Nuclear decay

1. Nuclear decay

2. The atom

3. Nuclei • A piece of the chart of nuclei carbon isotopes

4. Mass 1 unified mass unit: mass(12C)/12 Einstein: E=mc2 so: m=E/c2 1eV=1.60217733x10-19 J 1MeV=1.60217733x10-13 J 1u=931.494 MeV/c2

5. Binding energy The total energy (mass) of a bound system is less than the combined energy (mass) of the separated nucleons Example: deuteron 2H (1 proton + 1 neutron) mp =1.007825 u mn =1.008665 u mp+n =2.016490 u m2H=2.014102 u The deuteron is 0.002338 u lighter than the sum of the proton and the neutron. This is the binding energy and is the energy needed to break that nucleus apart

6. Binding energy M(A)=M(Z)+M(N)-B(N,Z) Most stable MeV per Nucleon Atomic mass

7. 222Rn: 222.017571 u 4He: 4.002602 u Sum: 226.020173 u difference: 0.005229 u (or 4.87 MeV) Now consider 226Radium 226Ra: 226.025402 u It is energetically more favorable for the 226Ra to emit an alpha particle (4He). 4.87 MeV in energy is gained. Where does this energy go to??? answer: kinetic energy of 222Rn and 4He

8. Radioactivity Spontaneous emission of radiation by unstable nuclei • 4He (alpha particles) •  particles (electrons or positrons) •  rays (energetic photons)

9. -decay Because of ‘tunneling’ the quantum mechanical wavefunction is not zero outside the nucleus and thus there is a small probability it can escape.

10. -decay --decay electron emission +-decay positron emission The atomic number is changed by 1, but the mass number remains constant

11. …not the complete story Besides the electron (positron) also an anti-neutrino (neutrino) is produced. if neutrinos would not exist all electron had this energy Kinetic energy of the electron (MeV)

12. -decay Just like in the case of electrons, the nucleus has different energy levels and going from one to another is associated with the release of a photon (MeV)

13. An example (226U)

14. radioactivity

15. Decay chain

17. In the lab: 137Cs (30.1 y) - 0.512 MeV (94.6%) - 1.174 MeV (5.4%) 137mBa(? min) The milk source  0.662 MeV(85%) 137Ba (stable)

18. decay R: decay rate or Activity : decay constant : decay time (=1/)

19. half-life N0 N0/2 1 Curie (Ci) = 3.7x1010 decays/s 1 Bq = 1 decay/s t1/2 For any given data point with N counts: error is N

20. 14C dating 14C is produced from 14N by Cosmic rays. While alive, organisms have a fixed 12C/14C ratio (1/1.3x10-12) (Carbon in CO2). After dying, no more 14C is absorbed and it decays away and the ratio of 12C/14C can be used for dating. Shroud of Turin Found to be 1320±60 years old

21. Radiation damage in matter Radiation damage: ionization effects in cells when radiation passes through it. 1 Roentgen (R): amount of radiation that will produce 2.08x109 ion pair in 1cm3 of air or the amount of radiation that deposits 8.76x10-3 J of energy into 1 kg of air. 1 rad (radiation absorbed dose): amount of radiation that deposits 10-2 J of energy into 1 kg of absorbing material

22. radiation safety The damage done by radiation also depends on the type of radiation: RBE (relative biological effectiveness): number of rad of gamma radiation that produces the same biological damage as 1 rad of the radiation being used: type RBE gamma-rays 1. beta-particles 1.0-1.7 Alpha particles 10-20 neutrons 4-10 dose in REM: dose in rad x RBE

23. Geiger counter

24. In the lab… Experiment with half-life, passage of radiation through matter etc.

25. Headline: Bush approval rating moves back up Retired general and former CNN consultant Wesley Clark remains at the top of the list of Democrats vying to replace Bush, with five Democrats in double digits and North Carolina Sen. John Edwards possibly getting a delayed announcement bounce as he rose from 2 percent to 6 percent. Example For any given data point with N counts: error is N

26. Types of errors • Statistical errors: Due to instrumental imprecision or statistical nature of observed phenomena (finite sample). • Systematic errors: Uncertainties in the bias of the data • Example: measurement of weight • Statistical error: scale imprecision (every measurement is slightly different) • Systematic error: The scale has an offset or is not properly calibrated • Example:Opinion poll Systematical error: poll group is biased (e.g. only people in one town etc) Statistical error: Sample error; if a Similar group was polled the results Would be different.

27. Average, Standard deviation and standard deviation of the mean • If we measure a quantity X N times, the best estimate for the value of that quantity is the average: • The spread in the measurements is given by the standard deviation: • The error in the determination of the mean is the standard deviation of the mean:

28. 2 i Example: mass on a spring We measure by how much a spring is stretched (d) if we hang a weight from it. The measurement is repeated 10 times. Measurement: 2.00±0.02 m

29. Standard deviation of the mean: error Average of 10 measurements Known weight Determination of a spring constant The same measurement is repeated with different weights (10 measurements per weight). We want to determine the spring constant k via the relation: F=kd weight=kd d(m) Slope=1/k w(N)

30. Fitting procedure Fit the data with the theoretical curve: x=(1/k)F [function type: y=ax] Use Kaleidagraph to fit (see details on LBS272L webpage)

31. Result from fitting Results: slope 1.0023±0.0067 so: (1/k)=1.002±0.007 Chisq=6.49 (2) How can we tell how good the fit was? Use the 2-value

32. 2-value and the goodness of fit • We have N measurements that we want to compare with theory (the data points with the fitted curve). • The data points are xi±dxi i=1,N • The theoretical values are ti=(1/k)fittedFi i=1,N • The the 2-value is: If a data point is close to the theoretical value, relative to the size of the error bar, it does not contribute strongly to the 2-value. If it is far from the theoretical value, relative to the error bar, the 2-value rises strongly

33. 2-value and degrees of freedom Degrees of freedom (D): Number of data points (N) minus the number of fitted parameters (Z). In spring example: N=6 Z=1 (1/k) So: D=N-Z=6-1=5 Degrees of Freedom 2-value D=5 2=6.49 g.o.f.=26.1% Goodness of fit: probability that the data matches the theory well (0-100%: use table or calculator on webpage)

34. Goodness of fit • From 2-value and D we find the goodness of fit (0-100%) • if g.o.f. is very low (<5%) the data does not match the theory well: • The theory could be wrong • The error bars are estimated too small • We were unlucky! • if g.o.f. is very high (>95%) the data matches the theory “unlikely” well. • This should only happen 1 in 20 measurements! Perhaps the error bars were estimated too large. • We were lucky! • A quick check of the goodness of the fit is the value 2/D. • 2/D1 for random error (I.e. not too small/large) • In the example: 2/D=6.49/5=1.3 okay!

35. Error propagation: addition/subtraction We have measured 2 quantities and their errors. What is the error in the sum/subtraction? example: A=5±1 B=8±2 C=A+B=5+8=13 dC=(12+22)=2.2 so: C=13±2 D=A-B=5-8=-3 dD=(12+22)=2.2 so: D=-3±2

36. Error propagation: product/division We have measured 2 quantities and their errors. What is the error in the product/division? example: A=5±1 B=-8±2 C=A*B=5*(-8)=-40 dC=|-40|[(1/5)2+(2/8)2]=12.8 so: C=-4·101±1·101 D=A/B=5/(-8)=-0.625 dD=|-0.625|[(1/5)2+(2/8)2]=0.2 so: D=-0.6±0.2

37. Error propagation: constants/polynomials example: A=5±1 k=3 C=kA=3*5=15 dC=|3|1=3 so: C=15±3 n=-2 D=An=5-2=0.04 dD=|0.04||-2|(1/|5|)=0.016 so: D=0.04±0.02

38. More info on error analysis • visit the webpage: • http://www.nscl.msu.edu/~zegers/lbs272l/lbs272l.html • link: help on error analysis • this lecture • extra explanations and examples • how to fit in kaleidagraph and get the fit parameters • 2-probability calculator Enjoy the lab!