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A box model of alkali halide crystals
J. Phys. Chem. Solids
Pergamon
Press 1967. Vol. 28, pp. 2053-2059.
Printed in Great Britain.
A BOX MODEL OF ALKALI HALIDE CRYSTALS P. A Department
of Physics,
(Received 6 February
SYSIU
University
1967;
in rtied
of Helsinki,
Finland
form 21 April 1967)
Ahatract-The model introduced in author’s first paper [An& Acad. Sci. fem. A 6, 110 (1962)J has been developed further, both theoretically and technically. It is based on the assumption that each ion is moving in its own, practically impenetrable sphere (or cube) in a constant coulombic potential. Then the eigenvalues of the Hamiltonian are easily found and the only experimental crystal parameter needed is the lattice constant d. Validity of the theory has been tested by comparing the computed values of the binding energy E, the molar heat C,, the coefficient of volume expansion a, and the adiabatic compressibility K, to the experimental ones; the mean (per cent) deviations are, respectively, 4.4 per cent, 5.1 per cent, 16.8 per cent and 34.3 per cent. The results are rather promising, so that this model seems to be worthy of further research.
INTRODUCTION
A GREAT deal of the theoretical research conceming thermal and elastic properties of crystalline solids, especially alkali halides, is based on the following methods : 1. Static potentials Most common are the classical potential functions of Born-Huang and Bom-Mayer.(1-7) In recent years also other types of potentials have been studied quite extensively.@-11) 2. Frequency analysis Basic examples are the well-known models of EINSTEIN, DEBYE, and BORN and VON KARMAN.‘~-5) Modern computing techniques have been used very successfully in this hind of work.(12-1g) Recently, PLENDLhas introduced an interesting mean-frequency method and RAMAN a promising normal mode system.(al) 3. Wave function synthesis Usually, the wave function for the whole crystal is calculated in the Heitler-London approximation with Hartree-Fock orbitals (LijwDIN et aZ.,(22-24) LUNDQVIST,(~~) ~QNSIKKA et al.(26) Very satisfactory results are given by a dynamical theory based on a tightly-bound electron approximation (MITSKJWICH(~~)). 2053
4. Solvable Hamihtaians Using creation and destruction operators, BAZAROV(~~)has developed a Hamiltonian to be applied to periodic crystal eigenfunctions. ODEN et aLcag) have used the energy levels of a particle in a box in connection with hard sphere statistics, and the specific heat of such particles has been calculated by ROSENSTOCK. Nevertheless, it seems that the box model has been applied to crystals only by the present author,c31) to whom Professor R. Niini suggested this topic in 1959. The present paper reports on refinements of the model and further numerical calculations, now concerning all alkali halides. An acceptable crystal theory must be capable of yielding the measurable quantities of a crystal also numerically. The static potential method above will give any thermodynamic quantity of a crystal, once the potential parameters are evaluated from the equilibrium conditions and experimental (thermal or elastic) data. If, especially, the molar heat and its temperature dependence are desired, frequency analysis is the proper method; it can be done on the basis of empirical elastic coefficients, or more directly from infrared measurements. Further, the method of wave function synthesis can predict purely theoretically any measurable crystal characteristic, including the lattice constant d. However, the procedure is
2054
P. A. SYSId
quite cumbersome; it involves several approximations, and has been applied numerically only to a few crystals. In the category of solvable Hamiltonians merely the box model is comparable to the preceding methods. Starting numerically from the experimental lattice constant, it yields other measurable crystal characteristics purely theoretically, most values with satisfactory accuracy despite the primitive nature of the model.
The quantum-mechanical treatment of the particle-in-box problem is a well-known procedure. Thus, without going into details, we shall merely take the energy needed in the following. In order to save computing time, the box has been taken to be spherical instead of cubic; in any case the shape of the box is not essential, as was shown in Ref. 31. For an ion pair the spherical-box energy levels are E
THE MODEL An alkali halide crystal is regarded here as simply as possible. The most prominent feature is the stable, almost incompressible cubic lattice with very low self-diffusion.(5) Thus, the crystal looks as though it were composed of rather hard ionic spheres, each bound to its own potential well. Further, the interaction of two ions consists, in first approximation, of the pure Coulomb attraction and an inverse-power or exponential repulsion.(1-5) Starting from a two-ion potential function like u(rl,) = of:l/rij+ l/~,~~,(~) the potential wells in a linear chain, qualitatively also in a cubic lattice, are seen to be approximately parabolic (harmonic forces). However, if the atomic nucleus together with the innermost electron shells is assumed to move relative to the outer shells (q.v. and Table 2), the potential is constant until overlapping of the shells causes a strong repulsion. The combined potential will thus be more square-well-like, and we can assume (see also Ref. 31) that
I_
SAe2 --) 4rc,d
where S = screening Madelung’s constant.
factor
fi2iln12+~2in 22 SAe2 2mIu12
2m2u22
--9
4q,d
(3)
where jl, 1 2 is a root of the equation J,, r ,2 = 0.(32) At equilibrium dE = 0, so that differentiating E adiabatically, i.e. jln 1.,2 = constant, with respect to the independent varrables a,, u2 and S = S(B) we get, with the aid of (2), u1.2 = and
hc,d2Tz2 jln 1,22 113 pmBSAe2 rnIs2
dS
S(u, +
-=
a2)
d
dB
(4)
(5)
’
The energy expression (3) can be simplified by multiplying its two first terms above and below by ur and u2, respectively. Then by denoting rr+r,
= do
and
B(u, +a,)
and taking (2) into account insertion (3) yields SAe2 1+2/2 E, = - -.p* 4mc,d, (1 +.z)~
= xd,
(6)
of (4) into
(7)
Similarly for z, using (2), (4) and (6), we obtain the third degree equation
The mass centre of each ion moves in a box with practically impenetrable walls and with constant coulombic potential UC
=
(1)
+(j,, 22/m2)1’3](1 +z)“’ and
s
A =
The ionic distance is d = r,+r, + B(a, + a,), where rl and Y, are the ionic radii, a, and a2 are the box dimensions and B is the coupling coefficient. The physical meaning of the parameters B will be discussed later.
I
x,(1 +x)2/3,
(8)
which is easily solved(33) to give x = xo3/3 +zo[(3+xo3/3 (2)
1 S and
+xs6/27
+ () +.~,,~/27)~‘~)~‘~ + (+ + xo3/3 +z,‘/27 -(t+,~~~/27)~‘~)~‘~].
(9) In principle, the problem is now solved except for the values of S and B. In order to evaluate
A BOX
MODEL
OF ALKALI
them, as well as for numerical applications, have to introduce next some aerodynamic.
We
HALIDE
2055
CRYSTALS
of U and V(34); this yields KS = - (aV~~p)~f V = (a~laT)~/ x [(aV/aT),(aav/aT2)s-(aU/aT),(a2V/aT2),J.
STATISTICAL QUANTITW
In the case of a large enough crystal, surface effects can be neglected. According to the model under discussion the crystal consists of a great number of independent boxes with mutually orthogonal spherical eigenfunctions. Those boxes can be recombined pair by pair, and thus, for a statistical quantity elr and for its temperature derivative, we find the expressions(34)
P), The thermal process in (15) is taken aa occurring in two steps. First, the crystal is compressed adiabatically at constant pressure without any change in the energy states. Secondly, the energy distribution is changed adiabatically at constant volume. Then, using (ll-13), we can insert the isochorie and isobaric derivatives of U and V into (16), which reduces to
where f, = (ZI, -!-1)(2&-i- l), and
SjaT
= (a?$-$)(~ -a)/kTa.
(11)
Both (11) and subsequent thermodynamic expressions can be simplified considerably by introducing an operator .D such that DPy”
n? (x - $P(y
-jQ”,
(121
where (pn, 4t B 0 and m+ n > 2). The two basic interrelations between the macroand the microscopic thermodynamic scopic quantities are lattice energy to binding energy, and molar volume to cell volume, namely, u=
-NJ!?
Y = gNP
to (13-15)
A = - (~~3J~~2
for x,
(191
and (17,181,
- lIEa V/~~V}/kT.
(20)
All the statistical expressions derived in the preceding section for the box system (7) are valid generally for any ensemble obeying the Maxwell-Boltzmann distribution law (10).(34)
(l&-13),
C, = ~~~~~T~~ = ~~Ez~kT2.
a = (WJ8T)JV
equation
(18)
(13)
where N = Avogadro’s number and g = 2 for NaCl-structure = (S/9)1/3 for CsCl-structure.
of volume
degree
x = &+(fi-A-)“2 where, according
= g?z&
Similarly, the coefficient takes the form’34)
x = CPjCV = &&/KS = 1 f VTa2&,rc,, This leads to a second which is solved to give
(&’ < 0) and
Thus, on the basis of equations molar heat C, will bef3*)
The theoretical isochoric molar heat C, (14) should be transformed into the isobaric value C,, if comparison with experimental C, data is required. This ~~sformation can be made starting from the general thermodynamic relations,f34)
the
(141
expansion
= DEV]kT=t?
a
(15)
In order to obtain an expression for tbe adiabatic compressibility Iir,, the derivative (aI’/@), must be transformed into the temperature derivatives
Ph3AMZTERSSAM)B An oversimplified picture of a physical
system can often be refined effectively by choosing suitable parameters. For example, in this model the coupling coefficient B in (2) takes into account the fact that the nucleus and its electronic shells can have different motions, because of their elastic intercoupling. Then also the effective charge of an ion as seen by ita neighbou~ will differ from unity. This shielding effect has been included explicitly in the theory by inserting the
2056
P. A. SYSId
screening factor S = S(B) into the coulombic potential in (1). On the basis of the equilibrium condition dE = 0, we found for S and B the differential equation (S), which transforms through (2) and (6) into dS S z -= dB >‘l+z’ (21) If equation
(8) is written
as
z = q(B2’3/S1’3)(1 +x)~‘~, substitution
(22)
into (21) and integration S1’3 = B2’“(4/2)/(l + integration
yield
C,
i.e. M 0~)
S = 0 B = 0,
(23)
B=l s = 1,
i.e.
no shielding
(29)
Thus, values of the parameters S and B are implicitly known on the basis of (26) and (29); in (26) the quantized z has been replaced by its mean value 5. In principle S and B are also quantized, but numerically we can find only a kind of statistical average for them.
completely E,,, have been numbered according to Table 1 by arranging first the quantum numbers jln(32) into increasing order of magnitude. Because the Table 1. { triangle
(24)
we find C = 0, so that, for a constant q value, according to (22) and (23), z does not depend on S and B. Then (21) can be integrated into account the boundary ,directly, taking conditions valid for any q or z value rigid coupling
= 0.
NUMERICAL PROCEDURE
when 4 and (1 + z) have been treated as constants. Introducing the “ideal gas boundary condition”, potential M 0 box dimensions
G(S, B) = 2-(l/kT)(DE3/DE2)
The energy levels (7), E, or more
+FZ)~‘~
constant
The derivative of E in that direction is simply dEJdS = EjS and, on the basis of (12, 14, 27), we get the extremum condition
11 12 13...
lfl
21 22.. . 2n 31...
3n
... (25)
?I1
whence we obtain S = B”/(l+“).
(26)
In Ref. 31 it was shown that the minimum of E corresponds to the maximum of C,. In order to find this extremum on the curve (26), i.e. to evaluate S and B, we shall need partial derivatives of the type aEZ
aw
Dw
aE
av
av
kT
av
-=
---*--,
(27)
where v stands for S or B and zZis some statistical quantity (10,12). A useful zero indicator for finding C,,,,,
turns
out to be the derivative d In C, G(S,B)=dln=--‘taken in the direction
S
dC,
C,
dS
dS/dB = 2SIB.
(28)
actual height of the box wall is finite, each box has a limited energy level number n determined here by the stability condition phonon energy kT2
I@-E)-(U,,-E,,)+kTj
mean free box energy i.e.
= [Al = min.
(30)
However, the change in energy values (7) due to the finiteness of the wall height is assumed to be insignificant. The computer program was written in ALGOL language (for an ELLIOTT 503), see Appendix. As well as the general physical constants(33) and quantum numbers jln, (32) the data input consists of the ionic masses,(33) the experimental lattice constant d,(18-35-36) and of the “guessed” starting values of B, do and n. Thus, the only experimental crystal characteristic needed in this model is d,
A
BOX
MODEL
OF
ALKALI
HALIDE
2057
CRYSTALS
Table 2. Model characteristics f
S
B
n
0*0503 0.0447 0.0379 0.0347
O-02104 0.02709 0.02982 0.03329
o-9499 0.9515 0.9558 0.9597
0.0825 0.1518 0*2102 0.2786
23
0.0054 0.0079 0.0089 0*0105
0.0370 0.0321 0.0266 0.0243
O-02386 0.02891 0.03084 0~03355
0.9561 0.9615 0.9679 0.9732
0.1457 0.2471 0.3358 0.4328
f: 28
0.2601 0.3049 0.3191 0.3410
0.0072 0.0098 0.0107 0.0123
0.0328 0.0281 0.0231 0.0208
0.02757 0.03206 0.03348 0.03601
0.9601 0.9677 0.9753 0*9819
0.2189 0.3478 0.4617 0*5905
23 23 24 25
0.2815 0.3290 0.3444 0.3671
0.2735 0.3183 0.3329 0.3541
O*OOSO 0.0107 0~0115 0.0130
0.0276 0.0233 0.0189 0.0168
0.02928 0.03357 O-03461 O-03680
0.9654 0.9750 0.9836 0.9910
0.2897 0.4591 0.6106 0.7760
26 24 23 23
0.3004 0.3571 0.3720 0.3956
0.2912 0.3447 0.3587 0.3808
0.0092 0.0124 0.0133 0.0148
0.0254 0.0213 0.0169 0*0151
0.03144 O-03612 0.03703 O-03897
0.9694 0*9815 0.9915 0.9995
0.3604 0.5856 0.7874 0.9860
30 26 23 23
d
do
d-do
-&+Q
LiF Cl Br I
0.2014 0.2570 0.2751 0*3000
0.1973 0.2502 0.2671 0.2903
0.0041 0.0068 0.0080 0.0097
NaF Cl Br I
0.2317 0.2820 0.2989 0.3237
0.2263 0.2741 0.2900 0.3132
KF Cl Br I
0.2673 0.3147 0.3298 0.3533
RbF Cl Br I CsF Cl Br I
which determines the ionic radii sum do, according to (2) and (6), or directly d,, = d/(1 +Z),
i.e.
d’ = d.
22
2. Boundary conditions (24) and (25) for S and B. 3. Dependence of box energy on wall height.
(31)
Values of the lattice constant d(1s*35*38)and some computed characteristics of the model are listed in Table 2 (lengths in nm = 10eQm; T = 298.15”K). Table 3 presents the computed and experimental values of E, C,, a and K,, together with their percentage differences. The computed values of the ratio C,/C,, are also listed in the Table 3.
author would like to express his gratitude to Professor R. NIINI for having suggested the basic idea behind this work, and for many fruitful discussions; to Drs. 0. INKINEN, K. KURKI-SUONIO and P. LIPAS as well as Mr. E. PAJANNEfor their helpful advice ; to Mr. E. PELTOLA for consultations and arrangements concerning the computing work; to Mrs. LORNA SUNDBTR~M for revising the English manuscript; and to The Finnish National Research Council for Sciences for financial support.
Acknowledgements-The
SUMMARY
The main aim of this work has been to determine whether the results of this rather crude model indicate the usefulness of further research into its capabilities. The results in Table 3 are fairly encouraging. In order to help future investigations, the weakest points of the theory are listed in their order of importance as follows: 1. Limiting criterion (30) for n.
E 42
REFERENCES 1. BORN M.
2. 3. 4. 5.
and HUANG K., Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford (1954). Dmuzzm A. J., Solid State Physics, Macmillan, London (1962). KITTEL C., Introduction to Solid State Physics, Wiley, New York (1961). SBITZ F.. Tke Modern T&my of Sol&a?, McGrawHill, New York (1940). ZHDANOV G. S., Crystal Physics, Oliver & Boyd, Edinburgh (1965).
10.26 8.46 8.03 744
8.92 7.62 7.32 6.87
8.51 7.34 7.07 6.67
8.00 6.85 6.64 6.29
NaF Cl Br I
KF Cl Br I
RbF Cl Br I
CsF Cl Br I
(37)
7.53 6.75 6.49 6.18 4.4
+6*2 +1.5 +2*3 +1*8
+5.3 +3.8 +4*3 +3.6
+5*8 +3.8 $4.3 $3.3
8.43 7.34 7.02 6.65
8.08 7.07 6.78 6.44
$8.2 $4.8 +4.6 $2.9
$11.5 $4.7 $3.7 +1.8
IO.54 8.78 8.31 7.78
9.48 8.07 7.68 7.23
%
exp
+ No recent data available.
Ref.
Mean I%1
11.75 9.19 8.62 7.92
LiF Cl Br I
camp
E eV
41.9 50.3 51.9 54.7 46.8 50.8 52.3 54.3 49.0 51.5 52.5 53.2 50.7 51.2 51.7 51.8 50.7 52.6 51.8 51.9
50.9 50.8 49.0 48.0 52.4 53.2 52.1 51.4 52.4 53.9 53.6 53.0 51.3 53.2 54.2 54.1 50.6 52.9 53.9 54.6
(38)
ew
camp %
5.1
-0.2 +0*6 +4*1 +5.2
+1*2 +3.9 +4.8 +4*4
+6*9 $4.7 +2.1 -0.4
+ 12.0 ++7 -0.4 -5.3
+21*5 +1*0 -5.6 - 12.2
C, J/m01 “K
126 153 161 172
121 145 153 161
118 141 146 153
103 126 130 138
88 111 114 121
camp
4.6 +5.0 +0+8 -48
g;
e
16.8
+10*9 + 14.2 + 17.0
* 138b 141b 147e
Ma (36)b (39)c
+18*6 +34.3 +342 +24*8
+7*3 +22*6 + 21.7 +13*3
-
-13.7 -15.9 -24.0 -31.6
%
102a 108d 114d 129d
1lOd 115d 12Od 135d
108d 120d 129d 145d
102c 132~ 150d 177d
exp
c( 10-B/aK
5.74 9.14 11.20 14.37
4.61 8.92 10.90 13.94
3.95 7.69 9.07 11.59
2.27 4.94 5.96 7.85
1.26 3.20 3.97 5.49
camp
K,
Table 3. Comparison of computed and experimental thermal data (298alS’K)
(42)a (43)b
4.25b 5.49a 6.30a P78a
3*61a 6*18a 7.25a 9*0Sa
3016a 5.51a 6eSla 8.21a
2.06a 4*00a 4*86a 6*28a
3.16a 3*91a 5.30b
1.44a
exp
%
34.3
+35-o + 66.6 +77*8 +84*7
+27*7 +444 +50.3 + 54.0
+25*0 +39*6 +39*3 +41*2
+10*2 f23.5 +22*6 +25*0
-12-s +1*3 +1-s +3*6
10 -6/bar
1.0536 1.0614 1.0615 1.0649
1.0495 1.0564 1.0580 1.0613
1.0462 1.0539 1.0566 1.0607
1.0398 1.0487 1.0521 1.0572
1.0352 1.0462 1.0498 1.0539
camp
GG
b;
2 _
* ;
*d
A BOX 6. 7. 8. 9. 10. 11.
12. 13. 14. 15.
MODEL
OF ALKALI
CUBICCIOTTI
D., J. &em. Phys. 31, 1646 (1959); 33. 1579 (19601: 34. 2189 (1961). Tosr M., &Ed &. ihys. 18, 1 {1964); J. Phys. Chem. Solids 24, 965 (1963). VARSI-INI Y. P. and SHUKLA R. C., J. them. Phys. 35. 582 (1961). CHA&RJ& S., kzdian J. Phys. 37, 105 (1963). SHARMAM. N. and MADANM. P.. Indian J. Phvs. 38, 231 (1964). KACHHAVA C. M. and SAXENAS. C., Phil. Mug. 8, 1429 (1963) ; Molec. Phys. 7, 465 (1964); Indian J. pure appl. Phys. 2, 336 (1964); Indian J. Phys. 38, 388 (1964); 39, 145 (1965). KARO A. M., J. them. Phys. 31, 1489 (1959). HARDY J. R., Phil. Mug. 7, 315 (1962). KARO A. M. and HARDY J. R., Phys. Rev. 129, 2024 (1963). WOODS A. D. B. et al., Phys. Rev. 119, 980 (1960);
131, 1025, 1030 (1963). 16. VERMA M. P. and DAYAL B., Phys. Status
HALIDE
CRYSTALS
2059
40. WEYL W. A., Of&e of Naval Res. Techn. Repts. 64-66, Pennsylvania (1955). 41. RYMBR T. B. and HAMBLINGP. G., Acta crystallogr. 4, 56.5 (1951). 42. HAUS~UHL S., 2. Phys. 159, 223 (1960); Acta crystallogr. 13, 685 (1960). 43. SPANGENBERG K., Naturwissenschaften 43, 394 (1956); SPANGW~BERG K. and HAUSSUHL S., 2. Kristallogr. 109, 422 (1957).
APPENDIX: COMPUTING SCHEME The main part of the ALGOL program consists of the statistical procedure BOLTZ(S,
B) =
(26) -+ (8,9) + (7)
+ (10,12) + (29,30).
Solidi 3,
(32)
901 (1963).
17. HOVI V. and PAUTAMO Y., AnnIs Acad. Sci. fem. A 6 68 (1960); 84 (1961). 18. PAUTAMOY., Annls Acad. Sci. fem. A 6,129 (1963). 19. ARENSTEINM., HATCHER R. D. and NEUBERGERJ., Phys.
Rev.
132,
73 (1963).
J. N., Phys. I& i23, 1172 (1961). 20. PL&L 21. RAMAN C. V., Proc. Indian Acad. Sci. A 56, 1 (1962). Fys. 35 A, 9, 30 22. L&DIN P. O., Ark. Mat. Astr.
The first Z value in (26) was obtained from the input data using (31), and the procedure (32) was then applied twice as follows: BOLTZ(&,
B,)
-+ G,;
BoLTz(S~,
B,) --f G,;
dm(31);
B, = & +(GJIG,I)-AB (33)
d&31).
(1947); Thesis, Uppsala 1948: Phys. Rew. 90. 120 (1953j; Adw. khy;.-5, 1 (1956). . 23. FREEMANA. J. and L~~WDINP. O., Phys. Rev. 111,
New values for B, do and G were found the basis of (33), according to
1212 (1958). 24. FR~MAN A. and L&WIN P. O., J. Phys. Chem. Solids
B = &
23, 75 (1962). 2.5. LUNDQVIST S. O.,
Ark.
Fys.
9, 435 (1955);
BOLTZ --f
26. MANSIKKA K., Quant. Chern. Groups 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
(1960). 38. LANDOLT-BERNSTEIN II 4, Kalorische grijssen (1961). 39. U.S. Bur. Stand. Circ. 500 (1952).
Zustands-
- B,)I(G,
do = do, - G,(do,
12,
263 (1957). Res. Rept. 107, Uppsala 1963; MANSIKKA K. and BYSTRAND F., Ibid. 149, Uppsala (1965). MITSKEVICH V. V., Sowiet Phys. solid St. 3, 2202 (1962). BAZAROVI. P., Physica 28, 479 (1962). ODEN L., HENDERSOND. and COLEMAN J., PYOC. Natn. Acad. Sci. U.S.A. 51, 629 (1964). ROSENSTOCKH. B., Am. J. Phys. 30, 38 (1962). SYSI~ P., Annls Acad. Sci. f&. A 6, 110 (1962). Nat1 Bur. Stand.. Tables of Spherical Bessel Functions II (1947). Handb. Chem. Phys. 43, Chem. Rubber Publ. Co., Cleveland 1961. HUANG K., B.I. Hochschultaschenbiicher 68-70, Mannheim 1964. Natl. Bur. Stand. Circ. 539 (1953-57). HIBTALA J., Annls Acad. Sci. fenn. A 6, 121 (1963). HOVI V. and MANSIKKA K.. Ann. Univ. Turhu A 40
- G,(B,
Thereafter
linearly on
- G,);
- do,)l(G,
- G,);’
G.
(34)
the iteration sequence
Bz-fB1; B+Ba;
&+4x; d,, -+doz;
Gz-+G G-tG,
*
(34)
1
(35)
was performed until 1G1l +I Gsl < 10m4 and GJGs -< 0. Near the point G(S, B) = 0 the sequence (35) was replaced by I--
-+
B = (B, + BJj2 --f BOLTZ --+ GG/G, > 0 else
B+B, G-tG,
1I 1
+
(36)
which was terminated by using the criterion 1GJ < 10 -B
or Il--B1/Bsl
< lo-‘.
The whole process (32-36) was repeated using the succeeding values 1z, n+l, etc., until the sign of A (30) was changed; the cycle having the smallest [Al was taken to be the correct one (30).
Pergamon
Press 1967. Vol. 28, pp. 2053-2059.
Printed in Great Britain.
A BOX MODEL OF ALKALI HALIDE CRYSTALS P. A Department
of Physics,
(Received 6 February
SYSIU
University
1967;
in rtied
of Helsinki,
Finland
form 21 April 1967)
Ahatract-The model introduced in author’s first paper [An& Acad. Sci. fem. A 6, 110 (1962)J has been developed further, both theoretically and technically. It is based on the assumption that each ion is moving in its own, practically impenetrable sphere (or cube) in a constant coulombic potential. Then the eigenvalues of the Hamiltonian are easily found and the only experimental crystal parameter needed is the lattice constant d. Validity of the theory has been tested by comparing the computed values of the binding energy E, the molar heat C,, the coefficient of volume expansion a, and the adiabatic compressibility K, to the experimental ones; the mean (per cent) deviations are, respectively, 4.4 per cent, 5.1 per cent, 16.8 per cent and 34.3 per cent. The results are rather promising, so that this model seems to be worthy of further research.
INTRODUCTION
A GREAT deal of the theoretical research conceming thermal and elastic properties of crystalline solids, especially alkali halides, is based on the following methods : 1. Static potentials Most common are the classical potential functions of Born-Huang and Bom-Mayer.(1-7) In recent years also other types of potentials have been studied quite extensively.@-11) 2. Frequency analysis Basic examples are the well-known models of EINSTEIN, DEBYE, and BORN and VON KARMAN.‘~-5) Modern computing techniques have been used very successfully in this hind of work.(12-1g) Recently, PLENDLhas introduced an interesting mean-frequency method and RAMAN a promising normal mode system.(al) 3. Wave function synthesis Usually, the wave function for the whole crystal is calculated in the Heitler-London approximation with Hartree-Fock orbitals (LijwDIN et aZ.,(22-24) LUNDQVIST,(~~) ~QNSIKKA et al.(26) Very satisfactory results are given by a dynamical theory based on a tightly-bound electron approximation (MITSKJWICH(~~)). 2053
4. Solvable Hamihtaians Using creation and destruction operators, BAZAROV(~~)has developed a Hamiltonian to be applied to periodic crystal eigenfunctions. ODEN et aLcag) have used the energy levels of a particle in a box in connection with hard sphere statistics, and the specific heat of such particles has been calculated by ROSENSTOCK. Nevertheless, it seems that the box model has been applied to crystals only by the present author,c31) to whom Professor R. Niini suggested this topic in 1959. The present paper reports on refinements of the model and further numerical calculations, now concerning all alkali halides. An acceptable crystal theory must be capable of yielding the measurable quantities of a crystal also numerically. The static potential method above will give any thermodynamic quantity of a crystal, once the potential parameters are evaluated from the equilibrium conditions and experimental (thermal or elastic) data. If, especially, the molar heat and its temperature dependence are desired, frequency analysis is the proper method; it can be done on the basis of empirical elastic coefficients, or more directly from infrared measurements. Further, the method of wave function synthesis can predict purely theoretically any measurable crystal characteristic, including the lattice constant d. However, the procedure is
2054
P. A. SYSId
quite cumbersome; it involves several approximations, and has been applied numerically only to a few crystals. In the category of solvable Hamiltonians merely the box model is comparable to the preceding methods. Starting numerically from the experimental lattice constant, it yields other measurable crystal characteristics purely theoretically, most values with satisfactory accuracy despite the primitive nature of the model.
The quantum-mechanical treatment of the particle-in-box problem is a well-known procedure. Thus, without going into details, we shall merely take the energy needed in the following. In order to save computing time, the box has been taken to be spherical instead of cubic; in any case the shape of the box is not essential, as was shown in Ref. 31. For an ion pair the spherical-box energy levels are E
THE MODEL An alkali halide crystal is regarded here as simply as possible. The most prominent feature is the stable, almost incompressible cubic lattice with very low self-diffusion.(5) Thus, the crystal looks as though it were composed of rather hard ionic spheres, each bound to its own potential well. Further, the interaction of two ions consists, in first approximation, of the pure Coulomb attraction and an inverse-power or exponential repulsion.(1-5) Starting from a two-ion potential function like u(rl,) = of:l/rij+ l/~,~~,(~) the potential wells in a linear chain, qualitatively also in a cubic lattice, are seen to be approximately parabolic (harmonic forces). However, if the atomic nucleus together with the innermost electron shells is assumed to move relative to the outer shells (q.v. and Table 2), the potential is constant until overlapping of the shells causes a strong repulsion. The combined potential will thus be more square-well-like, and we can assume (see also Ref. 31) that
I_
SAe2 --) 4rc,d
where S = screening Madelung’s constant.
factor
fi2iln12+~2in 22 SAe2 2mIu12
2m2u22
--9
4q,d
(3)
where jl, 1 2 is a root of the equation J,, r ,2 = 0.(32) At equilibrium dE = 0, so that differentiating E adiabatically, i.e. jln 1.,2 = constant, with respect to the independent varrables a,, u2 and S = S(B) we get, with the aid of (2), u1.2 = and
hc,d2Tz2 jln 1,22 113 pmBSAe2 rnIs2
dS
S(u, +
-=
a2)
d
dB
(4)
(5)
’
The energy expression (3) can be simplified by multiplying its two first terms above and below by ur and u2, respectively. Then by denoting rr+r,
= do
and
B(u, +a,)
and taking (2) into account insertion (3) yields SAe2 1+2/2 E, = - -.p* 4mc,d, (1 +.z)~
= xd,
(6)
of (4) into
(7)
Similarly for z, using (2), (4) and (6), we obtain the third degree equation
The mass centre of each ion moves in a box with practically impenetrable walls and with constant coulombic potential UC
=
(1)
+(j,, 22/m2)1’3](1 +z)“’ and
s
A =
The ionic distance is d = r,+r, + B(a, + a,), where rl and Y, are the ionic radii, a, and a2 are the box dimensions and B is the coupling coefficient. The physical meaning of the parameters B will be discussed later.
I
x,(1 +x)2/3,
(8)
which is easily solved(33) to give x = xo3/3 +zo[(3+xo3/3 (2)
1 S and
+xs6/27
+ () +.~,,~/27)~‘~)~‘~ + (+ + xo3/3 +z,‘/27 -(t+,~~~/27)~‘~)~‘~].
(9) In principle, the problem is now solved except for the values of S and B. In order to evaluate
A BOX
MODEL
OF ALKALI
them, as well as for numerical applications, have to introduce next some aerodynamic.
We
HALIDE
2055
CRYSTALS
of U and V(34); this yields KS = - (aV~~p)~f V = (a~laT)~/ x [(aV/aT),(aav/aT2)s-(aU/aT),(a2V/aT2),J.
STATISTICAL QUANTITW
In the case of a large enough crystal, surface effects can be neglected. According to the model under discussion the crystal consists of a great number of independent boxes with mutually orthogonal spherical eigenfunctions. Those boxes can be recombined pair by pair, and thus, for a statistical quantity elr and for its temperature derivative, we find the expressions(34)
P), The thermal process in (15) is taken aa occurring in two steps. First, the crystal is compressed adiabatically at constant pressure without any change in the energy states. Secondly, the energy distribution is changed adiabatically at constant volume. Then, using (ll-13), we can insert the isochorie and isobaric derivatives of U and V into (16), which reduces to
where f, = (ZI, -!-1)(2&-i- l), and
SjaT
= (a?$-$)(~ -a)/kTa.
(11)
Both (11) and subsequent thermodynamic expressions can be simplified considerably by introducing an operator .D such that DPy”
n? (x - $P(y
-jQ”,
(121
where (pn, 4t B 0 and m+ n > 2). The two basic interrelations between the macroand the microscopic thermodynamic scopic quantities are lattice energy to binding energy, and molar volume to cell volume, namely, u=
-NJ!?
Y = gNP
to (13-15)
A = - (~~3J~~2
for x,
(191
and (17,181,
- lIEa V/~~V}/kT.
(20)
All the statistical expressions derived in the preceding section for the box system (7) are valid generally for any ensemble obeying the Maxwell-Boltzmann distribution law (10).(34)
(l&-13),
C, = ~~~~~T~~ = ~~Ez~kT2.
a = (WJ8T)JV
equation
(18)
(13)
where N = Avogadro’s number and g = 2 for NaCl-structure = (S/9)1/3 for CsCl-structure.
of volume
degree
x = &+(fi-A-)“2 where, according
= g?z&
Similarly, the coefficient takes the form’34)
x = CPjCV = &&/KS = 1 f VTa2&,rc,, This leads to a second which is solved to give
(&’ < 0) and
Thus, on the basis of equations molar heat C, will bef3*)
The theoretical isochoric molar heat C, (14) should be transformed into the isobaric value C,, if comparison with experimental C, data is required. This ~~sformation can be made starting from the general thermodynamic relations,f34)
the
(141
expansion
= DEV]kT=t?
a
(15)
In order to obtain an expression for tbe adiabatic compressibility Iir,, the derivative (aI’/@), must be transformed into the temperature derivatives
Ph3AMZTERSSAM)B An oversimplified picture of a physical
system can often be refined effectively by choosing suitable parameters. For example, in this model the coupling coefficient B in (2) takes into account the fact that the nucleus and its electronic shells can have different motions, because of their elastic intercoupling. Then also the effective charge of an ion as seen by ita neighbou~ will differ from unity. This shielding effect has been included explicitly in the theory by inserting the
2056
P. A. SYSId
screening factor S = S(B) into the coulombic potential in (1). On the basis of the equilibrium condition dE = 0, we found for S and B the differential equation (S), which transforms through (2) and (6) into dS S z -= dB >‘l+z’ (21) If equation
(8) is written
as
z = q(B2’3/S1’3)(1 +x)~‘~, substitution
(22)
into (21) and integration S1’3 = B2’“(4/2)/(l + integration
yield
C,
i.e. M 0~)
S = 0 B = 0,
(23)
B=l s = 1,
i.e.
no shielding
(29)
Thus, values of the parameters S and B are implicitly known on the basis of (26) and (29); in (26) the quantized z has been replaced by its mean value 5. In principle S and B are also quantized, but numerically we can find only a kind of statistical average for them.
completely E,,, have been numbered according to Table 1 by arranging first the quantum numbers jln(32) into increasing order of magnitude. Because the Table 1. { triangle
(24)
we find C = 0, so that, for a constant q value, according to (22) and (23), z does not depend on S and B. Then (21) can be integrated into account the boundary ,directly, taking conditions valid for any q or z value rigid coupling
= 0.
NUMERICAL PROCEDURE
when 4 and (1 + z) have been treated as constants. Introducing the “ideal gas boundary condition”, potential M 0 box dimensions
G(S, B) = 2-(l/kT)(DE3/DE2)
The energy levels (7), E, or more
+FZ)~‘~
constant
The derivative of E in that direction is simply dEJdS = EjS and, on the basis of (12, 14, 27), we get the extremum condition
11 12 13...
lfl
21 22.. . 2n 31...
3n
... (25)
?I1
whence we obtain S = B”/(l+“).
(26)
In Ref. 31 it was shown that the minimum of E corresponds to the maximum of C,. In order to find this extremum on the curve (26), i.e. to evaluate S and B, we shall need partial derivatives of the type aEZ
aw
Dw
aE
av
av
kT
av
-=
---*--,
(27)
where v stands for S or B and zZis some statistical quantity (10,12). A useful zero indicator for finding C,,,,,
turns
out to be the derivative d In C, G(S,B)=dln=--‘taken in the direction
S
dC,
C,
dS
dS/dB = 2SIB.
(28)
actual height of the box wall is finite, each box has a limited energy level number n determined here by the stability condition phonon energy kT2
I@-E)-(U,,-E,,)+kTj
mean free box energy i.e.
= [Al = min.
(30)
However, the change in energy values (7) due to the finiteness of the wall height is assumed to be insignificant. The computer program was written in ALGOL language (for an ELLIOTT 503), see Appendix. As well as the general physical constants(33) and quantum numbers jln, (32) the data input consists of the ionic masses,(33) the experimental lattice constant d,(18-35-36) and of the “guessed” starting values of B, do and n. Thus, the only experimental crystal characteristic needed in this model is d,
A
BOX
MODEL
OF
ALKALI
HALIDE
2057
CRYSTALS
Table 2. Model characteristics f
S
B
n
0*0503 0.0447 0.0379 0.0347
O-02104 0.02709 0.02982 0.03329
o-9499 0.9515 0.9558 0.9597
0.0825 0.1518 0*2102 0.2786
23
0.0054 0.0079 0.0089 0*0105
0.0370 0.0321 0.0266 0.0243
O-02386 0.02891 0.03084 0~03355
0.9561 0.9615 0.9679 0.9732
0.1457 0.2471 0.3358 0.4328
f: 28
0.2601 0.3049 0.3191 0.3410
0.0072 0.0098 0.0107 0.0123
0.0328 0.0281 0.0231 0.0208
0.02757 0.03206 0.03348 0.03601
0.9601 0.9677 0.9753 0*9819
0.2189 0.3478 0.4617 0*5905
23 23 24 25
0.2815 0.3290 0.3444 0.3671
0.2735 0.3183 0.3329 0.3541
O*OOSO 0.0107 0~0115 0.0130
0.0276 0.0233 0.0189 0.0168
0.02928 0.03357 O-03461 O-03680
0.9654 0.9750 0.9836 0.9910
0.2897 0.4591 0.6106 0.7760
26 24 23 23
0.3004 0.3571 0.3720 0.3956
0.2912 0.3447 0.3587 0.3808
0.0092 0.0124 0.0133 0.0148
0.0254 0.0213 0.0169 0*0151
0.03144 O-03612 0.03703 O-03897
0.9694 0*9815 0.9915 0.9995
0.3604 0.5856 0.7874 0.9860
30 26 23 23
d
do
d-do
-&+Q
LiF Cl Br I
0.2014 0.2570 0.2751 0*3000
0.1973 0.2502 0.2671 0.2903
0.0041 0.0068 0.0080 0.0097
NaF Cl Br I
0.2317 0.2820 0.2989 0.3237
0.2263 0.2741 0.2900 0.3132
KF Cl Br I
0.2673 0.3147 0.3298 0.3533
RbF Cl Br I CsF Cl Br I
which determines the ionic radii sum do, according to (2) and (6), or directly d,, = d/(1 +Z),
i.e.
d’ = d.
22
2. Boundary conditions (24) and (25) for S and B. 3. Dependence of box energy on wall height.
(31)
Values of the lattice constant d(1s*35*38)and some computed characteristics of the model are listed in Table 2 (lengths in nm = 10eQm; T = 298.15”K). Table 3 presents the computed and experimental values of E, C,, a and K,, together with their percentage differences. The computed values of the ratio C,/C,, are also listed in the Table 3.
author would like to express his gratitude to Professor R. NIINI for having suggested the basic idea behind this work, and for many fruitful discussions; to Drs. 0. INKINEN, K. KURKI-SUONIO and P. LIPAS as well as Mr. E. PAJANNEfor their helpful advice ; to Mr. E. PELTOLA for consultations and arrangements concerning the computing work; to Mrs. LORNA SUNDBTR~M for revising the English manuscript; and to The Finnish National Research Council for Sciences for financial support.
Acknowledgements-The
SUMMARY
The main aim of this work has been to determine whether the results of this rather crude model indicate the usefulness of further research into its capabilities. The results in Table 3 are fairly encouraging. In order to help future investigations, the weakest points of the theory are listed in their order of importance as follows: 1. Limiting criterion (30) for n.
E 42
REFERENCES 1. BORN M.
2. 3. 4. 5.
and HUANG K., Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford (1954). Dmuzzm A. J., Solid State Physics, Macmillan, London (1962). KITTEL C., Introduction to Solid State Physics, Wiley, New York (1961). SBITZ F.. Tke Modern T&my of Sol&a?, McGrawHill, New York (1940). ZHDANOV G. S., Crystal Physics, Oliver & Boyd, Edinburgh (1965).
10.26 8.46 8.03 744
8.92 7.62 7.32 6.87
8.51 7.34 7.07 6.67
8.00 6.85 6.64 6.29
NaF Cl Br I
KF Cl Br I
RbF Cl Br I
CsF Cl Br I
(37)
7.53 6.75 6.49 6.18 4.4
+6*2 +1.5 +2*3 +1*8
+5.3 +3.8 +4*3 +3.6
+5*8 +3.8 $4.3 $3.3
8.43 7.34 7.02 6.65
8.08 7.07 6.78 6.44
$8.2 $4.8 +4.6 $2.9
$11.5 $4.7 $3.7 +1.8
IO.54 8.78 8.31 7.78
9.48 8.07 7.68 7.23
%
exp
+ No recent data available.
Ref.
Mean I%1
11.75 9.19 8.62 7.92
LiF Cl Br I
camp
E eV
41.9 50.3 51.9 54.7 46.8 50.8 52.3 54.3 49.0 51.5 52.5 53.2 50.7 51.2 51.7 51.8 50.7 52.6 51.8 51.9
50.9 50.8 49.0 48.0 52.4 53.2 52.1 51.4 52.4 53.9 53.6 53.0 51.3 53.2 54.2 54.1 50.6 52.9 53.9 54.6
(38)
ew
camp %
5.1
-0.2 +0*6 +4*1 +5.2
+1*2 +3.9 +4.8 +4*4
+6*9 $4.7 +2.1 -0.4
+ 12.0 ++7 -0.4 -5.3
+21*5 +1*0 -5.6 - 12.2
C, J/m01 “K
126 153 161 172
121 145 153 161
118 141 146 153
103 126 130 138
88 111 114 121
camp
4.6 +5.0 +0+8 -48
g;
e
16.8
+10*9 + 14.2 + 17.0
* 138b 141b 147e
Ma (36)b (39)c
+18*6 +34.3 +342 +24*8
+7*3 +22*6 + 21.7 +13*3
-
-13.7 -15.9 -24.0 -31.6
%
102a 108d 114d 129d
1lOd 115d 12Od 135d
108d 120d 129d 145d
102c 132~ 150d 177d
exp
c( 10-B/aK
5.74 9.14 11.20 14.37
4.61 8.92 10.90 13.94
3.95 7.69 9.07 11.59
2.27 4.94 5.96 7.85
1.26 3.20 3.97 5.49
camp
K,
Table 3. Comparison of computed and experimental thermal data (298alS’K)
(42)a (43)b
4.25b 5.49a 6.30a P78a
3*61a 6*18a 7.25a 9*0Sa
3016a 5.51a 6eSla 8.21a
2.06a 4*00a 4*86a 6*28a
3.16a 3*91a 5.30b
1.44a
exp
%
34.3
+35-o + 66.6 +77*8 +84*7
+27*7 +444 +50.3 + 54.0
+25*0 +39*6 +39*3 +41*2
+10*2 f23.5 +22*6 +25*0
-12-s +1*3 +1-s +3*6
10 -6/bar
1.0536 1.0614 1.0615 1.0649
1.0495 1.0564 1.0580 1.0613
1.0462 1.0539 1.0566 1.0607
1.0398 1.0487 1.0521 1.0572
1.0352 1.0462 1.0498 1.0539
camp
GG
b;
2 _
* ;
*d
A BOX 6. 7. 8. 9. 10. 11.
12. 13. 14. 15.
MODEL
OF ALKALI
CUBICCIOTTI
D., J. &em. Phys. 31, 1646 (1959); 33. 1579 (19601: 34. 2189 (1961). Tosr M., &Ed &. ihys. 18, 1 {1964); J. Phys. Chem. Solids 24, 965 (1963). VARSI-INI Y. P. and SHUKLA R. C., J. them. Phys. 35. 582 (1961). CHA&RJ& S., kzdian J. Phys. 37, 105 (1963). SHARMAM. N. and MADANM. P.. Indian J. Phvs. 38, 231 (1964). KACHHAVA C. M. and SAXENAS. C., Phil. Mug. 8, 1429 (1963) ; Molec. Phys. 7, 465 (1964); Indian J. pure appl. Phys. 2, 336 (1964); Indian J. Phys. 38, 388 (1964); 39, 145 (1965). KARO A. M., J. them. Phys. 31, 1489 (1959). HARDY J. R., Phil. Mug. 7, 315 (1962). KARO A. M. and HARDY J. R., Phys. Rev. 129, 2024 (1963). WOODS A. D. B. et al., Phys. Rev. 119, 980 (1960);
131, 1025, 1030 (1963). 16. VERMA M. P. and DAYAL B., Phys. Status
HALIDE
CRYSTALS
2059
40. WEYL W. A., Of&e of Naval Res. Techn. Repts. 64-66, Pennsylvania (1955). 41. RYMBR T. B. and HAMBLINGP. G., Acta crystallogr. 4, 56.5 (1951). 42. HAUS~UHL S., 2. Phys. 159, 223 (1960); Acta crystallogr. 13, 685 (1960). 43. SPANGENBERG K., Naturwissenschaften 43, 394 (1956); SPANGW~BERG K. and HAUSSUHL S., 2. Kristallogr. 109, 422 (1957).
APPENDIX: COMPUTING SCHEME The main part of the ALGOL program consists of the statistical procedure BOLTZ(S,
B) =
(26) -+ (8,9) + (7)
+ (10,12) + (29,30).
Solidi 3,
(32)
901 (1963).
17. HOVI V. and PAUTAMO Y., AnnIs Acad. Sci. fem. A 6 68 (1960); 84 (1961). 18. PAUTAMOY., Annls Acad. Sci. fem. A 6,129 (1963). 19. ARENSTEINM., HATCHER R. D. and NEUBERGERJ., Phys.
Rev.
132,
73 (1963).
J. N., Phys. I& i23, 1172 (1961). 20. PL&L 21. RAMAN C. V., Proc. Indian Acad. Sci. A 56, 1 (1962). Fys. 35 A, 9, 30 22. L&DIN P. O., Ark. Mat. Astr.
The first Z value in (26) was obtained from the input data using (31), and the procedure (32) was then applied twice as follows: BOLTZ(&,
B,)
-+ G,;
BoLTz(S~,
B,) --f G,;
dm(31);
B, = & +(GJIG,I)-AB (33)
d&31).
(1947); Thesis, Uppsala 1948: Phys. Rew. 90. 120 (1953j; Adw. khy;.-5, 1 (1956). . 23. FREEMANA. J. and L~~WDINP. O., Phys. Rev. 111,
New values for B, do and G were found the basis of (33), according to
1212 (1958). 24. FR~MAN A. and L&WIN P. O., J. Phys. Chem. Solids
B = &
23, 75 (1962). 2.5. LUNDQVIST S. O.,
Ark.
Fys.
9, 435 (1955);
BOLTZ --f
26. MANSIKKA K., Quant. Chern. Groups 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
(1960). 38. LANDOLT-BERNSTEIN II 4, Kalorische grijssen (1961). 39. U.S. Bur. Stand. Circ. 500 (1952).
Zustands-
- B,)I(G,
do = do, - G,(do,
12,
263 (1957). Res. Rept. 107, Uppsala 1963; MANSIKKA K. and BYSTRAND F., Ibid. 149, Uppsala (1965). MITSKEVICH V. V., Sowiet Phys. solid St. 3, 2202 (1962). BAZAROVI. P., Physica 28, 479 (1962). ODEN L., HENDERSOND. and COLEMAN J., PYOC. Natn. Acad. Sci. U.S.A. 51, 629 (1964). ROSENSTOCKH. B., Am. J. Phys. 30, 38 (1962). SYSI~ P., Annls Acad. Sci. f&. A 6, 110 (1962). Nat1 Bur. Stand.. Tables of Spherical Bessel Functions II (1947). Handb. Chem. Phys. 43, Chem. Rubber Publ. Co., Cleveland 1961. HUANG K., B.I. Hochschultaschenbiicher 68-70, Mannheim 1964. Natl. Bur. Stand. Circ. 539 (1953-57). HIBTALA J., Annls Acad. Sci. fenn. A 6, 121 (1963). HOVI V. and MANSIKKA K.. Ann. Univ. Turhu A 40
- G,(B,
Thereafter
linearly on
- G,);
- do,)l(G,
- G,);’
G.
(34)
the iteration sequence
Bz-fB1; B+Ba;
&+4x; d,, -+doz;
Gz-+G G-tG,
*
(34)
1
(35)
was performed until 1G1l +I Gsl < 10m4 and GJGs -< 0. Near the point G(S, B) = 0 the sequence (35) was replaced by I--
-+
B = (B, + BJj2 --f BOLTZ --+ GG/G, > 0 else
B+B, G-tG,
1I 1
+
(36)
which was terminated by using the criterion 1GJ < 10 -B
or Il--B1/Bsl
< lo-‘.
The whole process (32-36) was repeated using the succeeding values 1z, n+l, etc., until the sign of A (30) was changed; the cycle having the smallest [Al was taken to be the correct one (30).