THE CÆSIUM CHLORIDE ZERO POTENTIAL SURFACE IS NOT THE SCHWARZ P-SURFACE

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THE CÆSIUM CHLORIDE ZERO POTENTIAL SURFACE IS NOT THE SCHWARZ P-SURFACE

I. Barnes, S. Hyde, B. Ninham

To cite this version:

I. Barnes, S. Hyde, B. Ninham. THE CÆSIUM CHLORIDE ZERO POTENTIAL SURFACE IS NOT THE SCHWARZ P-SURFACE. Journal de Physique Colloques, 1990, 51 (C7), pp.C7-19-C7-24.

�10.1051/jphyscol:1990702�. �jpa-00231101�

THE CRSIUM CHLORIDE ZERO POTENTIAL SURFACE IS NOT THE SCHWARZ P-SURFACE

I.S. BARNES('), S.T. HYDE and B.W. NINHAM

Department of Applied Mathematics, Australian National University, Canberra ACT 2601, Australia

La surface "P" de Schwarz est l'exemple classique d'une surface minimale periodique. Elle a la mcme symktrie et topologie que la surface de potentiel zero de la distribution de charges du cristal Chlorure de Czesium et que la surface de Mackay qui y est relike. A partir d'une analyse de la position et de la courbure de ces trois surfaces, nous demontrons qu'elles sont differentes.

Abstract

The Schwarz P-surface is the standard example of a periodic m i n i surface. It shares the same symmetry and topology as the zero equipotential surface for the Czesium Chloride charge distribution and the related Mackay surface. It is shown here on the basis of analysis both of the position and the curvature of these surfaces, that they are distinct.

1 Introduction

Among the families of surfaces commonly thought of as related to periodic minimal surfaces are the periodic equipotential (and zero surfaces [l]. These surfaces are found to have significance in various chemical and biological structures. The existence of a pairing between known examples of the two families of surfaces and the striking similarity between corresponding surfaces has even led some authors to conjecture that the two classes are identical [2, 31 although more recently it seems to be accepted that they are not [4].

A third class of surfaces has been proposed by Mackay [5]. These surfaces are derived from the periodic zero potential surfaces by a bold approximation, replacing the full electrostatic potential by the lowest frequency terms in its Fourier expansion.

In this article we aim to clarify the relationship between these three classes of surfaces. First we set out an analysis of the electrostatic potential, essentially that of Ewald [6], which allows accurate numerical calculation of the potential and hence of the equipotential surfaces. Next we show how the curvature of equipotential and Mackay surfaces can easily be calculated. Finally we examine the LLcanonical" example of a periodic minimal surface, the Schwarz P-surface [7]. The zero potential surface for the Caesium Chloride charge distribution shares the same crystallographic symmetry and the same topology, as does the derived Mackay surface. On the basis of analysis of the position and the curvature of these surfaces it is shown that all three are distinct.

2 Analysis of the potential

Consider a periodic distribution of ideal point charges. Assuming cubic symmetry, the charge distribution is of the form,

N m

p ( x , y , z ) = C q j

C

S ( x - x j - m , y - y j - n , z - z j - p )

j=l m,n,p=-oo

C '

) Present address: Centre de Recherche Paul Pascal CNRS, Chsteau Brivazac, Avenue A. Schweitzer, 33600 Pessac, FRANCE

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990702

COLLOQUE DE PHYSIQUE

where there are N charges G in any unit cube, at the points ( z j + m , yj

+

^{n, zj +P).} This expression can be rewritten as a three-dimensional Fourier series

Now for any charge distribution p , the potential is

P(tl9,O d t d9 dC p(z>y7z) =

IJJ

^{d ( z}

^-

⁰²

⁺

^{- 9)2 +}

^-

the convolution of p with the l / r decay of the electrostatic potential about an isolated point charge. By the Fourier convolution theorem this gives

8 IY r COS 2?rh(z

-

^{zj) COS 2?rk(y -} yj) cos 2?rl(t

-

^{z j )}

= -Cqi

E

j=l h,k,2=0 h2

+

^k2

+

^l2

where the prime on the last sum indicates that terms are to be multiplied by for each zero.index.

This series is only conditionally convergent. To improve the convergence we first use the Jacobian 8- functions to obtain a simple and transparent representation of the potential and then use this to rewrite the potential as the sum of two rapidly convergent series which can be used for numerical computation.

Recall [8] that for

C

^E C

W

83(C, g) =

C

^{f 2 e2ni( = 93((1~)}

where g = ehr. Using the identity

allows the potential to be rewritten in the rather attractive form

We now break this integral into two parts, at t =

&/K

and treat each separately. Firstly

which is just the original series with a Gaussian factor added which accelerates the convergence.

For the small-t integral we make use of Jacobi's imaginary transformation of the 8-functions

In our case

C

⁼ ?r(z

-

z j ) and q = e-'l so that T = -t2/?ri. Writing s = l / t we have therefore

N

=

ce E

erfc ( f i K R )

j=l h,k,l=-W R

which sums over all charges in space (not just in the unit cell) but is extremely rapidly decaying with the distance

R = \/(z - x j

+

^h)2

+

^{(y - yj}

+ +

^{( z - t j}

+

and hence with h, k and l.

Now suppose we truncate these sums, discarding all terms with large values of the indices. For the sum we take only those terms with d h 2

+

^k2

+

¹²

^<

R1 and call this partial sum ~Y(')(R~). Writing Q =

zgl

^{we have [g], for R1}

^>

¹

^{+ 4}

Similarly, truncating the sum (p(2) by taking only those terms with R

<

R2 we have, for R2

>

1

+

^{3 4}

3 Curvature

We now turn to the calculation of the curvature of equipotential and Mackay surfaces. From Spivak [10], if a surface is defined implicitly as a level set of some (twice-differentiable) function ^U :

R3

-+ R then its mean and Gaussian curvatures are given by

where A, B and C are obtained from the determinant

and the subscripts indicate partial derivatives. This clearly applies to our situation, with ^U given either by p, the full electrostatic potential, or by its Mackay approximation. Through (almost) any point in space there is an equipotential surface, and the expressions above give the mean and Gaussian curvatures of that particular equipotential a t that point.

Expanding the determinant yields

and we have H in terms of the first and second derivatives of ^U

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Figure 1: One unit cell of the Mackay surface cos X

+

^cos Y

+

^{cos Z = 0.}

If we now set u equal to cp, the electrostatic potential, then the denominator is just

2E3,

where E is the electric field strength, and the mean curvature has the very simple form

where E =

E / E

is the unit or normalised electric field vector.

So to evaluate the curvature of the equipotentials, we only need to know the first and second partial derivatives of the potential cp. The rapidly convergent series cp(') and are easily differentiated term-by- term giving similarly rapidly convergent series for the partial derivatives. Error analysis for each of these series is similar to that for the potential itself. However the appearance of the electric field strength in the denominator of the expression for the mean curvature makes it impossible to give an error bound for the mean curvature which is valid over the whole of the unit cell. Essentially this is because the curvature can become infinite a t points where the electric field vanishes. But away from these points, it is no great problem to give a posteriori bounds on the error in a particular calculation.

For Mackay surfaces the calculation is simple and exact, as the "potential" is a sum of sines and cosines and the partial derivatives can be calculated analytically.

4 The Caesium Chloride structure

A unit cell in the crystal structure of Caesium Chloride has a positive charge at the origin and a negative charge at the body centre. In the notation used above, N = 2 and we have ql = $1 at (O,0,0) and 9, = -1 at ( f , f , f). Taking only the lowest frequency terms in the Fourier expansion of the potential, those of type (loo), gives the Mackay approximation

cp(z, y , z ) w - 16 (cos 2ax

+

cos 27ry

+

cos 2az), a

(where X = 2 ~ 2 and so on). One repeating unit of this surface is shown in Figure 1.

We have calculated the full potential with K = 2,

R1

= 6.5 and

R2

= 6.3, which gives a maximum error in calculated values of the potential of about 10-~. This involves summing over about 150 terms for and 1000 for In fact, most of these terms are orders of magnitude smaller than the chosen tolerance: while the convergence estimates given above are quite sharp asymptotically, their behaviour for small values of R1 and Rz is not too good. The surface resulting from this calculation is shown in Figure 2.

Both these surfaces have the same topology and symmetry as the P-surface of Schwarz [7]. Note the straight lines in the surfaces, which are also in the P-surface. These are twofold rotation axes which exchange positive and negative charges. An immediate consequence of this observation is that both surfaces have average mean curvature zero. The P-surface can be obtained by solving Plateau's problem for the skew quadrilateral drawn with heavy lines on Figure 1, and then repeating the resulting surface element throughout space by rotation about its edges.

There is one noticeable difference between these surfaces: the ends of the tunnels of the zero potential surface are distinctly more "diamond-shaped" than those of the Mackay surface. Now it is known that for the minimal surface the ends of the tunnels differ from perfect circles by very little-the variation in the radius is less than 0.4% [ l l ] . This gives us a first quantitative check on whether the three surfaces are in fact the same. In the co-ordinate system chosen, the unit cell shown occupies the cube [O, 113. For convenience consider the face z = 0. As the straight lines in the surfaces are symmetry axes, they-and all three surfaces-must pass through the points ( f ,

a), (4,

i ) , ( f , f ) and (a, i ) . So if the ends were perfect circles they would have radius

4.

A good check on the roundness of these ends is then to find where they cross the line y = z as it runs diagonally across this face of the cube. A perfect circle of radius

t

^crosses

C7-24 COLLOQUE DE PHYSIQUE

this line at X = y =

f

-

&

^E 0.3232. It is easily verified that the Mackay surface passes through the point

( B ,

f ,0), which means the radius varies by almost 6%. For the equipotential surface, calculation using the

method described above shows that the surface passes between the points z = y = 0.34 and X = y = 0.35, which gives a radius variation of at least 9.5%. That is, the minimal surface, the Mackay surface and the equipotential surface are all different.

We now turn to the question of the curvature. Consider again the line y = X , z = 0. Along this line the calculated mean curvature of the czsium chloride equipotentials is always negative, rising from -CO at

X = y = 0 to a maximum of about -2.1 at ^X = y G 0.3 and then going back to -00 at X = y =

i.

^{As any}

equipotential with the same topology as the P-surface must cross this line, this shows that none of them is a minimal surface. In fact they are quite a long way from being minimal surfaces: H = -2.1 is stronger curvature than that of a sphere which would just fit inside the unit cube.

For the Mackay approximation, the mean curvature of the level sets is given by

H = - sinZ X(cos Y

+

^{cos 2 )}

+

sin2 Y(cos X

+

^{cos 2 )}

+

sin2 Z(cos X

+

^cos Y)

?T (sin2 X

+

^sin2 Y

+

sin2 Z)3/2

At the point

(i,

$0) this gives H = -K/& E -1.28, confirming that the three surfaces are distinct.

References

[l] H.G. von Schnering & R. Nesper, Angewandte Chemie, International Edition in English 26 (1987) 1059 [2] S. Andersson, S.T. Hyde & H.G. von Schnering, Zeitschrift fiir Kristallogrphie 168 (1984) 1-17 [3] S.T. Hyde, Ph.D. Thesis, Monash University (1986)

[4] H.G. von Schnering & R. Nesper, personal communication

[5] A.L. Mackay, Angewandte Chemie, International Edition in English 27 (1988) 849 (61 P.P. Ewald, Annalen der Physik 64 (1921) 253

[7] H.A. Schwarz, "Gesammelte Mathematische Abhandlungen" Springer: Berlin (1890) [8] Whittaker & Watson, "A Course of Modern Analysis" Cambridge University Press (1927) [g] I.S. Barnes, Ph.D. Thesis, Australian National University (1990)

[l01 M.Spivak, "A Comprehensive Introduction to Differential Geometry" Volume 111, Publish or Perish:

Berkely (1979) pp 202-204

[l11 A.H. Schoen, "Infinite periodic minimal surfaces without self-intersections" NASA Technical Note No. D- 5541 (1970)

Note added in proof

What I have referred to as "Mackay surfaces" in this article are the same as the "nodal surfaces" of H.G.

von Schnering and R. Nesper. This is probably a better name for them.