(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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Since Li has only one core shell ", Cell[BoxData[ \(TraditionalForm\`\(\(\((1 s)\)\^2\)\(,\)\(\ \)\)\)]], "the sum reduces to a single term. Moreover, since the reduced matrix \ element \)\(\ \)\)\)\)]], "vanishes between two s states, the SMS vanishes for the 2s state. (This \ statement is true only in the independent particle approximation and is \ modified when correlation corrections are considered.)\n\nIntroduce the \ Coulomb wave function Pa[r] for the 1s core orbital and Pv[r] for the 2p \ valence orbital. Check that the wave functions are properly normalized." }], "Text"], Cell[BoxData[ \(Pa[r_]\ := \ 2\ Za^\((3/2)\)\ r\ Exp[\(-Za\)\ r]\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \(Pv[r_]\ := \ Zv^\((5/2)\)\ r^2\ Exp[\(-Zv\)\ r/2]/\((2\ Sqrt[6])\)\)], "Input", CellLabel->"In[2]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Na\ = \ Integrate[Pa[r]^2, {r, 0, Infinity}, Assumptions\ \[Rule] \ Za > 0]\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \(1\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Na\ = \ Integrate[Pv[r]^2, {r, 0, Infinity}, Assumptions\ \[Rule] \ Zv > 0]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \(1\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell["\<\ Work out the valence-core matrix element of the momentum operator Pva\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Pva\ = \ Integrate[\ Pv[r]\ \((D[Pa[r], r] - Pa[r]/r)\), {r, 0, Infinity}, Assumptions\ \[Rule] \ {Za > 0, Zv > 0}]\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \(\(-\(\(16\ \@6\ \((Za\ Zv)\)\^\(5/2\)\)\/\((2\ Za + Zv)\)\^4\)\)\)], \ "Output", CellLabel->"Out[5]="] }, Open ]], Cell["\<\ Check the above by working out the matrix element in the reverse order Pav. \ This sould be the adjoint of the above. Since we are ignoring the factor \ 1/i, the reverse order Pva ans Pav should have opposite signs.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Pav\ = \ \ Integrate[\ \ Pa[r]\ \((D[Pv[r], r] + Pv[r]/r)\), {r, 0, Infinity}, Assumptions\ \[Rule] \ {Za > 0, Zv > 0}]\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \(\(16\ \@6\ \((Za\ Zv)\)\^\(5/2\)\)\/\((2\ Za + Zv)\)\^4\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell["Substitute numerical values for Za and Zv", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Za\ = \ 3 - 5/16 // N\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(2.6875`\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Zv\ = \ 1\ + 1/8 // N\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(1.125`\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Pva // N\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \(\(-0.3489752842773039`\)\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Pav\)\(//\)\(N\)\(\ \)\)\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(0.3489752842773039`\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell[TextData[{ "Now evaluate the reduced matrix element ", "\)\)\)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RME\ = \ \(-\ Sqrt[3]\)\ ThreeJSymbol[{1, 0}, {1, 0}, {0, 0}]\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(1\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["\<\ Put the above together to obtain the SMS (up to a factor 1/Ma)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(T1\ = \ \(-\ N[3\^\(-1\)\ RME^2\ Pva^2]\)\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(\(-0.04059458301214169`\)\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell[BoxData[ \(Clear[Za, Zv]\)], "Input", CellLabel->"In[13]:="], Cell[TextData[{ "Now put in the isotope masses: ", Cell[BoxData[ \(TraditionalForm\`\(\ \^7\)\)]], Cell[BoxData[ \(TraditionalForm\`Li\)]], " \[RightArrow] 6.015 u and ", Cell[BoxData[ \(TraditionalForm\`\(\ \^7\)\)]], "Li \[RightArrow] 7.016 u. Note that the electron mass is 0.00054858 u. \ If we express electron mass in u, the conversion factor from a.u. to MHz is \ ", Cell[BoxData[ \(TraditionalForm\`3.60948\[Times]10\^6\)]], " For the 2p state, we find:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SMS6", "=", " ", RowBox[{ FormBox[\(\(-0.0405946\)\ 3.60948\ 10\^6\), "TraditionalForm"], " ", "/", "6.015"}]}]], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \(\(-24359.99946932668`\)\)], "Output", CellLabel->"Out[14]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SMS7", "=", " ", RowBox[{ FormBox[\(\(-0.0405946\)\ 3.60948\ 10\^6\), "TraditionalForm"], " ", "/", "7.016"}]}]], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \(\(-20884.463627137968`\)\)], "Output", CellLabel->"Out[15]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(SMS67\ = \ SMS6 - SMS7\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(\(-3475.535842188714`\)\)], "Output", CellLabel->"Out[16]="] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Normal Mass Shift", "Subsubsection"], Cell["\<\ The energy of the 2s state in the NIST data base is 0; therefore there is no \ NMS for this state. The 2p state energy is 0.067905 a.u. The corresponding \ NMS is:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NMS6", "=", " ", RowBox[{ FormBox[\(\(-0.067905\)\ 3.60948\ 10\^6\), "TraditionalForm"], " ", "/", "6.015"}]}]], "Input", CellLabel->"In[17]:="], Cell[BoxData[ \(\(-40748.41885286783`\)\)], "Output", CellLabel->"Out[17]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NMS7", "=", " ", RowBox[{ FormBox[\(\(-0.067905\)\ 3.60948\ 10\^6\), "TraditionalForm"], " ", "/", "7.016"}]}]], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \(\(-34934.68349486887`\)\)], "Output", CellLabel->"Out[18]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(NMS67\ = \ NMS6 - NMS7\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \(\(-5813.735357998958`\)\)], "Output", CellLabel->"Out[19]="] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Total Isotope Shift", "Subsubsection"], Cell["The resulting shift NMS+SMS in MHz is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ISMHz\ = \ SMS67 + NMS67\)], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \(\(-9289.271200187672`\)\)], "Output", CellLabel->"Out[25]="] }, Open ]], Cell["\<\ This value differs from the experimental result -10533 MHz primarily because \ the lowest-order calculation of the 2s SMS vanishes, whereas a higher-order \ MBPT calculation gives -1100 MHz. Adding this to the previous result gives \ -10389 MHz, in much closer agreement with experiment.\ \>", "Text"], Cell[TextData[{ "To convert the above energies to ", Cell[BoxData[ \(TraditionalForm\`cm\^\(-1\)\)]], ", simply divide by c. To convert energies expressed in ", Cell[BoxData[ \(TraditionalForm\`cm\^\(\(-\)\(1\)\(\ \)\)\)]], "to wavelengths, take the reciprocal." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c\ = \ 2.997525\ 10^10\)], "Input", CellLabel->"In[23]:="], Cell[BoxData[ \(2.997525`*^10\)], "Output", CellLabel->"Out[23]="] }, Open ]], Cell[TextData[{ "Thus, The isotope shift of the 2p-2s line in (", Cell[BoxData[ \(TraditionalForm\`cm\^\(-1\)\)]], ") is " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(IScm\ = \ ISMHz\ 10^6/\ c\)], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \(\(-0.30989803922194714`\)\)], "Output", CellLabel->"Out[27]="] }, Open ]], Cell["\<\ The corresponding wavlength difference in (cm) is the inverse of the previous \ vakue\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(d\[Lambda]\ = \ 1/Abs[IScm]\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \(3.226867786936225`\)], "Output", CellLabel->"Out[28]="] }, Open ]] }, Closed]] }, Open ]] }, Open ]] }, FrontEndVersion->"5.1 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{733, 701}, WindowMargins->{{Automatic, 244}, {Automatic, 29}}, Magnification->1.25, StyleDefinitions -> "ArticleModern.nb" ] (******************************************************************* Cached data follows. 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