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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:(