Lab moderna: add P1
Change-Id: Ibbbfa656bfcd29eef2afb9dbd78488e4484054b6
diff --git a/quad12/labmoderna/p1a/.gitignore b/quad12/labmoderna/p1a/.gitignore
new file mode 100644
index 0000000..f0de8a3
--- /dev/null
+++ b/quad12/labmoderna/p1a/.gitignore
@@ -0,0 +1 @@
+main.pdf
diff --git a/quad12/labmoderna/p1a/README.md b/quad12/labmoderna/p1a/README.md
new file mode 100644
index 0000000..55c81d5
--- /dev/null
+++ b/quad12/labmoderna/p1a/README.md
@@ -0,0 +1,8 @@
+# Pràctica 1a. Determinació de la semivida d'un radionúclid.
+
+Overleaf project (where the original report was written):
+https://www.overleaf.com/project/640d23c5debc8133b9e6c760.
+
+The overleaf project is guaranteed to compile. This folder contains the code for
+the graph generation (which is tested to work) and a copy of the LaTeX code
+(which is not guaranteed to compile).
diff --git a/quad12/labmoderna/p1a/dades/bari_a.txt b/quad12/labmoderna/p1a/dades/bari_a.txt
new file mode 100644
index 0000000..1158dea
--- /dev/null
+++ b/quad12/labmoderna/p1a/dades/bari_a.txt
@@ -0,0 +1,39 @@
+t ImpulsesPerSecond ProgAverage ProgVariance FitImpulsesPerSecond
+#Time Impulses / s Progressive average valu Progressive variance Fit(Impulses / s)
+#t/s A/#/s A/#/s A²/(#/s)² A/#/s
+10,0 45,37 45,367 0,000 43,78
+20,0 38,87 42,117 10,563 41,83
+30,1 40,77 41,667 7,447 39,96
+40,1 38,17 40,792 7,882 38,18
+50,2 37,17 40,067 8,408 36,48
+60,2 33,07 38,900 13,812 34,85
+70,2 34,37 38,252 14,356 33,30
+80,3 35,97 37,967 13,133 31,81
+90,3 26,67 36,711 24,285 30,39
+100,3 26,87 35,727 30,578 29,04
+110,4 33,57 35,530 28,184 27,74
+120,4 25,27 34,675 33,883 26,51
+130,4 21,97 33,697 42,744 25,33
+140,5 19,37 32,674 53,312 24,20
+150,5 22,27 31,980 56,497 23,12
+160,5 25,37 31,567 55,529 22,09
+170,6 21,67 30,984 57,689 21,10
+180,6 24,57 30,628 56,645 20,16
+190,6 18,67 29,998 60,797 19,26
+200,7 16,07 29,302 66,976 18,40
+210,7 17,87 28,757 69,717 17,58
+220,8 18,57 28,294 71,054 16,80
+230,8 17,67 27,832 72,662 16,05
+240,8 15,67 27,325 75,543 15,33
+250,9 12,47 26,731 80,999 14,65
+260,9 14,57 26,263 83,356 14,00
+270,9 14,07 25,811 85,574 13,37
+281,0 15,77 25,452 85,992 12,78
+291,0 13,47 25,039 87,810 12,21
+301,1 12,57 24,623 89,895 11,66
+311,1 12,87 24,244 91,310 11,14
+321,1 8,57 23,754 95,898 10,65
+331,1 9,07 23,309 99,331 10,17
+341,2 7,07 22,831 103,940 9,72
+351,2 7,47 22,392 107,523 9,28
+361,3 7,07 21,967 110,879 8,87
diff --git a/quad12/labmoderna/p1a/dades/fons_a.txt b/quad12/labmoderna/p1a/dades/fons_a.txt
new file mode 100644
index 0000000..4531288
--- /dev/null
+++ b/quad12/labmoderna/p1a/dades/fons_a.txt
@@ -0,0 +1,33 @@
+t ImpulsesPerSecond ProgAverage ProgVariance
+#Time Impulses / s Progressive average valu Progressive variance
+#t/s A/#/s A/#/s A²/(#/s)²
+10,0 0,47 0,467 0,000
+20,0 -0,53 -0,033 0,250
+30,1 0,57 0,167 0,247
+40,1 1,97 0,617 0,792
+50,2 1,27 0,747 0,702
+60,2 -0,23 0,583 0,718
+70,2 -0,63 0,410 0,797
+80,3 0,47 0,417 0,697
+90,3 1,07 0,489 0,662
+100,3 -0,43 0,397 0,672
+110,4 0,97 0,448 0,638
+120,4 -1,23 0,308 0,801
+130,4 1,47 0,397 0,834
+140,5 -0,13 0,360 0,794
+150,5 -1,03 0,267 0,861
+160,5 1,77 0,360 0,939
+170,6 0,07 0,343 0,889
+180,6 0,37 0,344 0,840
+190,6 -0,93 0,277 0,877
+200,7 -0,23 0,252 0,845
+210,7 -0,13 0,233 0,812
+220,8 -0,73 0,189 0,815
+230,8 0,67 0,210 0,789
+240,8 -1,33 0,146 0,852
+250,9 1,27 0,191 0,866
+260,9 0,97 0,221 0,855
+270,9 0,77 0,241 0,834
+281,0 0,37 0,245 0,805
+291,0 -1,63 0,180 0,894
+301,1 0,27 0,183 0,865
diff --git a/quad12/labmoderna/p1a/grafiques/README b/quad12/labmoderna/p1a/grafiques/README
new file mode 100644
index 0000000..8d55dbd
--- /dev/null
+++ b/quad12/labmoderna/p1a/grafiques/README
@@ -0,0 +1 @@
+Generated from the code in the "R" folder and copied here.
diff --git a/quad12/labmoderna/p1a/grafiques/bari.tex b/quad12/labmoderna/p1a/grafiques/bari.tex
new file mode 100644
index 0000000..e4e5535
--- /dev/null
+++ b/quad12/labmoderna/p1a/grafiques/bari.tex
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diff --git a/quad12/labmoderna/p1a/main.tex b/quad12/labmoderna/p1a/main.tex
new file mode 100644
index 0000000..bb86ff2
--- /dev/null
+++ b/quad12/labmoderna/p1a/main.tex
@@ -0,0 +1,132 @@
+\documentclass[a4paper, 11pt]{article}
+
+\input{preamble.tex}
+\addbibresource{references.bib}
+
+\rhead{Adrià Vilanova Martínez i Diana Tur Otero}
+\lhead{Pràctica 1a}
+\fancyfoot[C]{\thepage}
+
+\setlength{\droptitle}{-4em}
+\title{Pràctica 1a. Determinació de la semivida d'un radionúclid.}
+\author{Adrià Vilanova Martínez i Diana Tur Otero}
+\date{Divendres 10 de març de 2023}
+
+\begin{document}
+
+{
+ \parskip = 0pt
+ \maketitle
+}
+
+\section{Objectiu}
+Determinar el temps de semivida del \ch{^{137m}Ba} i comparar-lo amb un valor trobat a la literatura.
+
+\section{Metodologia}
+En desexcitar-se a l'estat fonamental, el \ch{^{137m}Ba} emet un fotó gamma, que detectem mitjançant un detector d'escintil·lacions. Així doncs, realitzant mesures d'aquesta radiació emesa en funció del temps i fent un ajust de les dades obtingudes a la llei de desintegració radioactiva
+\begin{equation}
+ N(t) = N_0 \exp( -t/\tau )
+ \label{eq:llei_desintegracio}
+\end{equation}
+–on $N$ és el nombre d'àtoms excitats– es pot obtenir el valor de la vida mitjana $\tau$.\footnote{Realment, nosaltres no mesurem tots els fotons emesos per les desintegracions gamma, només els que es dirigeixen cap el detector. Tot i així, com la direcció cap on surten és uniforme (no n'hi ha cap privilegiada), el que sí sabem és que el valor d'impulsos que mesurarem, $I$, serà proporcional a $N$ i, per tant, $\frac{N(t)}{N_0} = \frac{I(t)}{I_0}$.}
+
+La semivida $T_{1/2}$ és el temps en què la meitat dels àtoms han decaigut a l'estat fonamental, així que la seva expressió en funció dels paràmetres trobats a l'ajust és:
+\begin{equation}
+ \frac{N_0}{2} = N_0 \exp\left( \frac{-T_{1/2}}{\tau} \right) \implies \log\left( \frac{1}{2} \right) = \frac{- T_{1/2}}{\tau} \implies T_{1/2} = \tau \log(2).
+ \label{eq:semivida}
+\end{equation}
+
+Al llarg de l'experiment, degut al fet que aquest no està aïllat de la resta de l'Univers, el detector d'escintil·lacions rep radiació de fons que s'afegeix als fotons que volem detectar. Així doncs, abans de començar-lo hem mesurat el nivell del fons (com a una mitjana d'impulsos per segon) durant $\qty{5}{\minute}$ per tal de restar-lo de les mesures obtingudes a l'experiment. Hem utilitzat una opció del software de mesura \textit{measure} que ja ho resta automàticament per no haver-ho de fer durant el postprocessat de les dades.
+
+A més, per tal d'obtenir la mostra de \ch{^{137m}Ba} hem utilitzat un minigenerador, que és un recipient amb \ch{^{137}Cs}, que decau a \ch{^{137m}Ba} amb un temps de semivida de l'ordre de 30 anys. Fent passar pel minigenerador (molt poc a poc) una dissolució d'\ch{HCl}, a la sortida obtenim la dissolució amb àtoms de bari, ja que el bari és dissol amb l'àcid clorhídric però el cesi no.
+
+\section{Figures i valors obtinguts}
+D'acord amb el que hem explicat a la secció anterior, primer hem mesurat el fons durant $\qty{300}{\second}$. Això ens ha donat el següent valor mitjà d'impulsos per segon:
+\[ I_\text{fons} = \qty{19.333}{\#\per\second}. \]
+
+Posteriorment, tal com suggereix el guió de la pràctica, hem pres més mesures del fons amb l'opció activada de restar el valor mitjà del fons que hem obtingut anteriorment. Això ho hem fet en intervals de \qty{10}{\second} durant \qty{5}{\minute}. Els valors obtinguts es poden veure a la taula \ref{table:fons} i representats a la figura \ref{fig:fons}.
+
+\begin{figure}
+ \centering
+ \includegraphics[trim=0 0.45cm 0 1.5cm]{grafiques/fons_fluctuacio.tex}
+ \caption{Impulsos per segon rebuts sense cap font en cada finestra de 10 segons havent restat la mitjana d'impulsos per segon mesurada del fons. La mitjana està representada per la línia verda del mig i la desviació estàndard per les línies vermelles.}
+ \label{fig:fons}
+\end{figure}
+
+Aquesta darrera mesura ens permet calcular la incertesa deguda a la fluctuació del fons aproximant-la com la desviació estàndard dels valors obtinguts:
+\[ (\delta I)_\text{fluct. fons} = \qty{0.9}{\#\per\second}. \]
+
+La mitjana doncs és $\qty{0.2(9)}{\#\per\second}$, que és compatible amb el valor de 0 que idealment hauríem de veure.
+
+\begin{figure}
+ \centering
+ \includegraphics[trim=0 0.45cm 0 1.5cm]{grafiques/bari.tex}
+ \caption{Nombre de desintegracions gamma detectades per segon en cada finestra de 10 segons.}
+ \label{fig:bari}
+\end{figure}
+
+Finalment, després d'aquesta calibratge, hem munyit el bari i hem fet la mesura amb la mostra de bari. Els valors mesurats es poden veure a la taula \ref{table:bari} i a la figura \ref{fig:bari}.
+
+Ajustant l'equació $I = I_0 \exp(- t/\tau)$ a les mesures tal com hem explicat anteriorment obtenim els següents paràmetres:
+\[
+ \begin{cases}
+ I_0 = \qty{45.8(12)}{\#\per\second}, \\
+ \tau = \qty{220(10)}{\second}.
+ \end{cases}
+\]
+
+Per tant, utilitzant (\ref{eq:semivida}), el temps de semivida del \ch{^{137m}Ba} és:
+\[ T_{1/2} = \tau \log(2) = \qty{153(7)}{\second} = \qty{2.54(12)}{\minute}. \]
+
+A \cite{nubase} s'ha mesurat el temps de semivida com $\qty{2.552(1)}{\minute}$, que és un valor compatible amb el nostre resultat.
+
+Alternativament a l'ajust exponencial que hem realitzat mitjançant el mètode de mínims quadrats, també podem obtenir el valor del temps de semivida fent una regressió lineal si fem la següent transformació de l'equació (\ref{eq:llei_desintegracio}):
+\[
+ I(t) = I_0 \exp\left( \frac{-t}{\tau} \right) \implies \log(I(t)) = \log(I_0) - \frac{1}{\tau} t.
+\]
+
+\begin{figure}
+ \centering
+ \includegraphics[trim=0 0.45cm 0 1.5cm]{grafiques/bari_linealitzat.tex}
+ \caption{Regressió lineal amb les mesures de les desintegracions gamma del bari.}
+ \label{fig:bari_linealitzat}
+\end{figure}
+
+D'aquesta manera, fent la regressió de $t \sim \log(I)$ obtindrem que la pendent de la recta ajustada és la nostra estimació de $-1/\tau$. La constant de decaïment és $\lambda \equiv 1/\tau$, així que la pendent és directament $-\lambda$.
+
+En fer la regressió lineal (veure la figura \ref{fig:bari_linealitzat}), obtenim els següents paràmetres:
+\[
+ \begin{cases}
+ \log(I_0) = 3.85 \pm 0.04, \\
+ \lambda = \qty{0.0048(2)}{\per\second}.
+ \end{cases}
+\]
+
+\clearpage
+
+\section{Taules de dades}
+\begin{multicols}{2}
+ \begin{center}
+ \pgfplotstabletypeset[
+ columns={t, ImpulsesPerSecond},
+ columns/t/.style={read comma as period, column name={$t \pm 0.1$ (s)}, fixed, fixed zerofill, precision=1},
+ columns/ImpulsesPerSecond/.style={read comma as period, column name={$I \pm 0.01$ (\#/s)}, fixed, fixed zerofill, precision=2}
+ ]{dades/fons_a.txt}
+ \captionof{table}{Mesures del fons en impulsos per segon havent restat el valor mitjà del fons.}
+ \label{table:fons}
+ \end{center}
+
+ \begin{center}
+ \pgfplotstabletypeset[
+ columns={t, ImpulsesPerSecond},
+ columns/t/.style={read comma as period, column name={$t \pm 0.1$ (s)}, fixed, fixed zerofill, precision=1},
+ columns/ImpulsesPerSecond/.style={read comma as period, column name={$I \pm 0.01$ (\#/s)}, fixed, fixed zerofill, precision=2}
+ ]{dades/bari_a.txt}
+ \captionof{table}{Mesures del impulsos per segon rebuts de la font de bari.}
+ \label{table:bari}
+ \end{center}
+\end{multicols}
+
+\printbibliography
+
+\end{document}
diff --git a/quad12/labmoderna/p1a/preamble.tex b/quad12/labmoderna/p1a/preamble.tex
new file mode 100644
index 0000000..363f6d6
--- /dev/null
+++ b/quad12/labmoderna/p1a/preamble.tex
@@ -0,0 +1,81 @@
+% !TEX root = main.tex
+\usepackage[utf8]{inputenc}
+\usepackage[catalan]{babel}
+\usepackage{fancyhdr}
+\usepackage{graphicx}
+\usepackage[labelfont=bf, font=small]{caption}
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+\usepackage{geometry}
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+ heightrounded
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+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{booktabs}
+\usepackage{chemformula}
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+\usepackage[style=english]{csquotes}
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+\usepackage{adjustbox}
+\usepackage{array}
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+\usepackage{titling}
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+\algrenewcommand\algorithmiccomment[1]{{\color{CommentGray} \hfill\(\triangleright\) #1}}%
+
+% To prevent underfull/overfull messages in bibliography
+\usepackage{etoolbox}
+\apptocmd{\sloppy}{\hbadness 10000\relax}{}{}
+
+\usepackage{enumitem}
+\setlist{nolistsep}
diff --git a/quad12/labmoderna/p1a/r/.gitignore b/quad12/labmoderna/p1a/r/.gitignore
new file mode 100644
index 0000000..01564af
--- /dev/null
+++ b/quad12/labmoderna/p1a/r/.gitignore
@@ -0,0 +1 @@
+*.tex
diff --git a/quad12/labmoderna/p1a/r/main.r b/quad12/labmoderna/p1a/r/main.r
new file mode 100644
index 0000000..151853a
--- /dev/null
+++ b/quad12/labmoderna/p1a/r/main.r
@@ -0,0 +1,14 @@
+library(tikzDevice)
+
+table <- read.table('../dades/fons_a.txt', sep="\t", dec=',', header=TRUE)
+m = mean(table$ImpulsesPerSecond)
+m
+unc = sd(table$ImpulsesPerSecond)
+unc
+
+tikz("fons_fluctuacio.tex", width = 5, height = 4)
+plot(table$t, table$ImpulsesPerSecond, pch = 4, xlab = "$t$ (s)", ylim = c(-2.1, 2.1), ylab = "$I$ (\\#/s)")
+abline(h = m + unc, col = "red", lty = 2)
+abline(h = m - unc, col = "red", lty = 2)
+abline(h = m, col = "green", lty = 2)
+dev.off()
diff --git a/quad12/labmoderna/p1a/r/main2.r b/quad12/labmoderna/p1a/r/main2.r
new file mode 100644
index 0000000..76b4fff
--- /dev/null
+++ b/quad12/labmoderna/p1a/r/main2.r
@@ -0,0 +1,25 @@
+library(tikzDevice)
+
+table <- read.table('../dades/bari_a.txt', sep="\t", dec=',', header=TRUE)
+
+fit <- nls(ImpulsesPerSecond ~ ni*exp(-t/tau), start = list(ni = 40, tau = 20), data = table)
+summary(fit)
+
+tikz("bari.tex", width = 5.5, height = 3.5)
+#pdf('pdf.pdf', width = 6, height = 4)
+fit_coef <- coef(fit)
+plot(table$t, table$ImpulsesPerSecond, pch = 4, xlab = "$t$ (s)", ylab = "$I$ (\\# s\\textsuperscript{-1})")
+curve(fit_coef[1]*exp(-x/fit_coef[2]), col = "mediumorchid2", add = TRUE)
+legend("topright", inset = c(0.015, 0.03),col = c("black", "mediumorchid2"), legend = c("Mesures", "Ajust"), pch=c(4, NA), lty=c(NA, 1))
+dev.off()
+
+table$logI <- log(table$ImpulsesPerSecond)
+
+tikz("bari_linealitzat.tex", width = 5.5, height = 3.5)
+#pdf('pdf2.pdf', width = 6, height = 4)
+fit_lineal <- lm(logI~t, data = table)
+summary(fit_lineal)
+plot(table$t, table$logI, pch = 4, xlab = "$t$ (s)", ylab = "$\\log(I)$")
+abline(fit_lineal, col = "mediumorchid2")
+legend("topright", inset = c(0.015, 0.03),col = c("black", "mediumorchid2"), legend = c("Mesures", "Ajust"), pch=c(4, NA), lty=c(NA, 1))
+dev.off()
diff --git a/quad12/labmoderna/p1b/.gitignore b/quad12/labmoderna/p1b/.gitignore
new file mode 100644
index 0000000..f0de8a3
--- /dev/null
+++ b/quad12/labmoderna/p1b/.gitignore
@@ -0,0 +1 @@
+main.pdf
diff --git a/quad12/labmoderna/p1b/README.md b/quad12/labmoderna/p1b/README.md
new file mode 100644
index 0000000..fc02130
--- /dev/null
+++ b/quad12/labmoderna/p1b/README.md
@@ -0,0 +1,8 @@
+# Pràctica 1b. Absorció de radiació gamma.
+
+Overleaf project (where the original report was written):
+https://www.overleaf.com/project/6413a1f116995ac1f3863b3f.
+
+The overleaf project is guaranteed to compile. This folder contains the code for
+the graph generation (which is tested to work) and a copy of the LaTeX code
+(which is not guaranteed to compile).
diff --git a/quad12/labmoderna/p1b/dades/dades.txt b/quad12/labmoderna/p1b/dades/dades.txt
new file mode 100644
index 0000000..231ff4c
--- /dev/null
+++ b/quad12/labmoderna/p1b/dades/dades.txt
@@ -0,0 +1,10 @@
+lamines gruixMassic temps impulsosPerSegon
+# Unitats: -, g/cm^2, s, #/s
+E 15.88 600 3.06
+D 10.77 600 5.06
+A+C 7.93 300 7.04
+C 6.80 600 7.51
+A+B 4.53 300 9.53
+B 3.40 230 10.45
+A 1.13 180 12.01
+Cap 0.00 185 12.9
diff --git a/quad12/labmoderna/p1b/grafiques/README b/quad12/labmoderna/p1b/grafiques/README
new file mode 100644
index 0000000..8d55dbd
--- /dev/null
+++ b/quad12/labmoderna/p1b/grafiques/README
@@ -0,0 +1 @@
+Generated from the code in the "R" folder and copied here.
diff --git a/quad12/labmoderna/p1b/grafiques/grafica.tex b/quad12/labmoderna/p1b/grafiques/grafica.tex
new file mode 100644
index 0000000..4414370
--- /dev/null
+++ b/quad12/labmoderna/p1b/grafiques/grafica.tex
@@ -0,0 +1,133 @@
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diff --git a/quad12/labmoderna/p1b/grafiques/graficanormal.tex b/quad12/labmoderna/p1b/grafiques/graficanormal.tex
new file mode 100644
index 0000000..7bb3387
--- /dev/null
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+ (360.32, 72.09);
+\definecolor{drawColor}{RGB}{0,0,0}
+
+\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (297.83,269.57) rectangle (367.44,233.57);
+\definecolor{drawColor}{RGB}{209,95,238}
+
+\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (300.53,245.57) -- (318.53,245.57);
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+\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (307.28,255.32) -- (311.78,259.82);
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+\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (307.28,259.82) -- (311.78,255.32);
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+\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at (327.53,254.13) {Mesures};
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+\end{scope}
+\end{tikzpicture}
diff --git a/quad12/labmoderna/p1b/main.tex b/quad12/labmoderna/p1b/main.tex
new file mode 100644
index 0000000..83c31c5
--- /dev/null
+++ b/quad12/labmoderna/p1b/main.tex
@@ -0,0 +1,98 @@
+\documentclass[a4paper, 11pt]{article}
+
+\input{preamble.tex}
+\addbibresource{references.bib}
+
+\rhead{Adrià Vilanova Martínez i Diana Tur Otero}
+\lhead{Pràctica 1b}
+\fancyfoot[C]{\thepage}
+
+\setlength{\droptitle}{-4em}
+\title{Pràctica 1b. Absorció de radiació gamma.}
+\author{Adrià Vilanova Martínez i Diana Tur Otero}
+\date{Divendres 10 de març de 2023}
+
+\begin{document}
+
+{
+ \parskip = 0pt
+ \maketitle
+}
+
+\section{Objectiu}
+Estudiar l'absorció de radiació gamma pel plom i, en concret, determinar el seu coeficient d'atenuació màssic $\mu/\rho$.
+
+\section{Metodologia}
+Usarem una font puntual de \ch{^{137}Cs} per tenir un feix col·limat de raigs gamma, degut al decaïment del bari al seu estat fonamental, que alhora ha provingut del decaïment del cesi. La diferència amb la pràctica 1a és que en aquest cas aquests decaïments estan en el règim de l'equilibri secular. Per això la intensitat del feix serà constant.
+
+Hem fet incidir aquest feix en 8 combinacions de làmines de plom de diferents gruixos i hem mesurat la intensitat transmesa per unitat de temps. Això ens permetrà determinar el coeficient d'atenuació màssic $\mu/\rho$ perquè la intensitat transmesa és funció del gruix màssic $t := \rho x$ que ha de travessar el feix:
+\begin{equation}
+ I(t) = I_0 \exp\left( - \frac{\mu}{\rho} t \right).
+ \label{eq:intensitat_transmesa}
+\end{equation}
+
+\section{Figures i valors obtinguts}
+Abans de fer les mesures, teníem l'objectiu d'escollir un temps d'execució suficient gran per tal de garantir que la incertesa estadística en la intensitat mesurada sigui com a màxim del 2\%. Si anomenem $\delta(n)$ la incertesa en el nombre de decaïments observats, degut al fet que el decaïment segueix una distribució de Poisson tindrem $\delta(n) = \sqrt{n}$. Aleshores, per tal que $\delta(n)/n < 0.02$, s'ha de complir:
+\begin{equation}
+ 0.02 > \frac{\delta(n)}{n} = n^{-1/2} \iff n > \left( \frac{1}{0.02} \right)^2 = 2500.
+ \label{eq:incertesa_n}
+\end{equation}
+És a dir, hem de mesurar al menys 2500 impulsos per poder garantir que estem per sota del 2\% d'incertesa.
+
+Per tal de saber el temps que trigàvem en rebre al menys 2500 impulsos, hem agafat la làmina més gruixuda –és a dir, la que absorbeix més radiació– i hem mesurat durant $\qty{100}{\second}$ la intensitat mitjana transmesa per segon ($I_\text{mitj.} = \qty{3.06}{\#\per\second}$). Així, en el cas d'aquesta làmina havíem de fer la mesura durant al menys $n/I_\text{mitj.} = \qty{817}{\second}$.
+
+En el cas de les altres làmines, com són menys absorbents, aquest temps també és suficient. Tot i així, per les làmines més fines hem tornat a calcular el temps necessari mitjançant el procediment anterior (calculant-nos la $I_\text{mitj.}$ de cada combinació) per tal de reduir el temps de mesura.
+
+A més, també hem decidit truncar els temps d'execució a $\qty{600}{\second}$ en cas que el superin, degut al temps limitat que tenim al laboratori.
+
+Just abans de començar l'experiment pròpiament, hem tornat a fer un calibratge mesurant la radiació de fons de la mateixa manera que vam fer a la pràctica 1a. Això ens ha donat el valor
+\[ I_\text{fons} = \qty{18.5(9)}{\#\per\second}. \]
+
+\begin{figure}
+ \centering
+ \includegraphics[trim=0 0.45cm 0 1.5cm]{grafiques/graficanormal.tex}
+ \caption{Representació de la intensitat transmesa $I$ mesurada en funció del gruixos màssics $t$, amb l'exponencial ajustada a través de la regressió lineal de l'equació (\ref{eq:intensitat_transmesa_linealitzada}).}
+ \label{fig:directe}
+\end{figure}
+
+Els valors obtinguts durant l'experiment es poden trobar a la taula \ref{table:mesures} i a la figura \ref{fig:directe}. Per poder trobar el coeficient d'atenuació màssic a partir d'ells, podem linealitzar la relació (\ref{eq:intensitat_transmesa}) prenent el logaritme a ambdues bandes:
+\begin{equation}
+ \log(I(t)) = log(I_0) - \frac{\mu}{\rho} t.
+ \label{eq:intensitat_transmesa_linealitzada}
+\end{equation}
+
+\begin{figure}
+ \centering
+ \includegraphics[trim=0 0.45cm 0 1.5cm]{grafiques/grafica.tex}
+ \caption{Regressió lineal amb les mesures de les desintegracions gamma del bari.}
+ \label{fig:regressio}
+\end{figure}
+
+Així, fent una regressió lineal als valors obtinguts –que es pot veure a la figura \ref{fig:regressio}–, obtenim els següents paràmetres:
+\[
+ \begin{cases}
+ \log(I_0) = \qty{2.62(2)}{}, \\[0.5em]
+ \dfrac{\mu}{\rho} = \qty{0.092(4)}{\centi\meter\squared\per\gram}.
+ \end{cases}
+\]
+
+El coeficient màssic donat per la base de dades XCOM del NIST pel plom en el cas dels fotons que fèiem incidir ($E_\gamma = \qty{662}{\kilo\electronvolt}$) és de $\qty{0.1126}{\centi\meter\squared\per\gram}$. Per tant, veiem que en principi el valor que hem trobat no és compatible amb el donat a la base de dades donat que no està dins de 2 vegades la desviació típica (incertesa) de la mesura.
+
+Tot i això, si haguéssim fent un anàlisi de la propagació d'errors més acurat considerant també la incertesa deguda a la fluctuació del fons, és possible que haguéssim obtingut una incertesa molt més ampla que faria ambdós valors compatibles. Això és degut al fet que a l'informe curt només hem tingut en compte la incertesa estadística.
+
+\section{Taules de dades}
+\begin{center}
+ \pgfplotstabletypeset[
+ columns={lamines, gruixMassic, temps, impulsosPerSegon},
+ columns/lamines/.style={column name={Làmines}, string type},
+ columns/gruixMassic/.style={column name={\makecell[b]{Gruix màssic \\ $t \pm 0.01$ (\qty{}{\gram\per\centi\meter\squared})}}, fixed, fixed zerofill, precision=2},
+ columns/temps/.style={column name={\makecell[b]{Temps \\ $t$ (s)}}, fixed, fixed zerofill, precision=0},
+ columns/impulsosPerSegon/.style={column name={\makecell[b]{Intensitat transmesa \\ $I \pm 0.01$ ($\qty{}{\#\per\second}$)}}, fixed, fixed zerofill, precision=2}
+ ]{dades/dades.txt}
+ \captionof{table}{Mesures de la intensitat transmesa mitjana $I$ en funció del gruix màssic $t$. Les mesures s'han dut a terme durant el temps $t$ indicat.}
+ \label{table:mesures}
+\end{center}
+
+\printbibliography
+
+\end{document}
diff --git a/quad12/labmoderna/p1b/preamble.tex b/quad12/labmoderna/p1b/preamble.tex
new file mode 100644
index 0000000..c832f74
--- /dev/null
+++ b/quad12/labmoderna/p1b/preamble.tex
@@ -0,0 +1,82 @@
+% !TEX root = main.tex
+\usepackage[utf8]{inputenc}
+\usepackage[catalan]{babel}
+\usepackage{fancyhdr}
+\usepackage{graphicx}
+\usepackage[labelfont=bf, font=small]{caption}
+\usepackage{siunitx}
+\usepackage{geometry}
+\geometry{
+ margin=20mm,
+ heightrounded
+}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{booktabs}
+\usepackage{chemformula}
+\usepackage{hyperref}
+\usepackage[style=english]{csquotes}
+\usepackage{biblatex}
+\usepackage{pgfplotstable}
+\usepackage{adjustbox}
+\usepackage{array}
+\usepackage{float}
+\usepackage{stackengine}
+\usepackage{titling}
+\usepackage{algpseudocode}
+\usepackage{xcolor}
+\usepackage{multicol}
+\usepackage{gincltex}
+\usepackage{makecell}
+
+\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+\newcolumntype{R}[1]{>{\raggedleft\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+
+\pgfplotsset{compat=1.16}
+\pgfplotstableset{
+ empty cells with={--}, % replace empty cells with ’--’
+ every head row/.style={before row=\toprule,after row=\midrule},
+ every last row/.style={after row=\bottomrule},
+ set thousands separator={\,}%,
+}
+\sisetup{separate-uncertainty=true}
+\setlength{\parskip}{1em}
+\pagestyle{fancy}
+\fancyhf{}
+
+% Plantilles del notate
+\def\myupbracefill#1{\rotatebox{90}{\stretchto{\{}{#1}}}
+\def\rlwd{.5pt}
+\newcommand\notate[4][B]{%
+ \if B#1\else\def\myupbracefill##1{}\fi%
+ \def\useanchorwidth{T}%
+ \setbox0=\hbox{$\displaystyle#2$}%
+ \def\stackalignment{c}\stackunder[-6pt]{%
+ \def\stackalignment{c}\stackunder[-1.5pt]{%
+ \stackunder[2pt]{\strut $\displaystyle#2$}{\myupbracefill{\wd0}}}{%
+ \rule{\rlwd}{#3\baselineskip}}}{%
+ \strut\kern9pt$\rightarrow$\smash{\rlap{$~#4$}}}%
+}
+\newcommand\lnotate[4][B]{%
+ \if B#1\else\def\myupbracefill##1{}\fi%
+ \def\useanchorwidth{T}%
+ \setbox0=\hbox{$\displaystyle#2$}%
+ \def\stackalignment{c}\stackunder[-6pt]{%
+ \def\stackalignment{c}\stackunder[-1.5pt]{%
+ \stackunder[2pt]{\strut $\displaystyle#2$}{\myupbracefill{\wd0}}}{%
+ \rule{\rlwd}{#3\baselineskip}}}{%
+ \strut\kern-9pt$\leftarrow$\smash{\llap{$~#4\quad\;\,$}}}%
+}
+
+\newcommand{\Ham}{\mathcal{H}}
+
+\definecolor{CommentGray}{gray}{0.5}
+\algrenewcommand\algorithmiccomment[1]{{\color{CommentGray} \hfill\(\triangleright\) #1}}%
+
+% To prevent underfull/overfull messages in bibliography
+\usepackage{etoolbox}
+\apptocmd{\sloppy}{\hbadness 10000\relax}{}{}
+
+\usepackage{enumitem}
+\setlist{nolistsep}
diff --git a/quad12/labmoderna/p1b/r/.gitignore b/quad12/labmoderna/p1b/r/.gitignore
new file mode 100644
index 0000000..35f6a0e
--- /dev/null
+++ b/quad12/labmoderna/p1b/r/.gitignore
@@ -0,0 +1,2 @@
+*.tex
+*.pdf
diff --git a/quad12/labmoderna/p1b/r/main.r b/quad12/labmoderna/p1b/r/main.r
new file mode 100644
index 0000000..47cc204
--- /dev/null
+++ b/quad12/labmoderna/p1b/r/main.r
@@ -0,0 +1,39 @@
+library(tikzDevice)
+
+table <- read.table('../dades/dades.txt', sep="\t", header=TRUE)
+
+table$logI <- log(table$impulsosPerSegon)
+
+fit_lineal <- lm(logI~gruixMassic, data = table)
+summary(fit_lineal)
+
+fit_coef <- coef(fit_lineal)
+fit_coef
+
+plotall <- function() {
+ plot(table$gruixMassic, table$logI, pch = 4, xlab = "$t$ (g cm\\textsuperscript{-2})", ylab = "$\\log(I)$")
+ abline(fit_lineal, col = "mediumorchid2")
+ legend("topright", inset = c(0.015, 0.03),col = c("black", "mediumorchid2"), legend = c("Mesures", "Ajust"), pch=c(4, NA), lty=c(NA, 1))
+}
+
+plotallnormal <- function() {
+ plot(table$gruixMassic, table$impulsosPerSegon, pch = 4, xlab = "$t$ (g cm\\textsuperscript{-2})", ylab = "$I$ (\\#s \\textsuperscript{-1})")
+ curve(exp(fit_coef[1] + x*fit_coef[2]), col = "mediumorchid2", add = TRUE)
+ legend("topright", inset = c(0.015, 0.03),col = c("black", "mediumorchid2"), legend = c("Mesures", "Ajust"), pch=c(4, NA), lty=c(NA, 1))
+}
+
+pdf('grafica.pdf', width = 6, height = 5)
+plotall()
+dev.off()
+
+tikz("grafica.tex", width = 5.5, height = 4.5)
+plotall()
+dev.off()
+
+pdf('graficanormal.pdf', width = 6, height = 5)
+plotallnormal()
+dev.off()
+
+tikz("graficanormal.tex", width = 5.5, height = 4.5)
+plotallnormal()
+dev.off()
diff --git a/quad12/labmoderna/p1b/references.bib b/quad12/labmoderna/p1b/references.bib
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/quad12/labmoderna/p1b/references.bib