diff --git a/5.1.2/algos/Main b/5.1.2/algos/Main new file mode 100755 index 0000000..40f5e3c Binary files /dev/null and b/5.1.2/algos/Main differ diff --git a/5.1.2/algos/Main.hi b/5.1.2/algos/Main.hi new file mode 100644 index 0000000..0713469 Binary files /dev/null and b/5.1.2/algos/Main.hi differ diff --git a/5.1.2/algos/Main.hs b/5.1.2/algos/Main.hs new file mode 100644 index 0000000..6eb5158 --- /dev/null +++ b/5.1.2/algos/Main.hs @@ -0,0 +1,45 @@ +module Main where +import Text.Printf + +ns = [860, 847, 743, 672, 623, 521, 461, 427, 376, 343, 315, 288] +angles = take 12 $ [0, 10..] + +main :: IO() +main = do + let xs = (map (\x -> 1 - cos (pi * x / 180)) angles) + let ys = (map (\x -> 1 / x - 1 / (head ns)) ns) + print_data xs ys + + let k' = k xs ys + --let b' = b xs ys + + putStr "k=" + print k' + + let n0 = k' * (1 - cos 0) + 1/ (head ns) + print $ 1 / n0 + let n90 = k' * (1 - cos (pi / 2)) + 1 / (head ns) + print $ 1 / n90 + + +print_data :: [Double] -> [Double] -> IO () +print_data (x:xs) (y:ys) = do + printf "%f %f\n" x y + print_data xs ys +print_data _ _ = return () + +mean :: [Double] -> Double +mean lst = sum lst / n where + n = fromIntegral $ length lst + +k :: [Double] -> [Double] -> Double +k xs ys = xy / x' where + xy = mean $ zipWith (*) xs ys + x' = mean $ map (^2) xs + +b :: [Double] -> [Double] -> Double +b xs ys = (xy - x * y) / (x ^ 2 - x') where + xy = mean $ zipWith (*) xs ys + x = mean xs + y = mean ys + x' = mean $ map (^2) xs diff --git a/5.1.2/algos/Main.o b/5.1.2/algos/Main.o new file mode 100644 index 0000000..648305e Binary files /dev/null and b/5.1.2/algos/Main.o differ diff --git a/5.1.2/algos/plot.dat b/5.1.2/algos/plot.dat new file mode 100644 index 0000000..2b55da7 --- /dev/null +++ b/5.1.2/algos/plot.dat @@ -0,0 +1,12 @@ +0.0 0.0 +0.01519224698779198 0.00001784684659948926 +0.06030737921409157 0.00018310432251400673 +0.1339745962155613 0.0003253045404208194 +0.233955556881022 0.00044234573892269215 +0.35721239031346064 0.0007565950988706869 +0.4999999999999999 0.0010064066992887051 +0.6579798566743311 0.0011791296770328414 +0.8263518223330696 0.0014967837704106877 +0.9999999999999999 0.0017526611973693131 +1.1736481776669303 0.002011812476928756 +1.3420201433256687 0.0023094315245478034 diff --git a/5.1.2/algos/plot.plt b/5.1.2/algos/plot.plt new file mode 100644 index 0000000..6865f2c --- /dev/null +++ b/5.1.2/algos/plot.plt @@ -0,0 +1,13 @@ +set xrange[0:1.4] +set yrange[0:0.0025] +set grid +unset key + +set mxtics 5 +set mytics 5 + +set grid mxtics mytics + +set multiplot + +plot 1.7670420408213902e-3 * x diff --git a/5.1.2/algos/plot.png b/5.1.2/algos/plot.png new file mode 100644 index 0000000..f5b2881 Binary files /dev/null and b/5.1.2/algos/plot.png differ diff --git a/5.1.2/main-gnuplottex-fig1-eps-converted-to.pdf b/5.1.2/main-gnuplottex-fig1-eps-converted-to.pdf new file mode 100644 index 0000000..646bce9 Binary files /dev/null and b/5.1.2/main-gnuplottex-fig1-eps-converted-to.pdf differ 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stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on main.pdf (6 pages, 13145738 bytes). +PDF statistics: + 311 PDF objects out of 1000 (max. 8388607) + 87 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 38 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/5.1.2/main.pdf b/5.1.2/main.pdf new file mode 100644 index 0000000..eb39f52 Binary files /dev/null and b/5.1.2/main.pdf differ diff --git a/5.1.2/main.tex b/5.1.2/main.tex new file mode 100644 index 0000000..a572745 --- /dev/null +++ b/5.1.2/main.tex @@ -0,0 +1,211 @@ +\documentclass[a4paper, 12pt]{article}%тип документа + +%отступы +\usepackage[left=2cm,right=2cm,top=2cm,bottom=3cm,bindingoffset=0cm]{geometry} + +%Русский язык +\usepackage[T2A]{fontenc} %кодировка +\usepackage[utf8]{inputenc} %кодировка исходного кода +\usepackage[english,russian]{babel} %локализация и переносы + +%Вставка картинок +\usepackage{wrapfig} +\usepackage{graphicx} +\graphicspath{{pictures/}} +\DeclareGraphicsExtensions{.pdf,.png,.jpg} + +%оглавление +\usepackage{titlesec} +\titlespacing{\chapter}{0pt}{-30pt}{12pt} +\titlespacing{\section}{\parindent}{5mm}{5mm} +\titlespacing{\subsection}{\parindent}{5mm}{5mm} +\usepackage{setspace} + +%Графики +\usepackage{multirow} +\usepackage{pgfplots} +\pgfplotsset{compat=1.9} + +%Математика +\usepackage{amsmath, amsfonts, amssymb, amsthm, mathtools} + +\usepackage{graphicx,xcolor} +\usepackage[cleanup]{gnuplottex} +\usepackage{subcaption} + +\begin{document} +\begin{titlepage} + \centering + \vspace{5cm} + {\scshape\LARGE Московский физико-технический институт \par} + \vspace{4cm} + {\scshape\Large Лабораторная работа 5.1.2 \par} + \vspace{1cm} + {\huge\bfseries Исследование эффекта Комптона \par} + \vspace{12cm} + {\LARGE Гришаев Григорий С01-119} +\end{titlepage} + +\newpage +\paragraph{Цель работы:} С помощью сцинтилляционного спектрометра исследуется энергетический спектр $\gamma$-квантов, рассеянных на графите. Определяется энергия рассеянных $\gamma$-квантов в зависимости от угла рассеяния, а также энергия покоя частиц, на которых происходит комптоновское рассеяние. + +\section{Теория} +Рассеяние $\gamma$-лучей в веществе относится к числу явлений, в которых особенно ясно проявляется двойственная природа излучения. Волновая теория, хорошо объясняющая рассеяние длинноволнового излучения, испытывает трудности при описании рассеяния рентгеновских и $\gamma$-лучей. Эта теория, в частности, не может объяснить, почему в составе рассеянного излучения, измеренного Комптоном, кроме исходной волны с частотой $\omega_{0}$ появляется дополнительная длинноволновая компонента, отсутствующая в спектре первичного излучения. + +Появление этой компоненты легко объяснимо, если считать, что $\gamma$-излучение представляет собой поток квантов (фотонов), имеющих энергию $\hbar \omega$ и импульс $p=\hbar \omega / c .$ Эффект Комптона - увеличение длины волны рассеянного излучения по сравнению с падающим - интерпретируется как результат упругого соударения двух частиц: $\gamma$-кванта (фотона) и свободного электрона. + +Рассмотрим элементарную теорию эффекта Комптона. +Пусть электрон до соударения покоился (его энергия равна энергии покоя $m c^{2}$ ), a $\gamma$-квант имел начальную энергию $\hbar \omega_{0}$ и импульс $\hbar \omega_{0} / c .$ +После соударения электрон приобретает энергию $\gamma m c^{2}$ и импульс $\gamma m v,$ где $\gamma=$ $=\left(1-\beta^{2}\right)^{-1 / 2}, \beta=v / c,$ a $\gamma$-квант рассеивается на некоторый угол $\theta$ по отношению к первоначальному направлению движения. +Энергия и импульс $\gamma$-кванта становятся соответственно равным $\hbar \omega_{1}$ и $\hbar \omega_{1} / c$. +Запишем для рассматриваемого процесса законы сохранения энергии и импульса: +$$ +\begin{array}{c} +m c^{2}+\hbar \omega_{0}=\gamma m c^{2}+\hbar \omega_{1} \\ +\frac{\hbar \omega_{0}}{c}=\gamma m v \cos \varphi+\frac{\hbar \omega_{1}}{c} \cos \theta \\ +\gamma m v \sin \varphi=\frac{\hbar \omega_{1}}{c} \sin \theta +\end{array} +$$ +Решая совместно эти уравнения и переходя от частот $\omega_{0}$ и $\omega_{1}$ к длинам волн $\lambda_{0}$ и $\lambda_{1},$ нетрудно получить, что изменение длины волны рассеянного излучения равно +\begin{equation}\label{delta_lambda} +\Delta \lambda=\lambda_{1}-\lambda_{0}=\frac{h}{m c}(1-\cos \theta)=\Lambda_{\mathrm{K}}(1-\cos \theta) +\end{equation} +где $\lambda_{0}$ и $\lambda_{1}$ - длины волн $\gamma$-кванта до и после рассеяния, а величина +$$ +\Lambda_{\mathrm{K}}=\frac{h}{m c}=2,42 \cdot 10^{-10} \mathrm{cm} +$$ + +Основной целью данной работы является проверка соотношения (\ref{delta_lambda}). +Применительно к условиям нашего опыта формулу (\ref{delta_lambda}) следует преобразовать от длин волн к энергии $\gamma$-квантов. Как нетрудно показать, соответствующее выражение имеет вид +\begin{equation}\label{energy} +\frac{1}{\varepsilon(\theta)}-\frac{1}{\varepsilon_{0}}=1-\cos \theta +\end{equation} +Здесь $\varepsilon_{0}=E_{0} /\left(m c^{2}\right)-$ выраженная в единицах $m c^{2}$ энергия $\gamma$-квантов, падаюших на рассеиватель, $\varepsilon(\theta)$ - выраженная в тех же единицах энергия квантов, испытавших комптоновское рассеяние на угол $\theta, m-$ масса электрона. + + +\section{Экспериментальная установка} +Блок-схема установки изображена на Рис.1. Источником излучения 1 служит $137 \mathrm{Cs},$ испускаюший $\gamma$-лучи с энергией 662 кэВ. +Он помещен в толстостенный свинцовый контейнер с коллиматором. +Сформированный коллиматором узкий пучок $\gamma$-квантов попадает на графитовую мишень 2 (цилиндр диаметром 40 мм и высотой 100 мм). + +\begin{figure}[h] +\begin{center} +\includegraphics[width = \textwidth]{0.jpg} +\caption{Экспериментальная установка} +\label{setup} +\end{center} +\end{figure} + +Кванты, испытавшие комптоновское рассеяние в мишени, регистрируются сцинтилляционным счетчиком, принцип работы которого рассмотрен в работе 5.3. +Счетчик состоит из фотоэлектронного умножителя 3 (далее ФЭУ) и сцинтиллятора 4. +Сцинтиллятором служит кристалл NaI(Tl) цилиндрической формы диаметром 40 мм и высотой 40 мм, его выходное окно находится в оптическом контакте с фотокатодом ФЭУ. +Сигналы, возникающие на аноде ФЭУ, подаются на ЭВМ для амплитудного анализа. +Кристалл и ФЭУ расположены в светонепроницаемом блоке, укрепленном на горизонтальной штанге. +Штанга вместе с этим блоком может вращаться относительно мишени, угол поворота отсчитывается по лимбу 6. + +\section{Обработка результатов} + +Для обработки результатов используется формула (\ref{energy}) с замененными энергиями квантов на $N(\theta)$. +\begin{equation}\label{main_eq} +\frac{1}{N(\theta)} - \frac{1}{N(0)} = A(1-\cos\theta) +\end{equation} +\newpage +\section{Ход работы} + +Графики для углов $\theta = 0^0-110^0$ представлены на фотографиях ниже (Рис.\ref{graphs}). + +\begin{figure}[h] +\centering + +\begin{subfigure}{0.7\textwidth} +\includegraphics[width=\linewidth]{1.jpg} +\caption{Проверка для $\theta = 0-40^0$} +\end{subfigure} + +\hfill + +\begin{subfigure}{0.7\textwidth} +\includegraphics[width=\linewidth]{2.jpg} +\caption{Проверка для $\theta = 50-90^0$} +\end{subfigure} + +\hfill + +\begin{subfigure}{0.7\textwidth} +\includegraphics[width=\linewidth]{3.jpg} +\caption{Проверка для $\theta = 100-110^0$} +\end{subfigure} + +\caption{Графики для $\theta = 0 - 90^0$} +\label{graphs} +\end{figure} + +Коэффициент $A$ найдем с помощью графика зависимости $1-\cos \theta$ от $\frac{1}{N(\theta)} - \frac{1}{N(0)}$. В таблицу занесем все измеренные величины: + +\begin{table}[h!] +\begin{center} +\begin{tabular}{|c|c|c|c|} +\hline +$\theta$, $^0$ & $\sigma_{\theta}$, $^0$ & $N(\theta)$ & $\sigma_{N(\theta)}$ \\ \hline +0 & 1 & 860 & 20 \\ \hline +10 & 1 & 847 & 20 \\ \hline +20 & 1 & 743 & 20 \\ \hline +30 & 1 & 672 & 20 \\ \hline +40 & 1 & 623 & 20 \\ \hline +50 & 1 & 521 & 20 \\ \hline +60 & 1 & 461 & 20 \\ \hline +70 & 1 & 427 & 20 \\ \hline +80 & 1 & 376 & 20 \\ \hline +90 & 1 & 343 & 20 \\ \hline +100 & 1 & 315 & 20 \\ \hline +110 & 1 & 288 & 20 \\ \hline +\end{tabular} + \caption{Измерение $N(\theta)$} +\end{center} +\end{table} + +\begin{figure}[h] + \centering + \begin{gnuplot}[terminal=epslatex] + set xrange[0:1.4] + set yrange[0:0.0025] + set grid + unset key + + set mxtics 5 + set mytics 5 + + set xlabel '$1 - cos \theta$' + set ylabel '$\frac{1}{N(\theta)} - \frac{1}{N(0)}$' + set grid mxtics mytics + + set multiplot + + plot 'algos/plot.dat' pt 7 ps 1 + plot 1.7670420408213902e-3 * x + \end{gnuplot} + \caption{График зависимости $\frac{1}{N(\theta)} - \frac{1}{N(0)}$ от $1 - cos(\theta)$} + \label{graph} +\end{figure} + +В итоге: $ A = 0.001767 $. +Теперь, для уменьшения погрешности, мы можем использовать коэффициент $ A $ для вычисления $ N(0) $ и $ N(90) $: $ N(0) = 860 $, $ N(90) = 341 $. + +Получаем, что соотношение (\ref{main_eq}) справедливо и эффект Комптона в данном эксперименте имеет место быть. + +Теперь посчитаем энергию $\gamma$-кванта по формуле, полученной из формулы (\ref{energy}). Получаем формулы для энергии и погрешности +\begin{equation} + E_{\gamma} = mc^2 \frac{N(0) - N(90)}{N(90)} \approx 771.5 \text{ кэВ} +\end{equation} +\begin{equation} + \sigma_{E_{\gamma}} = \sqrt{(\frac{mc^{2}}{N(90)})^{2} \cdot \sigma_{N(90)}^{2} + (\frac{mc^{2}N(0)}{N^{2}(90)})^{2} \cdot \sigma_{N(90)}^{2}} \approx 80.6 \text{ кэВ} +\end{equation} + +Итого, +\[E_{\gamma} = (771.5 \pm 80.6) \text{ кэВ}\] + +\section{Вывод} +В работе мы исследовали спектр излучения $\gamma$-квантов на графите и получили, что соотношение (\ref{main_eq}), приведенное в теории, справедливо. +Также мы убедились в том, что эффект Комптона в принципе возникает в нашем опыте. +Мы измерили энергию $\gamma$-кванта: $(771.5 \pm 80.6) \text{ кэВ}$. +\end{document} diff --git a/5.1.2/pictures/0.jpg b/5.1.2/pictures/0.jpg new file mode 100644 index 0000000..fdddb91 Binary files /dev/null and b/5.1.2/pictures/0.jpg differ diff --git a/5.1.2/pictures/1.jpg b/5.1.2/pictures/1.jpg new file mode 100644 index 0000000..705ffe7 Binary files /dev/null and b/5.1.2/pictures/1.jpg differ diff --git a/5.1.2/pictures/2.jpg b/5.1.2/pictures/2.jpg new file mode 100644 index 0000000..9306f5a Binary files /dev/null and b/5.1.2/pictures/2.jpg differ diff --git a/5.1.2/pictures/3.jpg b/5.1.2/pictures/3.jpg new file mode 100644 index 0000000..29cfa3b Binary files /dev/null and b/5.1.2/pictures/3.jpg differ diff --git a/5.1.2/texput.log b/5.1.2/texput.log new file mode 100644 index 0000000..bd91bbb --- /dev/null +++ b/5.1.2/texput.log @@ -0,0 +1,21 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Arch Linux) (preloaded format=pdflatex 2023.8.26) 23 OCT 2023 10:44 +entering extended mode + \write18 enabled. + %&-line parsing enabled. +** + +! 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Сечение $ \sigma $ равно отношению числа $ N $ таких переходов в единицу времени к плотности потока рассеиваемых частиц $ n v $, падающих на мишень, т. е. к числу частиц, проходящих в единицу времени через единичную площадку, перпендикулярную к их скорости $ v $ ($ n $ -- плотность числа падающих частиц). +\begin{equation}\label{eq:sigma} + \sigma = \frac{N}{n v}. +\end{equation} +Таким образом, сечение имеет размерность площади. + + +\begin{figure}[h!] + \centering + \includegraphics[width=0.5\linewidth]{Screenshot_2} + \caption{Схема установки для измерения сечения рассеяния электронов в газах} + \label{fig:screenshot2} +\end{figure} +Качественно результат экспериментов Рамзауэра при энергии электронов порядка десятков эВ показан на рис. \ref{fig:screenshot1}. +По мере уменьшения энергии электрона от нескольких десятков электрон-вольт поперечное сечение его упругого рассеяния растет. Однако при энергиях меньше 16 эВ в случае аргона сечение начинает уменьшаться, а при $ E \sim 1 $ эВ практически равно нулю, т. е. аргон становится прозрачным для электронов. При дальнейшем уменьшении энергии электронов сечение рассеяния опять начинает возрастать. Это поведение поперечного сечения свойственно не только атомам аргона, но и атомам всех инертных газов. Такое поведение электронов нельзя объяснить с позиций классической физики. Объяснение этого эффекта потребовало учета волновой природы электронов. Схема эксперимента Рамзауэра показана, на рис. \ref{fig:screenshot2}. + + +С точки зрения квантовой теории, внутри атома потенциальная энергия налетающего электрона $ U $ отлична от нуля, скорость электрона изменяется, становясь равной $ v' $ в соответствии с законом сохранения энергии +\begin{equation*} + E = \frac{m v^2}{2} = \frac{m v'^2}{2}+ U, +\end{equation*} +а значит, изменяется и длина его волны де Бройля. Таким образом, по отношению к электронной волне атом ведет себя как преломляющая среда с относительным показателем преломления +\begin{equation*} + n = \frac{\lambda}{\lambda'} = \sqrt{1-\frac{U}{E}}. +\end{equation*} + +Коэффициент прохождения электронов максимален при условии +\begin{equation}\label{eq:at} + \sqrt{\frac{2 m (E+U_0)}{\hbar^2}}l = \pi n;\; n \in \mathbb{N}_1, +\end{equation} +где $ U_0 $ -- глубина потенциальной ямы. + +Это условие легко получить, рассматривая интерференцию электронных волн де Бройля в атоме. Движущемуся электрону соответствует волна де Бройля, длина которой определяется соотношением $ \lambda = h/m v $. Если кинетическая энергия электрона невелика, то $ E = m v^2/2 $ и $ \lambda = h/\sqrt{2 m E} $. При движении электрона через атом длина волны де Бройля становится меньше и равна $ \lambda' = h/\sqrt{2 m (E+U_0)} $ где $ U_0 $ — глубина атомного потенциала. При этом, волна де Бройля отражается от границ атомного потенциала, т. е. от поверхности атома, и происходит интерференция прошедшей через атом волны 1 и волны 2, отраженной от передней и задней границы атома (эти волны когерентны). Прошедшая волна 1 усилится волной 2, если геометрическая разность хода между ними $ \Delta = 2 l = \lambda' $, что соответствует условию первого интерференционного максимума, т. е. при условии +\begin{equation}\label{eq:condition} + 2 l = \frac{h}{\sqrt{2 m (E_1 + U_0)}} +\end{equation} +Прошедшая волна ослабится при условии +\begin{equation}\label{eq:condition2} + 2 l = \frac{3}{2}\frac{h}{\sqrt{2 m (E_1 + U_0)}} +\end{equation} + +Из \eqref{eq:condition} и \eqref{eq:condition2}, можно получить +\begin{equation}\label{eq:radius} + l = \frac{h \sqrt{5}}{\sqrt{32 m (E_2-E_1)}}. +\end{equation} +Оттуда же можно найти эффективную глубину потенциальной ямы атома: +\begin{equation}\label{eq:atomPit} + U_0 = \frac{4}{5}E_2-\frac{9}{5} E_1. +\end{equation} + + +Уравнение вольт-амперной характеристики тиратрона: +\begin{equation}\label{eq:VAH} + I_а = I_0 \exp (-C \omega (V));\; C = L n_а \Delta_а, +\end{equation} +где $ I_0 = e N_0 $ -- ток катода, а $ I_а = e N_а $ -- ток анода. +Отсюда определяется вероятность рассеяния электрона в зависимости от его энергии: +\begin{equation}\label{eq:probable} + \omega (V) = -\frac{1}{C} \ln \frac{I_а(V)}{I_0}. +\end{equation} + +\clearpage + +\section{Экспериментальная установка} + +Схема экспериментальной установки отображена на рис. \ref{fig:screenshot3}. + +\begin{figure}[h!] + \centering + \includegraphics[width=1.0\linewidth]{Screenshot_3} + \caption{Схема экспериментальной установки} + \label{fig:screenshot3} +\end{figure} + +В данной работе для изучения эффекта Рамзауэра используется +тиратрон ТГЗ-01/1.3Б, заполненный инертным газом. Электроны, эмитируемые катодом тиратрона, ускоряются напряжением $ V $, приложенным между катодом и ближайшей к нему сеткой. Затем электроны рассеиваются на атомах инертного газа (ксенона). Все сетки соединены между собой и имеют одинаковый потенциал, примерно равный потенциалу анода. Поэтому между первой сеткой и анодом практически нет поля. Рассеянные электроны отклоняются в сторону и уходят на сетку, а оставшаяся часть электронов достигает анода и создаёт анодный ток $ I_а $. Таким образом, поток электронов $ N(x) $ (т. е. число электронов, проходящих через поперечное сечение лампы в точке $ x $ в единицу времени) уменьшается с ростом $ x $ от начального значения $ X $ y катода (в точке $ x=0 $) до некоторого значения $ N_а $ у анода (в точке +$ x=L $). + +\clearpage + +\section{Результаты измерений и обработка данных} + +\subsection{Динамический метод} +По результатам измерений в динамическом режиме оценим размер электронной оболочки атома инертного газа по формулам \eqref{eq:condition} и \eqref{eq:condition2}. +Положение первого максимума $$ V_{max}^1 \approx 2.5 \;В.$$ +Положение первого минимума $$ V_{min}^1 \approx 6.5 \; В.$$ +Тогда +\begin{equation*}\label{key} + l = \frac{h}{\sqrt{2 m_e * 5}} \approx 2.8\; \text{\AA{}} +\end{equation*} +\begin{equation*} + l = \frac{3}{4} \frac{h}{\sqrt{2 m_e * 9}} \approx 3\; \text{\AA{}} +\end{equation*} +В данном случае оценка погрешностей не имеет смысла, так как точка $ V_{max}^1 $ указана неточно, а сам расчёт носит оценочный характер. + +Далее найдём радиус из формулы \eqref{eq:radius}: +\begin{equation*}\label{key} + l = (3.4 \pm 0.2)*10^{-10} \; \text{\AA{}} +\end{equation*} + +Эффективная глубина потенциальной ямы равна +\begin{equation*}\label{key} + U_0 = \frac{4}{5}*6.5 - \frac{9}{5}*2.5 = 1.1 \;эВ. +\end{equation*} + +Так как напряжение пробоя примерно равно $ 12 $ В, в колбу закачан ксенон. 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+\usepackage{multirow} +\setlength{\columnsep}{2cm} + +\usepackage{multicol} +\setlength{\columnsep}{2cm} +\usepackage{hyperref} +\usepackage{wrapfig} + +\begin{document} + +\begin{titlepage} + \centering + \vspace{5cm} + {\scshape\LARGE Московский физико-технический институт \par} + \vspace{4cm} + {\scshape\Large Лабораторная работа 5.10.1 \par} + \vspace{1cm} + {\huge\bfseries Электронный парамагнитный резонанс \par} + \vspace{12cm} + {\LARGE Гришаев Григорий С01-119} +\end{titlepage} + +\paragraph*{Цель работы:} Исследуется электронный парамагнитный резонанс в молекуле ДФПГ, определяется $g$-фактор электрона, измеряется ширина ЭПР. + +\section*{Теоретическое введение} + Энергетический уровень электрона в присутствии магнитного поля с индукцией $B$ расщепляется на подуровня, расстояние между которыми равно + \begin{equation} + \label{eq:dE} + \Delta E = E_2 - E_1 = 2\mu B. + \end{equation} + Здесь $\mu$ -- абсолютная величина проекции магнитного момента на направление поля. + + Между этими двумя уровнями возможны переходы. Эти переходы могут возбуждаться внешним высокочастотным электромагнитным полем, если оно имеет нужную частоту и нужное направление. + + Резонансное значение частоты определяется из очевидной формулы: + \begin{equation} + \label{eq:resonans_omega} + \hbar \omega_0 = \Delta E. + \end{equation} + + При переходе с нижнего на верхний уровень энергии электрон поглощает квант электромагнитной энергии, а при обратном переходе такой же квант излучается. Возбуждение электронных резонансных переходов электромагнитным полем, имеющим частоту, определяемую формулой~(\ref{eq:resonans_omega}), носит название электронного парамагнитного резонанса (ЭПР). + + В настоящей работе необходимо получить сигнал ЭПР на кристаллическом дифенилпикрилгидразиле (ДФПГ) и определить значение $g$-фактора для электрона. Как известно, связь между магнитным моментом $\mu$ электрона и его механическим моментом $\mathbf{M}$ выражается через гиромагнитное отношение $\gamma$ с помощью формулы + \begin{equation} + \label{eq:gyromagnit} + \mu = \gamma M. + \end{equation} + + А магнитный момент частицы, измеренный в магнитонах Бора, а механический - в $\hbar$, то их связь можно записать через $g$-фактор: + \begin{equation} + \label{eq:def_g} + \frac{\mu}{\mu_\text{Б}} = \frac{M}{\hbar} + \end{equation} + + Используя соотношения (\ref{eq:dE})-(\ref{eq:def_g}), нетрудно получить выражение для $g$-фактора через определяемые экспериментально величины: + \begin{equation} + \label{eq:g_is} + \tag{$\star$} + g = \frac{\hbar \omega_0}{\mu_\text{Б} B}. + \end{equation} + +\newpage +\section{Экспериментальная установка} + Образец (порошок ДФПГ) в стеклянной ампуле помещяется внутрь катушкииндуктивнсоти входящей в состав колебательного контура. Входящий в состав контура конденсатор состоит из двух платсин, разделенных воздушным зазором, одна из пластин может перемещаться поворотом штока. Колебания в контуре возбуждаются антенной, соединённой с генератором частоты (ВЧ) АКИП-3417. Амплитуда колебаний поля в катушке индуктивности измеряется по наводимой в петле связи ЭДС индукции. Высокочастотные колебания ЭДС индукции в приёмном контуре детектируются диодом, измеряемая при помощи осциллографа низкочастотная огибающая этого сигнала пропорциональна квадрату амплитуды колебаний поля в катушке. + \begin{figure}[h!] + \centering + \caption{Схема установки.} + \label{fig:equip} + \includegraphics[scale=0.17]{equip.png} + \end{figure} + + Постоянной магнитное поле создаётся пропусканием тока от источника постоянного тока через основные катушки. При этом при помощи вольтметра измеряется падение напряжения на резисторе в цепи основных катушек. Переменное поле небольшой амплитуды создаётся подачей на модуляционные катушки напряжения с регулируемого трансформатора ЛАТР. Для измерения амплитуды колебаний переменного поля используется пробная катушка известной геометрии, подключенная к вольтметру. + + + + \newpage + \section*{Ход работы} + Запишем параметры катушек в Таблицу \ref{coil}: + + \begin{table}[h] + \centering + \begin{tabular}{|l|l|l|} + \hline + \textbf{Катушка} & $N$ & $D$, см \\ \hline \hline + Основная & 6700 & $25$ \\ \hline + Модуляционная & 5000 & $30$ \\ \hline + Пробная & 45 & $1.52\pm0.01$ \\ \hline + \end{tabular} + \caption{Параметры катушек.} + \label{coil} + \end{table} + + \subsection*{Резонанс} + + + Настроим генератор на частоту колебательного конутра. Получаем резонансную частоту: + \begin{equation*} + f_0 = (162.79 \pm 5\times10^{-5}) \text{МГц}. + \end{equation*} + + Подберем величину постоянного магнитного поля в катушках так, чтобы наблюдался сигнал резонанского поглощения. Для этого подадим на катушки достаточное напряжение. + + Для более точной настройки и определения ширины линии резонасного поглощения будем наблюдать сигнал в $XY$-режиме. Запишем значение напряжения на резисторе в цепи основных катушек: + \begin{equation*} + U_0 = (128.47 \pm 0.39) \ \text{мВ}. + \end{equation*} + + Измерим и запишем $\varepsilon$ для пробной катушки: $\varepsilon = (14.35 \pm 0.04) $ мВ. + + \subsection*{Ширина линии поглощения} + + Определим ширину линии ЭПР (полуширина на на полувысоте линии резонасного поглощения): + \begin{equation*} + \Delta B = \frac{A_{1/2}}{A_{\text{полн}}}B_\text{мод}, + \end{equation*} + где $A_\text{полн}$ -- полный размах модулирующего поля, $A_{1/2}$ -- ширина кривой на полувысоте, $B_\text{мод}$ -- амплитуда модулирующего поля. + \begin{equation*} + \begin{gathered} + A_\text{полн} = (5.6 \pm 0.2 ) \ \text{дел}, \ A_{1/2} = (1 \pm 0.2) \ \text{дел}, \varepsilon = (1.4 \pm 4.2\times10^{-3}) \text{мВ} \\ + B_\text{мод} = \sqrt{2} \frac{2\varepsilon}{\pi^2d^2N\nu} = (0.77 \pm 0.02) \text{мТл}, + \end{gathered} + \end{equation*} + где $\varepsilon$ -- ЭДС индукции при внесении пробной катушки, $N$ -- число витков катушки, $d$ -- диаметр катушки, $\nu$ -- частота модулирующего напряжения (50 Гц). + + Имеем: + \[\boxed{\Delta B = (0.275\pm0.061) \ \text{мТл}}.\] + + \subsection*{Калибровка основной катушки} + + Определим связь между падением напряжения на резисторе в цепи основных катушек и магнитным полем в центре магнита. Результаты занесем в Таблицу \ref{table:field}: + + \begin{table}[h] + \centering + \begin{tabular}{|c|c|c|c|} + \hline + $U$, мВ & 111.64 & 128.47 & 142.75 \\ \hline + $\varepsilon$, мВ & 12.5 & 14.35 & 15.95 \\ \hline + \end{tabular} + \caption{Калибровочные измерения.} + \label{table:field} + \end{table} + + Методом наименьших квадратов найдем коэффициент пропорциональности между напряжением на основных катушках и напряжением на пробной катушке: + \begin{equation*} + k = 0.112 \pm 0.014 + \end{equation*} + + Рассчитав поле, создаваемое основными катушками, + + \begin{equation*} + B_0 = \frac{4 k U_0}{2\pi\nu N \pi d^2} = (5.6 \pm 0.7) \text{мТл}. + \end{equation*} + + Найдем $g$-фактор электрона: + + \begin{equation*} + g = \frac{hf_0}{\mu_BB_0} = 2.077 \pm 0.21 + \end{equation*} + + + + +\section*{Вывод} + В данной работе был исследован ЭПР в молекуле ДФПГ, определяется $g$-фактор электрона $\pmb{g = 2.077 \pm 0.21}$, а также измерена ширина линий ЭПР $\Delta B = 0.275 \pm 0.061~\text{мТл}$. + Измеренный $g$-фактор электрона совпадает с табличным значением для свободного электрона: $\pmb{g_{free} = 2,0}$. Это обусловлено тем, что ЭПР происходит на неспаренных электронах так же, как на свободных. + +\end{document} diff --git a/5.2.1/1.png b/5.2.1/1.png new file mode 100644 index 0000000..7449b76 Binary files /dev/null and b/5.2.1/1.png differ diff --git a/5.2.1/2.png b/5.2.1/2.png new file mode 100644 index 0000000..f07db7d Binary files /dev/null and b/5.2.1/2.png differ diff --git a/5.2.1/3.png b/5.2.1/3.png new file mode 100644 index 0000000..bd8af9a Binary files /dev/null and b/5.2.1/3.png differ diff --git a/5.2.1/4.dat b/5.2.1/4.dat new file mode 100644 index 0000000..11eb9d6 --- /dev/null +++ b/5.2.1/4.dat @@ -0,0 +1,29 @@ +0 0 +2.31 0.0138 +5 0.033 +9.35 0.0755 +14.27 0.1237 +19.74 0.1665 +18.72 0.1634 +17.37 0.1530 +23.69 0.0618 +20.75 0.1655 +21.86 0.1530 +22.61 0.0861 +23.05 0.0668 +36.23 0.2514 +23.93 0.0622 +27.72 0.1228 +32.04 0.1997 +34.8 0.2428 +35.77 0.2501 +37.2 0.2492 +38.09 0.2488 +46.04 0.1923 +42.18 0.2096 +44.5 0.1957 +45.11 0.1939 +46.97 0.1937 +48.31 0.1991 +54.14 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[] 0 setdash 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C5 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc + 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc} bind def +/C6 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C7 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C8 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C9 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 450 arc closepath fill + vpt 0 360 arc closepath} bind def +/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill + 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C11 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C12 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C13 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C14 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 360 arc closepath fill + vpt 0 360 arc} bind def +/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto + neg 0 rlineto closepath} bind def +/Square {dup Rec} bind def +/Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def +/S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def +/S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def +/S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def +/S3 {BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare} bind def +/S4 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def +/S5 {BL [] 0 setdash 2 copy 2 copy vpt Square fill + exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def +/S6 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare} bind def +/S7 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill + 2 copy vpt Square fill Bsquare} bind def +/S8 {BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare} bind def +/S9 {BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare} bind def +/S10 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill + Bsquare} bind def +/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill + Bsquare} bind def +/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def +/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy vpt Square fill Bsquare} bind def +/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def +/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def +/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def +/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def +/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def +/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def +/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def +/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def +/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def +/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def +/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def +/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def +/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def +/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def +/D12 {gsave translate 45 rotate 0 0 S12 stroke grestore} bind def +/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def +/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def +/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def +/DiaE {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke} def +/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke} def +/TriUE {stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath stroke} def +/TriDE {stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath stroke} def +/PentE {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore} def +/CircE {stroke [] 0 setdash + hpt 0 360 arc stroke} def +/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def +/DiaW {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V Opaque stroke} def +/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V Opaque stroke} def +/TriUW {stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V Opaque stroke} def +/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V Opaque stroke} def +/PentW {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + Opaque stroke grestore} def +/CircW {stroke [] 0 setdash + hpt 0 360 arc Opaque stroke} def +/BoxFill {gsave Rec 1 setgray fill grestore} def +/Density { + /Fillden exch def + currentrgbcolor + /ColB exch def /ColG exch def /ColR exch def + /ColR ColR Fillden mul Fillden sub 1 add def + /ColG ColG Fillden mul Fillden sub 1 add def + /ColB ColB Fillden mul Fillden sub 1 add def + ColR ColG ColB setrgbcolor} def +/BoxColFill {gsave Rec PolyFill} def +/PolyFill {gsave Density fill grestore grestore} def +/h {rlineto rlineto rlineto closepath gsave fill grestore stroke} bind def +% +% PostScript Level 1 Pattern Fill routine for rectangles +% Usage: x y w h s a XX PatternFill +% x,y = lower left corner of box to be filled +% w,h = width and height of box +% a = angle in degrees between lines and x-axis +% XX = 0/1 for no/yes cross-hatch +% +/PatternFill {gsave /PFa [ 9 2 roll ] def + PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate + PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse + clip + currentlinewidth 0.5 mul setlinewidth + /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def + 0 0 M PFa 5 get rotate PFs -2 div dup translate + 0 1 PFs PFa 4 get div 1 add floor cvi + {PFa 4 get mul 0 M 0 PFs V} for + 0 PFa 6 get ne { + 0 1 PFs PFa 4 get div 1 add floor cvi + {PFa 4 get mul 0 2 1 roll M PFs 0 V} for + } if + stroke grestore} def +% +/languagelevel where + {pop languagelevel} {1} ifelse +dup 2 lt + {/InterpretLevel1 true def + /InterpretLevel3 false def} + {/InterpretLevel1 Level1 def + 2 gt + {/InterpretLevel3 Level3 def} + {/InterpretLevel3 false def} + ifelse } + ifelse +% +% PostScript level 2 pattern fill definitions +% +/Level2PatternFill { +/Tile8x8 {/PaintType 2 /PatternType 1 /TilingType 1 /BBox [0 0 8 8] /XStep 8 /YStep 8} + bind def +/KeepColor {currentrgbcolor [/Pattern /DeviceRGB] setcolorspace} bind def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke} +>> matrix makepattern +/Pat1 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke + 0 4 M 4 8 L 8 4 L 4 0 L 0 4 L stroke} +>> matrix makepattern +/Pat2 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 0 8 L + 8 8 L 8 0 L 0 0 L fill} +>> matrix makepattern +/Pat3 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 8 M 8 -4 L + 0 12 M 12 0 L stroke} +>> matrix makepattern +/Pat4 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 0 M 8 12 L + 0 -4 M 12 8 L stroke} +>> matrix makepattern +/Pat5 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 8 M 4 -4 L + 0 12 M 8 -4 L 4 12 M 10 0 L stroke} +>> matrix makepattern +/Pat6 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L + 0 -4 M 8 12 L 4 -4 M 10 8 L stroke} +>> matrix makepattern +/Pat7 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L + 12 0 M -4 8 L 12 4 M 0 10 L stroke} +>> matrix makepattern +/Pat8 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L + -4 0 M 12 8 L -4 4 M 8 10 L stroke} +>> matrix makepattern +/Pat9 exch def +/Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def +/Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def +/Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def +/Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def +/Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def +/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def +/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def +} def +% +% +%End of PostScript Level 2 code +% +/PatternBgnd { + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse +} def +% +% Substitute for Level 2 pattern fill codes with +% grayscale if Level 2 support is not selected. +% +/Level1PatternFill { +/Pattern1 {0.250 Density} bind def +/Pattern2 {0.500 Density} bind def +/Pattern3 {0.750 Density} bind def +/Pattern4 {0.125 Density} bind def +/Pattern5 {0.375 Density} bind def +/Pattern6 {0.625 Density} bind def +/Pattern7 {0.875 Density} bind def +} def +% +% Now test for support of Level 2 code +% +Level1 {Level1PatternFill} {Level2PatternFill} ifelse +% +/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont +dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall +currentdict end definefont pop +% +Level1 SuppressPDFMark or +{} { +/SDict 10 dict def +systemdict /pdfmark known not { + userdict /pdfmark systemdict /cleartomark get put +} if +SDict begin [ + /Title (./main-gnuplottex-fig1.tex) + /Subject (gnuplot plot) + /Creator (gnuplot 5.4 patchlevel 10) +% /Producer (gnuplot) +% /Keywords () + /CreationDate (Mon Nov 27 02:43:21 2023) + /DOCINFO pdfmark +end +} ifelse +% +% Support for boxed text - Ethan A Merritt Sep 2016 +% +/InitTextBox { userdict /TBy2 3 -1 roll put userdict /TBx2 3 -1 roll put + userdict /TBy1 3 -1 roll put userdict /TBx1 3 -1 roll put + /Boxing true def } def +/ExtendTextBox { dup type /stringtype eq + { Boxing { gsave dup false charpath pathbbox + dup TBy2 gt {userdict /TBy2 3 -1 roll put} {pop} ifelse + dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse + dup TBy1 lt {userdict /TBy1 3 -1 roll put} {pop} ifelse + dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse + grestore } if } + {} ifelse} def +/PopTextBox { newpath TBx1 TBxmargin sub TBy1 TBymargin sub M + TBx1 TBxmargin sub TBy2 TBymargin add L + TBx2 TBxmargin add TBy2 TBymargin add L + TBx2 TBxmargin add TBy1 TBymargin sub L closepath } def +/DrawTextBox { PL PopTextBox stroke /Boxing false def} def +/FillTextBox { gsave PopTextBox fill grestore /Boxing false def} def +0 0 0 0 InitTextBox +/TBxmargin 20 def +/TBymargin 20 def +/Boxing false def +/textshow { ExtendTextBox Gshow } def +% +end +%%EndProlog +%%Page: 1 1 +gnudict begin +gsave +doclip +50 50 translate +0.050 0.050 scale +0 setgray +newpath +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +0.500 UL +LTa +LCa setrgbcolor +1210 704 M +5593 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +1210 704 M +63 0 V +5530 0 R +-63 0 V +stroke +LTb +0.500 UL +LTa +LCa setrgbcolor +1210 822 M +5593 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +1210 822 M +31 0 V +5562 0 R +-31 0 V +stroke +0.500 UL +LTa +LCa setrgbcolor 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ifelse} def +/Cshow {currentpoint stroke M dup stringwidth pop -2 div vshift R + Blacktext {gsave 0 setgray textshow grestore} {textshow} ifelse} def +/UP {dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def + /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def} def +/DL {Color {setrgbcolor Solid {pop []} if 0 setdash} + {pop pop pop 0 setgray Solid {pop []} if 0 setdash} ifelse} def +/BL {stroke userlinewidth 2 mul setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +/AL {stroke userlinewidth 2 div setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +/UL {dup gnulinewidth mul /userlinewidth exch def + dup 1 lt {pop 1} if 10 mul /udl exch def} def +/PL {stroke userlinewidth setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +3.8 setmiterlimit +% Classic Line colors (version 5.0) +/LCw {1 1 1} def +/LCb {0 0 0} def +/LCa {0 0 0} def +/LC0 {1 0 0} def +/LC1 {0 1 0} def +/LC2 {0 0 1} def +/LC3 {1 0 1} def +/LC4 {0 1 1} def +/LC5 {1 1 0} def +/LC6 {0 0 0} def +/LC7 {1 0.3 0} def 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sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke + Pnt} def +/Crs {stroke [] 0 setdash exch hpt sub exch vpt add M + hpt2 vpt2 neg V currentpoint stroke M + hpt2 neg 0 R hpt2 vpt2 V stroke} def +/TriU {stroke [] 0 setdash 2 copy vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath stroke + Pnt} def +/Star {2 copy Pls Crs} def +/BoxF {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath fill} def +/TriUF {stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath fill} def +/TriD {stroke [] 0 setdash 2 copy vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath stroke + Pnt} def +/TriDF {stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath fill} def +/DiaF {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath fill} def +/Pent {stroke [] 0 setdash 2 copy gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore Pnt} def +/PentF {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath fill grestore} def +/Circle {stroke [] 0 setdash 2 copy + hpt 0 360 arc stroke Pnt} def +/CircleF {stroke [] 0 setdash hpt 0 360 arc fill} def +/C0 {BL [] 0 setdash 2 copy moveto vpt 90 450 arc} bind def +/C1 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + vpt 0 360 arc closepath} bind def +/C2 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C3 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C4 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C5 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc + 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc} bind def +/C6 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C7 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C8 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C9 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 450 arc closepath fill + vpt 0 360 arc closepath} bind def +/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill + 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C11 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C12 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C13 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C14 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 360 arc closepath fill + vpt 0 360 arc} bind def +/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto + neg 0 rlineto closepath} bind def +/Square {dup Rec} bind def +/Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def +/S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def +/S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def +/S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def +/S3 {BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare} bind def +/S4 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def +/S5 {BL [] 0 setdash 2 copy 2 copy vpt Square fill + exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def +/S6 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare} bind def +/S7 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill + 2 copy vpt Square fill Bsquare} bind def +/S8 {BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare} bind def +/S9 {BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare} bind def +/S10 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill + Bsquare} bind def +/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill + Bsquare} bind def +/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def +/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy vpt Square fill Bsquare} bind def +/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def +/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def +/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def +/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def +/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def +/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def +/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def +/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def +/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def +/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def +/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def +/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def +/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def +/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def +/D12 {gsave translate 45 rotate 0 0 S12 stroke grestore} bind def +/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def +/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def +/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def +/DiaE {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke} def +/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke} def +/TriUE {stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath stroke} def +/TriDE {stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath stroke} def +/PentE {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore} def +/CircE {stroke [] 0 setdash + hpt 0 360 arc stroke} def +/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def +/DiaW {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V Opaque stroke} def +/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V Opaque stroke} def +/TriUW {stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V Opaque stroke} def +/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V Opaque stroke} def +/PentW {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + Opaque stroke grestore} def +/CircW {stroke [] 0 setdash + hpt 0 360 arc Opaque stroke} def +/BoxFill {gsave Rec 1 setgray fill grestore} def +/Density { + /Fillden exch def + currentrgbcolor + /ColB exch def /ColG exch def /ColR exch def + /ColR ColR Fillden mul Fillden sub 1 add def + /ColG ColG Fillden mul Fillden sub 1 add def + /ColB ColB Fillden mul Fillden sub 1 add def + ColR ColG ColB setrgbcolor} def +/BoxColFill {gsave Rec PolyFill} def +/PolyFill {gsave Density fill grestore grestore} def +/h {rlineto rlineto rlineto closepath gsave fill grestore stroke} bind def +% +% PostScript Level 1 Pattern Fill routine for rectangles +% Usage: x y w h s a XX PatternFill +% x,y = lower left corner of box to be filled +% w,h = width and height of box +% a = angle in degrees between lines and x-axis +% XX = 0/1 for no/yes cross-hatch +% +/PatternFill {gsave /PFa [ 9 2 roll ] def + PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate + PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse + clip + currentlinewidth 0.5 mul setlinewidth + /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def + 0 0 M PFa 5 get rotate PFs -2 div dup translate + 0 1 PFs PFa 4 get div 1 add floor cvi + {PFa 4 get mul 0 M 0 PFs V} for + 0 PFa 6 get ne { + 0 1 PFs PFa 4 get div 1 add floor cvi + {PFa 4 get mul 0 2 1 roll M PFs 0 V} for + } if + stroke grestore} def +% +/languagelevel where + {pop languagelevel} {1} ifelse +dup 2 lt + {/InterpretLevel1 true def + /InterpretLevel3 false def} + {/InterpretLevel1 Level1 def + 2 gt + {/InterpretLevel3 Level3 def} + {/InterpretLevel3 false def} + ifelse } + ifelse +% +% PostScript level 2 pattern fill definitions +% +/Level2PatternFill { +/Tile8x8 {/PaintType 2 /PatternType 1 /TilingType 1 /BBox [0 0 8 8] /XStep 8 /YStep 8} + bind def +/KeepColor {currentrgbcolor [/Pattern /DeviceRGB] setcolorspace} bind def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke} +>> matrix makepattern +/Pat1 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke + 0 4 M 4 8 L 8 4 L 4 0 L 0 4 L stroke} +>> matrix makepattern +/Pat2 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 0 8 L + 8 8 L 8 0 L 0 0 L fill} +>> matrix makepattern +/Pat3 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 8 M 8 -4 L + 0 12 M 12 0 L stroke} +>> matrix makepattern +/Pat4 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 0 M 8 12 L + 0 -4 M 12 8 L stroke} +>> matrix makepattern +/Pat5 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 8 M 4 -4 L + 0 12 M 8 -4 L 4 12 M 10 0 L stroke} +>> matrix makepattern +/Pat6 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L + 0 -4 M 8 12 L 4 -4 M 10 8 L stroke} +>> matrix makepattern +/Pat7 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L + 12 0 M -4 8 L 12 4 M 0 10 L stroke} +>> matrix makepattern +/Pat8 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L + -4 0 M 12 8 L -4 4 M 8 10 L stroke} +>> matrix makepattern +/Pat9 exch def +/Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def +/Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def +/Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def +/Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def +/Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def +/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def +/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def +} def +% +% +%End of PostScript Level 2 code +% +/PatternBgnd { + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse +} def +% +% Substitute for Level 2 pattern fill codes with +% grayscale if Level 2 support is not selected. +% +/Level1PatternFill { +/Pattern1 {0.250 Density} bind def +/Pattern2 {0.500 Density} bind def +/Pattern3 {0.750 Density} bind def +/Pattern4 {0.125 Density} bind def +/Pattern5 {0.375 Density} bind def +/Pattern6 {0.625 Density} bind def +/Pattern7 {0.875 Density} bind def +} def +% +% Now test for support of Level 2 code +% +Level1 {Level1PatternFill} {Level2PatternFill} ifelse +% +/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont +dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall +currentdict end definefont pop +% +Level1 SuppressPDFMark or +{} { +/SDict 10 dict def +systemdict /pdfmark known not { + userdict /pdfmark systemdict /cleartomark get put +} if +SDict begin [ + /Title (./main-gnuplottex-fig2.tex) + /Subject (gnuplot plot) + /Creator (gnuplot 5.4 patchlevel 10) +% /Producer (gnuplot) +% /Keywords () + /CreationDate (Mon Nov 27 02:22:52 2023) + /DOCINFO pdfmark +end +} ifelse +% +% Support for boxed text - Ethan A Merritt Sep 2016 +% +/InitTextBox { userdict /TBy2 3 -1 roll put userdict /TBx2 3 -1 roll put + userdict /TBy1 3 -1 roll put userdict /TBx1 3 -1 roll put + /Boxing true def } def +/ExtendTextBox { dup type /stringtype eq + { Boxing { gsave dup false charpath pathbbox + dup TBy2 gt {userdict /TBy2 3 -1 roll put} {pop} ifelse + dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse + dup TBy1 lt {userdict /TBy1 3 -1 roll put} {pop} ifelse + dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse + grestore } if } + {} ifelse} def +/PopTextBox { newpath TBx1 TBxmargin sub TBy1 TBymargin sub M + TBx1 TBxmargin sub TBy2 TBymargin add L + TBx2 TBxmargin add TBy2 TBymargin add L + TBx2 TBxmargin add TBy1 TBymargin sub L closepath } def +/DrawTextBox { PL PopTextBox stroke /Boxing false def} def +/FillTextBox { gsave PopTextBox fill grestore /Boxing false def} def +0 0 0 0 InitTextBox +/TBxmargin 20 def +/TBymargin 20 def +/Boxing false def +/textshow { ExtendTextBox Gshow } def +% +end +%%EndProlog +%%Page: 1 1 +gnudict begin +gsave +doclip +50 50 translate +0.050 0.050 scale +0 setgray +newpath +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +0.500 UL +LTa +LCa setrgbcolor +1078 704 M +5725 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +1078 704 M +63 0 V +5662 0 R +-63 0 V +stroke +LTb +0.500 UL +LTa +LCa setrgbcolor +1078 786 M +5725 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +1078 786 M +31 0 V +5694 0 R +-31 0 V +stroke +0.500 UL +LTa +LCa setrgbcolor +1078 869 M +5725 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +1078 869 M +31 0 V +5694 0 R +-31 0 V +stroke +0.500 UL +LTa +LCa setrgbcolor +1078 951 M +5725 0 V 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a/5.2.1/main.pdf b/5.2.1/main.pdf new file mode 100644 index 0000000..fb05fba Binary files /dev/null and b/5.2.1/main.pdf differ diff --git a/5.2.1/main.tex b/5.2.1/main.tex new file mode 100644 index 0000000..48a09bd --- /dev/null +++ b/5.2.1/main.tex @@ -0,0 +1,197 @@ +\documentclass[a4paper]{article} +\usepackage[left=2cm,right=2cm,top=2cm,bottom=3cm,bindingoffset=0cm]{geometry} +\usepackage{cmap} +\usepackage{mathtext} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage[russian]{babel} +\usepackage{indentfirst} +\usepackage[pdftex]{graphicx} +\usepackage{multirow} +%\usepackage{fancyhdr} +\bibliography{bib} + +\usepackage{graphicx,xcolor} +\usepackage[cleanup]{gnuplottex} +\usepackage{subcaption} + +\begin{document} +\begin{titlepage} + \centering + \vspace{5cm} + {\scshape\LARGE Московский физико-технический институт \par} + \vspace{4cm} + {\scshape\Large Лабораторная работа 5.2.1 \par} + \vspace{1cm} + {\huge\bfseries Эффект Франка--Герца \par} + \vspace{12cm} + {\LARGE Гришаев Григорий С01-119} +\end{titlepage} + +\newpage + +\section{Аннотация} +В данной работе проводится измерение энергии первого уровня атома гелия методом электронного возбуждения в динамическом и статическом режимах. + +\section{Теоретические сведения} + +Разреженный одноатомный газ (в нашем случае -- гелий) заполняет трёхэлектродную лампу. Электроны, испускаемые разогретым катодом, ускоряются в постоянном электрическом поле, созданным между катодом и сетчатым анодом лампы. Передвигаясь от катода к аноду, электроны сталкиваются с атомами гелия. Если энергия электрона, налетающего на атом, недостаточна для того, чтобы перевести его в возбуждённое состояние (или ионизовать), то возможны только упругие соударения, при которых электроны почти не теряют энергии, так как их масса в тысячи раз меньше массы атомов. + + +По мере увеличения разности потенциалов между анодом и катодом энергия электронов увеличивается и, в конце концов, оказывается достаточной для возбуждения атомов. При таких -- неупругих -- столкновениях кинетическая энергия налетающего электрона передаётся одному из атомных электронов, вызывая его переход на свободный энергетический уровень (возбуждение) или совсем отрывая его от атома (ионизация). + +При увеличении потенциала анода ток в лампе вначале растёт, подобно тому как это происходит в вакуумном диоде (рис. 2). Однако, когда энергия электронов становится достаточной для возбуждения атомов, ток коллектора резко уменьшается. Это происходит потому, что при неупругих соударениях с атомами электроны почти полностью теряют свою энергию и не могут преодолеть задерживающего потенциала между анодом и коллектором. При дальнейшем увеличении потенциала анода ток коллектора вновь возрастает: электроны, испытавшие неупругие соударения, при дальнейшем движении к аноду успевают набрать энергию, достаточную для преодоления задерживающего потенциала. + + +\begin{figure}[tbh] + \centering + \includegraphics[width=0.4\textheight]{1} + \caption{Характер зависимости $I (U)$} + \label{fig:pic1} +\end{figure} + + + +\section{Оборудование и инструментальные погрешности} + +Схема экспериментальной установки отображена на рис. \ref{fig:pic2} и \ref{fig:pic3}. + +\begin{figure}[h!] + \centering + \includegraphics[height=0.4\textheight]{2} + \caption{Принципиальная схема установки} + \label{fig:pic2} +\end{figure} + +\begin{figure}[h!] + \centering + \includegraphics[width=0.6\textheight]{3} + \caption{Блок-схема экспериментальной установки} + \label{fig:pic3} +\end{figure} + +\clearpage + +\section{Результаты измерений и обработка данных} + +\subsection{Динамический метод} + +По результатам, полученным на экране осциллографа (рис. \ref{fig:pic4}, \ref{fig:pic5}, \ref{fig:pic6}. Везде на фото $VOLTS/DIV(x) = 5; VOLTS/DIV(Y) = 5mV$): + + \begin{table}[h] + \centering + \begin{tabular}{|l|l|l|} + \hline + $V_з, \;В$ & $\Delta V, \; В$ & $E,\; эВ$ \\ \hline + 4 & $17\pm 2$ & $17\pm 2$ \\ \hline + 6 & $16\pm 2$ & $16\pm 2$ \\ \hline + 8 & $17\pm 2$ & $17\pm 2$ \\ \hline + \end{tabular} + \label{tab:1} + \caption{Результаты динамического измерения} + \end{table} + +По итогу, +\begin{equation*}\label{key} + E \approx 16.7 \pm 2 \; эВ. +\end{equation*} + +Итоговая погрешность равна инструментальной погрешности измерений, т.к. случайная погрешность много меньше. + + +\subsection{Статический метод} + +Полученные статическим методом результаты отобразим на графиках (Рис.\ref{graph_4_6_8}). + +\begin{table}[h] + \centering + \begin{tabular}{|l|l|l|} + \hline + $V_з, \;В$ & $\Delta V, \; В$ & $E,\; эВ$ \\ \hline + 4 & 16.49 & 16.49 \\ \hline + 6 & 17.34 & 17.34 \\ \hline + 8 & 17.04 & 17.04 \\ \hline + \end{tabular} + \caption{Результат измерения статическим методом} + \label{tab:my-table} +\end{table} + +\begin{equation*}\label{key} + E\approx 16.96\pm 0.25 \; эВ. +\end{equation*} + +Оценим погрешность: + +\begin{equation*}\label{eq:погр} + \sigma = \sqrt{\frac{1}{N(N-1)}\sum (x-\left\langle x\right\rangle )^2}, +\end{equation*} + +Итоговая погрешность равна случайной, т.к. инструментальная чрезвычайно мала ($0.03\%$). + +\section{Вывод} + +Полученное динамическим методом значение энергии 1 уровня: +\begin{equation*}\label{key} + E \approx 16.7 \pm 2 \; эВ. +\end{equation*} + +Полученное статическим методом значение энергии 1 уровня: +\begin{equation*}\label{key} + E\approx 16.96\pm 0.25 \; эВ. +\end{equation*} + +Референсное значение энергиии 1 уровня: +\begin{equation*}\label{key} + E = 19,5 \;эВ. +\end{equation*} + +И динамический, и статический методы являются достаточно грубыми, но второй метод в целом менее точен, так как следует заранее знать, в окрестности какой точки измерения надо проводить более тщательно. +Приближенное значение, которое мы получили, является достаточно адекватным. + +\newpage + +% Фото + +\begin{figure}[h!] + \centering + \includegraphics[width=0.7\linewidth]{4V} + \caption{Результат для $V_з = 4 \;В$} + \label{fig:pic4} +\end{figure} + +\begin{figure}[h!] + \centering + \includegraphics[width=0.7\linewidth]{6V} + \caption{Результат для $V_з = 6\; В$} + \label{fig:pic5} +\end{figure} + +\begin{figure}[h!] + \centering + \includegraphics[width=0.7\linewidth]{8V} + \caption{Результат для $V_з = 8\; В$} + \label{fig:pic6} +\end{figure} + +% Графики + +\begin{figure}[h!] +\centering +\begin{gnuplot}[terminal=epslatex] + set grid + set key left top + set mxtics 5 + set mytics 5 + + set xlabel 'V, В' + set ylabel 'I, А' + set grid mxtics mytics + set autoscale + + plot "4.dat" title "$V_{з} = 4В$" smooth unique with linespoints, "6.dat" title "$V_{з} = 6В$" smooth unique with linespoints, "8.dat" title "$V_{з} = 8В$" smooth unique with linespoints +\end{gnuplot} +\caption{Результаты статического измерения для V = 4,6,8 В} +\label{graph_4_6_8} +\end{figure} + +\end{document} diff --git a/5.2.1/missfont.log b/5.2.1/missfont.log new file mode 100644 index 0000000..bd7091d --- /dev/null +++ b/5.2.1/missfont.log @@ -0,0 +1,2 @@ +mktextfm rsfs10 +mktextfm rsfs10 diff --git a/5.2.1/photo_2023-11-26_22-20-00.jpg b/5.2.1/photo_2023-11-26_22-20-00.jpg new file mode 100644 index 0000000..dc5f018 Binary files /dev/null and b/5.2.1/photo_2023-11-26_22-20-00.jpg differ diff --git a/5.2.1/photo_2023-11-26_22-20-13.jpg b/5.2.1/photo_2023-11-26_22-20-13.jpg new file mode 100644 index 0000000..1867238 Binary files /dev/null and b/5.2.1/photo_2023-11-26_22-20-13.jpg differ diff --git a/5.2.1/photo_2023-11-26_22-20-16.jpg b/5.2.1/photo_2023-11-26_22-20-16.jpg new file mode 100644 index 0000000..d5f57ca Binary files /dev/null and b/5.2.1/photo_2023-11-26_22-20-16.jpg differ