\gdef\shorttitle{#1} \vglue4pc{\exhyphenpenalty=10000\hyphenpenalty=10000 \baselineskip=18pt \raggedright\parindent=0pt \headfonts#1\par}\futurelet\next\sh@rttitle
iopppt-modif.tex
\hsize=31pc 
\vsize=49pc 
% 
\lineskip=0pt 
\parskip=0pt plus 1pt 
\hfuzz=1pt   
\vfuzz=2pt 
\pretolerance=2500 
\tolerance=5000 
\vbadness=5000 
\hbadness=5000 
\widowpenalty=500 
\clubpenalty=200 
\brokenpenalty=500 
\predisplaypenalty=200 
\voffset=-1pc 
\nopagenumbers      
% 
\catcode`@=11 
% 
\newif\ifams 
\amsfalse %\amstrue si � suivant utilise
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%                                                           % 
%  The following section may be commented out and           % 
%  \ifams set to either \amstrue to use the AMS fonts       % 
%  or \amsfalse if they are not available                   % 
%                                                           % 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% 
%\def\Yesreply{Y } 
%\def\Noreply{N } 
%\def\yesreply{y } 
%\def\noreply{n } 
%\newif\ifnotyorn 
%\message{Do you want to use AMSfonts, msam and msbm? Y or N: }% 
%\loop 
%\read-1 to \reply 
%\ifx\reply\yesreply\global\amstrue\notyornfalse 
%\else\ifx\reply\Yesreply\global\amstrue\notyornfalse 
%\else\ifx\reply\noreply\global\amsfalse\notyornfalse 
%\else\ifx\reply\Noreply\global\amsfalse\notyornfalse 
%\else\notyorntrue 
%\message{Please type y or Y  (Yes) or n or N (No)}\fi\fi\fi\fi 
%\ifnotyorn\repeat 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% 
\newfam\bdifam 
\newfam\bsyfam 
\newfam\bssfam 
\newfam\msafam 
\newfam\msbfam 
% 
\newif\ifxxpt    
\newif\ifxviipt  
\newif\ifxivpt   
\newif\ifxiipt   
\newif\ifxipt    
\newif\ifxpt     
\newif\ifixpt    
\newif\ifviiipt  
\newif\ifviipt   
\newif\ifvipt    
\newif\ifvpt     
% 
% Headings in 20pt, 17pt or 14pt 
% 
\def\headsize#1#2{\def\headb@seline{#2}% 
                \ifnum#1=20\def\HEAD{twenty}% 
                           \def\smHEAD{twelve}% 
                           \def\vsHEAD{nine}% 
                           \ifxxpt\else\xdef\f@ntsize{\HEAD}% 
                           \def\m@g{4}\def\s@ze{20.74}% 
                           \loadheadfonts\xxpttrue\fi 
                           \ifxiipt\else\xdef\f@ntsize{\smHEAD}% 
                           \def\m@g{1}\def\s@ze{12}% 
                           \loadxiiptfonts\xiipttrue\fi 
                           \ifixpt\else\xdef\f@ntsize{\vsHEAD}% 
                           \def\s@ze{9}% 
                           \loadsmallfonts\ixpttrue\fi 
                      \else 
                \ifnum#1=17\def\HEAD{seventeen}% 
                           \def\smHEAD{eleven}% 
                           \def\vsHEAD{eight}% 
                           \ifxviipt\else\xdef\f@ntsize{\HEAD}% 
                           \def\m@g{3}\def\s@ze{17.28}% 
                           \loadheadfonts\xviipttrue\fi 
                           \ifxipt\else\xdef\f@ntsize{\smHEAD}% 
                           \loadxiptfonts\xipttrue\fi 
                           \ifviiipt\else\xdef\f@ntsize{\vsHEAD}% 
                           \def\s@ze{8}% 
                           \loadsmallfonts\viiipttrue\fi 
                      \else\def\HEAD{fourteen}% 
                           \def\smHEAD{ten}% 
                           \def\vsHEAD{seven}% 
                           \ifxivpt\else\xdef\f@ntsize{\HEAD}% 
                           \def\m@g{2}\def\s@ze{14.4}% 
                           \loadheadfonts\xivpttrue\fi 
                           \ifxpt\else\xdef\f@ntsize{\smHEAD}% 
                           \def\s@ze{10}% 
                           \loadxptfonts\xpttrue\fi 
                           \ifviipt\else\xdef\f@ntsize{\vsHEAD}% 
                           \def\s@ze{7}% 
                           \loadviiptfonts\viipttrue\fi 
                \ifnum#1=14\else 
                \message{Header size should be 20, 17 or 14 point 
                              will now default to 14pt}\fi 
                \fi\fi\headfonts} 
% 
% Text in 12pt, 11pt or 10pt  
% 
\def\textsize#1#2{\def\textb@seline{#2}% 
                 \ifnum#1=12\def\TEXT{twelve}% 
                           \def\smTEXT{eight}% 
                           \def\vsTEXT{six}% 
                           \ifxiipt\else\xdef\f@ntsize{\TEXT}% 
                           \def\m@g{1}\def\s@ze{12}% 
                           \loadxiiptfonts\xiipttrue\fi 
                           \ifviiipt\else\xdef\f@ntsize{\smTEXT}% 
                           \def\s@ze{8}% 
                           \loadsmallfonts\viiipttrue\fi 
                           \ifvipt\else\xdef\f@ntsize{\vsTEXT}% 
                           \def\s@ze{6}% 
                           \loadviptfonts\vipttrue\fi 
                      \else 
                \ifnum#1=11\def\TEXT{eleven}% 
                           \def\smTEXT{seven}% 
                           \def\vsTEXT{five}% 
                           \ifxipt\else\xdef\f@ntsize{\TEXT}% 
                           \def\s@ze{11}% 
                           \loadxiptfonts\xipttrue\fi 
                           \ifviipt\else\xdef\f@ntsize{\smTEXT}% 
                           \loadviiptfonts\viipttrue\fi 
                           \ifvpt\else\xdef\f@ntsize{\vsTEXT}% 
                           \def\s@ze{5}% 
                           \loadvptfonts\vpttrue\fi 
                      \else\def\TEXT{ten}% 
                           \def\smTEXT{seven}% 
                           \def\vsTEXT{five}% 
                           \ifxpt\else\xdef\f@ntsize{\TEXT}% 
                           \loadxptfonts\xpttrue\fi 
                           \ifviipt\else\xdef\f@ntsize{\smTEXT}% 
                           \def\s@ze{7}% 
                           \loadviiptfonts\viipttrue\fi 
                           \ifvpt\else\xdef\f@ntsize{\vsTEXT}% 
                           \def\s@ze{5}% 
                           \loadvptfonts\vpttrue\fi 
                \ifnum#1=10\else 
                \message{Text size should be 12, 11 or 10 point 
                              will now default to 10pt}\fi 
                \fi\fi\textfonts} 
% 
% Small sized material in 10pt, 9pt or 8pt 
% 
\def\smallsize#1#2{\def\smallb@seline{#2}% 
                 \ifnum#1=10\def\SMALL{ten}% 
                           \def\smSMALL{seven}% 
                           \def\vsSMALL{five}% 
                           \ifxpt\else\xdef\f@ntsize{\SMALL}% 
                           \loadxptfonts\xpttrue\fi 
                           \ifviipt\else\xdef\f@ntsize{\smSMALL}% 
                           \def\s@ze{7}% 
                           \loadviiptfonts\viipttrue\fi 
                           \ifvpt\else\xdef\f@ntsize{\vsSMALL}% 
                           \def\s@ze{5}% 
                           \loadvptfonts\vpttrue\fi 
                       \else 
                 \ifnum#1=9\def\SMALL{nine}% 
                           \def\smSMALL{six}% 
                           \def\vsSMALL{five}% 
                           \ifixpt\else\xdef\f@ntsize{\SMALL}% 
                           \def\s@ze{9}% 
                           \loadsmallfonts\ixpttrue\fi 
                           \ifvipt\else\xdef\f@ntsize{\smSMALL}% 
                           \def\s@ze{6}% 
                           \loadviptfonts\vipttrue\fi 
                           \ifvpt\else\xdef\f@ntsize{\vsSMALL}% 
                           \def\s@ze{5}% 
                           \loadvptfonts\vpttrue\fi 
                       \else 
                           \def\SMALL{eight}% 
                           \def\smSMALL{six}% 
                           \def\vsSMALL{five}% 
                           \ifviiipt\else\xdef\f@ntsize{\SMALL}% 
                           \def\s@ze{8}% 
                           \loadsmallfonts\viiipttrue\fi 
                           \ifvipt\else\xdef\f@ntsize{\smSMALL}% 
                           \def\s@ze{6}% 
                           \loadviptfonts\vipttrue\fi 
                           \ifvpt\else\xdef\f@ntsize{\vsSMALL}% 
                           \def\s@ze{5}% 
                           \loadvptfonts\vpttrue\fi 
                 \ifnum#1=8\else\message{Small size should be 10, 9 or  
                            8 point will now default to 8pt}\fi 
                \fi\fi\smallfonts} 
% 
\def\F@nt{\expandafter\font\csname} 
\def\Sk@w{\expandafter\skewchar\csname} 
\def\@nd{\endcsname} 
\def\@step#1{ scaled \magstep#1} 
\def\@half{ scaled \magstephalf} 
\def\@t#1{ at #1pt} 
% 
% For 14, 17 and 20 point fonts use \loadheadfonts 
% 
\def\loadheadfonts{\bigf@nts 
\F@nt \f@ntsize bdi\@nd=cmmib10 \@t{\s@ze}% 
\Sk@w \f@ntsize bdi\@nd='177 
\F@nt \f@ntsize bsy\@nd=cmbsy10 \@t{\s@ze}% 
\Sk@w \f@ntsize bsy\@nd='60 
\F@nt \f@ntsize bss\@nd=cmssbx10 \@t{\s@ze}} 
% 
% For 12 point fonts use \loadxiiptfonts 
% 
\def\loadxiiptfonts{\bigf@nts 
\F@nt \f@ntsize bdi\@nd=cmmib10 \@step{\m@g}% 
\Sk@w \f@ntsize bdi\@nd='177 
\F@nt \f@ntsize bsy\@nd=cmbsy10 \@step{\m@g}% 
\Sk@w \f@ntsize bsy\@nd='60 
\F@nt \f@ntsize bss\@nd=cmssbx10 \@step{\m@g}} 
% 
% For 11 point fonts use \loadxiptfonts 
% 
\def\loadxiptfonts{% 
\font\elevenrm=cmr10 \@half 
\font\eleveni=cmmi10 \@half 
\skewchar\eleveni='177 
\font\elevensy=cmsy10 \@half 
\skewchar\elevensy='60 
\font\elevenex=cmex10 \@half 
\font\elevenit=cmti10 \@half 
\font\elevensl=cmsl10 \@half 
\font\elevenbf=cmbx10 \@half 
\font\eleventt=cmtt10 \@half 
\ifams\font\elevenmsa=msam10 \@half 
\font\elevenmsb=msbm10 \@half\else\fi 
\font\elevenbdi=cmmib10 \@half 
\skewchar\elevenbdi='177 
\font\elevenbsy=cmbsy10 \@half 
\skewchar\elevenbsy='60 
\font\elevenbss=cmssbx10 \@half} 
% 
% For 10 point fonts use \loadxptfonts 
% 
\def\loadxptfonts{% 
\font\tenbdi=cmmib10 
\skewchar\tenbdi='177 
\font\tenbsy=cmbsy10  
\skewchar\tenbsy='60 
\ifams\font\tenmsa=msam10  
\font\tenmsb=msbm10\else\fi 
\font\tenbss=cmssbx10}%  
% 
% For 8 and 9 point fonts use \loadsmallfonts 
% 
\def\loadsmallfonts{\smallf@nts 
\ifams 
\F@nt \f@ntsize ex\@nd=cmex\s@ze 
\else 
\F@nt \f@ntsize ex\@nd=cmex10\fi 
\F@nt \f@ntsize it\@nd=cmti\s@ze 
\F@nt \f@ntsize sl\@nd=cmsl\s@ze 
\F@nt \f@ntsize tt\@nd=cmtt\s@ze} 
% 
% For 7 point fonts use \loadviiptfonts 
% 
\def\loadviiptfonts{% 
\font\sevenit=cmti7 
\font\sevensl=cmsl8 at 7pt 
\ifams\font\sevenmsa=msam7  
\font\sevenmsb=msbm7 
\font\sevenex=cmex7 
\font\sevenbsy=cmbsy7 
\font\sevenbdi=cmmib7\else 
\font\sevenex=cmex10 
\font\sevenbsy=cmbsy10 at 7pt 
\font\sevenbdi=cmmib10 at 7pt\fi 
\skewchar\sevenbsy='60 
\skewchar\sevenbdi='177 
\font\sevenbss=cmssbx10 at 7pt}%  
% 
%  For 6 point fonts use \loadviptfonts 
% 
\def\loadviptfonts{\smallf@nts 
\ifams\font\sixex=cmex7 at 6pt\else 
\font\sixex=cmex10\fi 
\font\sixit=cmti7 at 6pt} 
% 
% For 5 point fonts use \loadvptfonts 
% 
\def\loadvptfonts{% 
\font\fiveit=cmti7 at 5pt 
\ifams\font\fiveex=cmex7 at 5pt 
\font\fivebdi=cmmib5 
\font\fivebsy=cmbsy5 
\font\fivemsa=msam5  
\font\fivemsb=msbm5\else 
\font\fiveex=cmex10 
\font\fivebdi=cmmib10 at 5pt 
\font\fivebsy=cmbsy10 at 5pt\fi 
\skewchar\fivebdi='177 
\skewchar\fivebsy='60 
\font\fivebss=cmssbx10 at 5pt} 
% 
\def\bigf@nts{% 
\F@nt \f@ntsize rm\@nd=cmr10 \@step{\m@g}% 
\F@nt \f@ntsize i\@nd=cmmi10 \@step{\m@g}% 
\Sk@w \f@ntsize i\@nd='177 
\F@nt \f@ntsize sy\@nd=cmsy10 \@step{\m@g}% 
\Sk@w \f@ntsize sy\@nd='60 
\F@nt \f@ntsize ex\@nd=cmex10 \@step{\m@g}% 
\F@nt \f@ntsize it\@nd=cmti10 \@step{\m@g}% 
\F@nt \f@ntsize sl\@nd=cmsl10 \@step{\m@g}% 
\F@nt \f@ntsize bf\@nd=cmbx10 \@step{\m@g}% 
\F@nt \f@ntsize tt\@nd=cmtt10 \@step{\m@g}% 
\ifams 
\F@nt \f@ntsize msa\@nd=msam10 \@step{\m@g}% 
\F@nt \f@ntsize msb\@nd=msbm10 \@step{\m@g}\else\fi} 
% 
\def\smallf@nts{% 
\F@nt \f@ntsize rm\@nd=cmr\s@ze 
\F@nt \f@ntsize i\@nd=cmmi\s@ze  
\Sk@w \f@ntsize i\@nd='177 
\F@nt \f@ntsize sy\@nd=cmsy\s@ze 
\Sk@w \f@ntsize sy\@nd='60 
\F@nt \f@ntsize bf\@nd=cmbx\s@ze  
\ifams 
\F@nt \f@ntsize bdi\@nd=cmmib\s@ze  
\F@nt \f@ntsize bsy\@nd=cmbsy\s@ze  
\F@nt \f@ntsize msa\@nd=msam\s@ze  
\F@nt \f@ntsize msb\@nd=msbm\s@ze 
\else 
\F@nt \f@ntsize bdi\@nd=cmmib10 \@t{\s@ze}%  
\F@nt \f@ntsize bsy\@nd=cmbsy10 \@t{\s@ze}\fi  
\Sk@w \f@ntsize bdi\@nd='177 
\Sk@w \f@ntsize bsy\@nd='60 
\F@nt \f@ntsize bss\@nd=cmssbx10 \@t{\s@ze}}%  
% 
% Fonts for headings  
% 
\def\headfonts{% 
\textfont0=\csname\HEAD rm\@nd         
\scriptfont0=\csname\smHEAD rm\@nd 
\scriptscriptfont0=\csname\vsHEAD rm\@nd 
\def\rm{\fam0\csname\HEAD rm\@nd 
\def\sc{\csname\smHEAD rm\@nd}}% 
% 
\textfont1=\csname\HEAD i\@nd          
\scriptfont1=\csname\smHEAD i\@nd 
\scriptscriptfont1=\csname\vsHEAD i\@nd 
% 
\textfont2=\csname\HEAD sy\@nd         
\scriptfont2=\csname\smHEAD sy\@nd 
\scriptscriptfont2=\csname\vsHEAD sy\@nd 
% 
\textfont3=\csname\HEAD ex\@nd         
\scriptfont3=\csname\smHEAD ex\@nd 
\scriptscriptfont3=\csname\smHEAD ex\@nd 
% 
\textfont\itfam=\csname\HEAD it\@nd    
\scriptfont\itfam=\csname\smHEAD it\@nd 
\scriptscriptfont\itfam=\csname\vsHEAD it\@nd 
\def\it{\fam\itfam\csname\HEAD it\@nd 
\def\sc{\csname\smHEAD it\@nd}}% 
% 
\textfont\slfam=\csname\HEAD sl\@nd    
\def\sl{\fam\slfam\csname\HEAD sl\@nd 
\def\sc{\csname\smHEAD sl\@nd}}% 
% 
\textfont\bffam=\csname\HEAD bf\@nd    
\scriptfont\bffam=\csname\smHEAD bf\@nd 
\scriptscriptfont\bffam=\csname\vsHEAD bf\@nd 
\def\bf{\fam\bffam\csname\HEAD bf\@nd 
\def\sc{\csname\smHEAD bf\@nd}}% 
% 
\textfont\ttfam=\csname\HEAD tt\@nd    
\def\tt{\fam\ttfam\csname\HEAD tt\@nd}% 
% 
\textfont\bdifam=\csname\HEAD bdi\@nd  
\scriptfont\bdifam=\csname\smHEAD bdi\@nd 
\scriptscriptfont\bdifam=\csname\vsHEAD bdi\@nd 
\def\bdi{\fam\bdifam\csname\HEAD bdi\@nd}% 
% 
\textfont\bsyfam=\csname\HEAD bsy\@nd  
\scriptfont\bsyfam=\csname\smHEAD bsy\@nd 
\def\bsy{\fam\bsyfam\csname\HEAD bsy\@nd}% 
% 
\textfont\bssfam=\csname\HEAD bss\@nd  
\scriptfont\bssfam=\csname\smHEAD bss\@nd 
\scriptscriptfont\bssfam=\csname\vsHEAD bss\@nd 
\def\bss{\fam\bssfam\csname\HEAD bss\@nd}% 
% 
\ifams 
\textfont\msafam=\csname\HEAD msa\@nd  
\scriptfont\msafam=\csname\smHEAD msa\@nd 
\scriptscriptfont\msafam=\csname\vsHEAD msa\@nd 
% 
\textfont\msbfam=\csname\HEAD msb\@nd  
\scriptfont\msbfam=\csname\smHEAD msb\@nd 
\scriptscriptfont\msbfam=\csname\vsHEAD msb\@nd 
\else\fi 
% 
\normalbaselineskip=\headb@seline pt% 
\setbox\strutbox=\hbox{\vrule height.7\normalbaselineskip  
depth.3\baselineskip width0pt}% 
\def\sc{\csname\smHEAD rm\@nd}\normalbaselines\bf} 
% 
% Fonts for text 
% 
\def\textfonts{% 
\textfont0=\csname\TEXT rm\@nd         
\scriptfont0=\csname\smTEXT rm\@nd 
\scriptscriptfont0=\csname\vsTEXT rm\@nd 
\def\rm{\fam0\csname\TEXT rm\@nd 
\def\sc{\csname\smTEXT rm\@nd}}% 
% 
\textfont1=\csname\TEXT i\@nd          
\scriptfont1=\csname\smTEXT i\@nd 
\scriptscriptfont1=\csname\vsTEXT i\@nd 
% 
\textfont2=\csname\TEXT sy\@nd         
\scriptfont2=\csname\smTEXT sy\@nd 
\scriptscriptfont2=\csname\vsTEXT sy\@nd 
% 
\textfont3=\csname\TEXT ex\@nd         
\scriptfont3=\csname\smTEXT ex\@nd 
\scriptscriptfont3=\csname\smTEXT ex\@nd 
% 
\textfont\itfam=\csname\TEXT it\@nd    
\scriptfont\itfam=\csname\smTEXT it\@nd 
\scriptscriptfont\itfam=\csname\vsTEXT it\@nd 
\def\it{\fam\itfam\csname\TEXT it\@nd 
\def\sc{\csname\smTEXT it\@nd}}% 
% 
\textfont\slfam=\csname\TEXT sl\@nd    
\def\sl{\fam\slfam\csname\TEXT sl\@nd 
\def\sc{\csname\smTEXT sl\@nd}}% 
% 
\textfont\bffam=\csname\TEXT bf\@nd    
\scriptfont\bffam=\csname\smTEXT bf\@nd 
\scriptscriptfont\bffam=\csname\vsTEXT bf\@nd 
\def\bf{\fam\bffam\csname\TEXT bf\@nd 
\def\sc{\csname\smTEXT bf\@nd}}% 
% 
\textfont\ttfam=\csname\TEXT tt\@nd    
\def\tt{\fam\ttfam\csname\TEXT tt\@nd}% 
% 
\textfont\bdifam=\csname\TEXT bdi\@nd  
\scriptfont\bdifam=\csname\smTEXT bdi\@nd 
\scriptscriptfont\bdifam=\csname\vsTEXT bdi\@nd 
\def\bdi{\fam\bdifam\csname\TEXT bdi\@nd}% 
% 
\textfont\bsyfam=\csname\TEXT bsy\@nd  
\scriptfont\bsyfam=\csname\smTEXT bsy\@nd 
\def\bsy{\fam\bsyfam\csname\TEXT bsy\@nd}% 
% 
\textfont\bssfam=\csname\TEXT bss\@nd  
\scriptfont\bssfam=\csname\smTEXT bss\@nd 
\scriptscriptfont\bssfam=\csname\vsTEXT bss\@nd 
\def\bss{\fam\bssfam\csname\TEXT bss\@nd}% 
% 
\ifams 
\textfont\msafam=\csname\TEXT msa\@nd  
\scriptfont\msafam=\csname\smTEXT msa\@nd 
\scriptscriptfont\msafam=\csname\vsTEXT msa\@nd 
% 
\textfont\msbfam=\csname\TEXT msb\@nd  
\scriptfont\msbfam=\csname\smTEXT msb\@nd 
\scriptscriptfont\msbfam=\csname\vsTEXT msb\@nd 
\else\fi 
% 
\normalbaselineskip=\textb@seline pt 
\setbox\strutbox=\hbox{\vrule height.7\normalbaselineskip  
depth.3\baselineskip width0pt}% 
\everymath{}% 
\def\sc{\csname\smTEXT rm\@nd}\normalbaselines\rm} 
% 
% Fonts for small material (captions, footnotes etc) 
% 
\def\smallfonts{% 
\textfont0=\csname\SMALL rm\@nd         
\scriptfont0=\csname\smSMALL rm\@nd 
\scriptscriptfont0=\csname\vsSMALL rm\@nd 
\def\rm{\fam0\csname\SMALL rm\@nd 
\def\sc{\csname\smSMALL rm\@nd}}% 
% 
\textfont1=\csname\SMALL i\@nd          
\scriptfont1=\csname\smSMALL i\@nd 
\scriptscriptfont1=\csname\vsSMALL i\@nd 
% 
\textfont2=\csname\SMALL sy\@nd         
\scriptfont2=\csname\smSMALL sy\@nd 
\scriptscriptfont2=\csname\vsSMALL sy\@nd 
% 
\textfont3=\csname\SMALL ex\@nd         
\scriptfont3=\csname\smSMALL ex\@nd 
\scriptscriptfont3=\csname\smSMALL ex\@nd 
% 
\textfont\itfam=\csname\SMALL it\@nd    
\scriptfont\itfam=\csname\smSMALL it\@nd 
\scriptscriptfont\itfam=\csname\vsSMALL it\@nd 
\def\it{\fam\itfam\csname\SMALL it\@nd 
\def\sc{\csname\smSMALL it\@nd}}% 
% 
\textfont\slfam=\csname\SMALL sl\@nd    
\def\sl{\fam\slfam\csname\SMALL sl\@nd 
\def\sc{\csname\smSMALL sl\@nd}}% 
% 
\textfont\bffam=\csname\SMALL bf\@nd    
\scriptfont\bffam=\csname\smSMALL bf\@nd 
\scriptscriptfont\bffam=\csname\vsSMALL bf\@nd 
\def\bf{\fam\bffam\csname\SMALL bf\@nd 
\def\sc{\csname\smSMALL bf\@nd}}% 
% 
\textfont\ttfam=\csname\SMALL tt\@nd    
\def\tt{\fam\ttfam\csname\SMALL tt\@nd}% 
% 
\textfont\bdifam=\csname\SMALL bdi\@nd  
\scriptfont\bdifam=\csname\smSMALL bdi\@nd 
\scriptscriptfont\bdifam=\csname\vsSMALL bdi\@nd 
\def\bdi{\fam\bdifam\csname\SMALL bdi\@nd}% 
% 
\textfont\bsyfam=\csname\SMALL bsy\@nd  
\scriptfont\bsyfam=\csname\smSMALL bsy\@nd 
\def\bsy{\fam\bsyfam\csname\SMALL bsy\@nd}% 
% 
\textfont\bssfam=\csname\SMALL bss\@nd  
\scriptfont\bssfam=\csname\smSMALL bss\@nd 
\scriptscriptfont\bssfam=\csname\vsSMALL bss\@nd 
\def\bss{\fam\bssfam\csname\SMALL bss\@nd}% 
% 
\ifams 
\textfont\msafam=\csname\SMALL msa\@nd  
\scriptfont\msafam=\csname\smSMALL msa\@nd 
\scriptscriptfont\msafam=\csname\vsSMALL msa\@nd 
% 
\textfont\msbfam=\csname\SMALL msb\@nd  
\scriptfont\msbfam=\csname\smSMALL msb\@nd 
\scriptscriptfont\msbfam=\csname\vsSMALL msb\@nd 
\else\fi 
% 
\normalbaselineskip=\smallb@seline pt% 
\setbox\strutbox=\hbox{\vrule height.7\normalbaselineskip  
depth.3\baselineskip width0pt}% 
\everymath{}% 
\def\sc{\csname\smSMALL rm\@nd}\normalbaselines\rm}% 
% 
\everydisplay{\indenteddisplay 
   \gdef\labeltype{\eqlabel}}% 
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%                                                         % 
%  Macros to define extra maths symbols                   % 
%                                                         % 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% 
\def\hexnumber@#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or 
 9\or A\or B\or C\or D\or E\or F\fi} 
\edef\bffam@{\hexnumber@\bffam} 
\edef\bdifam@{\hexnumber@\bdifam} 
\edef\bsyfam@{\hexnumber@\bsyfam} 
% 
\def\undefine#1{\let#1\undefined} 
\def\newsymbol#1#2#3#4#5{\let\next@\relax 
 \ifnum#2=\thr@@\let\next@\bdifam@\else 
 \ifams 
 \ifnum#2=\@ne\let\next@\msafam@\else 
 \ifnum#2=\tw@\let\next@\msbfam@\fi\fi 
 \fi\fi 
 \mathchardef#1="#3\next@#4#5} 
\def\mathhexbox@#1#2#3{\relax 
 \ifmmode\mathpalette{}{\m@th\mathchar"#1#2#3}% 
 \else\leavevmode\hbox{$\m@th\mathchar"#1#2#3$}\fi} 
% 
\def\bcal{\fam\bsyfam\relax} 
\def\bi#1{{\fam\bdifam\relax#1}} 
% 
% If file amsmacro is not in current directory 
% or somewhere with set path add path before 
% file name in following line 
% 
\ifams\input amsmacro\fi 
% 
% Bold italic Greek characters 
% 
\newsymbol\bitGamma 3000 
\newsymbol\bitDelta 3001 
\newsymbol\bitTheta 3002 
\newsymbol\bitLambda 3003 
\newsymbol\bitXi 3004 
\newsymbol\bitPi 3005 
\newsymbol\bitSigma 3006 
\newsymbol\bitUpsilon 3007 
\newsymbol\bitPhi 3008 
\newsymbol\bitPsi 3009 
\newsymbol\bitOmega 300A 
\newsymbol\balpha 300B 
\newsymbol\bbeta 300C 
\newsymbol\bgamma 300D 
\newsymbol\bdelta 300E 
\newsymbol\bepsilon 300F 
\newsymbol\bzeta 3010 
\newsymbol\bfeta 3011 
\newsymbol\btheta 3012 
\newsymbol\biota 3013 
\newsymbol\bkappa 3014 
\newsymbol\blambda 3015 
\newsymbol\bmu 3016 
\newsymbol\bnu 3017 
\newsymbol\bxi 3018 
\newsymbol\bpi 3019 
\newsymbol\brho 301A 
\newsymbol\bsigma 301B 
\newsymbol\btau 301C 
\newsymbol\bupsilon 301D 
\newsymbol\bphi 301E 
\newsymbol\bchi 301F 
\newsymbol\bpsi 3020 
\newsymbol\bomega 3021 
\newsymbol\bvarepsilon 3022 
\newsymbol\bvartheta 3023 
\newsymbol\bvaromega 3024 
\newsymbol\bvarrho 3025 
\newsymbol\bvarzeta 3026 
\newsymbol\bvarphi 3027 
\newsymbol\bpartial 3040 
\newsymbol\bell 3060 
\newsymbol\bimath 307B 
\newsymbol\bjmath 307C 
% 
\mathchardef\binfty "0\bsyfam@31 
\mathchardef\bnabla "0\bsyfam@72 
\mathchardef\bdot "2\bsyfam@01 
\mathchardef\bGamma "0\bffam@00 
\mathchardef\bDelta "0\bffam@01 
\mathchardef\bTheta "0\bffam@02 
\mathchardef\bLambda "0\bffam@03 
\mathchardef\bXi "0\bffam@04 
\mathchardef\bPi "0\bffam@05 
\mathchardef\bSigma "0\bffam@06 
\mathchardef\bUpsilon "0\bffam@07 
\mathchardef\bPhi "0\bffam@08 
\mathchardef\bPsi "0\bffam@09 
\mathchardef\bOmega "0\bffam@0A 
% 
\mathchardef\itGamma "0100 
\mathchardef\itDelta "0101 
\mathchardef\itTheta "0102 
\mathchardef\itLambda "0103 
\mathchardef\itXi "0104 
\mathchardef\itPi "0105 
\mathchardef\itSigma "0106 
\mathchardef\itUpsilon "0107 
\mathchardef\itPhi "0108 
\mathchardef\itPsi "0109 
\mathchardef\itOmega "010A 
% 
\mathchardef\Gamma "0000 
\mathchardef\Delta "0001 
\mathchardef\Theta "0002 
\mathchardef\Lambda "0003 
\mathchardef\Xi "0004 
\mathchardef\Pi "0005 
\mathchardef\Sigma "0006 
\mathchardef\Upsilon "0007 
\mathchardef\Phi "0008 
\mathchardef\Psi "0009 
\mathchardef\Omega "000A 
% 
% Counter definitions 
% 
\newcount\firstpage  \firstpage=1  % start page no 
\newcount\jnl                      % journal no 
\newcount\secno                    % section number 
\newcount\subno                    % number of subsection 
\newcount\subsubno                 % number of subsubsection 
\newcount\appno                    % appendix number 
\newcount\tabno                    % table number 
\newcount\figno                    % figure number 
\newcount\countno                  % equation numbers 
\newcount\refno                    % reference number 
\newcount\eqlett     \eqlett=97    % equation letter 
% 
\newif\ifletter 
\newif\ifwide 
\newif\ifnotfull 
\newif\ifaligned 
\newif\ifnumbysec   
\newif\ifappendix 
\newif\ifnumapp 
\newif\ifssf 
\newif\ifppt 
% 
\newdimen\t@bwidth 
\newdimen\c@pwidth 
\newdimen\digitwidth                    %character width 
\newdimen\argwidth                      %argument width 
\newdimen\secindent    \secindent=5pc   %indentation of maths  
\newdimen\textind    \textind=16pt      %indentation of text 
\newdimen\tempval                       %temporary value 
\newskip\beforesecskip 
\def\beforesecspace{\vskip\beforesecskip\relax} 
\newskip\beforesubskip 
\def\beforesubspace{\vskip\beforesubskip\relax} 
\newskip\beforesubsubskip 
\def\beforesubsubspace{\vskip\beforesubsubskip\relax} 
\newskip\secskip 
\def\secspace{\vskip\secskip\relax} 
\newskip\subskip 
\def\subspace{\vskip\subskip\relax} 
\newskip\insertskip 
\def\insertspace{\vskip\insertskip\relax} 
% 
\def\sp@ce{\ifx\next*\let\next=\@ssf 
               \else\let\next=\@nossf\fi\next} 
\def\@ssf#1{\nobreak\secspace\global\ssftrue\nobreak} 
\def\@nossf{\nobreak\secspace\nobreak\noindent\ignorespaces} 
% 
\def\subsp@ce{\ifx\next*\let\next=\@sssf 
               \else\let\next=\@nosssf\fi\next} 
\def\@sssf#1{\nobreak\subspace\global\ssftrue\nobreak} 
\def\@nosssf{\nobreak\subspace\nobreak\noindent\ignorespaces} 
% 
\beforesecskip=24pt plus12pt minus8pt 
\beforesubskip=12pt plus6pt minus4pt 
\beforesubsubskip=12pt plus6pt minus4pt 
\secskip=12pt plus 2pt minus 2pt 
\subskip=6pt plus3pt minus2pt 
\insertskip=18pt plus6pt minus6pt% 
\fontdimen16\tensy=2.7pt 
\fontdimen17\tensy=2.7pt 
% 
% Labels etc for cross referencing macros 
% 
\def\eqlabel{(\ifappendix\applett 
               \ifnumbysec\ifnum\secno>0 \the\secno\fi.\fi 
               \else\ifnumbysec\the\secno.\fi\fi\the\countno)} 
\def\seclabel{\ifappendix\ifnumapp\else\applett\fi 
    \ifnum\secno>0 \the\secno 
    \ifnumbysec\ifnum\subno>0.\the\subno\fi\fi\fi 
    \else\the\secno\fi\ifnum\subno>0.\the\subno 
         \ifnum\subsubno>0.\the\subsubno\fi\fi} 
\def\tablabel{\ifappendix\applett\fi\the\tabno} 
\def\figlabel{\ifappendix\applett\fi\the\figno} 
% 
\def\gac{\global\advance\countno by 1} 
% 
% Redefinition of footnote macros to lose rule and remove indentation 
% 
\def\footnoterule{} 
\def\vfootnote#1{\insert\footins\bgroup 
\interlinepenalty=\interfootnotelinepenalty 
\splittopskip=\ht\strutbox % top baseline for broken footnotes 
\splitmaxdepth=\dp\strutbox \floatingpenalty=20000 
\leftskip=0pt \rightskip=0pt \spaceskip=0pt \xspaceskip=0pt% 
\noindent\smallfonts\rm #1\ \ignorespaces\footstrut\futurelet\next\fo@t} 
% 
% Redefinition of endinsert to give more controllable 
% space around  tables and figures 
% 
\def\endinsert{\egroup 
    \if@mid \dimen@=\ht0 \advance\dimen@ by\dp0 
       \advance\dimen@ by12\p@ \advance\dimen@ by\pagetotal 
       \ifdim\dimen@>\pagegoal \@midfalse\p@gefalse\fi\fi 
    \if@mid \insertspace \box0 \par \ifdim\lastskip<\insertskip 
    \removelastskip \penalty-200 \insertspace \fi 
    \else\insert\topins{\penalty100 
       \splittopskip=0pt \splitmaxdepth=\maxdimen  
       \floatingpenalty=0 
       \ifp@ge \dimen@=\dp0 
       \vbox to\vsize{\unvbox0 \kern-\dimen@}% 
       \else\box0\nobreak\insertspace\fi}\fi\endgroup}    
% 
% special macros for display equations 
% 
% for indentation of turned over lines in mathematics 
% 
\def\ind{\hbox to \secindent{\hfill}} 
% 
% for turned over equals sign to left of maths indent 
% 
\def\eql{\llap{${}={}$}} 
\def\lsim{\llap{${}\sim{}$}} 
\def\lsimeq{\llap{${}\simeq{}$}} 
\def\lequiv{\llap{${}\equiv{}$}} 
% 
% for other signs to left of maths indent 
% 
\def\lo#1{\llap{${}#1{}$}} 
% 
% displayed equation indented  
% 
\def\indeqn#1{\alignedfalse\displ@y\halign{\hbox to \displaywidth 
    {$\ind\@lign\displaystyle##\hfil$}\crcr #1\crcr}} 
% 
% displayed equation indented with alignments 
% 
\def\indalign#1{\alignedtrue\displ@y \tabskip=0pt  
  \halign to\displaywidth{\ind$\@lign\displaystyle{##}$\tabskip=0pt 
    &$\@lign\displaystyle{{}##}$\hfill\tabskip=\centering 
    &\llap{$\@lign\hbox{\rm##}$}\tabskip=0pt\crcr 
    #1\crcr}} 
% 
\def\fl{{\hskip-\secindent}} 
% 
\def\indenteddisplay#1$${\indispl@y{#1 }} 
\def\indispl@y#1{\disptest#1\eqalignno\eqalignno\disptest} 
\def\disptest#1\eqalignno#2\eqalignno#3\disptest{% 
    \ifx#3\eqalignno 
    \indalign#2% 
    \else\indeqn{#1}\fi$$} 
% 
% Roman small caps (if in Roman \sc gives small caps) 
% 
\def\rmsc{\rm\sc} 
% 
% Italic small caps (if in italic \sc gives italic small caps) 
% 
\def\itsc{\it\sc} 
% 
% Bold small caps (if in bold \sc gives bold small caps) 
% 
\def\bfsc{\bf\rm} 
% 
% Small caps in maths 
% 
\def\msc#1{\hbox{\sc #1}} 
% 
% Miscellaneous definitions 
% 
\def\ms{\noalign{\vskip3pt plus3pt minus2pt}} 
\def\bs{\noalign{\vskip6pt plus3pt minus2pt}} 
\def\ns{\noalign{\vskip-3pt}} 
\def\bk{\noalign{\break}} 
\def\nb{\noalign{\nobreak}} 
% 
\def\hfb{\hfill\break} 
% 
% Bold h bar 
% 
\def\bhbar{\rlap{\kern1pt\raise.4ex\hbox{\bf\char'40}}\bi{h}} 
% 
\def\case#1#2{{\textstyle{#1\over#2}}} 
\def\dash{---{}--- } 
\let\du=\d 
\def\d{{\rm d}} 
\def\e{{\rm e}} 
\def\etal{{\it et al\/}\ } 
\def\frac#1#2{{#1\over#2}} 
\ifams 
\def\lap{\lesssim} 
\def\gap{\gtrsim} 
\let\le=\leqslant 
\let\lequal=\leq 
\let\leq=\leqslant 
\let\ge=\geqslant 
\let\gequal=\geq 
\let\geq=\geqslant 
\else 
\let\lequal=\le 
\let\gequal=\ge 
\def\gap{\;\lower3pt\hbox{$\buildrel > \over \sim$}\;}% 
\def\lap{\;\lower3pt\hbox{$\buildrel < \over \sim$}\;}\fi 
\def\i{{\rm i}} 
\chardef\ii="10 
\def\tqs{\hbox to 25pt{\hfil}} 
\let\lc=\lowercase 
\let\uc=\uppercase 
\def\Or{\mathop{\rm O}\nolimits} 
\def\Tr{\mathop{\rm Tr}\nolimits} 
\def\tr{\mathop{\rm tr}\nolimits} 
\def\Cal#1{{\cal #1}} 
\def\Bbbone{1\kern-.22em {\rm l}} 
% 
% Primes to display summations and products  
% which also have sub or superscripts 
% 
\def\rp{\raise8pt\hbox{$\scriptstyle\prime$}} 
% 
% then use \sum^{...}_{...}\rp or \prod^{...}_{...}\rp. 
% 
% Shadow brackets 
% 
% Single brackets for normal size only 
% 
\def\lshad{\lbrack\!\lbrack} 
\def\rshad{\rbrack\!\rbrack} 
% 
% Variable size for display style 
% 
\def\[#1\]{\setbox0=\hbox{$\dsty#1$}\argwidth=\wd0 
    \setbox0=\hbox{$\left[\box0\right]$}\advance\argwidth by -\wd0 
    \left[\kern.3\argwidth\box0\kern.3\argwidth\right]} 
% 
% Variable size for text style 
% 
\def\lsb#1\rsb{\setbox0=\hbox{$#1$}\argwidth=\wd0 
    \setbox0=\hbox{$\left[\box0\right]$}\advance\argwidth by -\wd0 
    \left[\kern.3\argwidth\box0\kern.3\argwidth\right]} 
% 
\def\pounds{{\tenu \$}} 
% 
% Square for end of theorems 
% 
\def\sqr{\hfill\hbox{$\square$}} 
% 
\def\pt(#1){({\it #1\/})} 
\def\ts{{\thinspace}}% 
\def\co{$^{\rm c}\!/\!_{\rm o}$} 
% 
\let\dsty=\displaystyle 
\let\tsty=\textstyle 
\let\ssty=\scriptstyle 
\let\sssty=\scriptscriptstyle 
% 
% Definition for Nuclear Physics Keyword abstract 
% 
\def\reactions#1{\vskip 12pt plus2pt minus2pt%              
\vbox{\hbox{\kern\secindent\vrule\kern12pt% 
\vbox{\kern0.5pt\vbox{\hsize=24pc\parindent=0pt\smallfonts\rm NUCLEAR  
REACTIONS\strut\quad #1\strut}\kern0.5pt}\kern12pt\vrule}}} 
% 
% Definition for slashed characters 
% 
\def\slashchar#1{\setbox0=\hbox{$#1$}\dimen0=\wd0% 
\setbox1=\hbox{/}\dimen1=\wd1% 
\ifdim\dimen0>\dimen1%                         
\rlap{\hbox to \dimen0{\hfil/\hfil}}#1\else                                         
\rlap{\hbox to \dimen1{\hfil$#1$\hfil}}/\fi} 
% 
% Redefine \textindent for use in \item 
% 
\def\textindent#1{\noindent\hbox to \parindent{#1\hss}\ignorespaces} 
% 
% Symbols and curves for use in figure captions 
% 
\def\opencirc{\raise1pt\hbox{$\scriptstyle{\bigcirc}$}} 
\def\fullcirc{\hbox{\headfonts\rm$\scriptstyle\bullet$}} 
\ifams 
\def\opensqr{\hbox{$\square$}} 
\def\opentri{\hbox{$\vartriangle$}} 
\def\opentridown{\hbox{$\triangledown$}} 
\def\opendiamond{\hbox{$\lozenge$}} 
\def\fullsqr{\hbox{$\blacksquare$}} 
\def\fulldiamond{\hbox{$\blacklozenge$}} 
\def\fulltri{\hbox{$\blacktriangle$}} 
\def\fulltridown{\hbox{$\blacktriangledown$}} 
\else 
\def\opensqr{\vbox{\hrule height.4pt\hbox{\vrule width.4pt height3.5pt 
    \kern3.5pt\vrule width.4pt}\hrule height.4pt}} 
\def\opentri{\hbox{$\scriptstyle\bigtriangleup$}} 
\def\opentridown{\raise1pt\hbox{$\scriptstyle\bigtriangledown$}} 
\def\opendiamond{\hbox{$\diamond$}} 
\def\fullsqr{\vrule height3.5pt width3.5pt} 
\def\fulldiamond{\opendiamond}           %  These produce the 
\def\fulltri{\opentri}                   %  equivalent open character 
\def\fulltridown{\opentridown}           %  to be filled in. 
\fi 
\def\dotted{\hbox{${\mathinner{\cdotp\cdotp\cdotp\cdotp\cdotp\cdotp}}$}} 
\def\dashed{\hbox{-\ts -\ts -\ts -}} 
\def\broken{\hbox{-- -- --}} 
\def\longbroken{\hbox{--- --- ---}} 
\def\chain{\hbox{--- $\cdot$ ---}} 
\def\dashddot{\hbox{--- $\cdot$ $\cdot$ ---}} 
\def\full{\hbox{------}} 
% 
% Redefinition of \cases 
% 
\def\m@th{\mathsurround=0pt} 
% 
% Displaystyle now used for first term 
% 
\def\cases#1{% 
\left\{\,\vcenter{\normalbaselines\openup1\jot\m@th% 
     \ialign{$\displaystyle##\hfil$&\rm\tqs##\hfil\crcr#1\crcr}}\right.}% 
% 
% Original version of cases now called \oldcases 
% 
\def\oldcases#1{\left\{\,\vcenter{\normalbaselines\m@th 
    \ialign{$##\hfil$&\rm\quad##\hfil\crcr#1\crcr}}\right.} 
% 
% Cases with number at end each line (using automatic numbering) 
% 
\def\numcases#1{\left\{\,\vcenter{\baselineskip=15pt\m@th% 
     \ialign{$\displaystyle##\hfil$&\rm\tqs##\hfil 
     \crcr#1\crcr}}\right.\hfill 
     \vcenter{\baselineskip=15pt\m@th% 
     \ialign{\rlap{$\phantom{\displaystyle##\hfil}$}\tabskip=0pt&\en 
     \rlap{\phantom{##\hfil}}\crcr#1\crcr}}} 
% 
\def\ptnumcases#1{\left\{\,\vcenter{\baselineskip=15pt\m@th% 
     \ialign{$\displaystyle##\hfil$&\rm\tqs##\hfil 
     \crcr#1\crcr}}\right.\hfill 
     \vcenter{\baselineskip=15pt\m@th% 
     \ialign{\rlap{$\phantom{\displaystyle##\hfil}$}\tabskip=0pt&\enpt 
     \rlap{\phantom{##\hfil}}\crcr#1\crcr}}\global\eqlett=97 
     \global\advance\countno by 1} 
% 
% for equation numbers instead of \eqno 
% 
\def\eq(#1){\ifaligned\@mp(#1)\else\hfill\llap{{\rm (#1)}}\fi} 
\def\ceq(#1){\ns\ns\ifaligned\@mp\fi\eq(#1)\cr\ns\ns} 
\def\eqpt(#1#2){\ifaligned\@mp(#1{\it #2\/}) 
                    \else\hfill\llap{{\rm (#1{\it #2\/})}}\fi} 
\let\eqno=\eq 
% 
% Automatic numbering of equations 
% 
\countno=1 
\def\eqnobysec{\numbysectrue} 
\def\aleq{&\rm(\ifappendix\applett 
               \ifnumbysec\ifnum\secno>0 \the\secno\fi.\fi 
               \else\ifnumbysec\the\secno.\fi\fi\the\countno} 
\def\noaleq{\hfill\llap\bgroup\rm(\ifappendix\applett 
               \ifnumbysec\ifnum\secno>0 \the\secno\fi.\fi 
               \else\ifnumbysec\the\secno.\fi\fi\the\countno} 
\def\@mp{&} 
\def\en{\ifaligned\aleq)\else\noaleq)\egroup\fi\gac} 
\def\cen{\ns\ns\ifaligned\@mp\fi\en\cr\ns\ns} 
\def\enpt{\ifaligned\aleq{\it\char\the\eqlett})\else 
    \noaleq{\it\char\the\eqlett})\egroup\fi 
    \global\advance\eqlett by 1} 
\def\endpt{\ifaligned\aleq{\it\char\the\eqlett})\else 
    \noaleq{\it\char\the\eqlett})\egroup\fi 
    \global\eqlett=97\gac} 
% 
% abbreviations for Institute of Physics Publishing journals 
% 
\def\CQG{{\it Class. Quantum Grav.}} 
\def\IP{{\it Inverse Problems\/}} 
\def\JPA{{\it J. Phys. A: Math. Gen.}} 
\def\JPB{{\it J. Phys. B: At. Mol. Phys.}}        %1968-87 
\def\jpb{{\it J. Phys. B: At. Mol. Opt. Phys.}}   %1988 and onwards 
\def\JPC{{\it J. Phys. C: Solid State Phys.}}     %1968--1988 
\def\JPCM{{\it J. Phys.: Condens. Matter\/}}        %1989 and onwards 
\def\JPD{{\it J. Phys. D: Appl. Phys.}} 
\def\JPE{{\it J. Phys. E: Sci. Instrum.}} 
\def\JPF{{\it J. Phys. F: Met. Phys.}} 
\def\JPG{{\it J. Phys. G: Nucl. Phys.}}           %1975--1988 
\def\jpg{{\it J. Phys. G: Nucl. Part. Phys.}}     %1989 and onwards 
\def\MSMSE{{\it Modelling Simulation Mater. Sci. Eng.}} 
\def\MST{{\it Meas. Sci. Technol.}}                 %1990 and onwards 
\def\NET{{\it Network\/}} 
\def\NL{{\it Nonlinearity\/}} 
\def\NT{{\it Nanotechnology}} 
\def\PAO{{\it Pure Appl. Optics\/}} 
\def\PMB{{\it Phys. Med. Biol.}} 
\def\PSST{{\it Plasma Sources Sci. Technol.}} 
\def\QO{{\it Quantum Opt.}} 
\def\RPP{{\it Rep. Prog. Phys.}} 
\def\SST{{\it Semicond. Sci. Technol.}} 
\def\SUST{{\it Supercond. Sci. Technol.}} 
\def\WRM{{\it Waves Random Media\/}} 
% 
% Other commonly quoted journals 
% 
\def\AC{{\it Acta Crystallogr.}} 
\def\AM{{\it Acta Metall.}} 
\def\AP{{\it Ann. Phys., Lpz.}} 
\def\APNY{{\it Ann. Phys., NY\/}} 
\def\APP{{\it Ann. Phys., Paris\/}} 
\def\CJP{{\it Can. J. Phys.}} 
\def\GRG{{\it Gen. Rel. Grav.}} 
\def\JAP{{\it J. Appl. Phys.}} 
\def\JCP{{\it J. Chem. Phys.}} 
\def\JJAP{{\it Japan. J. Appl. Phys.}} 
\def\JP{{\it J. Physique\/}} 
\def\JPhCh{{\it J. Phys. Chem.}} 
\def\JMMM{{\it J. Magn. Magn. Mater.}} 
\def\JMP{{\it J. Math. Phys.}} 
\def\JOSA{{\it J. Opt. Soc. Am.}} 
\def\JPSJ{{\it J. Phys. Soc. Japan\/}} 
\def\JQSRT{{\it J. Quant. Spectrosc. Radiat. Transfer\/}} 
\def\NC{{\it Nuovo Cimento\/}} 
\def\NIM{{\it Nucl. Instrum. Methods\/}} 
\def\NP{{\it Nucl. Phys.}} 
\def\PL{{\it Phys. Lett.}} 
\def\PR{{\it Phys. Rev.}} 
\def\PRL{{\it Phys. Rev. Lett.}} 
\def\PRS{{\it Proc. R. Soc.}} 
\def\PS{{\it Phys. Scr.}} 
\def\PSS{{\it Phys. Status Solidi\/}} 
\def\PTRS{{\it Phil. Trans. R. Soc.}} 
\def\RMP{{\it Rev. Mod. Phys.}} 
\def\RSI{{\it Rev. Sci. Instrum.}} 
\def\SSC{{\it Solid State Commun.}} 
\def\ZP{{\it Z. Phys.}} 
% 
\headline={\ifodd\pageno{\ifnum\pageno=\firstpage\hfill 
   \else\rrhead\fi}\else\lrhead\fi} 
% 
\def\rrhead{\textfonts\hskip\secindent\it 
    \shorttitle\hfill\rm\folio} 
% 
\def\lrhead{\textfonts\hbox to\secindent{\rm\folio\hss}% 
    \it\aunames\hss} 
% 
\footline={\ifnum\pageno=\firstpage \hfill\textfonts\rm\folio\fi} 
% 
% 
\def\@rticle#1#2{\vglue.5pc 
    {\parindent=\secindent \bf #1\par} 
     \vskip2.5pc 
    {\exhyphenpenalty=10000\hyphenpenalty=10000 
     \baselineskip=18pt\raggedright\noindent 
     \headfonts\bf#2\par}\futurelet\next\sh@rttitle}% 
% 
\def\title#1{\gdef\shorttitle{#1} 
    \vglue4pc{\exhyphenpenalty=10000\hyphenpenalty=10000  
    \baselineskip=18pt  
    \raggedright\parindent=0pt 
    \headfonts\bf#1\par}\futurelet\next\sh@rttitle}  
\let\paper=\title 
% 
\def\article#1#2{\gdef\shorttitle{#2}\@rticle{#1}{#2}}  
% 
\def\review#1{\gdef\shorttitle{#1}% 
    \@rticle{REVIEW \ifpbm\else ARTICLE\fi}{#1}} 
\def\topical#1{\gdef\shorttitle{#1}% 
    \@rticle{TOPICAL REVIEW}{#1}} 
\def\comment#1{\gdef\shorttitle{#1}% 
    \@rticle{COMMENT}{#1}} 
\def\note#1{\gdef\shorttitle{#1}% 
    \@rticle{NOTE}{#1}} 
\def\prelim#1{\gdef\shorttitle{#1}% 
    \@rticle{PRELIMINARY COMMUNICATION}{#1}} 
\def\letter#1{\gdef\shorttitle{Letter to the Editor}% 
     \gdef\aunames{Letter to the Editor} 
     \global\lettertrue\ifnum\jnl=7\global\letterfalse\fi 
     \@rticle{LETTER TO THE EDITOR}{#1}} 
% 
\def\sh@rttitle{\ifx\next[\let\next=\sh@rt 
                \else\let\next=\f@ll\fi\next} 
% 
\def\sh@rt[#1]{\gdef\shorttitle{#1}} 
\def\f@ll{} 
% 
\def\author#1{\ifletter\else\gdef\aunames{#1}\fi\vskip1.5pc 
    {\parindent=\secindent   
     \hang\textfonts   
     \ifppt\bf\else\rm\fi#1\par}   
     \ifppt\bigskip\else\smallskip\fi 
     \futurelet\next\@unames} 
% 
\def\@unames{\ifx\next[\let\next=\short@uthor 
                 \else\let\next=\@uthor\fi\next} 
\def\short@uthor[#1]{\gdef\aunames{#1}} 
\def\@uthor{} 
% 
\def\address#1{{\parindent=\secindent 
    \exhyphenpenalty=10000\hyphenpenalty=10000 
\ifppt\textfonts\else\smallfonts\fi\hang\raggedright\rm#1\par}% 
    \ifppt\bigskip\fi} 
% 
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main.tex
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%           
%           
%	    Dragi Karevski
%	    Laboratoire de Physique des Mat\'eriaux
%           Universit\'e Henri Poincar\'e (Nancy I)
%           BP239, F--54506  Vand\oe uvre l\`es Nancy Cedex, France
%
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\overfullrule=0pt
\letter{Surface and bulk critical behaviour of the XY chain in a transverse field}[Letter to the Editor]

\author{Dragi Karevski}[Letter to the Editor]

\address{Laboratoire de Physique des Mat\'eriaux\footnote{\dag}{
Unit\'e Mixte de Recherche  CNRS No~7566},  Universit\'e Henri 
Poincar\'e
(Nancy~I), BP239, F--54506  Vand\oe uvre l\`es Nancy Cedex, France}

\abs
The surface magnetization of the quantum XY chain in a transverse field
is found for arbitrary nearest neighbour interactions in closed form.
This allows to derive the bulk phase diagram in a simple way. The
magnetic surface behaviour and the bulk correlation length are
found exactly.
\endabs
\vglue 1.5cm
\date
  
We consider the ferromagnetic anisotropic XY quantum chain in a
transverse field with hamiltonian~\cite{katsura} 
$$
{\cal H}=-\sum_{n=1}^{N}\left[
h_{n}\sigma^{z}_{n}+J_{n}\frac{1+\eta_{n}}{2}\sigma^{x}_{n}\sigma^{x}_{n+1}
+J_{n}\frac{1-\eta_{n}}{2}\sigma^{y}_{n}\sigma^{y}_{n+1}
\right]
\eqno(1)
$$
The $\sigma$s are the Pauli matrices, $h_{n}$ is the local transverse
field at site $n$, $|\eta_{n}|\le 1$ the local anisotropy between couplings in the $x$ 
and $y$ direction. When all the $\eta$s are equal to unity one
recovers the Ising quantum chain~\cite{Pfeuty}. 
With a constant anisotropy factor, $\eta_{n}=\eta$, the phase diagram and bulk 
critical behaviour  are known exactly~\cite{xy}. 
Here we focus on the surface critical behaviour of the chain with free boundary conditions.
To diagonalize the hamiltonian, one has first to perform a Jordan-Wigner~\cite{Jordan}
transformation, introducing fermionic creation and anihilation 
operators $c^{+}_{n},c_{n}$ which satisfy the usual fermionic anticommutation rules.
In terms of these new operators the hamiltonian~(1) is a quadratic form
$$
{\cal H}=-\frac{1}{\zeta}\sum_{n,m=1}^{N}\left[c^{+}_{n}A_{nm}c_{m}+\frac{1}{2}
\left(c^{+}_{n}B_{nm}c^{+}_{m}-c_{n}B_{nm}c_{m}\right)\right]
\eqno(2)
$$
where $\bf A$ is a symmetric matrix such that
$$
A_{jk}=2h_{j}\delta_{j,k}+J_{j-1}\delta_{j,k+1}+J_{j}\delta_{j,k-1}
\eqno(3)
$$
and $\bf B$ an antisymmetric matrix with elements
$$
B_{jk}=-J_{j-1}\eta_{j-1}\delta_{j,k+1}+J_{j}\eta_{j}\delta_{j,k-1}\; .
\eqno(4)
$$
The more efficient way to diagonalize the quadratic form~(2) is to
follow the procedure introduced by Lieb, Schultz and Mattis~\cite{lieb61}
which is valid for a general distribution of the coupling constants.
To proceed, we introduce a new set of diagonal fermionic operators 
$\varepsilon^{+},\varepsilon$ which are linear
combination of the $c^{+}$ and $c$ operators:
$$
\varepsilon^{+}_{q}=\sum_{n}\left[\phi_{q}(n)\frac{c_{n}^{+}+c_{n}}{2}+\psi_{q}(n)\frac{c_{n}^{+}-c_{n}}{2}\right]
\eqno(5)
$$
where the $\phi_{q}(n)$ and $\psi_{q}(n)$ are real.
The hamiltonian~(2) is diagonal
$$
{\cal H}=\sum_{q}\Lambda_{q}\left(\varepsilon^{+}_{q}\varepsilon_{q}-\frac{1}{2}\right)
\eqno(6)
$$
when the vectors $\Psi_{q}$($\Phi_{q}$) with components $\psi_{q}(n)$($\phi_{q}(n)$)
and the fermion excitation energies $\Lambda_{q}$ satisfy
the matrix relations:
$$
({\bf A-B})\Psi_{q}=\Lambda_{q}\Phi_{q}\cr
({\bf A+B})\Phi_{q}=\Lambda_{q}\Psi_{q}\; .
\eqno(7)
$$
One may combine the previous matrix equations to deduce $\Lambda_{q}^{2}$ from the eigenvalue problem
$$
({\bf A-B})({\bf A+B})\Phi_{q}=\Lambda_{q}^{2}\Phi_{q}
\eqno(8)
$$
In order to solve the problem, one has to find the eigenvalues and eigenvectors of
the excitation matrix $({\bf A-B})({\bf A+B})$. All the physical quantities, such as
magnetization, energy density or correlation functions can be expressed in terms of
the eigenvectors $\Phi_{q}$, $\Psi_{q}$ and corresponding excitation energies $\Lambda_{q}$.

We now concentrate on the surface magnetic behaviour. The $x(y)$ component of the surface magnetization 
is obtained from the autocorrelation function 
$G^{x(y)}_{s}(\tau)=\langle\sigma_{1}^{x(y)}(0)\sigma_{1}^{x(y)}(\tau)\rangle$ 
in imaginary time $\tau$  where
$\sigma_{1}^{x(y)}(\tau)=e^{\tau{\cal H}}\sigma^{x(y)}e^{-\tau{\cal H}}$.
In the limit of large $\tau$ we have 
$$
\lim_{\tau\rightarrow\infty}G^{x(y)}_{s}(\tau)=\left[m^{x(y)}_{s}\right]^{2}
\eqno(9)
$$
Using the diagonal basis of $\cal H$, we obtain
$$
m^{x(y)}_{s}=\left|\langle\sigma|\sigma_{1}^{x(y)}|0\rangle\right|
\eqno(10)
$$
where $|0\rangle$ is the ground state and $|\sigma\rangle=\varepsilon^{+}_{1}|0\rangle$ is the first
excited state with one fermion. Noticing that $\sigma^{x}_{1}=c_{1}^{+}+c_{1}$ and that
$\sigma^{y}_{1}=i(c_{1}-c_{1}^{+})$ and making use of~(5)
we obtain $m^{x}_{s}=\phi_{1}(1)$ and similarly 
$m^{y}_{s}=\psi_{1}(1)$.


Following Peschel~\cite{peschel84}, it is now possible to obtain a closed analytical formula for
the $x(y)$ component of the surface magnetization.
On the semi-infinite system ($N\rightarrow\infty$), in the ordered phase
the first gap $E_{\sigma}-E_{0}=\Lambda_{1}$ vanishes due to 
spontaneous symmetry breaking. In this case, equation~(7) simplifies into
$$
({\bf A-B})\Psi_{1}=0\cr
({\bf A+B})\Phi_{1}=0
\eqno(11)
$$
Noticing that changing $\eta$ into $-\eta$,  $m_{s}^{x}$ and $m_{s}^{y}$ 
exchange, in the following, we only concentrate on the $x$ component of the 
surface magnetization.

To find the eigenvetor $\Phi_{1}$ we can rewrite the matrix equation $({\bf A+B})\Phi_{1}=0$ 
under the form
$$
\pmatrix{\phi_{1}(n+1)\cr\phi_{1}(n)}={\bf T}_{n}\pmatrix{\phi_{1}(n)\cr\phi_{1}(n-1)}
\eqno(12)
$$
where ${\bf T}_{n}$ is the two by two matrix
$$
{\bf T_{n}}=-\pmatrix{\frac{2h_{n}}{J_{n}(1+\eta_{n})}& \frac{1-\eta_{n-1}}{1-\eta_{n}}\cr
&\cr
-1&0}
\eqno(13)
$$
Iterating equation~(12) one arrives at
$$
\pmatrix{\phi_{1}(n+1)\cr\phi_{1}(n)}=(-1)^n{\bf T}_{n}{\bf T}_{n-1}\dots{\bf T}_{2}{\bf T}_{1}\pmatrix{\phi_{1}(1)\cr0}\cr
\eqno(14)
$$
The ($n+1$)th component of the eigenvector $\Phi_{1}$ is then given by the expression
$$
\phi_{1}(n+1)=(-1)^{n}\phi_{1}(1)\langle 1|\prod_{j=1}^{n}{\bf T}_{j}|1 \rangle
\eqno(15)
$$
where $\langle 1|\dots|1\rangle$ stands for the $1,1$ component of the corresponding matrix.
The normalization of the eigenvector, $\sum_{n}\phi_{1}^{2}(n)=1$, leads to 
the surface magnetization:
$$
m_{s}^{x}=\phi_{1}(1)=\left[1+\sum_{n=1}^{\infty}
\big|\langle 1|\prod_{j=1}^{n}{\bf T}_{j}|1 \rangle\big|^{2}\right]^{-1/2}
\eqno(16)
$$
For $\eta=1$ we recover Peschel's formula~\cite{peschel84} for the Ising model in a transverse field.
This expression is valid for any distribution of the coupling constants of the chain
provided that the first gap vanishes, i.~e., in the ordered phase(s).
The transition to the paramagnetic phase is characterized by a divergence of the sum
in~(16). Since for a one-dimensional quantum system with short-range interactions,
the surface cannot order by itself, the surface transition is the signal of a transition in the bulk. 

We consider in the following the homogeneous chain, $\eta_{n}=\eta$,
$J_{n}=1$  and $h_{n}=h \; \forall n$. In this case, all the $\bf T$ matrices 
are the same and it is easy to evaluate~(16).
The eigenvalues of $\bf T$ are
$$
\lambda_{\pm}=\frac{1}{1+\eta}\left[h\pm\sqrt{h^{2}+\eta^{2}-1}\right]
\eqno(17a)
$$
for $h^{2}+\eta^{2}>1$ and complex conjugates otherwise:
$$
\lambda_{\pm}=\frac{1}{1+\eta}\left[h\pm{\rm i}\sqrt{1-h^{2}-\eta^{2}}\right]
\eqno(17b)
$$
On the line $h^{2}+\eta^{2}=1$ they become degenerate.
First we notice that the leading eigenvalue gives the behaviour of 
$\phi_{1}^{2}(n)\sim |\lambda_{+}|^{2n}$
which for an ordered phase implies that $|\lambda_{+}|<1$, 
the first mode is then localized near the suface. 
From equations~(17a,b) this corresponds to $h<h_{c}=1$.
When $h>1$ a vanishing surface magnetization is obtained and the bulk is in the 
paramagnetic phase.
The  line $h=1$ is the Ising transition line since 
for $\eta\ne 0$ it belongs to the Ising universality class. 
As it can be expected from~(17) and shown hereafter, 
the line $h^{2}+\eta^{2}=1$ 
separates an ordinary ferromagnetic phase ($h^{2}+\eta^{2}>1$) from an 
oscillatory one ($h^{2}+\eta^{2}<1$).
This line is known as the disorder line of the model. Finally, the line with 
vanishing anisotropy $\eta=0$ corresponds to a continuous transition 
(with a diverging correlation length) 
and is called the anisotropic transition line where 
the magnetization changes from the $x$- to the $y$-direction.
In the ordered phase, the decay of the eigenstate $\Phi_{1}$ gives the bulk
correlation length which is related to the 
leading eigenvalue $\lambda_{+}$ of the $\bf T$ matrix by
$$
\phi_{1}^{2}(n)\sim |\lambda_{+}|^{2n}\sim \exp\left(-\frac{n}{\xi}\right)
\eqno(18)
$$
so that 
$$
\xi=\frac{1}{2\ln |\lambda_{+}|}
\eqno(19)
$$
Using~(17), one obtains the correlation length in both the normal and 
oscillating phases~\cite{xy}. In particular, one recovers the known result that the correlation length 
exponent $\nu=1$ at both the Ising and anisotropic transitions.

Let us consider first the Ising transition line at a fixed anisotropy $\eta$. 
Then, close enough to the transition, we have
$h^{2}+\eta^{2}>1$. The eigenvalues $\lambda_{\pm}$ are real and it is easy to show that
$$
\langle 1|{\bf T}^{n}|1\rangle=\frac{\lambda_{+}^{n+1}}{\Delta\lambda}\left[1-\left(\frac{\lambda_{-}}{\lambda_{+}}\right)^{n+1}\right]
\eqno(20)
$$
where $\Delta\lambda=\lambda_{+}-\lambda_{-}$. This expression reduces for large 
$n$ to $\lambda_{+}^{n+1}/\Delta\lambda$.
Using equation~(16) we obtain the surface magnetization near the Ising transition
$$
m_{s}^{x}\simeq \left[ \left(\frac{1+\eta}{2\eta}\right)^{2}\lambda_{+}^{4}\left(1-\lambda_{+}^{2}\right)^{-1}\right]^{-1/2}
\simeq \frac{2}{1+\eta}\sqrt{2\eta} (1-h)^{1/2}
\eqno(21)
$$
Thus the surface critical exponent is $\beta_{s}^{I}=1/2$ on the Ising transition line.

Close enough to the anisotropic transition line ($\eta=0$), we are in the oscillatory phase.
The complex conjugate eigenvalues $\lambda_{\pm}$ of equation~(17b) can be parametrized as
$$
\lambda_{\pm}=\rho\exp(\pm {\rm i}\theta)
\eqno(22)
$$
with
$$
\rho=\sqrt{\frac{1-\eta}{1+\eta}}\; , \quad \theta=\arctan\left[\frac{\sqrt{1-h^{2}-\eta^{2}}}{h}\right]
\eqno(23)
$$
Using~(14) a straightforward calculation shows that
$$
\phi_{1}(n)=(-1)^{n}\frac{1+\eta}{\sqrt{1-\eta^{2}-h^{2}}}\left(\frac{\lambda_{+}^{n}-\lambda_{-}^{n}}{2}\right)\phi_{1}(1)\cr
=(-1)^{n}\rho^{n}\frac{1+\eta}{h\tan\theta}\sin(n\theta )\phi_{1}(1)
\eqno(24)
$$
We see from this expression the oscillatory nature of the phase with a 
wavelength given by $\theta^{-1}$\cite{xy}. Close to the transition
the normalization of the eigenvector leads to
$$
m_{s}^{x}=2\sqrt{1-h^{2}}\eta^{1/2}\; , \quad \beta_{s}^{a}=1/2
\eqno(25)
$$

Finally one can also consider the behaviour on the disorder line $h^{2}+\eta^{2}=1$ separating the 
oscillatory phase from the normal one.
On this line, the eigenvector reads
$$
\phi_{1}(n)=(-1)^{n}n\rho^{n-1}\phi_{1}(1)
\eqno(26)
$$
with $\rho$ defined in~(23). Thus we obtain
$$
m_{s}^{x}=\left[1+\rho^{-2}\sum_{n=1}^{\infty}n^{2}\rho^{2n}\right]^{-1/2}
=\left[1+\frac{1+\rho^{2}}{(1-\rho^{2})^{3}}\right]^{-1/2} \; .
\eqno(27)
$$
Near the special point $(\eta=0,h=1)$ we have simply
$
m_{s}^{x}\simeq 2\eta^{3/2}
$
with a surface magnetic exponent equal to $3/2$. This special value is due to the merging of 
the two relevant fields of the model. We may notice that this expression can be deduced from~(21)
or~(25) using the constraint $h^{2}+\eta^{2}=1$.

 

To summarize, we have determined the surface magnetic behaviour of the XY quantum chain 
using a very simple analysis. We have shown how to deduce the bulk phase diagram and the 
correlation length from the surface behaviour. 
This procedure may be of interest for other one-dimensionnal quantum systems 
where the surface behaviour is more tractable than the bulk one.
Finally one may notice that expression~(16),
cut off at some size $L$, may be used for finite-size scaling calculations with 
arbitrary couplings~\cite{IR}. 



\numreferences

\bibitem{katsura} {Katsura S 1962} {\PR} {\bf 127} 1508
\bibitem{Pfeuty} {Pfeuty P 1970} {\APNY} {\bf 27} {79} 
\bibitem{xy} {Barouch E and McCoy B 1971} {\PR\ {\rm A}} {\bf 3} {786} 
\bibitem{Jordan} {Jordan P and Wigner E Z 1928} {\ZP} {\bf 47} {631}
\bibitem{lieb61} {Lieb E H, Schultz T D and Mattis D C 1961} {\APNY} {\bf 16} {406}
\bibitem{peschel84} {Peschel I 1984} {\PR\ {\rm B}} {\bf 30} {6783} 
\bibitem{IR} {Igl\'oi F and Rieger H 1998} {\PR\ {\rm B}}  {\bf 57}  {11404}

\vfill\eject\bye

\end{document}

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% 28/10/93 Added \bitem definition for non numeric cross referencing  

cond-mat0009038.extracted_bib
\bibitem{katsura} {Katsura S 1962} {\PR} {\bf 127} 1508

\bibitem{Pfeuty} {Pfeuty P 1970} {\APNY} {\bf 27} {79} 

\bibitem{xy} {Barouch E and McCoy B 1971} {\PR\ {\rm A}} {\bf 3} {786} 

\bibitem{Jordan} {Jordan P and Wigner E Z 1928} {\ZP} {\bf 47} {631}

\bibitem{lieb61} {Lieb E H, Schultz T D and Mattis D C 1961} {\APNY} {\bf 16} {406}

\bibitem{peschel84} {Peschel I 1984} {\PR\ {\rm B}} {\bf 30} {6783} 

\bibitem{IR} {Igl\'oi F and Rieger H 1998} {\PR\ {\rm B}}  {\bf 57}  {11404}

\vfill\eject\bye

