(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 114815, 3134] NotebookOptionsPosition[ 110729, 2996] NotebookOutlinePosition[ 111155, 3014] CellTagsIndexPosition[ 111112, 3011] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["N-4: Rutherford Scattering", "Title", CellChangeTimes->{{3.436399157164626*^9, 3.4363991626214437`*^9}, { 3.436399200698379*^9, 3.436399203767496*^9}}], Cell["\<\ This is my data analysis for the N-4 Rutherford scattering lab for Physics \ 510. With a bit of luck this should be the last lab experiment that I ever \ have to do.\ \>", "Text", CellChangeTimes->{{3.436399207359894*^9, 3.4363992442127037`*^9}}], Cell[CellGroupData[{ Cell["Data Input", "Section", CellChangeTimes->{{3.43639942273493*^9, 3.4363994243460493`*^9}}], Cell[TextData[{ "Each data point in ", StyleBox["data", FontWeight->"Bold"], " is structured as{\[Delta]r", Cell[BoxData[ FormBox["2", TraditionalForm]]], ",\[Delta]r1,count}. ", StyleBox["baseline", FontWeight->"Bold"], " has values of r20, r10 (in that order). ", StyleBox["height", FontWeight->"Bold"], " contains the min and max value for the heights." }], "Text", CellChangeTimes->{{3.436399552213422*^9, 3.436399552810226*^9}, { 3.436400129896679*^9, 3.436400313428916*^9}, {3.436400774893257*^9, 3.436400777730208*^9}, {3.4364009719295597`*^9, 3.436400974957438*^9}, { 3.436401025631405*^9, 3.436401035335196*^9}, {3.43640112807323*^9, 3.436401137889471*^9}, {3.436401303564947*^9, 3.436401319862968*^9}, { 3.4364036295967503`*^9, 3.4364036318500443`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"mydata", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"20", ",", "20", ",", "63"}], "}"}], ",", RowBox[{"{", RowBox[{"20", ",", "40", ",", "64"}], "}"}], ",", RowBox[{"{", RowBox[{"40", ",", "40", ",", "65"}], "}"}], ",", RowBox[{"{", RowBox[{"40", ",", "60", ",", "65"}], "}"}], ",", RowBox[{"{", RowBox[{"60", ",", "60", ",", "60"}], "}"}], ",", RowBox[{"{", RowBox[{"60", ",", "80", ",", "56"}], "}"}], ",", RowBox[{"{", RowBox[{"80", ",", "80", ",", "66"}], "}"}], ",", RowBox[{"{", RowBox[{"80", ",", "100", ",", "64"}], "}"}], ",", RowBox[{"{", RowBox[{"100", ",", "100", ",", "85"}], "}"}], ",", RowBox[{"{", RowBox[{"100", ",", "120", ",", "70"}], "}"}], ",", RowBox[{"{", RowBox[{"120", ",", "120", ",", "72"}], "}"}], ",", RowBox[{"{", RowBox[{"120", ",", "140", ",", "92"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"baseline", "=", RowBox[{"{", RowBox[{"37", ",", "11"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"heights", "=", RowBox[{"{", RowBox[{"46.03", ",", "53.97"}], "}"}]}], ";"}]}], "Input", CellChangeTimes->{{3.436399579167547*^9, 3.436399840208173*^9}, { 3.436399945615521*^9, 3.4363999515484543`*^9}, {3.436400919992208*^9, 3.436400924033031*^9}, {3.436401066628854*^9, 3.436401067752275*^9}, { 3.436401103518409*^9, 3.436401103675568*^9}, {3.4364011686712837`*^9, 3.436401172932786*^9}, {3.436401357730234*^9, 3.436401359788804*^9}, { 3.4364015471177053`*^9, 3.436401549540851*^9}, {3.436401624273498*^9, 3.436401638531315*^9}, {3.436403975967853*^9, 3.4364039760455503`*^9}}], Cell["\<\ Now I define a function to give me the relevant values. The idea is to map \ the data list using the function Here's the info that I want: alpha(min), \ alpha(max), beta(min), beta(max), counts.\ \>", "Text", CellChangeTimes->{{3.4364021184213533`*^9, 3.4364021443801527`*^9}, { 3.436403809798643*^9, 3.436403813262782*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"angles", "[", "x_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"ArcTan", "[", FractionBox[ RowBox[{"heights", "[", RowBox[{"[", "1", "]"}], "]"}], RowBox[{ RowBox[{"baseline", "[", RowBox[{"[", "1", "]"}], "]"}], "+", RowBox[{"x", "[", RowBox[{"[", "1", "]"}], "]"}]}]], "]"}], ",", RowBox[{"ArcTan", "[", FractionBox[ RowBox[{"heights", "[", RowBox[{"[", "2", "]"}], "]"}], RowBox[{ RowBox[{"baseline", "[", RowBox[{"[", "1", "]"}], "]"}], "+", RowBox[{"x", "[", RowBox[{"[", "1", "]"}], "]"}]}]], "]"}], ",", RowBox[{"ArcTan", "[", FractionBox[ RowBox[{"heights", "[", RowBox[{"[", "1", "]"}], "]"}], RowBox[{ RowBox[{"baseline", "[", RowBox[{"[", "2", "]"}], "]"}], "+", RowBox[{"x", "[", RowBox[{"[", "2", "]"}], "]"}]}]], "]"}], ",", RowBox[{"ArcTan", "[", FractionBox[ RowBox[{"heights", "[", RowBox[{"[", "2", "]"}], "]"}], RowBox[{ RowBox[{"baseline", "[", RowBox[{"[", "2", "]"}], "]"}], "+", RowBox[{"x", "[", RowBox[{"[", "2", "]"}], "]"}]}]], "]"}], ",", RowBox[{"x", "[", RowBox[{"[", "3", "]"}], "]"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.436403317337895*^9, 3.436403329221511*^9}, { 3.436403379744955*^9, 3.436403437069409*^9}, {3.436403495588559*^9, 3.436403500114107*^9}, {3.4364035406829033`*^9, 3.436403546204072*^9}, { 3.436403578676097*^9, 3.436403582230205*^9}, {3.436403646908934*^9, 3.436403651310071*^9}, {3.436403714566514*^9, 3.4364037173273487`*^9}, { 3.436403849766362*^9, 3.436403927993814*^9}, {3.43640396480511*^9, 3.43640396598491*^9}}], Cell[TextData[{ "Now apply this to ", StyleBox["data", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{{3.4364038026301622`*^9, 3.4364038233647633`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"myangles", "=", RowBox[{"Map", "[", RowBox[{"angles", ",", "mydata"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.436403830116905*^9, 3.436403835366207*^9}, { 3.4364039695548353`*^9, 3.436403980466256*^9}, 3.436404069013702*^9}], Cell["\<\ Now define a function to write out theta(min), theta (max) and counts.\ \>", "Text", CellChangeTimes->{{3.436438375720804*^9, 3.436438400437644*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"thetas", "[", "x_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", RowBox[{"[", "1", "]"}], "]"}], "+", RowBox[{"x", "[", RowBox[{"[", "3", "]"}], "]"}]}], ",", RowBox[{ RowBox[{"x", "[", RowBox[{"[", "2", "]"}], "]"}], "+", RowBox[{"x", "[", RowBox[{"[", "4", "]"}], "]"}]}], ",", RowBox[{"x", "[", RowBox[{"[", "5", "]"}], "]"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.436438414950141*^9, 3.436438420559424*^9}, { 3.436438489681102*^9, 3.4364385505087957`*^9}}], Cell["\<\ Once again, apply this to the output of the previous function.\ \>", "Text", CellChangeTimes->{{3.4364385555985813`*^9, 3.4364385624650383`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mythetas", "=", RowBox[{"Map", "[", RowBox[{"thetas", ",", "myangles"}], "]"}]}]], "Input", CellChangeTimes->{{3.43643856403165*^9, 3.4364385730180807`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1.6574198311057553`", ",", "1.80751847589779`", ",", "63"}], "}"}], ",", RowBox[{"{", RowBox[{"1.413545575520392`", ",", "1.5717846842026733`", ",", "64"}], "}"}], ",", RowBox[{"{", RowBox[{"1.273016158595972`", ",", "1.42502024712576`", ",", "65"}], "}"}], ",", RowBox[{"{", RowBox[{"1.1139806969431527`", ",", "1.2612955533886825`", ",", "65"}], "}"}], ",", RowBox[{"{", RowBox[{"1.0182559079245823`", ",", "1.1576969797140442`", ",", "60"}], "}"}], ",", RowBox[{"{", RowBox[{"0.9113658399855868`", ",", "1.043050709671939`", ",", "56"}], "}"}], ",", RowBox[{"{", RowBox[{"0.8431161752725854`", ",", "0.9675098242094702`", ",", "66"}], "}"}], ",", RowBox[{"{", RowBox[{"0.7679212313466633`", ",", "0.8847562744657482`", ",", "64"}], "}"}], ",", RowBox[{"{", RowBox[{"0.7172366564160972`", ",", "0.8278326392379334`", ",", "85"}], "}"}], ",", RowBox[{"{", RowBox[{"0.6620339024762645`", ",", "0.7660678962756617`", ",", "70"}], "}"}], ",", RowBox[{"{", RowBox[{"0.6230909583139832`", ",", "0.72189835576217`", ",", "72"}], "}"}], ",", RowBox[{"{", RowBox[{"0.58107870476681`", ",", "0.6743704414089593`", ",", "92"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4364385751778193`*^9, 3.436548958815445*^9}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Integral", "Section", CellChangeTimes->{{3.4363995418085546`*^9, 3.436399544497718*^9}}], Cell["\<\ The main dependence of the differential cross section is an inverse \ sine-theta to the fourth behavior. The differential angular element has a \ factor of sine-theta, so the integral over theta is an inverse sine-theta to \ the third. We'll perform this integral for each data point above.\ \>", "Text", CellChangeTimes->{{3.436438626198956*^9, 3.436438654410434*^9}, { 3.436438736655365*^9, 3.436438800105188*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"thetapart", "[", RowBox[{"a_", ",", "b_"}], "]"}], ":=", RowBox[{ SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[{ FractionBox["1", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "3"]], RowBox[{"\[DifferentialD]", "\[Theta]"}]}]}]}]], "Input", CellChangeTimes->{{3.4363989077594557`*^9, 3.436398917304193*^9}, { 3.436399135307116*^9, 3.436399138599194*^9}, {3.4364386589481163`*^9, 3.436438665198502*^9}, 3.436438755081674*^9, {3.436438827422269*^9, 3.436438875493062*^9}, {3.4364389202402143`*^9, 3.436438936741794*^9}, { 3.43643898090506*^9, 3.436439031186874*^9}, {3.436440051499681*^9, 3.4364400659560947`*^9}, {3.436440112159116*^9, 3.43644011476303*^9}}], Cell["\<\ Now map these to a list containing the theta integral and the total counts. \ The ratio of these two numbers should be constant for each bin!\ \>", "Text", CellChangeTimes->{{3.43643943968927*^9, 3.436439441028475*^9}, { 3.43643947139244*^9, 3.4364394718849277`*^9}, {3.436439519464959*^9, 3.4364395398817577`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"takethetapart", "[", "x_", "]"}], ":=", RowBox[{"thetapart", "[", RowBox[{ RowBox[{"x", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"x", "[", RowBox[{"[", "2", "]"}], "]"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"ratio", "[", "x_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"takethetapart", "[", "x", "]"}], ",", RowBox[{"x", "[", RowBox[{"[", "3", "]"}], "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"myratios", "=", RowBox[{"Map", "[", RowBox[{"ratio", ",", "mythetas"}], "]"}]}]}], "Input", CellChangeTimes->{{3.436439558976824*^9, 3.436439673388617*^9}, { 3.436439899942479*^9, 3.436439924687853*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.1566155154874675`", ",", "63"}], "}"}], ",", RowBox[{"{", RowBox[{"0.16021012877023158`", ",", "64"}], "}"}], ",", RowBox[{"{", RowBox[{"0.16431444606024526`", ",", "65"}], "}"}], ",", RowBox[{"{", RowBox[{"0.1854685223154955`", ",", "65"}], "}"}], ",", RowBox[{"{", RowBox[{"0.20172940009764623`", ",", "60"}], "}"}], ",", RowBox[{"{", RowBox[{"0.23261235527036117`", ",", "56"}], "}"}], ",", RowBox[{"{", RowBox[{"0.2572917834518259`", ",", "66"}], "}"}], ",", RowBox[{"{", RowBox[{"0.2959185096665903`", ",", "64"}], "}"}], ",", RowBox[{"{", RowBox[{"0.32789421543810837`", ",", "85"}], "}"}], ",", RowBox[{"{", RowBox[{"0.37357631431175453`", ",", "70"}], "}"}], ",", RowBox[{"{", RowBox[{"0.4124188983363948`", ",", "72"}], "}"}], ",", RowBox[{"{", RowBox[{"0.4648669389460398`", ",", "92"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.436439927793667*^9, {3.4364400547345333`*^9, 3.43644007345492*^9}, 3.4364401184399137`*^9, 3.436548966337524*^9}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Check for consistency", "Section", CellChangeTimes->{{3.43643994637421*^9, 3.436439948607293*^9}}], Cell[TextData[{ "The ratio of each pair in ", StyleBox["myratios", FontWeight->"Bold"], " should be constant (representing all the non-angular coefficients in the \ Rutherford scattering cross section... with a 2pi or two for good measure). 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Notation: \ subscript '2' refers to the \"left side,\" i.e. moving the foil+source, \ subscript '1' refers to the \"right side,\" i.e. moving the source-only. \ \[Alpha] corresponds to the '2' coordinates, \[Beta] refers to the '1' \ coordinates. 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