Physics 3901 Intermediate Physics Lab

Blair Jamieson1, Jeﬀ Martin1

1 The University of Winnipeg, Winnipeg, MB, Canada

February 25, 2015

1 Contents

1 Radiation safety 9 1.1 Introduction ...... 9 1.2 Types and Sources of Radiation ...... 9 1.3 Radiation Dosages ...... 10 1.4 Calculation of Dosages ...... 10 1.4.1 γ-rays...... 11 1.4.2 β-particles ...... 11 1.5 Shielding, distance and time factors ...... 12 1.5.1 Shielding ...... 12 1.5.2 The Eﬀect of Distance ...... 13 1.5.3 The Eﬀect of Time ...... 14 1.6 Forms of Radioactive Sources ...... 14 1.7 Radio-toxicity ...... 14 1.8 Radioactive Half-Life ...... 14 1.9 Radioisotope Safety Practices ...... 15 1.9.1 Laboratory best practices ...... 16 1.9.2 Sealed Sources Leak Test ...... 18 1.9.3 Wipe Test - Radioactive Contamination Monitoring ...... 18 1.9.4 Storage and Waste Disposal ...... 18 1.9.5 Emergencies, Theft, Loss or Spills ...... 18

2 Latex Lab Reports 19 2.1 Scientific Writing ...... 19 2.2 The basic principles and elements of scientific writing ...... 19 2.3 The technical aspects and style of scientific writing ...... 20 2.4 How to write a lab report ...... 21 2.5 Using LATEX for PHYS 3901/4901 lab reports ...... 22 2.6 Latex writing exercise ...... 22

3 Error analysis introduction 24 3.1 Introduction to Statistical Uncertainties ...... 24 3.2 Signiﬁcant ﬁgures, Agreement, and Importance of Plotting Data ...... 25 3.3 Uncertainties in Calculated Quantities ...... 25 3.4 Systematic Errors ...... 26 3.5 Systematic Uncertainties ...... 26 3.6 The Normal Distribution ...... 28 3.7 Rejecting data ...... 30 3.8 Weighted averages ...... 30 3.9 The binomial distribution ...... 30 3.10 The Poisson distribution and Radioactive decays ...... 31 3.11 Error analysis problems ...... 33

2 4 Data Analysis with ROOT 34 4.1 C++basics ...... 34 4.1.1 Basic data types ...... 34 4.1.2 Making arrays of data ...... 35 4.1.3 Looping over data ...... 35 4.1.4 Object Oriented Concepts ...... 35 4.1.5 Reading data from ﬁle into our arrays ...... 36 4.1.6 Printing to the screen ...... 37 4.1.7 Branching and conditional statements ...... 37 4.1.8 Functions ...... 38 4.2 Using ROOT to Make Graphs and Analyze data ...... 38 4.3 ROOT Data Analysis Exercises ...... 40

5 Detection electronics lab 41 5.1 About NIM modules ...... 41 5.2 The oscilloscope ...... 41 5.2.1 Vertical controls and inputs ...... 41 5.2.2 Horizontal controls ...... 42 5.2.3 Triggering controls ...... 43 5.3 The Pulse Generator ...... 43 5.4 Detection Electronics Lab Tasks ...... 44 5.4.1 Equipment Used ...... 44 5.4.2 The Pulser and oscilloscope ...... 44 5.4.3 The pre-ampliﬁer and spectroscopy ampliﬁer ...... 46 5.4.4 The Single Channel Analyzer and Scaler ...... 47 5.4.5 The Linear Gate ...... 49 5.4.6 The Multi-Channel Analyzer ...... 51

6 Single Channel Analysis 52 6.1 Scintillation Counters ...... 52 6.1.1 Scintillating Materials ...... 52 6.1.2 Photomultiplier Tubes ...... 53 6.2 Response of Scintillator detectors to Gamma Rays ...... 54 6.3 Procedure ...... 56 6.3.1 SCA Informal Report ...... 57

7 Fitting Data 58 7.1 Fitting with Uncorrelated Errors ...... 58 7.2 Goodness of Fit ...... 59 7.3 Systematic Errors ...... 59 7.4 Propagation of Errors ...... 61 7.5 The meaning of Error Bars ...... 61 7.6 Relationship of Error Bar to Probability Distribution Function ...... 61 7.7 Covariance Matrices ...... 63 7.8 Example of correlated measurements ...... 64 7.9 Propagating Uncertainties ...... 65 7.10 Fitting Data Assignment ...... 66

3 8 Counting statistics lab 68 8.1 Objective ...... 68 8.2 Theory ...... 68 8.3 Equipment and Procedure ...... 71 8.4 Data Analysis ...... 72

9 Gamma Ray Spectroscopy 73 9.1 Gamma Ray Interactions ...... 73 9.2 Detector Resolution ...... 73 9.3 Gamma Ray Spectroscopy Experiment ...... 73

10 Beta Spectroscopy with Si(Li) Surface Barrier Detectors 77 10.1 Si(Li) Surface Barrier Detectors ...... 77 10.2 Theory ...... 78 10.2.1 Internal Conversion ...... 79 10.2.2 Fermi Theory ...... 79 10.3 Procedure ...... 81 10.4 Beta Spectroscopy Pre-Lab Homework ...... 83

11 Monte Carlo Techniques 85 11.1 Introduction ...... 85 11.2 How to do a Monte Carlo Simulation ...... 85 11.3 Generating random numbers ...... 85 11.4 Generating pseudo-random numbers from non-uniform distributions ...... 86 11.5 Monte-Carlo Assignment ...... 87

12 X-Ray Measurements with a Proportional Counter 90

13 Alpha Particle Energy Loss and Rutherford Scattering 98

14 Cosmic Ray Muon Lifetime 109 14.1 Cosmic Ray Muons ...... 109 14.2 Detector and Electronics Readout ...... 109 14.2.1 Scintillator and Photomultiplier Tube ...... 109 14.2.2 PMT High Voltage ...... 110 14.3 Readout Electronics ...... 110 14.4 Procedure ...... 113 14.5 Data Analysis ...... 115

15 Lock-in Ampliﬁer Measurements 119 15.1 About Lock-In Ampliﬁers ...... 119 15.2 Measuring a 1Ω Resistor ...... 120 15.3 Resistance of a Brass Rod ...... 121 15.4 Magnetic Shielding Measurements ...... 122

4 16 High Resolution Gamma Ray Spectroscopy 125 16.1 High purity germanium detector principles ...... 125 16.2 Experimental Method ...... 125 16.3 Energy Calibration ...... 125 16.4 Samples under study ...... 125

17 Magnetic Forces, Torque and Precession 126 17.1 Magnetic moment of a magnetic dipole ...... 126 17.2 Harmonic oscillation of a dipole about an applied field ...... 127 17.3 Precession of a rotating magnetic dipole about an applied field ...... 127 17.4 Force on a magnetic dipole in a gradient magnetic field ...... 127

5 List of Figures

1 The left plot shows a case where the data agree with a linear model, and the right plot shows disagreement with a linear model...... 25 2 The top left plot has small random error, and small systematic error. The top right plot shows small random error, and large systematic error. The bottom left plot shows large random error, and small systematic error. The bottom right plot shows large random error, and large systematic error...... 27 3 Gaussian distribution, and its 68% area around the mean...... 29 4 Binomial distribution for rolling a 3 or 4 k times in 10 rolls, compared to the Gaussian approximation Pgauss(µ = 3.33, σ = 1.49)...... 32 5 Poisson probability distribution function...... 32 6 Overview of Agilent oscilloscope controls. Figure from [1]...... 42 7 Example of a pulser pulse...... 44 8 Pulser pulse as seen on the oscilloscope, and connection of tee with termination to oscilloscope...... 45 9 Block diagram for amplifier test procedure...... 46 10 Pulse timing for logic pulses. The left plot shows that leading-edge (or threshold) timing leads to a time jitter, while the right plot shows that the zero crossing time will lead to better time resolution...... 48 11 Block diagram for SCA test procedure...... 48 12 Block diagram for linear gate test procedure...... 50 13 Diagram showing the main elements inside of a PMT. Image from Wikimedia commons. 53 14 Gamma ray interactions in a NaI(Tl) scintillator detector. Figure from Ortec [2]. . . . . 54 15 Typical gamma ray spectrum for a mono-energetic gamma ray detected with a NaI(Tl) scintillator...... 55 16 Block diagram showing the electronics and detector connections for the SCA lab. . . . . 56 17 Asymmetric Probability Distribution Function for some measured quantity x. If the distribution was symmetric the meaning of the mean and width (σ) would be well defined. For an asymmetric PDF it might be better to report the peak value, and have an asymmetric error bar...... 62 18 Measured length as a function of temperature for the two measurements. The horizontal dashed areas represent the two length measurement 1σ uncertainty bands, and the vertical dashed area represents the temperature measurement 1σ uncertainty band...... 65 19 Example of linear fit for an energy to channel calibration...... 66 20 Frequency of count of number of heads for 20 coin tosses repeated 10000 times is shown as the fine-dashed (green) line. The solid line is the binomial distribution. The smooth dot-dashed (blue) line is the Poisson distribution, and the histogram dot-dashed (red) line is sampling from the Poisson distribution...... 69 21 Chi-squared test probabilities, calculated using the root function TMath::Prob(χ2,ν) [3]. 71 22 Block diagram for counting statistics experiment setup...... 72 23 Gamma ray total cross section versus energy, showing the contribution to the cross section from various processes for carbon (top), and lead (bottom)[4]...... 74 24 Electronics block diagram for the gamma ray spectroscopy experiment...... 75 25 Cobalt-60 decay energy level diagram. Figure from wikimedia commons...... 76 26 Equilibrium charge carrier concentrations in a p-n junction diode. Figure from Wikimedia commons...... 77

6 27 Nuclear de-excitation energy levels after electron capture in 207Bi[5]...... 80 28 Kinetic energy spectrum of beta particles from the radioactive decay of 204Tl...... 81 29 Experimental set up...... 82 30 The solid line in the figure above is a PDF we wish to sample from. The black points are the (x, y) coordinates of uniform random values in the range 0, 1. The x value for the darker black points are accepted as random numbers with the distribution of the PDF. . 87 31 This picture shows one end of the wrapped scintillator and the PMT coupled to it. . . 110 32 The high voltage supply is shown in this figure. We will only be using channels one and two...... 111 33 The NIM and VME electronics used for this lab are shown in this picture...... 112 34 Electronics diagram showing how the PMT signals are sent to the digitizer, and processed to make a trigger based on a coincidence between the two PMTs...... 114 35 To check the signals from our LED pulser, and the signals from our PMTs, we will use three channels of an oscilloscope, as shown in this picture...... 115 36 Tree viewer window showing the variables in the table as leaves...... 116 37 Block diagram of a lock-in amplifier...... 119 38 Block diagram for the one ohm resistance measurement (left), and photograph of the physical setup (right)...... 120 39 Block diagram for the brass rod resistance measurement (left), and photograph of the physical setup (right)...... 121 40 Block diagram for the shielding measurements (top), conceptual diagram of connections for shielding measurements (left), and photograph of the physical setup (right). . . . . 123

7 List of Tables

1 The Quality Factor of Different Radiations ...... 10 2 Dosage Limits for Members of the General Public ...... 10 3 Linear Absorption Coefficients for a Few Materials in m−1 ...... 12 4 Maximum energy and average range of a few β sources ...... 13 5 Radio-toxicity of Selected Nuclides ...... 15 6 Half-Life of Selected Nuclides ...... 16 7 Data for standard error calculation example...... 24 8 Energy calibration data from an HPGe detector...... 40 9 Pulse amplitude [V] as a function of pulser pulse-height and attenuation setting . . . . . 45 10 Table of pulse amplitude at the output of the amplifier, as a function of amplifier gain settings...... 47 11 Example of operation of a Single Channel Analyzer. Refer to the lab manual text for a description of the example...... 50 12 Properties of Organic Fluorescent Materials ...... 52 13 Properties of Inorganic Scintillators ...... 53 14 Properties of several common gamma-ray sources...... 75 15 Sources used for the beta spectroscopy lab. The internal conversion (IC) process is described in Section 10.2.1...... 79 16 Fermi Function F (Z,P ) versus P for 204Tl...... 81 17 Energies in keV needed to calculate internal conversion electron energies...... 82 18 Properties of several common gamma-ray sources...... 126

8 1 Radiation safety

1.1 Introduction The use of radioactive materials warrants a careful and informed approach. This is so because of the potential hazards associated with their inappropriate and careless use. These hazards primarily centre around the damage of chromosomes within the human body by the ionizing effect of radiation (this may result in carcinogenic or congenital effects)[6]. This phenomenon is the primary one associated with low level radiation exposure and this document assumes that radiation exposures at the University would be in the low range due to the type and activity of radioactive material used. It should be realized that the ionizing effect of radiation is a naturally-occurring effect which happens continuously from cosmic radiation, natural terrestrial radiation and from radioactive isotopes occurring naturally in the human body. Although there is no threshold level of radiation below which the human body is always able to repair chromosome damage, the likelihood of repair increases with reduction of radiation intensity and ionizing effectiveness. Conversely, as the intensity and effectiveness of radiation exposure increases, this successful repair becomes less likely.

1.2 Types and Sources of Radiation Electromagnetic radiation may be emitted from nuclei when they naturally decay from one energy state to a lower one. Radiations so produced are γ-radiation (gamma-radiation). Nuclear decays may also produce charged α-particles (alpha-particles) which are helium nuclei and charged β-particles (beta- particles) which are electrons. The γ-rays cannot be distinguished from X-rays when they are of the same energy. The X-rays, however, are produced when atoms naturally decay from one electronic energy state to a lower one. The final form of radiation listed here is the neutrally-charged neutrons which are produced in a nuclear reaction called nuclear transformation in which high-energy γ-rays directly excite the nucleus, causing the ejection of a neutron. The relative hazards of these radiations varies with type of radiation. Energetic β− particles, neutrons and α-particles are more dangerous in most cases. However, another important factor here is whether the exposure to those radiations is external or internal. Internal exposure (due to ingestion) of an -emitter, for example, is an extreme hazard, although externally the alpha particles are stopped by the outer layer of the skin and pose little direct or immediate hazard. The energy of radiation is usually expressed in units of electron-volts (eV), kilo-electron volts (keV), or mega-electron volts (MeV). The electron volt is equal to the energy gained by an electron when accelerated through a potential difference of one volt. For example, a typical γ-ray emitter is 137Cs which emits a γ-ray of energy 0.662 MeV. (The symbol 137Cs refers to the isotope of Caesium with mass number 137.) Radiations are not all equally hazardous. The damage which is produced depends on the shortness of the trail of ionized atoms which is left as the radiations lose their energy. A shorter trail means a higher density of ionization and thus more effective damage, in that the body is less likely to effect a complete repair at this localized site. Based on this principle, it is possible to define a factor called the quality factor, Q, which is the best estimate of the relative biological effectiveness of the radiation or the RBE. The values of Q for the above radiations is listed in Table 1. Although the hazards associated with radiation depend on the type, it should be noted that the intensity and the energy also affect the hazard. The intensity of radiation (ie. Activity of the radioactive source) depends on the number of disintegrations per second of the atoms producing the radiation. The SI unit of intensity or activity is the Becquerel (Bq) which is equal to one disintegration per second.

9 Table 1: The Quality Factor of Diﬀerent Radiations Radiation type Q X-rays, γ-rays 1 β-rays with energy > 0.03 MeV 1 β-rays with energy < 0.03 MeV (ie. from tritium) 1.7 Low-energy neutrons 2.3 Neutrons up to 10 MeV and protons 10 α-particles (naturally-occurring) 20

The traditional unit of activity is the curie (Ci) which is equal to 3.7×1010 disintegrations per second or 3.7×1010 Bq. For example, a typical Physics student laboratory sealed calibration source would have an activity of 1.0 µCi (1.0 micro-curies) or 0.037 MBq (0.037 mega-becquerels).

1.3 Radiation Dosages The dosage of radiation can be expressed in terms of the amount of energy deposited in tissue by the radiation while it is absorbed. The unit of dosage is the gray (Gy) which is equal to 1 joule/kilogram. This unit correlates closely with the actual biological damage caused by the radiation in the case of αs and βs. Sometimes it is useful to consider the rate at which energy is being deposited and then the unit becomes Gy/s (grays per second). Not all dosages of radiation are equally effective in producing damage, due to the quality factor, Q (discussed above). Thus a dose equivalent unit is devised, which incorporates the Q into the gray. This unit is called the sievert (Sv). A frequently-used sub-multiple of the Sv is the mSy (milli-sievert) which is equal to 0.001 sievert. The Canadian Nuclear Safety Commission (CNSC) sets Canadian standards for maximum permissible dosages of radiation. These are set on an annual cumulative total basis in units of mSv/year. Two figures are published one for Nuclear Energy Workers and one for Members of the General Public. The figures used by the University of Winnipeg are those set for the general public, which are substantially lower than those for nuclear energy workers. These figures are considered to be well within the safe range of exposure with a negligible risk to the user. Table 2 lists the limits of dosage.

Table 2: Dosage Limits for Members of the General Public Organ, Tissue mSv/year Whole body, gonads, bone marrow 1 Bone, skin, thyroid 50 Tissue of hands, forearms, feet 50 Lens of the eye 15

1.4 Calculation of Dosages The dose may be calculated as the product of the dose rate and the exposure time. The rate will be affected by the distance from the source and any shielding between the source and the user (discussed in detail below). The dose calculation will be different for different sources. The calculations for γ-rays and s are detailed below.

10 1.4.1 γ-rays The dose rate from a γ source can be estimated by means of the following approximate expression:

D = 5.0 × 10−6CE/r2 (1) where D is the dose in mSv/h, C is the activity of the γ source in µCi, E is the total γ energy emitted per disintegration, in units of MeV, and r is the distance from the source in meters. The assumption in deriving the above equation is that the source of radiation is a point and that there is no buildup of ions.

Calculation A person holds a sealed 10 µCi 22Na source in his hand for 10 minutes while he discuses an experimental procedure with a collaborator. Assume that the thickness of the source backing is such that the distance to the ﬁngers is 1 mm. The 22Na emits one γray of 1.28 MeV and two annihilation γ-rays of energy 0.511 MeV. The total energy per disintegration is thus 2.30 MeV. The activity in mCi is 0.01 mCi and the distance is 10−3 m. The dose rate by the above equations is:

5.0 × 10−6 × 0.01 × 2.30/0.0012 mSv/h = 0.12 mSv/h (2)

And the dose is 0.02 mSv (since the time is 1/6 hour). If the person were to do this every day, ﬁve days a week, 50 weeks a year, then the total exposure would be 0.02 × 5 × 50 = 5 mSv. This would represent an exposure of 7% of the total annual allowable for tissue of the hands. Note: that holding a source in the hand for an extended period of time is an unsafe practice. It is preferable to use a pair of tweezers, and carry the source in a holder or tray.

1.4.2 β-particles The dose rates calculation for β-particles is somewhat diﬀerent than that for γ-rays. This is so because the βs generally have a spread of energies rather than sharply deﬁned energies like γs. The equation for βs is an estimation for energies in the energy range from 0.3 to 3.5 MeV. This excludes most β-emitters commonly used, except for 32P. The reason that lower energies of βs are not included is that they are easily stopped by a thin absorber, such as the glass walls of a vial or pipette. The dose rate for βs with energies from 0.3 to 3.5 MeV is:

0.065 mSv/minute at 1 cm from 1 µCi. (3)

The equation is mainly used to estimate the dosage when holding vials or pipettes in the hand. Note that latex gloves must be used when handling any vial or pipette.

Calculation Botanists frequently use 32P to trace a plants phosphate uptake from fertilized soils. Radio phosphate solution is drawn from a vial containing 500 µCi. The vial is steadied by holding it for 1 minute while a small quantity is drawn out. The dose to the ﬁngers during the minute will be:

0.065 × 500 = 32.5 mSv (4)

This procedure would deliver half a years maximum permissible dosage in one minute obviously an unsafe experimental practice. In such a case the vial must be held by a clamp.

11 1.5 Shielding, distance and time factors Shielding and maintaining a safe distance from the source are reliable methods for protection against external radiation exposure. Shields for γ- or X-rays should be constructed out of dense materials such as lead, steel, or concrete. Shields for βs are thinner and commonly made of glass or Plexiglas. The other important factor is the time of exposure.

1.5.1 Shielding The type of shielding required will depend on the type and energy of the radiation.

γ- and X-rays Shielding For the case of γ-rays and X-rays, the eﬀect of various density and thickness of materials can be calculated by the relation:

−µx I = Iαe (5)

Where Iα is the dose rate or intensity of γ- or X-rays hitting the shield, I is the dose rate or intensity (reduced) of γ- or X-rays transmitted through the shield, µ is the linear absorption coefficient (energy dependent) for the absorbing material of the shield, and x is the thickness of the shield. The linear absorption coefficient is listed for several different energies and absorbing materials in Table 3.

Table 3: Linear Absorption Coeﬃcients for a Few Materials in m−1 γ Energy (MeV) Pb Fe Concrete Water 0.5 164 65.5 20.4 9.7 1.0 77.1 47.0 14.9 7.1 1.5 57.9 38.3 12.2 5.6

Calculation 22Na source material is used in Positron research and is acquired in liquid (NaCl solution) form in a glass vial. The dose rate at the surface of a vial of 500 µCi (assuming a 1 cm distance) would be: 5 × 10−6 × 500 × 2.30/0.012 mSv/h = 11.5 mSv/h (6) What thickness of lead would be required to reduce the dose rate by 90%? Assume an average energy of 1 MeV. Then:

−77.1x I/I0 = 0.1 = e , (7) so that: x = − ln(0.1)/77.1 = 0.03 m. (8) Thus the shielding required is 3 cm thick.

β-source Shielding For β-particles travelling through a sold, a distance can be determined in which all of the β-particles are stopped. This is called the range. Frequently the range is given in units of mg/cm2 and the range in cm is determined by dividing the range ﬁgure (in units of mg/cm2) by the density of the material (in units of mg/cm2). A good set of data tables can be used to supply the range ﬁgures for various energy βs (see Table 4 for a list of some β ranges).

12 Table 4: Maximum energy and average range of a few β sources Nuclide Max β energy Range (mg/cm2) 3H 0.018 MeV Negligible 14C 0.156 MeV 28.3 32P 1.710 MeV 790 35S 0.167 MeV 31.5 45Ca 0.252 MeV 60.1

Calculation The radionuclide 14C is used frequently by Biologists in research. What thickness of glass (density 1500 mg/cm3) must be in the walls of the vial to stop all the βs? The range of the 14C β given as 28.3 mg/cm2. Thus the required thickness of the vial is:

(28.3 mg/cm2)/(1500 mg/cm3) = 0.02 cm. (9)

Thus a 0.2 millimetre thickness of glass will stop all βs. In practice the glass walls must be thicker than this for structural strength.

α-source Shielding α-particles, due to their size and charge are much more easily stopped than γs, X-rays, or βs. Any container material or a few millimetres of air will be eﬀective in stopping all the αs. For this reason, the danger of alphas is not in external exposure, but in internal exposure.

Neutron Source Shielding Neutrons are harder to stop than γs or βs because they are uncharged particles. The theory considered in determining effective shielding against neutrons is that of billiard- ball type collisions between the neutrons and atoms of the absorber. With this theory, it can be shown that the amount of energy absorbed per collision is greatest when the absorbing atoms are the most nearly equal in mass to neutrons. Thus, hydrogen atoms make effective absorbers. These must be contained in substances such as organic compounds (ie. Dense plastics or paraffin wax) or water.

1.5.2 The Eﬀect of Distance The distance of the user from the source strongly aﬀects the radiation dose. Assuming that the energy of the radiation is high enough that there is no appreciable absorption in air, the radiation at any point is inversely proportional to the square of the distance from the source. This principle can be expressed as the inverse-square law: I 2 2 = r0/r . (10) I0

Where I is the dose rate of radiation at a distance r and I0 is the dose rate at a distance r0.

Calculation The dose rate from a 57Co source is 1 mSv/h at a distance of 0.5 m. What is the dose rate at a distance of 5.0 m? Using the expression above,

2 2 2 2 I0r0/r = (1 mSv/h)(0.5 m) /(5 m) = 0.01 mSv/h. (11)

Note that increasing the distance by 10-fold reduces the dose rate by 100-fold.

13 1.5.3 The Eﬀect of Time The time of exposure is a factor which is sometimes neglected when considering the reduction of risk. This factor should be considered when the hazards are associated with external exposure to radiation from relatively strong sources. Then, simply stated, the radiation dose is cut in half for each reduction by a factor of two of the stay in the laboratory.

1.6 Forms of Radioactive Sources Radioactive sources can be categorized as closed or open sources. Closed sources are solids which are encapsulated securely so that they can be handled without any risk of contamination. The only risks posed by such sources is the exposure to the radiation dose from the source, or the accidental leakage of source material from the source capsule. Open sources can take on three forms: 1) liquid, 2) powder, and 3) deposited solid on a surface. Beyond the risk posed by exposure to the dose rate from such sources (which can easily be controlled), the risk of contamination is more dangerous. Accidental transfer of the material to the human body (directly, as dust inhalation or indirectly, as in hand to mouth after hand contamination) poses the greatest risk, as there are no counter-measures which can be taken after ingestion. The relative toxicity of various radioactive materials is discussed below. Diﬀerent safety guidelines apply to the use of each of the types of source materials. These are discussed below.

1.7 Radio-toxicity Radioactive sources are ingested without undue risk by medical patients in some medical procedures. The toxicity of such materials is considered to be low. On the other hand, the toxicity of other materials is high and even accidental ingestion of small amounts is undesirable. The principles which govern the determination of radio-toxicity of a substance relate to the form of radiation emitted and its Q value, the biological half-life of the material in the human body, and the organ, or localized site in which the substance tends to concentrate. For example, an α-emitter is known to cause significant harm internally. Internal exposure will continue until the radioactive half-life causes the radioactive nucleus to decay sufficiently, or until enough of the material is eliminated from the body. Toxicity of the material will not only depend on the nuclide which is the radioactive source, but on the compound to which the nuclide is chemically bound. For example, the radio-toxicity of 3H (tritium) is normally considered low. Tritiated water is readily absorbed into body fluids through lungs and skin, but is excreted with a half-life of only 10 days. (That is to say, the quantity of original material is reduced by 50% in 10 days). Thus the effect is not long-term. However, if 3H -thymidine is ingested, it would be permanently incorporated in DNA where is could deliver its dose almost entirely to the cell nucleus. Radioisotopes are categorized into four categories of toxicity: low, moderate, high, and very high. Some examples of radio-toxicity are listed in Table 5. Note that the toxicity can vary from the category listed if the compound in which the nuclide is contained has a dangerous affinity for a particular organ.

1.8 Radioactive Half-Life Nuclei in a radioactive sample disintegrate constantly, emitting various (characteristic) radiations as they move to a more stable form. The decay of the nuclei is an entirely random event, however the rate of the decay obeys the rules of a ﬁrst-order reaction. The decay rate is a unique characteristic of each

14 Table 5: Radio-toxicity of Selected Nuclides Radio-toxicity Radioisotope Critical Organ Low 3H Body ﬂuids Moderate 14C Fatty tissue 125I Thyroid 59Fe Spleen 32P Bone 35S Testis High 45Ca Bone 137Cs Body tissue 60Co Gastro. Tract 131I Thyroid 22Na Body tissue Very high 241Am Bone 210Pb Gastro. Tract 226Ra Bone radioactive nuclide. The decay rate can be simply related to the half-life of the nuclide. This is the 1 time in which the number of nuclei which may yet decay are reduced by a factor of 2 . The equations which describe the activity of a source as a function of time are:

−t/τ − ln 2t/t A = A0e = A0e 1/2 (12)

n A = A0/2 (13)

Where A0 is the activity at some time t0, A is the activity at some time t, t1/2 is the characteristic half-life and n is the number of integral half-lives which have elapsed. The ﬁrst equation above can be used to calculate the activity exactly when the activity at a particular time and the half-life are known. The second equation can be used if the time elapsed is a multiple of half-lives. The half-life of several nuclides is summarized in Table 6.

1.9 Radioisotope Safety Practices The Canadian Nuclear Safety Commission (CNSC) regulates the use of nuclear energy and materials to protect health, safety, security, and the environment, and to respect Canada’s international commit- ments on the peaceful use of nuclear energy[7][8][9][10][11]. The University of Winnipeg currently possesses a consolidated Nuclear Substances and Radiation Devices Licence issued by the CNSC. The University Radiation Safety Committee authorizes the pur- chase, use, and storage of radioisotopes at the University of Winnipeg. Authorization is provided to faculty and/or researchers in the form of an Internal Radioisotope permit issued under the University of Winnipeg Nuclear Substances and Radiation Devices Licence. Students must comply with all regulations and conditions outlined in both the consolidated licence and the Internal Permit, as well as the posted General Rules for Working with Radioisotopes in a Basic Laboratory in a laboratory.

15 Table 6: Half-Life of Selected Nuclides Radionuclide Half-life 241Am 458 y 210Pb 21 y 226Ra 1620 y 228Th 1.9 y 207Bi 30 y 45Ca 165 d 137Cs 30 y 60Co 5.3 y 131I 8 d 125I 60 d 54Mn 291 d 22Na 2.6 y 90Sr 28 y 7Be 54 d 109Cd 470 d 14C 5730 y 64Cu 13 h 55Fe 2.7 y 59Fe 45 d 32P 14 d 35S 87 d 65Zn 245 d 3H 12.3 y

1.9.1 Laboratory best practices The practices described below are required according to licence conditions and regulations. In all cases these represent good laboratory practices which promote the safe use of radioactive materials.

General Safety:

• Keep unauthorized persons out of the laboratory.

• Keep the laboratory locked when unoccupied.

• Keep external radiation exposure as low as reasonably achievable (ALARA) by using the principles of time, distance, and shielding.

• Use the minimum quantity of radioactivity possible.

• Use tongs, forceps, tweezers or other remote-handling equipment where appropriate.

• Dosimeters are to be worn at all times if required by the Radiation Safety Oﬃcer

• Dosimeters must be stored away from sources of radiation.

16 • In case of a radioactive spill or accident, follow emergency procedures and notify your supervisor and /or the Radiation Safety Oﬃcer immediately.

• Comply with the nuclear Safety and Control Regulations, Permit conditions and General Labo- ratory Rules for Working with Radioisotopes

Contamination Control:

• Do not eat, drink, store food, smoke or apply cosmetics while in the laboratory.

• Never work with unprotected cuts or with breaks in the skin.

• Clearly identify work surfaces used for handling radioactive materials.

Usage, Storage and Disposal:

• Store radioisotopes in a locked room or enclosure.

• Monitor radioisotopes at all times when in use.

• Place radiation safety warning symbols on radioactive containers and at the entrance to the storage room.

• Maintain up-to-date inventory, usage, and disposal records of all radioisotopes.

Additional practices to be followed while working with open sources:

• Wear lab coat and disposable gloves. Safety glasses should be worn whenever there is a potential for pressure build-up. Change gloves often.

• Work over a spill tray lined with absorbent paper.

• Work in a fume hood when working with dry powders or volatile substances.

• Monitor the laboratory for contamination at least weekly. Decontaminate as necessary. Keep a record of monitoring and decontamination results.

• Do not store contaminated materials at desk area.

• Glassware and other equipment used for radioactive work must be segregated until it has been monitored and, if necessary, decontaminated.

• Remove gloves, wash hands and monitor your clothing, hands and shoes when radioisotope work is ﬁnished and/or before leaving the lab.

17 1.9.2 Sealed Sources Leak Test A sealed source containing 50MBq or more must be leak tested:

• Where the source is being stored, every 24 months;

• Where the sealed source is located in a device, every 12 months;

• Where the sealed source is not located in a device, every 6 months.

The Radiation Safety Officer will take the leak test samples and send them to Cancer Care Manitoba for analysis. Leakage must not exceed 200 Bq. If leakage exceeds 200 Bq, the Radiation Safety Officer will advise your supervisor/instructor to discontinue using the source/device and the Radiation Safety Officer will contact the Canadian Nuclear Safety Commission.

1.9.3 Wipe Test - Radioactive Contamination Monitoring There are two methods for detecting and measuring radioactive contamination:

Direct - involves usage of a portable radiation contamination meter to measure removable and ﬁxed contamination.

Indirect - involves systematically collecting and counting wipe samples from work surfaces and measuring removable contamination. A cotton swab or ﬁlter paper may be used to wipe approximately 100 cm2 of the target area.

1.9.4 Storage and Waste Disposal Store radioisotopes in a secure container marked with a radiation warning symbol.Mark radioisotope laboratories or storage rooms/enclosures (e.g., refrigerator) with radiation warning symbol, the words Caution Radiation Hazard, list of radioisotopes and possession limit, emergency contact name(s) or job title, and 24 hr. emergency telephone numbers. Do not dispose of any waste without approval from the Radiation Safety Oﬃcer.

1.9.5 Emergencies, Theft, Loss or Spills Notify your lab instructor immediately if any radioactive material has been involved in an accident or fire. Theft or loss of any radioactive material shall be immediately reported to the Radiation Safety Officer. The Radiation Safety Officer must be notified of any occurrence of a radioactive spill. Spill procedures outlined in the University of Winnipeg Radiation Safety Policies and Procedures Manual shall be followed.

18 2 Latex Lab Reports

This section has some guidelines on how to write a physics review article, and provides some exercises to learn how to use latex. Latex is a powerful typesetting package which is used to write scientiﬁc articles.

2.1 Scientific Writing The final critical step in the scientific process is the effective communication of results and conclusions. Unfortunately, the importance of good scientific writing and communication is often overlooked and most students receive very little guidance on how to develop their writing and presentation skills. An important component of the PHYS 3901 and 4901 courses will be the development of good scientific writing. The basic principles of scientific writing are presented here and in several references [12, 13, 14, 15]. Students are encouraged to seek out the many other excellent resources on scientific writing that can be found in the library and on the internet (e.g. Refs. [16, 17, 18, 19]). Some of these references can be found on the course website at: http://t2kwinnipeg.uwinnipeg.ca/~jamieson/courses/ intermediate_lab/latex/Science-Writing-Handouts/.

2.2 The basic principles and elements of scientific writing The main purpose of scientific writing is to explain. As a result, scientific reports should be concise, complete, and accurate, written in clear, straightforward language, using words and phrases that have as precise a meaning as possible. Scientific reports should not be verbose, unduly long or complicated, and they should not contain jargon, colloquialisms, superfluous/emotive phrasing, or other distracting elaborations. While there is no single prescription for how to write well, the following ideas will certainly be of help when preparing your lab reports or any other scientific presentation. The following basic principles of scientific writing were excerpted and adapted from Barrass [16, pp. 28-30]:

• Explanation – Consider above all else the needs of your readers. What do they know or not know? What explanation is needed for them to understand and appreciate your work? Keep your reader in mind when preparing your report; put information where they expect to ﬁnd it and present material in a way that will help them grasp your message quickly and clearly.

• Clarity – Strive for clarity in your writing and illustrations. Just because you understand the message that you intend to convey does not mean that others will. Always ask yourself the question, “How will the reader (mis)interpret this?” Also, be concise. A short, straightforward report is always superior to one that is long, verbose and meandering.

• Completeness – Whether your work is a comprehensive treatise or a short report, be complete. Follow through on the promises you made in the introduction, covering all the topics and experiments mentioned there. Make sure that every statement is complete and that each line of argument is followed through to its conclusion.

• Impartiality – Make clear any assumptions underlying your arguments. Indicate how data were obtained, and specify the limitations of your work, the sources of error and probable errors, and the range of validity of conclusions.

• Order – The reader will ﬁnd your message easier to understand if information and ideas are presented in logical order.

19 • Accuracy – Accuracy and clarity depend on the care you take in the choice and use of words. Ask yourself, “Is this word/phrase ambiguous? Does it say what I want? How might the reader misinterpret it?”

• Objectivity – Every statement should be based on evidence and not on unsupported opinion. Never overstate your results.

Any written work should contain three main elements: introduction, body, and conclusion. Scientiﬁc reports and presentations typically also contain references, ﬁgures, tables, and appendices. The following descriptions of these elements were excerpted and adapted from Morrison [12]:

• Introduction – Here you provide the context of your report and give your reader a clear and succinct description of what it is all about. These introductory remarks are structurally embodied in the title, abstract and Introduction section of the report.

• Body – Here you develop a logical (but not necessarily chronological) sequence of ideas that will help your reader to understand the points you are trying to make. The body of a report typically spans the Introduction, Methods, Results and Discussion sections, and generally contains historical background, physical concepts, mathematical derivations, experimental details and measurements, data presentation, analysis, etc.

• Conclusion – Here you bring the report to closure, emphasizing what you most want your reader to remember. These concluding remarks are contained in a Conclusion or Summary section, and to a lesser extent in a preceding Discussion section.

• References – A Reference section is very important. Here you list the sources of information used in your report. Make sure to cite these sources within the report itself, especially where you have paraphrased or directly quoted from them – failure to do so constitutes plagiarism! A reference list also provides your reader with necessary background material and puts your report within the context of the larger scientiﬁc ﬁeld to which it belongs.

• Figures and Tables – Properly made figures and tables will help readers understand your ideas, data, and results quickly and easily. (A picture is worth a thousand words.) Be selective – show only the most important and necessary results. Figures are preferable to tables, but use the latter for archival purposes (documenting raw data in an appendix, say) or when a figure simply will not do. All figures and tables must be labelled and accompanied by a caption. The caption provides a brief explanation of the figure/table and identifies all of its elements (points, lines, columns, rows, etc.) Figure axes and table columns/rows must be labelled, including any appropriate units. You must refer to and discuss every figure and table within the report itself.

• Appendices – Appendices are optional. Use an appendix as a place to archive and describe ancillary information (data tables, computer code, derivations, etc.) that is pertinent to your work but would be disruptive to the reader if placed in the body of the report. You must refer to the appendix somewhere within the report, however.

2.3 The technical aspects and style of scientiﬁc writing While every journal has it own particular house style (on how to present references, for example), there are several technical aspects and tenets of style that one should observe in all scientiﬁc writing. Some of the more important points are detailed below. An excellent reference for physicists is the Physical

20 Review Style and Notation Guide [15], in particular Sections III and IV on typesetting mathematical material. • Write in standard English using complete sentences and paragraphs. Do not use point form. • Label/number every figure, table, appendix or displayed equation. Include a caption for every figure/table. Use horizontal lines only (to separate rows) in a table. • Make at least one explicit reference to each of the above within the body of the text. (Otherwise, delete it from the report as it appears to serve no purpose.) • Punctuate mathematical expressions as regular sentences or parts of sentences. Do not precede a displayed equation with a colon unless it is required by the grammatical structure of the sentence. See examples on page 13 and 18 of Ref. [15]. • Define every symbol, variable, parameter, etc. that is mentioned in the report. • Mathematical symbols, variables, parameters, etc. using the Latin alphabet are typeset in italic font. (The symbol for the speed of light is c.) • Do not surround a symbol by commas or parenthesis when it immediately follows the noun/phrase that defines it. (The speed of light c is known to a high degree of precision.) • Every number must include its appropriate unit. Units are typeset in roman, i.e. upright, font. (The accepted value for the speed of light rounded to four digits is 2.998 × 108 m/s.) • Every quantity found by measurement or derived from experiment must include an associated error or uncertainty. Observe the rules for significant figures. (We measured the speed of light to be (2.99 ± 0.02) × 108 m/s.) • Spell out any cardinal number that is less then ten or written as a single word. (We performed eleven trials. We interviewed 25 subjects.) • Do not begin a sentence with a symbol or a numeral.

2.4 How to write a lab report While there are no hard-and-fast rules about the specific sections to include in a lab report, the following arrangement is considered standard: Title, Abstract, Introduction, Methods, Results, Discussion, Conclusion, References, Appendix. An excellent strategy – excerpted and adapted from the LabWrite project [19] – is to compose the sections in the following order: 1. Methods – Using your lab manual, handouts, and notes taken during the lab as a guide, describe in paragraph form the experimental procedure you followed. Be sure to include enough detail about the materials and methods you used so that someone else could repeat your procedure. Do not re-write the manual – paraphrase the most important descriptions and derivations only. If you include figures from the manual, be sure to acknowledge this in the figure caption. 2. Results – Put your data into visual form first (graphs, figures, tables). Decide on the order in which you will present these results/visuals. Review all the data and write a one- or two-sentence summary of the main findings of this lab; this will form the opening of the Results section. In separate paragraphs, summarize the findings of each visual. Place these paragraphs in order, adding any additional text needed to smoothly link one paragraph/idea to the next.

21 3. Introduction – Clearly state the scientific concept of the lab and the main objectives, motivations and hypothesis of your work. Place your work in the context of the field in which it belongs. Also include in this section the historical context/significance and any necessary background theory. Two to three paragraphs should be sufficient to cover this material. (Feel free to create a separate Theory section if this part is a bit long.)

4. Discussion – State whether your results fully support, partially support, or do not support the hypothesis. Identify and refer to specific data that back up your claim. Using your understanding of the scientific concept of the lab, explain why the results did or did not support the hypothesis. In the latter case, discus how your understanding of the scientific concept has changed. Discuss, as appropriate, problems or uncertainties that may have affected your results, how your findings compare to others, and any suggestions for improving the lab. (Note: you may find it more effective in some instances to make a combined Results and Discussion section.)

5. Conclusion – Write a paragraph summarizing what you have learned about the scientiﬁc concept of the lab. Back up your statement with details from your lab experience. If there is anything else you have learned about from doing the lab, such as the lab procedures or kinds of analyses you used, describe it in a paragraph or two.

6. Abstract – Summarize each major section of the lab report – Introduction, Methods, Results, Discussion, and Conclusion – in a sentence (or at most two). Then string the summaries together in a block paragraph in the order the sections come in the ﬁnal report.

7. Title – Write a title that captures what is important about the lab, including the scientiﬁc concept of the lab, the variables involved, the procedure, or anything else that is important to understanding the message of the report.

8. References – List all the sources you referred to in writing the report, such as the lab manual, a textbook, a course packet, or scientiﬁc articles.

9. Appendix – Put any ancillary information (data tables, computer code, derivations, etc.) in an appendix. If more than one appendix is included, label as Appendix A, Appendix B, Appendix C, etc.

2.5 Using LATEX for PHYS 3901/4901 lab reports You lab reports should be written in LATEX, which is a document markup language that is particularly useful for typesetting scientiﬁc and mathematical content. You will be provided with a LATEX template in which to prepare your lab reports. LATEX is installed for your use on all computers in 2L14. It is also free to download and available for most platforms (Windows, Mac, Linux, Unix). There are several excellent books on LATEX in the library [20, 21, 22, 23], as well as resources online [24, 25, 26].

2.6 Latex writing exercise These are the instructions on what to do in this class.

1. Log in to a computer using your account.

2. Go to http://t2kwinnipeg..uwinnipeg.ca/~jamieson/courses/intermediate_lab/latex and have a look at the ﬁles there.

22 3. Copy the files in the directory LaTeX-Template-2012 to a new directory on your computer. Try opening the file LaTex-Template-2012.tex in Texniccenter, and try creating a pdf file from it. Show me the results.

4. In your newly created directory, copy the file LaTex-Template-2012.tex to a new file my-lab-template.tex. Ensure you can create a pdf file from that file. You can then use this technique of copying my-lab-template.tex to another file and editing the contents to write your lab reports.

5. Open up examples.tex and have a look at how to write the various equations, tables, sections, and so on. Implement the following items from examples.tex and examples.pdf into your newly created my-lab-template.tex(put them in the Results section): Equations (1), (9), Table 1, the two lists in Section 4, the list of greek symbols in Section 6.1, and equations (11) and (17) from Section 6.2. Show me a pdf ﬁle that has these examples properly displayed in it, with no reported errors from Texniccenter.

6. Download and unzip the LabManual.zip, and unzip it to a folder. This folder contains the current lab manual, which is longer latex example of how to use latex. It shows how sections can be included into a latex document, as can be seen in LabManual.tex. Each of the labs is broken into a separate latex ﬁle in the sections directory, and separate picture folders exist for each of the sections in the pictures directory.

7. Open LabManual.tex in Texniccenter, and have a look at how it includes various sections. Have a look at one or two of the latex ﬁles in the sections directory to see what appears in those ﬁles. Note that at the moment this example will not compile in the version of Texniccenter on the lab computer because some of the packages needed are missing. If you download a version of Texniccenter (the editor and GUI) on your own computer, along with MiCTex (the latex compiler), it will be able to download all of the required packages.

23 3 Error analysis introduction

3.1 Introduction to Statistical Uncertainties In order to understand if your measurement is signiﬁcant, the uncertainty on the measurement needs to be estimated. When reading scales, the uncertainty (often just called the error) can be estimated by exercising your judgement. For ﬁne scales a rule of thumb is that 1/2 the smallest increment is the uncertainty, while for coarse scales you can usually do better by estimating by eye. All measured quantities should include an estimate of the measurement uncertainty. For repeatable measurements one can get the best estimate as the average, and the uncertainty from the standard error in the measurements. For N repeated measurements xi, the average is:

N X xi x¯ = . (14) N i=1 The standard deviation in the measurements is: v u N u 1 X σ = t (x − x¯)2, (15) x N − 1 i i=1 and the standard error is: √ δx = σx/ N. (16)

Table 7: Data for standard error calculation example. i xi 1 2.1 2 2.4 3 2.3 4 2.2 5 2.3

The reported value in this case would bex ¯ ± δx. For example, consider the ﬁve measurements in Table 7. The average value of these measurements is: 1 x¯ = (2.1 + 2.4 + 2.3 + 2.2 + 2.3) = 2.26. (17) 5 The standard deviation is:

r1 σ = (2.1 − 2.26)2 + (2.4 − 2.26)2 + (2.3 − 2.26)2 + (2.2 − 2.26)2 + (2.3 − 2.26)2 = 0.114. (18) x 4 Finally, the standard error is: 0.102 δx = √ = 0.05. (19) 5 Which makes the measured valuex ¯ = 2.26 ± 0.05.

24 3.2 Significant figures, Agreement, and Importance of Plotting Data For the purposes of this lab, one or two significant digits should be kept in the uncertainty. For example 9.8213473 ± 0.0214 has too many significant digits, and should be rounded to 9.82 ± 0.02. Two numbers agree if their error bars overlap or touch. For example 7.4 ± 0.1 and 7.7 ± 0.2 agree. The values 6.5 ± 0.1 and 6.71 ± 0.02 do not agree. When reporting uncertainties it is best to use the absolute uncertainty δx. In some calculations it is useful to use the relative (fractional) uncertainty δx/|x|. Relative uncertainties may be reported as percentages. When comparing your data to a model, it is a good idea to not just fit the data, but to look at a plot of the data to visually see if the model and the data agree. Fig. 1 shows an example where the data agrees with a linear model, and another example where the data disagree with a linear model.

5 4 x(cm) x(cm) 4 3 3

2 2

1 1

0 200 400 600 800 200 400 600 800 m(grams) m(grams)

Figure 1: The left plot shows a case where the data agree with a linear model, and the right plot shows disagreement with a linear model.

3.3 Uncertainties in Calculated Quantities

For counting experiments, the square root rule can be used for the uncertainty in the number√ of events. For example if you count the occurrence of some event to be N, the uncertainty is N ± N. We will study this in more detail when we discuss the Poisson distribution. The rule for determining the uncertainty in sums and diﬀerences, is that they add in quadrature. For example if q = x + y, or q = x − y, where x ± δx and y ± δy are measured quantities, the uncertainty in q is given by: q 2 2 δq = δx + δy. (20) If instead we calculate a product q = xy, or a quotient q = x/y, the fractional uncertainties add in quadrature according to: s δ δ 2 δ 2 q = x + y . (21) |q| x y

25 For any function q(x, y, ...), the general method of determining the uncertainty in q is: s ∂q 2 ∂q 2 δ = δ + δ + .... (22) q ∂x x ∂y y

The above equations all assume that the uncertainties in x, and y are random statistical errors without any correlations. As an example of the above equation, consider the uncertainty in: y ln x q = . (23) x2 First we calculate the two derivatives: ∂q y 2y ln x = − (24) ∂x x3 x3 , and ∂q ln x = . (25) ∂y x2 So overall the uncertainty in q is: s (1 − 2 ln x) 2 ln x 2 δ = yδ + δ . (26) q x3 x x2 y

3.4 Systematic Errors 3.5 Systematic Uncertainties What is a systematic uncertainty? The most common definition is any error that is not a statistical error. A systematic uncertainty is a possible unknown variation in a measurement, or in a quantity derived from a set of measurements, that does not randomly vary from data point to data point. When reporting the uncertainty in a measurement, the statistical and systematic uncertainties are usually listed separately, such as: 6.3 ± 0.7(stat) ± 0.4(syst). Random errors estimated in the previous sections assume that there was no bias in the measurements. Any bias in the measurement that is not just a random deviation is a systematic error. Fig. 2 gives a pictorial representation of the difference between a random error and systematic error (bias). Some examples of systematic uncertainties from different types of measurements are: • You measure the length of an object, but worry that the ruler changes length with temperature.

• You try to measure the temperature, but miscalibrate the thermometer.

• You try to measure the brightness of a distant galaxy, but worry that intervening dust might make it dimmer than you expect.

• You try to fit an energy spectrum to a shape plus a background component to determine the size of a signal. There are different models for the background that result in different estimates for the signal size.

• A meter stick has the wrong scale. We can check this using a reference ruler, and change to using a ruler with a correct scale.

26 Random: small Random: small Systematic: small Systematic: large

Random: large Sysematic: small Random: large Systematic: large

Figure 2: The top left plot has small random error, and small systematic error. The top right plot shows small random error, and large systematic error. The bottom left plot shows large random error, and small systematic error. The bottom right plot shows large random error, and large systematic error.

27 • A rate changes for an unknown reason. This can be noticed by plotting the rate as a function of time. If a reason for the change cannot be determined, then the additional systematic error needs to be added to account for it.

• A source moves while doing your measurement. You can check the eﬀect of moving the source or shielding to diﬀerent locations to try to account for this.

• Someone else’s source may be nearby. This can be accounted for by changing the eﬀect of the nearby source as a function of distance.

To properly deal with systematics it helps if you can develop a model for how the systematic affects the measurement. For example does the systematic affect the measured data points themselves, or does it appear quantitatively in the calculations applied to the data? How do you determine a systematic uncertainty? So, you have an idea of a systematic effect giving you an uncertainty in your measurement, but how do you determine how large an uncertainty? There are various ways of making this estimate, such as:

• Take separate calibration measurements, that are separate from the main data

• Estimate the size of the eﬀect based on known parameters of the apparatus

• If data provides useful data about the systematic, ﬁt it from the data itself

• Theory. Some systematic uncertainties could be due to measurement uncertainties in theory parameters, or theorists’ estimates of errors due to approximations made.

• Data vs. Monte Carlo comparisons. Use calibration data to estimate how well Monte Carlo reproduces data, then use the Monte Carlo to predict the uncertainty in other quantities.

Even with these rough guidelines, estimating systematic uncertainties is a black art. Probably 90% effort in experimental physics comes from thinking of clever ways to reduce or measure the systematics. The challenge with systematic errors is that we need to consider what effects might affect our measurement, and try to do additional measurements which try to determine the size of the systematic effect. A usual way to look for a systematic effect is to make a change in the measurement settings in order to characterize the systematic. Once we have determined an uncertainty due to systematic effects, δsyst, it is often considered to be independent of the random error. In that case the total uncertainty in the measurement is just calculated using a quadrature sum of the statistical, δstat, and systematic uncertainties:

q 2 2 δtotal = δstat + δsyst. (27)

Alternatively, since the errors may not be uncorrelated, they can be reported separately as:

stat syst x ± δx ± δx . (28)

3.6 The Normal Distribution This section will introduce Probability Distribution Functions (PDFs), and in will study the normal (Gaussian) distribution as an example. The Gaussian distribution is the ideal model that is considered to describe the error bars we try to report on our measurements. A PDF, f(x) has the properties:

28 0.4

0.3

0.2 68% of area

0.1

0 -3 -2 -1 0 1 2 3 Number of σ from µ

Figure 3: Gaussian distribution, and its 68% area around the mean.

1. f(x)dx = the probability that any one measurement will give an answer between x and x + dx.

2. f(x) is normalized to have an area of 1: Z ∞ f(x)dx = 1. (29) −∞

The mean value of a PDF, also called the ﬁrst moment of a PDF is given by: Z ∞ x¯ = xf(x)dx. (30) −∞

The standard deviation of a PDF, σx, also called the second moment of a PDF is given by: Z ∞ 2 2 σx = (x − x¯) f(x)dx. (31) −∞ Note that the equations above are for continuous distributions. If the distribution is discrete (ie. only R P∞ takes on integer values), the relations are still true with the replacement of → ν=0. A Gaussian, normal, PDF is given by:

2 1 − (x−µ) f(x) = √ e 2σ2 , (32) σ 2π where µ is the mean of the Gaussian, and σ is the standard deviation. Figure 3 shows that 68% of the area of the Gaussian distribution lies within ±1σ of the mean. In other words we ﬁnd:

Z µ+σ 2 1 − (x−µ) I1σ = √ e 2σ2 dx = 0.68. (33) µ−σ σ 2π

29 (x−µ) To do this integral, we can do the substitution z = σ , which has diﬀerential dx = σdz. The integral becomes: Z 1 1 −z2/2 1 I1σ = √ e dz = erf(√ ), (34) 2π −1 2 where erf(z) is the “error function.” For reference erf(z) is deﬁned as:

z 2 Z 2 erf(z) = √ e−t dt. (35) π 0 Note that if we had been ﬁnding the fraction of times we would expect to measure something within 1.9σ of√ the mean, the integral would have been from −1.9 to 1.9, and I1.9σ would come out to be erf(1.9/ 2). The values for the integral above are tabulated in Appendix A of your text book[27]. Similarly one can show that 95.4% of the distribution falls within ±2σ of the mean, and 99.7% of it falls within ±3σ of the mean.

3.7 Rejecting data Never reject data based on statistical arguments. In order to reject a data point a real human error needs to be found, and the measurement re-done to conﬁrm the error.

3.8 Weighted averages If several measurements of the same quantity are made, the average value can be found, and the uncertainty updated. For example or N measurements xi ±δi, with standard deviations σi the weighted average is: PN x /σ2 P w x x = i=1 i i = i i . (36) wav PN 2 P w i=1 1/σi i 2 The rightmost part of the equation shows it in terms of weight factors wi = 1/σi . The uncertainty on the weighted average is given by: 1 σwav = pP . (37) wi

3.9 The binomial distribution The binomial distribution is a discrete PDF which describes the probability in games of chance, such as the odds of ﬂipping k heads in a row. These types of probability estimates can be reasoned out by carefully accounting for the diﬀerent combinations of ways that a particular outcome comes about. It can also describe the odds of rolling dice with a particular number several times in a row. One input to the binomial distribution is the probability, p, of attaining the result in a single toss. For example for an unbiased coin the probability of getting a heads is p = 0.5, for an unbiased die the probability of getting a six is p = 1/6, and in a deck of cards the probability of getting an ace is p = 1/13. The binomial distribution, for having k heads, sixes, or aces in n tosses, rolls or random draws is given by: n n! P (k, n) = pk(1 − p)n−k = pk(1 − p)n−l. (38) k k!(n − k)! The notation n!, is read as n factorial, and means n! = n × (n − 1) × ... × 1.

30 The average number of times we roll k sixes in n rolls is given by:

k=n X k¯ = kP (k; n) = np. (39) k=0 The standard deviation in the number of times we roll k in n rolls is given by:

k=n X ¯ 2 p σk = (k − k) P (k; n) = np(1 − p). (40) k=0 Note that as the number of tosses increases (as n → ∞), the binomial distribution looks more, and more like a Gaussian with: p Pbinomial(k, n) = Pgaussian(µ = np, σ = np(1 − p)). (41)

As an example for how to calculate odds using the binomial distribution, consider the following question. What is the probability of rolling a 3 or 4 on a 6 sided die k = 6 times in n = 10 rolls? The probability of a single event in this case (rolling a 3 or 4) is p = 1/3, and the probability for this occurring is: 10! 16 24 P (k = 6, n = 10) = = 0.0569 = 5.69%. (42) 6!4! 3 3 The average number of times that we would roll a 3 or 4 in ten rolls is: 1 k¯ = np = 10 × = 3.33. (43) 3 The standard deviation of the number of times we would roll a 3 or 4 in ten rolls is:

r10 2 σ = pnp(1 − p) = = 1.49. (44) k 3 3 The binomial distribution for P (k, n = 10) is shown in Fig. 4, and is compared to the Gaussian approximation Pgauss(µ = 3.33, σ = 1.49).

3.10 The Poisson distribution and Radioactive decays We will study this distribution, and radioactive decays in more detail when we do the counting statistics lab. For radioactive decays with a decay rate, R, in some time interval, t, we get an average number of counts µ = Rt =< N >. Since radioactive is a random process, in any time interval t, the number of counts we observe will vary. The probability of getting N = ν counts in a time interval t, is given by the Poisson probability:

e−Rt(Rt)N e−µµν P (ν = N, µ = Rt) = P (ν) = = . (45) µ N! ν! The Poisson distribution for diﬀerent values of mean expected numbers of counts, µ, is shown in Fig. 5. Like all PDFs, the Poisson distribution is normalized, so that:

∞ X P (ν) = 1. (46) ν=0

31 Pbinom(k,n=10)

P(k) 0.25 µ σ Pgauss( =3.33, =1.49) 0.2

0.15

0.1

0.05

0 0 1 2 3 4 5 6 7 8 9 10 k rolls

Figure 4: Binomial distribution for rolling a 3 or 4 k times in 10 rolls, compared to the Gaussian approximation Pgauss(µ = 3.33, σ = 1.49). )

ν µ = 1.0

P( µ = 2.5 0.3 µ = 4.0 µ = 5.5 µ = 7.0 0.2 µ = 8.5

0.1

0 0 2 4 6 8 10 12 14 16 18 20 ν

Figure 5: Poisson probability distribution function.

32 The mean value for the number of counts is given by:

∞ X µ =ν ¯ = νP (ν). (47) ν=0

The standard deviation in the number of counts, σν, can be found using the equation:

∞ 2 X 2 σν = (ν − ν¯) P (ν) = µ. (48) ν=0 What these distributions mean for us is that in our experiment, when we measure N counts, we can give the uncertainties: √ • in the number of counts is N ± N, and √ N N • in the rate is R = t ± t . Note that the fractional error decreases with count. In equations this is given by: √ δ δ N 1 R = N = = √ . (49) R N N N

3.11 Error analysis problems For homework, refer to your text book and do the problems[27]:

1. Do problem 2.4

2. Do problem 2.22

3. Do problem 3.24(a)

4. Do problem 3.28

5. Do problem 3.32

6. Do problem 4.2

7. Do problem 4.7

8. Do problem 11.9

9. Do problem 11.20

33 4 Data Analysis with ROOT

ROOT is an object-oriented C++ analysis package, developed at CERN[3]. ROOT uses a C++ interpreter called CINT, which on top of interpreting C++, includes commands which start with a “.”. ROOT C++ macros can be written, and loaded by the interpreter to produce 1-d, 2-d and 3-d graphics. It can also be used to analyze and ﬁt data. You can get a free copy of ROOT, and ﬁnd documentation at http://root.cern.ch. In addition to these notes, and the ROOT website, there are several ROOT tutorial slides available on the internet. A full set of tutorial slides can be found at: http://t2kwinnipeg. uwinnipeg.ca/~jamieson/courses/intermediate_lab/ROOT_onderwater/, which covers the capabil- ities of ROOT in more detail than we will have time to look at during class time.

4.1 C++ basics We will not need any fancy code to do our analysis. What you need to know are:

• What are the basic data types?

• How do we make arrays of data?

• How do we loop over our data?

• What are classes and objects in C++?

• How do we read our data into the basic data types?

• How do we print progress or information to the screen?

• How do we make decisions on which branch of commands to run?

• How do we make a function?

• How do we ﬁll the ROOT Histograms and graphs to do our analysis?

This section will address each of these very brieﬂy. We will be starting from a set of sample scripts that can be modiﬁed for the lab analysis, so often you will not even need to write an analysis from scratch. A copy of a useful tutorial which can be found online on how to program in C++ can be found at http:// t2kwinnipeg.uwinnipeg.ca/~jamieson/courses/intermediate_lab/Cpp_tutorial/Cpp_tutorial. pdf[28].

4.1.1 Basic data types The basic data types for real numbers are float and double for 4-byte and 8-byte lengths. The larger number of bytes allows for a larger range of real numbers to be represented. In ROOT there are also Float t and Double t which are essentially the same thing. The basic data types for integers are int and long for 4-byte and 8-byte integers. There is also a Boolean type called bool, which has values true = 1, and false = 0. Often an int type is used in a similar way to a Boolean, but in that case false = 0, and true is any other value. The only other catch with C and C++ data types is to learn about pointer types. For example in int* pvar, pvar represents a pointer to a location in memory that contains an int. If you want to get a pointer to a variable int var, then use an ampersand: pvar = &var. If you want to get the value of the variable which is pointed to by a pointer use the * to dereference it: var = *pvar.

34 4.1.2 Making arrays of data Square brackets are used to make an array of data. For example an array of 100 integers is declared as: int myarray[100]; To initialize the values in the array, you can do that using comma separated values inside curly brackets: int myarr2[5] = {1.0, 2.3, 4.5, 6.0, 7.0}; If the values are not initialized, you will need to set the values by some other means before using the array.

4.1.3 Looping over data There are two main C++ loops: the for loop, and the while loop. For loops repeat the commands inside the loop brackets for a fixed number of times, and while loops repeat until some condition is met. A for loop that loops with index i from zero to four to fill an array of data of length five looks looks like: int myarr3[5]; for (int i=0; i<5; i++){ // put your commands to execute 5 times inside these curly brackets myarr3[ i ] = 4.0*i; } Note that any line, or anywhere after a // on any line becomes a comment that is not considered to be part of the C++ command. A while loop is often used when reading in data from a file, where it is unknown beforehand how many items will be read in. In C++ the while loop is used to loop until some condition inside the brackets is met. A while(1) will continue to loop over the commands inside the curly brackets indefinitely, and a while(0) will break out of the loop and proceed to the next command after the end of the curly brackets. A break command inside a while loop will exit the loop. An example of a while loop will be seen when we discuss reading in data from a file.

4.1.4 Object Oriented Concepts In object oriented programming, a class is a description of a “thing” in the system. Class deﬁnitions include a way to create instances of the thing, data elements which describe the “thing”, and methods, which are functions which operate on the data contained in the class. An object is an instance of a class. We will not need to code any classes for our analysis, but will use several classes which are part of the ROOT data analysis package. ROOT is a collection of classes which are useful for data analysis. The classes we will most commonly use are: • TCanvas for a canvas to draw our graphs on

• TGraph for graphs without error bars

• TGraphErrors for graphs wit error bars

• TH1D for 1-dimensional histograms

35 • TF1 for 1-dimensional functions

• TLegend for adding legends to plots

• TAxis for modifying the axis on histograms and graphs

As an example of how C++ classes and objects work, you can for example make more than one graph object, in the example below. int n=5; double x[5]={1.0,2.0,3.0,4.0,5.0}; double y1[5]={0.7,0.9,1.3,1.9,2.7}; double y2[5]={1.7,1.9,2.3,2.9,3.7}; TGraph *tg1 = new TGraph(n,x,y1); TGraph tg2(n,x,y2);

In this example, the the new command in C++ is used to allocate new memory for the tg1 TGraph object, and tg1 is now a pointer to the object. A pointer is just a fancy way of saying the address in computer memory where the object is stored. The tg2 TGraph is the actual object itself, rather than a pointer to the object. Note that objects in a ROOT macro that are created using the new operator, rather than as an object, remain in existence after the macro ends. This is because of the way persistence works in C++. Anything created using the new operator stays in memory until it is deleted with the delete operator is called. This is useful because we want our graphs and canvas to stay there even when this script ends, so that we can see our graph.

4.1.5 Reading data from file into our arrays The ifstream class is a standard C++ class that defines objects which read in from a file. In the example below fin is an ifstream object (an instance of the ifstream class). Note that in this case we create an instance without using the new command, so the ifstream object will automatically be deleted when the script is done. In the example fin.open(...) calls the open is a method (function) of the ifstream class, which is used to define which file to read from. The two greater-than symbols after the fin object fin>> is the ifstream operator to read a value from the file opened by fin.open. Note that it tries to read in a value of the same type as x[i] and y[i] (floating point numbers). If the fin>> operator doesn’t get any data it returns zero, ending the loop. int MAX=1000; int i=0; double x[MAX],y[MAX],ex[MAX],ey[MAX]; ifstream fin; fin.open("SCA_data.dat"); while ( fin >> x[i] >> y[i] ) { ex[i] = 0; /// define error in x-axis value as zero ey[i] = sqrt(y[i]); /// y[i] is a count, error is sqrt(count) i++; /// ++ increments by 1 the value in i }

36 4.1.6 Printing to the screen In C++ the std::cout class is used, and to make a new line the std::endl command is used. Note that std part is actually a namespace, which is a way of packaging classes together. The following code: using namespace std; double xbar=100.0; double sigma_x=sqrt(xbar); double delta_x=sigma_x/5.0; cout << "xbar is " << xbar << endl; // endl is a carrage return. cout << "sigma_x is " << sigma_x << endl; cout << "delta_x is " << delta_x << endl; will print to the screen: xbar is 100 sigma_x is 10 delta_x is 2

4.1.7 Branching and conditional statements In order to decide which set of commands to run, we can use an if statement. The basic structure of an if statement is: if (condition) { // commands if condition is true

} else { // commands if condition is false

}

The condition can be made by using standard math condition checks such as less than x < 5, greater than x > 5, and so on. Combinations of conditions can be put together using “and” (&&) or “or” (||) operators. For example a conditional check might be: if ( x>5 && x<10) { // commands if x is between 5 and 10

} else { // commands if x is less than or equal to 5, or // x is greater than or equal to 10 if ( y>7 || y <0 ){ // commands if y greater than 7 or less than zero

} }

Note that the else statement can be left out, but it is often a good idea to have an else statement for every if statement. That way you know what happens if the condition in the if statement is not met.

37 4.1.8 Functions Functions are a way to put together several C++ commands that you want to run several times, or with diﬀerent arguments. A typical mathematical function could be written as a C++ function. As an example we could write the Poisson distribution in Eqn. 85 as a function. In this example, the Poisson function makes use of a factorial function, which recursively calls itself to calculate a factorial. int factorial(int n) { if ( n==1 || n==0 ) { return 1; } else { return n*factorial(n-1); } } double MyPoissonFcn( int aN, double aNavg){ return ( std::exp(-1.0*aNavg) * std::pow( aNavg, aN ) / factorial(aN) ); } In the example above, two more functions from the std namespace are used. Several math functions are built into the std namespace. Note that each of the functions above start with either int or double, to tell you what type of number the function returns. The variables in brackets after the function name tell you what values the function needs to be given in order to do a calculation. For example the MyPoissonFcn needs the number of events observed aN, and the average number of events expected aNavg. The value returned by the function is put in the function’s return statement. If your function does not return any values, then it should be prefaced with a void keyword, and the return statement is optional. To then call the function, and print the Poisson distribution value for several diﬀerent input values: std::cout<<"Poisson( N = 5, Navg = 2 )="<Set

When we hit “tab” after the →, we get a long list of methods, starting with:

SetBinContent SetBinsLength SetAxisColor SetAxisRange SetBarOffset SetBarWidth SetBinError

Again we can use the tabbing if we pick a particular method, and then hit tab after we put the opening bracket for the method to figure out what arguments it is expecting to be passed. If instead of “tabbing” to find the methods and arguments, you need to look up a class to figure out what it does, or read more details on what is done by a particular method, it is best to refer to the ROOT reference manual at http://root.cern.ch/drupal/content/reference-guide, and choose the “pro” (for production) version of the manual.

39 4.3 ROOT Data Analysis Exercises For the rest of the lab period we will look at several examples of ROOT macros that exist on the course website. In order to try out these ROOT macros, make a directory where you can put the macros. Then place a shortcut to ROOT in the same directory. In order to make that directory the working directory for ROOT, so it can find your macro files do the following. Right click on the ROOT shortcut, select Properties, and then select the shortcut tab. Then change the value inside the box next to “Start in:” to be the path to your directory with the root macros in it. What to do? Download all of the files from t2kwinnipeg.uwinnipeg.ca/~jamieson/courses/ intermediate_lab/root_macros to a single common directory. Try to run the macros, in the order in the following enumeration. After running them go through the files to see that you can understand how they are working. Write some notes about what each of the macros does in your lab notebook. This exercise will be marked as one of the informal labs, and will be usful for you try these in order to be able to do the data analysis for your labs.

1. MyPoissonFcn.C has the example shown in the introduction to functions in the previous section.

2. SCA example.C has an example of a TGraphErrors made from data in a ﬁle called SCA example.dat. It prints the mean standard deviation, and standard error in the data.

3. MCA example.C has an example of how to read data from a spectrum file to do a simple fit to the peak of a gamma ray spectrum. Note that the header information from the original spectrum file had to be stripped using notepad. A second MCA example 2.C exists. See if you can find the difference between the two. Do they work the same?

4. The Counting Stats examples directory contains two examples relevant for the counting statistics lab.

In addition to looking over the example macros, put the energy calibration data shown in Table 8 into a ﬁle called ecal.dat. Write a ROOT macro ecaldata.C with a function void ecaldata() that plots the channel versus energy data with error bars. E-mail your completed macro to bl.jamieson@ uwinnipeg.ca with the subject line “P3901 Root Data Analysis”.

Table 8: Energy calibration data from an HPGe detector. Energy (keV) Channel 140.511 238.8±0.5 181.068 308.2±0.5 366.421 625.0±0.5 739.500 1264.0±0.5 777.921 1330.2±0.5 822.972 1409.0±0.8 1091.35 1867.8±0.5 1173.237 2007.9±0.5 1200.23 2054.6±0.5 1332.501 2280.1±0.5

40 5 Detection electronics lab

The purpose of this lab period is to learn how to use an oscilloscope, nuclear instrument modules (NIMs), and a multi-channel analyzer.

5.1 About NIM modules NIM modules are designed in standard sizes and with standard connectors on the rear to plug into a rack (called a NIM bin) which contains the main power supply with all of the required voltages (±12 V, ±24 V). Newer NIM bins also provide ±6 V which is needed for some newer NIM modules. Interconnections between NIM modules are made with BNC type connectors and cables. As you learned in your electricity and magnetism class, the inner and outer conductor radii, and dielectric constant of the material between the conductors determines the impedance of a coaxial cable. Of the several types of coax cables made, the two types used with NIM modules are called RG-58, and RG62. RG-58 (50 Ω) is used for timing and logic signals, and RG-62 (93 Ω) is used for spectroscopy signals. NIM modules generally accept two kinds of signals. There are the analog signals that come from a detector, which can be either positive or negative pulses. These pulses may be unipolar (only on one side of the zero voltage reference), or bipolar where the voltage of the pulse crosses through the zero reference. The second type of signal is a logic signal, which is primarily a yes/no type of signal. The NIM standard speciﬁes the logic signal levels to be a current of -16 mA for true (-0.8 V through 50 Ω).

5.2 The oscilloscope The oscilloscope (or colloquially the scope) is commonly used to inspect the signals in electronics, and physics devices. Oscilloscopes plot the voltage versus time of an electrical signal. While all oscilloscopes knobs tend to be in slightly diﬀerent locations, and there may be diﬀerent menus of options, there are many common features. The main layout of the Agilent DSO-X-2004A oscilloscope being used for this lab is shown in Fig. 6. The oscilloscopes in this lab have a bandwidth of either 100 MHz, or 200 MHz. The bandwidth of the scope tells you roughly, what the maximum frequency you can get reliable results up to. In fact at the bandwidth of the scope, the amplitude of the signal will be decreased by -3 dB (70.7% of actual). Oscilloscopes have at least three groups of controls which will be described in the following subsec- tions. Additional buttons exist, which vary by brand and price range of the oscilloscope. A copy of the oscilloscope manual for this lab can be found on the lab computer desktop, on nexus, or on-line as indicated in the references section of this lab manual.

5.2.1 Vertical controls and inputs The vertical axis of the display has units of volts (or mV, depending on the scale setting for each input). There is often one knob, and set of buttons for each of the oscilloscope inputs. The vertical controls consist of:

• Analog channel on/off keys Use these keys to switch a channel on or off, or to access a channel’s menu in the soft-keys. There is one channel on/off key for each analog channel.

• Vertical scale knob There are knobs marked for each channel. Use these knobs to change the vertical sensitivity (gain) of each analog channel.

41 Figure 6: Overview of Agilent oscilloscope controls. Figure from [1].

• Vertical position knobs Use these knobs to change a channel’s vertical position on the display. There is one Vertical Position control for each analog channel.

• Label key use to label each trace on the oscilloscope display. See Chapter 9 of the Agilent manual[1] for details on how to add labels.

For more information on the vertical controls refer to Chapter 3 of the Agilent manual[1]., Labels, starting on page 117. For more information, see Chapter 3, Vertical Controls, starting on page 59.

5.2.2 Horizontal controls The Horizontal axis is the time, and to control the time scale the controls consist of:

• Horizontal scale knob Turn the knob in the Horizontal section that is marked to adjust the time/div (sweep speed) setting. The symbols under the knob indicate that this control has the eﬀect of spreading out or zooming in on the waveform using the horizontal scale.

• Horizontal position knob Turn the knob marked to pan through the waveform data horizontally. You can see the captured waveform before the trigger (turn the knob clockwise) or after the trigger (turn the knob counterclockwise). If you pan through the waveform when the oscilloscope is stopped (not in Run mode) then you are looking at the waveform data from the last acquisition taken.

• Horiz key opens the Horizontal Menu where you can select XY and Roll modes, enable or disable Zoom, enable or disable horizontal time/division ﬁne adjustment, and select the trigger time reference point.

42 • Zoom key splits the oscilloscope display into Normal and Zoom sections without opening the Horizontal Menu.

• Search key lets you search for events in the acquired data.

• Navigate keys can be used to move along the time base. See Navigating the Time Base” on page 56 of the Agilent users guide[1].

For more information on the time scale setting see Chapter 2, Horizontal Controls, starting on page 45 of the Agilent users guide[1].

5.2.3 Triggering controls In order to decide what time period to show the voltage versus time over, a trigger is needed. The trigger condition can either be set on one of the four inputs, or on a separate external input signal. The condition for a trigger is that the signal reaches a specified voltage (called the trigger level). The trigger can be either on the signal increasing above, decreasing below, or on either crossing of the trigger level. This method of triggering is called the normal mode of triggering. In the Auto trigger mode, if the specified trigger conditions are not found, triggers are forced and acquisitions are made so that signal activity is displayed on the oscilloscope. In the Normal trigger mode, triggers and acquisitions only occur when the specified trigger conditions are found. The Auto trigger mode is appropriate when:

• Checking DC signals or signals with unknown levels or activity.

• When trigger conditions occur often enough that forced triggers are unnecessary.

If you want to capture a waveform that only occurs occasionally, then it is more appropriate to use the normal trigger mode. Another trigger setting, is the coupling mode. The options are:

• DC coupling allows DC and AC signals into the trigger path.

• AC coupling places a 10 Hz high- pass ﬁlter in the trigger path removing any DC oﬀset voltage from the trigger waveform.

For further details on the triggering modes and settings refer to chapters 10 and 11 of the Agilent users guide[1].

5.3 The Pulse Generator The pulse generator, or pulser, is used to create a voltage pulse with a given amplitude, rise and fall time, and frequency of repetition. In this section you will get some practice using a pulser, and use the oscilloscope to measure the pulses from the pulser. Most pulsers have two variable output pulse controls. Sometimes one of these will be a screwdriver type, and the other a 10-turn potentiometer. Both of these controls adjust the attenuated output of the pulser. The pulser also has a second output called direct output. This output normally cannot be adjusted; it is always the same pulse height, and can be used to trigger the oscilloscope. Manuals for the diﬀerent models of pulser can be found in the ﬁling cabinet in the lab. The attenuated output has a pulse height that is a selected fraction of the direct output. The amplitude of these pulses can be varied between 0 V to a maximum of 10 V by using the attenuation

43 switches, or the pulse-height and calibration dials. Some versions of the pulser have controls to set the rise and fall time. A typical pulser signal is shown in Fig. 7, and is meant to mimic a signal that could come from a detector. In this example the rise time of the pulse is only a few ns, and the fall time has a lifetime of 40ns.

14 12 10 8 6

Amplitude (a.u) 4 2 0 0 50 100 150 200 250 300 350 time (ns)

Figure 7: Example of a pulser pulse.

5.4 Detection Electronics Lab Tasks The purpose of this mini-lab is to become familiar with several of the nuclear physics electronics used in the rest of the labs.

5.4.1 Equipment Used The equipment needed are: • NIM bin with power supply,

• pulser,

• oscilloscope,

• pre-ampliﬁer (Ortec 113 or Canberra 1405),

• spectroscopy ampliﬁer, and

• BNC cables, terminators, and tees.

5.4.2 The Pulser and oscilloscope Connect a BNC tee to one input of the oscilloscope, and connect a BNC cable from the direct output of the pulser to one end of the tee as shown in Fig. 8. Connect a ∼ 90 Ω terminator to the other end of the tee. A terminator is required, because the NIM pulses are all current pulses, which need to be dropped through a resistance before they can be measured as a voltage pulse. The value of 90 Ω matches the impedance of the transmission line, and is chosen to minimize electronic reﬂections. Recall that 50 Ω BNC cables are used for timing and logic signals, and 93 Ω BNC cables are used for

44 spectroscopy signals. Note that in some oscilloscopes (not the ones in this lab) there is an option to set the termination impedance, so that an additional termination resistor is not needed.

Figure 8: Pulser pulse as seen on the oscilloscope, and connection of tee with termination to oscilloscope.

Set the pulse height controls fully clockwise, turn oﬀ all attenuator switches (set them to 1×), set the pulser for positive output and obtain a trace on the oscilloscope similar to the one in Fig. 8. How do the controls aﬀect the outputs? Change the cable connection on the pulser to the attenuated output. With all of the pulse height fully clockwise and attenuation switches set to 1× this output should look the same as the direct output. The amplitude can now be adjusted by changing the attenuation, and by turning the pulse height knob.

Table 9: Pulse amplitude [V] as a function of pulser pulse-height and attenuation setting Attenuation setting Pulse height 1× 5× 20× 1000 / 1000 750 / 1000 500 / 1000 250 / 1000

Measure the pulse amplitude using the oscilloscope for various settings of the attenuation and pulse height. In your lab notebook, prepare a table like Table 9, ﬁll in the measurements and include an estimate of the uncertainty in each pulse height measurement. Describe in words below your table what the contributions to this uncertainty are. If there are any other controls on the pulser you are using, describe what they do. Remember to record the make and model number of the pulser you are using.

45 5.4.3 The pre-amplifier and spectroscopy amplifier Pre-amplifiers are connected to a radiation detector, and act as an interface between the detector and amplifier. Its main purpose is to standardize the pulse input to the amplifier, and prevent degradation of the pulse. An amplifier is never used without a pre-amplifier. Pre-amplifiers are designed to accept negative pulses coming from a detector. These negative pulses are called charge pulses, and they are negative because they are due to a flow of electrons from the detector. It is called a charge pulse, since the amplitude of the pulse is typically directly proportional to the amount of charge that was created in the detector by some type of radiation. Each type of detector has a different type of pre-amplifier which should be used. The pre-amplifier generally has two input terminals and one output terminal. The input terminals are labelled INPUT and TEST. The INPUT terminal is meant to connect to the output from a detector, and the TEST input is connected to a pulser to simulate the detector. Note that the pre-amplifier is powered by a special cable which connects it to the rear of an amplifier.

Figure 9: Block diagram for ampliﬁer test procedure.

Procedure

1. Connect the components as shown in the block diagram in Fig. 9. Note that the termination resistor is not needed at the scope because the modules are now connected in normal operating mode.

2. Set the pulser output to negative. You will get poor results if you pass positive pulses into the preamp.

3. If you are using the Ortec 113 preamp, connect the preamp output to the negative input of the amplifier. If you are using a Canberra 1405 preamp, use the positive input of the amplifier. (The two preamps provide different output polarities).

46 4. Set the amplifier output to unipolar and set the course gain to 1 (or as low as possible), fine gain to 0 (or low as possible). Do not force the potentiometers beyond their stop point. Set the input impedance of the amplifier to 93 Ω. Set the shaping to 1 µs. Note that each amplifier may have different settings, and you may need to adjust these to get a reasonable looking signal.

5. While observing the output of the preamp on the oscilloscope, adjust the pulser pulse-height until the amplitude of the pulse at the pre-amp output is 0.1 V.

6. For various fine gain settings from 0/1000 to 1000/1000, measure the amplitude of the pulses at the output of the amplifier. Prepare a table like Table 10, and fill in the measurements. What happens when the gain of the amplifier is too high?

7. Try to determine a functional relationship between the output voltage, input voltage, course gain, and ﬁne gain.

Table 10: Table of pulse amplitude at the output of the ampliﬁer, as a function of ampliﬁer gain settings. Coarse gain setting Fine gain 1000 / 1000 800 / 1000 600 / 1000 400 / 1000 200 / 1000

5.4.4 The Single Channel Analyzer and Scaler A Single Channel Analyzer (SCA) is a logic module that is used for pulse height analysis and pulse timing. The SCA produces a logic output pulse indicating the presence of an analog input pulse within the range determined by the E and ∆E settings (in differential mode), or merely exceeding the E setting (in integral mode). In SCA mode (sometimes called the WIN mode on some SCAs), both the baseline and window are set so that an input pulse of amplitude V will cause the SCA to generate a logic output only if baseline < V < baseline + window. In DISC mode (sometimes called INT mode on some SCAs), a logic pulse is output if the voltage of the analog input pulse is above the baseline. The timing function of the SCA determines when the output logic pulse is generated. There are two ways in which the timing of the logic pulse can be started. These are referred to as leading-edge timing, and cross-over timing. In leading-edge timing, the time-of-arrival of an input pulse is determined by the moment at which it passes through the threshold level. In cross-over timing, a bipolar input is required, and the time-of-arrival is determined by the moment at which the signal passes through zero volts in its transition from positive to negative. Cross-over timing is more accurate in establishing the time-of-arrival of a pulse, since the threshold position will vary with pulse height. The pulse timing for these two timing methods is shown in Fig. 10. Another form of cross-over timing used by some SCA modules is to try to find the peak of the pulse first, and then find the time when the pulse has reached some fraction of the total height. This type of time discrimination is called a constant fraction discriminator. Practically this is implemented inside the SCA module by subtracting a delayed version of the pulse with itself, and finding the cross-over timing of that internally generated signal.

47 Figure 10: Pulse timing for logic pulses. The left plot shows that leading-edge (or threshold) timing leads to a time jitter, while the right plot shows that the zero crossing time will lead to better time resolution.

Figure 11: Block diagram for SCA test procedure.

48 Procedure In this procedure you will use the SCA, and another NIM module called a scaler (or counter). The scaler is used to count the number of pulses, and thus can be used to determine the rate at which you are getting pulses. 1. Connect the equipment according to the block diagram in Fig. 11. Note that on some SCA modules the input pulse goes into the AC connector on the back of the module, while on others there is an input on the front of the SCA. 2. Set the SCA window (upper level) control to 1000/1000 and the baseline (lower level) to 0/100. Set the timing function of the SCA to CO (crossover), if this switch exists on the SCA you are using. When using the crossover mode of the SCA you need to give it the bipolar signal from the amplifier output. Set the SCA to SCA mode (or WIN mode, if available on the SCA being used). 3. Adjust the amplifier so that it provides a positive-first bipolar pulse of 4 V to the SCA. Continue to use the bipolar output in the SCA, but use the NORMAL output on the amplifier to set the pulse height. 4. Set the scaler to maximum time (if there is such a setting on the scaler being used), meaning that it will continue to increment until it is reset. Reset the scaler and start the timer. Counting of logic pulses should be observed. 5. Increase the baseline (lower level) control of the SCA until counting starts, and then stops. The baseline is now set at 4 V. 6. Increase the gain on the amplifier until it produces pulses with 4.5 V amplitude. Counting of logic pulses in the scaler should restart. 7. Decrease the window control until the counting stops. The window is now set to be 0.5 V wide, and the SCA will only make a logic pulse if the input bipolar pulse has a peak between 4 V and 4.5 V. Verify this by changing the input voltage from 3 V to 6 V. 8. Try changing the baseline without changing the window. The window will remain at 0.5 V wide and it will follow the baseline. Verify this. 9. Switch the SCA to DISC mode (or INT mode on some SCAs). This effectively sets the window to be infinitely wide, and only the baseline condition remains. Observe the effect using various input voltages.

5.4.5 The Linear Gate The linear gate allows the passage of an analog signal provided a logic signal arrives at the unit ∼ 10 ns before to open the gate. This logic signal that opens the gate is called the enable signal. This unit performs a coincidence function, in that the logic signal and analog signal need to be coincident in time. The resolving time of the coincidence is limited by the gate width (the length of time the gate is open after the arrival of the enable signal). Of course, the analog and logic signals must arrive simultaneously, so the linear signal will have to be delayed since the logic signal has passed through more electronics taking a longer time. In order to delay the analog signal, we can use the ampliﬁer’s delayed out, or a separate NIM module called a delay unit. Another form of delay that you might use in later labs is the cable length. As a rule of thumb the signals travel close to the speed of light so you can estimate that the delay should be about 3 ns/m of cable.

49 An illustration of the above concept is given in Tab. 11. In this example, a train of pulses arrive at the SCA and linear gate coincidentally in time, and these pulses vary from 1 V to 8 V. The SCA is in SCA mode, and set so that the baseline is 4 V and the window is 2 V.

Table 11: Example of operation of a Single Channel Analyzer. Refer to the lab manual text for a description of the example. Input to SCA and gate Output of SCA Gate Status Output of Gate 8 V none closed none 6 V yes open 5 V 5 V yes open 6 V 4 V yes open 5 V 3 V none closed none 1 V none closed none

Figure 12: Block diagram for linear gate test procedure.

Procedure

1. Connect the circuit shown in illustrated in the block diagram of Fig. 12.

2. As in the previous section, set the SCA baseline to 4 V and the window to 0.5 V. You will have to use the counter to check the SCA settings.

3. Set the gate width control to maximum, and the gate delay to minimum. Observe the gate output on the oscilloscope as you vary the gain of the ampliﬁer making the output pulse vary from 1 V to 8 V.

4. Describe this test in your lab notebook, and demonstrate that you have it working to your lab instructor or teaching assistant.

50 5.4.6 The Multi-Channel Analyzer The multichannel analyzer (MCA) is an important laboratory instrument which can measure the distributions of the pulse height of input analog pulses. The input pulses are sorted into bins (channels) according to their amplitude. The MCA provides a visual display of the resulting distributions on a computer, and usually can output the data to a a file further analysis. The x-axis of the histogram produced by the MCA is proportional to the pulse height of the signal coming in, and the y-axis is the count of the number of times a pulse of that height was observed. The MCA being used in this experiment is the Spectrum Techniques UCS-30[29]. This unit also has a built in high voltage supply, pre-amp, and amplifier. There is also the option of providing a logic gate signal to the MCA, and thus not need to use a separate NIM logic gate. Like all MCAs there is some software that is run to collect the data on the computer. Refer to the manual on the computer desktop [29], or try playing with the software a bit to become familiar with the different settings.

Procedure

1. Turn on the pulser, and adjust the pulser amplitude to produce an output of 1 V. Send the output of the pulser to the input of the MCA box sitting on top of the lab computers.

2. Start the MCA data collection program by double clicking on the icon on the computer desktop.

3. Collect data by pushing the green diamond button to see what the resulting MCA spectrum looks like. Stop the data collection when you are ready to do the next step.

4. Try setting the MCA to collect data for 10 s, and collect a spectrum.

5. Erase the MCA display, increase the pulser output by 1 V, and take another spectrum.

6. Explain in your own words the eﬀect of increasing and decreasing pulser voltage on the resulting pulse height spectrum.

7. Look through the menus, and see if you can set minimum and maximum thresholds.

8. Check what voltage these minimum and maximum thresholds are by adjusting the pulser pulse amplitude, and taking new MCA spectra.

9. Determine the relationship between the pulse height and channel number for your MCA. Are there any settings in the software to change this relationship?

10. Learn how to save a spectrum to a text ﬁle, so that if you collected data, you could later analyze the data with a diﬀerent software package.

51 6 Single Channel Analysis

In this lab you will investigate the energy spectrum for a mono-energetic gamma ray source measured with a scintillation counter.

6.1 Scintillation Counters A scintillation counter is a device used to measure the energy distribution of radiation. The device consists of two main components, a scintillating material, and a photomultiplier tube (PMT). The scintillating material produces a ﬂash of light when a gamma ray (or other ionizing radiation) passes through it. The light is converted to an electrical pulse using a PMT. This section will describe how scintillators work, how PMTs work, and what interactions ionizing particles have in scintillating materials.

6.1.1 Scintillating Materials A scintillating material is one that emits light when ionizing radiation passes through, or is stopped in it. A necessary requirement of useful scintillation materials is that they must be transparent to the scintillation light produced. There are several varieties of scintillating materials that work in different ways. The main types of scintillators include organic crystals, organic liquids, plastic scintillators, inorganic crystals, gaseous scintillators, and cerium activated glasses. Organic crystals are typically made from compounds which contain benzene rings. The π-bonds in the benzene rings allow for many vibrational energy levels in the energy level diagrams, which allows the light absorption and emission spectra of the crystals to be at different ranges of wavelength. The most common organic crystal is anthracene (C14H10), which has the highest light output of organic scintillators, and is taken as the reference light output to measure other scintillators against. The scintillation properties of several inorganic scintillator fluors are summarized in Table 12[30].

Table 12: Properties of Organic Fluorescent Materials Fluorescent material emission wavelength (nm) decay time (ns) Light Yield/NaI Napthalene 348 96 0.12 Anthracene 440 30 0.5 p-Terphenyl 440 5 0.25 PBD 360 1.2 0.35

Organic liquid scintillators consist of an organic solvent, with fluors dissolved into them, and often have additional wavelength shifting material to better match the spectrum of light produced to the sensitive range of a PMT. A typical fluor is PPO (C15H11NO), and two organic solvents are toluene, xylene and benzene. A commonly used wavelength shifter is POPOP (C24H16N2O). Plastic scintillators have their fluors suspended in a polymer matrix. Plastic scintillators typically have a very fast signals, which decay times of a few nanoseconds. They are also easy to form into different shapes, and have good mechanical strength properties. Inorganic scintillators typically have much higher light output than organic ones, and thus are often used for measuring the energies of gamma rays. Recall that the uncertainty in number of counts goes as the square-root of the number of counts, so having a higher light output means having a higher count of photons per amount of gamma ray energy deposited. The higher count of photons means a smaller

52 relative uncertainty in the measured energy of the gamma ray. The scintillation properties of several inorganic crystals are summarized in Table 13.

Table 13: Properties of Inorganic Scintillators

Scintillator NaI(Tl) CsI(Tl) Bi4Ge3O12 CeF3 Density (g/cm3) 3.67 4.51 7.13 6.16 Decay time (µs) 0.23 1.0 0.35 0.03 Emission λ (nm) 410 550 480 310 Light yield (photons/MeV) 40000 14000 8200 1000 Refractive index n 1.85 1.8 2.15 1.68 dE/dxmin (MeV/cm) 4.13 5.1 8.07 7.7 Temp. coeﬃcient (%/degC) −(0.22 − 0.9) < 0.2 −1.7 Hygroscopic yes weak no no

In this lab we will be using a NaI(Tl) crystal as our scintillating material.

6.1.2 Photomultiplier Tubes The PMT converts scintillation photons into an electrical pulse. The main elements making up a typical PMT are shown in Fig. 13. The volume inside of a PMT is evacuated, so that electrons are not absorbed or scattered as they are accelerated from dynode to dynode.

Figure 13: Diagram showing the main elements inside of a PMT. Image from Wikimedia commons.

When a scintillation photon comes into a PMT it first has to be converted into an electron by the photo-electric effect inside a photo-cathode material. The photo-cathode materials are semiconducting alloys of metals from the alkali group (Na, K, and Cs) with Antimony (Sb). The photo-cathode has to be transparent enough for the scintillation light to enter, and thin enough so that the photo-electrons are able to exit. The fraction of photo-electrons emerging from the photo-cathode per incident scintillation photon is referred to as the quantum efficiency (QE). The QE reaches a maximum value of ∼ 27% for scintillation photons with wavelengths of 370 nm. Electrons from the photo-cathode are then focused and accelerated towards a first dynode, which is at a positive potential with respect to the photo-cathode. The dynodes are made of materials that easily

53 give up electrons when bombarded by another electron (such as BeO, Mg-O-Cs). Typically three to five electrons per incident 100 to 200 eV electron can be liberated. The electrons from the first dynode are attracted to a second dynode which is at a positive potential with respect to the first dynode, and so on. PMTs typically have between eight and 14 dynodes, allowing a gain of up to 107 in the number of electrons. Typically the photo-cathode is at negative high voltage, and each successive dynode is a few hundred volts closer to zero volts, until the anode is reached at zero volts. The pulse of electrons finally reach the anode where they are read out with electronics. The time from when the first photo-electron is produced, until the pulse of electrons reaches the anode is around 30-40 ns, and the rise time of the pulse is about 2 ns.

6.2 Response of Scintillator detectors to Gamma Rays In this lab we will be using a 137Cs source which emits single 0.662 MeV gamma rays. In an ideal detector we would see this as a single line in an energy spectrum. The gamma ray can deposit energy in our scintillator by photoelectric absorption, electron-positron pair production, positron annihilation, and Compton scattering. The key features of the resulting spectrum are the full energy absorption (photo-peak), by scattering oﬀ an electron in the detector (Compton scatter), and scattering into the detector from outside (back-scatter). These processes are illustrated in Fig. 14.

Figure 14: Gamma ray interactions in a NaI(Tl) scintillator detector. Figure from Ortec [2].

In our scintillator the photo-peak has some width to it due to ﬂuctuations in the number of scintillation photons that reach the photo-cathode. This peak which represents the full energy of the gamma ray is called the photo-peak. In addition to the photo-peak there are lower energies deposited in the detector due to gamma rays which scatter (Compton scatter), and exit the detector before depositing their full energy.

54 Compton scattering can be considered as a pure kinematic collision between an incident photon and a free electron in the scintillator. Applying conservation of energy and momentum, one can ﬁnd that the wavelength of the scattered photon, λ0, in terms of the initial photon wavelength, λ, and the scattering angle, θ is: h λ0 − λ = (1 − cos θ). (50) mec The relationship between a photon energy, and its wavelength is E = h/λ, so we can write the Compton scattering equation as: 1 1 1 0 − = 2 (1 − cos θ). (51) E E mec The largest energy transfer to the scintillator ∆E occurs when θ = 180o, and is given by: E ∆E = E − 2 . (52) 1 + 2E/(mec )

2 2 The rest energy of the electron is mec = 0.511 MeV/c , so for a 1.5 MeV γ-ray, the maximum energy transfer is: 1.5 ∆E = 1.5 − MeV = 1.28 MeV. (53) 1 + 3.0/0.511 There is also the possibility of seeing a back-scatter peak in the spectrum which is due to gamma rays which Compton scatter outside of the scintillator, and then deposit their remaining energy in the scintillator. Ie. the back-scatter peak should appear at an energy, Ebs, of:

Ebs = E − ∆E. (54)

For our 1.5 MeV gamma ray, the back-scatter peak would appear at Ebs = 1.5 − 1.28 MeV = 0.22 MeV.

Photopeak 2500

2000

1500 Backscatter Compton Edge 1000 Counts / channel 500

0 200 400 600 800 1000 1200 Channel

Figure 15: Typical gamma ray spectrum for a mono-energetic gamma ray detected with a NaI(Tl) scintillator.

A typical NaI(Tl) scintillator response to a mono-energetic gamma ray is shown in Fig. 15. In this lab we will look at the spectrum shape using an SCA to selectively look at energy slices of the spectrum, and plotting count rate versus slice. This is done by setting the window on the SCA, and moving the baseline.

55 6.3 Procedure Note that the photomultiplier tube, and the NaI(Tl) crystal are relatively fragile, so make sure that you treat them gently. Also note that we will be working with a high voltage supply. If you need to connect or disconnect either the signal or HV cable from the PMT it is good safe practice to ﬁrst turn oﬀ the high voltage power supply.

Figure 16: Block diagram showing the electronics and detector connections for the SCA lab.

1. Connect the equipment as shown in Fig. 16. 2. Verify that the electronics are working, by using the pulser. 3. The high voltage for the photomultiplier tube on the NaI(Tl) crystal will be ramped up to somewhere between +800 V and +1300 V. Make sure that the HV power supply you are using is set to positive polarity, and that you start with the supply set to 0 V before powering on. Note that a special cable (diﬀerent from the BNC cables) with a SHV connector on one end, and a MHV connector on the other end is needed to connect the HV supply to the HV input on the photomultiplier tube. 4. Put a 137Cs source near your detector and observe the spectrum on the MCA output to the computer. Also save a copy of your MCA spectrum to a text ﬁle so that you can plot it in ROOT.

Now we will look at slices of the energy spectrum of the gamma source using the SCA, and compare it to the MCA output. The SCA window will be set to be very narrow (set it to 20), so that we measure pulse heights for a narrow energy window. We will adjust our system gain, so that the pulse height of the voltage pulses from the detector easily converts to an energy in keV. The SCA baseline dial can be varied from 0 to 1000, and we will adjust the gain so that as we change the baseline we change the start of our energy window to be from 0 keV to 2000 keV. Using this scale of 2 keV per SCA baseline setting, we want to set our photo-peak to sit at a baseline setting of 331 for a 137Cs source whose photo-peak is at 662 keV. With the settings above, we can now adjust the gain until the count in the scaler is a maximum. Now the energy scale has been adjusted so that the energy of gamma rays being measured is E = baseline × 2 keV. Now that you have your system calibrated, remember to record all of your settings. Scan the full spectrum with SCA baseline dial settings of 25, 50, 75, ..., and 350 (corresponding to energies of (50,

56 100, 150, ..., 700 keV). For each baseline setting, record the count rate in the scaler, and estimate the uncertainty in both the energy as well as in the rate.

6.3.1 SCA Informal Report In your report, include a description of how the scintillation detector works, what the Compton eﬀect is, and how the photo-electric process works. Use ROOT to plot your SCA spectrum with error bars, and identify the photo-peak, Compton edge, and back-scatter peak. Check that the energies of the Compton edge and photo-peak match what was expected. Also plot your MCA spectrum, and compare it to the SCA spectrum. Print your plots, cut them out and paste/tape them into your lab notebook.

57 7 Fitting Data

This section reviews the treatment of statistical and systematic errors when ﬁtting data. It also discusses correlated and uncorrelated uncertainties.

7.1 Fitting with Uncorrelated Errors

Suppose we desire to fit data in the form xi, yi, σi, where σi is the uncorrelated error in the measurement yi. Suppose the fit function is linear f(x) = mx+b, and by fitting we desire to determine the parameters m and b, and their errors. We first construct the variable χ2 to estimate the goodness of fit [31][27]:

2 2 X (yi − f(xi)) χ = 2 , (55) σi where the sum is over the data points. The procedure of least-squares ﬁtting is to minimize χ2. In the case of ﬁtting a line, we’d take derivatives of χ2 with respect to m and b and set them equal to zero, and then solve for the best m and b. Doing this gives: 2 ! 1 X xi X yi X xi X xiyi b = 2 2 − 2 2 (56) ∆ σi σi σi σi ! 1 X 1 X xiyi X xi X yi m = 2 2 − 2 2 (57) ∆ σi σi σi σi where 2 2 ! X 1 X xi X xi ∆ = 2 2 − 2 . (58) σi σi σi

Note that these reduce to the equations you used in 2nd year [32], if all the errors σi are equal. On top of this, by evaluating the second derivative of χ2, we can estimate the uncorrelated errors in the intercept and slope, respectively:

2 2 1 X xi σb = 2 (59) ∆ σi

2 1 X 1 σm = 2 (60) ∆ σi Finally, there are programs out there that can fit arbitrary functions f(x) to your data, with any number of parameters. The programs I normally use are available for free from CERN [3][33]. Common functions you might want to fit are polynomials, exponentials, or Gaussians, or any sum or combination of them. The way these programs work are to search the parameter space to find the minimum χ2. This gives the “best fit” for the parameters. In order to determine errors, they then evaluate the second derivative of χ2 numerically, by varying the parameters slightly around the minimum. It is therefore fine with me if you would prefer instead of using the equations above to use one of these programs, even for linear fits.

58 7.2 Goodness of Fit The minimum value of χ2 found by the minimization process can be interpreted in terms of a goodness of fit. For this, it is useful to define the number of degrees of freedom of the fit:

ν = (number of data points) − (number of ﬁt parameters), (61) which in turn enables deﬁnition of the reduced χ2:

χ2 χ2 = . (62) ν ν

2 The value of the reduced chi-square χν relates to the goodness of fit: 2 • If the value is χν ∼ 1, it means the fit is “good” or “reasonable”. 2 • If χν is much larger than unity, it means the quality of the fit is “bad” or “poor”, or that the function being fit is not an accurate representation of the data.

2 • If χν is much smaller than unity, it means the quality of the ﬁt is “too good”, or that the errors have been overestimated. Or that the data appear to agree suspiciously too well with the function begin ﬁtted.

2 With enough experience, the value of χν and goodness of fit can be estimated graphically from the data. If the best fit line, when plotted on top of the data with error bars, passes perfectly through each 2 point and well within the error bars, then clearly χν will be less than 1 and the fit is “too good”. If the 2 data points never touch the best fit line within the error bar, then clearly χν will be greater than 1 and the fit is “bad”. If, in a fit to say 10 points, a few of the data point have error bars that don’t touch the best-fit line, but the rest of them do touch the best fit line (some of them possibly lying perfectly 2 on the best fit line so that their contribution to χν would be close to zero) then the fit is “good” and 2 you can anticipate χν ∼ 1. Finally, there are varying degrees of “too good”, “good”, and “bad”, and some people (not me) prefer to state “the probability of exceeding χ2” which can be looked up in tables (see e.g. Appendix C of Ref. [34]). One way I like to interpret the meaning of this probability is: if I took these data points and their errors at face value, and then randomly moved the data points around within their error bars, what’s the likelihood I would get something close to this value of χ2 again? Ideally, this probability would be around 50%. The interesting thing about the probabilities is that, the larger the number of degrees of freedom, 2 the sharper this distribution becomes. So if you have 10 degrees of freedom and you measure χν = 1.05, the probability is about 40% (a slightly poor fit). But if you have 200 degrees of freedom and you 2 measure the same χν = 1.05, it turns out the probability is smaller 30%. It’s actually not that unlikely 2 to get χν = 1.5 for 10 data points (prob ∼ 15%), but for 200 data points, this would be a disaster (prob < 0.1%). 2 So the rule of χν ∼ 1 is a little more qualitative than looking up the probability in the table.

7.3 Systematic Errors Imagine you were taking data, and you made a random error each time. This error would be uncorrelated to the previous measurement you made. You could then assign an appropriate error and be conﬁdent

59 √ that the equations in the previous sections would work. This is indeed the case for the N error relating to counting experiments. However, imagine instead that you made the same error every single time you made the measurement. For example, say you measured the length of a number of lines using a ruler, and you knew the lengths should be multiples of one another, and got values 1 cm, 2 cm, 3 cm, etc. You then decide you fit your data to the function f(x) = mx + b where xi is the suspected number of multiples and yi is the length you measured. You arrive at the values m = 1 cm and b = 0. And you get errors σm and σb dependent on what vertical error bars σi you felt were reasonable based on how well you could read 2 the ruler. You also get χν = 1.0001. The fit is “good”. However, suppose you made one error consistently throughout this whole process. You misread the scale on the ruler. It was actually in inches! Therefore you made a factor of 2.54 error in every point you measured. Should you go back and increase the error on every single measurement you made to some huge 2 value? If you did that, χν would become very small. In this case, we would probably go back and correct each data point for the error we made (changing cm to inches). If there was some uncertainty associated with doing this, it would probably not be assigned as an uncorrelated error to be included in the σi used in the fitting process. The above example also relates to a form of a systematic error. You made a mistake about something systematically relating to each points. Imagine, in the previous example, that after taking all the measurements, you also realized you weren’t that careful about lining up the zero point on your ruler, and that you didn’t take this into account when making the measurements, or assigning the error to the measurements. This would at 2 first manifest itself as a poor χν, giving you a hint that you made some error. After realizing this, here’s a strategy you could take to address this systematic error. Measure just one of the lines using whatever technique you were using before. Then measure more carefully, specifically addressing the systematic error you made. Take the difference between the two measurements as the likely random error you made for each of the lines. This error could then be added in quadrature to the error you previously assigned, giving a revised random error for every data point. You could then 2 perform your fit again, and χν would hopefully improve. This extended example exemplifies two things about experimental physics, one positive and one negative.

• The positive: The best way to address systematic errors is to change something about the experiment you did and then investigate carefully what happens in the experimental result. Consider carefully whether you would have made the same error (correlated) or a diﬀerent error (uncorrelated) on each of the data points you measured.

2 • The negative: There can be a psychological eﬀect in experimental physics where, as soon as χν ∼ 1, you stop looking for uncorrelated errors. The example I gave above is an example of this. Doing a study in the order given above can lead you dangerously down this path, especially if you take 2 χν as some measure of success. This might, for example, lead you to overestimate your errors in one area, while completely neglecting the real error at hand.

One way to avoid the negative point is ﬁrst to make a list of all the systematic errors you think you 2 have made. Then do experiments to limit those errors, never considering χν. Then, after all is said and 2 done, analyze the data and determine χν. Basically, be aware of potential errors you could be making, 2 learn about them, but do not necessarily use χν as a real measure of success.

60 7.4 Propagation of Errors The formulae used in 2nd year [32] were correct, as long as they are used for uncorrelated errors. For example, suppose you measure the physical quantities a and b and assign uncorrelated errors δa and δb. You then desire to calculate the value of some function f(a, b). The error in f(a, b) is: b ∂f 2 ∂f 2 δf 2 = δa2 + δb2. (63) ∂a ∂b For correlated errors it is more complicated. Consider the case above where a and b are exactly the same physical√ quantity a. And the function is f(a) = a. You cannot magically reduce your errors by a factor of 2, as the above formula would imply! The correct error is obviously δa. In the case of correlated errors, the correct way to do things is to consider carefully the correlations between the measured quantities and their errors. If I change a, does b automatically change in some ∂f ∂f way? If the answer is yes, then clearly I cannot vary ∂b without necessarily varying ∂a . In this case, the error in a is manifested also as an error in b and they are not really independent. Their errors will therefore also not likely be uncorrelated. The answer “sort of” is also possible; b can be somewhat correlated with a. In such cases, a ∂2f full consideration of cross terms containing e.g. ∂a∂b would be necessary. We will not attempt such considerations in this term. Either it is fully correlated (b is a well-deﬁned function of a), or it is completed uncorrelated. Such strategies are also best followed in real life, too.

7.5 The meaning of Error Bars The proper treatment of uncertainties in measured quantities is a diﬃcult problem that could easily take a course to learn properly. The notes from previous term, and this term cover some of the basics, but more detail can be found by reading some of the references[35, 36, 37, 38, 39]. The sections on Probability and Statistics in the Particle Data group are fairly concise and can be found for free on their website[37]. To properly handle uncertainties and put the correct error bars on graphs we need to understand the meaning of the error bar. If someone hands you a graph with error bars on it, what do the error bars mean? If the meaning isn’t speciﬁed then you can’t be sure. Most commonly the error bars indicate the ±1σ uncertainty on the vertical scale. The horizontal scale error bar could either indicate the uncertainty in the x-coordinate, or the binning of the data.

7.6 Relationship of Error Bar to Probability Distribution Function As mentioned, the error bar on a graph is most often meant to represent the ±1σ uncertainty on a data point. If we were to repeat the measurement many times we could come up with a Probability Distribution Function (PDF) distributed about the “true value,” as shown in FIG. 17. If the data is normally distributed, then the meaning of the mean µ and width σ are clearly that from the Gaussian PDF: 1 − 1 (x−µ)2/σ2 P (x) = √ e 2 . (64) 2πσ The constant in front of P (x) is chosen so that the area of the function is normalized. In other R +∞ words it is chosen so that −∞ P (x)dx = 1. If the PDF distribution is asymmetric, then one needs to specify what is meant by the best value and uncertainty. The error bar is really just a shorthand approximation to the PDF. Most often this means pretending that the PDF is Gaussian, and reporting the mean and RMS.

61 Figure 17: Asymmetric Probability Distribution Function for some measured quantity x. If the distribution was symmetric the meaning of the mean and width (σ) would be well deﬁned. For an asymmetric PDF it might be better to report the peak value, and have an asymmetric error bar.

62 7.7 Covariance Matrices

Consider some multidimensional PDF p(x1, ..., xn), which describes a measurement and its uncertainty in n measured quantities. We deﬁne the covariance between any two variables by: Z cov(xi, xj) = dxp~ (~x)(xi − x¯i)(xj − x¯j). (65)

Note that in the case i = j, this is the equation for the variance (σ2). The set of all possible covariances between diﬀerent i, j deﬁnes the covariance matrix, often denoted as Vij:

 2  σ1 ρ12σ1σ2 ... ρ1nσ1σn ρ σ σ σ2 ... ρ σ σ  V =  21 1 2 2 2n 2 n . (66) ij  ......   2  ρn1σ1σn ρn2σ2σn ... σn

Note that the covariance matrix is always symmetric, square, and invertible. The values ρij are called the correlation coeﬃcients, and lie in the range(−1, 1) representing the correlation between the uncertainties (ρ = 0 meaning no correlation). The most common use of a covariance matrix is to invert it for use in calculating the χ2 when there are correlations between parameters:

2 X X −1 χ = (yi − f(xi))Vij (yj − f(xj)). (67) i j

In this equation the measurements are yi, at points xi, and the model being ﬁt to is a function f(x). Again like the uncorrelated case, the best value of parameters of f(x) can be found by minimizing the χ2. Another use of the covariance matrix is in averaging correlated measurements. For example the χ2 for several measurements with correlated uncertainties is:

2 X X −1 χ = (xi − µ)(xj − µ)(V )ij (68) i j

We can ﬁnd the best value of the average value of the measurement, µ, by minimizing this χ2. The best value of µ is given by: P −1 i,j xj(V )ij µ = P −1 , (69) i,j(V )ij and the uncertainty in the average value is:

2 1 σµ = P −1 . (70) i,j(V )ij In other words the uncertainty in the average is just given by summing the entries in the inverted covariance matrix.

63 7.8 Example of correlated measurements The example in this section is based on an example from Cowan’s book[35]. Say we measure an object’s length with two rulers, both calibrated to be accurate at one temperature T = T0, but otherwise have a temperature dependency. The true length y is related to the measured length Li(i = 1, 2) by:

yi = Li + ci(T − T0) (71)

We assume that we know the ci and that the uncertainties in the measured quantities are Gaussian. We measure L1 ± σL, L2 ± σL, and T ± σT , and calculate the object’s true length y. Now we wish to combine the measurements from the two rulers to get our best estimate of the true length of the object. We start by forming the covariance matrix of the two measurements:

cov(yi, yj) = cov(Li + ci(T − T0),Lj + cj(T − T0)) (72)

= cov(Li,Lj) + cov(Li, cj(T − T0)) + cov(ci(T − T0),Lj) (73)

+cov(ci(T − T0), cj(T − T0)) (74) 2 2 = σL + (ci + cj)σLσT + cicjσT . For the oﬀ diagonal terms, since the separate length measurements are independent, we will take the 2 covariance to just be due to the cicjσT term. The on-diagonal terms of the covariance matrix are for 2 2 2 2 one ruler, which leaves σi = σL + ci σT for the on-diagonal. The covariance matirx is therefore:

" 2 2 2 # σL + c1σT (c1 + c2)σLσT Vij = 2 2 2 . (75) (c1 + c2)σLσT σL + c2σT

eg. Suppose we found c1 = 0.1, c2 = 0.2, L1 = 2.0 ± 0.1, L2 = 2.3 ± 0.1, y1 = 1.8 ± 0.22, y2 = 1.90 ± 0.41, T0 = 25, and T = 23 ± 2. Then the covariance matrix is: " # " # 0.12 + 0.12 · 22 0.1 · 0.2 · 22 0.05 0.08 V = = (76) ij 0.1 · 0.2 · 22 0.12 + 0.22 · 22 0.08 0.17

Which, when inverted gives the matrix: " # 80.95 −38.10 V −1 = . (77) ij −38.10 23.81

The weighted average is then, using equation 69: y (V −1) + (y + y )(V −1) + y (V −1) y = 1 11 1 2 12 2 22 (78) avg −1 −1 −1 V11 + 2(V )12 + (V )22 1.8 · 80.95 + (1.8 + 1.9) · (−38.10) + 1.9 · 23.81 = (79) 80.95 + 2 · (−38.10) + 23.81 = 1.75.

The uncertainty on the weighted average, using equation 70, is: s 1 σ = (80) avg 80.95 + 2 · (−38.10) + 23.81 = 0.19.

T T0

L(cm) 2.5

L1 2 y 1 y 2

1.5

18 20 22 24 26 28 30 T (Celcius)

Figure 18: Measured length as a function of temperature for the two measurements. The horizontal dashed areas represent the two length measurement 1σ uncertainty bands, and the vertical dashed area represents the temperature measurement 1σ uncertainty band.

This may seem odd at first, since the weighted average in this case is lower than either of the measurements. What happens though, is that since there is such a large uncertainty in the temperature, having the two different length measurements (that disagree) tells us that the temperature we measured must be off. The fit for the average attempts to adjust the temperature to make the y1 and y2 agree better, as can be seen in FIG. 18. This is an example where the data itself constrains the systematic uncertainty due to the temperature.

7.9 Propagating Uncertainties

For a function f(x, y) of two variables x and y, whose uncertainties σx and σy are known and “small”, the uncertainty squared in the function is:

∂f 2 ∂f 2 ∂f ∂f σ2 = σ2 + σ2 + 2 ρ σ σ . (81) f ∂x x ∂y y ∂x ∂y xy x y

The last term in this equation is often ignored in earlier lab courses, but is essential in getting the correct uncertainty estimate. The equation above assumes that the covariances are known, and that the function f(x, y) is an approximately linear function of x and y over the span of x ± σx, and y ± σy. As an example, consider interpolating a straight line ﬁt y = mx + b, as shown in FIG. 19. As is typical of many ﬁtting programs the correlation between the parameter uncertainties is not reported by default. We now calculate the uncertainty in y (the energy) when we are at channel 800, which corresponds p 2 2 to an energy 1368.2 keV. If we neglect the correlation, then we get σy = 0.3411 + (800 · 0.000386) = 0.46 keV (where ∂f/∂m = x, and ∂f/∂b = 1). But this estimate is much too large, as we will see if we account for the correlation between the slope and intercept.

65 Figure 19: Example of linear ﬁt for an energy to channel calibration.

In fact redoing the ﬁt with the correlation reported, we ﬁnd ρ = −0.886. The correct uncertainty is therefore:

p 2 2 σy = 0.3411 + (800 · 0.000386) + 2(−0.886)(800 · 0.00036)(0.3411) = 0.19 keV. (82)

7.10 Fitting Data Assignment 1. You notice that by removing one data point from your dataset, the reduced χ2 drops from 3.00 to 0.50. Should you throw out the data point because of this statistical fact? Why or why not?

2. Do problem 7.2 from the text.

3. Do problem 8.5 from the text.

4. Do problem 8.25 from the text.

5. For problem 8.25, calculate the value of the reduced χ2 and thereby comment on the goodness of ﬁt. What is the associated probability corresponding to these parameters?

6. Suppose the dataset from question 1 contains three degrees of freedom without throwing the data point out. It therefore contains two degrees of freedom if the point would be thrown out. What are the probabilities corresponding to each value of χ2 for the given numbers of degrees of freedom?

7. Copy the files from http://t2kwinnipeg.uwinnipeg.ca/~jamieson/courses/intermediate_ lab/root_fitting/linear_fit/ into a directory, then try running the macro in ROOT. Go through the script to see how to do a linear fit in two different ways.

8. Copy the files from http://t2kwinnipeg.uwinnipeg.ca/~jamieson/courses/intermediate_ lab/root_fitting/spectrum_fit/ into a directory, then try running the macro in ROOT. Mod- ify the script to fit the 1335 keV peak to also fit it using a straight line, and a straight line

66 background model. Based on the results of the three diﬀerent ﬁts, what is a reasonable systematic uncertainty in the peak position, and in the peak resolution (width) due to the background?

67 8 Counting statistics lab

8.1 Objective The purpose of this lab session is to study the laws which apply to the frequency of decay of radioactive nuclei. The decay rate of 137Cs will be measured in a narrow energy range for ten second time intervals. These measurements will be used to study the statistical uncertainty in counting measurements.

8.2 Theory The nuclear decay of radioactive atoms is a random process with individual atoms decaying indepen- dently of each other. This statement can be tested by applying standard probability theory to predict the number of counts in a narrow energy range over a ten second period. If the predicted distribution of counts over many experiments matches those predicted by probabilistic arguments, we may conclude that the decay is indeed random. The radioactive decay of an isotope containing n parent atoms is given by: dn = −λn, (83) dt where n is the number of parent atoms, λ = 1/τ is the decay constant (related to the lifetime τ) of the isotope. Even though λ and n are precise quantities, the time at which a given isotope will decay is not known apriori. We find in practice, the number of decays of a radioactive source in a fixed time interval varies around some most probable number (usually the average). In order to be able to test this randomness, it is important that n remains constant during the experiment. That is to say, the half-life of the nuclide must be much longer than the duration of the experiment, or the analysis would need to account for the changing rate of decay. The variations result from the fact that λ is the probability of decay of a single atom, and (1 − λ) is the probability that the atom will not decay. For a large number of atoms, if we sample the decay rate we should find that on average it matches λ, but any individual decay rate is random. The above situation can be better understood by considering the analogous everyday situation of tossing coins. We know intuitively that the probability of getting either a head or tail when tossing an unbiased coin is 1/2 (i.e. λ = 1/2). If we toss 20 coins (n = 20) we would expect λn = 10 heads, but would not in reality get this result every time. If the experiment of tossing 20 coins were done many times, we would find that the value obtained most often would approach 10 heads. If we plot a histogram of the frequency of occurrence, N, versus the number of heads n in each toss, for many experiments we should find that the peak of the distribution is at 10. The distribution 10000 such experiments, normalized to an area of one was simulated as shown in Fig. 20, where it is compared to the Binomial distribution, and the Poisson distributions. Clearly, tossing coins follows a binomial distribution. In the binomial distribution, the probability for getting k heads in n tosses, where the probability of getting heads in one toss is p = 0.5 is given by:

n n! P (k; n, p) = pk(1 − p)n−k = pk(1 − p)n−k (84) k k!(n − k)!

Unlike coins tosses, radioactive decays follow a Poisson distribution. If the number of disintigrations, n, is counted in some time interval, where an average expected count isn ¯, then the most likely probability (P (n)) is: n¯n P (n) = e−n¯ (85) n!

68 Coin Toss Simulation 0.18

Binomial Distribution 0.16

Sampling Poisson Distribution 0.14

Poisson Distribution 0.12 Fraction of Experiments 0.1

0.08

0.06

0.04

0.02

0 0 2 4 6 8 10 12 14 16 18 20 Number of Heads

Figure 20: Frequency of count of number of heads for 20 coin tosses repeated 10000 times is shown as the ﬁne-dashed (green) line. The solid line is the binomial distribution. The smooth dot-dashed (blue) line is the Poisson distribution, and the histogram dot-dashed (red) line is sampling from the Poisson distribution.

69 wheren ¯ is the mean of the distribution. The mean of the distribution can be found from a histogram of the experimental data using: PNbins N n n¯ = i=1 i i (86) PNbins i=1 Ni where Nbins is the number of bins in the histogram, Ni is the count in bin i, and ni is the bin value (x- PNbins axis value). Note that i=1 Ni is the total number of trials (times that the experiment was repeated). Note thatn ¯ itself has an uncertainty on it. Also note then ¯ may not give the best correspondence between the experimental data and the theoretical distribution of Eqn. 85. If we let: N Xbins N = Ni, (87) i=1 P then we can calculate theoretical values, Ni for the experimentally observed counts in the i-th bin, Ni using:

P Ni = NP (ni) (88)

One method for testing how well the data obtained from the experiment ﬁts the theoretical distribution is by calculating the Chi-squared (χ2). The χ2 tells us the goodness of ﬁt between the experiment and the model. The χ2 is given by:

Nbins i 2 Nbins P 2 X (Nmodel(xi) − N ) X (N − Ni) χ2 = data = i , (89) σ2 N P i=1 i i=1 i where the first half of the equation could be used for an arbitrary data set with uncertainty σi in the 2 contents of bin i, and model Nmodel(x). The second half of the equation shows how to calculate the χ for the specific case of a counting experiment. In order to determine the goodness of fit from the χ2, one needs to calculate the Number of Degrees of Freedom (ν). The ν is the number of data points minus the number of parameters in the fit. Here the only parameter in the fit is the average number of countsn ¯ (i.e. P (n) is a one-parameter distribution function). After determining χ2 and ν, a table for the chi-squared test can be consulted to determine the goodness of fit. 2 2 A good fit has a reduced chi-squared χR = χ /ν ∼ 1. If the uncertainty you estimate in your 2 2 data is too large, the χR will be too close to zero. If the χR is too large compared to 1, then either the model being fit to does not correspond to the data, or the uncertainty in the data is under-estimated. The probability that the data matches the model versus reduced chi-squared for different numbers of degrees of freedom is shown in Fig. 21. If the number of disintigrations, n, which are counted in a given time interval are large enough, then the Poisson distribution approaches a Gaussian distribution. The Gaussian distribution is given by:

2 1 − (n−n¯) P (n) = √ e 2¯n . (90) 2πn¯ The above equation can be substituted for the Poisson equation, for large counts, when calculating the χ2. By comparing the χ2 obtained from these two models, you can say which of the distributions (if any) best ﬁts the data.

70 χ2 test fit probability Ndof=2 1 Ndof=4 Ndof=6

Probability Ndof=8 0.8 Ndof=10 Ndof=100 0.6

0.4

0.2

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 χ2 R

Figure 21: Chi-squared test probabilities, calculated using the root function TMath::Prob(χ2,ν) [3].

8.3 Equipment and Procedure For this experiment to test our theory for counting statistics, you will use a similar setup to the detection electronics lab, but will use a 137Cs source to investigate the random nature of radioactive decays.

Procedure Note that the background from the other groups’ sources may be an issue, since we are trying to get a relatively low counting rate. Place your source about 1.5 to 2 m from your detector, and try to leave its position ﬁxed for the lab period.

1. Connect the circuit as shown in Fig. 22.

2. Set the SCA to accept a narrow energy range, somewhere in the mid-range of voltage pulses obtained from the detector and the 137Cs source.

3. Set the counter timer interval to 10 s, and adjust the detector to source distance until you get a count of about 40 every 10 s.

4. Record the number of counts for 100 ten second intervals.

5. Adjust the counter time interval to 2 s, so that you get about 8 counts every 10 s.

6. Record the number of counts for 100 ten second intervals.

71 Figure 22: Block diagram for counting statistics experiment setup.

8.4 Data Analysis Prepare a histogram for each of the data-sets collected in the procedure. To make a histogram, tally the number of times, N5 which ﬁve counts occurred in the 10 s, and so on, for Ni,(i = 0, 1, 2, ..., Nmax). You may wish to use root to make the histogram. Once you have the histogram, the uncertainty in the count in each bin, as usual is the standard deviation in the counts: p σi = Ni (91)

The larger the number of counts Ni, the smaller the relative uncertainty σi/Ni in the number of counts. Plot a histogram of the counts Ni versus interval ni, and include error bars of length ±σi. Note that if you are using ROOT to make your histogram, it will automatically assume that the error bar on each bin is the square-root of the counts in the bin. Superimpose the best fit curve (theoretical) on top of this. To find the best fit curve, find the minimum χ2 between your histogram, and the theoretical curve asn ¯ is varied. When calculating the χ2 you should adjust the bin width of any bin with fewer than five counts, as the χ2 estimator becomes biased when there are low counts in bins. Note that the best fit mean of the poisson distribution should be very close to the average value of the histogram. If you are using ROOT, it automatically does this curve fitting using the Fit method of the histogram (TH1D). Note that if you are using ROOT, you can make a poisson probability distribution function using: TF1 * tf = new TF1("tf","[0]*ROOT::Math::poisson_pdf(x,[1])",0,100); tf->SetParameters(Ntrials, Navg); tf->FixParameter(0,Ntrials); where parameter [0] is the total number of trials, and parameter [1] is the average number of counts. The mean countsn ¯ is related to the width of the distribution at half√ its height. Ie. there is a relationship between the full-width-at-half-maximum (FWHM) and σ = n¯. Determine experimentally and theoretically what this relationship is. Note that you get the experimental FWHM by estimating graphically using the data points only, not the best fit curve.

72 9 Gamma Ray Spectroscopy

In the Single Channel Analysis lab we learned about the NaI(Tl) scintillator detector, and how it detects gamma rays. Refer to the SCA lab to review some of the theory that is relevant in this lab as well. In this lab we will extend this study to ﬁnd the energy of gamma rays from several sources, and measure some of the properties of this detector.

9.1 Gamma Ray Interactions A cross-section, σ, not to be confused with resolution, and standard deviation, is a measure of how likely a particle is likely to interact. The cross-section has units of area (the larger the area the more likely the target is to be hit), and is often measured in barns (1 barn = 1 × 10−24 cm2). Recall that gamma rays interact in materials by three main routes. In the photo-electric effect, the photon is absorbed and an electron is emitted. The electron deposits its energy in the scintillator, producing ultra-violet scintillation light. The photo-electric effect is dominant for gamma-ray energies below about 1 MeV. The second route gamma rays interact is via Compton scattering, which occurs when the gamma ray scatters off an electron in the scintillator. Compton scattering is dominant for photon energies around 1 MeV. At energies above twice the electron rest mass (1.022 MeV), electron- positron pair production can occur, and dominates at higher gamma ray energies. The total cross section for gamma rays in carbon and lead are shown in Fig. 23.

9.2 Detector Resolution The energy resolution of a gamma ray spectrometer is a measure of the width of the photo-peak, which tells us how well two different energy gamma rays can be resolved. The energy resolution of the NaI(Tl) scintillation detector is dominated by the counting statistics in the number of primary photo-electrons Npe liberated in the PMT. Neglecting less important effects of thermal motion, and nuclear recoil, the resolution, σR, is: p σR = Npe. (92) Experimentally, the resolution is found from the σ of a gaussian fit to the photopeak. The number of primary photo-electrons is proportional to the energy of the gamma ray (Npe ∝ E), so the resolution is: √ σR ∝ E. (93)

Often the resolution is written as a fraction, σf ; the resolution divided by the energy. In other words one might say that the resolution is 10% at a given energy. In this notation:

σR 1 1 σf = ∝ p ∝ √ . (94) E Npe E

This means that scintillator detectors have better fractional resolution for higher energy gamma rays.

9.3 Gamma Ray Spectroscopy Experiment For this lab we will collect the gamma ray spectrum from several sources to do an energy calibration, and to investigate the energy resolution as a function of energy. The properties of several gamma- ray sources that will be investigated in this experiment are summarized in Table 14. The gamma ray energies of a 152Eu source will be measured.

73 Figure 23: Gamma ray total cross section versus energy, showing the contribution to the cross section from various processes for carbon (top), and lead (bottom)[4].

74 Table 14: Properties of several common gamma-ray sources. Isotope half-life Gamma ray energies (keV) 133Ba 10.7 years 81.0, 276.3, 303.7, 355.9, 383.7 109Cd 453 days 88.0 57Co 270 days 122.1, 136.4 60Co 5.27 years 1173.2, 1332.5 137Cs 30.1 years 32, 661.6 54Mn 312 days 834.8 22Na 2.6 years 511, 1274.5 65Zn 244 days 1115.6 152Eu 13.5 Years Multiple energies

Figure 24: Electronics block diagram for the gamma ray spectroscopy experiment.

Procedure

1. Connect the NIM electronics as shown in Fig. 24.

2. Verify that the electronics are working by using the pulser, then turn the pulser oﬀ.

3. Put a 60Co source near the detector, and adjust the ampliﬁer gain until the 60Co spectrum in the MCA has its higher energy photopeak in the ﬁrst half of the cnannels (pulse height of the larger signals from the detector are around four volts).

4. Use the UCX-USB program on the computer to collect a spectrum from the MCA. Save the MCA spectrum to a text ﬁle to ﬁt the peaks to later.

5. For the 60Co spectrum, identify the photo-peaks, Compton edges, and back-scatter peaks.

6. Each time a radioactive decay happens in 60Co it emits a beta decay and the two gamma rays, as shown in the decay scheme of Fig. 25. If both gamma-rays deposit their energy in the scintillator detector, then a sum peak will appear. If each of the gamma rays is emitted in a random direction, calculate the probability of seeing a sum peak (you will need to measure the fraction of the total 4π solid angle, dΩ = A/r2, that your detector subtends). Does the relative rate of the sum-peak to the rate of the other two gamma rays make sense? If not, what other eﬀects might this be due to?

75 7. Still using the 60Co source near the detector, and adjust the ampliﬁer gain until the 60Co spectrum in the MCA has its higher energy photopeak near the hightest channels (pulse height of the larger signals from the detector are around eight volts).

8. Record all the settings, so that you could reproduce them next week, as closely as possible. Leave the gain and electronics settings the same for the remainder of the measurements in this experiment. This ensures that the energy calibration is the same for all of the sources.

9. Collect spectra for each of the sources in Table 14, and save them to a text ﬁle for later processing.

10. Fit each of the photo-peaks in the gamma-ray spectra. For your lab writeup, the ﬁts to the photo- peaks can appear in an Appendix. Your writeup should include a summary table that includes the photopeak mean channel, sigma, and ﬁt quality for each peak.

11. Make a table of measured gamma-ray energies with uncertainties in the measured channel, and associate the known energy for each photo-peak as given in Table 14.

12. Plot the energy versus channel number from all of the photo-peaks (except 152Eu), and ﬁt the data to determine the relationship between energy and channel.

13. Make a table of measured gamma-ray resolution with uncertainties versus energy, and plot the data to see if it behaves as expected.

14. Determine the energies of the gamma-rays from the 152Eu source.

Figure 25: Cobalt-60 decay energy level diagram. Figure from wikimedia commons.

76 10 Beta Spectroscopy with Si(Li) Surface Barrier Detectors

This lab is based on experiment 6 of the Ortec lab manual[2]. In this lab you will measure the spectrum of energies of electrons or positrons from various beta-decay sources. The measurements of the beta energies will be done using a surface barrier detector. There are two diﬀerent eﬀects that will be observed. One is internal conversion, in which the beta particle that is emitted is mono-energetic, and the other is from nuclear beta decay.

10.1 Si(Li) Surface Barrier Detectors A surface barrier detector is a radiation detector which consists of a semiconductor (such as Si) that has been doped with impurity atoms to form a p-n junction. A p-doped semiconductor is one which has conduction holes (positive Li ions are free to move around), and an n-doped semiconductor is one which has conduction electrons (electrons are free to move around). When these two doped semiconductors are in contact, they produce a space charge region with an electric field, as shown in Fig. 26. The space charge region forms because some of the conduction electrons from the n-doped side diffuse into the p-doped side, and some of the conduction holes from the p-doped side diffuse into the n-doped side. A p-n junction forms a Diode, which only allows current to flow in one direction, under a forward bias.

Figure 26: Equilibrium charge carrier concentrations in a p-n junction diode. Figure from Wikimedia commons.

The p-n junction used in a surface barrier detector, a reverse bias voltage is applied to widen the depletion region so that at reaches the front of the device[40]. To operate as a particle detector, the depletion region has to be wider than the range of the particle in the detector material (about 15 µm for 5 MeV α-particles in Si). When a charged particle passes through the depletion region, it loses energy by producing electron-hole pairs. A hole is a positively charged ion that is free to move around the semiconductor in a similar manner that an electron can freely move around in a conductor. In the depletion region the charge carrier concentration is greatly reduced, as is the possibility of recombination of the produced electron hole pairs. The liberated holes drift to the p-side contact of the diode where they recombine with electrons in the metal contact of the diode, and the electrons drift to the n-side contact where they can be collected. To continue the example of the 5 MeV α-particle, if it stops in

77 the depletion region it will produce about 1.7 × 106 electron-hole pairs. Due to the high electric ﬁeld across the diode, the electrons drift towards the p-doped side, and the holes will drift to the n-doped side, and the resulting current pulse can be collected with detection electronics. The energy resolution of surface barrier detectors is very good, since a large number of electron-hole pairs can be collected. There is an additional improvement in the resolution due to eﬀects that are not purely statistical. This non-statistical improvement factor is called a Fano Factor, which is named after Ugo Fano, an American theoretical physicist trained by Enrico Fermi in the 1930’s. The detector used for this lab has a depletion depth of 500 µm, when operated with a recommended bias voltage of 140 V. The detector has a resolution of about 12 keV for α particles. Note that the range of a 1 MeV electron in Si is 2.5 mm. For normal incident β particles on our detector, the full energy deposit only occurs for energies below about 300 keV. For the higher energy β particles, we will only see the full energy deposit for tracks that cross our depletion region diagonally.

10.2 Theory The basic beta decay processes are:

− − n → p + e +ν ¯e negative beta decay (β ) + + p → n + e + νe positive beta decay (β ) − p + e → n + νe orbital electron capture (EC) Free neutrons, and isotopes rich in neutrons often undergo β− decay, and isotopes rich in protons undergo β+ decay to move to a more stable nucleus with a higher binding energy per nucleon. The beta decay reactions for changing from one isotope, X, to another, Y, can be written as:

A A − Z X →Z+1 Y + e +ν ¯e negative beta decay A A + Z X →Z−1 Y + e + νe positive beta decay A − A Z X + e →Z−1 Y + νe Electron Capture (EC) Note that in the last reaction, the isotope produced, Y, is in an atomic excited state after capture. The atomic electron shell, for example the K shell, from which the electron was captured is vacant then, and an x-ray is emitted when an electron from a higher shell fills the vacancy. The last two reactions start and end with the same isotopes, but the electron capture tends to be favoured because the beta decay needs to have a Q value large enough to produce two electrons worth of rest mass in energy. In order to determine if it is favourable for an isotope to undergo β decay, the Q value for the reaction (mass difference expressed as an energy, between the initial and final isotope) is considered. The Q value is the sum of the energy that is then shared between the decay products (electron and neutrino). The Q value for the three types of beta decay process is calculated to be:

A A 2 − Q = (m(Z X) − m(Z+1Y))c for β decay A A 2 + Q = (m(Z X) − m(Z−1Y) − 2me)c for β decay A A 2 Q = (m(Z X) − m(Z−1Y))c − Bn for EC

The binding energy of the captured n-shell (n=K,L,...) electron, Bn, appears in the last equation. For EC the electron captured is the one from the atom itself, hence the need to account for the binding energy of the electron in the atom. In EC, the atom is left in an excited state, which then emits an x-ray to return to ground state. 90 90 − As an example calculation of a Q value, in the decay of Sr → Y + e +ν ¯e, the Q value is:

78 Q(90Sr) = m(90Sr) − m(90Y) = (89.907738u − 89.907152u) × 931.502 MeV/u = 0.546 MeV.

Since this Q value is positive, this isotope of Sr can undergo β− decay, with an endpoint energy of 0.546 MeV. In this lab, the beta decay sources that will be measured have some of their properties summarized in Table 15.

Table 15: Sources used for the beta spectroscopy lab. The internal conversion (IC) process is described in Section 10.2.1. Isotope Dominant decay mode(s) Half-life 137Cs β−, IC 30.07 y 207Bi EC, IC 31.55 y 152Eu β−, IC 13.522 y 14C β− 5730 y 204Tl β− 3.78 y 54Mn β+ 0.855 y 90Sr β− 28.79 y 65Zn EC 0.668 y

10.2.1 Internal Conversion As a contrast to the above beta decay processes, internal conversion is an electromagnetic process whereby an excited nucleus de-excites by knocking an electron out of the atom instead of a gamma ray. This process occurs by the quantum mechanical overlap of the nuclear wave function with the atomic shell wave functions (and not by a two step process of nucleus emitting a photon and that photon knocking out the electron). The kinetic energy, Te imparted to the ejected electron is:

Te = ∆E − B, (95) where B is the electron binding energy, and ∆E is the energy available from the nuclear de-excitation. As an example, consider the decay scheme of 207Bi shown in Fig. 27[5]. One of the excitation energies is 207 ∆E = 1063.662 keV. The binding energy for a K-electron of Bi is 90.521 keV, and for an L1-electron is 16.393 keV[41, 42, 43, 44], resulting in internal conversion electrons with energies of 973.141 keV, and 1047.269 keV.

10.2.2 Fermi Theory Enrico Fermi developed the four particle theory of beta decay in 1934, which treats the beta decay as a transition between states with a coupling constant between the initial, i, and ﬁnal, f, states. The transition probability, λif , is called Fermi’s golden rule: 2π λ = |M |2ρ , (96) if ¯h if f where |Mif | is the matrix element for the interaction, and ρf is the density of ﬁnal states. Fermi’s theory of beta decay leads to the shape of the electron energy distribution of:

2 N(P ) = K(W0 − W ) PEF (Z,P ), (97)

79 Figure 27: Nuclear de-excitation energy levels after electron capture in 207Bi[5].

where W is the reduced kinetic energy of the β, W0 is the reduced endpoint kinetic energy of the β spectrum, P is the reduced momentum of the β, K is an energy independent constant, and F (Z,P ) is the Fermi function which accounts for coulomb interactions in the nucleus. The reduced kinetic energy is defined as: T W = 2 , (98) mec 2 and mec = 0.511 MeV is the rest energy of the electron. Similarly the reduced endpoint is W0 = 2 Q/(mec ), and the reduced momentum is P = p/(mec). The Fermi function√ is tabulated in reference [45] for different nuclei, Z, and different reduced 2 momentum P = E − 1. Here E = Etot/me = W + 1 is the reduced total energy. The modified Fermi function for 204Tl is shown in Table 16. The form of the function published by E. Fermi in reference [46] is: 2(1 + S) F (Z,P ) = (2P ρ)2S−2eπη|Γ(S + iη)|2, (99) Γ(1 + 2S)2 √ where S = 1 − α2Z2, α is the fine structure constant, η = ±αZP/E (+ for electrons, - for positrons), ρ = rN /¯h, rN is the radius of the final state nucleus, and Γ is the gamma function. The beta decay spectrum, plotted using Equation 97 for 204Tl, is shown in Fig. 28. The shape of the spectrum is shown when plotted against momentum and versus kinetic energy. The effect of the Fermi Function on the shape of the spectrum is also shown.

80 Table 16: Fermi Function F (Z,P ) versus P for 204Tl. PF PF PF PF 0.0 28.26 0.8 25.09 2.2 19.10 5.0 14.13 0.1 28.19 0.9 24.53 2.4 18.54 6.0 13.17 0.2 27.99 1.0 23.98 2.6 18.03 7.0 12.40 0.3 27.67 1.2 22.95 2.8 17.55 8.0 11.77 0.4 27.25 1.4 22.01 3.0 17.12 9.0 11.24 0.5 26.76 1.6 21.17 3.5 16.18 13.0 9.718 0.6 26.23 1.8 20.41 4.0 15.39 15.0 9.182 0.7 25.66 2.0 19.72 4.5 14.71

Figure 28: Kinetic energy spectrum of beta particles from the radioactive decay of 204Tl.

10.3 Procedure First a few safety notes about the detector and sources used in this experiment. Most of the sources only have a thin 80 µg/cm2 of aluminized Mylar on them, so handle them carefully. A second note is that the surface barrier detectors are light sensitive, so need to be operated in the dark. Make sure that the detector high voltage is turned oﬀ before exposing it to light. Also, the shiny surface of the detectors are easily damaged, so do not touch or bang the surface of the detectors.

Electronics set-up, and data collection Connect the electronics according to Fig. 29. The pre- amplifer is an Ortec 142A, and the HV supply is a positive supply, which we will slowly ramp up to 140 V once we have placed a source in the chamber. Use either the 137Cs, 207Bi, or 152Eu sources to adjust the gain to make sure that the internal conversion lines can be resolved. The source should be placed as close to the detector as possible, to reduce the energy loss in air, and so that the detector subtends a larger solid angle, meaning that more of the beta rays will cross a longer distance in the depletion region. Adjust the threshold on the MCA to cut out electronics noise in the low pulse height region so that the dead time is less than about ﬁve percent. Once you have a reasonable spectrum, leave the electronics settings the same for all of the remaining spectra that you collect. Collect spectra for each of the sources 137Cs, 207Bi, 152Eu, 14C, 204Tl, 54Mn, 90Sr, and 65Zn. Save each of the spectra into a text ﬁle that can be read into ROOT for analysis.

81 Figure 29: Experimental set up.

Energy calibration In order to calibrate the energy scale for measurements of beta decay, we will use the MCA spectra from 137Cs, and 207Bi. While collecting spectra, you will need to calculate the expected energies of the internal conversion lines. Data relevant to this calculation are shown in Table 17. The data for the gamma-ray energies are from the IAEA Nuclear Data Services [47], [48], and [49]. Data for the atomic binding energies (xray absorption energies), Kab, L1ab, are from [50].

Table 17: Energies in keV needed to calculate internal conversion electron energies. 207Bi γ Energies 569.702, 1063.662 207 Bi Kab, L1ab 90.521, 16.393 137Cs γ Energy 661.657 137 Cs Kab, L1ab 35.959, 5.720 152Eu γ Energies 121.7817, 244.6975, 344.2785, 778.9040, ... 152 Eu Kab, L1ab 48.515, 7.754

Fit the peaks in the 137Cs, and 207Bi spectra to Gaussians, or Gaussians with a background model, and make a table of expected energy, peak height with uncertainty, mean channel with uncertainty, sigma in channels with uncertainty, and ﬁt quality. Plot the mean channel versus expected energy, and ﬁnd the energy calibration curve.

Internal conversion peaks of 152Eu Using the energy calibration constants already found, scale the axis on your 152Eu spectrum so that it is in units of keV. Fit the peaks to Gaussians, or Gaussians with a background model, and make a table of expected energy, peak height with uncertainty, mean channel with uncertainty, sigma in channels with uncertainty, and fit quality. Compare the fitted energies to the expected energies by calculating the pull, P , of the fit: E − E P = fit expected . (100) σfit Reasonable fits will have pull within ±1 from zero 65% of the time, and a pull within ±2 from zero 95% of the time. Comment on the detector resolution for the peaks. If any of the peaks are much wider than the detector resolution try to explain why that might be the case.

82 Kurie plots of beta spectra The most precise way of determining the endpoint energy of a beta decay source is to make a Kurie plot. The Kurie plot was developed by Franz Kurie, and is based on equation 97, which can be re-written as: s N(P ) = K(W − W ). (101) P 2F (Z,P ) 0 This can be re-written in terms of the reduced kinetic energy as: s N(W ) = K(W − W ). (102) EPF (Z,P ) 0 When the left side of the above equation is plotted against W, the spectrum will yield a straight line that can be extrapolated to the energy axis to give W0. Make a Kurie plot for each of the beta decay sources 14C, 204Tl, 54Mn, 90Sr, and 65Zn assuming that F (Z,P ) = 1, and determine the experimental end point energy of the highest energy beta for each of these sources. As usual make sure to determine a statistical uncertainty on this estimate, and consider any possible systematic eﬀects. These plots should use the energy calibration to convert the channel numbers into energy units. For the 204Tl source, overlay the plot which includes the F (Z,P ) in Table 16. Compare these to the energies (in keV) found in literature. Note that 65Zn is predominantly an EC source, so you should not expect to have any energetic electrons or positrons.

Some things to consider The thickness of the depletion region of our detector is roughly 500 µm. What effect might you expect in your spectra due to this limited range? Note that the ranges from the ESTAR data tend to be slightly higher than the average range, and that there is some range-straggling, meaning that some electrons will travel further or less far before stopping. Use the estimate found in the pre-lab for how much energy is lost in 5 mm of Air, and in 80 µg/cm2 of Mylar to comment on how these energy losses affect your results. As you change the source to detector distance, the solid angle that the detector subtends is changed. What effect might this have on the spectrum? What effect do you expect the detector resolution to have on the endpoint determination? ie. If the detector had much worse resolution, would that change your endpoint determination?

10.4 Beta Spectroscopy Pre-Lab Homework This assignment is to be completed, and handed in at the beginning of the ﬁrst lab session.

1. Look up, write down, and draw the decay schemes for 137Cs, 207Bi, 152Eu, 14C, 204Tl, 54Mn, 90Sr, and 65Zn. Only include the dominant decays (more than 5% of the intensity). 2. Starting from the atomic nucleus masses of the initial and ﬁnal nuclei, determine the Q value (which is the maximum energy of the β particle) for each of 137Cs, 207Bi, 14C, 204Tl, and 90Sr. Say whether β decay, and/or electron conversion should be possible based on the Q-value. 3. Use the data in Table 17 to calculate the expected energies of internal conversion electrons from 137Cs, 207Bi, and 152Eu. 4. Use ESTAR range data (http://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html) to determine the energy of an electron with a range of 500 µm in Silicon. Note you will have to convert the range in µm into a range in g/cm2 by multiplying by the density of Si.

83 5. Use ESTAR total stopping power data to estimate how much energy is lost by 100 keV, 300 keV, and 500 keV electrons in 5 mm of Air, and in 80 µg/cm2 of Mylar.

84 11 Monte Carlo Techniques

11.1 Introduction Often to understand a detector response, or systematic uncertainty in a physical measurement, a Monte Carlo technique can be used. This note reviews the role of simulation in data analysis, and the generation of random numbers. Data sets in physics often contain a random element, so we could use this randomness as a tool in our analysis. A Monte-Carlo simulation is the procedure of generating a random data set, that can be analyzed identically to the real data. Since the inputs to the simulation are programmed by the user, one can control the parameters, and can gain an understanding of the eﬀects of changes in the parameters. Suppose for example you want to understand how well you can measure the energy the photopeak and Compton edge of a gamma ray in you NaI detector. To do this you could generate 1000 fake data sets. Fit each of the 1000 data sets, and then look at the width of the distribution for the ﬁtted photopeak and Compton edge energies. If your simulation is good enough this will give you the uncertainty on the energy without doing any error propagation.

11.2 How to do a Monte Carlo Simulation If you want to write a simulation you need an accurate model for how your data relates to the underlying physics. The model needs to include any relevant physical eﬀects, including assumptions about the distributions of any random quantities. Note that a model is needed for data analysis anyway, so there is nothing new required for the simulation. An example of a simple Monte-Carlo for the transport of gamma rays in a medium can be found in [51]. The other ingredient needed to do Monte-Carlo is a means of generating random numbers. A random number is an unpredictable value with a known distribution (used to model a random variable). Many oﬀ the shelf tools exist for generating random numbers. In ROOT[3] for example

gRandom->Rndm() returns a random number from a uniform probability distribution function between 0 and 1.

11.3 Generating random numbers The random number generators available on computers are often of the pseudo-random number variety. The starting point to a random number generator is a routine that returns a random number drawn from a speciﬁed probability distribution function. Usually the base random number generator is for the uniform distribution on the interval [0, 1), and any other PDF is built from the uniform distribution. Useful properties of random number generators are:

• You cannot easily predict the next value that will be returned, even if you know the last N values returned.

• The frequency with which any value x comes up should be proportional to the underlying PDF (a uniform distribution)

• The number is not actually random.

85 You could get a true random number by using the Johnson noise in a resistor to generate a truly random number. It is actually useful, in order to reproduce a particular simulation result, to use pseudo-random numbers. The pseudo-random number is unpredictable, but reproducible. The simplest pseudo-random number generator is called a linear congruent generator, and the next random number is given by: Ij+1 = aIj + c (mod m). (103) In this equation, m is the modulus, a is called the multiplier, and c the increment (all positive constants). The number Ij/m will be a number between 0 and 1. This is a sequence of numbers that repeats after m calls, and all the numbers between 0 and m − 1 occur. In the worst case, the values of a and c are poorly chosen and the routine repeats much sooner. Numerical recipes ran0 algorithm uses a = 75, c = 0, m = 231 − 1 = 2.147 × 109, and has a period of 2.147 × 109[52]. To start the sequence an initial seed Ij is needed. Obviously if you are running a simulation 1000 times, and want to get a variation in your results you should start with different seeds (ie. read the seed values from a file or specify them as run time options). Some problems with the simple linear congruent generator are that sequential numbers are correlated, and least significant bits are less random than higher order bits. To fix this the output from ran0 can be shuffled, which fixes the correlation problems.

11.4 Generating pseudo-random numbers from non-uniform distributions There are four main methods of generating random numbers from non-uniform distributions: acceptance/rejection, transforming, hybrid methods, and using tricks. This section describes the acceptance/rejection, transform and trick methods. The hybrid method just means using a combination of the other three methods. The simplest method, though not the fastest, is to do what is called an acceptance/rejection method. If you have a PDF, P (x), then throw two uniform random numbers, one for x, and one for y. If y < P (x) then use this value of x, otherwise discard it and try again. This method is depicted in FIG. 30. To generate non-uniform distributions one can also use a transform method. The basic idea is to generate the uniform number F (x) between 0 and 1, and interpret it as the fraction of the PDF below a given value, and return that value of x. In other words, if f(z) is the PDF we want to generate random numbers from, then we do the transform: Z x f(z)dz = F (x) ∈ [0, 1). (104) −∞ For example, say we want to generate random numbers from the exponential PDF: 1 f(z) = exp (−z/τ). (105) τ Then, we do the integral to ﬁnd:

Z x dz Z x/τ F (x) = exp (−z/τ) = dy exp (−y) = 1 − exp (−x/τ). (106) 0 τ 0 So, if Y is a random number from 0 to 1, then set Y=F(x), and solve for x:

x = −τ ln (1 − Y ), or x = −τ ln Y. (107)

86 Figure 30: The solid line in the ﬁgure above is a PDF we wish to sample from. The black points are the (x, y) coordinates of uniform random values in the range 0, 1. The x value for the darker black points are accepted as random numbers with the distribution of the PDF.

The above two are equivalent since both Y and 1 − Y are uniform distributions ranging from 0 to 1. This method is very efficient, but requires that you are able to solve the integral. The final method is to stumble upon a nice method of finding the random numbers that follow the distribution you are interested in. This is the method used for generating random numbers that follow the Gaussian distribution. The trick is to notice that a 2D Gaussian can be written in polar coordinates:

x2 y2 r2 P (x, y)dxdy = exp (− − )dxdy = exp (− )rdrdθ. (108) 2 2 2 This is a product of integrable analytic PDFs over r and θ. Then we can use the transform method to generate a pair (r, θ). If we throw two uniform random numbers R1 and R2, then:

2 u = r /2 = −ln(R1), (109) √ r = 2u, (110)

θ = 2πR2, (111) x = r cos θ, and (112) y = r sin θ.

The values (x, y) are then two random numbers sampled from a Gaussian distribution. In other words to generate a Gaussian random number, you actually need to generate two random numbers at a time.

11.5 Monte-Carlo Assignment We will use the built in random generator in ROOT to write a simple Monte-Carlo to solve a problem for us. For this assignment, suppose we have a surface barrier detector operated in reverse bias with a depletion depth of 500 µm. The surface barrier detector has a radius of 5 mm, and the binding energy of

87 an electron-hole pair in the depletion region is 3.63 eV. We place a β− source with a 500 keV endpoint energy 1.0 cm from the detector in air. What we want to know is what the resulting spectrum will look like. There are a number of steps we need to take to simulate this situation. The sample scripts referred to in the steps below can be found at: http://t2kwinnipeg.uwinnipeg.ca/~jamieson/courses/intermediate_lab/mc_macros/. 1. Generate β− “events” with energies that follow the expected shape of the β− decay spectrum. For this problem we will use F (Z,P ) = 1, so that:

2 N(P ) = K(W0 − W ) PE (113) You can take this function definition (a TF1) from the mcbeta.C root script, and modify it to have a 500 keV endpoint. Once you have a function defined in root, it has a method (myfunction->GetRandom()) that uses the select-reject method to pick a random number that follows the shape of the (distribution) function. Make a histogram, and fill it with 10000 events that are generated with the GetRandom method. To add one count to a histogram, use the hist->Fill( value ) method. Also remember that a loop in a root macro can be made using:

for (int i=0; i<10000; i++){ commands to run 10000 times; }

Congratulations! You have just done your ﬁrst Monte-Carlo calculation. Save an overlay of your histogram overlaid with the function to a .png ﬁle. Note that you will have to rescale your histogram so that the curves are properly overlaid. You will need to get the area integral of a histogram, using the hist->GetIntegral(‘‘width’’) method, and to scale a histogram use the hist->Scale( scalefactor ) method. 2. Generate points on the surface of the detector that will be hit. Assume that the β will hit the face of the detector with equal probability of hitting any point on the detector. To generate a random point inside a circle, one could do a select-reject method, where one throws a uniform random X and a uniform random Y point, and then keep the result if if falls inside the circle. A more clever way is to use a transform method. In this method one generates a uniform random√ number for the angle, θ, between 0 and 2π around the circle, and a random number R = r Z, where Z is a √ uniform random number between 0 and 1, and r is the radius of the circle. The reason for the is that we want to have a uniform distribution, and the surface area at a given radius goes as r2. The random point that is hit in cartesian coordinates is then x = R cos θ, y = R sin θ. Make a 2D histogram:

my2dhist = new TH2D(‘‘my2dhist’’, ’’Title’’, numxbins, xmin, xmax, numybins, xmin, ymax );

with dimensions large enough to hold the face of the detector, and ﬁll it with 10000 random x, y points that are uniform on the surface of the circle. Draw your histogram with the option my2dhist->Draw(‘‘colz’’), and save it to a .png ﬁle. 3. Inside the loop to generate a point on the surface of the detector that is hit, throw a random β energy as in problem 1, and for that energy, determine the energy deposit in the detector by following these steps.

88 (a) Do a little geometry, and calculate how far through the air the beta travels, and using the routines from readestar.C calculate the energy of the β as it reaches the surface of the detector (assume no energy loss straggling). Again use a detector radius of 5 mm and point source centered on the detector a distance of 10 mm in air. (b) Do the geometry calculation to determine the thickness of the depletion region that the β could cross. For any edge events, assume that the detector shape is such that it would still cross 500 µm if it had been at normal incidence (ie. the edges aren’t squared oﬀ). (c) Do another calculation to determine the range of the β particle in the Si depletion region, using the TGraph from readestar.C. Make a histogram of the minimum of the range of the beta and the thickness of the depletion region that would be crossed. Save this histogram to a .png ﬁle. (d) Using the previous two calculations, determine the energy deposit in the detector. Ie. determine if the β stopped in the depletion region, and thus deposited all its energy, or assume (not quite correctly) that the β loses energy in proportion to the fraction of the distance travelled in the depletion region, to its total range.

Using the result from the last step above, make a histogram of the energy deposit in the detector, overlaid with the previous two energy histograms. Save this image to a .png ﬁle.

4. The actual signal in the detector is proportional to the number of electron-hole pairs created. The number detected should√ follow the ordinary counting statistics, meaning that the uncertainty on the number detected is N. Assuming it takes 3.63 eV to generate each electron-hole pair, calculate the number of expected electron-hole pairs, and use counting statistics to randomize this number (use gRandom→Poisson(N) ). Convert the randomized number of electron-hole pairs created back into an energy, and make another histogram of the measured β energy. Overlay this and save the image to a .png ﬁle.

5. Remake the last plot, but change the detector source distance to 5.0 mm.

Hand in this assignment by emailing a single root macro, with name (‘‘your name mc.C’’) to me ([email protected]). Also hand in your lab notebook with any of the .png ﬁles printed out and properly labelled.

89 12 X-Ray Measurements with a Proportional Counter

X-Rays are due to photon emission when electrons transition from an outer to inner shell of an atom. The spontaneous em-mission of X-rays from a radioactive source is often stimulated by a decay in the nucleus leaving the atom in an excited state. For this lab you will follow the Ortec experiment 11 title “The Proportional Counter and Low-Energy X-Ray Measurements”. The following changes to the lab need to be noted:

1. The isotopes 54Mn, 57Co, 65Zn, 109Cd, 137Cs, 55Fe, 152Eu, and 241Am are the X-Ray sources that will be used for this experiment.

2. The proportional counter used for this lab operates at a higher voltage than indicated in the Ortec lab. It will need to be brought to +2700 V to have suﬃcient gain.

3. Instead of using the 241Am source for the energy calibration, use as many of the other sources to do the energy calibration. Then using that energy calibration determine the x-ray energies of the 241Am and 152Eu sources. Also determine which atomic energy level transitions are made for these x-rays.

4. Instead of the Ortec MCA, the SpecTech MCA, and its software will be used to collect the X-Ray spectra. Save your spectra to a text ﬁle, so that they can be analyzed in Root.

5. For Ortec Experiment 11.2, use Aluminium foil only for the absorber. Measure the thickness of the foil used. Compare your measurements with photon mass attenuation lengths at http: //physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z13.html.

90 ® ORTEC Experiment 11 The Proportional Counter and Low-Energy X-Ray Measurements

Purpose In this experiment, the techniques for operating a thin-window proportional counter are explored, and some typical X-ray spectra are obtained. The proportional counter is employed to measure the mass absorption coefficients of aluminum and nickel at 22 keV.

CAUTION The thin beryllium window built into the proportional counter is very fragile. DO NOT allow any material to contact the window and DO NOT TOUCH IT.

Relevant Information As discussed in Experiments 3 and 7, the photoelectric interaction is the most pronounced type of photon interaction for energies below 100 keV. This experiment utilizes the proportional counter to detect X-ray energies below 60 keV. The typical proportional counter is basically a metal cylinder with a concentric wire stretched along the center of its longitudinal axis (ref. 1 and 8). The tube is filled with a counting gas mixture (e.g., 760 Torr of 90% Xenon and 10% CH4), and a positive high voltage of ~1700 V is applied to the central electrode. In the most common configuration, a thin beryllium window is built into the cylinder wall to allow low-energy X rays to enter into the counting region with minimum absorption. Proportional counters are also available with the beryllium window mounted at one end of the cylinder. Beryllium is used because it is a malleable metal, and it has a low atomic number (Z = 4) that minimizes absorption in the window. The low atomic number is advantageous, because the photoelectric absorption cross section varies as Z5 (see Experiments 3 and 7). When an X ray enters the tube, it ionizes the gas via the photoelectric interaction. The photon disappears, while transferring its energy to the electron ejected from the ionized atom. As the electron travels through the gas, it loses its energy by causing further ionization. At the end of the ionization process, the number of electron-ion pairs, n, is proportional to the original energy of the X-ray photon. E n = –––ε (1) Where E is the energy of the X ray, and є is the average energy required to create one electron-ion pair, typically 21.5 eV for Xenon.

Equipment Required • C-24-12 RG-62A/U 93-Ω Coaxial Cable with BNC Plugs, 12-ft. (3.7-m) length. • 4542PC Xenon-filled, End-Window Proportional Counter. • C-29 BNC Tee Connector. • PC-STAND-AX Proportional Counter stand with source and absorber foil holders. • Foil-Al-5: 10 each 1/2" diameter Aluminum foils, 0.005" thick (1.27 cm dia. x 0.0127 cm thick). • 556 High Voltage Power Supply. • Foil-Ni-3: 10 each 1/2" diameter Nickel foils, 0.003" thick • 142PC Preamplifier. (1.27 cm dia. x 0.00762 cm thick). • 575A Amplifier. • Sources: • 4001A/4002D NIM Bin and Power Supply. • GF-241-M-10 10 µCi 241Am (433-y half life). A license is • EASY-MCA-2K including USB cable and MAESTRO-32 required for this source. software (other ORTEC MCAs may be substituted). • GF-137-M-5 5 µCi 137Cs (30.2-y half life). A license is • PC-1 Personal Computer with USB port and a recent, required for this source. supportable version of the Windows operating system. • GF-109-M-10 10 µCi 109Cd (463-d half life). A license is • TDS3032C Oscilloscope with bandwidth ≥150 MHz. required for this source. • Cables and Connectors: • GF-057-M-20 20 µCi 57Co (272-d half life). A license is • C-36-12 RG-59A/U Coaxial Cable with SHV Female required for this source. Plugs, 12-ft. (3.7-m) length. • GF-055-M-10 10 µCi 55Fe (999-d half life). A license is • C-36-0.5-S RG-59A/U 75-Ω Coaxial Cable with SHV required for this source. Female Plugs, 0.5-ft. (15-cm) length. • ZN65S 1 µCi 65Zn (244-d half life). • Two C-24-4 RG-62A/U 93-Ω Coaxial Cables with BNC • Small, flat-blade screwdriver for tuning screwdriver-adjustable Plugs, 4-ft. (1.2-m) length. 91 controls. Experiment 11 The Proportional Counter and Low-Energy X-Ray Measurements

When the proper bias voltage is applied between the center wire (anode) and the outer cylinder (cathode), the cloud of electrons released in the initial ionization process is accelerated towards the center wire, while the positive ions drift towards the outer cylinder. As the electrons accelerate towards the anode, they acquire sufficient energy to ionize additional atoms. The result is an avalanche of electrons impinging on the center wire. The difference between a Geiger Counter and a Proportional Counter lies in the bias voltage. The voltage applied to a Geiger Counter is high enough so that the number of electrons reaching the anode wire is always the same, independent of the energy of the detected photon. For a Proportional Counter, the voltage is lowered to the point where the number of electrons reaching the anode, nA, is proportional to the energy of the detected photon. E nA = A n = A –––ε (2) Where A is the proportionality constant, which is the ratio of the number of avalanche electrons to the number of initial electrons. In other words, A is known as the gas gain. Happily, the gas gain significantly increases the amount of charge collected by the preamplifier for each detected X ray. That reduces the importance of the preamplifier noise, enabling the analysis of very low X-ray energies. The gas gain of the proportional counter is sensitive to the applied bias voltage. Consequently, a power supply that delivers a very stable bias voltage is essential. Also the bias voltage must be selected within the normal operating range of the detector. Figure 11.1 illustrates the operational characteristics of a cylindrical, gas-filled detector as a function of the applied bias voltage. Some voltage is required to separate the electron-ion pairs and drift the negative and positive charges to their respective electrodes, where they are collected by the preamplifier to form a pulse. Vi is the minimum bias voltage required to collect all the electron-ion pairs before some of them recombine. As the voltage is increased from Vi to Vp, the detector functions as an ionization chamber, i.e., the number of electron-ion pairs collected is independent of the voltage.

Beginning at Vp, the electrons pick up enough energy on their trip to the anode to cause further ionization. Thus the value of the gas gain is greater than unity. Because of the coaxial cylindrical geometry, the electric field strength is highest near the anode wire, and decreases towards the wall of the cylinder. Thus, it is only when the electrons get close to the anode that they pick up sufficient energy to cause an avalanche of ionization. As the bias voltage increases, the field strength increases, and electrons receive sufficient acceleration further from the anode to cause an avalanche of ionization. Thus the gas gain, A, increases with increasing bias voltage. The vertical scale in Figure 11.1 is logarithmic. Consequently, the parallel, diagonal lines for low-energy and high-energy photons reflect the increasing gas gain as the bias voltage increases. Above Vlp, the gas gain for high-energy photons does not increase as strongly as the gain for low-energy photons. Therefore, the pulse-height spectrum begins to compress at higher energies. This is the limited proportional region.

If the bias voltage is increased above Vg, all pulses will exhibit the same pulse height, and this is the Geiger Counter region. Even with a Geiger Counter, the region above Vc must be avoided because of the destructive continuous discharge in the detector. The normal operating range for a proportional counter is in the proportional region between Vp and Vlp. For the 4542PC Proportional Counter, these limits are from 1600 to 1850 Volts. Typically the charge collection time for electrons is less than 500 ns. However, the charge collection time for the positive ions is quite long, of the order of tens of microseconds. Therefore, a 0.5 µs shaping time constant is normally employed in the amplifier to emphasize the fast signal from the electrons, while suppressing the slow, positive-ion signal (ref. 8). Because the detector is filled with a gas at a pressure circa 760 Torr, the proportional counter does not have a high detection efficiency above 50 keV. Fig. 11.1. The Effect of Bias Voltage on the Operating Characteristics of a Gas Filled Detector.

2 92 Experiment 11 The Proportional Counter and Low-Energy X-Ray Measurements

Two common, alternative fill gases are Argon and Xenon. Because of its higher atomic number Xenon is the better choice when energies above 20 keV must be measured. Moreover, an end-window is preferred over the side window geometry when high energies are important, because the end-window geometry provides the full length of the cylinder for stopping the photons, whereas the side window presents the much smaller diameter of the cylinder. Figure 11.2 shows the typical intrinsic detection efficiency of an end- window proportional counter filled with Xenon at 760 Torr. The intrinsic detection efficiency specifies the number of photons that end up in the photopeak in the energy spectrum, expressed as a percent of the photons that are incident on the Beryllium window port. At Fig. 11.2. The Intrinsic Detection Efficiency of an End energies below 8 keV, the detection efficiency declines because of Window Proportional Counter Filled with Xenon Gas. The absorption in the 0.25 mm thick Beryllium window. The abrupt jump Beryllium Window Thickness is 0.25 mm. in detection efficiency at 34.579 keV corresponds to the K absorption edge of Xenon. Photons above that energy can ionize the K-shell of Xenon, causing much stronger absorption than is provided by only L-shell ionization at slightly lower energies. When electrons refill the vacancy in the K shell, Xenon K-series X rays are emitted. If those K X-rays escape the detector, there will be a deficit of energy recorded by the detector. This is the source of the escape peaks in the energy spectrum approximately 29.7 and 33.8 keV below the original energy of the photon entering the detector. References 1 and 8 delineate the factors that contribute to the Table 11.1. Recommended X-Ray and Gamma-Ray Calibration energy resolution of the proportional counter. Statistical Sources (1 to 10 µCi). variations in the number of original electron-ion pairs and Daughter Kα X-Ray KΒ X-Ray γ-Rays statistical variations in the gas gain are the major contributors. Radioisotope Half Life Isotope (keV) (keV) (keV) Electronic noise in the preamplifier is usually small compared 54Mn 312.3 d 54Cr 5.411 5.946 to these two contributions. For most applications, the energy 57Co 271.8 d 57Fe 6.399 7.057 14.41 resolution can be presumed to be proportional to the square root of the photon energy. A FWHM resolution of 2.2 keV is 65Zn 244.3 d 65Cu 8.040 8.921 typical at 22.1 keV. This is much worse than obtained with a 85Sr 64.8 d 85Rb 13.374 15.012 Si(Li) detector (Experiment 8). 88Y 106.6 d 88Sr 14.141 15.892

Because of the relatively coarse resolution, it is convenient to 109Cd 462.6 d 109Ag 22.102 25.062 88.03 use the weighted average of the Kα1 and Kα2 energies and 113Sn 115.1 d 113In 24.135 27.411 similarly for the Kβ1,3 and Kβ2 peaks. Table 11.1 shows the resulting energies for the radioisotopes typically used for 137Cs 30.17 y 137Ba 32.059 36.582 energy calibration.

EXPERIMENT 11.1. Energy Calibration Purpose In this experiment the detector and its supporting electronics will be set up and calibrated. The dependence of energy resolution on energy will be explored. The spectrum from an 241Am source will be examined to identify the X-ray lines and gamma-ray peaks. An unidentified source will be counted for the purpose of identifying the source via its energy spectrum.

Procedure 1. Connect the instruments as shown in Fig. 11.3, being careful to prevent touching the Be window on the proportional counter. More specifically: 2. Turn off the power switches on the 556 Power Supply and the NIM Bin power supply. Install the modules in the NIM Bin.

93 3 Experiment 11 The Proportional Counter and Low-Energy X-Ray Measurements

3. Connect the proportional counter to the INPUT of the 142PC Preamplifier using the shortest possible coaxial cable having the appropriate connectors. If the proportional counter employs an MHV connector, the cable will be 75-Ω RG-59A/U with an MHV connector on the end for the proportional counter, and an SHV connector to accommodate the Preamplifier INPUT. If the proportional counter utilizes an SHV connector, the RG 59A/U cable will have SHV connectors on both Fig. 11.3. Electronics Block Diagram for Experiment 11.1. ends. 4. On the 575A Amplifier, check that the switches accessible through the side panel have been set for a 0.5 µs SHAPING TIME. Set the front-panel input polarity switch to POSitive. 5. Connect the 142PC Preamplifier power cord to the PREAMP POWER connector on the rear panel of the 575A Amplifier. Using the 3.7 m, 93-Ω, RG-62A/U coaxial cable, connect the ENERGY output from the preamplifier to the INPUT of the 575A Amplifier. 6. On the rear panel of the 556 High-Voltage Power Supply, check that the CONTROL switch is set to INTernal. Verify that the POLARITY switch is set to POSitive. Using the 3.7 m, RG-59A/U coaxial cable with SHV plugs, connect one of the high voltage OUTPUTS to the BIAS input on the 142PC Preamplifier. 7. Set the voltage controls on the front panel of the 556 High-Voltage Power Supply approximately half way between the manufacturer’s recommended operating voltage and the minimum operating voltage specified by the manufacturer. For the 4542PC, the range of operating voltages is +1600 V to +1850 V, and the manufacturer’s recommended operating voltage is +1700 V. Using a slightly lower bias voltage permits higher counting rates before peak shifting caused by the positive ion cloud sets in (ref. 8). 8. Using a 93-Ω RG-62A/U cable, connect the UNIpolar OUTput of the 575A Amplifier to the analog INPUT of the EASY-MCA-2K. Ensure that the EASY-MCA-2K is interfaced to the supporting computer via the USB cable, and MAESTRO-32 is operating in the computer. 9. Via MAESTRO-32, set the MCA coincidence gate requirement to OFF. Select a 512-channel conversion gain. Set the Upper Level Discriminator to 512, and the Lower Level Discriminator to 10 channels. 10. Turn on the NIM Bin power and the 556 Power Supply. 11. Place the 241Am source approximately 5 cm from the Proportional Counter window on the axial centerline of the detector cylinder. 12. Observe the UNIpolar OUTput of the 575A Amplifier on the oscilloscope. 13. Adjust the Amplifier gain to achieve a +8.5 V pulse height for the 59.5 keV gamma ray. Once the adjustment is complete, lock the FINE GAIN dial so that the gain cannot be accidentally changed. 14. Employing the procedure taught in Experiment 3, adjust the Pole-Zero (P/Z) cancellation on the 575A Amplifier to make the output pulses return to baseline as quickly as possible without undershooting. NOTE: The slow positive-ion charge collection in the proportional counter will cause a slower return to baseline than is experienced with NaI(Tl), Si(Li), silicon surface-barrier or ion-implanted silicon detectors (see the pulse shape pictures in ref. 8). 15. Reconnect the 575A Amplifier UNIpolar OUTput to the EASY-MCA-2K analog INPUT. 16. Temporarily remove all radioactive sources from the vicinity of the Proportional Counter and start an acquisition on the MCS. Observe the percent deadtime. If the dead time is greater than 1% adjust the Lower Level Discriminator on the EASY-MCA-2K to be safely above the noise level. This can be accomplished by lowering the discriminator level until the percent dead time increases drastically. Next, slowly raise the discriminator threshold until the dead time is just less than 1%. If the dead time does not exceed 1%, there is no need to adjust the lower level discriminator. 17. Place the 241Am source on the axial centerline of the proportional counter, with a source to detector distance of approximately 5 cm. If the percent dead time exceeds 20% with any of the sources employed in this experiment, increase the source to detector distance to reduce the percent dead time below 20%.

4 94 Experiment 11 The Proportional Counter and Low-Energy X-Ray Measurements

18. Acquire a spectrum for a sufficient live time to obtain adequate statistical precision in the shape and location of the 59.5 keV gamma ray. If the gamma-ray is not centered on channel 426, adjust the amplifier fine gain to position the peak within ±10 channels of channel 426. Lock the FINE GAIN dial so that the gain cannot be accidentally changed. 19. Using the features available in MAESTRO-32, determine the channel numbers corresponding to the centroid of the 59.5 keV peak. Measure the FWHM resolution of the gamma-ray peak. Record both numbers and save the spectrum for possible later reference. 20. Replace the 241Am source with the 137Cs source. Acquire a spectrum, and record the channel positions for the centroids of the Kα and Kβ X-ray peaks. Measure and record the FWHM for each peak. Save the spectrum for possible later reference. 21. Repeat step 20 using the 57Co source. Record the channel positions

for the centroids of the K X-ray peak and the 14.4 keV gamma-ray Fig. 11.4 A Typical 57Co Spectrum Acquired with a peak. Measure and record the FWHM for each peak. Save the Proportional Counter. spectrum for possible later reference. Figure 11.4 illustrates a typical 57Co spectrum. 22. Repeat step 20 with the 109Cd source, recording the centroid channel positions and the FWHM resolutions of the silver K-lines, and saving the spectrum for possible later reference. 23. Using the spectra and centroids acquired in steps 19 through 22, and the calibration menus of MAESTRO-32 calibrate the MCA cursor to read directly in keV.

EXERCISES a. Plot an energy calibration curve for the data acquired above. Determine the slope of the calibration line in keV/channel. Using either the calibration curve or the MAESTRO-32 features, convert the measured FWHM values for the peaks from channels to keV, and plot them on a linear graph. b. Plot the FWHM in keV on a logarithmic scale versus the energy of the X ray or gamma ray on a logarithmic scale. Draw a straight line through the points and measure the slope. Does this line support the contention that the energy resolution is proportional to the square root of the energy? c. For cases where the Kα and Kβ peaks are not completely resolved, how does that complication affect the measured position and FWHM of the peak? Can you find any discrepancies in the energy calibration and the resolution graphs that are traceable to this resolution limitation?

24. Acquire a spectrum with the 241Am source long enough to enable identification of the weaker peaks. Save this spectrum on a transportable medium such as a CD, memory stick or portable hard disk for later incorporation in your report. It should be exported to the transportable medium as an ASCII file. This file can be imported into an Excel spreadsheet using tab and space delimiters for plotting and inclusion in your report. Alternatively, the computer display of the spectrum can be photographed with a high-resolution digital camera, for transfer of the image into your report.

95 5 Experiment 11 The Proportional Counter and Low-Energy X-Ray Measurements

EXERCISES d. Refer to Table 8.1 of Experiment 8, and identify the peaks in the 241Am spectrum. e. For the 59.5 keV gamma ray, forecast the location of the escape peaks caused by the Xenon Kα and Kβ X rays escaping from the detector. Identify those escape peak locations in your spectrum. Are these escape peaks detectable in your spectrum?

25. From your Laboratory Instructor, obtain an X-ray source whose identity has been hidden. Acquire a spectrum.

EXERCISE f. Identify the unknown X-ray source from its spectrum and the information in references 6, 9 and 10.

EXPERIMENT 11.2. Mass Absorption Coefficients for 22 keV X Rays Purpose

The mass absorption coefficients for the 22.1 keV Ag Kα X rays from 109Cd will be measured for aluminum and Nickel. The results for aluminum can be compared to the value measured in Experiment 8.

Procedure 1. Continue with the system as set up and calibrated in Experiment 11.1. 2. Place the 109Cd source on the axial centerline of the proportional counter approximately 5 cm from the detector window. 3. Acquire a spectrum for a preset live time of 100 seconds. Check the percent dead time. If the dead time is greater than 20%, increase the source-to-detector distance to reduce the dead time below 20%. If the percent dead time is less than 10%, reduce the distance to move the dead time into the 10% to 20% range. In any case, do not reduce the source-to-detector distance to less than 2.5 cm. 4. Once the distance has been adjusted, determine what preset live time is required to accumulate at least 4,000 counts in the area under the 22.1 keV peak. Use that preset live time for each acquisition in the remainder of the experiment. Table 11.2. Data for Absorption Measurement with Al Foils. 5. Accumulate a spectrum for the selected live time with no Aluminum Density = 2.70 g/cm3 metal foils between the 109Cd source and the proportional Absorber Thickness counter. Set a region of interest (ROI) across the full Line Live Time Counts in Ag Index, i Inches cm g/cm2 (sec) Kα Peak extent of the Ag Kα peak, and record the total counts in the ROI on line 0 in Table 11.2. 0 0.000 0.000 0.000 6. Being careful to not change the position of the source, 1 0.005 0.013 0.034 insert an aluminum foil between the source and detector, 2 0.010 0.025 0.069 using the thickness specified on line index 1 of Table 3 0.015 0.038 0.103 11.2. Repeat the measurement of the counts in the Ag Kα 4 0.020 0.051 0.137 peak using the same ROI and live time as in step 5. 5 0.025 0.064 0.171 Record the measurement in Table 11.2. 6 0.030 0.076 0.206 7. Repeat step 6 for each foil thickness specified in Table 7 0.035 0.089 0.240 11.2. 8 0.040 0.102 0.274 9 0.045 0.114 0.309 10 0.050 0.127 0.343

6 96 Experiment 11 The Proportional Counter and Low-Energy X-Ray Measurements

EXERCISES a. Make a plot on semilog paper of counts vs. absorber thickness (g/cm2) for aluminum. Figure 11.5 shows some typical data taken at a lower energy with thinner foils. b. From your plots and with reference to the discussion in Experiment 3, determine the half-value thickness and the mass attenuation coefficient for aluminum.

8. Repeat the sequence of measurements and the Exercises using the nickel foils instead of the aluminum foils. See Table 11.3 and Figure 11.6.

Table 11.3. Data for Absorption Measurement with Ni Foils. Nickel Density = 8.90 g/cm3 Fig. 11.5. Counts versus Aluminum Absorber Thickness for 65Zn X Rays. Line Absorber Thickness Live Time Counts in Ag Index, i Inches cm g/cm2 (sec) Kα Peak 0 0.000 0.000 0.000 1 0.003 0.008 0.068 2 0.006 0.015 0.136 3 0.009 0.023 0.203 4 0.012 0.030 0.271 5 0.015 0.038 0.339 6 0.018 0.046 0.407 7 0.021 0.053 0.475 8 0.024 0.061 0.543 9 0.027 0.069 0.610 10 0.030 0.076 0.678 References Fig. 11.6. Counts versus Nickel Absorber 1. G. F. Knoll,Radiation Detection and Measurement, John Wiley and Thickness for 65Zn X Rays. Sons, Inc., New York (1979). 2. R. W. Hendriks, Nucl. Instrum. Methods 102, 309 (1972). 3. M. W. Charles and B. A. Cooke, Nucl. Instrum. Methods 88, 317 (1970). 4. R. Gott and M. W. Charles, Nucl. Instrum. Methods 72, 157 (1969). 5. B. E. Fischer, Nucl. Instrum. Methods 141, 173 (1977). 6. C. M. Lederer and V. S. Shirley, Eds., Table of Isotopes, 7th Edition, John Wiley and Sons, Inc., New York (1978). 7. C. E. Crouthamel, Applied Gamma-Ray Spectrometry, Pergammon, New York (1960). 8. Ron Jenkins, R. W. Gould, and Dale Gedcke, Quantitative X-ray Spectrometry, Marcel Dekker, Inc., New York, 1981. 9. National Nuclear Data Base, Brookhaven National Laboratory, http://www.nndc.bnl.gov/. 10. X-Ray Critical-Absorption and Emission Energies in keV in the AN34 Library at www.ortec-online.com. Specifications subject to change 101411

® www.ortec-online.com ORTEC Tel. (865) 482-4411 • Fax (865) 483-039697 • [email protected] 801 South Illinois Ave., Oak Ridge, TN 37831-0895 U.S.A. For International Office Locations, Visit Our Website 13 Alpha Particle Energy Loss and Rutherford Scattering

In this lab you will be measuring the energy loss, energy loss straggling, and try to verify the Rutherford Scattering equation for an alpha particle source. This experiment is Ortec Experiment 15, which is appended in the following pages. Some changes to this experiment are noted on the remainder of this page.

1. The detector used will be a Si semiconductor detector (Ortec ULTRA alpha detector), model number BU-012-025-500. This detector has an alpha energy resolution of better than 12 keV, and noise less than 6 keV. These detectors should be operated with a bias voltage of +140 V, to have a 500 µm depletion depth.

2. Rather than the Ortec MCA and MAESTRO software, the SpecTech MCA and related software will be used to collect the alpha energy spectra.

3. Three alpha sources are available for use. An 0.1 µCi 241Am surface source will be used for the main energy calibration. A stronger 100 µCi 241Am source with an active diameter of 9.5 mm has a 100 µg/cm2 Au coating will be needed for the Rutherford scattering measurements. The third source is an 0.1 µCi 210Po source which has an 80 µg/cm2 coating of aluminized mylar over it. You should try to estimate the energy loss due to the coatings on these alpha sources using the data from ASTAR at http://physics.nist.gov/PhysRefData/Star/Text/ASTAR.html.

4. You can also use the β sources with conversion electrons to improve the energy calibration.

98 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils

NOTICE: This legacy experiment requires some equipment that is not available from ORTEC. See the Appendix 15-A for adaptation suggestions.

Equipment Required • 142B Preamplifier • 4001A/4002D Bin and Power Supply • 428 Detector Bias Supply • 480 Pulser • 575A Amplifier • ULTRA™ Ion-Implanted-Silicon, Charged-Particle Detector, Model BU-021-450-100 • EASY-MCA-8K System including USB cable and MAESTRO-32 software (other ORTEC MCAs may be substituted). • Personal Computer with USB port and recent, supportable version of the Windows operating system. • Coaxial Cables and Adapters: • One C-18-2 Microdot 100-Ω Miniature Cable with two Microdot male plugs, 0.61-m (2-ft) length; to connect the ULTRA detector in the Rutherford Scattering Chamber to the vacuum feedthrough. • One C-13 BNC to Microdot Vacuum Feedthrough with female BNC and male Microdot (may already be part of the Rutherford Scattering Chamber). • One C-30 Microdot to Microdot Connector with female Microdot on both ends to adapt the C-13 to the C-18-2 (may already be part of the Rutherford Scattering Chamber). • One C-24-1/2 RG-62A/U 93-Ω coaxial cable with BNC plugs on both ends, 15-cm (1/2-ft) length. Connects the vacuum feedthrough to the INPUT of the 142B Preamplifier. • Two C-24-12 RG-62A/U 93-Ω coaxial cables with BNC plugs on both ends, 3.7-m (12-ft) length. • One C-24-4 RG-62A/U 93-Ω coaxial cable with BNC plugs on both ends, 1.2-m (4-ft) length. • One C-36-12 RG-59B/U 75-Ω cable, with SHV female plugs on both ends, 3.7-m (12-ft) length. • TDS3032C Oscilloscope with bandwidth ≥150 MHz. • ALPHA-PPS-115 (or 230) Portable Vacuum Pump Station.

ORTEC does not sell the following equipment. It must be acquired from another source. • Small, flat-blade screwdriver for screwdriver-adjustable controls, or an equivalent potentiometer adjustment tool. • 0.5 to 1 mCi 241Am source with no absorbing window for the alpha emissions. (Half Life: 432 years). • Metal foils of the nominal, listed thicknesses: Au (1.57 mg/cm2), Al (0.62 mg/cm2), Ni (0.87 mg/cm2), Cu (0.86 mg/cm2), and Ag (1.1 mg/cm2). • Rutherford Scattering Chamber (Requires custom fabrication. See figures 15.2 and 15.3.)

Purpose In this experiment the scattering of alpha particles by a gold foil will be measured, and the results will be interpreted as experimental cross sections, which will be compared with theoretical equations. Optionally, the dependence of the scattering cross section on the atomic numbers of different metal foils can also be explored.

99 1 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils

Introduction No experiment in the history of nuclear physics has had a more profound impact than the Rutherford elastic scattering experiment. It was Rutherford’s early calculations based on the elastic scattering measurements of Geiger and Marsden that gave us our first correct model of the atom. Prior to Rutherford’s work, it was assumed that atoms were solid spherical volumes of protons and that electrons intermingled in a more or less random fashion. This model was proposed by Thomson and seemed to be better than most other atomic models at that time. Geiger and Marsden made some early experimental measurements of alpha-particle scattering from very thin, hammered-metal foils. They found that the number of alphas that scatter as a function of angle is peaked very strongly in the forward direction. However, these workers also found an appreciable number of scattering events occurring at angles >90°. Rutherford’s surprise at this observation is captured by his comment in one of his last lectures: “It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a fifteen-inch shell at a piece of tissue paper and it came back and hit you.” Rutherford tried to analyze this angular dependence in terms of the atomic model that had been proposed by Thomson. But, he observed that the Thomson model could not explain the relatively large back-angle cross section that had been found experimentally. Measurements by Geiger and Marsden revealed that 1 out of 8000 alpha particles incident on the platinum foil experienced a deflection >90°. This was in conflict with calculations based on the Thomson model which predicted that only 1 alpha in 1014 would suffer such a deflection. With intense effort, coupled with his unusual physical insight, Rutherford developed the nuclear model of the atom. His calculations, based on Coulomb scattering from the proposed hard central core of positive charge, produced the required 1010 increase in cross section found by Geiger and Marsden. Of course, the cross section was very difficult to determine experimentally with the equipment available to these workers (an evacuated chamber with a movable microscope focused on a scintillating zinc sulfide screen). The experimenters had to observe and count the individual scintillation flashes caused by each alpha particle impinging on the zinc sulfide screen. It was only through very careful and tedious measurements that the angular distribution was experimentally determined. The term, “cross section,” mentioned above is a measure of the probability for the scattering reaction at a given angle. From a dimensional standpoint, cross section is expressed by units of area. This seems reasonable since the relative probability of an alpha particle striking a gold nucleus is proportional to the effective area of the nucleus. The concept is similar to throwing a tennis ball at a basketball hanging from a string. The probability of hitting the basketball is proportional to the projected area of the basketball (π times the square of the basketball radius) as viewed from a distance by the person throwing the tennis ball. Cross sections are usually expressed in a unit called “barn,” where one barn is 1x10–24 cm2. This is a very small effective area. But, it is not unreasonable, when one considers the small size of the nucleus in comparison to the much larger size of the atom. The term, “barn,” originated during the WWII Manhattan project. Experimenters discovered cross sections for slow neutron interactions with atomic nuclei that seemed so large they were colloquially described to be “as big as a barn.” For a Rutherford scattering experiment it is most convenient to express the results in terms of cross section per solid angle. The solid angle referred to is the solid angle that the detector makes with respect to the target, and is measured in 2 steradians, (sr). The solid angle, ΔΩ, in steradians is simply Ad/R , where Ad is the sensitive area of the detector and R is the distance of separation between the detector and the target. The measurement of cross section is expressed in barns/steradian or more conveniently millibarns/steradian, i.e., mb/sr. The cross section defined here is referred to as the differential cross section, and it represents the probability per unit solid angle that an alpha will be scattered at a given angle θ. The theoretical expression for the Rutherford elastic scattering cross section can be simplified to the following formula: 2 2 dσ Z1 Z2 θ M ( )[ 4() () ] dΩ = 1.296 E csc 2 –2 A mb/sr (1)

Where Z1 and Z2 are the atomic numbers of the projectile and target, respectively, E is the energy of the projectile in MeV, M is the mass number of the projectile, and A is the mass number of the target nucleus. For our experiment, 197Au 4 4 197 ( He, He) Au, Z1 = 2, Z2 = 79, E = 5.48 MeV, M = 4, and A = 197. In alpha scattering from gold, it is quite difficult to

100 2 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils measure the cross section for scattering angles >90°. The reason for this difficulty is that it takes a prohibitively long period of time to make a measurement at the back angles. To limit the duration of this experiment, only the forward angles will be investigated. Even for the lightest foil (Al) used in experiment 15.2, the term 2(M/A)2 is always quite small compared to csc4(θ/2) and hence can be ignored without much error. To a good approximation, the differential cross section is then given by 2 θ dσ Z1 Z2 ( ) 4() dΩ = 1.296 E csc 2 mb/sr (2) Note that the expression in Eq. (2) varies 4 orders of magnitude from θ = 8° in the forward direction to θ = 90°. The purpose of this experiment is to show that the experimental cross section can be favorably compared with the theoretical expression in Eq. (2).

POST-EXPERIMENT EXERCISE The differential cross section in equation (2) tends toward infinity as the scattering angle, θ, approaches zero. Although this may initially seem counter-intuitive, there is a rational physical explanation for that asymptotic behavior. Explain the physical meaning of the infinite differential cross section at θ = 0°.

EXPERIMENT 15.1. Angular Dependence of the Rutherford Cross Section SAFETY ADVISORY: The 241Am radioactive source used in this experiment has no protective window over the source. Such a windowless configuration is necessary to emit alpha particles with their full energy. Do not touch the active surface of this radioisotope source. Follow all the safety precautions as outlined in “Safe Handling of Radioactive Sources” which can be found in the AN34 Library for Experiments in Nuclear Science on the ORTEC website. Procedure 1. Set up the electronics as shown in Fig 15.1 and the mechanical arrangement in Fig. 15.2 and Fig. 15.3. a. For the cable connections between the ULTRA™ charged-particle detector and the 142B preamplifier, see the description in the list of equipment. b. Turn off the power to the Bin and Power Supply. Verify that the 480 Pulser, 428 Bias Supply and the 575A amplifier are all adjacent to each other in the Bin. c. Check that the shaping time constant switches accessible through the side panel of the 575A are all set to 1.5 µs. d. Connect the power cable from the 142B Preamplifier to the Preamplifier Power connector on the rear panel of the 575A Amplifier. Set the input polarity switch on the 575A Amplifier to POSitive. e. Connect the E OUTPUT (Energy output) of the 142B Preamplifier to the INPUT of the 575A Amplifier using a C-24-12 RG-62A/U 93- Ω coaxial cable. f. On the 428 Bias Supply, set the POS/OFF/NEG switch to OFF, and turn both ten-turn dials completely counterclockwise (zero Volts). Connect the B OUTPUT of the 428 to the BIAS input of the 142B Preamplifier using the C-36-12 RG-59B/U 75-Ω cable with SHV plugs. g. On the 480 Pulser, set the OUTPUT polarity to NEGative. Select OFF with the ON/OFF switch. Set all ATTENUATOR toggle switches to their maximum value. Using a C-24-12 RG-62A/U 93-Ω coaxial cable, connect the ATTENuated OUTPUT of the 480 to the TEST input of the 142B Preamplifier. h. Connect the UNIpolar OUTput of the 575A Amplifier to the analog INPUT of the EASY-MCA using a C-24-4 RG- 62A/U 93-Ω coaxial cable. i. Turn on the Bin power. j. Turn on power to the computer supporting the EASY-MCA and activate the MAESTRO-32 MCA Emulator software.

101 3 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils

Fig. 15.3. An Example of a Rutherford Scattering Vacuum Chamber.*

Fig. 15.1. Electronics for the Rutherford Scattering Experiment.

*The Rutherford Scattering Chamber incorporates a mount for the 241Am source with source collimator, a centered holder for a thin gold foil, and a silicon, solid-state detector mounted on an arm suspended from the lid. The arm permits detector rotation through the desired scattering angles. The silicon detector mounted in a fixed position is optional, and can be used for coincidence experiments. The collimator aperture limits the solid angle for alpha-particle emission so that the surviving alpha particles (with the foil removed) fall within the sensitive area of the detector when it is located at a 0° scattering angle. The central foil holder permits rotation of the foil with respect to the incident direction of the alpha particles. The chamber requires a fore pump capable of achieving a vacuum below 200 µm. A vent/pump valve is recommended to make pumping and release of the vacuum convenient without turning off the fore pump. A baffle This is done in a vacuum of at least 200 µm. may be needed in the chamber to prevent rupturing the thin metal foils during rapid pump-down The detector should rotate from θ = 0° to θ = 90°. and venting.

Fig. 15.2. Experimental Arrangement for the Rutherford Scattering Experiment Using a Vacuum Scattering Chamber.

k. Via the Acquire menu and the ADC tab in the MAESTRO software that operates the EASY-MCA, select the Gate Off option. Choose the analog-to-digital conversion range to be either 2048 or 4096 channels for a 0 to +10 V input. Adjust the Upper Level discriminator to its maximum value and set the Lower Level discriminator as low as possible without causing excessive counting rate on the noise. Under the Preset tab, clear all data fields, and do the same for the MDA Preset option (if supported). Clearing those fields will default to manual control for starting and stopping spectrum acquisition. Familiarize yourself with the software controls for setting up, acquiring and erasing spectra. 2. To prepare for calibrating the system, remove the gold foil, and move the detector to θ = 0°. Pump a vacuum in the chamber to below 200 µm. On the 428 Bias Supply, turn the POS/OFF/NEG switch to POS and adjust the B dial to the voltage specified by the manufacturer of the ULTRA detector. Adjust the gain of the 575A Amplifier so that the peak from the 5.48-MeV alphas from 241Am accumulates in the top eighth of the analyzer. NOTE: All further measurements involving detection of the alpha particles requires a vacuum better than 200 µm. To protect the detector, the bias voltage should always be OFF when venting an opening the chamber. 3. Disconnect the 575A UNIpolar OUTput from the EASY-MCA, and connect it instead to the 1-MΩ input of the oscilloscope. Set the horizontal scale of the oscilloscope to 1 ms/cm and the vertical scale to 100 mV/cm. With a small, flat-blade screwdriver, adjust the PZ ADJ on the 575A Amplifier to make the pulses on the UNIpolar OUTput return to baseline as quickly as possible without undershooting the baseline between pulses. For further guidance on the Pole-Zero Cancellation adjustment, consult Experiment 3, the instruction manual for the amplifier, or the introduction to the amplifier product family on the ORTEC website at www.ortec-online.com. 4. Reconnect the 575A UNIpolar OUTput to the EASY-MCA INPUT. 5. Calibrate the MCA with the 241Am alphas and the 480 Pulser in the same manner as outlined for Experiments 4 and 5. Plot the analyzer calibration on linear graph paper, or construct a spreadsheet graph on your own PC. Alternatively, the energy calibration feature of MAESTRO-32 can be employed to make the cursor read directly in MeV. Make sure you turn the pulser off after this calibration.

102 4 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils

6. Turn off the detector bias and vent the vacuum. Insert the gold foil in the target position at an angle of 45° to the collimated alpha beam. Keep the detector at θ = 0°. Restore the vacuum and the detector bias voltage. Acquire the spectrum on the EASY-MCA. From the peak position, measure the energy, Ef, for the alpha particles that pass through the gold foil. Calculate the energy loss, ΔE, of the alphas in going through the foil (see Experiment 5); i.e.,

ΔE = E0 – Ef (3)

where E0 is 5.48 MeV for the source and Ef is the measured energy after the alphas pass through the gold foil.

EXERCISES a. From ΔE and the dE/dx information in reference 10, calculate the effective thickness of the gold foil in mg/cm2. Keep in mind that the orientation of the foil at 45° increases the alpha-particle path length through the foil by a factor of csc (45°). Consequently, the measured thickness will be larger than the specified mechanical thickness of the foil. Additional data on measured range-energy relationships for alpha particles in various materials is available on the Internet at the link in reference 10. b. The average energy of alphas experiencing scattering in the foil is

E0 + Ef (4) Eav = 2

Determine Eav for the measurements made in step 6. Use Eav Table 15.1. Comparison of the Theoretical and the Measured as the value for E in Eq (2), and calculate the values of dσ/dΩ Differential Cross Sections. (Theory) from Eq (2) for the angles specified in Table 15.1. Fill θ (degrees) dσ/dΩ (Theory) dσ/dΩ (Experimental) in the entire “Theory” column in the table. 10 15 c. Plot dσ/dΩ (Theory) versus θ on 5-cycle semilog paper, or create an equivalent spreadsheet graph on your PC. Because 20 of the extreme range of values, the differential cross section 25 requires a logarithmic scale, while the angle should be plotted 30 against a linear scale. 40 50 60 2 d. Calculate n0, the number of gold target nuclei per cm from the 70 following formula: 80 (g/cm2 of the foil) x 6.023 x 1023 90 n0 = –––––––––––––––––––––––––––– (5) A

The value of n0 will be used at a later time. e. Calculate ΔΩ from the formula ΔΩ = [the sensitive area of the detector in cm2] / R2 (6) Where R is the distance (in cm) from the detector to the gold foil. The sensitive area for the Model BU-021-450-100 ULTRA™ detector is nominally 450 mm2. WARNING: When measuring the diameter of the detector, do not touch the sensitive surface of the detector. Fingers often carry small dirt particles that can easily scratch the thin, ion-implanted window, rendering the detector permanently inoperative. Leaving a finger print on the window can increase the window thickness, thus increasing the energy loss experienced by the impinging alpha particles.

7. Remove the gold foil and check the alignment of the apparatus by measuring the counting rates for the values in Table 15.2. The counting rate is measured as follows:

103 5 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils

a. Acquire a spectrum on the EASY-MCA and set a Region of Table 15.2. Testing Apparatus Alignment at θ = 0° Interest (ROI) across the entire alpha-particle peak at 5.48 Angle θ (degrees) Counts/s Angle θ (degrees) Counts/s MeV. Read the integrated counts, N, for the peak from the 0 0 ROI. Read the elapsed Live Time, TL, displayed by the 1 –1 EASY-MCA software. The counting rate is calculated as 2 –2 N (7) I= 3 –3 T L 4 –4 b. The alpha particles emitted by the radioactive source are 5 –5 distributed randomly in time. Consequently, the number of 6 –6 alpha particles, N, counted in the live time, TL, is subject to 7 –7 a statistical variation. The expected uncertainty in N is characterized by the standard deviation

σN = √N (8) Consequently the expected standard deviation in the counting rate, I, is

σN √N σI = = (9) TL TL

The expected percent standard deviation in N is equal to the expected percent standard deviation in I, viz., σ σ N x 100% 100% I x 100% σ% σ%N = = = = I (10) N √N I

c. For the data in Table 15.2, count for a long enough time to achieve a 1% standard deviation in I. This will require at least 10,000 counts in the ROI set across the peak. 8. Plot the data in Table 15.2. If the instrument is properly aligned, the data in the table should be symmetric about 0°, with the maximum counts at 0°. If not centered on 0°, a correction for the offset should be applied to the angles in the remainder of the experiment.

9. The number of alphas per unit time, I0, that impinge on the foil can be calculated by one of the three following methods, depending on the specific geometry of the apparatus. a. If the source collimator limits the divergence of the alpha particles so that no particles are missing the sensitive area of the detector when the foil is absent, and no particles are missing the foil when it is present, then I0 is the counting rate measured when the detector is centered on the calibrated 0°, as determined from Table 15.2. b. If the source collimator does not satisfy both conditions in method “a”, but the exposed area of the foil defines the alpha particles that are able to scatter from the foil, then I0 can be calculated from

2 I0 = (activity of the source) (area of the foil) / 4π(R1) (11) Where the area of the foil is the area projected perpendicular to the beam of alpha particles from the source (see Fig. 15.2). The activity of the source can be determined by the methods outlined in Experiment 4. c. If none of the conditions in “a” and “b” are met, but the source collimator restricts the alpha particles to illuminating only a portion of the foil area, I0 must be calculated from equation (11) by replacing the area of the foil with the area of the restricting aperture of the collimator, while substituting the distance from the source to the restricting aperture for R1. Check with your laboratory manager to determine which of methods a, b, or c is appropriate for your scattering chamber. 10. You are now ready to measure the cross section. Replace the gold foil (at the angle φ = 45°). Set the detector at 10°, and count for a period of time long enough to get good statistics in the peak. Calculate the counting rate, I, at 10°. Repeat for all of the angles listed in Table 15.1. Keep a record of both the number of counts in the peak, N, and the

104 6 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils

related live time TL, in case you need to calculate the statistical uncertainty. It should be obvious from the theoretical cross section that the counting time will have to be increased as θ increases. For the smaller angles, you may be able to get close to a 1% standard deviation. For the larger angles, it will difficult to find enough time to achieve better than a 15% standard deviation. Plan your counting strategy to match the available time.

EXERCISE f. Calculate the experimental cross section for each of the points in step 10 by using the following formula:

dσ I [ ] 2 dΩ = I0 n0 ΔΩ cm /sr (12)

Since 1 barn = 10–24 cm2, the values calculated from Eq. (12) can be converted to millibarns per steradian and entered as dσ/dΩ (experimental) in Table 15.1. g. Plot the experimental cross section on the same graph as the theoretical cross section. h. Discuss the degree of agreement/disagreement between the theoretical and experimental cross sections in light of the uncertainties from counting statistics (equations 8, 9 and 10) and any other potential sources of measurement error. i. What quantitative effect does the angle, φ = 45°, have on the results?

2 EXPERIMENT 15.2. The Z2 Dependence of the Rutherford Cross Section 2 In this experiment alpha particles will be scattered from different foils to show the Z2 dependence in Eq. (2). The foils used are aluminum (Z2 = 13), nickel (Z2 = 28), copper (Z2 = 29), silver (Z2 = 47), and gold (Z2 = 79). The student will then plot this Z2 dependence and show that it does agree with the theory. Procedure 1. Ensure that steps 1 through 5 in Experiment 15.1 have been completed. Repeat step 6 of Experiment 15.1 for each of the foils. For each foil calculate n0 as in Experiment 15.1, Exercise d, Eq. (5). 2. For each foil set θ = 45°, and accumulate a pulse height spectrum for a period of time long enough to get at least 1000 counts in the scattered peak. Determine I (the number of scattered alphas per second) for each sample, per the methods used in Experiment 15.1.

EXERCISE 2 Plot I as a function of n0Z2 for each sample. The curve should be a straight line. The slope of the line can be determined 2 by equating the two expressions for cross section Eqs. (2) and (12) and solving for the product n0Z2. Therefore 2 θ Z1 Z2 I 1.296 ( ) csc4()x 10–27 = cm2/sr (13) E 2 I0 n0 ΔΩ

And hence θ –27 2 4 1.296 x 10 I0 ΔΩ Z1 csc () []2 2 I = E2 n0 Z2 (14)

105 7 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils

Since every term inside the [ ] brackets is a constant for all foils†: 2 I = K n0 Z2 (15) Where the constant, K, is equal to the contents of the [ ] bracket in Eq. (14). Therefore, the experimental intensity should plot as a straight line in this exercise. The slope of the curve is K. Explain any deviations of the data from a straight line by reference to the uncertainty from counting statistics or any other sources of error in the experiment.

References 1. A.C. Melissinos, Experiments in Modern Physics, Academic Press, New York (1966). 2. H. A. Enge, Introduction to Nuclear Physics, Addison-Wesley Publishing Co., Massachusetts (1966). 3. R. D. Evans, The Atomic Nucleus, McGraw-Hill, New York (1955). 4. B. L. Cohen, Concepts of Nuclear Physics, McGraw-Hill, New York (1971). 5. J. L. Duggan and J. F. Yegge, “A Rutherford Scattering Experiment”, J. Chem. Ed., 45 85 (1968). 6. J. L. Duggan, et.al., Am. J. Phys. 35, 631 (1967). 7. C. M. Lederer and V. S. Shirley, Eds., Table of Isotopes, 7th Edition, John Wiley and Sons, Inc., New York (1978). 8. G. Marion and P. C. Young, Tables of Nuclear Reaction Graphs, John Wiley and Sons, New York (1968). 9. Application notes, technical papers, and introductions to each product family at www.ortec-online.com. 10. Measured data on dE/dx and range-energy relationships for alpha particles in various materials is available from the US National Institute of Standards and Technology(NIST) Physics Laboratory at: http://physics.nist.gov/PhysRefData/Star/Text/ASTAR.html

Appendix 15-A. Finding/Developing the Missing Equipment The primary reason full equipment support for this experiment has been suspended is the difficulty in obtaining the 0.5 to 1 mCi 241Am source. For the laboratory that is able to find the appropriate alpha-particle source, this appendix provides some guidelines on procuring or developing the missing equipment.

Requirements for the 241Am Source Ideally, the source should have a construction similar to the Eckert and Ziegler Model AF 241 A2, but with 0.5 to 1 mCi activity. To allow effective collimation of the alpha particles, the diameter of the active area of the source must be restricted to a diameter ≤5 mm. The active material of the source must be thin enough to cause an energy loss that is negligible compared to 500 keV. Hence there must be no significant window thickness over the active surface of the source. These requirements are formidable to meet, and that is why it is difficult to procure an appropriate source. Lower activities are available in the desired configuration. But, the orders of magnitude lower activities require counting times that are orders of magnitude longer than the typical student lab period.

† This presumes that all the foils have been chosen with appropriate thickness so that E = Eav from Eq. (4) is approximately the same for all foils.

106 8 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils

Critical Collimator Dimensions Using the dimensions specified in Figure 15.2 and pictured in Figure 15.3, the distance from the active surface of the source to the center of the scattering foil should be 2 inches (50.8 mm). The distance from the center of the scattering foil to the front of the sensitive area of the detector should be 2.5 inches (63.5 mm). The collimator is designed to fulfill the condition in step 9a of Experiment 15.1. If the limiting aperture of the collimator has a 3.65 mm diameter and is located as close as possible to the scattering foil, at a distance of 43.0 mm from the active surface of the 241Am source, the collimated beam of alpha particles will illuminate an 18 mm diameter disk on the surface of the charged-particle detector. This leaves a 3 mm margin for misalignment on the 12 mm radius of the sensitive area of the detector. Thus, any misalignment of the collimator and detector must be controlled to less than 1 mm. Any metal that is convenient to machine is acceptable for fabricating the collimator. With the above design parameters, the angle of divergence of the collimated beam from any point on the active source area will be approximately 5.8°. The detector sensitive area defines a divergence angle of 21° at the center of the scattering foil. Because the two divergences combine as the square root of the sum of the squares, the effective angular resolution will be approximately ±11°.

Scattering Foils It is almost impossible to find Nickel foils in the desired thickness. However, adequate supplies of Aluminum, Copper, Silver and Gold foils can be procured from gilding supply retailers. One can search the Internet for convenient sources of supply using the “gilding” search term. Gilding materials suppliers typically specify the area (length and width) of each metal leaf and the weight per thousand leaves. From those numbers, one can calculate the thickness in mg/cm2 to compare with the desired thickness. Typically the thickness of the available leaves is below the target value for Experiment 15, and multiple leaves will have to be stacked to reach the desired thickness. Note that the individual leaf thicknesses typically vary by a factor of 1.6 above and below the average thickness for 1000 leaves. Consequently individual leaves will have to be evaluated for thickness, and combined accordingly to achieve the desired thickness. The desired thickness for each foil is the thickness that causes an energy loss of 354 keV when the 5.48 MeV alpha particle travels through the foil at 90° to the surface of the foil. This energy loss and foil thickness can be checked by inserting the foil in the scattering chamber and measuring the energy loss with the foil set at 90° to the collimated beam, and with the detector located at the 0° position. An energy loss of 354 keV at normal incidence leads to a 500 keV energy loss when the foil is at the 45° angle in the scattering experiment. That offers a reasonable compromise between maximizing the counting rate and minimizing the energy loss. Plastic frames for mounting 35 mm slides offer a convenient means for supporting and positioning the foils. To secure the ultra-thin foils, it may be necessary to use a piece of conventional 35 mm photographic film (0.14 mm thickness) as a gasket in the frame. To create the gasket, the 35 mm film is mounted in the frame, and the 34 mm x 23 mm window is cut out with a knife. Then, the slide holder is re-assembled with the thin metal foil held in place by the film gasket. The outside dimensions of the slide frame are 50 mm x 50 mm with a thickness in the range of 1 to 3.2 mm. The thin foils are quite fragile. Therefore, it is advisable to build a considerable stock of replacement foils already mounted in the slide frames and certified regarding thickness. It is also important to limit the vacuum pumping speed on the scattering chamber to avoid rupturing the foil with a sudden rush of air. An orifice in the pumping/venting port is an effective way to provide such protection.

107 9 ORTEC ® AN34 Experiment 15 Rutherford Scattering of Alphas from Thin Gold Foil and Other Optional Metal Foils

Forecasted Counting Rates The theoretical equations in Experiment 15 can be used to predict the measured counting rates. With the parameters outlined above, and a 0.50 mCi 241Am source activity, the counting rate with the detector at 0° with no foil present should be circa 7,100 counts/second. The total counting time for Experiment 15.1 is forecasted to be 3.1 hours to achieve 10% counting statistics. Most of this time is consumed at angles between 45° and 90°. The counting times for smaller angles is insignificant. For Experiment 15.2, the total counting time for 5% counting statistics is expected to be 1.8 hours, if a 30° angle is used. Most of this counting time is consumed on the lighter element foils (Cu and Al). The counting time is much longer, if the 45° angle recommended in Experiment 15.2 is employed. The 30° angle is a more productive option. Considering the counting times, two 4 hour lab sessions will be required for the two segments of Experiment 15.

Specifications subject to change 062911

® www.ortec-online.com ORTEC Tel. (865) 482-4411 • Fax (865) 483-0396 • [email protected] 801 South Illinois Ave., Oak Ridge, TN 37831-0895 U.S.A. For International Office Locations,108 Visit Our Website 14 Cosmic Ray Muon Lifetime

In this lab you will measure the rate, and lifetime of cosmic ray muons using a plastic scintillator viewed by two photomultiplier tubes.

14.1 Cosmic Ray Muons Cosmic ray muons were discovered by Carl D. Anderson and Seth Neddermeyer in 1936. They were originally thought to be the Yukawa particle, which was theorized to be the force carrier in the strong interactions. It was later determined that the muon was in fact another lepton, like a heavy electron. At the time this was a very puzzling particle, and a famous theorist I.I. Rabi famously questioned, “Who ordered that?” The cosmic ray muons that we will be observing come from muons produced in the upper atmosphere by cosmic ray protons striking the upper atmosphere at high energies. The protons are stopped in the upper atmosphere, and in their interactions produce pions π±, which are quark-antiquark pairs, and are the closest particle to being the Yukawa particles. Pions only have a mean lifetime of 29 ns, and decay to muons and muon-neutrinos. The muons produced from the decay of these pions, are the ones we can observe in this experiment. If the muons produced int the upper atmosphere were not moving at relativistic speeds, their 2.1969811(22) × 106 µs lifetime would have been to short for them to reach Earth[53]. In the Standard Model (SM) of particle physics, positive muons decay via the weak (V −A) interaction into positrons plus neutrinos: µ → e ν ν through a virtual state involving W vector bosons. Recall that the decay rate, R(t), can be expressed as an exponential scaling of a rate at time zero R(0), of the form: R(t) = R(0)e−t/τ = R(0)e−λt. (114) The decay rate λ is related to the particle lifetime by λ = 1/τ. In this experiment we will be measuring the rate of muon decays, by looking at the time diﬀerence between when a muon stops in our scintillator, and when an electron appears.

14.2 Detector and Electronics Readout This section describes the detector that will be used, and how it is connected to make measurements of muons, and their associated decay electrons. The method of calibrating the time scale will also be described.

14.2.1 Scintillator and Photomultiplier Tube The detector used for this lab is a large slab of plastic scintillator. The scintillator used is actually four layers of 20 × 2 × 100 cm plastic scintillator. Each of the four layers is wrapped in aluminized Mylar to help reflect the scintillation light to either end of the scintillator. The scintillator light is converted to an electrical signal using a photomultiplier tube on either end of the scintillator. The PMTs used are large area PMTs (five inch diameter), to allow collecting light from the large area of scintillator. One end of the wrapped scintillator and PMT are shown in FIG. 31. The red cable is the high voltage cable for the PMT, and the black cable is the signal from the PMT. Note that the PMT has a mu-metal shield around it to shield it from Earth’s magnetic field.

109 Figure 31: This picture shows one end of the wrapped scintillator and the PMT coupled to it.

14.2.2 PMT High Voltage The PMTs need to be supplied a negative high voltage of −1000 V. The HV supply is a CAEN based high voltage mainframe shown in FIG. 32. In order to turn on the high voltage, first turn the key on the bottom left to local. Let the mainframe boot up, and then press any key to get the login menu. The login and password are both admin. To bring up the high voltage settings use the keyboard to select Channels from the menu. We are using channels one and two, which are already set to ramp up to 1000 V when turned on. To turn on the high voltage for a channel, use the arrow keys to move over to the channel and column that says Pw, then hit the space bar to toggle the power on or off. Remember when leaving for the day to power off the high voltages, and turn off the HV mainframe.

14.3 Readout Electronics A V1720 digitizer will be used to save waveforms from the signals from the PMTs. A digitizer saves samples of voltage versus time for some number of time bins around a trigger signal. The digitizer can also self trigger, but to make the trigger more explicit, we will generate a trigger that is the coincidence of the two PMT signals. That is, we will only trigger if we get a signal above a certain size in both PMTs at the same time. The digitizer is in a Versa-Module Europa (VME) crate, that is used to power the board. To collect data from the digitizer on a computer, an optical link from the digitizer to a PCIE card on the computer is connected. A photograph of the electronics is shown in FIG. 33. A circuit diagram for this electronics is shown in the procedure section. For the ﬁrst step in our electronics, we will make a copy of each of the signals from the PMTs, so that we can send one copy to be used in the coincidence, and another that we send directly to the digitizer. This is done using a model 740 fan-in fan-out unit. To make a coincidence, we ﬁrst turn our analog pulse into a digital pulse with a known width for pulses that go below an adjustable threshold level. The device used to turn the analog pulse into a

110 Figure 32: The high voltage supply is shown in this figure. We will only be using channels one and two. digital one is called a discriminator (which is a form of the SCA we used previously). The discriminator we are using is a model 705 discriminator, whose threshold can be adjusted using a screw driver to access a potentiometer through a hole labelled threshold. The setting of the threshold can be checked using a multimeter to measure the voltage on a pin near the threshold setting screw. The multimeter reading will be ten times the threshold setting. You should check the threshold, the input and the output of the discriminator to make sure you are only discriminating on real pulses, and not noise. The discriminator output is then used to make the coincidence, but is also sent to a visual scaler. The visual scaler just counts the number of times you get a logic pulse. The N1145 scaler being used has been set up with a timer, so that it is cleared every second. In this way you can easily look to see what rate in Hz that you are getting pulses on the PMTs. Once the PMT signals have been turned into digital pulses, a logic unit can be used to only forward a pulse if both PMTs have a pulse. The electronics unit used for this is a N405 logic unit, which is set to AND mode. This logic unit has multiple outputs, one of which is sent to a third channel of the scaler, and another is used as the trigger signal for the digitizer. Another component of the electronics setup is a tail pulse generator, which will be used to check the time scale of your readings. The tail pulse generator generates pulses with a tunable separation in time, and repeats that pair of pulses at a known rate. The unit used is a BNC tail pulse generator, and it has been connected to a green light emitting diode, which has been placed at one end of the scintillator near PMT A. In this way we can send light flashes to the PMTs with known time separations, and use that to check our time calibration. To turn on the tail pulse generator there is a toggle switch to change from REF (which means it would only fire on some reference input trigger), to INT (which means to send a pair of pulses at a rate of the current frequency setting). Note that diodes have a polarity, so the polarity switch has to be set to positive to have the LED light up. Also we want fairly narrow pulses so that we can do time calibration, so the rise time and fall time should be set to their minimum values. The amplitude of the pulses can also be varied. Make sure that the amplitude is large enough

111 Figure 33: The NIM and VME electronics used for this lab are shown in this picture.

112 to light the LEDs, but not so high that it produces too much light for the PMT. This can be adjusted until the pulses from the PMT are a few 100 mV.

14.4 Procedure First check the electronics is connected as shown in FIG. 34. A description of each of the elements of the electronics is described in the previous sections.

1. Following the instructions in section 14.2.2, turn on the high voltage.

2. Power on the NIM crate (the top crate in FIG. 33. Power on the VME crate (the bottom crate).

3. In order to check the pulses from the PMT you can take another copy of the signal from the fan-in fan-out unit to the oscilloscope. The signal needs to be terminated with a 50 Ω resistor, as shown in FIG. 35.

4. To check the time scale on our digitizer connect the tail pulse generator to the LED, and through the oscilloscope (also shown in FIG. 35. The tail pulse generator should be set on frequency range 100 Hz, and the delay should be set on the range 10 µs. To turn on the tail pulse generator change the toggle switch from REF to INT, and you should get traces similar to those in FIG. 35. Trigger on the tail pulse signal (which you send to channel 3) on the oscilloscope.

5. In order to collect the waveforms with the digitizer, log into the computer next to the setup. The username is midas, and the password is pipe daq. If the computer is already on and logged in then you are ready to take data. If the computer has to be rebooted, then a kernel module needs to be loaded. To load the kernel module open a terminal (the black box icon), and type the following commands:

> cd CAENdirver/A2818Drv > sudo ./a2818_load pipe_daq

6. To start the waveform collection software open a terminal, and change to a working directory, and start the program by typing:

> cd wavedump-3.5.3/src > ./wavedump WaveDumpConfig.txt

The program has various commands, and the capitalization of the commands matters. To start the digitizers from the program window, type S. To show waveforms being collected from the digitizers type P. To write the waveforms being collected to the ﬁles wave0.dat and wave1.dat, type W.

7. Collect a few minutes of time calibration data with pulse separation times of 0.5 µs, 1 µs, 2 µs, 3 µs, 4 µs, 6 µs, 8 µs, and 10 µs. Measure with the oscilloscope, as best as you can, the actual time separation between these pulses. Between each run, you will have to copy wave0.dat and wave1.dat to a directory under your name so that they are not overwritten when you start a new run with new settings.

113 Figure 34: Electronics diagram showing how the PMT signals are sent to the digitizer, and processed to make a trigger based on a coincidence between the two PMTs.

114 Figure 35: To check the signals from our LED pulser, and the signals from our PMTs, we will use three channels of an oscilloscope, as shown in this picture.

8. Once you have your time calibration data, turn of the tail pulser by changing its toggle switch from INT back to REF. Change the oscilloscope trigger, so that you are triggering on one of the PMT signals from channel 1 or 2.

9. Now you will start a long run (overnight), to collect cosmic ray muon data. While the data is being collected, set the oscilloscope scale to 1 µs per division, and move the trigger time to the left of the display (around 1 µs). Now set the oscilloscope display persistence to inﬁnite, and any pulses after the trigger are either random coincidences, or electrons from the cosmic ray muon decays.

10. Also while collecting the long overnight run, you can begin the analysis of your calibration data.

14.5 Data Analysis A root script to convert the binary waveform data into a root TTree can be copied from http:// t2kwinnipeg.uwinnipeg.ca/~jamieson/courses/intermediate_lab/cosmu_macros/plotwave.C.A TTree is like a table of numbers, that we will use to save for each waveform the time, amplitude, rise time, fall time, and fit quality information. The script reads the binary data, and saves the waveforms into root histograms (TH1D). Each 100 waveforms a new directory of waveforms is created. The script then fits the waveform to a function that is like a capacitor charging, then discharging. The waveform fit information is then stored into the TTree, along with a difference in time between the waveform being fitted, and the previous waveform. For reference the function that the waveform is fitted to has the form:  0 : t < t0  −(t−t )/t V (t) = A(1 − e 0 rise ) + V0 : t0 < t < t0 + 5trise) (115)  −(t−t −5t )/t A(e 0 rise fall + V0 : t > t0 + 5trise

115 To run this script, copy it to a directory along with your waveform data files (for example if your data files are wave0.dat and wave1.dat. Then from inside root you would run: root> .x plotwave.C( "wave0.dat", "wave1.dat", "outputname.root", 0, 100000) where the first 0 is for the number of waveforms to skip, and the 100000 is the number of waveforms from the binary file to process. You can change the last number in the argument list to −1 to process all of the events in the binary file. Note that if you have a larger file to process, there is also a compiled version of the code, that does not save each waveform to a histogram. Instead it just saves the TTree. To run the compiled version, copy the file plotwaveall.C and Makefile from the website. To build the executable, from a terminal, in the directory you have saved the files above: cd my/directory/name make Then to run this version of the executable on your data files are wave0.dat and wave1.dat: cd my/directory/name plotwaveall wave0.dat wave1.dat outputname.root 0 -1 Once you have your output file outputname.root you can open it in root, and start a TTree viewer to make histograms of the quantities saved in the TTree. The TTree in the file is called Twf, and the command to start the viewer are: root> tf = new TFile("outputname.root","read"); root> Twf->StartViewer() This will bring up a window that looks like FIG. 36.

Figure 36: Tree viewer window showing the variables in the table as leaves.

Each of the green leaves in the window that appears is one variable in the TTree. To make a histogram of the values in that variable, you can double click on the leaf icon. The set of buttons of interest on the TTreeViewer canvas allow you to perform the following operations:

116 Button Function Command User commands executed via interpreter Option Histogram graphics options Histogram Name of the histogram created by command Hist Redeﬁnes the default histogram (default = htemp) Draw Draw the current selection Stop Abort current operation Update Update the tree viewer X To select the X variable Y To select the Y variable Z To select the Z variable

As an example of how to use the viewer, lets make a two dimensional lego view of waveform zero’s Amplitude, Amplitude0, versus peak time, T00, with a selection on successful fits Status0 >= 0. To do this, in the Option field, type in: LEGO2. Left mouse click on Amplitude0 and drag and drop it to the X button. Left mouse click on T00 and drag and drop it to the Y button. Left mouse click on Status0 and drag and drop it to the scissors button. Right mouse click on the menu’s Edit and select Cut. An option box appears on your screen. In the Selection field, type in: Status0>=0, and then right mouse click on ”DONE”. Click on the Draw button, and voila. The variables in the tree are stored once for each event, which is made up of a waveform on each of two channels. The variable names have the following meanings:

117 Type Variable name Meaning int Event event counter int Sec0 ch 0 time from daq cpu in seconds int NSec0 ch 0 time from daq cpu in nsec (to add to above in seconds) float deltaT0 ch 0 difference in time since last pulse (ns) float deltaT1 ch 1 difference in time since last pulse (ns) int Status0 ch 0 status of pulse fit to waveform (0 good, negative bad) int Time0 ch 0 trigger time (ns since last reset) float Chi20 ch 0 chi2 of pulse fit to waveform float Ndf0 ch 0 number of degrees of freedom in waveform fit float Prob0 ch 0 prob of chi2/ndof for waveform fit float A0 ch 0 amplitude of waveform fit (ADC counts) float T00 ch 0 time of pulse peak from fit (ns relative to trigger time) float TRise0 ch 0 rise time of pulse (ns) from peak fit float TFall0 ch 0 fall time of pulse (ns) from peak fit float Ped0 ch 0 pedestal (zero voltage) level (ADC counts) int Sec1 ch 1 time from daq cpu in seconds int NSec1 ch 1 time from daq cpu in nsec (to add to above in seconds) int Status1 ch 1 status of pulse fit to waveform (0 good, negative bad) int Time1 ch 1 trigger time (ns since last reset) float Chi21 ch 1 chi2 of pulse fit to waveform float Ndf1 ch 1 number of degrees of freedom in waveform fit float Prob1 ch 1 prob of chi2/ndof for waveform fit float A1 ch 1 amplitude of waveform fit (ADC counts) float T01 ch 1 time of pulse peak from fit (ns relative to trigger time) float TRise1 ch 1 rise time of pulse (ns) from peak fit float TFall1 ch 1 fall time of pulse (ns) from peak fit float Ped1 ch 1 pedestal (zero voltage) level (ADC counts)

Now that you have some brief introduction to the format of the data. The variables in the tree we are interested in plotting are the time diﬀerence variables deltaT0, and deltaT1. These are the time diﬀerences between successive waveforms on channel zero and channel one respectively. Fit histograms of these variables to an exponential decay with constant background to determine the muon lifetime.

118 15 Lock-in Ampliﬁer Measurements

In this lab you will measure the resistance of a conducting brass rod, and investigate the magnetic shielding of various conducting shells. In order to measure very small resistances, the voltage drop across the resistor is very small, and using ordinary techniques would be lost in the noise. By using a lock-in ampliﬁer to select the signal around a known frequency, the signal to noise can be greatly improved.

15.1 About Lock-In Ampliﬁers Figure 37 shows a block diagram of a lock-in ampliﬁer, described in this section.

Figure 37: Block diagram of a lock-in ampliﬁer.

Consider a sinusoidal input signal V (t) = Vs sin (ωt + φ), and a reference signal VR(t) = Vr sin Ωt. The product of these two gives beats at the sum and diﬀerence frequencies:

V (t)VR(t) = VsVr sin (ωt + φ) sin Ωt 1 (116) = 2 VsVrcos (ω − Ω)t + φ − cos (ω + Ω)t + φ

If this signal is passed through a low-pass filter, the output voltage, VPSD, has its AC signals are removed. In the case where the reference and input frequencies are the same, a DC voltage remains: 1 V = V V cos φ (117) PSD 2 s r Now consider the case where the input also includes noise. Only the amplitude of the signal with frequency very close to the lock-in frequency is passed by the low-pass filter. This means that any noise at frequencies other than the reference frequency is also filtered. Suppose we also multiply our input signal by the reference signal with an additional 90 degree phase 0 o shift, VR(t) = Vr sin (Ωt + 90 ). In this case the voltage at the output of the low-pass filter is: 1 V 0 = V V sin φ (118) PSD 2 s r With these two signals one can define:

VPSD ∼ X = Vs cos φ 0 (119) VPSD ∼ Y = Vs sin φ

119 √ 2 2 −1 with magnitude R = Vs = X + Y , and phase φ = tan (Y/X). The outputs X and Y represent the real and imaginary parts of the signal input to the lock-in amplifier. The output from the lock-in amplifier is displayed as an RMS voltage, and any angle in degrees. Review the document about Lock-in amplifiers from Sanford Research Systems for more details about how the Lock-in amplifier works[54].

15.2 Measuring a 1Ω Resistor In order to get familiar with the lock-in amplifier we will use the lock-in amplifier to measure the resistance of a ∼ 1Ω resistor. The circuit used for this measurement is shown in Fig. 38. In this circuit Ra ∼ 10 kΩ is already connected inside of the BNC cable, and Rb ∼ 1Ω is the resistor being measured. The voltage across the resistor is measured using input a of the lock-in amplifier. Refer to the SRS Users Manual for more information on the different controls on the lock-in amplifier[55].

Figure 38: Block diagram for the one ohm resistance measurement (left), and photograph of the physical setup (right).

Procedure

1. Connect the circuit as shown in Fig. 38.

2. Set the reference voltage to Vs = 1 Vrms, and frequency to 1 kHz.

3. Find an equation for Rb in terms of Vs, Vb, and Ra.

4. Adjust the lock-in setting to get a measurement of Vb, and determine the resistance Rb and its measurement uncertainty.

5. Change the input frequency to 20 kHz and repeat the measurement. Is there any diﬀerence in the result?

120 15.3 Resistance of a Brass Rod We will set up a simple circuit to measure the resistance of a brass rod using the lock-in amplifier. The circuit we will use is shown in Fig. 39, where Ra ∼ 10 kΩ is already connected inside of the BNC cable, and Rb is the resistance of the brass rod. For the voltage source, Vs, the reference signal for the lock-in amplifier will be used. The voltage measured, Vb, is to be measured across the brass rod using input A of the lock-in amplifier.

Figure 39: Block diagram for the brass rod resistance measurement (left), and photograph of the physical setup (right).

Procedure

1. Measure the dimensions of the brass rod, and using a guess of 6.6 × 10−6 Ω cm for the resistivity, what resistance, Rb, do you predict for the brass rod. 2. Connect the circuit as shown in Fig. 39.

3. What voltage do you expect to measure across the brass rod?

4. Set the reference voltage to 1 Vrms, and frequency to 1 kHz.

5. Adjust the lock-in setting to get a measurement of Va, and determine the resistance Ra and its measurement uncertainty. In this case the resistance being measured is very small, so you will need to set a fairly long integration time (narrow bandwidth).

6. Measure the resistance as a function of frequency (between 1 kHz and 50 kHz).

At DC the current in a conductor is distributed evenly over the cross-section of the wire. The AC current density, J, at a depth, d, in a conductor decreases exponentially from its value at the surface, Js, according to: d J = Jse δ . (120)

121 The skin depth of the conductor is δ, and can be approximated as:

r 2ρ δ = , (121) ωµrµ0 where ρ is the DC resistivity of the conductor, ω is the angular frequency, µr is the relative permittivity of the conductor, and µ0 is the permeability of free space.

15.4 Magnetic Shielding Measurements Electromagnetic radiation is shielded by conductors. In this part of the lab you will measure the magnetic shielding from three conductors: a copper pipe, an aluminium pipe, and a mu-metal pipe. The ratio of the magnetic ﬁeld inside of a conducting pipe, H~i, relative to the ﬁeld outside the conduction pipe, H~o can be determined by solving Maxwell’s equations. For a cylindrical geometry the solutions follow Bessel’s functions, which have fairly simple forms for higher frequencies. At higher frequencies the limiting form is given by:

Hi = ρeiφ Ho     q  −k (R −R )  ρ = 2 R2 e 0 2 1 R1 r 2 2  R1k0   1+R1k0+ 2  (122) φ = k (R − R ) + atan R1k0 0 2 1 2+R1k0

2π q ωσµ k0 = δ = 2

In these equations R1 is the inner diameter of the conducting pipe, and R2 is the outer diameter. The skin depth is δ, the conductivity of the metal is σ, the angular frequency is ω, and the permeability of the metal is µ. The limiting form is valid for k0(R2 − R1) < 1. Determine what frequency this corresponds to for each of the conductors, and see if this holds experimentally.

Procedure

1. Measure the resistance of the solenoid, and pick-up coil.

2. Connect the circuit according to Fig. 40.

3. Without the shielding pipe in the solenoid, measure the voltage across the pick-up coil as a function of frequency (100 Hz to 50 kHz). Note that the voltage is proportional to the magnetic ﬁeld inside the solenoid (this is the magnetic ﬁeld outside the pipe).

4. Repeat the measurements, at the same frequencies but with the pick-up coil inside of the Al pipe, which in turn is inside the solenoid. Note that this voltage is proportional to the magnetic ﬁeld inside the conducting pipe.

5. Repeat the last step for the copper pipe.

6. Repeat the last step for the mu-metal pipe.

122 Figure 40: Block diagram for the shielding measurements (top), conceptual diagram of connections for shielding measurements (left), and photograph of the physical setup (right).

123 Plot the ratio of the magnetic ﬁelds (magnitude, and phase) as a function of frequency for each of the pipes. Try to determine the skin depth of each of the materials. You may need to look up values for the conductivity of the materials, and the permeability of the materials. Which of the shields was the most eﬀective?

124 16 High Resolution Gamma Ray Spectroscopy

In the high resolution gamma ray spectroscpy lab you will use a High Purity Germanium Detector (HPGe) to investigate the gamma ray spectra of several samples.

16.1 High purity germanium detector principles HPGe detectors have resolutions about 20-30 times better than the NaI detectors used in term one. This high resolutions makes them better suited to determining the radioisotopes present in various samples. For more details on HPGe detectors refer to notes in the lab from a McMaster medical physics course here[56].

16.2 Experimental Method The HPGe detector is cryogenically cooled, and will have been cooled down by the lab instructor before you arrive. The electronics and detector readout will already be connected. The detector signal and high voltage is connected through a pre-amp to a 2700 V power supply and a spectroscopy amplifier. The output of the spectroscopy amplifier goes to an 8196 channel multi-channel analyzer (MCA). The MCA is connected to a USB port on a linux based laptop for readout. The software for collecting the MCA spectra is on a laptop in the lab. The instructor will show you how to collect a spectrum into a text file which can be processed in root. Draw a diagram showing the connections from detector to readout. Before taking any spectra you will need to ramp up the high voltage. First turn on the NIM bin containing the power supply and spectroscopy amplifier. Then turn the power supply to positive voltage, press the reset button, and ramp the voltage to 2700 V. Red bars on the power supply should light up as you ramp the voltag up. If the voltage is not ramping it may indicate a problem with the detector cooling, as there is an interlock signal to avoid damaging the detector if it is not sufficiently cold.

16.3 Energy Calibration Use the sources in Table 18 (mostly the same table as from the Gamma ray spectroscopy experiment) to do an energy calibration. No further details of how to do this are provided, as this should be fairly routine for you after having done the Gamma ray spectroscopy lab and the beta spectroscopy lab. For Eu, as an exercise, you will have to look up the gamma ray energies. We can obtain a high energy gamma ray from an AmBe neutron source. The Am source gives oﬀ α particles which, when the bombard Be cause it to undergo the reaction:

9Be + α → 12C + n. The 12C is produced in an excited state that decays by giving oﬀ a gamma ray of 7.656 MeV.

16.4 Samples under study Several commercially available products, such as kitty litter, ceramic tiles, old camera lenses, and fertil- izer have some naturally ocurring radioactive material that can be identified. Depending on availability test samples of some of these materials to try to identify any radioactive isotopes. We also have samples of various Uranium mine tailings that can be studied to identify the isotopes present in these samples. You could also bring in a piece of your furnace filter (in a plastic bag) to see if any isotopes can be identified from dust and gases in the basement.

125 Table 18: Properties of several common gamma-ray sources. Isotope half-life Gamma ray energies (keV) 133Ba 10.7 years 81.0, 276.3, 303.7, 355.9, 383.7 109Cd 453 days 88.0 57Co 270 days 122.1, 136.4 60Co 5.27 years 1173.2, 1332.5 137Cs 30.1 years 32, 661.6 54Mn 312 days 834.8 22Na 2.6 years 511, 1274.5 65Zn 244 days 1115.6 12C fast decay 7656.0 152Eu 13.5 Years Multiple energies

17 Magnetic Forces, Torque and Precession

In this lab you will do several mini-labs to investigate and measure the magnetic dipole moment of a ball containing a bar magnet using four different techniques using both a constant and in a gradient magnetic fields. The magnetic field for these experiments is produced by a set of coils N = 195, in a Helmoltz coil configuration. A Helmoltz coil configuration has coils separated by the same distance as their radius, and one can show that the magnetic field near the mid-way point between the coils is:

3 4 2 µ NI B = 0 . (123) 5 R You may find that the set of coils that we are using are separated by a distance, L, which is different than the radius of the coils. In this case we can approximate the magnetic field along the axis of the coils using the superposition of the field from each individual set of coils. The result is: " # µ R2NI 1 1 B = 0 ± . (124) 2 (z + L/2)2 + R2)3/2 (z − L/2)2 + R2)3/2

The + sign is for when the currents in the two coils has the same sense, and the − sign is for when the currents in the two coils have opposite sense. Notice that for the case where the currents are in the same sense, at the origin this expression simplifies to: µ R2NI B = 0 . (125) ((L/2)2 + R2)3/2 A sample ROOT macro to plot this magnetic field as a function of the coordinate along the axis of the coils, z, can be found on the course website at http://t2kwinnipeg.uwinnipeg.ca/~jamieson/ courses/intermediate_lab/magnetictorque/plotfield.C. This macro also plots the gradient of the magnetic field for the case where the current in the coils have opposite sense.

17.1 Magnetic moment of a magnetic dipole Do experiment one from the attached Teach Spin experiments. You will make a plot of the radial distance ~r of a known mass that applies a torque to a ball, as a function of an applied ﬁeld, B = |B~ |

126 that exactly balances the mass. Measure for at least seven diﬀerent radii, and determine the magnetic moment of the ball. Remember to evaluate the statistical uncertainty, and consider any possible sources of systematic uncertainty in this measurement.

17.2 Harmonic oscillation of a dipole about an applied field Do experiment two from the attached Teach Spin experiments. Again you will determine the magnetic moment of the ball. In this case you will do so by measuring the period of oscillation of a magnetic moment (ball) in an external magnetic field. Measure the period for at least seven different applied magnetic field strengths, and make a plot to determine the magnetic moment of the ball (with an estimate of the measurement uncertainty). Compare your result with your previous measurement from experiment one.

17.3 Precession of a rotating magnetic dipole about an applied field Do experiment three from the attached Teach Spin experiments. In this experiment you will measure the precession frequency of the magnetic dipole with a spin axis that is not aligned with in an external magnetic field. Measure the precession frequency for at least seven different applied magnetic fields. Once again from the measurements you will determine the magnetic moment of the ball (with uncertainty), which you should compare with the results from experiments one and two. You will need to calculate the angular momentum L = Iω, from your previous determination of the ball’s moment of inertia, and rotation frequency, ω (measured by strobe light).

17.4 Force on a magnetic dipole in a gradient magnetic ﬁeld Do experiment four from the attached Teach Spin experiments. In this experiment you will show that there is only a force on a magnetic dipole, when there is a gradient magnetic ﬁeld. Yet again you will determine the magnetic moment of the ball with systematic uncertainty. Compare your result with experiments one trough three.

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