On Calculation of Lattice Energy in Spatially Confined Domains

Transcription

1 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 On Ccution of Lttic Engy in Sptiy Confind Domins Yvgn Biotsky Dptmnt of Mti Scinc nd Engining, Ato nivsity Foundtion Schoo of Chmic Tchnoogy,.O. Box 6, FIN-76 AALTO, Finnd Abstct-Evution of intn ngy nd th int-tomic o ionic intctions in cyst ttic usuy quis pcis ccution of ttic sums. This in th cs of sm nno-ptics (s spc-imitd domins) psnts sv chngs, s convntion mthods usuy vid ony fo infinit ttics, tiod fo spcific potnti. In this wok, nw mthod hs bn dvopd fo ccution of tomic intctions bsd on th di dnsity function with th gomtic pobbiity ppoch, xtndd to bity fixd ttics nd potntis in nno-ptic. Th divd di dnsity function (DF) combins tms fo unifom ptics distibution, fo non-unifom sphic symmty nd th st on fo n ddition, ng-dpndnt tm. Th scond tm oigints fom Wfisz-ik fomu fo ttic sums. Th DF with ths th tms is xpicity intgtd fo sphic ttic domins suting in th intn ngy of th systm with pscibd intction potnti. Th ppiction of th mthod ws dmonsttd fo Wign mod of ctons ttic intcting with compnsting positiv jy in finit ttic sph, which intcting ngy btwn ttic nd jy ws vutd. Th xcss of this ngy cusd by spc-imittion of th ttic ws xpicity xpssd in th tms of bsouty convgnt ttic sums. Kywods- Nnocysts; Engy; Gomtic obbiity; Sufc; Convgnt Lttic Sums I. INTODCTION Mny poptis of nno-siz sc mtis th distinct fom thos of buk mtis. Howv, in th ccution of thmodynmic poptis such s intn ngy of nno-ptics, th sm ppoch s fo buk (infinit) systms is usd, wh ony th xt sufc ngy contibution is bing ddd []. Th tnt ppoch fo ccution of ong-ng Couomb intction in g finit cyst in systms, consisting of piodicy ptd pics of idntic unit cs, hs bn poposd in [] (togth with usfu cittions on tht subjct). In tht tic, by using som ddition gb, th fin sut xpssd th s six-dimnsion intg cn b ngd into much simp fom givn by D intg. Mthmtic difficutis, ising fom condition convgnc of th Couomb sis in finit cysts, hv bn nysd in []. It is woth noting tht using potntis oth thn Couomb potnti quis ssnti modifiction fo ccution of ttic sums in convntion ppoch. In this wok, w suggst nw nytic ppoch fo th ccution of th ngy of finit (spc-imitd) nno-ptics with n bity typ of cystin ttic. Th intction ngy btwn two points (toms, chgs tc.), usd h, is bsd on th nytic cnt intction ( ). In this ppoch, w dicty ccut th numbs of pis of intcting points s th function of distncs btwn ths points in th domin. In this cs, ddition of th sufc ngy tm is not quid, insmuch s th sufc contibution is utomticy xpicity incudd in th who ngy. It is impotnt tht, in this mthod, kinds of intctions btwn points (ttic-ttic, ttic-continuum nd continuum-continuum) cn b vutd spty. This is of pticu intst fo ccution of th ngy of sttic cton ttics [4]. It is known tht th ngy of intction of cton with oth ctons, octd in th ttic sits, cn b stimtd by counting ths cton pis, but this ngy divgs fo infinit ttics. Th coct ttic ngy coud b obtind ony ft subtcting th intction ngy of n cton with th compnsting positiv bckgound chg ( jy ). Th sm pocdu impid fo finit domin, fcs som difficutis, s th intction ngy of cton with bckgound chg (ccution of which is simp fo infinit ttics) is much mo compictd fo finit ttics. Appying th tot nutity condition fo th sphic ionic ttic domin ds to cncing th divgnt chg misbnc tm in th sum of th intction ngy, but not th conditiony convgnt sum, ssocitd with th dipo momnt of th sph. W suggst tht th domin shoud b nut not ony goby, but so ocy, fo vy spcific gomtic sph insid th domin, which utomticy ds to zo dipo momnt. Th fin xpssion fo intction ngy in this cs contins ony bsouty (i.. not conditiony) convgnt sums. W ccut this ngy with nw ppoch nd dmonstt how this ngy dpnds on domin siz, which is of gt impotnc fo stimtion of th ngy dpndnc on siz of sm domins, spciy fo nnocysts. Dspit th fct tht th pubictions dvotd to ccution of Couomb intction in finit cysts, oftn using piodic boundy conditions, but to ou knowdg, th is no such nytic ccution fo ttic-bckgound intction ngy fo finit cysts. H (fo ny nytic cnt potnti) ony D intg nds to b ccutd, ft th di dnsity function is found

2 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 This function dpnds on gomty of th domin, but not on n intction potnti. Th mthod psntd h ows ccution of diffnt kinds of intction ngy btwn ttic nods nd btwn ttic nod nd point in jy-ik distibutd chg. Th co of th mthod is on th gomtic pobbiity thoy (GT) modifid fo finit ttic domin. Th GT tditiony hs bn usd fo ccution of th pobbiity dnsity () to find two ndomy chosn points to b distnc pt insid unifom domin o in domin with som continuous point s distibutions [5-], but h it ws so xtndd to disct distibutions. II. THE INTENAL ENEGY EQATION FOMLATION Lt s consid th systm of toms in finit sphic cyst of dius intcting by two-body cnt potnti ( ). Th intction ngy of N toms of this sptiy confind ttic systm is gny wittn s: H N N i j ( i j ). () In this study w xtnd th gomtic pobbiity tchniqus [5-], which is usuy ppid fo th systms with continuous points distibution, fo vution of this ttic sum (). Th ngy Expssion () in th continuous fom my b wittn s: N N ( i j ) ( ) ( ) ( ) i j () VV H d d wh ( ) is th micoscopic toms dnsity distibution function []: () ( ) n n n ninf with bing th Dic dt-function, i th bsis vctos of th -D ttic, ni - intgs, nd voum of th domin. V 4 is th tot III. GEOMETIC OBABILITY AOACH Accoding to GT [5], fo ny function () its vg vu tkn ov voum of dius is: ( ) ( ) d (4) wh () s is th pobbiity dnsity to find th distnc btwn two points insid th domin [5-]. This function fo unifomy distibutd dnsity points insid th sph [5] is ( x y ) dxdy ( ) (5) dxdy Th xtnsion of (5) fo non-unifom dnsity distibution ws considd by sv uthos. H w us on divd by Schf t. [6]: with ( ) d d ( ) ( ) ( ) d d ( ) ( ) ( ) d G( ;, ) ( ) d,, d d (7) (6) - 8 -

3 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 nd G( ;, ) d. (8) is th tot numb of pis of nod points insid th sph (Fig. ). Th sphic-symmtic dnsity function () usd in [6], is ng-indpndnt, so G ( ;, ) in (6) tnsfoms into G (, ) 6 ( ) ( ) d d /. (9) Fig. An xmp of ttic domin of dius in th -D pn. Th cicumfnc of dius fo sctd point ony pty fs insid th domin nd thfo numb of toms on this in (which intct with th sctd tom insid th domin) diffs fom such numb in th buk o infinit ttic In divtion of (9), th symmty of th intgs, ting inquity (whn ) nd th w of cosins cos () w usd. This DF fom kps th vctos, nd ntiy insid th domin, s vidnt fom Fig.. Th N N Expssion (6) givs th vg ngy p on pi of intcting toms (points), nd s th toty intcting pis in th domin with N toms, hnc th tot intn ngy is N N H G( ;, ) ( ) d. () IV. THE ADIAL DENSITY FNCTION FO A SHEICAL LATTICE DOMAIN Th Fomu (9) is ppicb ony fo sphic symmtic dnsity (), which dpnds ony on th distnc, whi th dnsity function ( ) in () dpnds on th who vcto. Thfo th ppoch hs to b modifid to so ccount ttic points distibution. By pting stps, which d fom Eqs. (6) to (9), but without initi intgtions ov zimuth ngs nd, th foowing xpssion is obtind: G(, ) ( ) d d ( ) d d d. () To pfom th intgtions ov th ngs in (), th micoscopic dnsity hs to b wittn xpicity in tms of ths ngs. It might b pfomd fom () with w-known quity: - 9 -

4 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 ( ) i k b k b k b i n n n, () ninf v k, k, kinf v inf wh k, k nd k numtd by intg vus, v is th voum of th unit c of th cyst nd k b k b k b. Th symbo v (4) inf mns tht th componnt with is xcudd fom summtion. Th bsis vctos b i of th cipoc ttic xpssd vi th bsis vctos of th ttic by known tions: b, b, b. (5) In th cs of th othogon ttic, th xs of th cipoc ttic coin to thos of th cyst ttic [], simpifying (5) to: nd spctivy th voum of th unit c of th cyst to: with xpicit vu of th modu: b, b, b, (6) v, (7) k k k. (8) Th sum in () coud b wittn using known yigh fomu: with i m * m m 4 i j ( ) Y (, ) Y (, ) (9) cossin, sinsin, cos, cos sin, sin sin, cos, k cos k k k () th sphic hmonic function m m m! m im Y (, ) (cos ), 4! () m th ssocitd Lgnd poynomis (cos ) nd sphic Bss functions j ( ) tnsfomtion d to th sut s:. This substitution nd m * m ( ) 4 i j ( ) Y (, ) Y (, ). v kinf m Nxt, intgtion of () ov th zimuth ng givs xpssion: () - -

5 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 m m inf m im m im (cos ) 4! m! m ( ) d 4 i j( ) (cos ) v v 4 m!! d v ij( ) (cos ) (cos ) inf which is vid fo ny ttic typ. Futhmo, th tm with my b sptd fom th sum in () nd ngd m using th tion btwn th ssocitd Lgnd poynomis (cos ) nd th Lgnd poynomis () In this wy w civ fo (): H (cos ) (cos ). (4) ( ) d s( ) (,cos ). (5) nd sin( ) s ( ), (6) v inf (,cos ) i j ( ) (cos ) (cos ), (7) v inf Th tm is th unifom dnsity distibution on th sphic sufc of dius, th isotopic tm () nd u nisotopic on (,cos ) th coction fo th non-unifom ttic dnsity distibution on th sphic sufc which iss fom th nxt tms of ( ) xpnsion xpssd s sum of sphic wvs. With ths dfinitions, th intgtion ov in () convgs to 4 sin( ) ( ) d sind v v inf. (8) This qution, ccoding to Wfisz-oisson fomu [], is th xpssion fo th dnsity of th ttic points octd on th sphic sufc fo dius. Th tot numb of points (toms o ions) insid th sphic domin coms by intgting of (8) by dius coodint: 4 cos( ) sin( ) ( ) sin. N d d d v v inf v kinf (9) Th Eq. (8) shows th symmtic pt (6) of th dnsity function () cocty tks into ccount th numb of th ttic points insid th domin, octd on th distnc fom ny ttic point insid th domin. Th numb of th ttic points ying on th sphic sufcs ony pty octd insid th domin (Fig. ) is dtmind by th dnsity function () which incudd symmtic (6) nd symmtic (7) tms. In th nxt sction w us ony symmtic tm ppoximtion fo ccution xmp, i.. ony th fist tm in si xpnsion () (with ). In oth wods, th numb of th ttic points ying which bongs to th intsction of th domin sufc nd sphic sufc of th dius (Fig. ) hs bn countd by vging of th ov st sufc. Th diffnc 4 cos( ) sin( ) v v inf v kinf N( ) N () is cusd by th pcuiitis of th sphic sufc coss-sction of th cyst. This oigints fom gomtic s - -

6 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 impossibiity to fit cubic ttic into ny sphic domin with xct numb of toms quivnt to th stting cubic domin, s discussd in []. In th cs of th cyst fomd by positiv nd ngtiv chgs (ionic ttics), this diffnc cuss dic fuctutions of sufc chg (vn if th infinit cyst hs nut symmty []) nd ds to th convgnc pobms, spciy fo ccution of th Mdung constnt. As J. F. Dod suggstd (s not in []), summing ov sphs woks if th missing chg wi b ddd bck nd th who sph wi b nut. It shoud b noticd tht vn in th cs of th cubic domin th ssumption of th unifom positiv chg distibution (which is vid fo infinit ttic) hs to b coctd by considing th diffnt numb of th nst ttic points insid th domin nd on th sufc. Th nutity of ionic ttic mns tht non-compnstion in ctonic chg shoud b quiibtd by th spctiv xt ionic chg: N( ) N ( ). () This is th tot nutity condition fo ionic ttics. W so suggst tht th sph is nut not ony in tot but so ocy, fo vy sph with dius insid th domin. Tht ds to th condition tht ddition chg fuctutions insid must b compnstd by th distibutions of positiv chgs nd s consqunc I N() N I () () N() N () d I sin( ) sin( ) NI () d, v v 4 inf inf () wh N I () is th vition in positiv chgs numbs. This condition mns tht th sph hs zo dipo momnt nd zo qudupo momnt s w. It is woth noting tht th ctosttic ngy of cyst with non-zo dipo momnt (i.. whn th Eq. () is not fufid) convgs ony conditiony. W c th Eq. () s oc nutity cition. sing th dsigntion sin( ) sin( ) N() N() d. (4) v v th divging sum cos( ) cn b wittn s: v inf inf 4 inf inf cos( ) sin( ) N() N( ). v v (5) 4 inf This sut cots with Eu s [4, 5] viw on divgnt sis: Lt us sy tht th sum of ny infinit sis is finit xpssion fom which th sis cn b divd. tuning to () nd combining Expssions (), (5) nd (8), w cn obtin th DF xpssion fo sphic ttic domin: 6 s s / G(, ) ( ) d ( ) (, ) d s G (, ) G (, ), which contins th sphic symmtic pt of th pobbiity dnsity (6) 6 6 v v inf k / / s G (, ) d d d sin( k ) d d d inf / 6 sin( ) v ) d inf sin( ) d / inf 6 sin( v (7) - -

7 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 nd symmtic on 6 s / (8) G (, ) ( ) d (, ) d. Th fist tm in (7) givs fo w-known sut fo unifom distibution of dnsity points ( jy mod [5], Fig. ): G ( u 6, ) v p d / d (9) Fig. Th pot of th pobbiity dnsity (to find th distnc btwn two points insid sph with unifomy distibutd dnsity points s th function of ) fo sph with unity dius Th subtction of Eqs. (9) fom (6) givs th diffnc btwn th tot numb of pis points (sptd by th distnc ) in th ttic nd jy mods. Hving th xpssion fo G(, ) fom (6) on cn ccut th intction ngy vi intgtion in (), not ony btwn th ttic points, but so btwn th ttic points nd th jy points. This tchniqu is dmonsttd bow by ccution of th intction ngy btwn ctons, which occupy thi ttic sits nd compnsting positiv bckgound chg in simp cubic ttic cyst, imitd by sph. V. CALCLATION EXAMLE: THE INTEACTION ENEGY OF STATIC ELECTON LATTICES IN SHEICAL DOMAIN H is n xmp of th ccution of th ngy of th sttic ctons ttic with compnsting positiv bckgound chg. Th intction ngy btwn th cton nd compnsting positiv bckgound chg fom th ngy of intction ctons octd in thi ttic sits sh b subtctd fom th who ngy fo pop ccution of th ttic ngy (s so th xmp in [4]). Th intction ngy btwn cton nd compnsting positiv bckgound chg fo infinit ttic hs simp fom: b x y z im dxdydz x, y, z (4) N V x y z owing to tnstion symmty of th infinit ttic. In (4) th tio N V is th vg tomic dnsity n d, nd is th cton chg. This ngy divgs in infinit domin [4], nd th intction ngy of ctons on thi ttic sits m n (4) so divgs in infinit domin s w [4]. Th pmt is th distnc btwn th ttic sits (th ttic piod in simp cubic ttic). Th diffnc btwn two ngis is, howv, finit: b.8797 (4) - -

8 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 nd is ppoching th cofficint known s Mdung constnt (in this cs M SC fo simp cubic ttic [4]). Th Eq. (4) hs such simp fom ony fo infinit ttic, whi in finit ttic th ngy dpnds mo stongy on th ttic typ nd on th position in th domin occupid by th cton, s it is shown in Fig.. Thfo, Eq. (4) must b coctd. With this nw ppoch, it coud b don dicty, using Eqs. () nd (7): Ou mthod divd bov ows quit simp ccution of such ngy fo finit ttic domin. This ttic-bckgound Couomb ngy (p ptic), coms fom th gn Expssion () s N fb G d ( ;, ), (4) with G ( ;, ) s th DF fo positivy-ngtivy chgd pis. This DF is th subst of Eq. (6) nd w us ony symmtic pt of this qution (s th xpntion ft Eq. (9)) 6 6 v v inf k / / G (, ) d d d sin( k ) d 6 sin( ) d d. v inf / (44) Th fist is th DF fo points continuousy distibutd insid th domin (jy mod). Th contibution of this givs th intction ngy btwn cton nd compnsting positiv bckgound chg fo finit ttic sph (s [6]) 6 / 5 sph x y z N dxdydz N d d d N. V v (45) Th scond nd th thid tms in Eq. (44) kp on of th points in vy pi xcty in th nod of th ttic nd th contibution into th ngy is N / inf sin( ) d d / d inf sin( ) d d. (46) Th st tm in Eq. (7) bongs to th ttic-ttic pis, thfo is not incudd into Eq. (44). It is woth noting tht th ow imit in th intg ov is zo ony fo ttic-bckgound pis, but fo ttic-ttic pis such ow imit is qu to th minimum distnc btwn th ttic nods. Aft ccution of intgs in (46) w hv N ~ v / inf inf / sin( ) d d d d sin( ) d inf cos( ) 4 inf 4 inf inf ~ sin ( ) 4. sin ( ) Th ight-hnd sid of this qution is th min contibution fo th domin siz 4. As it ws mntiond bfo, th sum cos( ) divgs, but by using Eq. (5) w cn wit Eq. (47) in mo suitb fom inf fb b v inf sin ( ) 4 inf 4 inf (47) sin( ) N ~ ( ) N( ), 5 (48) 4-4 -

9 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 wh th sums in figu bckts convg bsouty, not conditiony. Th dipo momnt nd sufc-oiginting tms N() N( ) must b compnstd by positiv bckgound chg fo cocty dfind mod. Thfo, th xpssion ( ) v inf 4 inf inf sin ( ) 4 4 inf sin( ) dscibs xpicity th xcss of th intction ngy btwn cton nd compnsting positiv bckgound chg fo finit sphic ttic, in compison with infinit ttic. As sut of this ccution, Eq. (4) fo th Mdung constnt fo infinit simp cubic ttic tnsfoms into (49) fb.8797 δ(). (5) Fig. shows th tio δ(), which dpicts th ddition ngy contibution to th cyst ngy oiginting fom th M SC spc imittion. It is sn this xt contibution by od of.6-.% of th Mdung constnt vu is in th ng btwn (5 4). Fo ptic with ~.4 nm this mns < < 6 nm. This contibution incss ngy of sm ptic (i.. dds xt sufc ngy), which gs with w-known tndncy of sm ptics to cosn by minimizing thi ngy (dcsing sufc). Expssion (6) is potnti-indpndnt nd numic vus ony function of th ttic typ nd its siz. ( ) M SC 4 () Fig. Th pot of th tio of M SC - th xcss of th intction ngy btwn cton nd compnsting positiv bckgound chg to th Mdung constnt fo finit sphic ttic comping with infinit ttic s th function of th sph dius fo th simp cubic cyst in ngth units This xmp shows tht xtndd gomtic pobbiity dnsity ppoch mks it possib cocty ccut intction btwn ttic sits in spc-confind systms such s sm nnoptics. It so incuds th ddition compnsting chg, which ws pviousy known cusing pobms in ttic sums ccutions []. Ou ppoch cn b tiod fo ccution ttic ngy in non-sphic domin s w, n xmp fo n obt sphoid domin. Th qution fo th sufc of n obt sphoid domin (n obt sphoid) is x y z (5) mjo wh mjo is th mjo smi-x nd mino mino smi-x. In th sphic coodints, th qution fo n obt sphoid is [8] mino wh ( ) mjo sin (5) - 5 -

10 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 mino. (5) mjo Th Eq. (6), which is vid fo sph, sh b substitutd by th qution ( ) (54) d d ( ) ( ) ( ) d ( ) d d ( ) ( ) fo n obt sphoid domin. In this qution th dnsity function ( ) coms fom Eq. (). Th Tyo xpnsion of Eq. (54) in th pow of ows simpifying this qution fo sm (sighty dfomd sph). Simi xpssion cn b wittn fo n bity convx ttic domin, but th function (, ), which dscibs th sufc of th domin, dpnds on th two ngs nd d d ( ) ( ) ( ) (, ) (, ) d ( ) d d ( ) ( ). (55) VI. CONCLSIONS In this wok, nw mthod fo ccution of th di dnsity function nd gomtic pobbiity dnsity is dvopd fo th cyst ttics in finit sphic shp domin. Extndd gomtic pobbiity tchniqu ows ppiction of n bity fixd ttic point distibutions, ding to xpicit xpssion fo th DF, which is psntd s function of th ttic sums. Ths sums bsouty convging nd suitb fo numic computtions. Th intn ngy of th systm of toms octd t th nods of ttic ws wittn s th function of th DF (6), nd it coud b ppid fo ny istic nnoptics siz nd intction potntis. In th xmp fo simp cubic ttic, th xt contibution to th ngy ws ccutd s n ddition to th Mdung constnt, psnting xpicit ffct of sufc ngy nd ws numicy ssssd fo diffnt dius of th ptic. Th xcss of th intction ngy btwn cton nd compnsting positiv bckgound chg fo finit sphic ttic, comping with infinit ttic hs bn obtind in th fom of bsouty convgnt ttic sums. Th sut obtind opns possibiity to vut xpicity ny istic ttic typ nno-ptics, fo xmp, god nnoptics, fom th point of viw of thi intction with nvionmnt. ACKNOWLEDGMENTS I woud ik to thnk of. Mich Gsik, Ato nivsity Foundtion, fo vy hpfu discussions. Finnci suppot fom th Foundtion of Hsinki nivsity of Tchnoogy nd Finnish Ntion Tchnoogy nd Innovtion Agncy (Tks) is gtfuy cknowdgd. EFEENCES [] T. L. Hi, Thmodynmics of Sm Systms, Nw Yok: Dov ubictions, 994. [] L. N. Kntoovich nd I. I. Tupitsyn, J. hys.: Condns. Mtt, Couomb potnti insid g finit cyst, vo., pp [] L.. Buh,. E. Cnd, J. hys. A: Mth. Gn., On th convgnc pobm fo ttic sums, vo., pp. 5-58, 99. [4] D. Bowin, J. M. Bowin,. Shi nd I. J. Zuck, Engy of sttic cton ttics, J. hys. A: Mth. Gn., vo., pp. 59-5, 988. [5]. Dthi, 99 Ann. Fc. Sci. niv. Touous., Su théoi ds pobbiités géométiqus, vo., pp. -64, 99. Avib: [6] D. Schf, M. y, Tu Shu-Ju, B. Woodh nd E. Fischbch, Appiction of gomtic pobbiity tchniqus to th vution of intction ngis ising fom gn di potnti, J. Mth. hys., vo. 4, pp. -, 999. [7] A. M. Mthi, An Intoduction to gomtic pobbiity, Godon nd Bch Scinc ubishs, 999. [8] M. y, nd E. Fischbch, J. Mth. hys., obbiity distibution of distnc in unifom ipsoid: Thoy nd ppictions to physics, vo. 4, pp ,. [9] Shu-Ju, nd E. Fischbch, J. hys. A: Mth. Gn., ndom distnc distibution fo sphic objcts: gn thoy nd ppictions to physics, vo. 5, pp ,. [] A. Mzzoo., J. Mth. hys., optis of chod ngth distibutions of nonconvx bodis, vo. 44, pp ,. []. Gci-yo, J. hys. A: Mth. Gn., Distibution of distnc in th sphoid, vo. 8, pp , 5. [] Guini A., X-y Diffction in Cysts, Impfct Cysts, nd Amophous Bodis, Sn Fncisco nd London: W. H. Fmn nd Compny,

11 Advncs in Mtis Scinc nd Appictions Dc., Vo. Iss. 4,. 7-7 [] d Fits nd A. N. Chb, J. hys. A: Mth. Gn., Th Wfisz-ik fomu fom oisson s summtion fomu nd som ppictions, vo. 6, pp. 5-5, 98. [4] G. H. Hdy, Divgnt sis, Oxfod Cndon ss, 949. [5] L. Eu, Diffnti ccuus-ey woks to 8, Sping-Vg Nw Yok, Inc.,