T h e T ak agi {T aupin equa ti ons, the fun damenta l equations for X -ra y di˜ra ctio n deduced from the Ma x w ell equations, are considered . T he connection b etw een the T akagi{T aupin equations and the K lein{ Gordo n equation is show n. A metho d of solution of these equations using external di˜erential form formalism is prop osed. T he solution s for both a narrow and a wide incident beams as a function of boundary conditions is analy zed. T he so-called spherical and quasi-plan e w aves used in the X -ray exp erimental metho ds are derived from.PAC S numb ers: 61.10.Dp, 02. 30.J r, 61. 72. {y
I n t r o d u ct io nIn the present pa p er the Kl ei n{ G ordo n form ul ati on of the T akagi { Taupi n equati ons (TTE) i s descri b ed. The TTE, ha vi ng the form of a set of two pa rti al di˜erenti al equati ons for two i ndep endent functi ons, ha ve b een deri ved by T akagi and T aupi n [1, 2] for the electro m agneti c Ùelds i n crysta l s fro m the Ma xwel l equati ons. The TTE are pa rti al di˜erenti al equati ons of the hyp erb ol i c typ e. These equati ons can b e converted to the Kl ei n{ G ordo n equati on (KG E) [3] descri bi ng the scalar Ùeld wi th a m ass i n rel ati vi sti c qua ntum Ùeld theory . Natura l l y sol uti ons of the KG E are well kno wn and for thi s reason sol uti ons of the TTE i n the La ue [4] geom etry ha ve b een presented earl i er [5]. Ho wever, the present way of obta i ni ng these sol uti ons i s very conveni ent for further appl i cati ons for two reasons. Fi rstl y, the use of externa l di˜erenti al form s form ali sm and the general i zed Stokes theo rem [6] m ake the deri va ti on of sol uti ons clearer. Thi s i s very useful i n the La ue case and even more i n the Bra gg case [4], where the mi xed b ounda ry conditi ons m ake the sol uti ons much m ore com pl icated. Secondl y, the form ul atio n in the Kl ein{ G ordo n f orm gi ves the p ossibi l ity to use the very well -known mathem ati cal ( 767)