ground state

through a series of gamma decays.

c) Iremsstrahlung -radiation is

produced "hen high energy electronsare progressively decelerated by a

material

d) nnihilation$ e! 8 e86

lectric 2ipole 9oment$ d = qz

Electromagnetic Radiation


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9agnetic 2ipole 9oment$ = iA

Electromagnetic RadiationField b varing t!e di"olemoment:#llo$ c!arge to oscilate along

%&a'is so t!at

d (t)= qzcoswt

(t) = iA coswt

Electromagnetic Radiation Field bvaring t!e magnetic di"olemoment: (ar t!e current so t!at :


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L&* ; Dipole Transition

L&; 9uadrupole Transition

L&>; Jctapole Transition

L&ehadecapole Transition

Question:How can we identify experimentally the

multipolarity of a particular radiation eld?

An5,lar ?oent, and arity Selection B,le


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2uring a gamma-ray transition bet"een states of initial spin & i"ith parity i

and final state spin &f "ith parity f, a gamma photon is emitted "hich caries

an energy e*ual to the difference in energy bet"een the t"o states (Ki K)anda definite angular momentum MQ governed by conservation of angularmomentum "hich re*uires that$

Ii-I 4M

orRIi+ I R M RIiO IR

xample$ Ii& >%, I& =%

N>% 8 =%N O L O N>% P =%N

and

M= 1 2 3 < Q

L&* ; Dipole Transition

L&; 9uadrupole TransitionI=3/2


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L&; 9uadrupole Transition

L&>; Jctapole Transition

L&ehadecapole Transition

The parity selection rule re*uires$

4 no for een Mtransition is lectric, for odd M9agnetic( ?1 K2 ?3 K< )

4 yes for odd Melectric transition, for een Mmagnetic transition( K1 ?2 K3 ?< )

/ote that there are no monopole (M=0) transitions

The parity of the radiation field is$

( KM)& (!*)M (?M) & (!*)M+1

/ote that electric and magnetic multipole of the same order al"ays have opposite

parity.

I =!/2

L

&*,,>,

i l i h di i i i d i h di l


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@ne unit of angular momentum in the radiation is associated "ith dipole

transitions (a dipole consists of t"o separated e*ual charges, plus and minus).

&f there is a change of nuclear parity, the transition is designated electric dipole(1) and is analogous to the radiation of a linear half-"ave dipole radio

antenna.

&f there is no parity change, the transition is magnetic dipole (91) and is

analogous to the radiation of a full-"ave loop antenna.

ith t"o units of angular momentum change, the transition is electric

*uadrupole (%), analogous to a full-"ave linear antenna of t"o dipoles out-of-

phase, and magnetic *uadrupole (9%), analogous to coaxial loop antennas

driven out-of-phase. 6igher multipolarity radiation also fre*uently occurs "ith

radioactivity.


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(of the probability of

emission

) :

nternal!onversion

xample$

F "/#$ i i i ! %/#$


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From statei= "/#$ transition to state "it! f = %/#

$) = no

&fA =% and 'bet"een states is ' = & e then

s can be seen the lo"er order multipoles are more probable

(A*) (E) ( A>) (E)

1 1.=x13-' %.1x1-13 1.'x13-1'

(E*)

(A)

( E>)

(A)

1 1.=x13-' %.1x1-13 1.'x13-1'

From statei= "/#* transition to state "it! f = %/#

$) = yes

J@nly 1 contributes to the transition

ow is the multipolarity of

!radiation determined Q

"ngular distribution measurementsof the radiation to distinguish bet"een


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electric and magnetic transitions.

et us consider a dipoletransition &i41 O &f43

+lace nuclei in a very strong

magnetic field I O orient &i

O split of that level to misubstates ( 4 I)

For

mi43 O mf43

-radiation varies as sin%

For

mi4^1 O mf43

-radiation varies as_ (1Lcos%)

&f "e could pic out only, for

example the component "ith


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example, the component "ith

K+K, then "e "ould observe theJ (1+co2) contribution only.

6o"ever Kis too small J13-!eD.

2etectors unable to distinguish

anything belo" J1.5 eD.

:o "e observe a mixture of all

possible i (+10 0 0-1 0).

W()is the observed angulardistribution$

W() = T (i) Wmi 6 mf()i

W( ) T ( ) W ( )here,$(i) is the population of the initial


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W() = T $(i) Wmi 6 mf()i

sub-state +i, the fraction of the nuclei that

occupies each level.

Wnder normal circumstances all the populations are e*ual$

"1( & "0( & "-1( & 1/3

and

W() +1/3 U J (1+co2) ] + 1/3 (in2) + 1/3 U J (1+co2) ]

Thus the angular distribution becomes isotropic - the radiation intensity isindependent of direction


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&n order to create une*ual populations$(i)and introduce anisotropic W() $

a) Low temperature #uclear Jrientation;

+lace nuclei in a strong magnetic field "#D cool them to very lo"

temperature so that the populations are made une*ual by the IoltPman

distribution$

$(i) ' e i (K / k8)

To have K ' k8 , 8must be 8=0.01 V


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9easuring the angular distribution for dipole radiation "ould give W() sho"n above

b) "ngular -orrelations;

>reate an une*ual mixture of populations $( ) by observing a previous


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>reate an une*ual mixture of populations $(i) by observing a previous 1radiation at a specific direction.

The second radiation%is observed in a

direction %"rt the

first direction.

m34 3 O mi 43 $

proportional to*in21

m34 3 O mi 4^1

+roportional to

_ (1Lcos%

1)

:ince "e deine the P-axis by the direction of 1, it follo"s that 1 = 0and 3 O3 cannot be emitted in that direction That is the nuclei for


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and 3 O3 cannot be emitted in that direction. That is, the nuclei for

"hich % is observed follo"ing 1 must have population $(i) = 0 ori=0

Thusthe angular distribution of relative to * is $

="( ~1/2 > "1co*2) ] + 0"*in2) + 1/2 > "1co*2) ]

="( ~"1co*2)

ngular >orrelation functions for multipole

radiations of higher order than dipole $

M

W() = 1 + T a2kco2k k=1

herea2k depend onIi IandM


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2erturbed "ngular -orrelations !

2"-

+> can discriminate among local

environments of probe atoms in

solids. &nternal fields in solids exert

tor*ues on nuclear moments$-- a

magnetic field exerts a tor*ue on the

magnetic dipole moment and anelectric!field gradient (EH) exerts

a tor*ue on the electric *uadrupole

moment.

These nuclear hyperfineinteractions lead to fre*uencies of

precession of probe nuclei that are

proportional to the internal fields and

are characteristic of the probeVs lattice

location. .

To determine hether the radiation is electric or magnetic e need linear


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To determine "hether the radiation is electric or magnetic "e need linear

polarisation distribution experiments to determine the direction of

emission of the radiation in relation to the direction of K.

(maes use of the polarisation dependence of >ompton scattering)


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#uclear Electromagnetic Aoments


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2istributions of >harges

,, ,, >urrents

}produces

*

lectric field

9agnetic field

Darying "ith distance in characteristic fashion

Multipolemoment

Angulrmomentum

!ptil"epen"ence

+ono"ole ,=- ./r2

0i"ole ,=. ./r1

uadru"ole ,=2 ./r3

lectric and magnetic multipole moments behave similarly (9agnetic

monople does not exist).

+arity of electric moment is (!*)L

+arity of magnetic moment is (!*)L8*

Aonopole Electric Aoment

spherical charge distribution


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spherical charge distribution

/et nuclear charge$ Be

Electric 9uadrupole Aoment

non-spherical charge

distribution

Aagnetic Dipole Aoment

>lassically, circular loop area ,

current i


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xpectation value of a moment$

Aagnetic Dipole Aoment

>lassically, circular loop areaA, current i$

&n the case of protons "ith charge Le "e have$


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5l l N

+rotons 5l = 1

/eutrons 5l = 0 }@rbital motion l

$pin g!factors

= 5s s N

#$eory %&periment

electron '.( '.((')proton '.( 5.5591'

neutron (.( ,noc$rge

-).'()


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e9 = eXY (32 r2) X d

&f X2spherically symmetric 9 = 0

I4 X2concentrated in$y-plane,

&f X2concentrated along P-axis,P4r , J L% 0r%[


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Light nuclei

6eavy nuclei

e9= " # 10-30 e2 6 !0 # 10-30e2

e9= 0.0" e 6 0.! e

10-22 = 1 arn = 1

arge for rare earths, collective effects, that single particle model

does not explain.

"ngular Aomentum

>lassical +hysics $ l = r $ $


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>lassical +hysics $ l r $$

uantum +hysics $

#eplace$ "ith operator e*uivalents valuate cross product

>ompute l2S & l2 + ly2 + lz2 ;

0l2[ 4 5 l "l 1(

lbecomes constant of motion for central potential "hich gives"avefunction B(r)Zl

l(,)

Iy measuring lz thel and ly are indeterminate due to uncertaintyprinciple


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&ntrinsic spin of a nucleon s &

sS & (+1) ; = = [ J

otal angular +o+entu+ D e!ave* lie l an: *

L = l + , FG2A & G"G1( , FG HA & F lH *HA & + D


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N\4MKAB ?]FKMS

N,clear ?odel


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The semiempirical liUuid drop model binding-energy formula gives a

good overall picture of the trends of nuclear binding energies. 2oes not

account for finer details.

/ote$to+ic in:ing energie* ;luctuate erio:ically Iit! @ . !ey get

a#ia Den electron ill a cloed ell an: dro$ tee$lyDu*t eyon: eac!clo*e: *!ell.

Wltimately, "e "ant to try to model the behavior of the nucleus$

` hat ind of potential do the nucleons feelN 7

` >an "e reproducepredict the important nuclear parameters such as$

` /uclear spin

` /uclear magnetic moments

` /uclear *uadrupole moments

` 9agic numbers

Nuclear C!arge 0ensit is


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a""ro'imatel constant 64rom e&&scattering e'"eriments7

Free electron gas: "rotons and neutrons moving 8uasi&4reel $it!in t!el l


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nuclear volume

9 2 dierent "otentials $ells 4or "rotons and neutrons;

9


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Hermi as Aodel


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#uclear 2otentialsV..&n atomic physics an Jn:een:ent

Larticle Mo:el "L( for B electrons in

h l i d


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an atom each electron is assumed to move

independently in the field produced by the

other (B-1) electrons.

e can apply a similar &+ approach by

taing the nucleon-nucleon potential

energy and averaging it over the nuclear

volume sphere "ith # J (1.3? fm) 1'.

&n the case of protons of course the

potential W(r) has an additional term, the

>oulomb potential $

"r( & nuc"r( oul"r(

Form of nuclear potentials $

&nfinite "ell (Fig. .5)

6armonic oscilator (Fig. .5)

#ealistic form (Fig. 5.5)

#ealistic form L spin-orbit potential (Fig. 5.!)


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The +auli +rinciple, W#o two electrons can have the same set of Uuantum

numbers (n l l ) is applicable to identicalparticles. /o restriction onthe states occupied by particles of different type Therefore a proton and a


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the states occupied by particles of different type. Therefore, a proton and a

neutron can both occupy the same state.

&gnoring >oulomb repulsion, the

"ells in "hich the protons and

neutrons move are identical.

mong any set of isobars the nucleus "ith B closest to / has the lo"est energy

8e Sell ?odel


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tomic shell structure /uclearlevels also exhibit shell- lie structure

2ouble shell arrangement for t"o classes

of nucleons (protons and neutrons)

/uclei "ith B or / values that correspond

to complete shells (completely filled) arehighly stable. This is in analogy "ith

atoms "ith complete electronic shells

(inert gases).

These B and / values are called

G ?a5icN,er

, ?, ', ?, =', ? and *@

Aaria Aayer #obel 2ri:e *C@>

Gohannes Gensen #obel 2ri:e *C@>

8e Sell ?odel9agic number nuclei


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ag c u be uc e

sho" an abnormally

high first excitation

energy i$lyin5 lar5eener5y 5a$ etDeente lat illed ener5yleel or ell and tene#t e$ty tate.

(e*uivalent to atomicelectron shells).

Si$le Sell ?odelThe basic assumption $ 2escribed by a single-particle potential. @ne might thin

that "ith very high density and strong forces, the nucleons "ould be colliding all

the time and therefore cannot maintain a single-particle orbit. Iut, because of +auli

exclusion the nucleons are restricted to only a limited number of allowed

orbits.

/uclei "ith magic number of neutrons or protons, or both, are found

to be particularly stable as can be seen from the follo"ing data


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to be particularly stable, as can be seen from the follo"ing data.

The figure belo" sho"s the abundance of stable isotones (same /) is

particularly large for nuclei "ith magic neutron numbers.

The neutron separation energy :nis particularly lo" for nuclei "ith one more

neutron than the magic numbers, "here


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This means that nuclei "ith magic neutron numbers are more tightly bound

Variation o; neutron

*earation energy Iit!

neutron nu+er o; t!e ;inal

nucleu* M",@(.

The neutron capture cross sections for magic nuclei are small,indicating a wider spacing of the energy levels Kust beyond a closed

shell


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typical shell-model potential is

"here typical values for the parameters are DoJ 5? 9ev, # J 1.%51' F,

J 3.!5 F.

For a given spherically symmetric potential D(r), one can examine the

bound-state energy levels that can be calculated from radial "ave

e*uation for a particular orbital angular momentum l,

The energy

/ig.1&


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The energy

levels of the

nucleons for an

infinite spherical

"ell and a

harmonic

oscillator

potential, V "r( &

+O 2 r 2/ 2 .

@ne can see from these results that a central force potential is able to account forthe first three magic numbers, %, 8, %3, but not the remaining four, %8, 53, 8%, 1%!.

This situation does not change "hen more rounded potential forms are used. The

implication is that something very fundamental about the single-particle

interaction picture is missing in the description.

9eyer and Rensen in order to explain the magic numbers suggested that, in

addition to the average central force, there is a strong $in-orit interaction


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g , g $"hich acts on each nucleon.

The $in-orit interaction is proportional toM^S."*trongly *uorte: y e$eri+ental evi:ence(.

:inceScan be either parallel or anti-parallel to the total angular momentumLeach "n,l( level is separated into t"o levels by the spin orbit interaction,"ith the lo"er energy corresponding toM andSparallel.To tae into account this interaction "e add a term to the 6amiltonian 6,

"here @o is another central potential (no"n to be attractive). Thismodification means that the interaction is no longer spherically symmetricU

the amiltonian now depends on the relative orientation of the spin and

orbital angular momenta.

&n labelling the energy levels in Fig. .! "e had already taen into account thefact that the nucleon has an orbital angular momentum (it is in a state "ith a

specifiedl ), and that it has an intrinsic spin of . For this reason the numberof nucleons that "e can put into each level has been counted correctly


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of nucleons that "e can put into each level has been counted correctly.

For example, in the 1s ground state

one can put t"o nucleons, for Peroorbital angular momentum and t"o

spin orientations (up and do"n).

>onventional spectroscopic

notation "ith the value of

is sho"n as a subscript For


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is sho"n as a subscript. For

the energy levels of the

nucleons in the shell model

"ith spin-orbit coupling

2 d 3/2

no.of times

lstate hasoccurred

_

nucleon state is described by a set of *uantum numbers n l _ "hereD&l B >

For a given l the state "ith D&l > has


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g D

lo"er energy thanD&l - >

The separation bet"een t"o states "ith the samelbut different _ is propotional to2l + 1, and therefore increases "ith l.

2egeneracy for each n, l, D state is$ 2D1. "2D1 possible orientations of L (

:ince

M^S = J ( L2 M2 S2)

& >" D"D1( - l"l1( - P( ?2

& { >l ?2 D & l >

->"l1( ?2 D & l - >


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&n general, for a state of given l, the number of nucleons that can go into that

state is %(% lL1).

(the eigenfunctions of the system$ :iagonaliHe the s*uare of the orbital angular

momentum operator M2, its P-component,


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_ = l [ J_ = 1/2 3/2

"2D1possible orientations

ofL o 2_+1 dierent$article in tee tate(

Jl Q2 - U-J(l+1) Q2` = ( l + J) Q2

+auli exclusion principle$


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max number of protons or neutrons in a given staten,l, D is2D 1%

that is$

0 1' )' 5' ' 9' 11'1)' 2.

+a'; number o4"rotons orneutrons

2 3 > .-.2 .3 ?;

(2_+1 possible orientations ofL o 2_+1dierent $article in tee tate )

rrangement of single-particle

energy levels for protons and

neutrons


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neutrons .

ach (n, l) level due to the averagecentral force is split by the spin-

orbit interaction.

The energy gaps at the magic

numbers are clearly sho"n. Theyoccur every time a ne" high value

of l appears, producing a largeM^Ssplitting.


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rediction o te Sell odel

a."ngular Aomentum$ s "ith all single particle energy level models, the :hell

9odel predicts that all even even nuclei "ill have Pero angular momentum n


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9odel predicts that all even-even nuclei "ill have Pero angular momentum. n

odd nucleus "ill have the angular momentum of the odd nucleon. For example $

+redicts ground state spinsN of nuclei $ 2b'?

?


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Ioth / and B areMagic(N &1%!,@ & 8% ) corresponding to completely filled

sub-shells for both p and n.

xclusion +rinciple$ spin and orbital angular momenta are pairedN and therefore

I = 0

hen B or / 4 9G&> ^ 1

then according to the

xclusion +rinciple$

the e/tra nucleon (or XholeY)

determinesI.


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S$in-]rit otential Well or Ne,tron and rotone have so far considered only

a spherically symmetric nuclear


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potential "ell. e no" there is

in addition a centrifugal

contribution of the form

l,l8*)h % r

and a spin-orbit contribution. s

a result of the former the "ellbecomes narro"er and

shallo"er for the higher orbital

angular momentum states.

:ince the spin-orbit coupling isattractive, its effect depends on

"hether S is parallel or anti-parallel to M. The effects areillustrated in the figure. /otice

that for

l = 0both are absent.

&n addition to the boundstates in the nuclear

potential "ell there exist

also virtual states (levels)


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virtual level is therefore not a bound stateU on the other hand, there is a

non-negligible probability that inside the nucleus a nucleon can be found

in such a state

also virtual states (levels)

"hich are positive energy

states in "hich the "avefunction is large "ithin the

potential "ell. This can

happen if the de Iroglie

"avelength is such that

approximately standing"aves are formed "ithin

the "ell. (>orrespondingly,

the reflection coefficient at

the edge of the potential is

large.)

Aagnetic Aoments; :hell model provides good agreement

9agnetic moment from expectation value of magnetic moment operator for

i h i i f l


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state "ith maximum P proection of angular momentum.

= N( 5l l+ 5) / -#

Iut l and not precisely defined. Wse G & l 8 s

= NU5l L+ ( 5 5l) ` / -#

xpectation value "hen _= _ -

0 ; = NU5l _ + ( 5 5l) ;` / -#2etermine

;_ is only vector of interest instantaneous value variesLcomponent along L constant

Wnit vector along _ is _ / R _ R

The component of along Kis N _N/ R _ R


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These are the so called Scidt Miit for nuclear magnetic moments and mostactual values lie bet"een them. This can be explained in terms of a mixing of

states such as in the case of the deuteron

The vector _ is * L = _ R _ R / R _ R 2

expectation values $ 06; = U _ / 2_ (_+1) ` U _ (_+1) - l (l+1) + (+1) ` -#

"here N _N& (l + ) "*ee calculation G & "l *( (or _ = l + 1/2 ; = -/#or _ = l - 1/2 ; = -- . /# (. $ &)

mgnetic "ipole moment3:or _ = l + 1/2 ; = U (_ - J )5l+ J5` N

or _ = l - 1/2 ; = U _ (_ +3/2 )5l - J5` N/(. $ &)

5l= 1 for the proton, 0for the neutron5= !.!> for the proton, -3.3 for the neutron. (5nucleon in nucleus @ gsfree due

to meson cloud )


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Electric 9uadrupole Aoments;

&f this moment "ere ust due to the odd proton it should be given by 9 'B2(2_ 1)/(2(_ + 1))


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-B2(2_ - 1)/(2(_ + 1))"here_is the angular momentum *uantum number of the odd particle. For even-even nuclei it should be about Pero and should change sign on going through theshell closure. :ome examples are

The 9in (fm)thus

the numbers have to

be multiplied by the

electronic charge to

obtain the actual

*uadrupole

moment.

There are clear deviations from the :hell 9odel predictions (:9)$

i.@dd neutron nuclei have about the same as odd proton ones H

1./uclei "ith atomic mass number in the ranges 153 to 13 and greater than %33 have

very large *uadrupole moments. This is a serious failure of the model.

"pplication to #uclei

&t is possible to determine g.s. spin, parity and magnetic moment using follo"ing$


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1. >omplete filled shells have

L

ell= 0+

ell= 0 2o not contribute toL of nucleus /eed to consider only ;valence< nucleons, not ;core< nucleons.

%. &n partially occupied shell individual spins for lie-nucleons pair "ith

opposite sign

L$air= 0+ $air= 0

L

of nucleus due to ,n$aired n,cleon

'. "llD/ / S D/ B nuclei have no unpaired nucleons

L= 0+ = 0

=. For JDD "nucleus, odd nucleon shell_


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L = _ = (-1)l and

= for single nucleon in the shell

5. For @22 / - @22 B nucleus, spin is the vector sum of Ns for last

neutron and last proton.

L = _n + _$

R _n - _$ R b L b _n + _$

and n,cle,= n $

-ollective E/citations of #uclei

The discussion is limited to even-even nuclei (i.e. even protons-even neutrons).

The :hell model then predicts a 3Lground state (all nucleons are paired)


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>onsider the nucleus as a collective entity, as

opposed to being composed of nucleons.

xample$ *>'='$n?'

>onstruct a shell-model level diagram.

53 protons fill the g% shell and 83

neutrons need another % to fill completely

the h11% shell to complete the magic

number /48%.

To form an excited state "e can breaone of the pairs and excite a nucleon to a

higher level. The coupling then bet"een

the t"o odd nucleons determines the spin

and parities of the t"o levels.

&f "e assume that the g.s. of *>'$n

consists of filled 1/2 and d3/2subshells and 13 neutrons occupy

h b h ll h ld


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the 11/2 subshell, then "e couldform an excited state by breaing

the 1/2 pair and promote one ofthe 1/2 neutrons to the h11%subshell.

>oupling angular momenta

_1and

_2gives values bet"een_1+ _2and NK*!KN and for the above 11/2 + J ="and 11/2 J =! .

This results in states predicted atabout K#= 2 ?e@. This energy ischaracteristic of "hat is needed to

brea a pair and excite a particle

"ithin a shell (experimentally).


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>oupling angular momenta5ae al,ebet"een11/2 + J = "and 11/2 J =! .

Anoter $oiility* Irea one of the d'%pairs andplace a n in h11%. >oupling angular momenta_1and_2gives values bet"een_1+ _2and N K*!KN and for theabove 11/2 + 3/2 =


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fter examining hundreds of even-even nuclei a remarable finding$

ach one has an ;anomalous< %L state at an energy at or belo" one-half of the

energy needed to brea a pair. &n most cases this %L state is the lo"est excited

state.

There are other properties that are identified that are not related to the motion

of fe" valence nucleons but rather "ith the entire nucleus$ 4ollectie$ro$ertie

$ystematics

1. Fig. 5.15a nergy of first excited %L state


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15( 4 A 4 19(

%. Fig. 5.15b #atio (=L) (%L)


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1; Fig; A;.a +agnetic moments o4 2B states


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. Hig. =.*@b Electric Uuadrupole moments


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8Do ty$e o collectie tr,ct,re *1. A 1!0 * iration ao,t $erical e,iliri, a$e2. 1!0 A 1>0 * rotation o a non-$erical yte

#uclear Bibrations

>onsider a li*uid drop

vibrating at high


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vibrating at high

fre*uency, about a

spherical e*uilibriumshape. e can describe

instantaneous shape by

radius vector B( , )of the surface, at time t.

!erical *ur;ace

e see the modes of vibration for l 4 1U %U '. *uantum vibration is called aphonon dding Z ( ) term into nuclear "avefunction introduces l h units of


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phonon. ddingZl(, ) term into nuclear "avefunction introduces l h units ofangular momentum, "ith parity (-1)l . The vibrations are seen as bosons carrying

angular momentum l h.

>onsider the follo"ing possibilities for an even-even nucleus "ith g.s. 3L, 1%3Te

(fig 5.1)$

dding anl 4 1, dipole phonon "ith &4 1-U and is e*uivalent to a net displacementof the centre of mass of an isolated system. 6ence, this is not observed.

dding an l 4 %, *uadrupole phonon "ith &4 %L. That is, it creates an excited %Lstate. Wsually the lo"est excited state of even spherical nucleus.

dding an l 4 ', octupole phonon "ith a &4 '-state. This state is commonly seenat higher excitation energies than %L *uadrupole vibrations.

#uclear Level $chemes

>onsider an even-even nucleus "ith a 3L

ground state, e.g. 1%3Te (fig. 5.1)


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- dd one *uadrupole phonon (l4 % )

creates a l4 % state, %dependence. +arity (-1)l 4 (-1) % 4 L

3L O %L

- dd t"o *uadrupole phonons (l4 % )

Forms triplets of states 3L, %L, =L at t"ice theenergy of the first %L state.

/ote that t"o identical phonons carry t"ice as

much energy as one. (fig.5.1).

- dd three *uadrupole phonons (l4 %)

- dd one octapule phonon l4 ').Forms '- state.

Table 5.% for

resulting total

component of


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2redictions of the vibrational model

1. spherical e*uilibrium shape for the first %L state has a vanishing

*uadrupole moment. :een for 0 153

%. 9agnetic moments of first %L excited states given as % ( B ) "ith

range 3.8 S 1.3. :een in fig. 5.1!a

'. (=L) (%L) 4 %.3 :een in fig. 5.15b

component of .ssociate a value

for l "ith eachtotal P-

component.

#uclear Rotations

/uclei "hich lie close to the magic numbers are roughly spherical, and therefore

cannot rotate.


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6o"ever, in certain regions of the nuclear chart$ *=' Z " Z*C'or " S '[ that

is, regions near middle of shell gapsU nuclei have substantial distortions from

spherical shape. e.g. 15% !!2y8!.

9any nucleons participate in the motion$ hence, a collective effect collective

rotation.

Types of Deformed $hape

The most common deformed shape of rugby-ball shaped$ axially symmetric,

prolate shape. e describe the surface by$

B(t) = BaU 1 + 2co t Zl(, ) `

2is the *uadrupole deformation parameterU

2[3 , prolate ellipsoidU

20 3, oblate ellipsoid.

:table deformation O large electric *uadrupole moment

a) 2efine intrinsic *uadrupole moment


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a) 2efine intrinsic *uadrupole moment 3

90 = (3/ !) B2a6 (1 + 0.1 )

@bserved in a reference frame in "hich nucleus at rest (body-fixed ref frame).

b) rotating deformed nucleus in lab frame has different *uadrupole moment

#otating prolate distribution6 time averaged oblate distribution3 [ 3 leads to observed 0 3

2epends on nuclear angular momentum6 %L states 4 - %? 3

O region 153 13 stable permanent deformation

J -%b J L?b b J 3 % lar e

Energies of Deformed $hapes \ Rotational ]ands

deformed nucleus has a rotational degree of freedom. The classical expression

for rotational inetic energy is ust$


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gy

8 = J 2

"hereis the moment of inertia.

uantum mechanically, energy of rotating obect$

K = -#/ # I ( I + 1 )

#otational band even / S even B E "0( & 0E "2( & 6 "-#/ # (E"#( & 20 "-#/ # (E"6( & #2 "-#/ # (A.. A. A.

xcited states of 1!=r (fig 5.%%)

"+ea*ure:(

Documents

Nucl Phys ESH2012 Honours B