Simulation of γ spectrum-shifting based on the parameter adjustment of Gaussian function space

Simulation of γ spectrum-shifting based on the parameter adjustment of Gaussian function space 1 first-author 2 1 corresp Corresponding author. E-mail address: dingweicheng@cdut.cn Corresponding author. E-mail address: dingweicheng@cdut.cn dingweicheng@cdut.cn 1 1 Applied Nuclear Techniques in Geosciences Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China Applied Nuclear Techniques in Geosciences Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China Drilling Engineering Research institute of southwest Petroleum Engineering Company, SINOPEC, Deyang 618000, China Drilling Engineering Research institute of southwest Petroleum Engineering Company, SINOPEC, Deyang 618000, China Based on the statistical characteristics of energy spectrum and the features of spectrum-shifting in spectrometry, the parameter adjustment method of Gaussian function space was applied in the simulation of spectrum-shifting. The transient characteristics of energy spectrum were described by the Gaussian function space, and then the Gaussian function space was transferred by parameter adjustment method. Furthermore, the spectrum-shifting in measurement of energy spectrum was simulated. The applied example shows that the parameters can be adjusted flexibly by this method to meet the various requirements in simulation of energy spectrum-shifting. This method was one parameterized simulation method with good performance for the practical application. Gaussian function space parameter adjustment simulation of energy spectrum -shifting 1 Introduction 1 Nuclear signal generator and relevant generation method are one of the most important tools in the research of digital nuclear instruments. Using nuclear signal that meets the demand of diversity, flexibility, variability and repeatability can verify the performance of the digital nuclear instrument directly. In the measurement of energy spectrum, the peak position shift problems occur usually. Such as the magnification of photomultiplier tube and its spectral response, the photon yield of scintillator (for the fixed energy rays) changes with temperature, which will cause the output pulse amplitude of detector change and the peak position shift. In addition, the magnification of linear pulse amplifier that changes with temperature will cause peak position shift. Furthermore, the magnification of photomultiplier in detector will be influenced by the changes of count rate and high pressure, which will also cause the peak position shift [ 1 1 , 2 2 ] . This article combines the statistical characteristics of energy spectrum with the features of spectrum-shifting in the radioactivity measurement, and then gives the simulation method of gamma spectrum-shifting based on the parameter adjustment of Gaussian function space. 2 Gaussian function space 1 The Gaussian function space is constructured by multiple Gaussian basis functions [ 3 3 ‒ 7 7 3‒7 ] . Assume a Gaussian function as Φ (t), Φ (t) is given as <math display="block" id="M1"><mrow><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mtext>2π</mtext></mrow></msqrt><mi>σ</mi><mi>λ</mi></mrow></mfrac><mi>exp</mi><mo stretchy="false">[</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mtext>2</mtext><msup><mi>σ</mi><mtext>2</mtext></msup></mrow></mfrac><msup><mi>t</mi><mtext>2</mtext></msup><mo stretchy="false">]</mo></mrow></math> Its integer translation function can be expressed as <math display="block" id="M2"><mrow><msub><mi>φ</mi><mrow><mn>0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mtext>2π</mtext></mrow></msqrt><mi>σ</mi><mi>λ</mi></mrow></mfrac><mi>exp</mi><mo stretchy="false">[</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mtext>2</mtext><msup><mi>σ</mi><mtext>2</mtext></msup></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mtext>2</mtext></msup><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mo>,</mo><mtext> </mtext><mi>k</mi><mo>∈</mo><mi>Z</mi></mrow></math> where <math display="block" id="M3"><mrow><mi>λ</mi><mo>=</mo><msqrt><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mrow><mo>+</mo><mi>∞</mi></mrow></msubsup><mrow><msup><mrow><mo>{</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mtext>2π</mtext></mrow></msqrt><mi>σ</mi></mrow></mfrac><mi>exp</mi><mo stretchy="false">[</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mtext>2</mtext><msup><mi>σ</mi><mtext>2</mtext></msup></mrow></mfrac><msup><mi>t</mi><mtext>2</mtext></msup><mo stretchy="false">]</mo><mo>}</mo></mrow><mtext>2</mtext></msup><mtext>d</mtext><mi>t</mi></mrow></mrow></mstyle></mrow></msqrt></mrow></math> , σ is the standard variance, and k is translation value. Assume the telescopic translation function of Φ ( t ) is Φ jk ( t ), Φ jk ( t ) is given as <math display="block" id="M4"><mrow><msub><mi>φ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mtext>2</mtext><mrow><mfrac><mrow><mo>−</mo><mi>j</mi></mrow><mtext>2</mtext></mfrac></mrow></msup><mi>φ</mi><mo stretchy="false">(</mo><msup><mtext>2</mtext><mrow><mo>−</mo><mi>j</mi></mrow></msup><mi>t</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mtext> </mtext><mi>k</mi><mo>∈</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow></math> And <math display="block" id="M5"><mrow><msub><mi>φ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mtext>2</mtext><mrow><mo>−</mo><mn>0.5</mn><mi>j</mi></mrow></msup><mfrac><mn>1</mn><mrow><msqrt><mrow><mtext>2</mtext><mi>π</mi></mrow></msqrt><mi>σ</mi><mi>λ</mi></mrow></mfrac><mi>exp</mi><mo stretchy="false">[</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mtext>2</mtext><msup><mi>σ</mi><mtext>2</mtext></msup></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><msup><mtext>2</mtext><mrow><mo>−</mo><mi>j</mi></mrow></msup><mi>t</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mtext>2</mtext></msup><mo stretchy="false">]</mo><mtext> </mtext></mrow></math> where j is the telescopic size. The following inner product can be acquired. <math display="block" id="M6"><mtable columnalign="left"><mtr><mtd><mo>〈</mo><msub><mi>ϕ</mi><mrow><mo stretchy="false">(</mo><mi>j</mi><mi>＋</mi><mn>1</mn><mo stretchy="false">)</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>ϕ</mi><mrow><mo stretchy="false">(</mo><mi>j</mi><mi>＋</mi><mn>1</mn><mo stretchy="false">)</mo><msup><mi>k</mi><mo>′</mo></msup></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>〉</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><msup><mn>2</mn><mrow><mfrac><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mi>t</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><msup><mn>2</mn><mrow><mfrac><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mi>t</mi><mo>−</mo><msup><mi>k</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow></mrow></mstyle><mi>d</mi><mi>t</mi></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><msup><mn>2</mn><mrow><mfrac><mrow><mo>−</mo><mi>j</mi></mrow><mn>2</mn></mfrac></mrow></msup><mi>ϕ</mi><mo stretchy="false">(</mo><mfrac><mrow><msup><mn>2</mn><mrow><mo>−</mo><mi>j</mi></mrow></msup></mrow><mn>2</mn></mfrac><mi>t</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><msup><mn>2</mn><mrow><mfrac><mrow><mo>−</mo><mi>j</mi></mrow><mn>2</mn></mfrac></mrow></msup><mi>ϕ</mi><mo stretchy="false">(</mo><mfrac><mrow><msup><mn>2</mn><mrow><mo>−</mo><mi>j</mi></mrow></msup></mrow><mn>2</mn></mfrac><mi>t</mi><mo>−</mo><msup><mi>k</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow></mrow></mstyle><mi>d</mi><mi>t</mi></mtd></mtr><mtr><mtd><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><msup><mn>2</mn><mrow><mfrac><mrow><mo>−</mo><mi>j</mi></mrow><mn>2</mn></mfrac></mrow></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mrow><mo>−</mo><mi>j</mi></mrow></msup><msup><mi>t</mi><mo>′</mo></msup><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><msup><mn>2</mn><mrow><mfrac><mrow><mo>−</mo><mi>j</mi></mrow><mn>2</mn></mfrac></mrow></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mrow><mo>−</mo><mi>j</mi></mrow></msup><msup><mi>t</mi><mo>′</mo></msup><mo>−</mo><msup><mi>k</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow></mrow></mstyle><mi>d</mi><msup><mi>t</mi><mo>′</mo></msup><mtable><mtr><mtd><mrow/></mtd><mtd><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mo>′</mo></msup><mo>=</mo><mfrac><mi>t</mi><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><msub><mi>ϕ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><msub><mi>ϕ</mi><mrow><mi>j</mi><msup><mi>k</mi><mo>′</mo></msup></mrow></msub><mo stretchy="false">(</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow></mrow></mstyle><mi>d</mi><msup><mi>t</mi><mo>′</mo></msup><mo>=</mo><mo>〈</mo><msub><mi>ϕ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>ϕ</mi><mrow><mi>j</mi><msup><mi>k</mi><mo>′</mo></msup></mrow></msub><mo stretchy="false">(</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo>〉</mo><mn>…</mn></mtd></mtr></mtable></math> When σ takes the minimum value, < Φ jk ( t ), Φ jk ′ ( t ) > ( k ≠ k ′) is near zero, for example, when σ =0.15, the consequence of < Φ 0 k ( t ), Φ 0 k ′ ( t ) > ( k ≠ k ′) is the order of magnitude of 10 –5 , when σ =0.1, the consequence of < Φ 0 k ( t ), Φ 0 k ′ ( t ) > ( k ≠ k ′) is the order of magnitude of 10 –11 , which can be regarded as zero. In addition, it is easy to get that < Φ 0 k ( t ), Φ 0 k ( t ) > =1 from formula (2). The following relationship can be established by formula (5). <math display="block" id="M7"><mrow><mo><</mo><msub><mi>φ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mi>φ</mi><mrow><mi>j</mi><mi>k</mi><mo>'</mo></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>k</mi><mo>'</mo><mo stretchy="false">)</mo></mrow></math> where δ ( k ‒ k ′) = 0 ( k ≠ k ′), δ ( k - k ′) = 1( k = k ′). So, it can be considered that when σ take the minimum value, Φ jk ( t ) has the orthogonality relationship . The space V j = span { Φ jk ( t )}( j , k ∈ Z ), which formed from Φ jk ( t ), can be regarded as the Gaussian function space and then Φ jk ( t ) can be regard as the orthogonal basis in space V j . Using P j f ( t ) to present the projection of f ( t ) in the Gaussian function space V j : <math display="block" id="M8"><mrow><msub><mi>p</mi><mi>j</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mo>−</mo><mi>∞</mi></mrow><mrow><mo>+</mo><mi>∞</mi></mrow></munderover><mrow><msub><mi>c</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>φ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></math> If f ( t )∈ V j , signal f ( t ) can be expressed further with basic functions in space V j as follow: <math display="block" id="M9"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mi>j</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mo>−</mo><mi>∞</mi></mrow><mrow><mo>+</mo><mi>∞</mi></mrow></munderover><mrow><msub><mi>c</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>φ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></math> where c jk is the weight of linear combination, and is given by <math display="block" id="M10"><mrow><msub><mi>c</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><mo><</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>ϕ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo></mrow></math> In radioactivity measurement, energy spectrum signal f ( n ) can be regarded as the discrete form of some function like f ( t )∈ V 0 . 3 The expression of the γ energy spectrum by Gaussian function space 1 Multiple small Gaussian functions can be used to get the good approximation and simulation of radioactive energy spectrum [ 8 8 , 9 9 ] , and the multi-scale analysis can extract characteristics of signal effectively [ 10 10 , 11 11 ] . The short transient characteristics of energy spectrum can be described by using Gaussian function space, and the parameter adjustment for the state transition of Gaussian function space combined with statistical fluctuation characteristics of energy spectrum can be employed to simulate the spectrum-shifting in energy spectrum measurement. In fact, the projection of energy spectrum on the function space V j made up of Gaussian functions is expressed linearly by multiple Gaussian functions, and each Gaussian function can affect the count of all channels by its weight, namely the count of any channel will be affected by adjacent channels, moreover, the more adjacent channel has, the greater effect it will cause. This method is accord with the statistical characteristics of actual spectrum. Take the example of γ energy spectra, the representation of Gaussian function space can be expressed in details as below. Assume the original energy spectrum is f ( n ) ( n =1… N ), where N is the total number of channels, and the gross count is N total . Using Gaussian function to show it in following steps: Regard the original spectrum f ( n ) as a continuous function f ( t ), and choose a Gaussian function space V j , which is chosen as j ≥0 usually, then the projection of f ( t ) in the V j is given by <math display="block" id="M11"><mrow><msub><mi>p</mi><mi>j</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><msup><mtext>2</mtext><mi>j</mi></msup></mrow></munderover><mrow><msub><mi>c</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>φ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></math> According to formula (9), c jk = < f ( t ), Φ jk ( t ) > , for convenience, it can be expressed as <math display="block" id="M12"><mtable columnalign="left"><mtr><mtd><msub><mi>c</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><mo><</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>φ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>≈</mo><mi>f</mi><mo stretchy="false">(</mo><msup><mtext>2</mtext><mi>j</mi></msup><mi>t</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mrow><mo>+</mo><mi>∞</mi></mrow></msubsup><mrow><msub><mi>φ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>t</mi></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mtext>2</mtext><mrow><mfrac><mi>j</mi><mtext>2</mtext></mfrac></mrow></msup><mi>f</mi><mo stretchy="false">(</mo><msup><mtext>2</mtext><mi>j</mi></msup><mi>k</mi><mo stretchy="false">)</mo><mo>/</mo><mi>λ</mi></mtd></mtr></mtable></math> Parameter λ is the same as the λ in formula (2), f (2 j k ) can take the average of local area where t =2 j k. In the actual calculation, Φ jk ( t ) can take several or dozens of discrete values at local area where t =2 j k , which will reduce the calculation amount of formula (10) greatly. The final energy spectrum f′ ( n ) can be obtained according to the rounding and discretization of following formula. <math display="block" id="M13"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>N</mi><mrow><mtext>total</mtext></mrow></msub><msub><mi>P</mi><mi>j</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>/</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow><msub><mi>P</mi><mi>j</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mtext>1</mtext><mo>⋅</mo></mrow></mstyle><mo>⋅</mo><mo>⋅</mo><mi>N</mi><mo stretchy="false">)</mo></mrow></math> Formula(12) can be regarded as the revised formula for the nonorthogonality of Gaussian function space, f′ ( n ) according to formula (12) is exactly the space expression of energy spectrum. 4 Simulation of γ spectrum-shift within the Gaussian function space 1 The original energy spectrum can be decomposed to the combination of uniform distribution, Gaussian distribution, exponential distribution and polynomial distribution, which can be used to the effective simulation of nuclear energy spectrum [ 6 6 ] . Primitive preceding analysis shows that the peak position shift often happens in the energy spectrum measurement. Therefore, in order to solve the simulation problem of peak position shift, the further research is very necessary. If the shift of the energy spectrum can be simulated by effective methods, undoubtedly, it is of great significance to the study of energy spectrum. Peak position shift may be caused by many factors, but the generally available model is expressed as in Fig.1 Fig.1 . The gamma ray excitation signal is determined by the radioactive nuclide. Observable parameters of the Gaussian function space are very sensitive to the external environment and test conditions, such as the temperature, the counting rate, the change of high voltage and other factors. The principle of Gaussian function space to simulate the gamma energy spectrum shift is: using a Gaussian function space to describe the instantaneous or short time gamma spectrum; Gaussian function space parameters (e.g., mean, standard variance) will be adjusted over time, which has the adjustment pattern made of requirements to simulate the external environment and test conditions, such as the changing temperature, the diverse counting rate, the changing high pressure and other factors. Finally, the random numbers will be generated to simulate energy spectrum shift and statistical fluctuation process. The generation of random numbers can take random sampling method of the Gaussian mixture function or discrete direct sampling method. 4.1 The random sampling of Gaussian mixture function. 2 The random sampling of Gaussian mixture function was realized by the additive sampling method [ 12 12 , 13 13 ] . first of all, normalize the energy spectrum in Gaussian function space into the form of following formula <math display="block" id="M14"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mtext>1</mtext></mrow><mi>M</mi></munderover><mrow><msub><mi>P</mi><mi>n</mi></msub><msub><mi>f</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></math> where, P n ≥ 0, ∑ P n = 1( n = 1… M ), f n ( x ) is a Gaussian density function which is related to parameter n ( n = 1,2,… M ), and M is the number of Gaussian density functions. Secondly, determine n by random sampling to get the random number ( x ) by random sampling according to function f n ( x ) . 4.2 Discrete direct sampling method 2 Firstly, normalize the energy spectrum in Gaussian function space into a density function ρ ( x ), the discrete distribution function of F ( x ) is given as <math display="block" id="M15"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><msub><mi>x</mi><mi>i</mi></msub><mo><</mo><mi>x</mi></mrow><mrow/></munderover><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></math> where x i is the discrete point of density function ρ ( x ), that is also the serial number of the channel of energy spectrum, ρ ( x i ) is the corresponding probability, and ∑ ρ ( x i )=1 ( I =1… N ). With the random number x calculated according to the sampling of (15), simulation of F ( x ) distribution on energy spectrum can be obtained, ε is the random number of uniform distribution within [0, 1] [ 14 14 , 15 15 ] . <math display="block" id="M16"><mrow><msub><mi>x</mi><mi>F</mi></msub><mo>=</mo><msub><mi>x</mi><mi>I</mi></msub><mo>,</mo><mtext> </mtext><mtext> </mtext><mi>w</mi><mi>h</mi><mi>e</mi><mi>n</mi><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>I</mi><mo>−</mo><mn>1</mn></mrow></munderover><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mstyle><mo>≤</mo><mi>ε</mi><mo><</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>I</mi></munderover><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mrow></math> where x I is just the random number x F of distribution function F ( x ). 4.3 Example analysis 2 With the model shown in Fig.1 Fig.1 , the shift of 40 K γ spectrum shown by S 1 (.) in Fig.2 Fig.2 can be simulated. The γ spectrum is measured by the 1024-channel NaI (Tl) scintillation spectrometer, with the gross count of 1.9247×10 5 , namely N total =1.9247×10 5 . The selected Gaussian function space is V 3 , which has 128 Gaussian functions, and has the standard variance σ ( σ =1).The 40 K γ spectrum curve in the Gaussian function space is shown by S 2 (-) in Fig.2 Fig.2 . For simplicity, only the steady spectrum shift to the right (that is, the energy spectrum shift at a constant speed and in the direction of high energy) was simulated, the simulation method of shift to left is similar. Actually, the transfer process of Gaussian function space can be designed as linear or non-linear mode to simulate the complex spectrum shift by the influence of temperature and humidity. Assumed the number of shift channels is 40, then the continuous shift process can be dispersed into 80 Gaussian function spaces, namely, the interval between the two adjacent Gaussian function spaces is 0.5 channel. In fact, interval can get 0.1, 0.2, 0.3, ..., 1, 1.1, 1.2, etc., the smaller the interval is, the more accurate the shift process of energy spectrum can achieve, which is also the main advantage of this method. In addition, because the channel is an integer, so it must get the integer energy spectrum after rounding operation on the sampled channels of Gaussian function space finally. Figure 3 Figure 3 shows the shift process of the entire energy spectrum. The dashed line(--) represents each entire energy spectrum in different time in the spectrum measurement, shifting from left to right, the shorter the time is, the lower the curve is. The solid line(-) represents the entire energy spectrum after shifting for 40 channels, with a gross count of 1.9247×10 5 . Figure 4 Figure 4 shows the ultimate energy spectrum without shifting, and one with shifting. Figure 5 Figure 5 shows the statistical fluctuation process by generating random numbers to simulate energy spectrum shift, which has peak position shift to the right way for 40 channels. From this example, it is clear that the Gaussian function space can be used to simulate the gamma energy spectrum shift as a state transition method conveniently, realistically and visually. In addition, the model can be easily used to simulate the fluctuation process of energy spectrum when measuring environment or condition, such as detector resolution, external temperature and humidity, changes. Actually, the state transition of Gaussian function space is completed by parameter adjustment of the Gaussian function, such as mean value adjustment and standard variance adjustment. This article only gives a simulation instance of spectrum shift in the direction of high energy. The simulation method of spectrum shift in the direction of low energy is similar. The parameters can be flexibly adjusted to satisfy the requirements of the diversity for the simulation of spectrum shift. For example, the transition state can be designed as linear or non-linear function to simulate the spectrum shift in a complex environment, the number of Gaussian function spaces can be increased to simulate the spectrum shift with slow speed, and the parameter j ( j = 0, 1, 2, 3...) in formula (3) can be used to increase or decrease the dimension of Gaussian function space in order to improve the accuracy of the energy spectrum, or improve the ability of inhibiting statistical fluctuation and increase computing speed. Comparison of theoretical spectrum peak and sampling spectrum peak after shifting is showed in Table 1 Table 1 and Fig.6 Fig.6 . Two curves in Fig.6 Fig.6 almost coincide with each other, and most of errors in these channels are 0.07%‒2%, as showed in Table 1 Table 1 , indicating that this method is practicable. Where, the gross count is increased to N total =1.9247×10 6 in order to improve the simulating accuracy of the energy spectrum. In the energy spectrum measurement, γ-ray spectrum-shifting occurs usually when the conditions change, such as the high voltage, temperature, count rate, etc, and the forms of shift are very different. However, with our proposed method, random nuclear signals, which meet the demand of diversity, flexibility, variability and repeatability under complex conditions, can be easily generated to simulate the spectrum -shifting, because these conditions can be described collectively, not respectively, by the state transition of Gaussian function space. In the course of studying on the digital nuclear instrument and algorithm, the spectrum- shifting is usually considered under all conditions, not one condition. In fact, it is not necessary and impractical to measure the spectrum-shifting under a special condition. Therefore, random nuclear signals generated by this method can meet the demand of spectrum-shifting processing in the studying on the digital nuclear instrument and algorithm, and this method can improve the performance of instrument. 5 Conclusion 1 This article proposed an approach that combines the statistical characteristics of energy spectrum with the features of spectrum-shifting in the radioactivity measurement, and gives the simulation method of gamma spectrum-shifting based on the parameter adjustment of Gaussian function space. The spectrum- shifting in energy spectrum measurement can be simulated by the description of short transient characteristics of energy spectrum in Gaussian function space, and the parameter adjustment for the state transition of Gaussian function space. The instances show that, by this method, the parameters can be flexibly adjusted to satisfy the requirements of the diversity for the simulation of spectrum shift. The parameterized simulation method has a good performance in actual application. In Table 1, Ch: channel; Theo-val: theoretical value of spectrum after shifting; Samp-val: sampling value of spectrum after shifting; Error (%): relative error, [(Samp-val)-(Theo-val)]/(Theo-val). 1 1 Ch 1 1 Theoval 1 1 Sampval 1 1 Error(%) 1 1 Ch 1 1 Theoval 1 1 Sampval 1 1 Error(%) 1 1 Ch 1 1 Theoval 1 1 Sampval 1 1 Error(%) 1 1 Ch 1 1 Theoval 1 1 Sampval 1 1 Error(%) 1 1 Ch 1 1 Theoval 1 1 Sampval 1 1 Error(%) 1 1 400 1 1 1445 1 1 1464 1 1 1.3 1 1 420 1 1 1920 1 1 2021 1 1 5.26 1 1 440 1 1 2707 1 1 2789 1 1 3.03 1 1 460 1 1 2488 1 1 2579 1 1 3.66 1 1 480 1 1 1283 1 1 1287 1 1 0.31 1 1 401 1 1 1450 1 1 1511 1 1 4.2 1 1 421 1 1 1961 1 1 1949 1 1 –0.61 1 1 441 1 1 2730 1 1 2752 1 1 0.81 1 1 461 1 1 2442 1 1 2463 1 1 0.86 1 1 481 1 1 1223 1 1 1245 1 1 1.8 1 1 402 1 1 1456 1 1 1508 1 1 3.6 1 1 422 1 1 2004 1 1 2067 1 1 3.14 1 1 442 1 1 2749 1 1 2695 1 1 –1.96 1 1 462 1 1 2392 1 1 2367 1 1 –1.05 1 1 482 1 1 1164 1 1 1185 1 1 1.8 1 1 403 1 1 1465 1 1 1469 1 1 0.27 1 1 423 1 1 2047 1 1 2077 1 1 1.47 1 1 443 1 1 2765 1 1 2849 1 1 3.04 1 1 463 1 1 2340 1 1 2330 1 1 –0.42 1 1 483 1 1 1107 1 1 1099 1 1 –0.72 1 1 404 1 1 1476 1 1 1517 1 1 2.8 1 1 424 1 1 2091 1 1 2107 1 1 0.77 1 1 444 1 1 2778 1 1 2897 1 1 4.28 1 1 464 1 1 2286 1 1 2352 1 1 2.89 1 1 484 1 1 1051 1 1 1059 1 1 0.76 1 1 405 1 1 1489 1 1 1510 1 1 1.4 1 1 425 1 1 2136 1 1 2115 1 1 –0.98 1 1 445 1 1 2787 1 1 2746 1 1 –1.47 1 1 465 1 1 2229 1 1 2178 1 1 –2.29 1 1 485 1 1 997 1 1 972 1 1 –2.51 1 1 406 1 1 1504 1 1 1467 1 1 –2.5 1 1 426 1 1 2180 1 1 2215 1 1 1.6 1 1 446 1 1 2793 1 1 2911 1 1 4.22 1 1 466 1 1 2171 1 1 2209 1 1 1.75 1 1 486 1 1 945 1 1 955 1 1 1.06 1 1 407 1 1 1521 1 1 1658 1 1 9 1 1 427 1 1 2224 1 1 2234 1 1 0.45 1 1 447 1 1 2795 1 1 2688 1 1 –3.83 1 1 467 1 1 2111 1 1 2150 1 1 1.85 1 1 487 1 1 895 1 1 886 1 1 –1.01 1 1 408 1 1 1540 1 1 1601 1 1 4 1 1 428 1 1 2269 1 1 2186 1 1 –3.66 1 1 448 1 1 2793 1 1 2735 1 1 –2.08 1 1 468 1 1 2050 1 1 2003 1 1 –2.29 1 1 488 1 1 847 1 1 833 1 1 –1.65 1 1 409 1 1 1561 1 1 1587 1 1 1.7 1 1 429 1 1 2313 1 1 2364 1 1 2.2 1 1 449 1 1 2788 1 1 2727 1 1 –2.19 1 1 469 1 1 1987 1 1 1969 1 1 –0.91 1 1 489 1 1 800 1 1 798 1 1 –0.25 1 1 410 1 1 1584 1 1 1693 1 1 6.9 1 1 430 1 1 2356 1 1 2368 1 1 0.51 1 1 450 1 1 2779 1 1 2794 1 1 0.54 1 1 470 1 1 1924 1 1 1940 1 1 0.83 1 1 490 1 1 756 1 1 764 1 1 1.06 1 1 411 1 1 1610 1 1 1620 1 1 0.62 1 1 431 1 1 2398 1 1 2351 1 1 –1.96 1 1 451 1 1 2766 1 1 2834 1 1 2.46 1 1 471 1 1 1859 1 1 1882 1 1 1.24 1 1 491 1 1 714 1 1 694 1 1 –2.80 1 1 412 1 1 1637 1 1 1677 1 1 2.4 1 1 432 1 1 2440 1 1 2485 1 1 1.84 1 1 452 1 1 2750 1 1 2830 1 1 2.91 1 1 472 1 1 1795 1 1 1756 1 1 –2.18 1 1 492 1 1 674 1 1 726 1 1 7.72 1 1 413 1 1 1666 1 1 1647 1 1 –1.14 1 1 433 1 1 2480 1 1 2551 1 1 2.87 1 1 453 1 1 2729 1 1 2687 1 1 –1.54 1 1 473 1 1 1730 1 1 1714 1 1 –0.92 1 1 493 1 1 637 1 1 573 1 1 –10.05 1 1 414 1 1 1698 1 1 1725 1 1 1.59 1 1 434 1 1 2519 1 1 2505 1 1 –0.56 1 1 454 1 1 2705 1 1 2721 1 1 0.59 1 1 474 1 1 1664 1 1 1655 1 1 –0.54 1 1 494 1 1 601 1 1 576 1 1 –4.16 1 1 415 1 1 1731 1 1 1765 1 1 1.96 1 1 435 1 1 2556 1 1 2524 1 1 1.25 1 1 455 1 1 2677 1 1 2587 1 1 –3.36 1 1 475 1 1 1599 1 1 1604 1 1 0.31 1 1 495 1 1 567 1 1 553 1 1 –2.47 1 1 416 1 1 1766 1 1 1769 1 1 0.17 1 1 436 1 1 2591 1 1 2605 1 1 0.54 1 1 456 1 1 2646 1 1 2600 1 1 –1.74 1 1 476 1 1 1535 1 1 1503 1 1 –2.08 1 1 496 1 1 536 1 1 515 1 1 –3.92 1 1 417 1 1 1802 1 1 1840 1 1 2.1 1 1 437 1 1 2623 1 1 2671 1 1 1.83 1 1 457 1 1 2612 1 1 2672 1 1 2.3 1 1 477 1 1 1471 1 1 1422 1 1 –3.33 1 1 497 1 1 506 1 1 541 1 1 6.92 1 1 418 1 1 1840 1 1 1904 1 1 3.48 1 1 438 1 1 2654 1 1 2583 1 1 –2.68 1 1 458 1 1 2574 1 1 2542 1 1 –1.24 1 1 478 1 1 1407 1 1 1446 1 1 2.77 1 1 498 1 1 478 1 1 498 1 1 4.18 1 1 419 1 1 1879 1 1 1962 1 1 4.42 1 1 439 1 1 2682 1 1 2621 1 1 –2.27 1 1 459 1 1 2533 1 1 2559 1 1 1.03 1 1 479 1 1 1345 1 1 1346 1 1 0.07 1 1 499 1 1 453 1 1 432 1 1 –4.64 References Wang Z R . 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Computer Random Simulation 10.13538/j.1001-8042/nst.2013.06.009 Supported by National Natural Science Foundation of China (41204133) , Scientific Reserch Fund of Sichuan Provincial Education Department (13ZA0066) , Cultivating programme of excellent innovation team of Chengdu University of technology (KYTD201301) 2013-04-20 2013-12-01 2021-04-11 2013 Nuclear Science and Techniques 6 24