LATTICE-THEORY OF ALKALINE EARTH CARBO NATES. Part I. Lattice-Energy of Crystals of Aragonite Type and Their Thermo-Chemical Applications. BY B. Y. OKE. (From the Department of Physiis, Indian Institnte o] Science, Bangalore.) Received June 2, 1936. (Communicated by Sir C. V. Raman, xt., r.R..% N.I,.) Introduction. MEASVR~:M~~~St of the elasticity coefficients of Aragonite show a strong anisotropy. The two elasticity constants in perpendicular directions are in the ratio of 1:2. An atternpt is made in this work to explain this anisotropy. According to the suggestion of Prof. M. Born the substance is assumed to be a polar compound composed of Ca+ + and (CO3)--. The forces are th› electrostatic attractions and repulsions of the charges and b some other forces of the forro ~ which are usually assumed to account for the stability of the crystal. Before taking up the actual calculations of the elasticity constants, ir is necessary to calculate the total lattice-energy and to derive from this, with the help of the compressibility measurements, the value of n. In this paper we restrict ourselves to the evaluation of this n. The lattice energy of the crystal is not a directly measurable quantity. Ir is, however, possible to construct chemical cyclic processes; after the mamze.r of Born~ and Haber, 3 in which this quantity enters along with other' chemical quantities such as heat of sublimation, heat of formation, heat of ionisation, etc. The cyclic processes can be used to verify the theoretically calculated value of the lattice-energy, as well as to estimate the heats of certain chemical reactions. 1 W. Voigt, Ann. de Phy., 1907, 24, 290. -~ M. Born, Verh. d. Deutsch Phys., 1919, 21, 679. a F. Haber, Verh. d. Deutsch Phys., 1919, 21, 750. Al 2 B.Y. Oke w Lattice-Energy of a Crystal. ...> The electrostatic potential of the ]attice at a point rk' of the lattice is --> called the " Selbs-potential " of the lattice at the point rk' and is given by r (rk') = S 27 ek l k rlkk' " ....... ( 1' 1) where S denotes the sunlmation over all the three cell-indices 11, l~, la from 1 -- ~ to oo and the dash over the summation sign 2~ denotes that the index combination li =12,-13 = 0 and k =k' does not occur, ek denotes the charge at the point k and rkk' the radius-vector from k to k'. The electro-static potenfial r at the points in a unir cell of the crystal is given by r = 27 ek' r (rk') k, = S Z eke," z kk, rlkk' ......... (1, 2) ...> --> -9, -.> -9. ...> I,et a, b, c be the " Grund-vectors" of the lattice anda', b', c' be the " Grund-vectors" of the reciprocal lattice so that a,= 1 [ -~ -~ -~ -> xc]'b'=, ~1 [c• , ~1 [a • .. (1,3) where ax a v a~ b~ by b, = 3 3= A ..... (1,4) Cx Cy Cz ,.-), We define a vector qZ by -~ --> --> --> q~ = 2~ (l~ a' + 13 b' +/3 c') ...... (1, 5) Aecording to the method of Ewald ~ Ck' (r#) can be expressed as the sum of two expressions as follows Ck' (rk') = ek' ~(0) + k27' ek ~b (rkk') ...."" (1, 6) where the dash over the summation sign denotes that the term k = k' does not occur. (0) and ,& (rkk')ate given by .-> 4,~ S' e"(~'. ~kk') r [q~l ~ ........ (1,7) 4 M. Born, "Ency. der Math. Wissen.," Physik, 3, 727. La[Ÿ Theory of ,4/ka/iue Earth Caz3onates 3 and ~(0) = ~b~ (0) + ~b; (0) ........ (1, 8) 1 - i-Z~ [ q Z 12 4rr S' e 227 where .... (1, 9) (o) = -s ~ i q~l 2 27r1 2 _ ~~ f e-a~ da and ~ (0) = s' o ~r z # AZ~ .... (1, 10) The introduction of this " Partition-factor " (Trennungstelle) 27 is the chief advantage of Ewald's method. The final resnlt is independent of the choice of 27 ; which may, therefore, be chosen in any suitable manner. Substituting from (1, 6) in (1, 2) r = ~b (0) 27 ek ,~ + 27 eke//r (rk~') ...... (1, 11) k" kk" 1 The electro-static energy density Uo is given by U0 = ~-~ ,6o (1, 12) w Electro-Static Energy De~~sity of Aragonite and Similar Crystals. The crystals of calcium carbonate in the Aragonite forro, as well as the crystals of Strontium and Barium-Carbonates, are ortho-rhombic. The "Grund-vectors " have the components ax - a, br ~ b Cz = c ........ (2, 1) all other components being zero. The values £ of a, b, c, as found by X-ray analysis, are tabulated below in ~ngstrSm units. TABLE I. Substance a b c CACO3 4 "94 7.9~t 5 '72 SI'CO a 5.13 8.42 6.10 BaCO 3 5.29 8 "38 6 "41 The only surviving components of the " Grund-vectors " of the reciprocal lattice are 1 1 1 a'x = a' b'y ~ , c'z c (2, 2) 5 Wyckoff, The Structure of Crystals, 2nd edition, p. 273. 4 B.Y. Oke The .rector qz wii1 have the componen~s q~ ' b' c. " ....... (J.) :' Evaluation of ,/J (0) .-- Substituting the values of the coinponents of ~7 from (2, 3) in (1, ge~ _ ~-~ (~ ,~~- ,~~~ e 2:-' + ~ + ~~~ (o) = Z- SI l~~ 13~ Taking Z -- ~rb .......... -- (~I~ .... b l,a -F la 2 b2 ) b'~ S' e a" " ~a" 2 4;; ~~ (o) - ,,~ , ( m- ~~~ + 12~ + l~"., ~~)c~ b "" The expression under the sign of summation is rapidly converging we need consider the first few terms only. We • below the results in detail for the case of Aragonite this we have (~Y ---.2.58 and (7"c = 1"926" We write 2.58 ll 2 + 1. 926 la 2 = Azz, :2, z~. TABLE II. J e-~/a,,l,,,13 12 13 e-'t6, z:, h P( L,, la) j e_)t r 12, la P--X 0 0 2.582 -0757 .0293 -0585 1 0 1 3678 .3678 .7356 0 1 1-926 14C8 -0706 .1412 0 0 10.328 ..,. 2 0 4 .0183 .0046 "0092 0 2 7-704 .... 1 0 3.58 .0278 -0078 -0312 1 I 2.926 -0533 -0182 .0728 0 1 4.508 .0111 -0024 .0096 Sum 1.0582 LaHice- Fheory of A lkcdi~~e s 243 The 7th column gives the number of different combinations for the same numerieal values of/i, la, la ; thus for //1 [ ---- 1 ; I 12 ] ---- 1 ; and t~ ----"0 we have the fotlowing 4 combinations : (1,1,0); (--1,1,0); (L--l, 0); (--1, --1,0). Hence ~~ (0) = ~ b ~ 1.0582 ~T-62~ ...... (% 7) ='71 x 1.0582 -- 3.544 . (2, 8) b ., .. Now b 1- ~/--~f_2 e-~~da r (o) -- S' o b-o z r/ ~abc .... (2, 9) The contdbution of the first ~erm containing the integral is negligible. ~~(o)= ~/ax b/~ ~b 7~ ........ (2, 10) b F~om (2, 8) and (2, 10) (0) = .~1 • .05s2 - 3.544 b ...... (2,11) 3. 503 -- b .......... (2, 12) (ii) Evaluation of Z ek ek' ~b (rk~').--- kk' A uŸ cell of the crystal consists of ,t: molecules of the carbonate. The calcium atoms are situated at and the CO~ radicals at (0,2b a b, ab5_dc ) li From (1, 7) and (2, 3) 27ri ( llXk~__~' .~ 5'kk'~ + z"3~ ,,~~') 4~r S' e .... (2, 15) . Z -~! r where x~k', y~~' s.nd z~k' are the project{oas of rkK on the three co-ordŸ axes. 6 B.Y. Oke Henee 2rri ( 1i xkk' + lo Ykk" la Zkk" b 2 e --~ "~+ c . 2 e~ ek" ~b (r/~› = trabe S'z ›I ekez." b~ ba ( . .. (2, 16) Expressing the exponential as a eosine funetion this can be written as b2 2 cos (/1 "xk› + l~'~~Ÿ + l'~:-')a S' Y" b2 b2 kk" .. (2, ~7) The dash over thel's denotes that ii a eertain set of vatues say (t], l-~, -/~/is taken, we need not oonsider the set ( -- l~, -- 1,, -- L). Any other set like (11, --l~, -- ~) of ({1, -- l,, la) has however to be considered. The co-ordinates of the 8 points in the eell ate given in (2, 11) and (2, 12) ; from whieh the values of x~~,, etc., are obtained. Substituting these and assoeiating a charge 2e with eaeh calcium atom and a eharge - 2e with each COa group, we obtain I ek e/g ~ (rkk') -- b/a b/c 4e 2 x 4 S' --F {cos (11', l~', l;) ,,} x.k' ~r b z" b2 b2 112 a-~ + 122 + la c~ (2, lS) where the function F is given by F --- 2cos (ll+l~)~r+2c~ cos l~+13 rr+2 cos ll--l~+ ~ rr + cos (/1 +~-51'+l~)rr+cos(lt +l'~- la) rr +cos(11--l~+ -~)~"4la' + cos (l,--l, -- ~2 13)rr _ {~~o~(~,~ + ~~)o+~~o~(~_ ~~)~+~o~(~~ + ~~)~+co~ (~ ~~+~5 13)~ +~~o~(,~+ ~~g + ~~) ~r 2 cos ( 11 ....~~~ ~)~+cos (p +~)~ +cos (~~,+~~~)o + cos (lz+~+ 513'-~-)rr+cos(/1 ~~ ~~~~o ~)~ As before we need eonsider only the first few terms of the series. Carrying out the summation we obtain 71 x 16e ~ 2 eh. ek" ~b (rkk') = b x 5. 598 .... (2, 20) kk' Laltice. Theory of A lkaliJze L'arlk Car£ 7 Substituting the values from (2, 12) and (2, 20) in (1, 11) we get g2 r = -- ~- 173.608 ........ (2, 21) From (1, 4) and (2, ]) we get 33 = abc ............ (2, 22) Substituting the values for Aragonite from Table (I) 3 = .765 b .......... (2, 23) whence e 2 r = -- 132.8 .......... (2, 24) which may be written as e 2 r = - 2~ ~ .......... (2; 25) where a = 66.4 .........