INVESTIGATION OF RADIOACTIVE EQUILIBRIUM AND MEASURING THE HALF-LIFE OF 137m Ba

Transcription

1 Vilius Uiversity Faculty of Physics Departmet of Solid State Electroics Laboratory of Applied Nuclear Physics Experimet No. 5 INVESTIGATION OF RADIOACTIVE EQUILIBRIUM AND MEASURING THE HALF-LIFE OF 37m Ba by Adrius Poškus ( adrius.poskus@ff.vu.lt)

2 Cotets The aim of the experimet 3. Tasks 3. Cotrol questios 3 3. Theoretical backgroud The law of radioactive decay Nuclear gamma radiatio Radioactive equilibrium i a system of two uclides 6 4. Experimetal setup ad procedure 8 5. Aalysis of measuremet results 0 6. The Geiger-Müller couter user s maual 7. Liear fittig 5

3 3 The aim of the experimet Ivestigate the pheomeo of radioactive equilibrium, test the law of radioactive decay.. Tasks. Usig a 37 Cs / 37 Ba isotope geerator, obtai the eluate cotaiig atoms of the uclide 37m Ba.. Usig a Geiger-Müller couter, measure the time depedece of the eluate activity ad the time depedece of the isotope geerator activity after creatig the eluate. 3. Plot the measured depedeces. 4. Calculate the half-life ad average lifetime of the secod excited state of the 37 Ba ucleus, ad their errors. 5. Compare the measuremet results with theoretical predictios.. Cotrol questios. Explai the law of radioactive decay.. Explai the origi of uclear gamma radiatio ad the cocept of uclear eergy levels. 3. Explai the cocept of radioactive equilibrium. 4. Explai the operatio of a isotope geerator ad its applicatios i medical diagostics.

4 3. Theoretical backgroud The law of radioactive decay It is kow that the umber of atoms of a radioactive uclear species (uclide) decreases expoetially with time: λt Nt () = N0e, (3.) where N(t) is the umber of atoms (uclei) at time t, N 0 is the iitial umber of uclei (at time t = 0), ad λ is the decay costat. The decay costat determies the rate at which the umber of radioactive uclei decreases: the larger the decay costat, the faster the radioactive decay. Aother way to specify the rate of decrease of the umber of radioactive uclei is by specifyig the half-life, which is related to the decay costat as follows: l T/ =. λ (3.) The half-life is the time eeded for the umber of radioactive uclei to decrease by half. Yet aother quatity that ca be used to describe the decay rate is the average lifetime of the uclide. It is equal to the average time that has to pass util a give ucleus udergoes the decay. The average lifetime (τ) is the iverse of the decay costat: τ =. λ (3.3) Equatio (3.) is the formulatio of the so-called law of radioactive decay. A equivalet way to formulate it is to take time derivatives of both sides of the equatio: dn = λn. dt (3.4) This is the first-order differetial equatio whose solutio correspodig to iitial coditio N(0) = N 0 is (3.). The left side of this equatio is opposite to the rate of decay (the umber of atoms decayig per uit time). The rate of decay is the so-called activity of the sample. Sice N decreases with time, from Eq. (3.4) it follows that activity also decreases expoetially with time: dn λt Φ() t = λn() t = λn0e. dt (3.5) The logarithm of this fuctio is a straight lie (see Fig. ). Its itercept gives the logarithm of the iitial activity l(λ N 0 ), ad its slope is the opposite of the decay costat ( λ). Thus, aalysis of the decay curve is a simple method of determiig the decay costat λ ad the half-life T /. lφ l(λn 0 ) l(λn 0 e λt ) = l(λn 0 ) λt 0 t Fig.. Time depedece of the atural logarithm of activity 3.. Nuclear gamma radiatio Gamma rays or gamma radiatio (deoted as γ) are electromagetic radiatio of high frequecy (very short wavelegth). As a rule of thumb, the term gamma radiatio is usually applied whe the wavelegth is much less tha the distace betwee atoms of solid matter (i. e. much less tha 0 8 cm). From the poit of view of quatum mechaics, electromagetic radiatio ca be iterpreted as a flux of elemetary particles called the photos. The photos of gamma radiatio are frequetly called γ quata (here, quata is the plural form of the ou quatum ). The wavelegth rage of gamma radiatio partially overlaps with the wavelegth rage of X-ray radiatio (e. g., see Table i Sectio 3 of descriptios of Experimets No. ad No. ). This is because i practice the distictio betwee those two types of electromagetic radiatio is based maily o their physical origi ad ot o their wavelegth. If the electromagetic radiatio is caused by quatum trasitios betwee uclear eergy

5 levels or by particle aihilatio reactios, the it is called gamma radiatio. If the electromagetic radiatio is caused by quatum trasitios betwee atomic eergy levels correspodig to iermost electro shells of a atom or by deceleratio of fast electros i matter, the it is called X-ray radiatio (a more detailed iformatio about X-ray radiatio is i Sectio 4 of the descriptio of Experimet No. ). The physical mechaism of uclear gamma radiatio is the followig. A ucleus that is formed as a product of radioactive decay or of a uclear reactio may be i a excited eergy state (this meas that the ucleus has excess iteral eergy). I such a case it evetually udergoes a quatum trasitio to the mai (lowest-eergy) state. Durig this trasitio, the excess eergy is usually emitted i the form of a photo (γ quatum). This trasitio oly causes a decrease of iteral eergy of the ucleus; it is ot accompaied by a chage of uclear compositio (i. e. of the umbers of protos ad eutros iside the ucleus). The eergies of γ quata emitter by a ucleus form a discrete sequece (i. e. there are several well-defied eergy values istead of a cotiuous eergy spectrum). This discrete ature of uclear eergy spectrum is a quatum pheomeo caused by localizatio of the costituet particles of a system i a small regio of space (the uclear costituet particles are protos ad eutros). I eergy diagrams, those discrete eergy values are show as horizotal lies (e. g. see Fig. ) ad are called eergy levels. The lowest eergy level is called the groud eergy level. The correspodig state of the ucleus is called the groud state or pricipal state. All other levels are called excited levels ad correspod to the excited states of the ucleus. The typical iterval betwee eergy levels of a quatum system is maily determied by spatial dimesios of the systems: the smaller the system, the larger typical differeces of eergy levels. That is why the typical itervals betwee uclear eergy levels (00 to 000 kev) are 0 to 0 6 times larger tha the typical itervals betwee atomic eergy levels, which vary from a few ev (for valece electros) to a few kev (for ier-shell electros). Thus, uclear γ radiatio is electromagetic radiatio caused by quatum trasitios of excited uclei to lower eergy levels (i Fig., oe such quatum trasitio is show by a vertical arrow). Sometimes the trasitio of a ucleus to the groud level proceeds i stages, as a series of trasitios to lower excited levels. The opposite trasitios (from lower to higher eergy levels) are also possible. However, sice those trasitios ivolve a icrease of the iteral eergy of the ucleus, they ca ot happe spotaeously (ulike the trasitios to lower levels, which do ot require ay exteral ifluece). They ca oly happe as a result of a exteral evet that supplies eergy to the ucleus, such as a collisio with a eergetic subatomic particle. A process that is the iverse of the process leadig to uclear γ radiatio is absorptio of a photo. 7/+ 37 Cs (30.04 m. y. ) 5 β MeV (94.6%) β -.74 MeV (5.4%) / - 37m Ba (.6 mi) MeV / MeV 3/ + 37 Ba 0 Fig.. The diagram of the radioactive decay 37 Cs 37 Ba (cesium-37 barium-37). The diagram shows the decay half-lives, the maximum eergies of the β particles (i. e. of the electros emitted from the ucleus), the probabilities of the two β-decay chaels, the lowest eergy levels of the 37 Ba ucleus ad the most probable quatum trasitio betwee 37 Ba eergy levels.

6 The eergy of the photo that is emitted whe the ucleus drops from the higher eergy level E to a lower eergy level E (or absorbed whe the ucleus udergoes the opposite trasitio) is equal to the differece of the two eergy levels: hν = E E, (3.6) where ν is the frequecy of the radiatio, ad h is the Plack costat: h = J s. Thus, i order for a ucleus to be able to absorb a photo, the photo s eergy must be equal to the differece of two eergy levels. The average time util the spotaeous trasitio of a excited ucleus to the groud level is called the lifetime of the excited state of the ucleus. If is usually of the order of ( ) s. However, some uclei stay i a excited state for a much loger period of time. For example, the average lifetime of a 37 Ba ucleus i the secod excited level (which is formed as a result of the β decay of 37 Cs) is s. Such uusually log-lived excited states are called the metastable states ad deoted by a letter m after the uclear mass umber (for example, 37m Ba ). Emissio of a photo is usually the most probable mechaism by which a excited ucleus loses the excess eergy, but it is ot the oly possible oe. Two other mechaisms of eergy loss are possible, too: iteral coversio ad iteral pair creatio. Iteral coversio is a process whereby a excited ucleus trasfers the excess eergy directly to a electro of the atom to which is belogs. Sice the eergy trasferred to the electro is much larger tha the bidig eergy of a atomic electro, the electro is ejected from the atom with a kietic eergy that is approximately equal to the uclear excitatio eergy. I the case of the previously-metioed secod excited level of 37 Ba, there is about 0 % probability of eergy loss by meas of iteral coversio. Iteral pair creatio is process whereby the excitatio eergy of a ucleus is trasformed ito total relativistic eergy of two particles that did ot exist prior the uclear trasitio to the groud state. Those two particles are the electro ad the positro (the positro is the electro s atiparticle). From the defiitio of the total relativistic eergy E = mc, (3.7) where m is the relativistic mass of the particles, it follows that iteral pair creatio oly becomes possible whe the excitatio eergy is larger tha m 0 c =.0 MeV, where m 0 is the rest mass of the electro. Gamma radiatio is also emitted durig some aihilatio reactios. Aihilatio is process that occurs whe a subatomic particle collides with its respective atiparticle. As a result of this process, both iitial particles disappear (hece the term aihilatio, which is defied as "total destructio" or "complete obliteratio" of a object). Sice eergy ad mometum must be coserved, the particles are ot actually made ito othig, but rather ito ew particles. Some or all of the ew particles may be photos. Sice their eergies are usually of the order of hudreds of kev or more, those photos are also classified as γ radiatio. The aihilatio reactio that is easiest to achieve i practice is the aihilatio of a electro ad a positro Radioactive equilibrium i a system of two uclides Let us discuss a mixture of two radioactive uclides, where oe uclide (the primary uclide ) decays ito aother radioactive uclide (the secodary uclide ). A example of such a system is a mixture of 37 Cs ad 37m Ba (see Fig. ). I this case, the primary uclide is 37 Cs, which decays ito 37m Ba (the secodary uclide), which the decays ito the stable groud state of 37 Ba. This decay chai ca be writte as follows: λ λ N N N 3, (3.8) where N is the umber of the uclei of the primary uclide (i this case, 37 Cs), N is the umber of the uclei of the secodary uclide (i this case, 37m Ba), ad N 3 is the umber of the uclei of the secodary uclide s decay product (i this case, groud state of 37 Ba). The chage of the umbers N ad N with time is described by a system of two differetial equatios: dn dn = λn ad = λn λ N. (3.9) dt dt The first equatio is simply the law of radioactive decay for the primary uclide (see Eq. 3.4). The secod equatio icludes a additioal positive term (λ N ) o the right-had side. This term reflects the fact that, i additio to the radioactive decay of the secodary uclide (this process decreases N ), there is aother process that iflueces N : productio of the secodary uclide due to radioactive decay of the primary uclide (this process icreases N, therefore the correspodig term o the right-had side of the secod equatio is positive). The solutio of the system of differetial equatios (3.9) depeds o iitial 6

7 coditios, i. e. o iitial values of N ad N. For example, if values of N ad N at the iitial momet of time (t = 0) are N (0) ad 0, respectively, the the solutio is λt N( t) = N(0)e, (3.0a) t N() t N(0) λ ( e λ = e λ t λ λ ). (3.0b) If (λ > λ ), the the secod expoetial fuctio i paretheses decreases faster tha the first oe. After a time that is much larger tha / (λ λ ), the secod term becomes much smaller tha the first oe, therefore the secod term may be igored. The N (t) becomes proportioal to a sigle expoetial λ fuctio e t, as is N (t). Thus, the ratio of quatities of both uclides becomes costat: N λ =. (3.) N λ λ Such a state of a mixture of radioactive uclides belogig to a sigle decay chai, whe the ratios of uclide quatities remai costat, is called radioactive equilibrium. If the decay costat of the secodary uclide (λ ) is much larger tha the decay costat of the primary uclide (λ ), the Eq. 3. ca be approximately re-writte as λ N λn. (3.) I. e., the decay rate of the secodary uclide (λ N ) is practically equal to its productio rate λ N, which, i tur, is equal to the decay rate of the primary uclide (i. e. to activity of the primary uclide). This coditio is satisfied i the system of 37 Cs/ 37m Ba, because the 37m Ba decay half-life (53 s) is about times smaller tha decay half-life of 37 Cs (30 years), i. e. the 37m Ba decay costat is about times larger tha decay costat of 37 Cs (see the relatio betwee decay costat ad decay half-life, Eq. 3.). If the radioactive equilibrium is artificially disturbed by removig a part of the secodary uclide from the system, the the amout of the secodary uclide begis to grow util the equilibrium is restored. This icrease of N ca be derived from Eq. 3.0b, takig ito accout that λ >> λ : λ λt λt λ t λ ( λ λ ) t λ λt N( t) = N(0) ( e e ) = N(0)e e N ( e ). (3.3) λ λ λ λ λ Sice i practice it is much easier to cout the umber of particles emitted by radioactive uclei per uit time tha to measure the umber of those uclei directly, is it more useful to formulate the latter result i terms of activities of the two uclides. This trasformatio is doe usig the fact that curret activity of a particular uclide is equal to the curret umber of its uclei multiplied by its decay costat (see Eq. 3.5). Thus, a removal of 37m Ba uclei from the mixture 37 Cs/ 37m Ba causes a decrease of the umber γ photos emitted per secod (because all photos are emitted due to decay 37m Ba 37 Ba), but the it begis to grow accordig to the equatio λt λt A ( t) λ N ( t) = λ N e A, (3.4) Number γ kvatų of skaičius γ photos per laiko per uit vieetą time A A (-e - ) ( ) ( ) e /λ A (t) Laikas time t Fig. 3. Time depedece of γ radiatio itesity after removig a part of the secodary uclide from the mixture cosistig of a log-lived primary uclide ad a short-lived secodary uclide (it is assumed that oly the secodary uclide emits γ photos). A is activity of the primary uclide, which is practically costat t 7

8 where A is activity of the primary uclide 37 Cs (see Fig. 3). At equilibrium, the umber of photos emitted per secod is equal to the primary uclide s activity A, which is practically costat. Therefore, A should be measured either before removig 37m Ba from the mixture or after complete restoratio of radioactive equilibrium. Havig measured A, the decay costat of 37m Ba ca be determied from the measured values of the differece A A (t), because λt A A ( t) = A. (3.5) e 8 4. Experimetal setup ad procedure The equipmet used for this experimet cosists of the followig devices:. A 37 Cs/ 37m Ba isotope geerator.. A Geiger-Müller couter (the dead time of that couter is τ d 0.00 s mi). 3. Isotrak ratemeter (its user s maual is give i Sectio 6). A geeral view of the equipmet is show i Fig. 4. Fig. 4. Geeral view of the equipmet A isotope geerator is a device for chemical separatio of a short-lived secodary uclide from a log-lived primary uclide. The separatio of the two uclides is achieved by passig a special solutio through the geerator. I the case of a 37 Cs/ 37m Ba isotope geerator, the metioed solutio cosists of 0.9 % NaCl ad % HCl. I Fig. 4, the isotope geerator is see as small cylider placed i frot of a Geiger-Müller detector (also called a Geiger-Müller tube ). The electros ( beta particles ) emitted durig the decay of the primary uclide (e. g. see Fig. ) are much less peetratig tha γ rays, therefore those electros are completely absorbed i the plastic cover of the isotope geerator. Oly the γ photos produced by quatum trasitios of the secodary uclide escape from the isotope geerator. Iside the 37 Cs/ 37m Ba isotope geerator, a material cotaiig 37 Cs is deposited oto a special orgaic polymer layer (called the substrate ), which facilitates trasfer of Ba ios ito a solutio cotaiig a sufficietly high cocetratio of Na (sodium) ios. I. e., the Na ios take up the positios o the substrate previously occupied by Ba ios, whereas Ba ios eter the solutio. Such a chemical reactio is called the io exchage. I this way, a majority of 37 Ba atoms ca be removed from the 37 Cs/ 37m Ba isotope geerator with the solutio. The same priciple is used i 99 Mo/ 99m Tc (molybdeum- 99 / techetium-99m) geerators for medical applicatios of radioactivity. I those geerators, the primary uclide is 99 Mo, which has a decay half-life of 66 hours, ad the secodary uclide is 99m Tc (a metastable excited state of 99 Tc), which has a decay half-life of 6 hours. Molibdeum-99 (as well as cesium-37) is a product of uclear fissio (which takes place, e. g., i uclear reactors). The solutio

9 cotaiig short-lived secodary uclide obtaied from such a geerator is ijected ito blood of a patiet, ad the its γ radiatio is recorded. I this way, the uclide s distributio iside the patiet s body ca be determied, which, i tur, may provide iformatio about various medical coditios (such as tumors). Oe of the reasos why the isotope geerators are useful i medicie is that the log half-life of the primary uclide makes trasportatio over log distaces easy, whereas the short half-life of the secodary uclide makes the itraveous ijectios less dagerous (the activity of the secodary uclide decays rapidly ad therefore does ot do ay sigificat damage to the biological tissues). 9 The measuremet procedure. Switch o ad prepare the ratemeter with the Geiger-Müller tube. I this experimet, the automatic coutig mode must be used, with the gate time of 60 s (see the ratemeter s user s maual i Sectio 6). Sice the measuremet results must be stored i the ratemeter s memory, which ca cotai o more tha 50 couts, it is importat to erase ay data stored i the memory before startig the experimet (this is doe by pressig the Start/Stop ad Reset buttos simultaeously). The state of the ratemeter after preparig it for the measuremets must be such that it is eough to press the Start/Stop butto to start a sequece of couts. Warig! Wear disposable gloves whe maipulatig the radioactive eluate. After the experimet, drop the gloves ito a bi.. Extract the uclide 37m Ba is from the 37 Cs/ 37m Ba geerator. This must be doe i the followig way: a) attach a plastic tube to a syrige, the take approximately ml of the elutig solutio ito the tube, the detach the tube from the syrige, b) uscrew the protective cap of the isotope geerator (it is at the side of the geerator where the label Isotope geerator is) ad screw the syrige with the solutio i its place, c) take off the protective cap that is at the opposite side of the isotope geerator (there is o screw thread there, so that it is eough to pull the cap) ad place the isotope geerator together with the syrige above a empty flask, d) push the pluger of the syrige util all solutio passes through the geerator ad is collected i the flask (this step should take about 0 or 0 s), e) put both protective caps back o the isotope geerator ad place it i frot of the Geiger-Müller tube as show i Fig. 4, f) place the flask with the radioactive eluate at a distace of m or more from the couter (so that it does ot ifluece the measuremet results), g) start the sequece of couts by pressig the butto Start/Stop o the ratemeter. 3. Do 30 couts, each cout miute-log. The results must be writte dow i the form of a table. 4. Empty the flask with the eluate (which is o loger radioactive at this poit) ito a sik. Repeat Step, but ow place the flask with the radioactive eluate i frot of the detector, whereas the isotope geerator must be placed at a distace of at least m from the detector. 5. Do 0 couts, each cout miute-log. Agai, write dow the results i the form of a table. 6. Place the flask at a distace of at least m from the detector. Do 0 couts of the backgroud. 7. Empty the flask with the eluate ito a sik, rise the flask with water. 8. Show the measuremet results (i table format) to the laboratory supervisor for sigig.

10 5. Aalysis of measuremet results. Correct the couts takig ito accout the so-called dead time of the detector (it is the time that has to pass after detectig a particle util the ext particle ca be detected). I. e., if N' is the measured umber of particles i mi ad τ d is the dead time (i miutes), the the true umber of particles i mi is. Calculate the average backgroud. N N =. N τ 3. Subtract the average backgroud from the corrected results correspodig to the radioactive eluate. 4. Plot the time depedece of the atural logarithm of the umbers obtaied i previous step (it should be plotted as a scatter graph). The calculate the decay costat λ of 37m Ba ad its error, usig the method of liear fittig, ad plot the fittig lie i the same graph. Calculate the average lifetime ad decay half-time of 37m Ba ad their errors (use Eq. 3. ad 3.3). Liear fittig ca be doe usig various computer programs, or it ca be doe by had, usig a calculator (see Sectio 7). 5. Calculate the average value of the last 0 couts of the 37 Cs/ 37m Ba isotope geerator (i. e. measuremets No. 30). At this stage of the measuremets, itesity of γ radiatio of the isotope geerator has bee completely restored, therefore this average is equal to the iitial γ radiatio itesity of the geerator (prior to uclide separatio). 6. Subtract each of the first te couts of the 37 Cs/ 37m Ba isotope geerator from the iitial itesity obtaied i previous step. The plot ad aalyze the resultig te umbers i the same way as i Step 4 of this sectio. 7. Compare the calculatio results of Step 4 ad Step 6 with each other ad with the exact value of 37m Ba half-life (53 s). 8. Discuss the geeral shape of both measured time depedeces. d 0

11 6. The Geiger-Müller couter user s maual Below are four pages scaed from a Isotrak booklet that is icluded with the measuremet equipmet. The Geiger-Müller couter cosists of a detector (the Geiger-Müller tube) ad a ratemeter. The ratemeter has a built-i the adjustable power supply for the detector, a LCD display for showig the couts, ad buttos for selectio of various coutig modes.

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15 7. Liear fittig The aim of liear fittig is determiatio of the least squares estimates of the coefficiets A ad B of the liear equatio y = A + B x, (7.) The essece of the method of least squares is the followig. Let us assume that a data set cosists of the argumet values x, x,, x -, x ad correspodig values of the fuctio y(x). A typical example is a set of measuremet data. I such a case, is the umber of measuremets. The measured fuctio values will be deoted y, y,, y. The theoretical value of y at a give argumet value x k is a fuctio of the ukow coefficiets A ad B (see (7.)), hece we ca write y(x k ) = y(x k ; A,B) (k =,,, ). The problem of estimatig the coefficiets A ad B is formulated as follows. The most likely values of A ad B are the values that miimize the expressio 5 ( k ) k. (7.) k = F( AB, ) y x; AB, y Expressio (7.) is the sum of squared deviatios of theoretical values from the measured oes (hece the term least squares ). That sum is also called the sum of squared errors (SSE). This expressio always has a miimum at certai values of A ad B. However, eve if the form of the theoretical fuctio y(x) correctly reflects the true relatioship betwee the measured quatities y ad x, those optimal values of A ad B, which correspod to the miimum SSE, do ot ecessarily coicide with the true values of A ad B (for example, because of measuremet errors). The method of least squares oly allows estimatio of the most likely values of A ad B. Everythig that was stated above about the method of least squares also applies to the case whe the theoretical fuctio is oliear. Regardless of the form of that fuctio ad of the umber of ukow coefficiets, a SSE expressio of the type (7.) must be miimized. However, whe y(x) is the liear fuctio (7.), this problem ca be solved aalytically (i.e., A ad B ca be expressed usig elemetary fuctios), but whe y(x) is oliear, this problem ca oly be solved umerically (applyig a iterative procedure). If y(x) is the liear fuctio (7.), the the SSE expressio (7.) ca be writte as follows: k k k k k k k k k= k= k= k= k= k= F( A, B) ( A + Bx y ) = A + B x + y + AB x A y B x y. (7.3) It is kow that partial derivatives of a fuctio with respect to all argumets at a miimum poit are zero. After equatig to zero the partial derivatives of the expressio (7.3) with respect to A ad B, we obtai a system of two liear algebraic equatios with ukows A ad B. Its solutio is x y x y B = k k k k k= k= k= xk xk k= k=, (7.4) B A = yk k k= x. (7.5) k= The B coefficiet is called the slope of the straight lie, ad the A coefficiet is called the itercept. The stadard deviatios (or errors ) of those two coefficiets are calculated accordig to formulas F x, (7.6) ( ) Dx mi Δ A = + Fmi Δ B =, (7.7) ( ) Dx where F mi is the miimum value of the SSE (7.3), i.e., the value of SSE whe A ad B are equal to their optimal values (7.4) ad (7.5), x is the average argumet value: xk k = ad D x is the variace of the argumet values: Dx = ( xk x) xk x = x =, (7.8). (7.9) k= k=