## Efficiency of XR-100T-CdTe Detectors

Application Note (AN-CdTe-001 Rev. 1)

Amptek’s XR-100T-CdTe is a high performance x-ray and gamma ray detector system. Like our other XR-100 products, a detector element and preamplifier components are mounted on a thermoelectric cooler. The CdTe products replace the Si diodes used in the other XR-100 products with cadmium telluride, a wide bandgap, compound semiconductors as the detector element. The primary advantage of CdTe is its much greater efficiency, due to its higher atomic number, Z. The photoelectric cross-section scales as Z^{5}. For Si, Z=14, while for CdTe, Z=50. The efficiency of Amptek’s 500 µm thick Si detectors begins to fall above 10 keV, while for 1 mm CdTe, efficiency is high to 100 keV.

The detection efficiency is a very important consideration, but due to charge transport effects, defining it is somewhat subtle. The purpose of this note is to provide general information on efficiency, for estimating system performance, and to recommend procedures for measuring the actual efficiency of a given detector. The charge transport is discussed in more detail in another Amptek Application Note, “Charge Trapping in XR-100T-CdTe and -CZT Detectors ” AN-CZT-002.^{3-a}

## Introduction

As is well known^{1}, when a beam of energetic photons, X-rays or g-rays, passes through a material the result is a simple exponential attenuation of the primary beam. Each of the possible interaction processes can be characterized by a probability of occurrence per unit path length in the absorber. The sum of the probabilities for the individual processes is the total probability per unit length that the photon is removed from the beam. This is termed the linear attenuation coefficient, is denoted µ, and has units of inverse length (cm-1). The number of primary photons transmitted through a thickness t is

where I_{0} is the flux of incident photons, t is the thickness of the attenuator, µ is the linear attenuation coefficient, and Itrans is the flux of transmitted primary photons. The number of primary photons interacting in a thickness t is obviously

The linear attenuation coefficient obviously depends strongly on energy, since the interaction mechanisms are energy dependent. The attenuation is often described using the mass attenuation coefficient, µ_{m}=µ/p where p is the density of the medium. This can be written in units of cm^{2}/g, with the density in g/cm^{3}, or in units of barns (1 barn = 10^{-24} cm^{2}) with the density in atoms per cm^{3}.

There are several different processes by which photons interact. In the energy range most often measured with the XR-100T-CdTe, the most important processes are the photoelectric interaction and Compton scattering. In a photoelectric interaction, the entire incident energy of the interacting photon is deposited in the detector, while in Compton scattering, only a portion of the incident energy will generally be deposited in the detector. Photoelectric interactions contribute to the full energy, which is usually of primary interest. The probability of a photoelectric interaction is usually of primary interest.

## Application to the XR-100T-CdTe

Amptek’s standard XR-100T-CdTe consists of a 1 mm thick CdTe located behind a 4 mil (100 µm) Be window. The probability of a photon interaction somewhere in the thickness is the product of (1) the probability of transmission through Be and (2) the probability of interaction in the material,

Figure 1 and Figure 2 show results which have been computed for our standard 1 mm CdTe detector. Also shown, for comparison, are results for a 0.5 mm Si detector and for a CdTe stack detector, 2.25 mm thick. The calculation includes the effects of transmission through the Be window and of stopping in the detector, but neglect the effects of trapping and hole tailing. Both total and photoelectric probabilities are given.

Figure 1. Linear plot of interaction probability.

Figure 2. Log-log plot of interaction probability between 1 keV and 1 MeV.

The data plotted in Figure 1 and Figure 2 are listed in ASCII text files. For more information on these efficiency calculations, please refer to the Efficiency Package package.

Energy (keV) | Total Interaction | Photoelectric Interaction | Total | Photoelectric |

1 | 0.00% | 0.00% | 4.82E+04 | 4.82E+04 |

1.006 | 0.00% | 0.00% | 4.76E+04 | 4.76E+04 |

1.006001 | 0.00% | 0.00% | 4.88E+04 | 4.87E+04 |

1.5 | 0.02% | 0.02% | 2.08E+04 | 2.07E+04 |

2 | 2.99% | 2.99% | 1.05E+04 | 1.05E+04 |

3 | 36.80% | 36.80% | 3.87E+03 | 3.83E+03 |

3.5375 | 52.76% | 52.76% | 2.55E+03 | 2.52E+03 |

3.537501 | 52.76% | 52.76% | 4.67E+03 | 4.64E+03 |

3.727 | 58.39% | 58.39% | 4.11E+03 | 4.07E+03 |

3.727001 | 58.39% | 58.39% | 5.11E+03 | 5.07E+03 |

4 | 66.49% | 66.49% | 4.29E+03 | 4.26E+03 |

4.018 | 66.76% | 66.76% | 4.24E+03 | 4.21E+03 |

4.018001 | 66.76% | 66.76% | 4.70E+03 | 4.66E+03 |

4.3414 | 71.60% | 71.60% | 3.87E+03 | 3.84E+03 |

4.341401 | 71.60% | 71.60% | 5.62E+03 | 5.59E+03 |

4.612 | 75.64% | 75.64% | 4.90E+03 | 4.87E+03 |

4.612001 | 75.64% | 75.64% | 5.73E+03 | 5.70E+03 |

4.9392 | 80.53% | 80.53% | 4.86E+03 | 4.83E+03 |

4.939201 | 80.53% | 80.53% | 5.27E+03 | 5.25E+03 |

5 | 81.44% | 81.44% | 5.12E+03 | 5.09E+03 |

6 | 88.80% | 88.80% | 3.22E+03 | 3.20E+03 |

8 | 94.86% | 94.86% | 1.52E+03 | 1.50E+03 |

10 | 97.01% | 97.01% | 8.76E+02 | 8.60E+02 |

12 | 97.79% | 97.79% | 5.38E+02 | 5.23E+02 |

15 | 98.57% | 98.57% | 2.96E+02 | 2.85E+02 |

20 | 98.95% | 98.95% | 1.36E+02 | 1.28E+02 |

25 | 98.95% | 98.95% | 7.47E+01 | 6.85E+01 |

26.711 | 99.09% | 99.09% | 6.25E+01 | 5.69E+01 |

26.71101 | 99.09% | 99.09% | 1.74E+02 | 1.68E+02 |

30 | 99.16% | 99.16% | 1.29E+02 | 1.24E+02 |

31.814 | 99.16% | 99.16% | 1.11E+02 | 1.06E+02 |

31.81401 | 99.16% | 99.16% | 2.13E+02 | 2.08E+02 |

35 | 99.16% | 99.16% | 1.66E+02 | 1.62E+02 |

40 | 99.23% | 99.23% | 1.18E+02 | 1.14E+02 |

45 | 99.21% | 99.21% | 8.61E+01 | 8.31E+01 |

50 | 99.12% | 99.08% | 6.51E+01 | 6.24E+01 |

55 | 98.63% | 98.46% | 5.04E+01 | 4.80E+01 |

60 | 97.47% | 97.03% | 3.99E+01 | 3.78E+01 |

70 | 92.20% | 90.73% | 2.64E+01 | 2.45E+01 |

80 | 83.60% | 80.90% | 1.84E+01 | 1.68E+01 |

90 | 73.55% | 69.54% | 1.35E+01 | 1.20E+01 |

100 | 63.55% | 58.64% | 1.02E+01 | 8.92E+00 |

125 | 44.02% | 37.23% | 5.85E+00 | 4.69E+00 |

150 | 30.85% | 24.13% | 3.71E+00 | 2.78E+00 |

175 | 23.22% | 16.25% | 2.66E+00 | 1.78E+00 |

200 | 17.96% | 11.38% | 1.99E+00 | 1.22E+00 |

250 | 12.69% | 6.24% | 1.36E+00 | 6.48E-01 |

300 | 9.50% | 3.78% | 1.00E+00 | 3.87E-01 |

350 | 7.95% | 2.51% | 8.32E-01 | 2.55E-01 |

400 | 6.81% | 1.76% | 7.08E-01 | 1.78E-01 |

500 | 5.56% | 0.99% | 5.75E-01 | 1.00E-01 |

600 | 4.84% | 0.64% | 4.98E-01 | 6.39E-02 |

800 | 4.00% | 0.33% | 4.09E-01 | 3.31E-02 |

1000 | 3.49% | 0.21% | 3.56E-01 | 2.06E-02 |

1022 | 3.45% | 0.20% | 3.51E-01 | 1.96E-02 |

1250 | 3.08% | 0.13% | 3.13E-01 | 1.32E-02 |

1500 | 2.80% | 0.09% | 2.85E-01 | 9.49E-03 |

2000 | 2.49% | 0.06% | 2.52E-01 | 5.78E-03 |

3000 | 2.23% | 0.03% | 2.26E-01 | 3.07E-03 |

4000 | 2.16% | 0.02% | 2.18E-01 | 2.04E-03 |

5000 | 2.15% | 0.02% | 2.17E-01 | 1.51E-03 |

6000 | 2.17% | 0.01% | 2.19E-01 | 1.19E-03 |

7000 | 2.20% | 0.01% | 2.23E-01 | 9.81E-04 |

8000 | 2.25% | 0.01% | 2.28E-01 | 8.33E-04 |

9000 | 2.30% | 0.01% | 2.33E-01 | 7.22E-04 |

10000 | 2.35% | 0.01% | 2.38E-01 | 6.38E-04 |

Table 2. Table of linear attenuation coefficients, along with interaction probabilities for the 1 mm CdTe thickness. This table does not reflect the effective depth due to hole tailing, as discussed in the text.

Figure 3. Log-log plot of interaction probability between 1 keV and 1 MeV for 1 mm CdTe.

Figure 4. Linear plot of interaction probability between 10 keV and 250 keV for 1 mm CdTe.

Efficiency Package: A ZIP file of coefficients and a FAQ about efficiency. This pacakge is provided for general information. It should not be used as a basis for critical quantitative analysis.

## Consequences of Trapping and Hole Tailing

CdTe is wide bandgap, high-Z, compound semiconductor material. It is used for X-ray and gaama-ray spectroscopy because it has a very high linear attenuation coefficient, permitting high efficiency in a small volume, and low leakage currents, permitting low electronic noise without cryogenic cooling^{2}. However, like other compound semiconductors, it exhibits significant spectral distortions due to hole trapping. As is discussed elsewhere^{4}, the trapping length of holes in CdTe is smaller than the linear dimensions of the detector. For interaction occurring near the anode, virtually all of the signal is due to electrons and so the full charge is collected. For interactions occurring near the cathode, virtually all of the signal is due to holes and so a smaller charge is collected.

The result is that the measured signal, the measured “energy”, depends upon the depth of the interaction on the detector, decreasing with increasing depth. In the output spectrum, one observes a tail of counts towards lower amplitudes, an effect known as “hole tailing”. Figure 5 is a plot of the pulse height as a function of depth in the detector, computed from the Hecht relation^{5} for two different values for the hole lifetime.

Figure 5. Plot showing the induced signal size as a function of depth, computed for two different values of the hole lifetime.

Note that in Figure 5, for the blue curve, about 40% of the depth of the detector produces a full signal. The rest of the depth produces a smaller signal. The effective volume of the detector, which contributes to the primary, full energy peak, is only 40% of the physical volume of the detector. For the red curve, about 20% of the detector volume contributes to the full energy peak.

For CdTe and other detectors in which charge trapping is important, it is critical to distinguish between the total geometric efficiency and the efficiency of the full energy peak. The total geometric efficiency, which is due to the total physical volume of the detector, can be used to compute the total rate of counts in the detector. The efficiency of the full energy peak, which is due to the volume of the detector leading to “full charge collection”, can be used to compute the total rate of counts in the primary peak. The phrase “full charge collection” is in quotation marks because it is not well defined. The charge collection efficiency decreases smoothly with increasing depth. Different users may define the full energy peak differently, depending on the specific application.

In Amptek’s XR-100T-CdTe, risetime discrimination (RTD) is used to minimize spectral distortions due to hole tailing. Figure 5 showed that the induced signal size is correlated with depth in the detector. The risetime of the pulse from the preamp is also well correlated with depth in the detector. Therefore, ours processors measure the risetime of the pulses and rejects those with a long risetime. This leads to a significant improvement in the quality of the spectrum.

For the 1 mm thick CdTe detectors, RTD is not as required although it still improves the spectrum. This is due to the improved charge transport of the device (see application note ANCZT-2). Figure 7 is a plot of a CdTe detector with RTD on and off for ^{57}Co. One advantage of CdTe is that a higher bias voltage can be applied. When the high bias is applied the charge transport becomes much better and therefore RTD is is not needed at all.

Figure 7. ^{57}Co Spectrum taken with RTD on and off with 1 mm thick CdTe.

## Measuring the Efficiency

In many applications it is important to know the detection efficiency at a particular energy. Because there exists a wide variation in the effective depth of Amptek’s XR-100T-CdTe detectors, if a user requires precise knowledge, the best solution is to measure the actual efficiency at the energy of interest from some well-known standard. Since this can be difficult, another approach is to measure the effective depth of the particular detector. There are two approaches to this measurement.

If one has a calibrated source, with a very well known strength and an energy high enough to be only partially absorbed in the material, then the effective depth can be readily computed by inverting the relation above:

For example, from Figure 1 the 122 keV line from ^{57}Co should be detected with 70% efficiency and from Table 1 that the linear attenuation coefficient is 6.22 cm-1. Assume that a lab measurement shows that this line is detected with 35% efficiency. This gives t=0.7 mm. This effective depth can be used to computed the efficiency at any other energy (above the energy where attenuation in the Be window is significant).

In the absence of a calibrated source, then a single source which emits g-rays at two distinct lines with a well known ratio can be used, if at least one of the lines is high enough in energy to be detected with <100% efficiency and both are above the energy where the Be window is significant. The source must not attenuate either line significantly. For example, ^{57}Co emits at 14.4 keV with 9.8% efficiency and 122 keV with 85.6% efficiency. We define P1 and P2 as the probabilities of emission of the two lines, N1 and N2 and the measured counts of the two lines, and µ1 and µ2 are the linear attenuation coefficients of the two lines. The ratio of the measured counts will obviously be

Applying some algebra, this implies

AN-CdTe-1 Application Note written by Robert Redus, Amptek Inc.

## References

1.) a) Knoll, Glenn F., Radiation Detection and Measurement, John Wiley & Sons, New York. 1989.

b) Tsoulfanidis, Nicholas, Measurement and Detection of Radiation, Hemphire Publishing Corporation, New York, 1983.

2.) Jordanov, V.T., J.A. Pantazis, and A.C. Huber, “Thermoelectrically-Cooled Cadmium Zinc Telluride Detectors (CZT) for X-Ray and Gamma-Ray Detection,” Radiation, Vol. 43, No. 1, July 1996.

3.) a) Charge Trapping in XR-100T-CZT Detectors, Amptek Application Note by Bob Redus, 2000

b) Semiconductors and Semimetals vol. 43, Semiconductors for Room Temperature Nuclear Detector Applications, section on Characterization and Quantification of Detector Performance by Vernon M. Gerrish, Volume Editors T.E. Schlesinger and R.B James, Academic Press, San Diego, 1995.

4.) a) Hecht (1932)

b) Semiconductors and Semimetals vol. 43, Semiconductors for Room Temperature Nuclear Detector Applications, section on Characterization and Quantification of Detector Performance by Vernon M. Gerrish discusses Hecht relation, Volume Editors T.E. Schlesinger and R.B James, Academic Press, San Diego, 1995.

5.) Squillante, M.R., Presentation at 11th International Workshop on Room Temperature Semiconductor X-Ray and Gamma-Ray Detectors and Associated Electronics, 1999 Vienna Austria.

Additional Reference: Redus, R.H., J. Pantazis, A. Huber, T.Pantazis, “Improved Sensitivity X-ray Detectors for Field Applications” Presented at 12th International Workshop on Room Temp. Det. Nov ’01, Submitted to IEEE Trans. Nucl. Sci.

XR-100T-CdTe Efficiency Application Note AN-CdTe-001 in PDF 300 k)

Efficiency Package: A ZIP file of coefficients and a FAQ about efficiency. This pacakge is provided for general information. It should not be used as a basis for critical quantitative analysis.