Atoms of a given substance with differing of isotopes atomic masses …salfordphysics.com/gsmcdonald/pp/PPLATOResources/h-flap/... · 2016. 10. 9. · Nuclei, like atoms, can exist

FLAP P9.2 Radioactive decayCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Atoms of a given substance with differingatomic masses are said to be isotopes ofthat substance. The various isotopes of anelement all contain the same number ofprotons but different numbers of neutrons.


Marie Curie (1867–1934), who had twodaughters, was responsible for coining theterm ‘daughter’ for the product of aradioactive decay.


The total energy of a particle of rest mass

m0 and speed v is E = m0c2

1 − ( v2 c2 ).

The rest energy of such a particle isE0 = m0c2 and the

relativistic kinetic energy is

Ekin = E – E0 = m0c2

1 − 1

1 − (v2 c2 )

.

Throughout this module all references tomass will be to rest mass, and such masseswill generally be denoted by m rather thanm0.


The familiar quantity m0v2/2 provides anapproximate value for Ekin when v is smallcompared with the speed of light, c, but notin general.


The atomic nucleus and thenuclear binding energy graph is discussedmore fully elsewhere in FLAP, as is anexplanation of the stability line on Figure 1in terms of thestrong interaction between nucleons.


X-rays are also a type of ionizing radiation,but they are not a direct result ofradioactivity. They differ from γ-rays onlyin having a lower frequency.


The charge on the protone = 1.6022 × 10−19

1C

The atomic mass unitu = 1.6606 × 10−27

1kg = 931.5021MeV/c2


The binding energy of a nucleus is thedifference between the rest energy of theseparate nuclear constituents (N neutronsand Z protons) and the rest energy of thenucleus. The binding energy of a nucleusmultiplied by c2 is known as themass defect of the nucleus.


In this example we use K for kineticenergy, rather than Ekin, in order to avoidhaving too many subscripts.


The Newtonian formula for kinetic energyis an adequate approximation in situationswhere speeds are much less than that oflight (and hence the rest energy is muchgreater than the kinetic energy).This condition is satisfied by α-particles,since their rest energy is about 411GeV.


We have used a bold typeface p torepresent momentum to remind you that itis a vector quantity with magnitude anddirection.


According to Einstein’s theory of relativitythe momentum of a particle of mass mmoving with speed v is of magnitude

p = mv

1 − ( v2 c2 )

However, mv provides an approximatevalue for p in this case in the same waythat 1

2 mv2 provided an approximate valuefor K.


The symbol ≈ is read as ‘is approximatelyequal to’.


The transfer of kinetic energy in anelastic collision is greatest if the collidingparticles have comparable mass. Such acollision can be analysed in terms ofenergy conservation andlinear momentum conservation.These ideas are discussed elsewhere inFLAP, see the Glossary for details.


1 1µm (micrometre) = 10−61m


The weak interaction is not the same as theweak (van der Waals) intermolecular force,which is electromagnetic in origin.It should also be noted that the terminteraction is used to describe not only theforces that subatomic particles exert oneach other but also the way they decay.


Electron antineutrino is often abbreviatedto antineutrino.

ν is the Greek letter nu.


There are two other pairs of neutrinos(particle and antiparticle) which completethe family of six. Just as neutrinosinvolved in β-decay are associated with theelectron, each of the other pairs isassociated with another particle (the muonand the tauon) which in many respectsbehaves like the electron, but which isunstable and much heavier than theelectron.


The term half-life here is equal to the timefor half a sample of free neutrons to decayinto protons. The meaning of half-life willbe discussed more fully in Section 3 of thismodule.


It has been verified experimentally that theproton, if it were unstable, would have ahalf-life of at least 1031

1yr, but it has beenpostulated to be unstable by sometheorists.


Spin is a property of particles which youmay have met first in relation toelectron spin. Many other particles havespin and this is associated with theirmagnetic interactions, but a generaldiscussion of spin is beyond the scope ofFLAP.


Remember, the rest masses of theneutrinos are negligible or zero.


501pm = 50 × 10−121m = 0.051nm.


Nuclei, like atoms, can exist in a variety ofdifferent discrete states that may bedistinguished from one another by theircharacteristic energy. For any particularnucleus or atom the state of lowest energyis called the ground state. States of higherenergy are called excited states.


A photon can also be called a quantum(plural quanta) of electromagneticradiation. Both terms are used widely inFLAP.


1 1eV = 1.602 × 10−191J


Exponential functions are covered in themaths strand of FLAP. The origin of thiskind of exponential behaviour is discussedin the block on differential equations.


It is worth pointing out that the ratio of theabsorption coefficient for 21MeV γ-rays inlead and in water is approximately equal tothe ratio of the densities of the twomaterials, as implied above.


Although the first step in the decay of

92238U involves only α-emission, the

subsequent steps involve β−- and

γ-emission and hence uranium cannot

be used as a pure α-source.


The curie was named after Marie Curie1—another early investigator of radioactivity,and discoverer of radium.


Fluctuations in random events, such asradioactive decay or random errors in anexperiment are described elsewhere inFLAP.


If there is a mixture of nuclides, theactivity will be the sum of thecontributions from the various nuclides.


Note that the decay constant λ has no

connection with the wavelength λ ofelectromagnetic radiation.


For further discussion of units anddimensions, refer to the Glossary.


The calculus notation is introduced in themaths strand of FLAP.


The symbols t1/2 and T1/2 are alsocommonly used for half-life.


Differential equations, exponential decayand exponential functions are discussedmore fully in the maths strand of FLAP.See the Glossary for details.


1 1µg (microgram) = 10−61g = 10−9

1kg


1 1u = 1.66 × 10−271kg


41.51MBq = 1.11mCi


This is only an approximation, because N0is only approximate.


Cosmic rays are mainly very high energyprotons originating from the Sun or fromoutside our solar system. The particlesenter the Earth’s upper atmosphere and,because of their extreme energies, caninduce nuclear processes1—1such as areneeded in the production of 6

14C .The full topic lies beyond the scope ofFLAP.


Note that we are using the same symbol, R,for activity and for specific activity.The meaning should be clear from thecontext.


✧

Because of the difference in energies, theatomic electron will be ejected from theatom which, as a result, will become apositive ion.3❏


✧

It does so because originally there aremore neutrons than protons in the heavynuclei that are potential α-decaycandidates. The loss of two protons isproportionally a larger change than the lossof two neutrons and therefore emitting anα-particle increases the ratio of neutronnumber to proton number and thusincreases the stability.3❏


✧

If there were just two particles as productsof the decay then the argument wouldparallel that for α-decay. Since the masses

of the β−-particle and the daughter nucleusare fixed, the available energy(the Q-value) would split in a definiteproportion between the two and only onedefinite kinetic energy for the β−-particlewould be possible. If there is an additionalthird particle involved then the energy canbe split between the three in an infinitenumber of ways, limited only by the totalenergy available1—1and all kinetic energiesfor the β−-particle, up to some maximum(total energy available), become possible.


The maximum β−-particle kinetic energycorresponds to those decays in which theβ−-particle takes almost all the availableenergy with the third particle having anegligible amount1—1but the converse canalso happen, with the β−-particle havinglittle or no kinetic energy. All values ofβ−-particle kinetic energy between these

two extremes are possible.3❏


✧

Changes in the neutron/proton ratio withinthe nucleus result in changes in the nuclearbinding energy. In β+-decay the netpositive charge on the nucleus decreasesand so the electrostatic repulsion isreduced and this tends to increase thebinding energy. So there can still be a netgain in nuclear binding in β+-decay despitethe required increase in the rest mass of thenuclear constituents as a proton becomes aneutron.3❏


✧

The activity R(t) is the number of decaysper second and the number of nuclei N(t) isdimensionless, so λ will have the unit s-1.

☞3❏


✧

R(t) is directly proportional to N(t), so thegraph would have the same shape (thoughthe scales on the vertical axes would differby a factor of λ).3❏


✧

N(t) decreases to one-half its initial value.If R(t) halves in a given time interval, thenEquation 11

R(t) = λN(t) (Eqn 11)

shows that N(t) will also halve in this sametime interval.3❏


✧

Since 57301years is the half-life of 614C ,

you would expect the average specificactivity to be 7.51Bq1g−1.3❏

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Atoms of a given substance with differing of isotopes atomic masses …salfordphysics.com/gsmcdonald/pp/PPLATOResources/h-flap/... · 2016. 10. 9. · Nuclei, like atoms, can exist