Quantum Mechanics of Alpha Decay Lulu Liu Partner: Pablo Solis 8.13 Junior Lab Experiment #4 December 5, 2007

Classical Mechanics: What Do We Expect to See? image from nu.phys.laurentian.ca r 0 ~ fm, V(r 0 ) is energy minimum Less than energy min! It happens at all! What is alpha decay? V(r) = ZZ  e 2 / r We see V(r 0 ) > E 

Quantum Tunneling image from hyperphysics.phy-astr.gsu.edu T is transmission coefficient, R is radius of nucleus. derived in Yung-Kuo Lim (originally by Gamow) Geiger-Nuttall Law / Gamow Relation: where = 1/  WKB Approximation

Verifying the Geiger-Nuttall Law Measure energy of emitted alphas Measure half-lives,  1/2 Plot Ln(1/  ) vs. Z*E -1/2 (look for linear relation) Errors Conclusions Equipment Radioactive Series and Bateman Equations

Naturally Occurring Radioactive Series

Bateman Equations Governs time evolution for A decays into B: solve:

Can Detector to MCA Setup to MCA... Decay chains start with Po

Rn-222 Half-Life Using Scintillator Result: Equilibrium established in several hours Assumption: Initially Pure Rn-222

Plot of Rn-222 Activity Background ~1/s Measured Value:  1/2 = 3.99 § 0.36 days Accepted Value:  1/2 = 3.84 days

Energy Spectrum from Can Detector Calibrated to Po-212 alpha energy of 8.78 MeV

Evolution of Peaks E

Po-218 Half-Life - Method - Integrate for 20s every minute - Assume constant background - Po-212 (.3  s halflife) daughter isotope of Bi-212 Po-218 Po Account for Bi-212 activity - Halt supply of Po atoms - Po-218/Bi-212/Rn-220 etc

Po-218 Half-Life Fit Measured Value:  1/2 = 3.14 § 0.33 min Accepted Value:  1/2 = 3.10 min

Bi-211 and Bi-212 Half-Lives Initial Conditions: assume equilibrium dA/dt = dB/dt = dC/dt... A, B, C are amounts of different isotopes in decay chain A/  A = B/  B = C/  C follows from lab guide eq. 4.1 Obtain ratios of isotope abundance. Conduct Bateman analysis – decay under no voltage

Bi-212 Fit to Bateman Equations Measured Value:  1/2 = 1.13 § 0.13 hr Accepted Value:  1/2 = 1.01 hr Po-215 -> Pb-211 -> Bi-211 Bi-211

Geiger-Nuttall Fit

Errors Half-Lives: statistical – Lack of actual equilibrium in cans – Peak widening (degrading detectors) Systematic Corrections: – background subtraction, bateman determinations Additional Effects:

Error- Continued Bismuth points Geiger-Nuttall Relation an approximation. Can be fit per decay chain, element, etc. Dependence on mass and atomic numbers, atomic radius

Conclusions Geiger-Nuttall Law verified Quantum mechanics offers explanation for alpha decay.

Po-218 Transient Behavior

Geiger-Nuttall Derivation Yung-Kuo Lim, 2000

Lack of Rise Time in Po min

Comparing Parameters m theoretical: -1.3 m measured: -3.3