On Certain Manifestations of Ionic Forces in the Liquid and Solid State Citation Evjen, Haakon Muus (1929) On Certain Manifestations of Ionic Forces in the Liquid and Solid State. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/c9s4-7b05. https://resolver.caltech.edu/CaltechTHESIS:08192024-211113532 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1929 Record Number: CaltechTHESIS:08192024-211113532 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08192024-211113532 DOI: 10.7907/c9s4-7b05 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16651 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 20 Aug 2024 20:34 Last Modified: 20 Aug 2024 20:47 Thesis Files PDF - Final Version See Usage Policy. 11MB Repository Staff Only: item control page

ON CERTAIN MANIFESTATIONS OF IONIC FORCES I N THE LIQ,UID AND SOLID ST.ATE THESIS BY Haakon Muus Evjen In Partial Fulfillment of the Requirements for t he DEGREE OF DOCTOR OF PHILOSOPHY California Institute of Technology Pasadena California 1929 -1- ON CERI'AIN MANIFESTATIONS OF IONIC FORCES IN THE LIQUID AND SOLID STATE. Introduction and Summary. The first part of this paper was worked out in collaboration with Dr. F. Zwicky and represents a continuation of some preyious work by the co-author. It is shown that the difference in thermal properties of dilute solutions ag~inst those of the pure solvent is due to the superposition of two effects: 1) A physical effect of the ion on the solvent which is manifested as an internal pressure varying from point to point in the solution. This pressure distribution is calculated for aqueous solutions of monovalent salts by an approximation theory in which the effect of the so-called ion atmosphere is estimated by means of the Debye-Huckel potential. In this as in ensuing calculationljl, the thermal properties of the solvent are assumed to be known. The effect of this internal pressure on the thermal coefficient of expansion and on the compressibility of dilute binary solutions is calculated and the results are found to be in satisfactory agreement with experimental values. 2) Definite evidence is obtained that the second effect determining the thermal properties of solutions is due to a chemical action of the ion on the solvent. creases linearly with the molal concentration. This effect -in- As a result of this paper and previous work by the co-author, to which reference is given, we can say that the thermal as well as the . caloric behaviour of dilute solutions is duly accounted for. The second part of the paper was worked out in consultation -2- with Dr. Zwicky and represents a first attempt at a rigorous treatment of the so-called mosaic structure in heteropolar crystals of the facecentered type. Certain variations from the crystal lattice of perfect symmetry are pointed out,which would lead to a decrease in potential energy. Certain energetic quantities,closely related to these variations, are calculated. An expression is obtained for the change in energy by the contraction of a 100-plane in patches, against a fixed space lattice, in terms of the end disph:cement ( S~), and the number of ions in the linear patch (p). This expression is found to have a simultaneous minimum with regard to b~ and pat a displacement and a number of ions in the patch which is in close agreement with previous phenomenological estimates. The minimum change in energy comes out positive but considerably less than the energy of thermal agitation. This then shows that at least there will be a considerable probability for patches of definite size, although with the variations here consideredJthese patches apparently do not represent the configuration of maximum stability. A difference in size of the patches at the surface and in the interior is also indicated by this work, a possibility which, so far as known, has not previously been pointed out in the literature. ON (3) • CERTAIN MANIFESTATIONS . OF IONIC FORCES IN THE LIQUID AND SOLID STATE. I. Internal Pressure in Strong electrolytes of Moderate Concentrations. 1) In a previous paper by one of the authors the pressure distribution in a strong electrolyte of moderate concentration was calculated in connection with a detailed analysis of the specific heat of solutions. It is the purpose of the present paper to improve somewhat on this calculation, in that account will be 2) taken of the so-called ion atmosphere . As applications we shall consider the effect of this pressure distribution on the compressibility and on the thermal coefficient of expansion of aqueous solutions of monovalent salts. 1) As in the paper already referred to, we shall choose as our starting point for the calculation of pressure the general expression for ponderornotive 3) force on a dielectric, derived by Dallenbach: ( 1) where ki is the i-component of force density, (force per unit volume), _p is the true charge density, Ei is the i-component of electric field, and Pi is the i-component of electric polarization. ~e expression is to be summed over double indices. For the derivation of Eq. (1) the assumption has to be made that, in the region considered, our medium is isotropic. of the regular system. The formula also holds for a crystal Otherwise the expression is perfectly general, and not restricted to any special functional relation between electric polarization and electric field. it. For small electric fields the polarization is proportional to For strong electric fields the polarization becomes more and more independ- ent of the field strength due to saturation of the dielect:ic. Consequently, the dependency of Pon E in our field will vary with the radial distance from the ion. It has been shown before 1) that a very good approximation is obtained by -1- -4considering two regions, one outside a certain critical radius, rk, and one inside. In the first region the polarization is calculated by means of the ordinary dielectric constant, whereas in the second region we use its saturation value. The latter is essentially obtained by multiplying the dipole mom- ent by the number of molecules per cc. The critical radius, rk, will be defined as the value of r for which the polarization as computed by the two methods is the same. Region 1 ( r > rk). Let the field be some function E{r) of r which will be specified later. Then: (2) P = (D - l)E(r)/4~ where Dis the dielectric constant. Referring to rectangular coordinates, we can establish the following relations for the components: E = ~x,jr; 'c £ ,· == /c) /=.1r;, 'c)l 1't cl X,· Substituting these quantities into (1) one obtains af'ter some easy transformations: I re I = and since, div (D·E)=4'1i'p; div E=4'1T'p/D We get: /kl == /e/ { !_pl (1 • (o-1) -f 2 D-1 / d£/( s-o ) + roii tl.tt ,, 1 J 3 .,z D1 (3) The absolute values can be used as shown because .J> and dE/dr always have the same sign. For the potential of the field we shall use the value derived by 4) Debye and Huckel. -D-vc -2- - 5And consequently, j o = - _Q_ t72.JI/ = - J2 X' ,I/ 41r 41i 'f' i £= -;t=~t ✓,+1] ( 4) dE = - T JU { l ~ -f ..i:_ ] X + k A'~ d/t_ In these expressions and f is the elementary charge = 4.77 x 10 -10 e.s.u. V is the valency of the ion. x.. = [B?T m Cvt.)1/D{< T] 1/z. where k=l.37 x 10 -16 ergs, (Boltzmann's constant), and n= the number of ions per cc. Substituting the values (4) into (3), one gets after some transformations: _ (5) where oC-= .302.+ 4D -2. The pressure is obtained by integrating k over r: (6) Region 2 (r (rk). The second region turns out to be comparatively small, and it is easily seen that here one may safely neglect th~ effect of the space charge. The error introduced in the pressure by so doing is not :rwre than a few percent. In applications there is a :rwre serious source oferror in this region due to the uncertainty about the thermal properties of the solvent at the extremely high pressures in the neighborhood of the ion. These sources of error, however, are offset by the smallness of the region, i.e. in integrating for the thermal properties of the solution as a whole, the contribution from the region of uncertainty is small. -3- -6- We shall therefore neglect the effect of space charge . in this region, and write: dP d..lC = 0 ) E. =: so that, 81T , /-k' I 5 where N is the number of molecules per cc of solvent. If by pkwe denote the value of the pressure at the critical radius, then, where (7) As we have already mentioned, the value of the critical radius is given by: Q.:l f (Jc ) = D-i 41T k 41f -D i) E. Expanding and neglecting terms of the order (,0rk) 3 and higher, this gives: (8) In order to carry through the numerical computations we must make an assumption as to the value of f- . The dipole moment of a single water molecule can be deduced from I!Easurements of the dielectric constant of water vapor. In -18 e.s.u. This value, however, cannot be used on this way one find /\= 1.8 ::x: 10 l) account of polymerization. In the paper already referred to, it was shown by a very general consideration that the average value of JJ'- in water must be less -19 than 9 x 10 e.s.u. It is found that the best quantitative agreement with experimental results is obtained by adjusting f- to about 5 x 10 -19e.s.u. -4- -7- With this value we find for aqueous solutions of monovalent salts the pressure distribution given in Table I. I' stands for concentration in mols per 1000 grams of solvent. These values must be further mdified because our theory, while it takes account of the space charge, paradoxically so to speak, neglects the presence of the individual ions making up that space charge. It is clear that at some point in between a positive and negative ion the pressure effect nnist be zero, while by our theory it extends to infinity. In other words, the integration (6) should have been carried out to some finite upper limit rather than to infinity. As a somewhat arbitrary correction which, however, cannot be very far off, we propose the following: Depress the whole pressure scale by an amount equal to the pressure, p, at a distance r . 0 0 from the ion, and let this distance be given by the relation: (9) where V is the vo_lume of a solution containing 1000 gr solvent, and Ii is the 0 23 Loschmidt number= 6.06 x 10 . Table II gives the :i;ressure distributions corrected as here stated. With increasing concentrations the reliability of the figures decreases. Aside from the yarious approximations made in this development, there are essentially two sources ofarror. In the first place, no account is taken of the chemical action of the ion on the solvent, an effect which so far does not permit of quantitative treatment. In the second place the effect of the ion atmosphere at high concentrations is not rigorously given by the DebyeHuckel potential. For a rigorous treatment of this effect statistical methods 5) would have to be used . The mathematical difficulties of such a theory however, so far have been insurmountable. As another, probably small, source of error, it should be mentioned that this theory does not - take account of any pressure dependence of the di6) electric constant. It has been shown by Roentgen, 7) Falckenberg , and - 8- others that the pressure dependence of the dielectric constant of water is 2 very small up to 200 kg/cm. No experimental data are available for pres- sures higher than this. 2 TABLE I. Pressures in kg/cm. Water at 20°0. r 0.1 0.15 o.3 o.5 1.0 2.0 r=lA 2 3 4 5 6 7 8 9 10 76000 76000 75000 74800 '74580 '72350 16000 16000 15330 15036 13880 121'70 5229 5020 4628 4288 3205 2000 1702 1579 1214 980 68'7 380 522 474 406 294 199 237 217 169 124 117 106 78 52 66 56 38 38 32 23 18 TABLE II. Pressures in kg/cm. Water at 20°0. r r 0.1 0.15 0.3 o.5 1.0 2.0 12.5A 10.9 8.66 7.3 5.8 4 .6 2 r=lA 2 3 '76000 76000 '75000 '74800 74500 72200 16000 16000 15300 14977 13800 12020 5220 5010 4604 424'7 3124 1850 4 5 6 7 8 9 10 230 206 145 83 110 95 54 11 5'7 45 14 29 20 14 0 1695 515 1568 463 1190 382 940 253 606 118 227 '7 I.B. APPLICATIONS (a). The Thermal Coefficient of Expansion. When a solution is heated from T to T+ d Tat constant external pressure, the internal µirts of the liquid expand at variable pressures. If we consider a spheri cal shell of radius rand thickness dr around the ion, the expansion per degree rise i n temperature will be: d{Ll.1{T = dl'J?.d~T if be the number of mols in ' the spherical shell. Total expansion: ,to f!Y dT tn. J I -;;; 0 -6- d?r dT 411 )(' e,vc.. L 2 dtn = 4 7T,t dt/rir -9- and = Now, I cLV == V clT ~ clT So that, 3 c(s-= 1(_3 0 0 where o(( r) is the thermal coefficient of expansion at the pressure f (r), and X(r) is the compressibility at this pressure, or JCc:, ex, - ((0 = t, Jl c((11.) - l( 0 - xc .. )(tt2 1 (10) /(~A 0 where C( 0 is the coefficient of expansion of the pure solvent under the given conditions of external pressure and temp~r.ature. It turns out that the iso- thermal expansion represented by the third member of the integrand is quite small compared with the isobar expansion o( (r). This isothermal expansion therefore may be neglected in first approximation. be calculated from graphical integration. The quantity (10) is to The variation in thermal coeffi2 cient of expansion for pure water for pressures varying from Oto 8000 kg/cm 8) are given by Bridgman and reproduced in Fig. 1. Beyond this upper pressure one must be satisfied by rather dubious extrapolations, but these high pressures occur only in very small regions of the solution, and contribute little to the total effect. The pressure effect, calculated, as here outlined, is shown graphically as a functionat' concentration in Fig. 2. The experimental values are taken 9) from the critical tables. The good agreement exhibited may be accidental to some extent but certainly not altogether. There is a definite qualitative -10- agreement, and that is all we can expect, since, (by necessity), no account is taken of hydration effects in the present treatise. The experimental values for the thermal coefficient of expansion do not permit of a closer analysis of the hydration effect. In the case of the compressibility of solu- tions, however, we shall see definite evidence of the part played by chemical action of the ion on the oo lvent in determining the thermal properties of solutions. (b). Compressibility. P0 When the external pressure, p 0 , is changed from to p0 + 11 p0 the pressure in the interior is changed by say an amount b. p which we must determine. Consider again a spherical shell of radius rand thickness dr, containing dn mls of water. When the external :rressure is in- creased, the change in volume of this shell will be: Total change : .f'o L1 V == J~ 2 411ll dt 0 Consequently: ](.s = Now, as the external pressure is increased, the radius of the shell under consideration is decreased, thereby making the internal pressure change by an aI1X>unt: Hence and () -8- -11- An upper limit for the second term of the integrand may be obtained by the following consideration: Let X{r) be the compressibility of that part of the liquid which is inside a sphere of radius r around the ion. Then X{r) , , X(t) And f -t xc~, = /i - By evaluating this upper limit it is seen that the second term of the integrand must be small, and that a sufficiently close approximation is obtained by using the upper limit multiplied by a correction factor )Cs/x. 0 where a first approximation value of Xs is used. 'dp(tt.) For we can write, to the same order of approximation, O IL. r~P';:! = - 4 fC11.)/4 Finally, in order to carry out the graphical integration it is more convenient to write: ..to JlXo - [ 1 + ~~A) f,;J (11) X(1t>] /(•J.x_ 0 The values of )(( r) are taken from Bridgman's curve, reproduced in Fig. 3. The result of such integrations for various concentrations is plotted in Fig. 4. In the same figure we have plotted some experimental data due to 10) Roentgen and Schneider. It will be seen that the experimental compressibili- ties are higher than those computed from the pressure effect alone. Some pertinent evidence that this may be due to hydration is given in Fig. 5. The data from which this family of curves was plotted also were taken from 11) Roentgen and Schneider and show, within the limits of experimental accuracy, that there is a linear relation between the difference in compressibility of any two binary solutions and concentration. In other words, the difference in compressibility is due to an effect which increases linearly with the number of ions of the same kind in solution. We know that as long as the supply of water molecules is sufficient each ion of the same kind will produce, qualitatively and qnantitatively, the same chemical effect on the water molecules. Therefore whatever the influence of this chemical effect on the thermal properties of the solution, we should expect a linear increase in the effect with concentration. This linear dependency has previously been pointed out also for a 1) caloric property of electrolytes; namely, the specific heat. If, in Fig. 4, one takes the difference between the experimental curve for KCl and the theoretical curve, one obtains a linear relation up to a concentration of about 0.15 rnols per liter. Beyond this point then, our approximation theory yields results which have only a qualitative, but definitely no quantitative significance. The reason for this probably is to be looked for in our method of obtaining the pressure distribution as well as in secondary hydration effects. 1) From this and previous work we can draw the conclusion that the difference in thermal and caloric properties of dilute solutions against those of the pure solvent is due to the suparposi tion of two effects: First, a physical effect of the ion on the solvent which is predominantly manifested as an internal pressure varying from point to point in the solution. Secondly, a chemical effect of the ion on the solvent, which for dilute solutions increases linearly with the molal concentration. -10- -13- - oSO x10 5 cj / 4E 45 (!.) ...... ~ 40 <..... Q) 8 35 C .s 30 (f) C I V / __ v / V / I R2s >< I µ..l 20 1 'c 3 4 Pressure, p, 5 0 (kg/cm 2) 7 x10 3 Fig.l. Coefficient of thermal expansion of water at 20° Cent. 150 x1Q -G ~ - - - - , - - - r - - - - r - - , t::. NaCl L KCl □ HI --l---+----+----r.:--i . 100 o HCl !::. 2'.: 5Ql...----l--=L4-=+-____..::._+-----t----i Theoretical curve at 20°c 0.4 0.8 1.2 1.0 2.0 r (mols per liter) Fig. 2. Llc( for solutions of NaCl, KCl, HI, HCl, at av-erage temperature of 22.5° C as a functi cn of concentra tion. 500 x10-7 400 I\ 300 \ I'\. ""~ 'Xo 200 r--.... ---r--- r-- r-- 100 1 2 3 4 5 7. 0 Pressure, p, (k3/cm 2) r-- 6x1O 3 Fig 3. Compressibility of water· at 2 G°C. (Br·idgman, 1 Sl4) ~ l~" . . . 'O 420 ~- ...., ...... .... 1:9:_ ~ 350 .... ~ ...... _r---o . . . ...... r-- r- I~r--... - ...._ K~J:?- 'o, rNaCl :'Ir-, X ~ - -.....- · Pressure effect--........___, (computed) -..... --........___ 340 I 300 1 0.4 O.B I I 1.2 I 1.0 r (mols per liter) 2.0 Figil:.Compressibilities of aqueous solutions at 2 0°C -15- u (0 /201e-+--+--,l'.---,4--7'f'----r?'9-o~,- 1.c: 1.0 per lit er) Fig.5. 'c.0 Showing the linear relation betwe t n the difference in comptessibility of any two binary solutions and concentration. -16- REFERENCES. 1) F. Zwicky, Phys. Zeits. 26, 664 (1925}; 2?, 271 (1926). F. Zwicky, Proc. Nat. Acad. 12, 86 (1926). 2) S. R. Milner, Phil. Ji..iag. 23,551 (1912). Debye and Huckel, Phys. Zeits, 24, 185 (1923}~ 3) w. Dallenbacli', Phys. Zeits. 27, 632, (1926). It is doubtful whether the term Jl Ei should be left in the exp!Bession, as the ions undoubtedly are somewhat accelerated by the field, and thus do not contribute by the full amount _p E. to k. . We have shown, however, that leaving out this terrn ~akes 1 a difference of only about 1% in the computed pressures for aqueous solutions. 4) Reference 3. This will give us an estimate of the influence of the space charge effect on the pressure, at least for small concentrations. 5) s. R. Milner, Phil. Mag. 23,551 (1912). 6) W. c. Roentgen, Wied. Ann-:-52, 593 (1894). 7) G. Falckenberg, Ann. d. Phys~k 72, 191 (1923). 8) Bridgman, Proc. Am. Ac. Arts andSciences, 48, 309 (1912-13); Bridgman, Critical Tables III, 40 {1928). 9) Critical Tables III, 54 {1928). 10) W. C. Roentgen and J. Schneider, Wied. Ann,~' 531 (1888). 11) W. c. Roentgen and J. Schneider, Wied. Ann, 29, 165 (1886). -11- -17- II. On the Possibility of Mo saic Structure in Heteropolar Crystals of the Face-centered Type. A. Descriptive. The potential energy of two ions may be written on the form: ei. A _ E_ ::: ::___:;;;___ °t _~ _••L __ ____ ( 1) Where the first term represents the potential energy due to the inverse square Coulomb forces and the second term phenomenologically stands for the energy due to the repulsive forces between ions, which are nee~ essary for static equilibrium. A and 0(. in general depend on the kind 1) of ions considered. In the following, however, we shall confine our attention to crystals of the rock salt type and shall assume that these constants are the same for each of the three combinations of ions which occur. This greatly simplifies the calculations and is permissible in first approximation. In the case of rock salt then, the best value of O<. will be that corresponding to a Na-ion against a Cl-ion; namely., l) 9 • With this constant fixed, and the value of the lattice constant known, the value of A may be computed from the condition that the energy 2) shall be a minimum. With Bragg's value for the lattice constant of rock salt; namely, 2.81 x 10-8 cm, the value A= 25.8 x 10-82 is obtained. Details of this as of ensuing computations will be found in the mathematical appendix. This calculation exemplifies the usual method of attack, whereby one assumes an infinite space lattice, IB rfectly symmetrical, with a lattice constant everywhere the same. It is the purpose of the present paper to investigate certain departures from this perfect symmetry which will decrease the average energy of an ion. One such 3) variation has already been pointed out by Zwicky. easily seen by the following consideration: Its reality may be Consider any 100-plane -18- in the infinite space lattice of perfect symmetry. The average energy per ion in this plane may be ascribed to two sources: 1) Energy due to the rest of the ions of this ~lane= Ep, and 2) Energy due to the ions outside of this plane= Ee. If we let this plane contract by an infinitessimal amount against the rest of the crystal, there will be a first oro.er decrease in Ep, whereas, the first order change in Ee will be zero. Hence there must be a resultant decrease in energy and there- fore a tendency for this plane to contract. This, of course, is due to the fact that the lattice constant of an isolated plane is smaller than that of the space lattice. In the particular case of rock salt, as shown in the appendix, the contraction amounts to 5%, and the liberation of energy by this contraction from the lattice constant of the three dimensional crystal to that of the plane amounts to 67 ergs per crrf- af the uncontracted surface. This is of the same order of magnitude as the surface tension of the crystal which is a fact of considerable im3) portance in the theory of sub-microscopic cracks. Similarly there will be a tendency for any nuxed linear point succession in the plane or in the infinite space lattice to contract. The equilibrium condition gives a lattice constant for the line which is only 87% of that of the three-dimensional crystal. The energy liber~ ated by the contraction is 0.908 x l0- 5ergs per cm which is of the same order of magnitude as the edge energy of the crystal. In fact, it is 4) somewhat larger, indicating that there will be a considerable tendency for edge formation. Similar conclusions may be drawn from a comparison of the energy c£ contraction of a dipole and the corner energy of the crystal. By analogous considerations, it may be shown that certain planes -19and lines will have a tendency .to expando We s hall, however, for the present, confine ourselves to the phenomenon of contraction. In the case of the line or the plane respectively a gainst the infinite space lattice, it is evident from inspection that there will be certain obstacles against the contraction of the unit as a whole. It is seen that in order to get a contraction, there nmst be a certain rearrangement of the ions in the lattice, or the contraction nmst truce place in patches, or both. For the present we shall neglect the possibility of a rearrangement of the ions, and shall consider only the change in energy due to the forma t ion of patches. This change does not necessarily have to be ne gative, for, superposed on this variation in any case, will be a contraction of the plane as a whole which certainly gives a considerable ne gative cont~bution. Even if t he total change in energy should not be negative there r:.ight be a considerable probability for mosaic structure if the functional relation giving the change in energy in terms of the end displacement, with the number of ions (p) as a parameter, should exhibit a definite minimum at an end displacement less than the lattice constant. If, moreover, this relation should have a minimum with respect top, we could conclude that the patches formed would have a preference as regards size. For simplicity we shall consider only surface 100-planes, although the same considerations with only sli ght modifications applies also to any 100-plane in the interior. We shall start with the crystal of perfect symmetry and let the surface plane contract, keeping the rest of the ions fixed. After a certain displacement there will be an opposing force due to the rest of the crystal which will keep the ions forming the edge of the surface plane from further displacement. We shall then have a state of affairs analogous to the following simple exawple which permits of a rigorous solution: -20- 0 X GX+ b GJ tE X 0 Fig . l d ---...t Consider the four ions shown in Fig.l, at first displaced at the lattice constant of the three-dimensional crystal . Keep the end ions fixed and leave the two central ions to find their equi l ibrium positions, which, of course, will be sym.-tetrical. By a graphical solution of the 10th degree equation which arises, it is found that the energy is a minimum when 8 = 0.15d. In words, equilibrium is reached with two patches of considerable ' crack' in between. two ions each and a A similar me thod of approach to the problem of the contracting infinite plane would lead to an infinite number of equations and an infinite number of unknovms. It is therefore necessary to make certain simpl i fying assumptions. We shall assume that the contraction truces place in patches of perfect symmetry, the contraction of two neighboring ions being everywhere be /p-1 ,:. ~ except at the edges of patches where the displ a cement will be Se . We shall next attempt to determine the first, second, and hi gher order changes in energy at the start of the process, and in this way obtain an expansion analogous to the Taylor series in terms of 6e_ • pas a parameter. This expansion, of course, will contain The change in energy then will be ex-pressed as follows: (2) Un. fortunately this series alternates and converges somewhat s lowly. It is found, hovrever, that after t he t hird term the higher orders may be expressed to good approximation by a simple remainder, so tha t we get: (3) and If here we impose t he conditions we Obtal .· n two equat1·ons 1·n 0(' and - p• Th1·s pa1· r wh ich may be determined graphically. The value of p thus obtained checks 01~ eq uat1·ons has a solution 1 -21closely with previous phenomenological estimates by Zwicky 3) and Smekal 5) The change in energy corresponding to th i cJ solution i s pos itive for the variation considered, but only about 5.5 x 10- 15 ergs/ion, against an energy of thermal agitation of about 20 x 10 -15 ergs / ion. Thus we have established that there will be a great probability for patches of de f inite size in the crystal, although with the variations considered, we have failed to show that this is a more stable configuration than the space lattice of perfect symmetry. In order to obtain the negative change, which, as we have shown, mus t exist for certain variationat, it may be necessary to take account of the fact that the next neighboring planes also will take part i n the contraction. Or it may be nece ssary to consider a different kind of vari- ation which, for exrunple, takes account of the tendency, (which we have pointed out ) , of certain 1ines to expand . n Te h possibi 1 ity of a rearrage- ment of the ions in the lattice also must be considered. Although this seems to promise considerable additional complications, this will be the object of a future treatise. B. Mathematical Appendix. 1) Computation of A. Let the potential energy of any two ions be written on the form: ( 4) Then the energy of two ions in the infinite perfect crystal lattice will be: -22where s1 = 1.745 and s 2 = 6.63 represent certain summations over the crystal which have been evaluated in the literature. 6 ) By the minimum energy condition: s, For rock salt 2 ) q 5:i.,¾: = 0 e.4 - a 0 = 2.81 x 10-8ciNn_ A == so that (5) 2) ·contraction in an isolated 100-plane. The energy of two ions in such a plane will be: ez. A £ == - .53 ,1: where f- S4- .ll 9 ,. - - - s 3 = 1.616 and s4 = 4.19 are summations evaluated elsewhere. 7) By the equilibrium condition: 9.5,,_ A = o 5 :3 e 2. - a._ a ~ so that 0,0 / 7al° .::: [ Sz. • 53 ] Ye,== s, . s,,. I. o41 Energy change by contraction: i1 £ =-- - ~ L7. 04-7 - I 7 -t : :4~O J d [i. ~ 9- I] == - 61 ~ / 4 7 / ' l . , -i. 3) Contraction in an isolated mixed point succession: Energy of two ions: e,. ~= -Ss ../2:: - £_ -f' where s 5 = 2.ln2 and s 6 = 2.004 By the equilibrium condition: Ss-e 2 - so that /, 12 'j (8) Change in energy by contraction fro m the lattice constant of the crystal to the lattice constant of the line: ( 7 -23- 4) Patch formation in a simple case: Referring to Fig. 1, the energy of two ions may be expressed as follows: _ 2e 2 I::. = - -,i,, .Ze -f- - z. _ _e 2,f,tr.f ei z. {'t"S - - d ,,., 2 ,,..., -f - 9 -1- ';fe.,t,,tfv ~ ,4 -r {?:.,-t-a)9 ;;{9 ., A /) c:. H - c -79 -f - 3X1 -f J:: d == .3 '<-o There is also the relation: The conditions for equilibrium then may be written: Hence we get the equation: which, when solved graphically gives: 6 =0 .15 cl Change in energy by the formation of patches of a surface 100-plane. 5) The following classification will be found convenient in carrying out this calculation: (A} 7 According to the order of the change, or the power of (B) a ) Change in energy (by contraction in :i;atches ) of linear mixed point succe ssion. b) Change in energy by patch contraction of isolated 100-plane. c) Change in energy of ions in line due to the rest of the ions in the plane defined by that line. d) Change in energy of ions in line due to the rest of the ions of the crystal except as computed under c ) . (C) Change in energy due to polarization. 6) Second order change in energy . The first order effects vanish throughout; so we shall proceed with the second order change : (a) Linear mixed point succession: es . . .. - - - -- ® : «-' '' - - - - - e' ~ ' ~ = s e ~~ Ee 2.s _.,,""\ lo- e .do~ ' @- - - ...I 4s ~ I I ffi 0 Et) e ~ - - - - -- 1 2. 3 4 5 UND!5Plf/C£0 D!SPLACE.. D Fi~. 2. -24- Referring to Fig. 2 we can write the general Coulomb energy of the contracted line gitter: (10) where k = kq(n,p) is a function of the periodicity p, (or number of ions in the pitch), and also of the serial number q of the particular ion considered. The ·. average second order change in energy per ion due to a displacement S of :i;ariodici ty p, therefore, will be: Ll [<z.J = - 2 1 _!_Z/r. 2 (-1),yz.,., ? 1=1 1 !YI= -oo /m/ 3 :i~ L r2= (11) turns out to be an even function of n, so we can write: where it is easily seen that: (12) {The symbol r (n/p) stands for remainder in n/p} or with the abbreviation: S has been evaluated by direct summation for a number of different p values of p. By a somewhat different method of attack , however , (as exemplified under the computation of polarization energy), the following summation formula for SP was obtained: (13) - 25 - where ~P even ( 1 4 ) and P odd Here. -rtx +1)= d.2. ,lf(it -t-1 ) dx.. 'I.. 't wh i ch wi l l be found tabulated in 8) the l iterature. The above expression for Spmay be further simplified if one observes that : U.s t I == UH\ This gives: Sr=± t I 5p = z: L I f /) 5 - - - . p even p odd = -Up-st, fti. -/ 1'/i. ~ t u (-!) 5 ~tI P..=1 z. L\, - St \ -t L. c-f's(f-s)Us·tr3 p even I (16) scr -s) U St / - - . - - - - p odd (17) For reasonably large values of p (p >10 ), it is f ound from this, that the second order change in the Coulomb energy may be represented as . follows: (18) The second order change in the energy of repulsion similarly may (19) (b) Cont raction of isolated 100-plane. r.• 0 ,,,,J ., The energy of an average ion in t he plane may be thought of as due to the fo l lowing sources: 1) Energy due to two intersecting perpendicular lines. These contribute equally by the amount computed under (a ) . 2 ) Energy due 'i> the rest of the ions of t he pl ane . This will be given by: (15 -2 6- ( 20 ) Noticing that cross-products of k and k2 must vanish in the summa1 tion, we can write the second order change: This expression is symmetrical in m and 1, so that in the summation the two members of the bracket contribute equally. Hence, we get for the second order change in the Coulomb energy : (21) And for the second order change in the energy of repulsion: L1 t ( 22 ) ?, These series were evaluated by direct summation to fair accuracy. The result fbr p >10: ( 23 ) ( 24) ( C) Change in energy of ions of contracting line due to the rest of the ions in the plane defined by _this line. Referring to Fi g. 3, we can write the energy of an ion . - - - - ---© C. /t. -= j ::: ffi I e ® () I ff) e ffi e & e I 2. ~ 4 B $ e $ p e EB e e ffi z. $ ao-6 -im. ! 1. EB e tEB - - - - e) e $ k-- ao-1 e ffi .j. e EE> - Fig.3. -27- in the boundary line w.r. to the rest of the ions in the plane: (25) where h = hq (m) From this the second order change in energy i s easi ly seen to be: (26) From Fig. 3, it appears that { f '-/ (27) 12. Hence: :::: oeA S' (28) which by adjustment of the constants gives: Change in Coulomb energy: ll 2 - = Change in repulsive energy : il 2+ = (29) 9A 2:/ce~-1) t'r /2 a_ I/ !_ Ss - z I (: J ....J--1 T (30) where I Z _i'- : : / c,Q _51-- = I I/ , tJooS- These results still must be corrected for the Lennard-Jones contraction perpendicular to the surface. ing 1) That is, i n the terms contain- L = 1, this must be replac~by 0.95. As a final result, we obtain: (31) -28- ( d) Change of energy of ions in line due to the rest of the ions of the space lattice except as computed under (c). SIDE VIEW' 63 t ;- FRONT VIEN EB -'- h& - - ..:r-' ,.,. : + -- + + i fyYL ;-- + -· - -t- -r Fig . 4 . + -rn.. -1 - - -t- ' -- + Referring to Fig. 4, we may write the energy under consideration as follows: (32) Where the limits of variability of l, m, and n are: l= m= -oa--- _ ___ +o0 1 .- --- -- .f_ *0 +o0 From this, we can write the second order change in energy as follows: Repulsion: Coulomb: where (33) (34) -29 s , s , and s 3 were evaluated by direct summation. s 4 , on the other hand, 2 1 was derived from a known series by a method wh ich will be reproduced below. With the correction for the Lennard-J ones contraction of the terms containing m = l, the final result is: ( 35 ) (d') Derivation of the series s 4 9) The value of the series S is given in the literature • 0 I t is : f Im / -,-/f/ =/:- o] z z. 5 = 4 0 I Hence: L. I ::, 4- Now: 54- ZI.. So that: Hence: So "° :;- = 4L, t-41/o I - flo r (-l)m".e. oO I [m-,l-t-.f:}:J/1. = 2_z - Z,3 2. = "° ~ £mi "+-f._1-3/,_ ~ ~ Now: I t-4Z_£3 rrJ ,40 =- ,,z_s-;79,t.~. and [0zZ~~jJ0_ I Where: where: = DO 00 Then: z-3 =:: I oe ~ ~ ?- ?- IJJ2 -f .t ~ II odd I [1n<.f.€'-j3/1. // -30- (e) Second order effects sum,.~arized. The total change in energy due to Coulomb and repulsive forces now may be written: (36) For future use it is more convenient to express this in terms of the end displacement be= S (p-1)/a. Hence: (37) (f) .Some general remarks about the second order change. At this stage of the calculations may we interrupt to call attention to the physical meaning of the results obtained so far. The fact that the first order change in energy is zero and the second order positive must be taken to mean that at the start of the process of contraction there is no tendency for patch formation, since the energy is a minimum against the variation. To see whether the energy does not become negative or does not possess definite minima at finite end displacements we shall proceed to calculate the higher order changes in energy. First, however, may we also call attention to the fact that the contribution of the space lattice to the change in energy (AEc) contains p to a higher power t han the contribution of the plane (AEP) • This indicates that the limit to the size of the patches will be set by .Ll Ee, which again tells us that there is going to be a difference in size between the patfhes in the interior and at the surface, since LI.Ee, aside from the effect of the Lennard-Jones contraction, is twice as large in the former case as in the latter. -31- ?) Third Order changes in energy. a) Linear mixed point succession. The change in energy will be: e2. Coulomb: - Repulsion: 83 a_,4- £9. _:.. ,!_ k 3 2._ 1. 1 _ (-! i111.. I IJ1_4I !!:.!!!:.!!. f,3JI ,z. 3 a 12. t:>O ) f- ..J... (38) ± -k3 1-:___,_1_ (39) m 1 2. (40) where For p >lO this gives to fair approximation: Coulomb: (41) Repulsion: (42) b) Contraction of isolated 100-plane. The change in energy except as computed under (a) will be: 6 e.?. tS 3 Lex> Lo0 Coulomb : a 4- Repulsion: 2-t;'·//, 11 3 .t._ It_/ ..,( [ ;-t 1 1 o2 a Ii!.. ...L t' 0,0 / L CO I I [ ,.e -z.-r --n-z / ~ J?.£.J 1-n:e y t_ tyn 1 on.,.~ (-/) 7,/;z_ /J [10 /J'I.. ·A. T-<- (43) 't<. -,m _) £ ~ 2-,,. mi y 1-½_ (44) For p >10 this gives to fair approxi:rmtion: Coulomb: (45) Repulsion: (46) c) Third order effect against crystal. It is evident that this effect nru.st vanish throughout. d) Total third order ch~ge: e 2 83 LJ£3 = -7.6G(f-t)(p-z) (2<10r, in terms of the end displacement: (47) (48) -32- 8) Fourth and higher order changes in energy. 4~h order The;Coulomb contribution of two intersecting perpendicular linear mixed pO.'!iintt .,·suocessions will be: 4-2. ..!.Z--ft4- 00 ~ > (:-/)IYI. -p,nSa_s L 2. (49) I 1 For p :::,- 10 this may be expressed approxirmtely as: E- = _z fp-1)4 &4-e , _ iJ f> 4 (50) a_r or in terms of the end displacement: ~ 4- il.£.-=-.J,.,. "-1f ee 'l. {51) a_ In first approximation, this may be considered as the total Coulomb contribution to the fourth order change. per cent smaller than this. The actual value will be a few An estimate of the higher order change in Coulomb energy then -may be obtained as foilows: t2 f-= - 2. e '2. f1 q_ [.-!-;. - I -r de - de' S/_jl -f- If- c>e {52) The energy Df repulsion may be represented to fair approximation by the first term in the expansion, for the fourth and higheI· orders • .: , That is : ,4 / -1)0 ~ y lf.(9-1-1}·· • •• {9-to-t) 1° ~ Y.' a. r:;+ r sa (;::,3) ,J which will be seen to represent a general term in the binominal expansion of: I ) ,4 I (54) j:7 c:z9 (l-r&'e)9 Consequently, the fourth and higher orders of the energy of repulsion (55) which may be taken to mean that these higher orders arise entirely from the change in energy of the ions forming the boundaries of the patch. 9) Energy of polarization. We shall first consider the energy of pblarization created by patch con- traction of a linear mixed point succession although the result of this calculation will not be directly applicable to the plane due to the vectorial -33- nature of the field stre~gth. For this purpose we shall consider the displaced point succession as a number of interposed linear gitters, the number of such gitters being p, and the gitter constant of these gitters pa. If pis odd, it is seen that these elementary point successions will be mixed, whereas, if pis even, all the charges making up the elementary linear gitter will have the same sign. The following notation will be used: Frs= field strength at ion #r due to the elementary gitter of which sis a member. We can then write the following symmetry relations: 1) Fqq = 0 2) Fqs = Fsq if q-s = ± 2n-l 3) Fqs =-Fsq if q-s = -+ 2n 4) Fqs = Frt 11' q-r = s-t= :2n 5) Fqs = -Frt if q-r = s-t= :!:2n-l l!,or s )' q we obtain from this q-1 Fqs=(-1) F1, s-q +Ir For s <q, we obtain: Fqs=(-1) q F1 , q-s + 1 Hence we can write for - P. ---1: Fqs S=t - - - - - _. - - =- 0 - - - - if q~ p/2 if q = (p+l)/2 - . if q > (p+l)/2 -34- Only the first of these relations interest us since if the field strength is known at the first half of the ions the field strength at the remainder follows by syrmnetry. Now we can write: = _ c( F d.X. 1L.-+1,1 _ d v] v] X:=/i{ci.-b) - - -t A: S d 2 'it _ (56) cf_x_1.JX...=k.a.. dX X=A.<Z.. 6) where vazil(= - ;'(_["41G;ci ti) ~o/c2--;ci)-1f~ tf)-rCt -2r~) p odd - t~] 2 ~ - ~ + r~ x c2.pa..-x .x.tra.. ra.-x. ( 57) (58) and p even So that (See er14) and So that the change in energy by . polarization become·s: p odd !J £ ~ f-:,-1 .z. - "'-a~'r / ; p-( /--;- l~ fir u,.,,T (59) (60) p even (61) where These expressions reach the same asymptotic value. The negative change, however, in general is higher for an even number of ions in the patch than for an:. odd number . 10·) With Born's values for the polarizabili ties the asymptoti c -35- change in energy becomes: (62) 10) Second order polarization in plane. No convenient general expression has as yet been developed for the potential due to a plane gitter, so we shall resort to the method of direct expansion. With our previous notation, the x-compnnent of the field at ion #q (63) The first summation has been worked out under (9), in fact: LJ/\c, I½ f o<7 yf ==- . /Y,_t( J > - 2a.J"e _.tA;{-;) = ~L; Ji!~$> f-1)/f IJ +f (-) f1 La_ (__L "I};_ -1-/ -= 3 J '/ 3 ! °?tft so that: ( 64) ( p even) (65) (66) where ( 67) (68) etc. These summations have been worked out previously. (69) -3611. Results of the calculations. By eq_uations (37}, (&8}_, (52), (55}, and (69) we can write for the (70) Hence r.),1£. 1,q42. -=---1' ~ ~ l''l. Or, to good approximation, for p fairly large: so that ~ J o.~.re,3 ( 1-1- ~e) 9 L IS:4-<le - tJ//()6oe +l,47S"de -/,9¥2 + / ~ r::; o,o:,- <f"e 3 - 0.1 ?~le1.-f/.47.rJ"e-f,9'1-Z -t /:,f'e O,OS-83 (/-t-tfe)9 For different values of ~e this gives: be_ = 1 0.9 o.s 0.7 = 37 34 30 2$ Pp 0.6 0 22 2 5 By the other minimum condition: t>@ /4 2 f~{t') ==-O=-l, f-75--tZ[!.14-f!:;-o.6J-0.366(~) Giving l.47S-0, 288ae +O.t8Je p= .,_ ) -t0,0!1¢. z. o.szs- - ([;&e)~ -t (I~ ~e2 ,o (72) 2.. x 0,0/14 Oe ? or different values of Se. this gives: s~ = 1 0.9 o.s 0.7 0.6 p 37 39 40 43 4$ = 2.. 3 O.S2.> S 1, p(('-:z C. 82. +. - ---),a p- t jt3z7. 6 -7,(, (j,-!)~J e '(uJ;,f'-(!fSe ~2-1, 0.5 0.3 0 42 41 42 -37- Equations (71) and (72) are therefore seen t o have a simultaneous solution at p = 37 and Se= 1. The value of the change in energy at this minimum is given by equation (70). It is: (73) Compare: (74) - REFEHENCES - 1) Lennard-Jones & Dent, Proc. Roy. Soc. 121, 247, 1928. Lennard-Jones e., Taylor , Proe. Royl Soc:-i'09, 476 9 1926. Lenna.rd-Jones & Inghruit,;'·;P roc. Roy. Soc. 107, 636, 1925. Leri. nard--Jones, Trans. Farad~ Soc . Feb. 1928 ., 2) Bragg, X-Reys & Crystal St:ru.ctu:re - 306, 1924. !3) F. Zwicky, Nat., Ac. of' Se • .!§_, #3 0 253 0 i&irch 1929. 4) Born & Stern, Sitzu ker. ·-a. Pr. Ak .. d. Wissenschaften 9 90, 1919. -t.ech?1.. 5) A. Smelml, Zeits f./\ Physik, '1 0 535, 1926;-8, 561, 1927. a) See f. ox .. Made lung, Phyaik 7...ei t. ~' 524, 1918. 7) See Reference l 8) Jahnke & Emde , Fu.nktionenta:t'eln, 30, 1909 .. 9) Topping, Proc. Roy. Soc. 114, 67, :1927. 10) Born & Heisenberg, Z.e it. f. Phys ik, 23, 388, 1924.