Department of Physics, Middlebury College | 1992-93 |

Modern Physics Laboratory |

The speed of light in vacuum is now a defined fundamental constant, with the value c = 299,792,458 m/s in the SI unit system.

^{1}At the undergraduate level it is possible to determine the speed of visible light to four or five significant figures with simple apparatus. Measurement of the speed of higher energy electromagnetic quanta is an entirely different matter. Although we have theoretical justification for believing that gamma rays should have exactly the same propagation speed in vacuum as visible light, precise experimental measurements of the speed of gamma rays are inherently more difficult to perform. The main reason for this difficulty is the short wavelength (10^{-10}- 10^{-13}m) of gamma radiation and our inability to produce gamma ray interference phenomena that can be compared directly to optical realizations of the meter.In this experiment you will determine the speed of gamma rays to an accuracy of about 10% using the simple time-of-flight apparatus shown in Fig. 1.

^{2}The apparatus should be largely familiar to you from your work on the angular correlation of pair annihilation gamma rays. Briefly, two NaI(T) detectors lie diametrically opposite a^{22}Na radioactive source that provides time-correlated gamma ray pairs from positron-electron annihilation events. One detector remains a fixed distance from the^{22}Na radioactive source; the other detector is free to move away from the radioactive source along a straight track, as shown in Fig. 1.Gamma-gamma coincidences from pair annihilation events are detected electronically with the coincidence detection system shown in Fig. 3 of the pair annihilation experiment. In this experiment you will acquire time-to-amplitude converter (TAC) spectra for different values of x, where x is the distance from the radioactive source to the front face of the movable NaI(T) detector. When the TAC spectra for increasing values of x are compared, one sees that the centroid of the TAC peak has shifted to the right in the TAC spectrum, as in Fig. 2. An increase in time delay t between the Stop pulse (from the movable detector) and the Start pulse (from the fixed detector) may be attributed directly to the increased distance x that a gamma ray must travel to the movable detector. It is then a simple matter to compute the gamma ray speed from these measurements using c = x/t. The major experimental challenge is to detect the small, approximately 1 ns shift of the TAC peak for each foot that the movable detector is displaced away from the radioactive source.

**Procedure**

Obtain five TAC spectra for x = 30, 60, 90, 120 and 150 cm. The TAC Range setting should be set to 50 ns to give a timing resolution of 50 ns/1024 ch = 0.049 ns/ch in the TAC spectra accumulated with the Quantum 8 multichannel analyzer. You should adjust the coaxial delay cable at the Stop input of the TAC so that the TAC peak is placed near the center of the TAC spectrum.

You are trying to measure a time shift of only a few ns as the detector is moved from x = 30 cm to x = 150 cm. In that same distance, the coincidence counting rate will drop by a factor of about 25 because of the geometric reduction of the coincidence gamma ray intensity. This experiment is therefore very susceptible to small changes in environmental conditions. To avoid erroneous results you must (a) not change any of the signal or high voltage cables, and (b) not adjust any of the amplifier, discriminator or TAC settings, during the period in which you take your five TAC spectra. Don't bump the table or even open the windows for a cool breeze.

Because of the sharp reduction in coincidence counting rate as x is increased, you will have to increase your total counting time with increasing x. Your counting time at x = 150 cm will be at least two hours, so plan your schedule accordingly. Be sure to record the total counting time at each distance in your laboratory notebook.

Store your five TAC spectra on disk using the Q8 program.

**Analsis**

(1) Using the Centroid option in the Q8 program, determine the centroid channel of the gamma-gamma coincidence TAC peaks in each of your five TAC spectra. Determine a reasonable estimate of the uncertainty in your centroid determinations. Convert all channel data to time data using the conversion factor 50 ns/1024 ch = 0.049 ns/ch. Put the channel and time results, along with associated uncertainties, in a table.

(2) In a figure, plot the time data given in the table of step (1). Plot TAC peak time t, in ns, along the vertical axis and distance, x, in cm, along the horizontal axis. Your five TAC peak times should be plotted with error bars that indicate associated uncertainties.

(3) Use a calculator to determine the slope and intercept of a linear least squares fit to the data of your table in step (1).

^{3}Draw this line on the figure of step (2). From the reciprocal of the slope of the least squares fit to your data, determine the speed of pair annihilation gamma rays.(4) By considering lines of lesser and greater slope through your data, estimate the uncertainty in the the gamma ray speed determination of step (3). You may wish to review the method that was used to establish an uncertainty for half-life determinations in an earlier experiment. State your final value and associated uncertainty for the speed of pair annihilation gamma rays.

(5) Use the ROI option in the Q8 program to determine the number of valid gamma-gamma coincidences obtained at the five distances. This means that you should apply a simple correction for the presence of random coincidences lying underneath the main TAC peak in the five spectra. Divide each of these valid coincidence totals by the respective total times required to accumulate them. Compute a valid gamma-gamma coincidence counting rate, in counts per second (Hz), at each of the five distances.

Plot the five counting rates versus detector distance x. Draw a curve corresponding to the approximate 1/x

^{2}fall off in intensity expected for the coincidence counting rate.

**References**

1. *Table of Isotopes*, 7th ed., edited by C.M. Lederer and V.S. Shirley (John Wiley & Sons, New York, 1978).

2. T.G. Ferris, *Three Coincidence Experiments*, B.A. thesis, Middlebury College, 1985.

3. H.D. Young, *Statistical Treatment of Experimental Data* (McGraw-Hill, New York, 1962), pp. 101-115.