The Atomic Scattering Factors. of the Potassium Ion in Trigonal Potassium Nitrate

Transcription

1 457 Acta Cryst. (1957). 10, 457 The Atmic Scattering Factrs. f the Ptassium n in Trignal Ptassium Nitrate :BY U. KORHON~,N nstitute f Physics, University f Helsinki, Finland (Received 20 August 1956 and in revised Jrm 4 February 1957) ~fhe writer has previusly develped a methd fr the determinatin f the atmic scattering factr f an in in a crystal f which the structure factrs are knwn. Applying this prcedure t trignal KNO 3, he has determined the atmic scattering factr values f the K in fr the reflexins (hhh), (h00), 0~h0) and (hh0). When the atmic scattering factr values btained in this way are presented graphically against sin 0/2, it is fund that the values calculated fr the different reflexins differ systematically. This difference is rtatinally symmetrical with regard t the trignal axis and increases at small sin 0/). values, attains its largest value at sin 0/~t = 0-24, decreases thereafter and has whlly vanished by sin 0/). = This cannt be explained slely by the heat mtin but requires that an electrnic anistrpy f the K in be taken int accunt. A critical errr analysis shws that this phenmenn cannt be due t the errrs arising frm the methd f calculatin; mrever, the experimental values used in the calculatins seem t be reliable. ntrductin The writer has shwn (Krhnen, 1953, 1955) that if the structure factrs F(hkO f a crystal are knwn, the atmic scattering factrs fl,(hk0 f an in 1, belnging t this crystal and having the psitin crdinates (xl, Yl, zl), can be calculated as fllws: 4zR a 1 + 2: F~h'k'z') ~ --cs h'kt F~lhk0 = FChk0. exp [- 2~i (h (xl/a) + k(yl/b) + l(zl/c))], 0 = 2zR/d(h_h,, ~-k', l-v) Here d(h-h,, ~-k',z-v) dentes the spacing f the lattice planes (h-h', k-k', 1-1'). F(hk0 t is the structure factr fr the reflexin (hkl) when the pint (xl, Yl, zl) is taken as the rigin. V is the vlume f the unit cell and R the inic radius f in 1. The writer has als shwn (Krhnen, 1956a) that, using series (1) t determine the atmic scattering factrs frm the always limited number f structure factrs experimentally measured, the errr due t the breaking-ff can be remved as far as pssible by using the 'difference series methd'. Artificial structure factrs Fart.,(hk/)fr the crystal under cnsideratin are cnstructed by the aid f the theretical atmic scattering factrs and f the apprpriate values fr the heat mtin t crrespnd as nearly as pssible t the real structure factrs F(hz.O at thse values f sin 0/2 at which the last structure factrs have been measured. The difference, AfL(hh.a, between f~,(hk0 and fa, ~t,(h~.0 is then calculated by the aid f series (1) (l') and by using the structure factr differences AA,(hk,) = --F 1 1. A F(hkO = F(z, h O- Fart., (hko Af (~.k~) 1 1 AF ~,ko = F(h~'O -Fart., (hko, ~ = 2~R/d(h-j,,,k-k,,z_r). h,~.,raf(h,k,r)--~ --COS ~, (1') Prcedure The trignal unit cell with dimensins r = /~ and a = ' is used in the fllwing calculatins. Table 1, clumn (3), shws the Fq, ko values measured experimentally by Tahvnen (1947a). Fr the K in the atmic scattering factrs calculated by Hartree's methd, and fr the atms N and 0 thse measured experimentally by Tahvnen (1947b), were used as the artificial atmic scattering factrs. The heat mtin f the atms was taken int cnsideratin by multiplying the atmic scattering factrs by the Debye factr D: D = 10 -~ sin, 01~' (2) Fr the K in the value c~=2.054 A 2 and fr :N and 0 the value c~, = )~2, fund by trial and errr, were used. The psitin f the NO3 grup in the lattice is nt knwn accurately; the grup has been suppsed t rtate (Kracek, Barth & Ksanda, 1932). By using the methd f trial and errr it was fund cnvenient t divide each grup int tw psitins in the lattice s that the crystal symmetry

2 458 THE ATOMC SCATTERNG FACTORS OF THE POTASSUM ON Table 1 (hkl) sin 0/2 (A -1) F(hkO Far~.,(hk0 AF(hkO (1) (2) (3) (4) (5) (100.66) lo (-) ilo ' ] 12" ~" " " " " " " " "2" i " ' "65~ 7"01 8"76 ] i " : (.) Tl l was fulfilled at the same time. Thus, the fllwing crdinates fr the atms were used: 2K- (0, 0, 0; ~-, ~, ½). 4 (½N)" (¼-~x, ¼-~x, k-a; ¼+c~, ~+a, ¼+a; ~-~, ~-~,, ~-~,; ~+,, ~+~, ~+.~,). 12 (½0): (~-cx-fl, k-~x-fl, k-a+2fl; k+.~, +~, k+~ +fl, ¼+,-2fl; k-~,-~, k-~+efl, k-,-~; k+~+~, k+:,-2/~, ¼+~+~; k-~ +2t~, ¼-,-/~, k-~- ~; k+.~-2fl, k+a+fl, k+~+(fl), and, in additin, six psitins which can be btained by adding ½ t the abve crdinates. (3) The structure factrs Fart..(t,~. culd then be calculated by the frmula: Fart., (hi't) fK art., (~k0-t-2fn art., (hkl) COS 2ze(h+k+l) -}- 2f art., (hk/) ( cos 2ze[(h+k+l)~x+ (h+k-21)fl] +cs 2~[(h+k+l)s + (h-2k+l)fl] +cs 2ze[ (h + k + l) + (- 2h + k + l)fl]}. (4) The plus signs are valid when (h+k+l) is divisible by 4 and the minus signs when it is divisible by 2. Fr the parameters a and fl the fllwing values were used : 2g~ = 6 30', 2~fl =47. (5) The artificial structure factrs btained in this way are seen in Table 1, clumn (4); they have been calculated fr Cu Ka radiatin (H6nl, 1933a, b). The differences AF(],~. 0 = F(hk0-Fart., (hk/) are given in clumn (5). We see that the differences AF(h~z) are rather small, and when calculated fr ne mlecule nly they must, in additin, be divided by 2. The methd we have used in the abve calculatins seems t apply well t the crystal--at least as well us the rtatin cupled with vibratin (Tahvnen, 1947a). Als, the electrn distributin f the NO a grup n the plane (ll0) btained here crrespnds very well with the experimental electrn distributin given by Tahvnen (1947a). All this may mean that the NOa grup has tw psitins f equal value in this crystal and that the crystal symmetry crrespnds t the statistical average. Having nw available the values AF hkz), we need nly knw the value fr the inic radius R in rder t use series (1'). The value 1.33.~ was used fr R. The values f a can be btained by calculating the spacings dq,-1,,,~-~.,,t_r) frm the dimensins f the unit cell. Results Table 2, clumn (3), presents the differences/lfk ' (1,~t) btained by calculatins. n clumn (4) there are the values f the artificial atmic scattering factrs f the Table 2 (hkl) sin 0]2 (A -1) A fk. (:~.t.t) fk art.. fht't) fie, (:,ks (l) (2) (3) (4) (5) ') ll " (.) q

3 U. KORHONEN 459 _.0- - ". --O (3-- _ j.u_ "-C', OX E.< -'0 ta) ~ ~ ~.,,.,v -..." ~'Mm "t + "0._~ 0-0 O,,! (-~ ~ D X E-1 "0,x X (b) 6 - e,,- m ~.~_ 0"0.< O.. -Q.-- " ~ Q=r, 6 A v ~j i e % 0 x -1.Ok- (c) &.. O "9-" --- -O E0"0 < ~- -," "0~',,_. --"~" ~..~ ~. (d) "- " "~ -1" "1 0"2 0"3 0"4 0"5 sin ~/,'. Fig.. (a) Differences between the experimental and theretical atmic scattering factrs f the K in in the trignal KNO 3 crystal. The circles stand fr reflexins (hhh), the slid circles fr (h00), the crsses fr (hho), and the triangles fr (hho). The brken curves shw the crrespnding atmic scattering factr curves. (b) Differences between the experimental and theretical atmic scattering factrs f the K in in the trignal K~TO 3 crystal. The small symbls are the same as in (a). The large symbls stand fr the differences when the artificial crystal needed in the calculatins cntains nly tw K ins in the psitins (0, 0, 0; ½, ½, ½). (c) Differences between the experimental and theretical atmic scattering factrs f the K in in the trignal KNO 3 crystal. The small symbls are the same as in (a). The large symbls stand fr the differences btained frm series (l') when the value A 2 is used in the Debye factr fr the K ins in the artificial crystal. The brken line stands fr' the theretical atmic scattering factrs f the K in in the artificial crystal. (d) The circles (hhh), slid circles (h00), and crsses (/~h0) stand fr the atmic scattering factrs f the K in having ellipsidal heat mtin. The hrizntal axis stands fr thse f the K in having spherically symmetrical heat mtin. The values c~culated back frm series (') have been dented by small pints when they deviste frm the crrespnding initial values. A C 10 32

4 460 THE ATOMC SCATTERNG FACTORS OF THE POTASSUM ON K in, f~ art., (hkl), and in clumn (5) the final atmic scattering factr values fk, (hk0 fr Cu K~ radiatin. The differences AfK,(h~z) are presented als graphically against sin 0/~t in Fig. 1 (a). Examinatin f the results The accuracy f the results will be discussed first. Errrs may arise bth frm calculatins and frm the errrs in the experimental structure factrs used in the calculatins. The writer has shwn earlier (Krhnen, 1955) that the value used fr R has nly a slight influence n the results, and therefre n nticeable errrs can arise frm this surce. The artificial structure factrs needed in the calculatins are used nly as temprary apprximate values (cf. Krhnen, 1956a, p. 7) and therefre they als have nly a slight influence n the results. n rder t shw this mre clearly the authr calculated the new structure factr differences fr the KNOa crystal frm the measured structure factrs by using as the artificial crystal, this time, a crystal cntaining nly the K ins in the psitins (0, 0, 0; ½, ½, ½). (hkl) Z].Fne w Table 3 z~.fl d (hkl) z~.fnew AFld "00 0" "16 --0" "91 ~- 1" ~- 0" " "51 --2" "41 --1" "71 ~- 1" "73 --0"14 il0 --1"78 --4" "60 --0" "76 --0" "26 +0" "74 --0" "47 --1" "51 --0" "78 --0" "44 +0" "45 --0" "30 ~-- 1"22 2]1 --2"83 --1" " "91 +0" "08 --0" " "29 --0" "26 --0" "27 --0" " "27 --0" "47 --1" "23 --0" " "10 --0" " " "80 --2" "97 --0" "17 --0" "51 --0" " " "14 +0"68 Table 3 shws the new ZJFne w values, which are quite different frm the ld values A Fld. The new results calculated by the aid f series (1'), hwever, differ nly very slightly frm the ld nes, as seen frm Fig. l(b). t is, therefre, sure that the crdinates and the temperature factrs f the NOa grup have n prfund influence upn the btained results. Their influence n the results is s small because the NO 3 grup is situated utside the sphere (x2+ye+z 2 = R e) used at integratin. The temperature factr f the K in has a little greater influence n the results. n rder t shw the influence f this factr the authr calculated nce mre the new structure factr differences fr the KNOa crystal by using, this time, the value A 9 instead f A s fr c~ in the Debye factr (cf. frmula (2)). Using these structure factr differences in series (1') he btained the results seen in Fig. l(c) (the large pints). They deviate a little frm the ld results (the small pints), but the deviatins axe large nly at the beginning and end pints, as ne can expect (cf. Krhnen, 1956b, p. 18). The brken line in Fig. 1 (c) shws the values f the atmic scattering factrs fr the K in when the value J~2 is used in its Debye factr. We see clearly frm Fig. l(c) that we can expect better values fr the beginning and end pints if we lwer the artificial atmic scattering factr values frm the brken line twards the hrizntal axis. This is due t the fact that the Af l, ko differences calculated this last time by the aid f series (1') are differences between the large pints and the brken line and therefre the last pints are bent twards the brken line (the s-called 'Gibbs phenmenn'). We see, hwever, that the temperature-factr value f the K in als has n prfund influence n the results and that the temperature-factr value A 2 is a prper chice. AF(hk values are unknwn beynd sin 0/~t = 0-5. This causes errrs t the results, as seen, fr instance, frm Fig. l(c), where the last pints are bent twards the brken line. Here it is imprtant t knw hw much errr can arise frm this surce. The aly way t determine this questin is t take a knwn case where we can expect a greater terminatin errr than in ur actual crystal, and t determine the terminatin errr in that case. The authr has earlier (Krhnen, 1956a) treated an artificial crystal having the same dimensins as ur trignal KNO a crystal and having nly the theretical K ins in the psitins (0, 0, 0; ½, ½, ½). He made these K ins vibrate s that the mean square f the displacement frmed a flattened rtatinal ellipsid with the trignal axis as its rtatinal axis. Then he calculated the structure factrs fr this crystal up t the reflexin (521), i.e. t the same limit as in Table 1. After this he calculated anew the structure factrs fr this crystal, nw letting the K ins vibrate with spherical symmetry, and frmed the structure factr differences by subtracting the latter frm the frmer values. By the aid f these structure factr differences, and using series (1'), he calculated back the crrespnding atmic scattering factr differences fr the K in. The results f these calculatins are seen in Fig. 1 (d). We see that nly the three last values shw large deviatins. These three last pints, hwever, differ clearly frm each ther in the crrect directins (twards the crrespnding crrect values) and they als differ clearly frm the hrizntal axis. When we nw return t Fig. 1 (a) we see that in this case all these last three pints have nearly the same value, falling nearly n the hrizntal axis. This

5 phenmenn cannt be due slely t the terminatin errr: a great part, at least, cmes frm the experimental structure factrs. Thus we can cnclude that the terminatin errr in Fig. 1 (a) is even smaller than in Fig. l(d) because the discntinuity prduced by equating all AF(hk values t zer at sin 0/~ > 0-5 is greater in the latter case. The errrs due t the experimental values are mre difficult t estimate. Hwever, n knwn errr in experimental values leads t a dependence n the crystallgraphic directins like that seen in Fig. 1 (a), s we have n particular reasn t suspect the experimental values. When, in additin, we cnsider the smth curse f the curves in Fig. l(a), cmpared with each ther, and the fact that they all begin at zer level, indicating thus the right electrn number, it must be admitted that the experimental values used in the calculatins seem accurate enugh. These cnclusins cannt f curse prve that the experimental values are certainly crrect, but they shw we have reasns t think s. Cnsequently, the nly remaining pssibility seems t be t interpret the curves in Fig. l(a) as due t the K in itself. Let us first try t find such a heat mtin f the K in, which, accrding t the frmula fk,(hko = fkther.,fhko exp [--8g2u~hk0sin20/22], (6) shuld lead back t ur results when fkther., (hkl)is the atmic scattering factr f the K in calculated theretically and U(hk02 is the mean square f the displacement perpendicular t the lattice plane (hkl). This, hwever, is impssible, as can be seen frm Fig. 1 (a), where the differences between the atmic scattering factr curves fr different reflexins increase at first but then decrease and finally vanish (cf. Fig. 1 (a) with Fig. l(d), where the atmic scattering factr curves bey the frmula (6)). We arrive, therefre, at the result that we must, in additin, take a pssible electrnic anistrpy f the K in int accunt (cf. Krhnen, 1956b, where the authr als btained results like this). t may be that the last experimental structure factrs have greater experimental errrs than the first structure factrs and that this phenmenn causes the vanishing f the differences between the different atmic scattering factr curves at large sin 0/)l values. This assumptin, hwever, is nt very likely, fr the experimental accuracy is essentially the same fr all reflexins in the methd used by Tahvnen. U. KO RHONEN 461 With the aid f the differences between the atmic scattering factr values seen in Table 2 the differences between the electrn distributins f the same ins als can be calculated. Fig. 2 presents the difference d 1-0 0,0. ]. ~ "".~. J... -~ (e A 1),,,'"""" -1.~.,,, "0 1 "5 2"0 (h) Fig. 2. Differences between the electrn distributin f the K in in the trignal KNO a crystal and that f the theretical K in prjected n the trignal axis (slid line) and n the directin perpendicular t the trignal axis (brken line). between the electrn distributin f the K in in the trignal KN0a crystal and that f ur artificial K in prjected n the trignal axis (slid line) and n the directin perpendicular t the trignal axis (brken line). The frmer difference is slight all the way, while the latter is much greater and shws that electrns have mved frm the centre utwards in the directins perpendicular t the trignal axis. References H6~rL, H. (1933a). Z. Phys. 84, 1. H6~L, H. (1933b). Ann. Phys., L~pz. 18, 42, 625. KORHONE~, V. (1953). Ann. Acad. Sci. Fennicae, (A), :N KOR~O~EN, U. (1955). Ann. Acad. Sci. Fennicae, (A), N KORHONEN, U. (1956a). Ann. Acad. Sci. Fennicae, (A), N KOR~ONE~, U. (1956b). Ann. Acad. Sci. Fennicae, (A), N KRACEK, F. C., BARTH, W. F. W. & KSANDA, C. J. (1932). Phys. Rev. 40, TAHVO~rE~, P. E. (1947a). Ann. Acad. Sci. Fennicae, (A), N. 44. TAHVONEN, P. E. (1947b). Ann. Acad. Sci. Fennicae, (A), :N *