1 Introduction

With the continuous development of space science, space industry attracts great attentions all over the world. The aerospace X/γ detectors, which are commonly installed on spacecraft, are of great importance in, for example, gamma-ray burst (GRB) observation^{[1, 2]}, elemental composition detection of planetary surface^{[3, 4]}, black hole investigation^[5-7], study of the flare and solar energetic particle (SEP) events^{[8, 9]}, nuclear detonation inspection^[10]_, and other fundamental or application researches. All these require the X/γ detectors of high energy resolution in energy region from dozens of keV to several MeV.

At present, radioactive isotopes, such as ²⁴¹Am, ⁶⁰Co, ¹³⁷Cs, ⁴⁰K and ²⁰⁸Tl are often used as aerospace X/γ detector calibration sources^[11]. However, they provide isolated monoenergetic X/γ-rays and the detectors are calibrated through linear interpolation or extrapolation. This is not suitable for energy calibration over a wide energy region since the linearity of detectors’ response would become deteriorated^[12-14]. Besides, radioactive isotopes can scarcely produce γ-rays of several MeV ^{[15, 16]}. Despite its good energy tunability from the infrared to the X-ray region, synchrotron radiation is not suitable for this task either, because it can hardly reach over 300 keV^[17-19]. Bremsstrahlung can produce high energy photons, but its intensity changes slowly over a wide spectral range, making it not suitable for energy calibration^[20].

Laser Compton Scattering (LCS) γ source is a potential solution to overcome the above difficulties. It uses high-power short-pulse laser beam with high-brightness relativistic electron beam to achieve Compton scattering and produce high-flux, short-pulse, quasi-monochromatic X/γ-ray. The past decades, with advances in accelerator and laser technology, witnessed rapid development of the LCS X/γ-ray source, which is rated as one of the most potential ultra-short pulse light sources^[21]. Currently, LLNL^[22], PLS^[23], CLS^[24], ELI-NP^[25], ALBA^[26], MIT^[27], Spring-8^{[28, 29]}, JAEA^[30], SSRF^{[31, 32]}, INFN^[33], TUNL^[34] and other research institutions are committed to the construction of the experimental devices of LCS.

With an energy-calibrated high-purity germanium (HPGe) detector, the exact energy and shape of the high energy edge of the measured Compton spectrum were used to determine the electron energy and the electron beam energy spread in BESSY I ^[35]. C. Sun et al. improved this beam diagnostics method with a more comprehensive model and successfully determined the electron beam energy of the HIGS’s storage ring^{[36, 37]}. This technique can be applied to detector energy calibration with the electron beam of known energy, i.e., using a series of Compton edges of the X/γ-ray spectra produced by LCS to calibrate the detectors. Recently, at NewSUBARU, Hiroaki et al. attempted with a similar idea and calibrated LaBr₃(Ce) detector at 10.19, 9.14 and 8.19 MeV with LCS Compton edges by changing the electron beam energy ^[14].

The proposed SINAP III ^{[38, 39]} facility is such a kind of LCS γ source: by changing continuously the colliding angle between the electron beam and laser beam, the steep Compton edge of scattered photon energy spectrum is thus continuously adjustable from 25 keV to 740 keV. In this paper, we present an energy calibration method developed to perform consecutive calibration over a wide energy region based on this facility.

In this article, the principle of consecutive energy calibration method and the calibration steps are introduced. The process to precisely determine location of the Compton edge is described in detail The simulation setups to study the dependence of the calibration accuracy on relevant systematic uncertainties are demonstrated. The simulation results and the calibration accuracy are discussed. Finally, we summarize this consecutive detector energy calibration method and propose the future application of this method on other LCS facilities.

2 Principle of consecutive detector energy calibration based on LCS

For relativistic electron, energy expressions of γ ray generated from laser electron Compton scattering ^{[31, 40]}is as following:

where E_L and E_e are the incident photon and electron energy, respectively, in laboratory reference frame; θ_in is the angle between the incident laser and electron movement direction, referred to as the laser incident angle; and θ is the angle between the direction of movement of the scattered photons and electrons, referred to as the γ-ray scattering angle. When θ_in, E_L, E_e are fixed, E_γ reaches the theoretical maximum value E_max when θ=0.

Energy spectrum of Compton scattering drops rapidly at E_max, generating a steep Compton edge. According to Eq. (2), when E_L, E_e are fixed, E_max and θ_in is one-to-one corresponded^[41]. SINAPIII is designed based on this principle. The simulated γ spectra of different laser incident angles θ_in generated by SINAPIII are shown in Fig.1(a). The E_max - θ_in relationship is shown in Fig.1(b).

Fig.1Full-screen View

Gamma spectra (a) at different laser incident angles, and the E_max - θ_in relationship (b), at E_e= 180 MeV and λ= 800 nm, on SINAP III

Thus, a consecutive energy calibration method based on the Compton edge can be carried out in following steps,

(1) Measure the LCS energy spectrum at a laser incident angle θ_in-i to obtain the E_max(i) by Eq. (2);

(2) Find the Compton edge channel address, i.e., Channel #(i);

(3) Change θ_in and repeat Steps (1) and (2);

(4) With the array of [Channel #(i), E_max(i)] occupied, a map between channel address and E_max can be generated, corresponding to a series of points on the plane where the channel address is the x coordinate and the γ-ray energy is the y coordinate. Finish the calibration by fitting the points with a polynomial equation just like what calibration is done in conventional way^[11].

The advantage of this consecutive energy calibration method is that the points can be generated over a wide range of energy with very small intervals. On SINAP III, we can generate a series of Compton edges in energy steps of no larger than 0.07 keV by changing incident laser angle θ_in with 0.01°. This enables the calibration of a specific range near any particular energy of interest with tiny rotation of the incident laser and avoids the error brought by the extrapolation from full energy peaks of radioactive isotopes. The experiments with various radioactive isotope sources can be omitted, and the calibration process is easy to be automated.

3 Determination of the Compton edge location

In the above calibration steps, the key issue of this method is how to determine the channel address of the Compton edge E_max precisely. In actual situation, the systematic uncertainties or variables, such as energy spread ΔE/E and the emittance of electron bunches ε, would smear the Compton edge (Fig.2). Fortunately, as will be discussed later in Section 4, the corresponded energy at midpoint of the edge is quite stable if the systematic uncertainties do not vary too much during the calibration, and is approximately equal to E_max, with a difference of <1.6% (denoted as ΔE_max).

Fig.2Full-screen View

Simulated energy spectra near the Compton edge, at laser wavelength of λ=800 nm in collision angle of θ_in= 90°, and electron beam energy of E_e=180 MeV in beam bunch emittance of ε=6 mm·mrad with energy spread of ΔE/E=0.1% (a) and ΔE/E=0.3% (b)

So, the key to this method now turns into finding the midpoint of Compton edge E_mid or channel number C_mid (E_mid and C_mid are equivalent in this scenario, and we will always use E_mid from now on) which is approximate enough to E_max. Two fitting methods are applied to the simulated energy spectra. The fitting function derived in Refs.[35, 36] is also applied as comparison. The results show that all the three methods can solve the problem with satisfactory accuracy.

The linear fitting method is of five steps:

(1) Find the channel C_peak with the highest count N_peak in the spectrum. Start from C_peak and search towards the right side of C_peak, stop at the first channel C_low where the count is less than 20%N_peak.

(2) Choose the low platform (P_L) of the energy spectrum near the Compton edge. The P_L is in width of k_L channels, with C_low being the initial left endpoint and N_L being the platform height averaged from the counts of the k_L channels. Move the platform one channel towards the right side and calculate N_L again, denoted as N_L'. If |N_L−N_L'|>0.01N_L, i.e. if the platform is not well chosen, keep on moving the platform towards the right side channel by channel until |N_L−N_L'|<0.01N_L. Then, the left and right endpoints of the final low platform are denoted as C_downL and C_downR, respectively.

(3) Choose the high platform (P_H) of the energy spectrum near the Compton edge. Start from C_low and search towards the left side of C_low, stop at the first channel C_high whose count is larger than 80%N_peak. The P_H is in width of k_H channels, with C_high being the initial right endpoint, and N_H being the platform height averaged from the counts of the k_H channels. Move the platform one channel towards the left side and calculate N_H again, denoted as N_H'. If |N_H−N_H'|>0.001N_H, keep on moving the platform towards the left side channel by channel until |N_H−N_H'|<0.001N_H . Then, the left and right endpoints of the final chosen high platform are denoted as C_upL and C_upR, respectively.

(4) Perform a linear fitting to the intercepted energy spectrum between C_high and C_low

where p₁ and p₀ are the coefficients to be determined, N is the count of channel, C is the channel address number.

(5) Substitute N with N_mid=50%N_H+50%N_L to obtain E_mid:

The R. Klein model method is:

where a₁–a₅ are the coefficients to be determined. Here, a₁ corresponds to E_mid. Another fitting model in Ref.[36] is not applied as we neglect the collimation effect yet. The fitting range is the intercepted energy spectrum between C_upL to C_downR.

The error function method is:

where erf is the standard error function, C₀, C₁, E_mid and σ are the coefficients to be determined. This one is equivalent to the fitting function used in [42], and can be regarded as a simplified model of Eq. (5) with a₄=0.

4 Simulation setup

The energy calibration method is tested with the data generated by an updated 4D Monte Carlo simulation code^[43]. Main parameters of the laser and electron beams of SINAP III for the simulation are listed in Table 1.

Ten system variables affecting the γ-ray spectrum are listed in Table 2, where the input value of each variable changes within its scan range, so that ΔE_max caused by systematic uncertainties of SINAP III can be simulated.

5 Results and discussion

We will first demonstrate a rough estimation of ΔE_max at a certain incident angle, and then extend it to the whole adjustable angle range of SINAP III, i.e. [20°,160°]. Finally, consider the deviation correction and give a more accurate ΔE_max.

5.2. Estimation of ΔE_max from 20° to 160°

In order to test if E_mid is always consistent with E_max in the other cases, the operations are repeated at θ_in= 20°–160°. As shown in Fig.3, the δE_max values are less than 1.6%, with the same scan ranges of systematic uncertainties as in θ_in=90°.

Fig.3Full-screen View

Relationship between ΔE_max (δE_max) and θ_in.

5.3 Deviation correction

The above ΔE_max and δE_max need correction because E_max(q_i) in Eq.(7) changes when different θ_in, λ and E_e are sampled due to Eq.(2). Take the deviation of E_max(q_i) into consideration, a more precise ΔE_max expression can be obtained as:

$\Delta E_{\max }=\sqrt{{\displaystyle \sum _{q_{\text{im}}}{(\Delta E_{\max }^{q_{\text{im}}})}^2}+{\displaystyle \sum _{q_{\text{ex}}}\left[{(\Delta E_{\max }^{q_{\text{ex}}})}^2+{(|\partial E_{\max }/\partial q_{\text{ex}}|\Delta q_{\text{ex}})}^2\right]}},\delta E_{\text{max}}=\sqrt{{\displaystyle \sum _{q_{\text{im}}}{(\delta E_{\max }^{q_{\text{im}}})}^2}+{\displaystyle \sum _{q_{\text{ex}}}\left[{(\delta E_{\max }^{q_{\text{ex}}})}^2+{(|\partial E_{\max }/\partial q_{\text{ex}}|\frac{\Delta q_{\text{ex}}}{E_{\max }})}^2\right]}}$ (9)

where q_ex denotes the explicit variables of θ_in, λ and E_e, q_im denotes the other implicit variables regarding E_max, and ∂E_max/∂q_ex is the partial derivative of E_max with respect to q_ex, as shown in Fig.4.

Fig.4Full-screen View

The partial derivatives of ∂E_max/∂θ_in, ∂E_max/∂λ and ∂E_max/∂E_e as a function of θ_in =20°–160° on SINAPIII

It is worth mention that at θ_in =160° we can estimate the following rates from Fig.5(b) and 5(c),

Fig.5Full-screen View

The ΔE_max and δE_max as a function of θ_in after correction

$\frac{\Delta E_{\max }/E_{\max }}{\Delta \lambda /\lambda }\approx \frac{(\partial E_{\max }/\partial \lambda )\lambda }{E_{\max }}=\frac{-1\times 800}{740}\approx -1,\frac{\Delta E_{\max }/E_{\max }}{\Delta E_e/E_e}\approx \frac{(\partial E_{\max }/\partial E_e)E_e}{E_{\max }}=\frac{8\times 180}{740}\approx 2.$ (10)

The two rates are consistent with the equation δE_ma/ΔE_max ≈ [(2σ_Ee/E_e)²+(σ_Ep/E_p)²]^1/2 derived under head-on (θ_in=180°) LCS geometry^[36]. According to design of the SINAPIII, typical values of δE_e=0.1%, Δλ=1 nm and Δθ_in=0.01° are used. The ΔE_max and δE_max after correction are shown in Fig.5.

Compared with Fig.3, we can see that when the laser incident angle θ_in gets larger, δE_e and Δλ contribute more to ΔE_max and δE_max. In fact, the two systematic uncertainties are almost dominating when θ_in> 60°. Over all, the δE_max is lower than 1.6% in the energy region of 25−300 keV, lower than 0.5% in the energy region of 300–740 keV, when the systematic uncertainties can be limited within the scan range in the calibration process.

6 Summary

More precise calibration of X/γ detectors in a wide energy range demands new tunable X/γ sources and new calibration methods. LCS light source is a suitable γ source candidate. Continuously changing the collision angle between laser and electron beam is one of the most effective ways to continuously change γ energy spectra’s high energy edges^[44]. It takes advantage of the one-to-one correspondence of the laser-electron collision angle and the energy of the generated gamma spectrum’s Compton edge. In this work, an energy calibration method based on this technique is carried out. It can perform consecutive energy calibration with small energy gap in the neighborhood of a specific energy. This would eliminate the error from extrapolation, solve the problem of the deterioration of detector’s linearity and improve calibration accuracy. The uncertainty of this method for the X/γ detectors on SINAPIII facility is tested, which is lower than 1.6% in the energy region of 25–300 keV, and is lower than 0.5% in the energy region of 300–740 keV.

This energy calibration method could also be applied on the other LCS light sources with continuously variable laser incident angle. SINAP III is the prototype of the Shanghai Laser Electron Gamma Source ^{[31, 32, 45-47]} (SLEGS) on the storage ring of SSRF ^[48]. Once constructed, the larger energy region and higher repetition rates with lower systematic uncertainties of SLEGS will enable a faster and more precise energy calibration process for X/γ detectors.