nuclear masses and decays

8/17/2019 Nuclear Masses and Decays

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Phys 3033 – Nuclear Masses and Decays - 1 of 29

Labelling Nuclei

Elements contain a particular number of protons usually denoted !"# and called the atomicnumber $

%a &calcium' – 20 protons → "(20% &carbon' – ) protons → "()* &uranium' – 92 protons → "(92

+he number of neutrons in a nucleus is usually denoted by !N#,

+he mass number is the total number of nucleons usually denoted !#, .b/iously ( N ",

i/en nucleus is labelled$ X N Z

A

n eneral e 4ust use 5 because 6noin the element 5 defines the atomic number " and thenthe neutron number is deri/ed from N(-",

7ome commonly used terms &memorise them8'$

isotopes$ nuclei ith the same "isotones$ nuclei ith the same Nisobars$ nuclei ith the same


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Geiger and Marsden

+he eistence of the nucleus as demonstrated by:eier and Marsden in the famous !;utherford#

eperiment performed in 1909,

Expt$ beam of alpha particles &helium nuclei' isincident on a old foil,Results$ .nly a small number of alpha particles arescattered and a /ery fe are bac6-scattered,

n ;utherford


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Nuclear Charge Density

+he cur/es on the left sho the chare density e>ui/alent to the proton density as measured byelectron scatterin eperiments,+his >uantity saturates i,e, all nuclei ha/e the same interior

&proton' density, We assume neutrons are the same! &?o could e chec6@' +his assumptionresults in the total nuclear matter density distributions shon on the riht that for all nucleisaturate at a nucleon density of A0,1) nucleonsBfm3,

Empirically e find that the nuclear radius ; ( r0 1B3 ith r0(1,2 fm, ;emember this8

Eercise$ Chat is the density of the nucleus in !nucleonsBfm3# in Bcm3 and in 6Bm3@f the Earth had the density of a nucleus it ould fit inside ondon %ircuit in %i/ic8


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Phys 3033 – Nuclear Masses and Decays - of 29

Masses in Nuclear and article hysics

Fecause of the familiar mass-enery e>ui/alence E(mc2 as ell as the importance of bindin

eneries and masses in understandin nuclear stability it is /ery common in Nuclear Physics &andalso in other fields of physics' to refer to masses in units of GeneryBc2H,

7ome etremely useful masses to remember include the folloin$

mass of a proton$ mp ( 93I,2J20K MeLBc2

mass of a neutron$ mn ( 939,K)K3I MeLBc2

atomic mass unit$ u ( 931,90) MeLBc2 &1B12 the mass of an unbound neutral 12% atom'

mass of an electron$ me ( K10,99I93 6eLBc2 or 0,K1099I93 MeLBc2

Ce ill often nelect to include the c2 and ill simply tal6 about masses in terms of their enerye>ui/alents, Note that in nearly all cases any calculation in this course that uses an enery or amass ill be more easily sol/ed if you use units of MeL for enery and MeLBc2 for mass,

7o hen e say the mass of a proton is 93I MeL don


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Phys 3033 – Nuclear Masses and Decays - K of 29

Nuclear inding Energy

+he mass of a nucleus is smaller than the sum of the mass of its nucleons$

M&"' "mp &-"'mn

+he difference is used to bind the nucleons toether and is called the bindin enery i/en by$

M&"'c2 ( GM&"' – "mp – &-"'mnHc2 0

Ce need an enery -M&"'c2 to transform the nucleus into free nucleons,

useful >uantity is the a/erae eneryneeded to release one nucleon from thenucleus, t is denoted FB defined byFB ( -M&"'c2B,

Oor nuclei hea/ier than carbon FB is rouhly

constant at around I or 9 MeL B nucleon,

Note that ?e is much more hihly bound thanituence of nuclear shell structure,


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Phys 3033 – Nuclear Masses and Decays - ) of 29

Re"isiting inding Energy

;ecall that the mass of a nucleus issmaller than the sum of the mass of

its constituent nucleons$

M&"' "mp &-"'mn

+he difference is the enery thatbinds the nucleons toether and iscalled the bindin enery i/en by$

M&"'c2 ( GM&"' – "mp – &-"'mnHc2

+he fiure at riht ta6es themeasured masses &bindin eneries'of all 6non nuclei and plots them/ersus the total number of nucleons,t is readily apparent that this is an

approimately linear cur/e,

+his is a conse>uence of the fact of the shape of the nucleon-nucleonpotential &stron nuclear force',


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Phys 3033 – Nuclear Masses and Decays - J of 29

Nucleon#Nucleon otential

+his as mentioned in pre/ious lectures sohere is a >uic6 reminder of the important

features of the nucleon-nucleon potential,&Note that this as discussed by %edric,'

• +o first order it is independent of nucleon type so pp pn and nninteractions are !identical#

• t has a hihly repulsi/e corepre/entin nucleons from o/erlappin in

space,• t has a finite rane and is stron,

+hese features mean that in nuclei thea/erae distance beteen ad4acent nucleonsis almost a constant on the order of a fm,

+his also means that e/ery nucleon added toa nucleus occupies an almost constant/olume &see chare densities and radiidiscussed earlier' and also contributes an approimately constant amount of etra bindin enery&pre/ious slides',


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Phys 3033 – Nuclear Masses and Decays - I of 29

Nuclear inding Energy $ Nucleon

+he fact that the bindin enery /ersusnucleon number cur/e is almost linearmeans that bindin eneryBnucleon orFEB is almost constant,

ain ta6in all the measured nuclear

masses and plottin FEB /s e obtainthe picture on the riht, +he fact thatmost nuclei lie on an almost smoothcur/e is the reason that

a' n approimate simplified cur/e isoften used to describe this beha/iour as

shon on pK and in almost e/ery otherdiscussion concernin nuclear bindinenery

b' +he !7emi-Empirical Mass Oormula#&7EMO' can be used as a first startin point to !predict# nuclear masses,


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ethe#Wei%s&c'er ( )emi#Empirical Mass *ormula

M A , Z c2≈−a1 Aa

2 A

2/3a3

Z Z −1

A

1 /3a

4

N − Z 2

A

±a5 A

−1/2

+his model assumes the nucleus beha/es li6e a !li>uid drop#, ;ecall also that the nuclear radius isproportional to 1B3 and that more bound nuclei ha/e larger negati"e "alues of M&"'c2,

+he first term is the volume term and is neati/e because it fa/ours the bindin of more nucleonsith e/ery nucleon increasin the bindin by a constant amount hence proportional to ,

+he second term is the surface term, Nucleons at the surface are less bound so it is positi/e,+he surface area oes as r2 hence 2B3,

+he third term is the Coulomb term, +here are "&"-1' pairs of interaction protons and their%oulomb interaction oes as 1Br i,e, as 1B 1B3, t is repulsi/e so the term is positi/e,

+he fourth term is the symmetry energy term, t fa/ours nuclei ith the same number of protonsand neutrons, Ce ill loo6 at hy this is so in more detail later in the course,

+he last term is the pairing energy term, +his has its oriin in >uantum mechanics and is relatedto the fact that e/en numbers of li6e nucleons are more bound than hen there are an oddnumber of li6e nucleons, Chen there are e/en numbers of both protons and neutrons &an e/en-e/en nucleus' this term is neati/e hen there are odd numbers of both protons and neutrons&an odd-odd nucleus' this term is positi/e, Oor odd-e/en nuclei this term is =ero,

n empirical fit i/es a1(1K,K) a2(1J,23 a3(0,J a(23,) and aK(11,2 all in units of MeL,


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Contribution of different terms to the mass formula

+he fiure on the left shos the shape of the first three terms of the mass formula as ell as thesum of the these first three terms,

+he fiure on the riht shos the cumulati/e sum after addin the terms one by one,

+hese first three terms are sufficient to describe the ross shape of the bindin enery cur/e,

uestion$ Chat does it mean to plot /ersus @ ?o do e define N and " in this case@


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+he Nuclear Chart

%ertain nuclei are stable and occur naturally, .ther nuclei are unstable and undero radioacti/edecay, +he locus of the stable nuclei in a plot of N /s " is 6non as the /alley of stability and is aconse>uence of a balance beteen the %oulomb and symmetry eneries,


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Nuclear inding Energy $ Nucleon

t is often instructi/e to plot nuclear properties as a function of N and ", +his is sometimes called a7ere chart named after Emilio 7ere &Nobel Pri=e 19K9',

+he plot of FEB at rihtshos that the mostbound nuclei are centredaround K)Oe and thatmo/in aay in anydirection in the NB"

plane ill result in anucleus ith less bindinenery B nucleon,

+his feature helpseplain many naturalphenomena ranin

from the spontaneousfission of a sinle hea/ynucleus all the ay upto the beha/iour of superno/a eplosions,


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Modes of radioacti"e decay

f a process that chanes one nucleus into another is eneretically possible there is usually someprobability that it can happen, +hus there are many different modes of radioacti/e decay ith

the most common bein alpha decay beta decay positron decay and electron capture$

: X N Z

A Y N −2 Z −2

A−4 4 He

− : X N Z

A Y N −1 Z 1

A e− e

: X N Z

A Y N 1 Z −1

A e e

EC : X N Z

A e− Y N 1 Z −1

A e

Qou ha/e already discussed some aspects of beta-decay ith %edric in the particle physics part o

the course,


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Main decay modes of nuclei

Note that any i/en nucleus can decay /ia more than one mode the abo/e fiure simplysummarises hich is the most probable mode of decay,


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Phys 3033 – Nuclear Masses and Decays - 1K of 29

Nuclear Lifetimes and Radioacti"e Decay

Ce said that nuclei that are notstable can decay bac6 toards stable

nuclei by radioacti/e processes,

+he fiure at left shos the lifetimesof nuclei and it can be readily seenthat in eneral as nuclei mo/e aayfrom the line of stability theirlifetimes decrease,

Chate/er the nuclear decay mode&see the net slide' each radioacti/enucleus has a constant decay probability per unit time denotedby R sometimes also called thedisintegration constant,

f the number of nuclei at time t is N then the number that decay in time t is i/en by N(RNt,%hanin to differentials and interatin e can sho that N(N 0e

-Rt here N0 is the number ofnuclei at t(0, &Qou should sho this yourself,'

Ce define the meanlife or lifetime as S(1BR, +he halflife is defined as the time it ta6es for thenumber to reduce by a factor of 2, Qou should sho that the halflife is i/en by +1B2(S ln2


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Phys 3033 – Nuclear Masses and Decays - 1) of 29

roton and Neutron )eparation Energies

+he minimum enery re>uired to release one neutron &or proton' from the nucleus is 6non as theneutron &or proton' separation enery and is defined by$

S n=[ M A−1, Z mn− M A , Z ] c2

S p=[ M A−1, Z −1m

p− M A , Z ]c2

+he pictures at leftsho that it isdifficult to remo/e aproton surrounded

by neutrons and /ice/ersa, +his is aresult of thesymmetry enery inthe 7EMO,

Chen 7n(0 or 7p(0this defines the

upper and loerlimits of stability forparticular isotopicchains – hen addina ne nucleon it illnot be bound,


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Phys 3033 – Nuclear Masses and Decays - 1J of 29

De"iations from the )EM* ( E"idence for shell structure

+his fiure shos a plot of the difference beteen the semi-empirical mass formula and theeperimentally measured bindin enery across the nuclear chart, +he lines that appear aloncertain mass numbers for instance at N or " e>uallin 2I K0 I2 12) are e/idence that certainmagic numbers of neutrons and protons are especially fa/oured, ;eminiscent of the noble asesin atomic physics this is e/idence that there is shell structure in nuclei,


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Phys 3033 – Nuclear Masses and Decays - 1I of 29

,tomic masses "ersus nuclear masses

+he 7EMO defines nuclear masses hoe/er you ill often also use tables of atomic masses,

M atom( Z , A) ==

M nucl ( Z , A)+ Z me− Belec / c2

Z m p

+ ( A− Z ) mn−Δ M ( A , Z )+ Z m

e− B

elec/c2

Note that in many situations you are dealin ith mass differences in hich case the electronmasses often cancel,

7ince the bindin enery of the electrons is typically much smaller than the nuclear bindineneries &especially if you are ta6in differences beteen masses' you can &in almost all cases'inore the electron bindin eneries Felec,

tomic mass tables are sometimes i/en in atomic mass units, +his is defined to be eactly 1B12the mass of a neutral unbound 12% atom, t is usually abbre/iated as !u# and to con/ert to MeLyou use$

1 u ( 931,9 MeLBc2


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-#"alues

+a6e the eample of alpha-decay,

: X N Z A

Y N −2 Z −2 A−4

4

He

%onser/in enery for this process$

M x

c2= M

Y c

2T Y M

c2T

+his process can only occur if the 6inetic eneries are reater than =ero and hence e define the

-/alue$

Q= M xc2− M

Y c

2− M

c2

and e must ha/e T0 for alpha-decay to occur,

More enerally$

Q=[initialmasses− finalmasses] c2

and any process for hich T0 is eothermic and can proceed &in principle' immediately hereasany process for hich 0 is endothermic and re>uires the input of enery to ma6e it happen,


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-#"alues for alpha#decay

.nly hea/y nuclei ha/e positi/e -/alues and can spontaneously alpha-decay, et


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-#"alues for alpha#decay cont.

+he points in the fiure at left sho the -/alues deri/ed

from eperimental masses for nuclei lyin alon the line of the /alley of stability, +he line is a cur/e deri/ed from the7EMO,

&1' +he 7EMO describes the o/erall beha/iour rather ell,

&2' De/iations near Pb are associated ith shell structure

and the especially stable nature of

20I

Pb as discussed in thesections about the nuclear shell model,

&3' +he nuclei beyond "()K ha/e T0 and could allpotentially alpha-decay hoe/er eperimentally theydon


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-uantum tunnelling in alpha#decay

f an alpha particle ere to eist atthe ede of the parent nucleus it

ould eperience a %oulomb force,

t the radius that corresponds to analpha particle and 20K+l 4ust touchin&about I fm' the %oulomb enery ismuch larer than 3,11 MeL,

t is not until the alpha particle isJK fm from the nucleus that the%oulomb enery reduces to 3,11 MeL,

?ence there is a %oulomb barrier thatinhibits the emission of the alphaparticle,

+he only ay for the alpha particle to escape is to >uantum tunnel throuh the barrier, 7uch aprocess is LE;Q stronly dependent on the enery of the alpha particle,

Eercise$ *sin ;(1,21B3 fm and the %oulomb force /erify the shape of the cur/e abo/e,


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Geiger ( Nuttall rule for #decay

ln R A -1B2

nitially an empirical obser/ation by:eier and Nuttall in 1911 theeponential beha/iour as eplained by:amo in 192I as a conse>uence of>uantum tunnellin, &%onsult atetboo6 for a deri/ation if you areinterested in the details',

Note the lo scale spannin MNQorders of manitude on the lifetime &y'ais, # small chan!e in the $-value for alpha-decay has hu!e consequences for the decay lifetime.

+he blue lines connects the points fordecay of different isotopes &i,e, ha/inthe same number of protons', +his is

further e/idence that the phenomenon arises from >uantum tunnellin throuh a %oulomb barrier,&ncreasin " increases the %oulomb barrier and hence increases the decay lifetime,'

+he :eier-Nuttall rule applies ith different offsets for each separate isotopic chain,


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Energetics of beta decay

+he simplest beta-decay is the decay of a free neutron, +his is possible because the proton mass&93I,2I MeLBc2' is less than the neutron mass &939,KJ3 MeLBc2', +he halflife of a free neutron is

about 10 minutes hile the eperimental lifetime limit for the decay of a proton is T1030

yr,

Chen the protons are bound in a nucleus the chane in bindin enery beteen different nucleimeans that it is eneretically possible for both protons and neutrons to decay leadin to toforms of !normal# beta-decay as ell as electron capture hich is somehat similar to β decay,

− : X N Z

A Y N −

1 Z

1

A e− e

: X N Z

A Y N 1 Z −1

A e e

EC : X N Z

A e− Y N 1 Z −1

A e

+he beta decay processes connect nuclei alon isobaric chains i,e, constant /alues of , et us

ta6e a closer loo6 at the 7EMO and ho it beha/es alon an isobaric chain,

M A ,Z c2≈−a

1 Aa

2 A

2 /3a

3

Z Z −1

A1/3

a4

N − Z 2

A±a

5 A

−1/2


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Phys 3033 – Nuclear Masses and Decays - 2K of 29

Nuclear masses along isobaric chains

et us loo6 at ho the nuclear mass chanes alon a chain of constant , ;emo/e N from the 7EMOby usin N(-", ?ence the term &N-"'2 becomes &-2"'2,

M A , Z c2≈−a

1 Aa

2 A

2/3a

3

Z 2− Z

A1/3

a4

A2−4AZ4Z2

A±a

5 A

−1/ 2

No e can define the mass-enery e>ui/alent for an atom to be$

M atom

c2= A− Z m

nc2 Z m

pc2 Z m

ec2−a

1 Aa

2 A

2/3a

3

Z 2

− Z

A1/3

a4

A2

−4AZ4Z2

A±a

5 A

−1/ 2

Epandin and combinin the terms in poers of " e obtain$

M atom

c2= A mn c

2−a1 Aa

2 A

2/3a4 A±a

5 A

−1/2 − a3 A1/3 4a4mn−m p−me c2 Z 4a

4

A

a3

A1/3 Z 2

Define each of the three terms as , , respecti/ely to obtain$

M atom

c2=− Z Z 2

nd hence e see that for a constant the mass follos a parabolic beha/iour as a function of "

ith the minimum mass occurrin at the inteer " /alue lyin closest to Z min

= /2 ,


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Phys 3033 – Nuclear Masses and Decays - 2) of 29

Nuclear masses along an odd#mass isobaric chain

+he fiure at riht shos the masses of atoms ith(12K, ncreasin mass is hiher on the pae and the

atomic number &"' increases from left to riht,t is eneretically possible for 12Kn ith "(9 todecay to 12K7n ith "(K0, +his is a β- decay

con/ertin a neutron into a proton &plus an electronand an anti-neutrino',

Ourther β- decays are possible until 12K+e is reached

hich is stable and cannot decay further,

Nuclei ith " hiher than 12K+e can decay /ia a seriesof β decays or electron capture processes to also

e/entually reach 12K+e, &+here are some limits onthese processes dependin on the mass differences asdescribed on the net pae,'

Note that the further aay from stability the larerthe enery emitted in the decay, %his !enerally leadsto shorter lifetimes for nuclei further from stability.

&or all odd-mass isobaric chains, there is only onestable nucleus.


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Phys 3033 – Nuclear Masses and Decays - 2J of 29

Mass limits for beta#decay processes

Chen ""min β- decay can occur and the limit for this to occur in terms of nuclear masses is$

M nuc A , Z M

nuc A , Z 1m

em

+he limit on the atomic masses is found by addin the mass of " electrons on each side and hence$

M atom

A , Z M atom

A , Z 1 &for β- decay to be possible'

n contrast hen "T"min β decay can occur and the limit in terms of nuclear masses is$

M nuc A , Z M

nuc A , Z −1m

em

No hen you con/ert to atomic masses &aain inore the neutrino' by addin the mass of "electrons to each side you ha/e some etra electron masses left o/er$

M atom

A , Z M atom

A , Z −12me &for β

decay to be possible'

?oe/er for electron capture the folloin holds true$

M nuc A , Z m

e M

nuc A , Z −1m

ddin &"-1' electron masses to each side and inorin the neutrino i/es the atomic mass limit$

M atom

A , Z M atom

A , Z −1 &for EC decay to be possible'

7o for /ery proton-rich nuclei both E%Bβ can compete but for proton-rich nuclei close to stability

ith a mass difference less than 2me it may be that only E% is eneretically possible,


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Phys 3033 – Nuclear Masses and Decays - 2I of 29

Mass parabolas for e"en#mass isobaric chains

Qou can ma6e an e/en- nucleus from either the number ofneutrons and protons both bein odd or them both beine/en, +he e/en-e/en nuclei are fa/oured compared to the

odd-odd nuclei because of the pairin &aK' term in the 7EMO,+he result is to parabolas that are split apart in enery,

+his means that hen folloin the isobaric chain donthrouh a series of decays the -/alues increase anddecrease often leadin to the decay lifetimes chaninirreularly, &+he decays of odd-odd nuclei are often fasterthan the decays of their e/en-e/en dauhters,'

+he lihtest odd-odd nucleus in a i/en chain can usuallydecay in to directions to to different dauhter nuclei,+his means that essentially all odd-odd nuclei are unstableto beta-decay ith 2? )i 10F and 1N bein the only stableodd-odd nuclei,

n any e/en-mass chain there are usually to e/en-e/ennuclei that are stable, Chile a !double# beta-decay li6ethat shon by the dotted line from 12I+e to 12I5e iseneretically possible it is /ery unli6ely, .nly 12 casesha/e been obser/ed across the entire nuclear chart and thelifetime limits are all T 1019 years88


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Nuclear Decay )chemes

Eamples of decay schemes are i/en here, mportant thinsto note$

• ifetimes for the parent isotopes,• Different modes of decay$ beta alpha etc,• Different branches are possible and these can feed

ecited states in the dauhter nucleus,• n ecited state in the dauhter nucleus can

subse>uently amma decay &emit a hih enery photon',

nuclear masses and decays

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