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Kelly, 1996-2002" }], "Text", TextAlignment->Center, TextJustification->0, FontSize->14, FontSlant->"Italic"], Cell[TextData[{ "The Debye model for lattice vibrations postulates that the density of \ states is proportional to ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\^2\)]], " for phonon frequencies up to a maximum value, ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_D\)]], ", and relates the cut-off frequency to the elastic properties of the \ material. This excitation spectrum is more realistic than the Einstein model \ and provides a better description of the thermodynamic properties of solids. \ The thermal expansivity is derived also from the Gr\[UDoubleDot]neisen \ model." }], "1ColumnBox"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "\tThe contribution of lattice vibrations to the heat capacity of a \ crystalline solid is determined by its spectrum of normal modes. For a \ system of ", Cell[BoxData[ \(TraditionalForm\`N\)]], " atoms there will be ", Cell[BoxData[ \(TraditionalForm\`3 N\)]], " normal modes. The simplest approximation, due to Einstein, is to assume \ that all modes have the same frequency. A better approximation due to Debye \ assumes that the density of states is proportional to the square of the \ frequency up to a maximum frequency determined by the total number of modes. \ This assumption for the excitation spectrum (density of states) is often \ called the", StyleBox[" Debye interpolation formula", FontSlant->"Italic"], " because it provides a method for interpolating the temperature dependence \ between low and high temperature limits which can be derived from rather \ general considerations. This model of the excitation spectrum is fairly \ accurate for long wavelengths, but the behavior of real crystals at short \ wavelengths is much more complicated. Nevertheless, the Debye interpolation \ formula remains useful because many thermodynamic properties, such as the \ specific heat, are not terribly sensitive to details of the excitation \ spectrum." }], "Text"], Cell[TextData[{ "\tThe quantized normal modes of vibration are represented by phonons which \ obey the dispersion relation ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\ = \ k\ v\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`k\)]], " is the wave number and ", Cell[BoxData[ \(TraditionalForm\`v\)]], " is the propagation velocity. If we assume that the sound velocity is \ independent of wave length and that the system is isotropic, the density of \ states ", Cell[BoxData[ \(TraditionalForm\`g[k]\ \ \[DifferentialD]k\ = \ V\ 4 \[Pi]\ k\^2\ \[DifferentialD]k\/\((2 \[Pi])\)\^3\)]], " can be expressed in terms of frequency as ", Cell[BoxData[ \(TraditionalForm\`g[\[Omega]]\ \ \[DifferentialD]\[Omega]\ = \ A\_\[Omega]\ \[Omega]\^2\ \[DifferentialD]\[Omega]\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(A\_\[Omega]\)\)\)]], " is a normalization factor. The normalization of the density of states is \ determined by the condition that the total number of states below the \ cut-off, ", Cell[BoxData[ \(TraditionalForm\`k\_D\)]], " or ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_D\)]], ", must be equal to the total number of normal modes, ", Cell[BoxData[ \(TraditionalForm\`3 N\)]], ", such that" }], "Text"], Cell[BoxData[{ \(\[Integral]\_0\%\(k\_D\)V\ \(\(k\^2\) \[DifferentialD]k\)\/\((2 \[Pi])\ \)\^3\ = \ \(3 N\ \ \[DoubleLongRightArrow]\ \ k\_D\^3 = 18 \( \[Pi]\^2\) N\/V\)\), "\[IndentingNewLine]", \(\[Integral]\_0\%\(\[Omega]\_D\)\(A\_\[Omega]\) \(\[Omega]\^2\) \ \[DifferentialD]\[Omega]\ = \ \(3 N\ \ \[DoubleLongRightArrow]\ \ A\_\[Omega] = \(\(9 N\)\/\[Omega]\ \_D\^3 = \(9 N\)\/\((\(k\_D\) v)\)\^3\)\)\)}], "DisplayFormula", TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell[TextData[{ "For a crystal with a simple, nearly isotropic, structure, we can classify \ the vibrational modes into two types: longitudinal waves with density \ variations parallel to the direction of propagation or transverse waves with \ density variations perpendicular to the direction of propagation. We assume \ that the sound velocity is independent of frequency but permit different \ velocities for longitudinal and transverse modes so that the dispersion \ relations for these types become ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_\[ScriptL]\ = \ k\ v\_\[ScriptL]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_t\ = \ k\ v\_t\)]], ". Since there are two independent transverse polarization directions but \ only one longitudinal polarization direction, transverse modes should be \ normalized to ", Cell[BoxData[ \(TraditionalForm\`2 N\)]], " and longitudinal to ", Cell[BoxData[ \(TraditionalForm\`N\)]], ". Provided that the propagation speeds are not drastically different, it \ is useful to interpret ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Omega]\_D\ = \ \(k\_\(\(D\)\(\ \)\)\) v\&_\)\(\ \)\)\)]], " as a weighted average over polarizations, such that" }], "Text"], Cell[BoxData[ \(3\/v\&_\^3 = 1\/v\&_\_\[ScriptL]\^3 + \ 2\/v\&_\_t\^3\)], "DisplayFormula", TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`v\&_\)]], " is the weighted average propagation speed. Therefore, combining these \ results, we find that the Debye frequency, " }], "Text"], Cell[BoxData[ \(\[Omega]\_D = \(\((18 \( \[Pi]\^2\) N\/V)\)\^\(1/3\)\) v\&_\)], "DisplayFormula", TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell[TextData[{ "is proportional to the cube root of the density or the average interatomic \ spacing. The density dependence of ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_D\)]], " can be used to estimate the expansivity of the crystal." }], "Text"], Cell[TextData[{ "Glossary:\n\t\[ScriptCapitalN]\t= number of atoms\n\t", Cell[BoxData[ \(TraditionalForm\`V\)]], "\t= volume of crystal\n\t", Cell[BoxData[ \(TraditionalForm\`k\)]], "\t= wave number\n\t", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], "\t= vibrational frequency\n\t", Cell[BoxData[ \(TraditionalForm\`v\)]], "\t= speed of sound (averaged over polarization)\n\t", Cell[BoxData[ \(TraditionalForm\`n[y]\)]], "\t= phonon occupation probability\n\t", Cell[BoxData[ \(TraditionalForm\`g[k]\)]], " \t= density of states in terms of wave number\n\t", Cell[BoxData[ \(TraditionalForm\`g[\[Omega]]\)]], " \t= density of states in terms of frequency\n\t", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_D\)]], "\t= Debye frequency (max. ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], ")\n\t", Cell[BoxData[ \(TraditionalForm\`k\_B\)]], "\t= Boltzmann constant\n\t", Cell[BoxData[ \(TraditionalForm\`T\)]], "\t= temperature\n\t", Cell[BoxData[ \(TraditionalForm\`y\)]], "\t= ", Cell[BoxData[ \(TraditionalForm\`\[HBar]\ \ \[Omega]\ /\ \((k\_B\ T)\)\)]], " = ratio of phonon/thermal energies\n\t", Cell[BoxData[ \(TraditionalForm\`T\_E\)]], "\t= Einstein temperature\n\t", Cell[BoxData[ \(TraditionalForm\`T\_D\)]], "\t= Debye temperature\n\t", Cell[BoxData[ \(TraditionalForm\`U\)]], "\t= internal energy \n\t", Cell[BoxData[ \(TraditionalForm\`C\_V\)]], "\t= heat capacity" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Initialization", "Section"], Cell[CellGroupData[{ Cell["Defaults, packages, and labels", "Subsection"], Cell[BoxData[{ \(\(ClearAll["\"];\)\), "\[IndentingNewLine]", \(\(Off[General::spell, General::spell1];\)\)}], "Input", CellLabel->"In[1]:="], Cell[BoxData[{ \(\($DefaultFont = {"\", 12};\)\), "\n", \(\($TextStyle = {FontFamily \[Rule] "\", FontSize \[Rule] 12, FontSlant \[Rule] "\"};\)\)}], "Input", CellLabel->"In[3]:="], Cell[BoxData[{ \(\(Needs["\"];\)\), "\n", \(Needs["\"]\)}], "Input", CellLabel->"In[5]:="], Cell[BoxData[ RowBox[{ RowBox[{"Symbolize", "[", TagBox[\(k\_B\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(C\_V\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(T\_x_\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(\[Omega]\_x_\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(Z\_m_\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";"}]], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(SetAttributes[{k\_B, \[HBar]\ }, Constant]\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(\(label[C\_V]\ = \ StyleForm[\*"\"\<\!\(C\_V\)/(3\!\(\[ScriptCapitalN]k\_B\))\>\""];\)\)]\ , "Input", CellLabel->"In[9]:="] }, Closed]], Cell[CellGroupData[{ Cell["Simplification assumptions", "Subsection"], Cell[BoxData[ \(\(MyAssumptions = {m > 0, V > 0, k\_B > 0, T > 0, \[HBar] > 0, \[Beta] > 0, \[ScriptCapitalN] > 0, k\_D > 0, \[Omega]\_D > 0, T\_D > 0, T\_E > 0};\)\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[{ \(\(MySimplify = Simplify[#, MyAssumptions] &;\)\), "\[IndentingNewLine]", \(\(MyFullSimplify = FullSimplify[#, MyAssumptions] &;\)\)}], "Input", CellLabel->"In[11]:="] }, Closed]], Cell[CellGroupData[{ Cell["Rules for changing variables", "Subsection"], Cell[BoxData[{ \(\(\[Beta]ToT\ = \ {\[Beta] \[Rule] 1\/\(\(k\_B\) T\)};\)\ \), "\[IndentingNewLine]", \(\(Tto\[Beta]\ = \ {T \[Rule] 1\/\(k\_B\ \[Beta]\)};\)\), "\n", \(\(yTo\[Omega]\ = \ {y \[Rule] \(\[HBar]\ \[Omega]\)\/\(\(k\_B\) \ T\)};\)\), "\n", \(\(\[Omega]Tox\ = \ {\[Omega] \[Rule] x\ \[Omega]\_D};\)\), "\n", \(\(\[Omega]ToT\ = \ {\[Omega]\_D \[Rule] \ \(\(k\_B\) T\_D\)\/\[HBar]\ \ };\)\), "\n", \(\(TDto\[Tau]\ \ = \ {T\ \[Rule] \ \[Tau]\ T\_D};\)\ \), "\ \[IndentingNewLine]", \(\(TDtoy\ = \ {T\ \[Rule] \ T\_D/y};\)\), "\n", \(TEto\[Tau]\ \ = \ {T\ \[Rule] \ \[Tau]\ T\_E}; \ TEtoy\ = \ {T\ \[Rule] \ T\_E/y};\)}], "Input", CellLabel->"In[13]:="] }, Closed]], Cell[CellGroupData[{ Cell["Density of states", "Subsection"], Cell[TextData[{ "The phonon occupation probability was derived and studied in the \ notebook", StyleBox[" ", FontSlant->"Italic"], StyleBox[ButtonBox["hotherm.nb", ButtonData:>{"hotherm.nb", None}, ButtonStyle->"Hyperlink"], FontSlant->"Italic"], StyleBox[".", FontSlant->"Italic"], " The density of states assumes that wave numbers are distributed \ isotropically in momentum space. The normalization condition assumes that \ there is a maximum wave number that is based upon the interatomic spacing of \ the crystal." }], "Text"], Cell[BoxData[ \(n[y_] := 1\/\(Exp[y] - 1\)\)], "Input", CellLabel->"In[21]:="], Cell[BoxData[ \(\(g[k]\ = \ V\ \(4 \[Pi]\ k\^2\)\/\((2\ \[Pi])\)\^3\ ;\)\)], "Input", CellLabel->"In[22]:="], Cell[CellGroupData[{ Cell[BoxData[ \(kDsol\ = \ \(Solve[\[Integral]\_0\%\(k\_D\)\ g[k]\ \[DifferentialD]k == \ 3 \[ScriptCapitalN], \ k\_D]\)[\([3]\)]\ // MySimplify\)], "Input", CellLabel->"In[23]:="], Cell[BoxData[ \({k\_D \[Rule] 2\^\(1/3\)\ \((3\ \[Pi])\)\^\(2/3\)\ \ \((\[ScriptCapitalN]\/V)\)\^\(1/3\)}\)], "Output", CellLabel->"Out[23]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Omega]Dsol\ = \ \[Omega]\_D\ \[RightArrow] \ k\_D\ v\ /. kDsol\)], "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(\[Omega]\_D \[RightArrow] 2\^\(1/3\)\ \((3\ \[Pi])\)\^\(2/3\)\ v\ \ \((\[ScriptCapitalN]\/V)\)\^\(1/3\)\)], "Output", CellLabel->"Out[24]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(g[\[Omega]]\ = \ A\ \[Omega]\^2\ /. \(Solve[\[Integral]\_0\%\(\[Omega]\_D\)A\ \ \(\[Omega]\^2\) \[DifferentialD]\[Omega]\ == \ 3 \[ScriptCapitalN], \ A]\)[\([1]\)]\)], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \(\(9\ \[ScriptCapitalN]\ \[Omega]\^2\)\/\[Omega]\_D\%3\)], "Output", CellLabel->"Out[25]="] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Einstein model", "Subsection"], Cell[TextData[{ "To better understand the Debye interpolation formula, it will be helpful \ to compare with the Einstein model. The relevant results are quoted here. \ For further information, refer to the notebook", StyleBox[" ", FontSlant->"Italic"], StyleBox[ButtonBox["hotherm.nb", ButtonData:>{"hotherm.nb", None}, ButtonStyle->"Hyperlink"], FontSlant->"Italic"], StyleBox[".", FontSlant->"Italic"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(uEinstein = \(\(3\)\(\ \)\(\[ScriptCapitalN]\)\(\ \)\(k\_B\)\(\ \ \)\(T\_E\)\(\ \)\(n[T\_E\/T]\)\(\ \)\)\)], "Input", CellLabel->"In[26]:="], Cell[BoxData[ \(\(3\ k\_B\ T\_E\ \[ScriptCapitalN]\)\/\(\(-1\) + \ \[ExponentialE]\^\(T\_E\/T\)\)\)], "Output", CellLabel->"Out[26]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(cvEinstein = \[PartialD]\_T uEinstein\)], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \(\(3\ \[ExponentialE]\^\(T\_E\/T\)\ k\_B\ T\_E\%2\ \[ScriptCapitalN]\)\/\ \(\((\(-1\) + \[ExponentialE]\^\(T\_E\/T\))\)\^2\ T\^2\)\)], "Output", CellLabel->"Out[27]="] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Heat capacity for isotropic Debye crystal", "Section"], Cell[TextData[{ "\tThe internal energy is obtained by integrating the energy for each mode \ times its occupation probability against the density of states. Although \ this integral can be performed in closed form, the ", ButtonBox["PolyLog", ButtonStyle->"RefGuideLink"], " function is probably not familiar. Therefore, we compare the high and \ low temperature behaviors of the Debye model with those of the simpler \ Einstein model below. The following section then compares the Debye and \ Einstein models graphically." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(uDebye = \(\((\[Integral]\_0\%\(\[Omega]\_D\)n[\(\[HBar]\ \ \[Omega]\)\/\(\(k\_B\) T\)] \[HBar]\ \[Omega]\ g[\[Omega]]\ \[DifferentialD]\ \[Omega])\)\ /. \[Omega]ToT // Simplify\) // Collect[#, {\[ScriptCapitalN], k\_B, T}] &\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \(k\_B\ \[ScriptCapitalN]\ \((\(-\(\(9\ T\_D\)\/4\)\) + 9\ T\ Log[ 1 - \[ExponentialE]\^\(T\_D\/T\)] + \(27\ T\^2\ PolyLog[2, \ \[ExponentialE]\^\(T\_D\/T\)]\)\/T\_D - \(54\ T\^3\ PolyLog[3, \ \[ExponentialE]\^\(T\_D\/T\)]\)\/T\_D\%2 - \(3\ T\^4\ \((4\ \[Pi]\^4 - 360\ \ PolyLog[4, \[ExponentialE]\^\(T\_D\/T\)])\)\)\/\(20\ T\_D\%3\))\)\)], "Output",\ CellLabel->"Out[28]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(cvDebye = \(\[PartialD]\_T uDebye\ // Simplify\) // Collect[#, {\[ScriptCapitalN], k\_B, T}] &\)], "Input", CellLabel->"In[29]:="], Cell[BoxData[ \(k\_B\ \[ScriptCapitalN]\ \((\(9\ \[ExponentialE]\^\(T\_D\/T\)\ T\_D\)\/\ \(T - \[ExponentialE]\^\(T\_D\/T\)\ T\) + 36\ Log[1 - \[ExponentialE]\^\(T\_D\/T\)] + \(108\ T\ PolyLog[2, \ \[ExponentialE]\^\(T\_D\/T\)]\)\/T\_D - \(216\ T\^2\ PolyLog[3, \ \[ExponentialE]\^\(T\_D\/T\)]\)\/T\_D\%2 - \(12\ T\^3\ \((\[Pi]\^4 - 90\ \ PolyLog[4, \[ExponentialE]\^\(T\_D\/T\)])\)\)\/\(5\ T\_D\%3\))\)\)], "Output",\ CellLabel->"Out[29]="] }, Closed]], Cell[TextData[{ "\tThe prediction that the specific heat capacity depends only upon a \ simple reduced temperature variable, ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\)]], ", is an example of the ", StyleBox["law of corresponding states", FontSlant->"Italic"], " in which the thermodynamic functions assume universal forms. This \ principle often applies, to a reasonable approximation, when the dynamics of \ a system are governed by a single identifiable scale, in this case the Debye \ temperature. However, because the excitation spectrum for most crystals is \ more complicated than the Debye interpolation formula, real systems exhibit \ departures from this simple model. For example, the normal modes for layered \ systems may have quite different propagation speeds for vibrations within a \ layer compared with vibrations between layers. In such cases there will be \ at least two temperature scales and the heat capacity will be a more \ complicated function of temperature. Nevertheless, the Debye interpolation \ formula remains useful because the heat capacity is not particularly \ sensitive to the details of the excitation spectrum. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cvDebye\ /. 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This exponential behavior occurs because the excitation \ spectrum of the Einstein crystal has an energy gap which leads to a heat \ capacity with the characteristic two-state form when ", Cell[BoxData[ \(TraditionalForm\`T \[LessLess] T\_E\)]], ". The low temperature behavior of the Debye model is more realistic \ because its excitation spectrum is continuous \[LongDash] although the range \ of excitations which contribute strongly decreases as temperature decreases, \ modes are always available at energies less than ", Cell[BoxData[ \(TraditionalForm\`k\_B\ T\)]], " such that the heat capacity exhibits a power law instead of an \ exponential suppression at low temperatures. In particular, its prediction \ that at low temperature ", Cell[BoxData[ \(TraditionalForm\`C\_V\ \[Proportional] \ T\^3\)]], " is in good agreement with the data for most solid electrical insulators. \ Furthermore, the value of ", Cell[BoxData[ \(TraditionalForm\`T\_D\)]], " obtained by fitting the temperature dependence of the specific heat is \ usually in good agreement with that obtained from elastic properties (sound \ speed and density)." }], "Text"], Cell[TextData[{ "\tSimilar considerations can also be applied to liquids, with several \ important differences. First, fluids do not support sheer stress and hence \ do not support transverse modes of vibration; therefore, sound waves in \ fluids are longitudinal and only that contribution to the normalization of \ the Debye frequency should be considered. At low temperature the heat \ capacity is then ", Cell[BoxData[ \(TraditionalForm\`C\_V\ \[TildeTilde] \(4 \[Pi]\^4\)\/\(\(5\)\(\ \ \)\)\ N\ k\_B\ \((T\/T\_D)\)\^3\)]], ". Second, there are other modes of excitations, such as vortices, that \ can mask the phonon contribution at modest temperatures. Only liquid \ helium-4 remains liquid at sufficiently low temperature for the phonon \ contribution to clearly dominate such that the ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(T\^3\)\)\)]], " behavior becomes evident. Nevertheless, one finds that the specific heat \ for liquid-4 predicted from its density and sound velocity is in good \ agreement with calorimetry for ", Cell[BoxData[ \(TraditionalForm\`T\ \[LessTilde] \ 0.6\)]], " kelvin." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Gr\[UDoubleDot]neisen model", "Section"], Cell[TextData[{ "\tIn addition to the lattice vibrations described by the Debye model, the \ energy of a crystal also depends upon the mean interatomic spacing. This \ effect can be represented by a potential energy ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi][V]\)]], " which depends upon the volume of the crystal. Furthermore, the frequency \ for each mode also depends upon volume. The density dependence of phonon \ energies is parametrized by" }], "Text"], Cell[BoxData[ \(\[Gamma] = \(\(-\(\(\[PartialD]ln\ \[Omega]\)\/\(\[PartialD]ln\ V\)\)\)\ \ = \ \(-\(V\/\[Omega]\)\) \[PartialD]\[Omega]\/\[PartialD]V\ \[TildeTilde] \ \ 1\/3\)\)], "DisplayFormula", TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell["\<\ which is assumed to be constant over the relevant ranges of temperature and \ pressure and is known as the Gr\[UDoubleDot]neisen constant. 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Furthermore, we assume that \ ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi][V]\)]], " can be expanded around the equilibrium density, such that" }], "Text"], Cell[BoxData[ \(\[CapitalPhi][ V]\ \[TildeTilde] \ \[CapitalPhi]\_0\ + \ \((V - V\_0)\)\^2\/\(2\ \ \[Kappa]\ V\_0\)\)], "DisplayFormula", TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`N\/V\_0\)]], "is the equilibrium density and ", Cell[BoxData[ \(TraditionalForm\`\[Kappa]\)]], " is an elasticity parameter. 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Finally, we employ a general \ thermodynamic relationship between the principal heat capacities to obtain \ the isobaric heat capacity for this model.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(C\_p\ = \ C\_V\ + \ T\ V\ expansivity\^2\/compressibility\ // Simplify\)], "Input", CellLabel->"In[60]:="], Cell[BoxData[ \(\(-\(\(C\_V\ V\ \((V\ \((2 + \[Gamma])\) + V\_0\ \((1 + \[Gamma])\)\ \((\(-1\) + p\ \[Kappa])\))\)\)\/\(\(-V\^2\)\ \((2 + \[Gamma])\) + C\_V\ T\ V\_0\ \[Gamma]\^2\ \[Kappa] - V\ V\_0\ \((1 + \[Gamma])\)\ \((\(-1\) + p\ \[Kappa])\)\)\)\)\)], "Output", CellLabel->"Out[60]="] }, Closed]], Cell[TextData[{ "In the limit that the pressure is small (compared to the internal forces \ which determine ", Cell[BoxData[ \(TraditionalForm\`V\_0\)]], "), the isobaric and isochoric heat capacities are nearly equal." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(C\_p\ /\ C\_\(\(V\)\(\ \)\) /. {p \[Rule] 0, V \[Rule] V\_0}\ // Series[#, {C\_V, 0, 1}] &\) // Simplify\)], "Input", CellLabel->"In[61]:="], Cell[BoxData[ InterpretationBox[ RowBox[{"1", "+", \(\(T\ \[Gamma]\^2\ \[Kappa]\ C\_V\)\/V\_0\), "+", InterpretationBox[\(O[C\_V]\^2\), SeriesData[ C\[UnderBracket]Subscript\[UnderBracket]V, 0, {}, 0, 2, 1]]}], SeriesData[ C\[UnderBracket]Subscript\[UnderBracket]V, 0, {1, Times[ T, Power[ V\[UnderBracket]Subscript\[UnderBracket]0, -1], Power[ \[Gamma], 2], \[Kappa]]}, 0, 2, 1]]], "Output", CellLabel->"Out[61]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\(C\_p\ - \ C\_\(\(V\)\(\ \)\) /. {p \[Rule] 0, V \[Rule] V\_0}\ // Series[#, {C\_V, 0, 2}] &\) // Simplify\)], "Input", CellLabel->"In[62]:="], Cell[BoxData[ InterpretationBox[ RowBox[{\(\(T\ \[Gamma]\^2\ \[Kappa]\ C\_V\%2\)\/V\_0\), "+", InterpretationBox[\(O[C\_V]\^3\), SeriesData[ C\[UnderBracket]Subscript\[UnderBracket]V, 0, {}, 2, 3, 1]]}], SeriesData[ C\[UnderBracket]Subscript\[UnderBracket]V, 0, { Times[ T, Power[ V\[UnderBracket]Subscript\[UnderBracket]0, -1], Power[ \[Gamma], 2], \[Kappa]]}, 2, 3, 1]]], "Output", CellLabel->"Out[62]="] }, Closed]], Cell[TextData[{ "Therefore, recognizing that ", Cell[BoxData[ \(TraditionalForm\`C\_V\ \[Proportional] \ T\^3\)]], " for small ", Cell[BoxData[ \(TraditionalForm\`T\)]], ", we conclude that ", Cell[BoxData[ \(TraditionalForm\`C\_p\ - \ C\_V\ \[Proportional] \ \ T\^7\)]], " and is quite small." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Problems", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "Heat capacity for ", Cell[BoxData[ \(TraditionalForm\`\[Omega] \[Proportional] k\^s\)]] }], "ExerciseTitle"], Cell[TextData[{ "Consider a system for which the excitation spectrum satisfies a dispersion \ relation of the form ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\ \[Proportional] \ k\^s\)]], ". Show that such excitations give a contribution to the heat capacity \ that at low temperatures is proportional to ", Cell[BoxData[ \(TraditionalForm\`T\^\(3/s\)\)]], ". Compare graphically the temperature dependence of the heat capacities \ for ", Cell[BoxData[ \(TraditionalForm\`s = 2\)]], ", which applies to spin waves propagating in a ferromagnetic system, to \ those for ", Cell[BoxData[ \(TraditionalForm\`s = 1\)]], ", which applies to elastic waves in the lattice." }], "ExerciseText"] }, Closed]], Cell[CellGroupData[{ Cell["Layered crystal", "ExerciseTitle"], Cell[TextData[{ "Graphite exhibits a layered crystalline structure for which the restoring \ forces parallel to each layer are much larger than those perpendicular to the \ layers. Hence, lattice vibrations can be separated into modes in which the \ atoms vibrate within the layer, characterized by Debye frequency ", Cell[BoxData[ \(TraditionalForm\`\(\[Omega]\_ \[DoubleVerticalBar] \)\)]], ", and modes in which atoms vibrate perpendicular to the layers, \ characterized by Debye frequency ", Cell[BoxData[ \(TraditionalForm\`\(\[Omega]\_ \[UpTee] \)\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`\(\[Omega]\_ \[DoubleVerticalBar] \) \ \[GreaterGreater] \ \(\[Omega]\_ \[UpTee] \)\)]], ". Vibrations within the plane are two-dimensional, whereas vibrations \ perpendicular to planes are one-dimensional. \na) Show that the heat capacity \ for lattice vibrations in an ", Cell[BoxData[ \(TraditionalForm\`n\)]], "-dimensional Debye crystal can be expressed in the form ", Cell[BoxData[ \(TraditionalForm\`\(C\_V\)( T)\ \[Proportional] \ \[Tau]\^n\ \(\(f\_n\)(\[Tau])\)\)]], " where ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\)]], " is an appropriate reduced temperature and ", Cell[BoxData[ \(TraditionalForm\`\(f\_n\)(\[Tau])\)]], " is a definite integral with ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\)]], " as one of its limits.\nb) Obtain explicit expressions for the temperature \ dependence of the heat capacity of graphite in each of the following regimes:\ \n\ti) ", Cell[BoxData[ \(TraditionalForm\`\(k\_B\) T\ \[LessLess] \ \[HBar]\ \(\[Omega]\_ \[UpTee] \)\)]], ";\n\tii) ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[HBar]\ \(\[Omega]\_ \[UpTee] \)\ \ \[LessLess] \(k\_B\) T\ \[LessLess] \ \[HBar]\ \(\[Omega]\_ \[DoubleVerticalBar] \)\)\)\ \)]], ";\n\tiii) ", Cell[BoxData[ \(TraditionalForm\`\(k\_B\) T\ \[GreaterGreater] \ \ \[HBar]\ \(\[Omega]\_ \ \[DoubleVerticalBar] \)\)]], ".\nc) Sketch ", Cell[BoxData[ \(TraditionalForm\`\(C\_V\)(T)\)]], " assuming that ", Cell[BoxData[ \(TraditionalForm\`\(\[Omega]\_ \[DoubleVerticalBar] \)/\(\[Omega]\_ \ \[UpTee] \)\ \[Tilde] \ 10\)]], "." }], "ExerciseText"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Integrated difference in heat capacity for Dulong-Petit and Debye models\ \>", "ExerciseTitle"], Cell[TextData[{ "Show that the integrated difference between the heat capacities for the \ Dulong-Petit and Debye models, ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%\[Infinity]\((\(C\_V\)(\[Infinity])\ \ - \ \(C\_V\)(T))\) \[DifferentialD]T\)]], ", is equal to the zero-point energy of the solid. 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For somewhat larger momenta, excitations he \ called ", StyleBox["rotons", FontSlant->"Italic"], " satisfy the dispersion relation" }], "ExerciseText"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[HBar]", " ", SubscriptBox["\[Omega]", "roton"]}], " ", "=", " ", RowBox[{"\[CapitalDelta]", " ", "+", " ", FractionBox[ SuperscriptBox[ RowBox[{ SuperscriptBox["\[HBar]", "2"], "(", RowBox[{"k", "-", SubscriptBox["k", "0"]}], ")"}], "2"], RowBox[{"2", "\[Mu]"}]]}]}], TraditionalForm]]], " " }], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(\(\[CapitalDelta]\/k\_B \[TildeTilde] \ 8.65\)\(\ \)\)\)]], " kelvin, ", Cell[BoxData[ \(TraditionalForm\`k\_0\ \[TildeTilde] \ 1.92\ \[Angstrom]\^\(-1\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\[Mu] \[TildeTilde] 0.16\ m\_He\)]], ". For intermediate momenta, one finds that there is a maximum value, ", Cell[BoxData[ \(TraditionalForm\`\(\[HBar]\ \[Omega]\_1\)\/k\_B \[TildeTilde] 13.92\)]], " kelvin , at ", Cell[BoxData[ \(TraditionalForm\`k\_1\ \[TildeTilde] \ 1.11\ \[Angstrom]\^\(-1\)\)]], ".\na) Deduce the separate phonon and roton contributions to the heat \ capacity.\nb) Sketch the excitation spectrum and construct a simple function \ which describes both the phonon and roton portions in an approximate manner.\n\ c) Obtain either an analytical or a numerical approximation to the \ temperature dependence of the heat capacity for superfluid helium valid for a \ range of temperatures that includes both the phonon and roton regimes." }], "ExerciseText"] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowToolbars->{"RulerBar", "EditBar"}, WindowSize->{1016, 668}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, Visible->True, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->False, ShowCellTags->False, InputAliases->{"notation"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation>"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation<"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "symb"->RowBox[ {"Symbolize", "[", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], "]"}], "infixnotation"->RowBox[ {"InfixNotation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], ",", "\[Placeholder]"}], "]"}], "addia"->RowBox[ {"AddInputAlias", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], ",", "\[Placeholder]"}], "]"}], "pattwraper"->TagBox[ "\[Placeholder]", NotationPatternTag, TagStyle -> "NotationPatternWrapperStyle"], "madeboxeswraper"->TagBox[ "\[Placeholder]", NotationMadeBoxesTag, TagStyle -> "NotationMadeBoxesWrapperStyle"]}, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, Magnification->1.25, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. 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"Printout"], CellMargins->{{2, 10}, {2, 30}}, FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{12, Inherited}, {2, 2}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->24, FontColor->RGBColor[0, 0, 0.6]], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{2, 10}, {2, 4}}, FontSize->18, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellMargins->{{12, Inherited}, {2, 12}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subsubtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Helvetica", FontSize->14, FontSlant->"Italic", FontColor->RGBColor[0, 0, 0.6]], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {2, 8}}, FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellFrame->{{0, 0}, {2, 0}}, CellMargins->{{10, 4}, {2, 50}}, CellElementSpacings->{"ClosedGroupTopMargin"->18}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->18, FontWeight->"Bold"], Cell[StyleData["Section", "Printout"], CellMargins->{{2, 4}, {2, 80}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{24, 4}, {2, 18}}, CellElementSpacings->{"ClosedGroupTopMargin"->12}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CellFrameLabelMargins->6, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->15, FontWeight->"Bold"], Cell[StyleData["Subsection", "Printout"], CellMargins->{{2, 4}, {2, 18}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellMargins->{{10, 4}, {2, 18}}, CellElementSpacings->{"ClosedGroupTopMargin"->12}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subsubsection", FontFamily->"Helvetica", FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{2, 4}, {2, 18}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Text", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{10, 4}, {0, 8}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, Hyphenation->True, ParagraphSpacing->{0, 8}, CounterIncrements->"Text"], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["MathCaption"], CellFrame->{{4, 0}, {0, 0}}, CellMargins->{{47, 62}, {0, 14}}, CellFrameMargins->5, CellFrameColor->RGBColor[0, 0.2, 1], Hyphenation->True, LineSpacing->{1, 1}, ParagraphSpacing->{0, 8}, FontColor->RGBColor[0, 0, 0.6]], Cell[StyleData["MathCaption", "Printout"], CellMargins->{{34, 62}, {0, 14}}, CellFrameColor->GrayLevel[0.700008], FontSize->10, FontColor->GrayLevel[0]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Input/Output", "Subsection"], Cell["\<\ The cells in this section define styles used for input and output to the \ kernel. Be careful when modifying, renaming, or removing these styles, \ because the front end associates special meanings with these style names. \ \ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{56, 4}, {3, 9}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->StandardForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Printout"], ShowCellBracket->False, CellMargins->{{42, 4}, {3, 8}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{57, 4}, {5, 2}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->StandardForm, CounterIncrements->"Output"], Cell[StyleData["Output", "Printout"], ShowCellBracket->False, CellMargins->{{42, 4}, {4, 2}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{56, 4}, {3, 8}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontColor->RGBColor[0, 0.2, 1]], Cell[StyleData["Message", "Printout"], ShowCellBracket->False, CellMargins->{{42, 4}, {4, 2}}, FontSize->10, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{56, 4}, {3, 8}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Printout"], ShowCellBracket->False, CellMargins->{{42, 4}, {4, 2}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{56, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", StyleMenuListing->None], Cell[StyleData["Graphics", "Printout"], CellMargins->{{40, 4}, {4, 2}}, ImageSize->{250, 250}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0.2, 1]], Cell[StyleData["CellLabel", "Printout"], FontSize->7, FontSlant->"Oblique", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Unique Styles", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["Abstract"], CellFrame->False, CellMargins->{{45, 75}, {Inherited, 30}}, Hyphenation->True, LineSpacing->{1, 1}], Cell[StyleData["Abstract", "Printout"], CellMargins->{{36, 67}, {Inherited, 50}}, FontSize->10] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["TextTop"], CellFrame->{{0, 0}, {0, 2}}, CellMargins->{{10, 4}, {2, 80}}, CellHorizontalScrolling->True, CellFrameMargins->4, ShowSpecialCharacters->Automatic, ParagraphSpacing->{0, 8}, CounterIncrements->"Text"], Cell[StyleData["TextTop", "Printout"], CellMargins->{{2, 4}, {2, 80}}, FontSize->10] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["ItemizedText"], CellMargins->{{20, 4}, {0, 8}}, ShowSpecialCharacters->Automatic, Hyphenation->True, ParagraphSpacing->{0, 8}, ParagraphIndent->-15, CounterIncrements->"Text"], Cell[StyleData["ItemizedText", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ItemizedTextNote"], CellMargins->{{35, 4}, {0, 4}}, ShowSpecialCharacters->Automatic, Hyphenation->True, ParagraphSpacing->{0, 4}, CounterIncrements->"Text"], Cell[StyleData["ItemizedTextNote", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["IndentedText"], CellMargins->{{20, 4}, {0, 6}}, ShowSpecialCharacters->Automatic, Hyphenation->True, ParagraphSpacing->{0, 8}, CounterIncrements->"Text"], Cell[StyleData["IndentedText", "Printout"], FontSize->10] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Note"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, ShowSpecialCharacters->Automatic, ParagraphSpacing->{0, 8}, CounterIncrements->"Text", FontFamily->"Helvetica", FontSize->10], Cell[StyleData["Note", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->8] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["1ColumnBox"], CellFrame->1, CellMargins->{{30, 30}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.900008], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnSpacings->1}], Cell[StyleData["1ColumnBox", "Printout"], CellMargins->{{30, 30}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["2ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->{0.39, 0.59}}], Cell[StyleData["2ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["2ColumnSmallBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnSpacings->1.5, ColumnWidths->0.35, ColumnAlignments->{Right, Left}}], Cell[StyleData["2ColumnSmallBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["3ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->0.325}], Cell[StyleData["3ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["3ColumnSmallBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnSpacings->1.5, ColumnWidths->0.23, ColumnAlignments->{Right, Center, Left}}], Cell[StyleData["3ColumnSmallBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["4ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->{0.145, 0.345, 0.145, 0.345}}], Cell[StyleData["4ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["5ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->0.195}], Cell[StyleData["5ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["6ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->{0.13, 0.23, 0.13, 0.13, 0.23, 0.13}}], Cell[StyleData["6ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Picture"], CellMargins->{{10, Inherited}, {0, 8}}, CellHorizontalScrolling->True], Cell[StyleData["Picture", "Printout"], CellMargins->{{2, Inherited}, {0, 8}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Caption"], CellMargins->{{10, 50}, {0, 3}}, PageBreakAbove->False, Hyphenation->True, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["Caption", "Printout"], CellMargins->{{2, 50}, {2, 4}}, FontSize->7] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ExerciseTitle"], CellDingbat->"\[FilledDownTriangle]", CellMargins->{{23, Inherited}, {4, 18}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->13, FontWeight->"Bold", FontColor->GrayLevel[0]], Cell[StyleData["ExerciseTitle", "Printout"], CellDingbat->"\[FilledDownTriangle]", CellMargins->{{9, 0}, {6, 20}}, FontSize->11, FontColor->GrayLevel[0]] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["ExerciseText"], CellMargins->{{32, 10}, {5, 5}}, LineSpacing->{1, 3}, ParagraphSpacing->{0, 8}, CounterIncrements->"ExerciseText", FontFamily->"Times"], Cell[StyleData["ExerciseText", "Printout"], CellMargins->{{32, 0}, {4, 4}}, ParagraphSpacing->{0, 6}, FontSize->10] }, Open ]], Cell[StyleData["Problem"], CellDingbat->"\[FilledDownTriangle]", CellMargins->{{23, Inherited}, {4, 18}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->13, FontWeight->"Plain", FontColor->GrayLevel[0]], Cell[StyleData["Problem", "Printout"], CellFrame->False, CellDingbat->"\[FilledDownTriangle]", CellMargins->{{9, 0}, {6, 20}}, FontSize->11, FontColor->GrayLevel[0]] }, Open ]], Cell[CellGroupData[{ Cell["Tables", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["2ColumnTable"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, GridBoxOptions->{ColumnWidths->{0.39, 0.59}, ColumnAlignments->{Left}}], Cell[StyleData["2ColumnTable", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["3ColumnTable"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, StyleMenuListing->None, GridBoxOptions->{ColumnWidths->0.325, ColumnAlignments->{Left}}], Cell[StyleData["3ColumnTable", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["ChemicalFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", AutoSpacing->False, ScriptLevel->1, ScriptBaselineShifts->{0.6, Automatic}, SingleLetterItalics->False, ZeroWidthTimes->True], Cell[StyleData["ChemicalFormula", "Printout"], CellMargins->{{34, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, SpanMaxSize->Infinity, UnderoverscriptBoxOptions->{LimitsPositioning->True}, GridBoxOptions->{ColumnWidths->Automatic}], Cell[StyleData["DisplayFormula", "Printout"], CellMargins->{{34, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", FontFamily->"Courier"], Cell[StyleData["Program", "Printout"], CellMargins->{{2, Inherited}, {Inherited, Inherited}}, FontSize->9.5] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Subsection"], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. 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