# Determination of the attenuation coefficient for 662 keV gamma rays in lead

### Abstract

The attenuation coefficient μ for a 662 keV gamma ray has been determined using four different methods: 1) count rates, 2) count rates at optimal measurement conditions, 3) least-squares-method and 4) linearization. Lambert-Beers law of exponential attenuation has been examined. The relation between the signal-to-noise ratio and the thickness of the absorber has been considered.

### 1. Introduction

#### 1.1 Attenuation of gamma radiation in an absorber

Gamma rays are monochromatic electromagnetic radiations that are emitted from the nuclei of excited atoms after radioactive transformations. Gamma radiation (photons) can not be completely absorbed and stopped in an absorber, but the intensity of the rays is reduced by increasingly thicker absorbers. The relation between the thickness of an absorber and the intensity from a source is given by Lambert-Beers law of absorption:

Here $$I(x)$$ is the gamma-ray intensity transmitted through an absorber of thickness $$x$$, $$I_0$$ is the gamma-ray intensity at zero absorbator thickness, $$μ$$ is the attenuation coefficient and $$x$$ is the absorber thickness. The attenuation coefficient is characteristic for a specific absorbator and photon energy, and is an expression for the probability of interaction. There are two processes that can take place when 662 keV gamma radiation from $$^{137}Cs$$ go through an absorber, namely photoelectric effect and Compton scattering. Photoelectric effect is an interaction where the whole photon is absorbed and transferred to an electron. In Compton scattering, the photon is scattered and lose some of its energy to an recoiling electron. A cesium source also emits photons with energies around 32 keV, so the counting apparatus must have a threshold that does not take photons with this energy into account. $$^{137}Cs$$ has a half-life of over 30 years. Therefore it can be assumed that the intensity of the source is constant.

When the activity of a source is low, the background radiation has to be accounted for. Lambert-Beers law can then be modified to

where $$I_B$$ is the background intensity from the surroundings.

#### 1.2 Calculation of the linear attenuation coefficient μ with use of count rates

Formula (2) kan be solved for μ to obtain the following expression:

The three intensities can be found by determining the count rates in three different situations: with source and without absorber ($$r_1$$), with source and with absorber ($$r_2$$), and without source ($$r_3$$). The count rates have to be corrected for downtime τ.

The attenuation coefficient μ can then be written as a function of the three count rates and the thickness of the absorber:

The uncertainty of the count rates are given by

where $$N$$ is the number of counts and $$t$$ is the count time. The measurement uncertainty of the attenuation coefficient is then given by

#### 1.3 Determination of the optimal measurement conditions

The best starting point for measuring and calculation of the linear attenuation coefficient in a metal is at optimal measurement conditions. The optimal thickness of the absorbator is in the point where the relative uncertainty of the attenuation coefficient is as low as possible. The relative uncertainty is given by the following formula:

It is here assumed that the uncertainties for the count rates $$r_1$$ and $$r_3$$ are less important for estimation of the optimal condition than $$r_2$$, and therefore negligible. The derivative of the expression with respect of the product μx is then given by

where another simplification has been done. The uncertainty due to the thickness $$x$$ is eliminated to make the expression nicer. Equation (11) shows that the relative uncertainty is at its lowest value when $$μx$$ is 2. Optimal thickness of the absorber for finding the attenuation coefficient is therefore:

The uncertainty of the optimal absorber thickness is given by

#### 1.4 Validation of Lambert-Beers law

The intensity, or count rate, from a source when the background counts are removed, is given by

where $$N$$ is the total number of counts, $$N_b$$ is the background counts, $$t$$ is the count time and $$τ$$ is the downtime of the detector. The uncertainty of the intensity is given by the following expression:

By conversion of Lambert-Beers lav with background radiation, equation (2), the following expression can be obtained:

This equation can be used to make a plot of $$ln(I-I_B)$$ vs $$x$$. The line will then have a slope of $$-μ$$. The total uncertainties in both x- and y-direction for the points are given by

If the experimental points in the plot of $$ln(I-I_B)$$ vs $$x$$ lie on a straight line, this validates Lambert-Beers law. By using least-squares-method it is possible to find a value for the slope of the best fitted line, and thus finding the attenuation coefficient.

#### 1.5 The method of linearization

The attenuation coefficient can also be determined with the method of linearization by the following formula:

Here $$(x_1,ln(I_1))$$ and $$(x_2,ln(I_2))$$ is two points far apart in the plot of $$ln(I-I_B)$$ vs $$x$$, and $$x_2>x_1$$. The uncertainty of μ for this method can be found by drawing alternative lines that could have fitted the two points used in the calculation.

#### 1.6 The scintillation-detector

The figure under shows a typical longitudinal section for a scintillation-detector with a crystal and a photomultiplier-tube.

The purpose of a electronic detector for radioactive radiation is to transform the radiation energy to an electric signal. For a scintillation detector, the amplitude of the output signal is proportional to the energy deposited in the detector. When a gamma particle interacts with a crystal in the detector, one or more secondary electrons are created. These electrons loses all its energy by ionizing and exciting the molecules in the crystal. When these molecules de-excite, a flash of light with energy proportional to the energy of the in-going particle is generated. Because of a reflecting and opaque encapsulation around the the crystal, all the photons in this flash are guided through an optic link to a semi-transparent photocathode, where they strike electrons. The photocathode is inside a vacuum tube, where it is high voltage between the photocathode and the anode. This voltage makes the photoelectrons accelerate towards the anode. Before a photoelectron reaches the anode, it hits dynodes, which gives the photoelectron enough energy to kick out more electrons, and thus increase the number of photoelectrons and reinforce the signal. After the photomultiplier tube, the signal goes to the amplifier. Here it is a filter that that ensures an optimal signal-to-noise relationship, and the signal gets amplified. The amplitude of the output-signal from the amplifier is proportional to the energy deposited in the scintillation-crystal. Detectors with a discriminator or a simple-channel-analyser (SCA) give out a TTL-signal (Transistor-Transistor Logic) every time the signal from the amplifier is over a given threshold value. The downtime per pulse for a scintillation-detector is around τ = 22 μs.

### 2. Materials and methods

The equipment used in this experiment is listed in table 1.

The scintillation-detector consists of a NaI-crystal, an embedded high voltage generator, an amplifier and a pulse height discriminator. The $$^{137}Cs$$ gamma-source emits 662 keV gamma-particles. The oscilloscope is used for visualisation of the signals from the analog output, while the counter measures the the intensity of the digital output of the detector. Absorbers of lead are used for attenuation of gamma-rays, and the caliper is used for measurement of the absorber thickness. The figure under shows the setup of the detector for this experiment.

The voltage source was set to +12V. The detector was linked to the voltage source. The oscilloscope was liked to the analog exit of the detector, and the counter was linked to the digital (TTL) exit of the detector. The threshold was set to be 1V to avoid counts due to noise. The background radiation was measured with $$t_3=300s$$, and the count rate without the source, $$r_3$$ was determined by use of equation (6). τ = 22 μs was used as the downtime. The uncertainty of $$r_3$$ was found by equation (8). The source was placed in the holder, as shown in the figure over. The radiation of the source without absorbers was measured with $$t_1=300s$$, and the count rate, $$r_1$$ was determined by use of equation (4). The uncertainty of $$r_1$$ was found by equation (8). The calpier was then used to measure the thickness of two absorbers. The measurement uncertainty for the caliper was assumed to be $$±0,05mm$$. The radiation from the source that went through these two absorbers was measured with $$t_2=60s$$, and the count rate, $$r_2$$ was determined by use of equation (5). The uncertainty of $$r_2$$ was found by equation (8). The attenuation coefficient μ with uncertainty was then determined by equation (7) and (9).

The first estimate of μ was then used to calculate the optimal thickness of the absorber, $$x_{optimal}$$ with uncertainty , given by equation (12) and (13). The number of absorber-boards that gave the total thickness closest to the optimal thickness, was placed in the holder. Then another measurement with $$t_2=60s$$ was completed, and a new count rate, $$r_2$$ was determined by use of equation (5). The uncertainty of the new $$r_2$$ was found by equation (8). This new value for $$r_2$$ was then used together with the values of $$r_1$$, $$r_3$$ and $$x_{optimal}$$ in equation (7) and (9), to find a value of $$μ_{optimal}$$ with uncertainty.

Measurements with more than one absorber thickness are needed to check the validity of Lambert-Beers law experimentally. Therefore, six measurements was done with different numbers of absorber-boards in the holder (0 to 5). Since the statistical uncertainty in the six measurements should be approximately equal, the amount of counts in each of them should also be approximately equal. The count time for each absorber thickness was therefore calculated so that the number of counts was almost the same for all thicknesses. Used the caliper to measure the five absorber-boards. Measured the background counts, $$N_b$$, and the counts, $$N$$, for the six different absorber thicknesses with the assigned count time, $$t$$. The six intensities with uncertainties was calculated by use of equation (14) and (15). Determined $$ln(I-I_B$$ and plotted it in against $$x$$ for all six distances in ROOT. Used the script log_of_int_vs_thickness.C to make the plot with uncertainties along each axis. Used least-squares-method to find the slope of the line -μ, based on the uncertainties obtained by formula (17). Used the plot to validate Lambert-Beers law.

The plot of $$ln(I-I_B)$$ vs $$x$$ was also used together with the method of linearization to obtain a fourth value of μ. Selected the two points furthest away from each other and used equation (18) to find this value. Estimated the uncertainty by drawing alternative lines that also fitted these two points. Only the uncertainty in the y-direction were considered here.

Determined the signal-to-noise ratio $$I/I_B$$ and plotted it in against $$x$$ for all six distances in ROOT. Used the script signal_to_noise.C to make the plot. This ratio is a measure that compares the level og the signal to the level of background noise. The ideal situation to detect radiation is when the signal-to-noise ratio is high, so that the signal stands out compared to the background. Used the plot to consider how the signal-to-noise ratio is dependent on the thickness of the absorber.

Made a last plot of the different values of μ obtained by the different methods: 1) count rate, 2) optimal measurement conditions, 3) least-squares-method and 4) linearization. Used the script values_of_mu.C to make this plot. Used the plot to evaluate which of the methods that are the best ones for determining the attenuation coefficient, and compared the values obtained to the value from the litterature.

### 3. Results

Measurement of background radiation for five minutes gave a number of counts equal to

By use of formula (6) and (8), the count rate was calculated te be

The radiation from the source without an absorber was measured for five minutes, which gave a number of counts equal to

Formula (4) and (8) was used to find the count rate. That gave the following result:

The thickness of the absorber, consisting of two absorber-boards was measured to be

With an absorber in the holder, the radiation from the source that went through the absorber was measured for one minute. That gave a number of counts equal to

The count rate was with use of formula (5) and (8) calculated to be

Formula (7) and (9) was then used to find the first estimate of the attenuation coefficient μ. The result was the following:

The optimal absorber thickness was by formula (12) found to be

Since a single absorber-board has a thickness of 0,37 cm, seven boards will give a thickness closest to the optimal thickness. Therefore, seven absorber-boards was placed in the holder, and the radiation from the source that went through the absorber was measured for again one minute. That gave a number of counts equal to

A new value for the count rate was by formula (5) and (8) found to be

A new estimate for the attenuation coefficient μ was then found by the same procedure as the first estimate with formula (7) and (9), but now with a new value for $$r_2$$. That gave the value of

The total thickness of the absorber consisting of $$0≤i≤5$$ boards was measured, and is presented with uncertainties in table 2. Used the value of μ obtained at optimal measurement conditions to calculate the count time for the different thicknesses, which is also presented in table 2. The number of counts, number of background counts, intensity, logarithm of intensity and total uncertainty of the logarithm of intensity are also shown in table 2. The intensity with uncartainty was calculated with use of formula (14) and (15). The total uncertainty of the logarithm of intensity was found by formula (17).

The figure under shows a plot made in ROOT of the logarithm of the intensity $$ln(I-I_B)$$ against the total thickness $$x$$ of the absorber. The points plotted come from table 2, and the code used to plot is log_of_int_vs_thickness.C.

Observe that the points in the figure over lie approximately on a straight line. This indicates that gamma radiation attenuate exponentially, and that Lambert-Beers law is valid. Used least-squares-method with the total uncertainty on the plot to find the slope$$-μ$$ to the best fitted line. Found that the value of the attenuation coefficient found by least-squares-method is given by

For the linearization, the two points furthest apart were chosen, so that

By linearization with formula (18), it was found a fourth value of μ, given by

The figure under shows a plot made in ROOT of the signal-to-noise ratio for different absorber thicknesses. The points plotted come from table 2, and the code used to plot is signal_to_noise.C.

The signal-to-noise ratio decreses when the total thickness of the absorber increases. This makes sense, since the intensity of the signal decreses when it goes through the absorber. The decrease seems to be exponential, which corresponds well to the fact that the attenuation of gamma radiation in an absorber is exponential.

The figure under shows a comparison of the values of the attenuation coefficient μ obtained by four different methods. Method 1 is by use of count rates. Method 2 is by use of count rates at optimal measurement conditions. Method 3 is by use of least-squares-method on the plot of the logarithm of the intensity vs the absorber thickness. Method 4 is by linearization of points in the plot of the logarithm of the intensity vs the absorber thickness. The code used to plot is values_of_mu.C.

### 4. Discussion and conclusion

The validity of Lambert-Beers law was examined in this exercise. The points in the plot of the logarithm of the intensity against the absorber thickness lied almost on a straight line. This indicates that that the attenuation is exponential and that Lambert-Beers law is valid. The plot of the signal-to-noise ratio for several thicknesses also indicates this exponential decreasing relationship between the intensity of a ray and the thickness of an absorber.

The value from the litterature of the attenuation coefficient in lead is $$1.41 cm^{-1}$$ for 600 keV and $$1.01 cm^{-1}$$ for 800 keV (Johnson, 2017). Since the gamma rays in this experiment had an energy of 662 keV, it was expected that the calculated value of μ would lie between the values for 600 keV and 800keV. Instead, the values calculated lie around $$0.8 cm^{-1}$$, which is considerable lower than the values from the litterature. The reason for this could be scattered radiation from the source. Some of the photons that interact in the lead-boards get scattered towards the detector and thus become registered although they have interacted in the absorber. This leads to a higher count rate, which in turn leads to a smaller value for the attenuation coefficient μ.

All the obtained values of μ lie around $$0.8 cm^{-1}$$, and can therefore be said to be consistent. There are no significant deviations between the values. Method 1 with count rates and method 3 with least-squares-method gives almost the same value of μ, but these are the methods that give values that correspond the least with the values from the litterature. The uncertainty is smaller for method 3 than for method 1, so method 3 is considered the best method of these two. Method 4 with linearization has a small uncertainty, and also gives a higher value for μ than method 1 and 3. Method 2 with count rates at optimal measurement conditions gives the value of μ that is closest to the value from the litterature, and the uncertainty is acceptable. Method 2 is therefore evaluated to be the best method for determining the attenuation coefficient for 662 keV gamma rays in lead.

### 5. References

Johnson, Thomas E.. Introduction to Health Physics, Fifth Edition. New York: McGraw-Hill Education, 2017, p.891.

Sætre, Camilla. Laboratorieoppgave 8 - Oppgave. Bergen, 2020.

Sætre, Camilla. Laboratorieoppgave 8 - Veiledning. Bergen, 2020.