A new consecutive energy calibration method for X/γ detectors based on energy continuously tunable laser Compton scattering light source

A new consecutive energy calibration method for X/γ detectors based on energy continuously tunable laser Compton scattering light source 1 2 first-author 1 1 1 1 corresp Corresponding author. E-mail address: wangdong@sinap.ac.cn Corresponding author. E-mail address: wangdong@sinap.ac.cn wangdong@sinap.ac.cn Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China In this paper, we present an energy calibration method based on steep Compton edges of the Laser Compton Scattered (LCS) photon energy spectra. It performs consecutive energy calibration in the neighborhood of certain energy, hence the improved calibration precision in the energy region. It can also achieve direct calibration at high energy region (several MeV) where detectors can only be calibrated by extrapolation in conventional methods. These make it suitable for detectors that need wide range energy calibration with high precision. The effects of systematic uncertainties on accuracy of this calibration method are studied by simulation, using the design parameters of a LCS device –SINAP III. The results show that the SINAP III device is able to perform energy calibration work over the energy region of 25–740 keV. The precision of calibration is better than 1.6% from 25 keV to 300 keV, and is better than 0.5% from 300 keV to 740 keV. Laser Compton scattering (LCS) Energy calibration method Gamma-ray application Monte Carlo simulation 1 Introduction 1 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 1 , 2 2 ] , elemental composition detection of planetary surface [ 3 3 , 4 4 ] , black hole investigation [ 5 5 - 7 7 5-7 ] , study of the flare and solar energetic particle (SEP) events [ 8 8 , 9 9 ] , nuclear detonation inspection [ 10 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 241 Am, 60 Co, 137 Cs, 40 K and 208 Tl are often used as aerospace X/γ detector calibration sources [ 11 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 12 - 14 14 12-14 ] . Besides, radioactive isotopes can scarcely produce γ-rays of several MeV [ 15 15 , 16 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 17 - 19 19 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 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 21 ] . Currently, LLNL [ 22 22 ] , PLS [ 23 23 ] , CLS [ 24 24 ] , ELI-NP [ 25 25 ] , ALBA [ 26 26 ] , MIT [ 27 27 ] , Spring-8 [ 28 28 , 29 29 ] , JAEA [ 30 30 ] , SSRF [ 31 31 , 32 32 ] , INFN [ 33 33 ] , TUNL [ 34 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 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 36 , 37 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 LaBr 3 (Ce) detector at 10.19, 9.14 and 8.19 MeV with LCS Compton edges by changing the electron beam energy [ 14 14 ] . The proposed SINAP III [ 38 38 , 39 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 1 For relativistic electron, energy expressions of γ ray generated from laser electron Compton scattering [ 31 31 , 40 40 ] is as following: <math display="block" id="M1"><mrow><msub><mi>E</mi><mtext>γ</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>E</mi><mtext>L</mtext></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>β</mi><mi>cos</mi><msub><mi>θ</mi><mrow><mtext>in</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>β</mi><mi>cos</mi><mi>θ</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><msub><mi>E</mi><mtext>L</mtext></msub></mrow><mrow><msub><mi>E</mi><mtext>e</mtext></msub></mrow></mfrac><msup><mi>γ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>cos</mi><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>β</mi><mi>cos</mi><msub><mi>θ</mi><mrow><mtext>in</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo></mrow></math> 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. <math display="block" id="M2"><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi>γ</mi><mn>2</mn></msup><msub><mi>E</mi><mtext>L</mtext></msub><msub><mi>E</mi><mtext>e</mtext></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>β</mi><mi>cos</mi><msub><mi>θ</mi><mrow><mtext>in</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>E</mi><mtext>e</mtext></msub><mo>+</mo><mn>2</mn><msup><mi>γ</mi><mn>2</mn></msup><msub><mi>E</mi><mtext>L</mtext></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>β</mi><mi>cos</mi><msub><mi>θ</mi><mrow><mtext>in</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></mrow></math> 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 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) Fig.1(a) . The E max - θ in relationship is shown in Fig.1(b) Fig.1(b) . 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 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 1 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 Δ E / E and the emittance of electron bunches ε , would smear the Compton edge ( Fig.2 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 Δ E max ). 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 35 , 36 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 '| > 0.01 N 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 '| < 0.01 N 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 '| > 0.001 N H , keep on moving the platform towards the left side channel by channel until | N H − N H '| < 0.001 N 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 <math display="block" id="M3"><mrow><mi>C</mi><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><mi>N</mi><mo>+</mo><msub><mi>p</mi><mn>0</mn></msub><mo>,</mo></mrow></math> where p 1 and p 0 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 : <math display="block" id="M4"><mrow><msub><mi>E</mi><mrow><mtext>mid</mtext></mrow></msub><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><mn>50</mn><mi>%</mi><msub><mi>N</mi><mtext>H</mtext></msub><mo>+</mo><mn>50</mn><mi>%</mi><msub><mi>N</mi><mtext>L</mtext></msub></mrow><mo>)</mo></mrow><mtext> </mtext><mo>+</mo><msub><mi>p</mi><mn>0</mn></msub><mo>.</mo></mrow></math> The R. Klein model method is: <math display="block" id="M5"><mrow><mi>N</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>γ</mtext></msub><mo>,</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><msub><mi>a</mi><mn>4</mn></msub><mo>,</mo><msub><mi>a</mi><mn>5</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mn>3</mn></msub><mrow><mo>{</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>a</mi><mn>4</mn></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>γ</mtext></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mi>e</mi><mi>r</mi><mi>f</mi><mi>c</mi><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>E</mi><mtext>γ</mtext></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub></mrow><mrow><msqrt><mn>2</mn></msqrt><msub><mi>a</mi><mn>2</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mrow><msub><mi>a</mi><mn>2</mn></msub><msub><mi>a</mi><mn>4</mn></msub></mrow><mrow><msqrt><mrow><mn>2</mn><mtext>π</mtext></mrow></msqrt></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>γ</mtext></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msubsup><mi>a</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>}</mo></mrow><mo>+</mo><msub><mi>a</mi><mn>5</mn></msub><mo>,</mo></mrow></math> where a 1 – a 5 are the coefficients to be determined. Here, a 1 corresponds to E mid . Another fitting model in Ref.[ 36 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: <math display="block" id="M6"><mrow><mi>N</mi><mo>=</mo><msub><mi>C</mi><mn>0</mn></msub><mo>−</mo><msub><mi>C</mi><mn>1</mn></msub><mtext>erf</mtext><mrow><mo>(</mo><mrow><mi>C</mi><mo>/</mo><mi>σ</mi><mo>−</mo><msub><mi>E</mi><mrow><mtext>mid</mtext></mrow></msub><mo>/</mo><mi>σ</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> where erf is the standard error function, C 0 , C 1 , E mid and σ are the coefficients to be determined. This one is equivalent to the fitting function used in [ 42 42 ], and can be regarded as a simplified model of Eq. (5) with a 4 =0. 4 Simulation setup 1 The energy calibration method is tested with the data generated by an updated 4D Monte Carlo simulation code [ 43 43 ] . Main parameters of the laser and electron beams of SINAP III for the simulation are listed in Table 1 Table 1 . 1 1 Parameters 1 1 Value 1 1 Electron energy /MeV 1 1 180 1 1 Energy spread 1 1 0.1% 1 1 Emittance /mm·mrad 1 1 6 1 1 Bunch length(rms) / mm 1 1 σ l e = 0.72 1 1 RMS beam size /mm 1 1 σ w e = σ h e = 20 1 1 Laser wavelength /nm 1 1 800 1 1 Energy/pulse /mJ 1 1 1.75 1 1 Repetition rate /Hz 1 1 1000 1 1 Pulse length(rms) /ps 1 1 σ lp =1 1 1 RMS beam size /mm 1 1 σ wp = σ hp = 20 1 1 Incident angle /° 1 1 20, 45, 67.5, 90, 112.5, 135, 160 Ten system variables affecting the γ-ray spectrum are listed in Table 2 Table 2 , where the input value of each variable changes within its scan range, so that Δ E max caused by systematic uncertainties of SINAP III can be simulated. * θ C is the center value of the scan range of the laser incident angle. It changes every time over the input settings of 20° θ C ≤160°. 1 1 System variables 1 1 Scan range 1 1 Laser incident angle θ in /° 1 1 [ θ C −0.05, θ C +0.05]* 1 1 Laser wavelength λ /nm 1 1 [797.5,802.5] 1 1 The center value of electron Energy E e /MeV 1 1 [179.9,180.1] 1 1 Emittance ε /mm·mrad 1 1 [6,36] 1 1 Electron energy spread Δ E / E 1 1 [0.0005,0.0055] 1 1 Spot size of laser waist σ l /μm 1 1 [20,70] 1 1 RMS beam size of electron σ e /μm) 1 1 [20,70] (σ e =σ w e =σ h e ) 1 1 Deviation between laser pulse center and electron bunch in horizontal ( D x ),vertical ( D y ) and electron beam ( D z ) directions 1 1 D x (μm): [−50,50] 1 1   1 1 D y (μm):[−50,50] 1 1   1 1 D z (mm): [−1.5,1.5] 5 Results and discussion 1 We will first demonstrate a rough estimation of Δ E 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 Δ E max . 5.1. Estimation of Δ E max at a certain incident angle 2 Let us demonstrate the calculation process in which laser incident angle is 90° as an example. All the system variables are set equal to the center values at the beginning. In order to evaluate the deviation between E max and E mid caused by the variation of one particular system variable q , which is denoted as Δ E q max , the scanned range of the concerned system variable is evenly divided into ( m −1) intervals. Then the input value of the concerned system variable in the simulation will switch through the m endpoints of the intervals while the other system variables remain unchanged. Next, apply the three fitting methods mentioned in the previous section to the simulation data to get E mid . Finally, we obtained Δ E max which is defined as the average deviation between E max and E mid of the m samples. <math display="block" id="M7"><mrow><mi>Δ</mi><msubsup><mi>E</mi><mrow><mi>max</mi></mrow><mi>q</mi></msubsup><mo>=</mo><mover accent="true"><mrow><mo>|</mo><msub><mi>E</mi><mrow><mtext>mid</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>q</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>E</mi><mrow><mtext>max</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>q</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>|</mo><mo>+</mo><mi>e</mi><mi>r</mi><msub><mi>r</mi><mrow><msub><mi>E</mi><mrow><mtext>mid</mtext></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>q</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">¯</mo></mover><mo>=</mo><mfrac><mn>1</mn><mi>m</mi></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mrow><mrow><mo>[</mo><mrow><msub><mi>E</mi><mrow><mtext>mid</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>q</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>E</mi><mrow><mtext>max</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>q</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>|</mo><mo>+</mo><mi>e</mi><mi>r</mi><msub><mi>r</mi><mrow><msub><mi>E</mi><mrow><mtext>mid</mtext></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>q</mi><mi>i</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mo>,</mo></mrow></mstyle></mrow></math> where q ∈( ε , Δ E / E , σ 1 , σ e , D x , D y , D z , θ in , λ , E e ) and err E mid ( q i ) is the fitting error of E mid ( q i ). Repeat the above steps through every system variable, the Δ E q max caused by 10 system variables are collected and the δ E q max =Δ E q max / E max is calculated. The results are given in Table 3 Table 3 . 1 1 System variables 1 1 Center value 1 1 Scan range 2 1 Linear fit method 2 1 Error function method 2 1 R.Klein model 1 1   1 1   1 1   1 1 Δ E q max /eV 1 1 δ E q max    /×10 −5 1 1 Δ E q max /eV 1 1 δ E q max /×10 −5 1 1 Δ E q max /eV 1 1 δ E q max /×10 −5 1 1 Ε /mm·mrad 1 1 6 1 1 [6, 36] 1 1 44.7 1 1 11.6 1 1 107.4 1 1 28.0 1 1 55.1 1 1 14.4 1 1 Δ E / E 1 1 0.001 1 1 [0.0005, 0.0055] 1 1 125.4 1 1 32.7 1 1 65.7 1 1 17.1 1 1 146.5 1 1 38.2 1 1 σ l /μm 1 1 20 1 1 [20, 70] 1 1 20.1 1 1 5.23 1 1 31.4 1 1 8.17 1 1 25.6 1 1 6.68 1 1 σ e /μm 1 1 20 1 1 [20, 70] 1 1 30.2 1 1 7.88 1 1 46.7 1 1 12.2 1 1 46.4 1 1 12.1 1 1 D x /μm 1 1 0 1 1 [−50, 50] 1 1 20.2 1 1 5.27 1 1 29.5 1 1 7.70 1 1 30.8 1 1 8.01 1 1 D y /μm 1 1 0 1 1 [−50, 50] 1 1 29.2 1 1 7.61 1 1 54.3 1 1 14.1 1 1 55.7 1 1 14.5 1 1 D z /mm 1 1 0 1 1 [−1.5, 1.5] 1 1 28.0 1 1 7.31 1 1 62.6 1 1 16.3 1 1 63.5 1 1 16.6 1 1 θ in    /° 1 1 90 1 1 [89.95, 90.05] 1 1 19.8 1 1 5.17 1 1 31.7 1 1 8.25 1 1 32.8 1 1 8.54 1 1 λ /nm 1 1 800 1 1 [797.5, 802.5] 1 1 20.5 1 1 5.35 1 1 32.7 1 1 8.53 1 1 32.7 1 1 8.52 1 1 E e /MeV 1 1 180 1 1 [179.9, 180.1] 1 1 19.4 1 1 5.06 1 1 32.1 1 1 8.36 1 1 34.2 1 1 8.92 The total error Δ E max and relative error δE max can now be represented as the following equations, <math display="block" id="M8"><mrow><mi>Δ</mi><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>=</mo><msqrt><mrow><msup><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>q</mi></munder><mrow><mo stretchy="false">(</mo><mi>Δ</mi><msubsup><mi>E</mi><mrow><mi>max</mi></mrow><mi>q</mi></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mrow><mn>2</mn></msup></mrow></msqrt><mo>,</mo><mtext> </mtext><mi>δ</mi><msub><mi>E</mi><mrow><mtext>max</mtext></mrow></msub><mo>=</mo><msqrt><mrow><msup><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>q</mi></munder><mrow><mo stretchy="false">(</mo><mi>δ</mi><msubsup><mi>E</mi><mrow><mi>max</mi></mrow><mi>q</mi></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mrow><mn>2</mn></msup></mrow></msqrt><mo>.</mo></mrow></math> The values of δ E q max in Table 3 Table 3 are all less than 0.05%, so δ E max | θ c=90° < 10 1/2 ×0.05%=0.16%, which infers that E mid coincide well with E max at θ C =90°. 5.2. Estimation of Δ E max from 20° to 160° 2 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 Fig.3 , the δE max values are less than 1.6%, with the same scan ranges of systematic uncertainties as in θ in =90°. 5.3 Deviation correction 2 The above Δ E 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 Δ E max expression can be obtained as: <math display="block" id="M9"><mrow><mi>Δ</mi><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>=</mo><msqrt><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>q</mi><mrow><mtext>im</mtext></mrow></msub></mrow></munder><mrow><msup><mrow><mo stretchy="false">(</mo><mi>Δ</mi><msubsup><mi>E</mi><mrow><mi>max</mi></mrow><mrow><msub><mi>q</mi><mrow><mtext>im</mtext></mrow></msub></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mstyle><mo>+</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>q</mi><mrow><mtext>ex</mtext></mrow></msub></mrow></munder><mrow><mrow><mo>[</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mi>Δ</mi><msubsup><mi>E</mi><mrow><mi>max</mi></mrow><mrow><msub><mi>q</mi><mrow><mtext>ex</mtext></mrow></msub></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mo>|</mo><mo>∂</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>/</mo><mo>∂</mo><msub><mi>q</mi><mrow><mtext>ex</mtext></mrow></msub><mo>|</mo><mi>Δ</mi><msub><mi>q</mi><mrow><mtext>ex</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow></mrow></mstyle></mrow></msqrt><mo>,</mo><mtext> </mtext><mi>δ</mi><msub><mi>E</mi><mrow><mtext>max</mtext></mrow></msub><mo>=</mo><msqrt><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>q</mi><mrow><mtext>im</mtext></mrow></msub></mrow></munder><mrow><msup><mrow><mo stretchy="false">(</mo><mi>δ</mi><msubsup><mi>E</mi><mrow><mi>max</mi></mrow><mrow><msub><mi>q</mi><mrow><mtext>im</mtext></mrow></msub></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mstyle><mo>+</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>q</mi><mrow><mtext>ex</mtext></mrow></msub></mrow></munder><mrow><mrow><mo>[</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mi>δ</mi><msubsup><mi>E</mi><mrow><mi>max</mi></mrow><mrow><msub><mi>q</mi><mrow><mtext>ex</mtext></mrow></msub></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mo>|</mo><mo>∂</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>/</mo><mo>∂</mo><msub><mi>q</mi><mrow><mtext>ex</mtext></mrow></msub><mo>|</mo><mfrac><mrow><mi>Δ</mi><msub><mi>q</mi><mrow><mtext>ex</mtext></mrow></msub></mrow><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow></mrow></mstyle></mrow></msqrt></mrow></math> 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.4 . It is worth mention that at θ in =160° we can estimate the following rates from Fig.5(b) and 5(c) Fig.5(b) and 5(c) , <math display="block" id="M10"><mrow><mfrac><mrow><mi>Δ</mi><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>/</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><mrow><mi>Δ</mi><mi>λ</mi><mo>/</mo><mi>λ</mi></mrow></mfrac><mo>≈</mo><mfrac><mrow><mo stretchy="false">(</mo><mo>∂</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>/</mo><mo>∂</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>λ</mi></mrow><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>1</mn><mo>×</mo><mn>800</mn></mrow><mrow><mn>740</mn></mrow></mfrac><mo>≈</mo><mo>−</mo><mn>1</mn><mo>,</mo><mtext> </mtext><mfrac><mrow><mi>Δ</mi><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>/</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow><mrow><mi>Δ</mi><msub><mi>E</mi><mi>e</mi></msub><mo>/</mo><msub><mi>E</mi><mi>e</mi></msub></mrow></mfrac><mo>≈</mo><mfrac><mrow><mo stretchy="false">(</mo><mo>∂</mo><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub><mo>/</mo><mo>∂</mo><msub><mi>E</mi><mi>e</mi></msub><mo stretchy="false">)</mo><msub><mi>E</mi><mi>e</mi></msub></mrow><mrow><msub><mi>E</mi><mrow><mi>max</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>8</mn><mo>×</mo><mn>180</mn></mrow><mrow><mn>740</mn></mrow></mfrac><mo>≈</mo><mn>2.</mn></mrow></math> The two rates are consistent with the equation δE ma /Δ E max ≈ [(2 σ Ee / E e ) 2 +( σ E p / E p ) 2 ] 1/2 derived under head-on ( θ in =180°) LCS geometry [ 36 36 ] . According to design of the SINAPIII, typical values of δE e =0.1%, Δ λ =1 nm and Δ θ in =0.01° are used. The Δ E max and δE max after correction are shown in Fig.5 Fig.5 . Compared with Fig.3 Fig.3 , we can see that when the laser incident angle θ in gets larger, δE e and Δ λ contribute more to Δ E 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 1 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 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 31 , 32 32 , 45 45 - 47 47 45-47 ] (SLEGS) on the storage ring of SSRF [ 48 48 ] . 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Chinese Science Bulletin , 54 ( 22 ), 4171 ( 2009 ). doi: 10.1007/s11434-009-0689-y http://doi.org/10.1007/s11434-009-0689-y Shanghai Synchrotron Radiation Facility The authors would like to thank Professors Yu-Gang MA, Hong-Wei WANG and Xing-Tao WANG for helpful discussions. 10.1007/s41365-017-0272-1 **This work was supported by the National Key Research and Development Program of China (No. 2016YFA0401901) and the National Natural Science Foundation of China (No. 11405427). 2017-03-09 2017-04-05 2017-04-19 2017-08-05 2017-09-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2017 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-017-0272-1 http://dx.doi.org/10.1007/s41365-017-0272-1 . 2017 Nuclear Science and Techniques 09 28