14.5 Draft: Ra­dioac­tiv­ity

Nu­clear de­cay is gov­erned by chance. It is im­pos­si­ble to tell ex­actly when any spe­cific nu­cleus will de­cay. There­fore, the de­cay is phrased in terms of sta­tis­ti­cal quan­ti­ties like spe­cific de­cay rate, life­time and half-life. This sec­tion ex­plains how these are de­fined.

14.5.1 Draft: Half-life and de­cay rate

As a generic ex­am­ple of an un­sta­ble nu­cleus, con­sider tri­tium, an iso­tope of hy­dro­gen. The ${}\fourIdx{3}{1}{}{}{\rm {H}}$ tri­tium nu­cleus, the tri­ton, con­sists of one pro­ton and two neu­trons. The tri­ton suf­fers beta de­cay. Even­tu­ally it will eject an elec­tron and an an­ti­neu­trino. This turns one of the two neu­trons into a pro­ton. So the de­cay turns the tri­ton into the ${}\fourIdx{3}{2}{}{}{\rm {He}}$ he­lium nu­cleus iso­tope called the he­lion. The orig­i­nal tri­ton is lost.

Un­like the nor­mal ${}\fourIdx{4}{2}{}{}{\rm {He}}$ he­lium nu­cleus, the he­lion con­tains only one neu­tron. How­ever, it is sta­ble; there will not be any fur­ther spon­ta­neous de­cays.

(Note that the tri­ton de­cays even though it has two neu­trons, a magic num­ber. But the he­lion it de­cays to has two pro­tons, also magic. And just like a lone neu­tron de­cays into a less heavy pro­ton, two neu­trons in the tiny tri­ton is just too much of a neu­tron ex­cess com­pared to the neg­li­gi­ble ad­di­tional Coulomb re­pul­sion in the he­lion.)

The big ques­tion in this sec­tion is, when will an un­sta­ble nu­cleus like the tri­ton de­cay? Un­for­tu­nately, there is no com­plete an­wer to that ques­tion. A given tri­ton might last for 10 years, or it might last for 20 years or what­ever. It could last less than a year, though that is not very likely. It could last for 100 years, even though that is much less likely still. But there is no way to tell for sure.

How­ever, sup­pose you take a very large num­ber of tri­tons and record for each how long the tri­ton lives. Then you can av­er­age all these times and you get a num­ber that is called the “life­time” $\tau$ of the tri­ton. The cor­rect term would be ex­pected or av­er­age life­time, but this is physics, re­mem­ber. Cor­rect terms are not al­lowed. (To be fair, some physi­cists do use the proper term “mean life­time” in­stead of just life­time.)

If you av­er­age over enough tri­tons, you will find this mean life­time of tri­tons to be al­most 18 years. But not a sin­gle tri­ton will de­cay af­ter ex­actly 18 years. It is much like the ex­pected life­time of a new­born baby in the USA is, say, 80 years. De­spite that, al­most no one dies on their 80th birth­day. Some die at birth or as kids.

Still, there is one big dif­fer­ence be­tween peo­ple and nu­clei. If you have a per­son who is 80 years old, surely you do not ex­pect them to live un­til they be­come 160 years old. But if you have a bunch of tri­tons, on av­er­age this bunch of tri­tons will last for an­other 18 years. That is re­gard­less of how long these tri­tons have sur­vived al­ready when you start ob­serv­ing them. Nu­clear de­cay is a com­pletely ran­dom process that oc­curs out of the blue; it does not de­pend on any pre­vi­ous his­tory of the nu­cleus.

There is an­other is­sue. Un­less you are an ac­coun­tant by call­ing, why would you want to sit down, mea­sure life­times of nu­clei, and av­er­age them? What is the use?

A phys­i­cally much more rel­e­vant sce­nario is that you have man­aged to cre­ate a large num­ber of tri­tons, and you would like to know how long they will last for do­ing ex­per­i­ments. In par­tic­u­lar, you might want to know how long it takes be­fore half of the tri­tons you cre­ated with blood, tears, and tax-payer money, are gone. This phys­i­cally more mean­ing­ful time pe­riod is called the half-life $\tau_{1/2}$. It is re­lated to the mean life­time by a sim­ple fac­tor $\ln2$:

\tau_{1/2} = \tau \ln 2
\end{displaymath} (14.3)

Note that $\ln2$ is less than one. Half-life is some­what shorter than mean life­time.

The half-life of the tri­ton is 12.32 years. So if you ini­tially have a large col­lec­tion of tri­tium nu­clei, af­ter 12.32 years only half will be left. Af­ter 24.64 years, only a quar­ter will re­main, and af­ter a cen­tury only 0.4%. Af­ter a mil­len­nium only 4 10$\POW9,{-23}$% will re­main. (Since a gram of tri­tium con­tains about 2 10$\POW9,{23}$ atoms, af­ter a mil­le­nium, not a sin­gle atom would be left of a gram of tri­tium, if you man­aged to cre­ate that many of them!)

Now the earth is over 4 mil­lion mil­lenia old. So you will ap­pre­ci­ate that al­most none of the tri­tium ever present on earth still ex­ists. Some new tri­tium is con­tin­u­ously be­ing cre­ated in the at­mos­phere by high-en­ergy cos­mic rays, but be­cause of the ge­o­log­i­cally short half-life, there is no mea­sur­able ac­cu­mu­la­tion. The to­tal amount of tri­tium present on earth is vir­tu­ally zero.

Fig­ure 14.3: Nu­clear half-lifes. [pdf][con]
...ebox(0,0)[l]{$\fourIdx{209}{83}{}{}{\rm {Bi}}$}}

As in an ear­lier sec­tion, fig­ure 14.3 shows again the de­cay processes of the nu­clei. But un­like in the ear­lier fig­ure 14.2, this time the square size of each nu­cleus has been ad­justed to il­lus­trate its half-life.

For the full size squares in the fig­ure, the half-life is 10$\POW9,{18}$ sec­onds or longer. Now 10$\POW9,{18}$ sec­onds is about twice the es­ti­mated age of the uni­verse since the Big Bang. So for nu­clei that re­ally have full size squares in fig­ure 14.3, most of these nu­clei that the uni­verse ever cre­ated is likely to be still around.

On the other hand, if the square size of a nu­cleus is even slightly smaller than full size, then most of these nu­clei that the uni­verse ever cre­ated will have de­cayed. You may note that the square size of the ${}\fourIdx{3}{1}{}{}{\rm {H}}$ tri­ton is not­i­ca­bly smaller than full size.

On the other hand, all sta­ble green nu­clei have full size squares. You might also note bis­muth-209, ${}\fourIdx{209}{83}{}{}{\rm {Bi}}$. While not ac­tu­ally sta­ble, for all prac­ti­cal pu­poses it is. Its half-life of 6 10$\POW9,{26}$ sec­onds ex­ceeds the age of the un­verse by a fac­tor of over a bil­lion. In fact, it took physi­cists un­til 2003 to ob­serve that bis­muth-209 did ac­tu­ally suf­fer al­pha de­cay; un­til then it was be­lieved sta­ble. (Note that bis­muth 209 has 126 neu­trons, a magic num­ber.) The var­i­ous dou­ble-beta-de­cay light green nu­clei live even longer, on the or­der of 10$\POW9,{30}$ sec­onds.

Based on var­i­ous ar­gu­ments, it was de­cided to take the min­i­mum half-life shown in fig­ure 14.3 to be one nanosec­ond. Clearly the fig­ure needs some lower limit. And a nanosec­ond is re­ally fast for al­pha de­cay, and much faster than any beta de­cay. (Note that you do not see any re­ally small red or blue squares in fig­ure 14.3, and only a few yel­low ones.) And for gamma de­cay, a nanosec­ond is of­ten used as a cut-off be­tween prompt” and “iso­meric de­cay.

Still, some nu­clear de­cay processes are much quicker than a nanosec­ond. For ex­am­ple, you might note that the light nu­clei that de­cay though pro­ton or neu­tron emis­sion in fig­ure 14.2 have dis­ap­peared in fig­ure 14.3. Such a de­cay process may have a half life on the or­der of 100 ys, (i.e. 100 10$\POW9,{-24}$ sec­onds). That is much faster than the nor­mal de­cays. In fact, in terms of quasi-clas­si­cal physics, these de­cay times are com­pa­ra­ble to the time for a nu­cleon with a speed of say a tenth of that of light to move just once through a nu­cleus with a size of the or­der of fem­tome­ters.

An­other no­table nu­cleus that has dis­ap­peared is ${}\fourIdx{8}{4}{}{}{\rm {Be}}$, beryl­lium-8. Berylium-8 falls apart in two ${}\fourIdx{4}{2}{}{}{\rm {He}}$, he­lium-4, al­pha par­ti­cles. The half-life of that process is just 67 as (i.e. 67 10$\POW9,{-18}$ sec­onds). While it is tech­ni­cally called al­pha-de­cay be­cause an al­pha par­ti­cle is emit­ted, it is phys­i­cally very dif­fer­ent from the nor­mal al­pha de­cay of heavy nu­clei. In the al­pha de­cay of heavy nu­clei, a heavy nu­cleus is left, not a sec­ond al­pha par­ti­cle. In par­tic­u­lar, the berylium-8 de­cay is many or­ders of mag­ni­tude faster than the nor­mal al­pha de­cays dis­cussed in sec­tion 14.11.

Be­sides (mean) life­time and half-life, there is one more re­lated term that is com­monly used in de­scrib­ing nu­clear de­cays. It is called the “spe­cific de­cay rate” $\lambda$. The spe­cific de­cay rate is the rel­a­tive amount of nu­clei in a large sam­ple that is lost per unit time. Math­e­mat­i­cally, the spe­cific de­cay rate is just the in­verse of the mean life­time $\tau$:

\lambda \equiv \frac{1}{\tau}
\end{displaymath} (14.4)

To bet­ter un­der­stand the var­i­ous vari­ables math­e­mat­i­cally, it may be worth­while to see how the men­tioned re­la­tion­ships be­tween them arise. First, ac­cord­ing to the very de­f­i­n­i­tion of the de­cay rate $\lambda$ above, if the cur­rent num­ber of nu­clei is $I$, then the num­ber of nu­clei that are lost, call it $-{\rm d}I$, in an in­fin­i­tes­i­mally small time in­ter­val ${\rm d}{t}$ is given by

- {\rm d}I = \lambda I {\,\rm d}t
\end{displaymath} (14.5)

This can be in­te­grated af­ter mov­ing the $I$ to the left-hand side. The re­sult shows that if the amount of nu­clei at time zero is $I_0$, then at an ar­bi­trary later time $t$ the amount of re­main­ing nu­clei $I$ is
I = I_0 e^{-\lambda t}
\end{displaymath} (14.6)

(To check this ex­pres­sion, just dif­fer­en­ti­ate it.) To find the half-life, you can set $I$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12I_0$and $t$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\tau_{1/2}$; that shows that $\tau_{1/2}$ must be $\ln2/\lambda$. Also, from the ex­pres­sion above, you can com­pute the (av­er­age) life­time as

\tau=\left.\int_0^{\infty}t\frac{-{\rm d}I}{{\rm d}t}{\,\rm d}t \right/
\int_0^{\infty}\frac{-{\rm d}I}{{\rm d}t}{\,\rm d}t

giv­ing $\tau=1/\lambda$. That then gives $\tau_{1/2}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ln2\tau$.

14.5.2 Draft: More than one de­cay process

One very im­por­tant point must be em­pha­sized. Many nu­clei un­dergo more than one de­cay process. In that case, each de­cay process has its own de­cay rate, in­de­pen­dent of the other de­cay processes. In such cases,

Al­ways add spe­cific de­cay rates, never life­times or half-lifes.
The sum of the spe­cific de­cay rates gives the to­tal spe­cific de­cay rate of the nu­cleus. The rec­i­p­ro­cal of that to­tal is the ac­tual life­time. Mul­ti­ply by $\ln2$ to get the ac­tual half-life.

14.5.3 Draft: Other de­f­i­n­i­tions

You prob­a­bly think that hav­ing three dif­fer­ent names, the spe­cific de­cay rate $\lambda$, the life­time $\tau$, and the half-life $\tau_{1/2}$, for es­sen­tially the same phys­i­cal quan­tity, is no good. You want more! Physi­cists are only too happy to oblige. How about us­ing the term “de­cay con­stant” in­stead of spe­cific de­cay rate? Its re­deem­ing fea­ture is that con­stant is a much more vague term, max­i­miz­ing con­fu­sion. Even bet­ter, how does “dis­in­te­gra­tion con­stant” sound? Es­pe­cially since the nu­cleus clearly does not dis­in­te­grate in de­cays other than spon­ta­neous fis­sion? Why not call it “spe­cific ac­tiv­ity,”come to think of it? Ac­tiv­ity is an­other of these vague terms that the hated non­spe­cial­ists can­not make heads or tails of.

How about call­ing the prod­uct $\lambda{I}$ the “de­cay rate” or“dis­in­te­gra­tion rate” or sim­ply the “ac­tiv­ity?”

You prob­a­bly want some units to go with that! What is more log­i­cal than to take the de­cay rate or ac­tiv­ity to be in units of “curie,” with sym­bol Ci and equal 3.7 10$\POW9,{10}$ de­cays per sec­ond. (Of course you guessed that straight away. If you add 3 and 7 you get 10, not?) There is also the “bec­querel,” Bq, equal to 1 de­cay per sec­ond, de­fined but al­most never used. Why not “dpm,” dis­in­te­gra­tions per minute, come to think of it? Why not in­deed. The minute is just the ad­di­tional unit the SI sys­tem needs, and us­ing an acronym is great for cre­at­ing con­fu­sion.

Of course the ac­tiv­ity only tells you the amount of de­cays, not how bad the gen­er­ated ra­di­a­tion is for your health. The “ex­po­sure” is the ion­iza­tion pro­duced by the ra­di­a­tion in a given mass of air, in SI units of Coulomb per kg. Ex­po­sure is very im­por­tant for all peo­ple made of air. Of course, a bet­ter unit than a blasted SI one is needed, so the “roent­gen” or “rönt­gen” R is de­fined to 2.58 10$\POW9,{-4}$ C/kg. Why not?

But health-wise you may be more in­ter­ested in the “ab­sorbed dose” or “to­tal ion­iz­ing dose” or “TID.” That is the ra­di­a­tion en­ergy ab­sorbed per unit mass. That would be in J/kg or “gray,” Gy, in SI units, but peo­ple re­ally use the “rad” which is one hun­dredth of a gray.

If an or­gan or tis­sue ab­sorbs a given dose of ra­di­a­tion, it is likely to be a lot worse if all that ra­di­a­tion is con­cen­trated near the sur­face than if it is spread out. The “qual­ity fac­tor” $Q$ or the some­what dif­fer­ently de­fined “ra­di­a­tion weight­ing fac­tor” $w_{\rm {R}}$ is de­signed to cor­rect for that fact. X-rays, beta rays, and gamma rays have ra­di­a­tion weight­ing fac­tors (qual­ity fac­tors) of 1, but en­er­getic neu­trons, al­pha rays and heav­ier nu­clei go up to 20. Higher qual­ity means worse for your health. Of course you al­ready guessed that.

The bad ef­fects of the ra­di­a­tion on your health are taken to be ap­prox­i­mately given by the “equiv­a­lent dose,” equal to the av­er­age ab­sorbed dose of the or­gan or tis­sue times the ra­di­a­tion weight­ing fac­tor. It is in SI units of J/kg, called the “siev­ert” Sv, but peo­ple re­ally use the “rem,” equal to one hun­dredth of a siev­ert. Note that the units of dose and equiv­a­lent dose are equal; the name is just a way to in­di­cate what quan­tity you are talk­ing about. It works if you can re­mem­ber all these names.

To get the “ef­fec­tive dose” for your com­plete body, the equiv­a­lent doses for the or­gans and tis­sues must still be mul­ti­plied by “tis­sue weight­ing fac­tors and summed. The weight­ing fac­tors add up to one when summed over all the parts of your body. The ICRP de­fines “dose equiv­a­lent” dif­fer­ent from equiv­a­lent dose. Dose equiv­a­lent is used on an op­er­a­tional ba­sis. The per­sonal dose equiv­a­lent is de­fined as the prod­uct of the dose at a point at an ap­pro­pri­ate depth in tis­sue, (usu­ally be­low the point where the dosime­ter is worn), times the qual­ity fac­tor (not the ra­di­a­tion weight­ing fac­tor).