Nuclear Decay kinetics : Transient and Secular Equilibrium. What can we say about the plot to the right?

Transcription

1 uclear Decay kineics : Transien and Secular Equilibrium Wha can we say abou he plo o he righ?

2 IV. Paren-Daugher Relaionships Key poin: The Rae-Deermining Sep. Case of Radioacive Daugher (Paren) 1/2 () 21 i 5.1d - (Daugher) 1/2 () C (Granddaugher) 21 Po 138.4d 26 Pb (sable) Same problem as a sepwise chemical reacion!

3 . Mahemaics of he Problem 1. Paren: Decay: d d ( ) e ohing new here 2. Daugher: Formaion: Decay: ET: d /d = d /d = d e d This is a linear firs-order differenial equaion

4 3. Soluion e e e If he sample is pure iniially, second erm vanishes since 4. Daugher: C C Conservaion of aoms 5. Special Cases: Long ime soluions classificaion ime relaionship Rae-deermining sep a. o Equilibrium > > C b. Transien Equilibrium > > c. Secular Equilibrium >> > (c. is case of U-Th decay series)

5 C. o Equilibrium 1. Figure 3-4 (on righ) Daugher is rae-deermining sep 1/2 () > 1/2 () a. Curve a: Toal aciviy of iniially PURE sample of OTE: resembles wo-componen independen decay curve. OT!!! (oal) = () + () b. Curve b: Paren aciviy, () c. Curve d: Daugher aciviy, () ; noe growh curve

6 2. Curve c: Long-erm behavior: > > 1/2 () Therefore, / 1/2 () = e e oe: < ; ()() = + 3. Long erm curve:ll of has disappeared ; his defines 1/2 ()

7 D. Transien Equilibrium Paren decay is rae-deermining sep 1/2 () > 1/2 () 1. Fig. 3-2 (on righ) a. Curve a: Toal aciviy of iniially pure sample of (oal) = () + () decay curve ha iniially increases wih ime is a signaure of ransien or secular equilibrium. b. Curve b: Paren civiy: () c. Curve d: Daugher civiy: () d. Iniial growh of (oal) and hen decay according o he half-life of he paren.

8 2. Maximum Daugher civiy a. Maximum occurs when d /d = Therefore, differeniae equaion for and se his equal o zero; his defines max as ln( / ) max is maximum when = max b. Imporance Medical isoopes Milking a cow how long mus one wai before exracing he daugher aciviy again? c. Example: 55 Cs 3 y 137m 5 a 2. 6m 137 a (sable) max ln( / ) ln[ 1/ 2( ) / 1/ 2( )] m 3y ln[(3y / 2.6 min) / 2.6 min 5 min/ y] max = 58 min

9 3. Long-Time Soluion: Curve e a. For iniially pure, > > 1/2 () e e i.e., a long ime, raio / is COSTT wih ime SYSTEM PPERS TO E I EQUILIRIUM

10 4. Consequences a. Long erm decay is governed by paren b. civiy: muliply equaion in 3a. above by c c c = = = - c. Half-life of can be deermined by combining: long erm behavior 1/2 () aciviy raio above

11 E. Secular Equilibrium (Fig. 3-3) 1/2 () > > > > 1/2 () Special Case of Transien Equilibrium; e.g., U-Th Decay Series Rn gas problem; naural background from U and Th decay producs; daing 1. a. Curve a Toal aciviy of iniially PURE Sample OTE: a long ime, CTIVITY IS COSTT ; signifies special case b. Curve b Paren aciviy ; since / 1/2 ~, / = = consan c. Curve d Daugher aciviy growing in

12 2. General Soluion a. ssume = (i.e., pure ) ; 1/2 () > > 1 e e e 1 Growh curve b. Long-ime soluion >> 1/2 () ; e e = c. If c = c = = oal/2

13 d. Example: How many aoms of 222 Rn are presen in an iniially pure sample of 226 Ra afer 3 monhs? ssume 226 mg of 226 Ra; Wha is he aciviy? 1/2 ( 222 Rn) = 3.82 d ; 1/2 ( 226 Ra) = 162 y

14 Soluion : Rn = Ra Rn) (Ra) 1 (3.82 d) / 2( (Ra) = ( Ra) 162 y (365 d / y) Rn 1/ g 226 g / mole Rn = aoms = moles = STP d d Rn aoms = 3.82 d 144 m/ d d/min

15 3. Poins o keep in mind a. cons b. oal c. Half-life of Weigh and deermine from = c. Measure couns weigh d. Half-life of a = 1/2 () ; e = 1/2 = = ( 1 1/ 2), or corresponds o 1/2 () Since (oal) =, 1/2 () is also ime a which (oal) = (3/2)

16 F. Several Successive Decays C D, ec. 1. aeman Soluions d c d = c c

17 IV. ranching Decay Compeiive Decay Modes for he same nucleus EC 2 O 4 2O+O 2. Toal Probabiliy = = or ; = parial half-lives i 1/ 2 1/ 2(1) 1/ 2(2) 1/ 2(3). Parial Half-Life Definiion: The half-life a nucleus would have if he compeing decay modes were swiched off. (bu O swich).

18 C. Deerminaion of Parial half-lives 1. ranching Raio: R (ssume wo branches, 1 & 2) R = 1 1 1/ 2 oal oal (oal) 1/ 2( 1 ) 2. Measuremen; if c 1 = c 2 R = c c oal oal 3. Example: 4 K ; 1/2 = y 1/ 2 R(EC) = (.17) = EC ; EC = y R( ) =.893) = 1/ 2 ; = y OTE: 1/2 < 1/2 ( ) < 1/2 (EC)

19 V. Deerminaion of Half-lives Measurable range: 1 23 s o ~ 1 3 y (6 orders of magniude). T 1/2 1 year: Specific civiy 1. = c = couns/uni ime.693 = weigh of sample Measure = c = deecion coefficien e.g., 238 mg of 238 U has a specific aciviy of ~3 dps 1/2 c = 1 2. Limi: aural background radiaion; when (sample (bkg), large errors. Decay Curves: 1 y 1/2 1 s 1n 1.5 1/2 C. Elecronic Techniques D. Doppler Shif E. Channeling in Crysals F. Heisenberg Uncerainy Principle G. ngular Disribuions of Emied Paricles H. Small ngle Correlaions