A&C_task 1.4

Albert Cheung
  2024-09-10 15:03:23 2024-09-10 15:03:23 Created   2024-09-18 18:28:57 2024-09-18 18:28:57 Updated      8.4k Words  44 Mins

Tasks

共享文件中所有A_B格式的TH1D直方图(A:0-191; B:0-5)为1152个320微米厚硅微条探测器对宇宙线粒子的响应(可以认为相同的输入)，选用合适的函数拟合，并计算不同探测器的增益系数(使不同探测器对相同输入实现相同的输出幅度)

分享文件:9.09科创[文件夹]
云盘链接:https://pan.cstcloud.cn/s/bv3DxWHbQVE
密码:he3y 过期时间:2025-09-30 19:43:08

One Possible Solution

首先弥补之前内容的缺失，在这里简单补充硅微条探测器（Silicon Strip Detector）的知识。这里是硅微条探测器的示意图。

img1

由图可知，硅微条探测器在N型硅的表面上制作了高掺杂的P+微条，于是在每个微条与N型硅基底形成了一个PN结；另外，它在微挑上附加了铝条作为电极（阴极）。在硅条之间使用了SiO_2进行了隔离，底部的背面还有高掺杂的N+层附上铝电极作为阳极。

当带电粒子穿过耗尽层时，会产生电子-空穴对，分别向正极与阴极移动，并且和其它电离传感器类似，在电极上感应出电压信号。由于我们在铝电极上施加了反向电压，PN结形成的耗尽层在外电场的作用下随偏压的增加而进一步变厚；最终，耗尽层由于外电场的作用几乎扩展到整个N型硅材料，导致死区被挤压得特别小。由于耗尽层具有高电阻率，漏电流相对很小，导致几乎所有的电压都被施加在耗尽层上，大大提高了耗尽层内的电场强度，进一步导致耗尽层成为灵敏区。另外，由于载流子迁移率较高，且灵敏区较薄（如常约300μm），信号感应时间较短（约5ns）。

容易看出，感应信号通过微条上的电极将电信号导出并传输到电子学系统，并且微条的编号可以相对准确地反映入射粒子的位置或者重现它们的轨迹。类似地，如果采用P型硅作为基体，并将P+和N+掺杂位置互换就可以得到P型硅微条探测器。

因此，感应信号的分辨其实由硅条的间距决定。事实上，它满足：

需要注意的是，这里提到的硅微条探测器实际上用的是直流电；对于交流电的改进版本，我们使用交流硅微条探测器。交流硅微条探测器的原理和滞留硅微条探测器类似，都是基于PN结形成耗尽层进行电荷收集。不过，在交流耦合探测器中，微条电极和读出电子学系统之间使用了电容进行隔离，形成“交流耦合”。因此，它的工作机制事实上依赖于感应信号在电极和读出系统之间通过电容耦合进行传输，而非普通的硅微条探测器的直接连接。

二者的结构有一些区别。直流硅微条探测器事实上几乎没有或只有非常薄的绝缘层，而交流耦合硅微条探测器使用使用单层的SiO2或其他材料构成的绝缘层将电极与硅基体隔离起来。具体说来，交流硅微条探测器具有以下结构特点：

  • 电容耦合电极和读出系统
  • 在电极和基体之间有绝缘材料（例如SiO2），使感应信号通过交流耦合电容传输

因此，交流硅微条探测器具有和直流相比的优势。在直流中，信号通过直流电流直接被传递；而在交流硅微条探测器中，耦合电容传递了交流信号，使直流噪声被去除。在直流中，虽然漏电流很少，但是偶尔还是需要进行特殊处理；在交流中，漏电流被电容隔离，因此不会产生电子学噪声。因此，直流硅微条探测器相对来说应用在较简单的情况中，此时我们对信号纯度的要求并不高；而交流硅微条探测器能够有效屏蔽部分噪声，适合对低噪声与高纯度有要求的系统，特别是在高粒子流环境下表现相对较好。所以，直流探测器成本较低，适用于对噪声不特别敏感的例如中低能粒子探测器中；交流耦合探测器更适用应用于高能物理实验，如粒子对撞机、宇宙射线探测等，尤其在高技术率的情况下表现更卓越。

不过需要注意的是，直流硅微条探测器的电荷收集效果相对较好，虽然有时存在电荷积累影响系统产生的测量结果；相比而言交流硅微条探测器的电荷收集效果就较差。

另外，需要简单介绍一下题目中需要我们计算的”增益系数“。增益系数（Gain Coefficient）事实上是粒子探测器中用来描述探测器对输入信号放大能力的一个参量。具体来说，增益系数通常表示为探测器输入信号与输出信号之间的比值；在这里，给定相同的输入，可以通过拟合探测器输出的响应分布计算出探测器的增益系数。

在附件va.root中，我们有1152个TH1D类型的直方图，分别对应320微米厚硅微条探测器对粒子的响应。其中，横坐标代表输出信号，纵坐标代表事件的频率或概率。在代码中，我们采用了郎道卷积高斯函数进行拟合（参考cern.root上的教程代码进行卷积计算），其中有四个参数：

  • par[0]: 郎道分布的宽度参数
  • par[1]: 郎道分布的最可能值
  • par[2]: 卷积的总面积
  • par[3]: 高斯分布的宽度

在拟合直方图后，我们可以提取出郎道分布中的最可能值作为增益系数；因为它其实是输出响应的最可能值，能够代表探测器在接受到相同输入时，输出信号的最大值位置，能够代表每个探测器的增益系数的大小。

另外，如果希望得到更确切的MPV值，可以选择计算郎道卷积高斯分布最大值的横坐标作为MPV。在这里，我们选择这一种做法。

以下是代码：

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// 有一个需要注意的点：在ROOT中，直方图默认与当前目录关联（例如TFile）。当关闭TFile时，与其关联的所有直方图都会被删除，如果之后需要调用的话就会产生无效的内存访问。这可能是我在运行中偶尔出现 crash 的原因。

gSystem -> Load("libMathCore"); // 似乎有时会产生崩溃，并且提示可能问题出在 libMathCore 上。因此添加这行代码防止崩溃（虽然可能是我电脑的问题）。

// 这部分是根据指导网页 https://root.cern/doc/master/langaus_8C.html 中的教程，定义朗道卷积高斯函数。**好像有直接调用的方法？**
double langaufun(double *x, double *par) {

   // Fit parameters:
   // par[0]=Width (scale) parameter of Landau density
   // par[1]=Most Probable (MP, location) parameter of Landau density
   // par[2]=Total area (integral -inf to inf, normalization constant)
   // par[3]=Width (sigma) of convoluted Gaussian function

   // Numeric constants
   double invsq2pi = 0.3989422804014;   // (2 pi)^(-1/2)
   double mpshift  = -0.22278298;       // Landau maximum location 应该是Landau分布峰值位置的修正

   // Control constants
   double np = 100.0;      // number of convolution steps
   double sc = 5.0;      // convolution extends to +-sc Gaussian sigmas

   // Variables
    double xx;
    double mpc;
    double fland;
    double sum = 0.0;
    double xlow,xupp;
    double step;
    double i;

   // MP shift correction
   mpc = par[1] - mpshift * par[0];

   // Range of convolution integral
   xlow = x[0] - sc * par[3];
   xupp = x[0] + sc * par[3];

    step = (xupp-xlow) / np;

   // Convolution integral of Landau and Gaussian by sum
    for(i=1.0; i<=np/2; i++) {
      xx = xlow + (i-.5) * step;
    fland = TMath::Landau(xx,mpc,par[0]) / par[0];
      sum += fland * TMath::Gaus(x[0],xx,par[3]);

      xx = xupp - (i-.5) * step;
    fland = TMath::Landau(xx,mpc,par[0]) / par[0];
      sum += fland * TMath::Gaus(x[0],xx,par[3]);
    }

   return (par[2] * step * sum * invsq2pi / par[3]);
}


void task1_4() {
    TVirtualFitter::SetDefaultFitter("Minuit"); // **在上一次程序运行中发现无法使用 Minuit 改用 Minuit2，这一次发现无法使用 Minuit2 改用 Minuit。由于每个A_B都要提示一次（1152次），于是预先在这里声明。但是为什么？**
    TFile *file = TFile::Open("va.root", "READ");
    TFile *outFile = new TFile("fit_results.root","RECREATE");

    std::ofstream outputFile("chi2Ndf_and_gainCoefficients.txt");

    TH2D *gainMap = new TH2D("gainMap", "GainCoefficients; A; B", 192, 0, 192, 6, 0, 6);
    TH2D *chi2Map = new TH2D("chi2Map", "ChiSquared/NDF; A; B", 192, 0, 192, 6, 0, 6);

    double totalMPV = 0; // 用于累加MPV
    int numDetectors = 0; // 探测器数量

    std::vector<double> mpvList; // 每个探测器的MPV
    std::vector<TH1D*> correctedHists; // 输出的（校正后的）值方图
    std::vector<TH1D*> originalHists; // 由于出现了程序崩溃，存储一个原始直方图。应该是程序开头的注释中提到的问题

    for (int A=0; A < 192; ++A){
        for (int B=0; B < 6; ++B){
            TString histName = TString::Format("%d_%d", A, B);
            TH1D *hist = (TH1D*)file -> Get(histName);

            TH1D *histClone = (TH1D*)hist -> Clone();
            histClone -> SetDirectory(0); // 克隆后解除关联（也是为了防止读取空内存）
            histClone -> Sumw2(); // 计算误差，方便后续误差条的显示（虽然并不明显？）
            originalHists.push_back((TH1D*)hist -> Clone()); // 再次存储一个直方图

            TF1 *landauGausFit = new TF1("landauGausFit", langaufun, 35, 150, 4); // 选择35到150的原因在文章中有说明
            landauGausFit -> SetParameters(1.8, 50, 50000, 3);
            landauGausFit -> SetParNames("Width", "MP", "Area", "Sigma");

            landauGausFit -> SetNpx(1000); // 由于初步拟合时发现峰值附近的拟合函数看起来太尖锐，手动增加绘制精度 

            hist -> Fit(landauGausFit, "Q");

            double gainCoefficient = landauGausFit -> GetMaximumX(35, 150);

            double chi2 = landauGausFit -> GetChisquare();
            int ndf = landauGausFit -> GetNDF();
            double chi2NDF = chi2 / ndf;

            gainMap -> SetBinContent(A + 1, B + 1, gainCoefficient);
            chi2Map -> SetBinContent(A + 1, B + 1, chi2NDF);

            outputFile << A << "_" << B << " " << chi2NDF << " " << gainCoefficient << "\n";

            mpvList.push_back(gainCoefficient);
            totalMPV += gainCoefficient;
            numDetectors ++;

            hist -> Write();

        }
    }

    double targetMPV = totalMPV / numDetectors; // 计算平均值用来对齐

    for (int A = 0; A < 192; ++A){
        for (int B = 0; B < 6; ++B){

            TH1D *hist = originalHists[A * 6 + B];
            double gainCoefficient = mpvList[A * 6 + B];
            double correctionFactor = targetMPV / gainCoefficient; // 定义一个修正（对齐）因子

            TH1D * correctedHist = (TH1D*)hist -> Clone(TString::Format("corrected_%d_%d", A, B));
            correctedHist -> SetDirectory(0); // 也是解除关联
            correctedHist -> Sumw2();
            correctedHist -> Scale(correctionFactor);
            correctedHists.push_back(correctedHist);

            correctedHist -> Write();
        }
    }

    // 这是一个用于合并的直方图
    TH1D *mergedHist = (TH1D*)correctedHists[0] -> Clone("merged_hist");
    mergedHist -> Reset();
    mergedHist -> SetDirectory(0); // 也是解除关联
    mergedHist -> Sumw2(); // 计算误差

    for (size_t i = 0; i < correctedHists.size(); ++i){
        mergedHist -> Add(correctedHists[i]);
    }

    TF1 *mergedFit = new TF1("mergedFit", langaufun, 35, 150, 4);
    mergedFit -> SetParameters(1.8, 50, 50000, 3);
    mergedFit -> SetParNames("Width", "MP", "Area", "Sigma");
    mergedFit -> SetNpx(1000);
    mergedHist -> Fit(mergedFit, "Q");

    mergedFit -> Write();
    mergedHist -> Write();
    gainMap -> Write();
    chi2Map -> Write();
    
    file -> Close();
    outFile -> Close();
    outputFile.close();

    
    TFile *inFile = TFile::Open("fit_results.root", "READ");

    TH2D *gainHist = (TH2D*)inFile -> Get("gainMap"); // 增益系数热力图
    TH2D *chi2Hist = (TH2D*)inFile -> Get("chi2Map"); // 卡方分布热力图
    TH1D *mergedHistFromFile = (TH1D*)inFile -> Get("merged_hist"); // 合并后的直方图
    TF1 *mergedFitFromFile = (TF1*)inFile -> Get("mergedFit"); // 拟合后的合并后的直方图

    TCanvas *c1 = new TCanvas("c1", "Gain Coefficient Heatmap", 800, 600);
    gStyle -> SetOptStat(0);
    gainHist -> Draw("COLZ");

    c1 -> SaveAs("gain_heatmap.pdf");

    TCanvas *c2 = new TCanvas("c2", "ChiSquared/NDF Heatmap", 800, 600);
    gStyle -> SetOptStat(0);
    chi2Hist -> Draw("COLZ");

    c2 -> SaveAs("chi2Ndf_heatmap.pdf");

    TCanvas *c3 = new TCanvas("c3", "Merged Spectrum", 800, 600);
    mergedHistFromFile -> Draw("HIST");
    
    // 这里的 merged_spectrum 实际上是对每个原始直方图进行郎道卷积高斯函数拟合，得到MPV，计算平均MPV后以 correctionFactor = targetMPV / gainCoefficient 作为校正因子对齐得到的合并的直方图
    c3 -> SaveAs("merged_spectrum.pdf");

    TCanvas *c4 = new TCanvas("c4", "Fitted Merged Spectrum", 800, 600);
    gStyle -> SetOptFit(1111); 
    mergedHistFromFile -> Draw("E1");
    mergedFitFromFile -> SetLineColor(kRed);
    mergedFitFromFile -> Draw("SAME");

    gPad -> Update();
    TPaveStats *stats = (TPaveStats*)mergedHistFromFile -> FindObject("stats");
    if (stats){
    stats -> SetX1NDC(0.65);
    stats -> SetY1NDC(0.65);
    stats -> SetX2NDC(0.90);
    stats -> SetY2NDC(0.90);
    stats -> SetBorderSize(1);
    }

    c4 -> SaveAs("fitted_merged_spectrum.pdf");

    inFile -> Close(0);
    c1 -> Close(0);
    c2 -> Close(0);
    c3 -> Close(0);
    c4 -> Close(0);
}

以下是输出的图片，分别为

  • 增益系数热力图，用于粗略地检查增益系数的分布与区间，对增益系数总体上的分布情况有初步的概念
    pic4
  • 卡方热力图，是使用郎道卷积高斯函数拟合后得到的结果，用于粗略地估计拟合的情况及优劣
    pic5
  • 对齐合并后的数据，使用郎道卷积高斯函数在区间（50,135）范围内对原始数据进行拟合，分别得到MPV；然后，计算平均MPV后以 correctionFactor = targetMPV / gainCoefficient 作为校正因子对齐得到的合并的直方图
    pic6
  • 使用郎道卷积高斯函数对对齐合并后的数据拟合，得到结果
    pic7

首先解释一下选择拟合区间为（35,150）的原因。首先打开所给的数据文件，对数据有一个直观的理解。在初步随机寻找中我们发现在数据30_4，40_0，173_3等文件中，大约在20和240的位置分别有一个明显的峰值。为了准确提取关键数据或者得到更精确的拟合结果，我们选择从35到150进行拟合。

pic1

pic2

pic3

总体上，拟合结果依然存在许多问题。首先还是上一次任务中出现的峰值与拟合结果重合程度较低的问题。在图4中可以直观地发现，拟合结果与上一次十分相似，都在峰值位置出现了过高的情况。这依然可能是由于数据点数量和噪声的影响，但是为何在峰值最为明显依旧值得考虑（甚至噪声总产生正影响，说明未必是白噪声）。

其次，这次的数据相较上一次而言，在主峰的两侧有两个更明显的副峰，在上面的三个截图中可以发现，而右侧的副峰在修正合并后的图像中更为明显。注意到右侧副峰似乎有一种高斯分布或者略偏态分布的情况，如果再次使用郎道卷积高斯函数对225-350的范围进行再次拟合，可能也能得到另外一个结果。

[Question：为什么误差棒的大小这么小？]

在程序中，我们使用了mergedHistFromFile -> Draw("E1");的代码为第四副图绘制了一个误差棒。但是误差棒似乎特别小，我甚至不确定它是否绘制成功。

另外，我们可以快速从下方的.txt文件中读出各探测器的卡方/自由度对分别的拟合优劣性进行评判，也可以计算上方总卡方/自由度进行计算。事实上，计算对齐后数据的平均卡方/自由度约为73.44。考虑到峰值附近具有明显的误差，这样的误差说明数据在峰值之外的位置相对拟合地较好。另外，也可以考虑去除峰值附近的数据进行再计算。

Attached Files

这是输出的.txt文件，分别为序号、卡方/自由度、增益系数

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  • Title: A&C_task 1.4
  • Author: Albert Cheung
  • Created at : 2024-09-10 15:03:23
  • Updated at : 2024-09-18 18:28:57
  • Link: https://www.albertc9.github.io/2024/09/10/2024A-Ctask4/
  • License: This work is licensed under CC BY-NC-SA 4.0.
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A&C_task 1.4
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