ATOMISTIC CALCULATIONS ON EDGE DISLOCATIONS IN IONIC CRYSTALS OF ROCK SALT STRUCTURE

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ATOMISTIC CALCULATIONS ON EDGE

DISLOCATIONS IN IONIC CRYSTALS OF ROCK SALT STRUCTURE

F. Granzer, V. Belzner, M. Bücher, P. Petrasch, C. Teodosiu

To cite this version:

F. Granzer, V. Belzner, M. Bücher, P. Petrasch, C. Teodosiu. ATOMISTIC CALCULATIONS ON

EDGE DISLOCATIONS IN IONIC CRYSTALS OF ROCK SALT STRUCTURE. Journal de Physique

Colloques, 1973, 34 (C9), pp.C9-359-C9-365. �10.1051/jphyscol:1973962�. �jpa-00215437�

J O U R N A L D E PHYSIQUE Colloque C9, supp/ii?~er?t au

11-12, Tome 34, Nouei?lbre-Dkceinbre 1973, page (29-359

ATOMISTIC CALCULATIONS ON EDGE DISLOCATIONS IN IONIC CRYSTALS OF ROCK SALT STRUCTURE

F. G R A N Z E R , V. B E L Z N E R , M. BUCHER a n d P. P E T R A S C H Institut fiir Angewandte Physik, Universitat Frankfurt/Main

a n d C. T E O D O S I U

MPI fiir Metallforschung, Stuttgart

R6sum6. - Nous avons calcult la configuration des ions au cceur des dislocations coin dans les cristaux ioniques du type NaCl utilisant, pour la premiere fois, la theorie elastique de troisitme ordre et des potentiels d'interaction amCliores.

Pour obtenir une transition continue entre la region exterieure, traitee comme un continuum Clastique, et la region interieure (le cceur de la dislocation), nous avons adapte tous les parametres qui se presentent dans les divers terrnes des potentiels d'interaction aux constantes ilastiques de 2e et 3 e ordre. De cette faqon, on peut Cviter la formation d'une pseudo-configuration dans le cceur de la dislocation.

Abstract. - F o r the first time atomistic calculations of the core configuration of edge dislo- cations in NaCI-type crystals have been carried out using nonlinear elasticity theory together with improved interaction potentials.

Fitting the parameters appearing in the different potential terms to the elastic constants of second and third order a smooth transition between the elastically and atomistically calculated regions is obtained thus avoiding artifacts in the core configuration.

Introduction. - Some important properties of dislocations, like the core-energy, the Peierls-stress, t h e structures a n d energies of kinks a n d jogs, the tendency o f a complete dislocation t o dissociate into partials a s well a s the interaction o f dislocations with point defects depend o n the atomic configuration o f the core of the dislocation. In this region elastic c o n t i n u u m theory breaks down and the atomic posi- tions must be calculated o n the basis of a discrete atomistic model.

S o f a r the usual procedure in such calculations has been t o divide t h e crystal into two regions (Fig. I), a n inner cylindrical region with the dislocation line as axis - the core region

a n d an outer region - the rest o f t h e crystal

where the atomic positions a r e given by well known formulas of linear elasticity theory. Using boundary conditions

supplied by the latter - a s wcll a s appropi-iate interaction potentials the equilibrium positions of the a t o m s in the core region may be calculated by minimizing the interaction energy in this region which then automatically yields the core-energy o f the dislocation.

Such calcul;~tions have been performed so far mainly f o r metals and t o a ccrtuin extent also for semi- conductors. rare gas and ionic crystals. In the following only edge (lisloc:itions

ionic crystals with

salt structure \vill bc consider.etl iind thc progress ;~cliieved in this field by the Fl-;11lcl'ort-gl-oup i n c e the first publication by I-luntington

01. [I]. ['I ^{will be}

shortly summarizrd.

FIG. I .

Division of the crystal into an outer region (hatched), where continuum elasticity is applicable, and an inner region (dots), whcre the discrete nature of the crystal must be taken

into account.

I . Problems concerning the determination of a true (realistic) core radius. - In the early work o f Huntington ot (11. [lo(,. (.it.], and Kurosawa [3] o n atomistic calctllations of' core-energies a n d Peierls- stresses of disloc:~tions in alkali halides high speed computers bci ng not yet avi~il;lule, the radius of the core had t o be restricrcd to onl:: some Angstroms corresponding ro about 20 lattice ro\ils in the core regioil. Latcl- Granzer

(11. [4), starting with

radius

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973962

C9-360 F. GRANZER, V. BELZNER, M. B ~ ~ C H E R , P. PETRASCH AND C. TEODOSIU

of ten Burgers vectors (corresponding to 640 rows) have shown by physical arguments, that the region where linear elasticity theory is no more applicable

i. e. the true core region -extends to 3b from the dislocation line, containing 56 rows. But there still remains some difficulty in joining smoothly the discrete core region to its continuum elastic surrounding, when the latter has been treated on the basis of linear elasticity theory. The ionic positions given by linear isotropic or anisotropic elasticity theory, which itself is based on a kar177onic interaction potential, abruptly change at the boundary of the core into ionic positions which have been computed using anliar~nonic interac- tion potentials (see part 2 !).

This situation is clearly demonstrated in figure 2, where the initial elastic

ionic configuration at the

FIG. 2. - Relaxation of a

linear elastic

ionic configuration (dotted circles) into a configuration based on an anharmonic interaction potential (f~111 circles) around the center of a < 1 l o > ,

{1 10)-edge dislocation.

center of a < ¹¹⁰ >, ^{( l} 10 ).-edge dislocation (dotted circles) has relaxed (full circles) under the influence of realistic anharmonic interionic potentials during the minimization process. The initially existing compres- sion of the crystal lattice above the glide plane decreases when the weak harmonic (parabolic) interaction potential is replaced by a harder anharmonic interionic potential. This effect is schematically sketched in figure 3 where, while an edge dislocation is being formed, an ion j at first is elastically displaced from its initial equilibrium position ( I ) it occupied in the ideal crystal towards the position (2) along a parabola (dotted curve). Under the action of the harder anhar- monic potential ( f u l l curve) the interaction energy between two ions i and j increases from ~ " ( r ; ) to

~ " ( r ; ) . After relaxing the ion j has moved into its final equilibrium position 6, decreasing both the interaction energy from uU(ryj) to U1'(G) and the amount of compression above the glide plane.

FIG. 3.

Change of the eq~~ilibrium distance

of two ions i and j when the harmonic parabolic potential h is replaced

by an anharmonic interaction potential

But even such an ionic rearrangement cannot lead to an overall dilatation of the crystal lattice of about on atomic volume per atom plane as it might be expect- ed from experiments and theoretical considerations.

And that's why the core region as a ~vliole, embedded in a linear elastic continuum (where the expansion below the glide plane is cancelled by the compression above the glide plane) can never show any compression or dilatation at all provided that there are neither voids nor overlapping. Therefore each ionic configu- ration of the dislocation core having been calculated with boundary conditions derived from linear elasticity theory will consist of a narrow dilated region around the dislocation line, which, as shown in figure 4, would

...

& A n n dilatotton below gltde

FIG. 4. - The artificially compressed shell appearing in the outer region of a 10 b-core, when

elastic boundary condi-

tions have been used.

ATOMISTLC CALCULATIONS ON EDGE DISLOCATIONS IN IONIC CRYSTALS C9-361

be compensated by an artificiallj- compressed shell separating the inner core region from the elastic continuum. As repeatedly emphasized by Seeger [ 5 ] ,

[6] such an unrealistic core configuration can be avoid- ed only if, right from the beginning, third orrler elastic terms were incorporated into the continuum elastic treatment of the outer region of the dislocation.

Moreover tlie core radius should be included into the variational procedure minimizing the total energy (atomistically as well as elastically calculated energy) of the dislocation yielding not only the equilibrium positions of the ions but also tlie true core radius.

2. The choice of realistic interaction potentials.

Contrary to metals and semiconductors, where the reliability of interatomic potentials used in atomistic calculations of the dislocation core is still a matter of discussion, the structures and energies of defects in ionic crystals may be handled to a high degree of approximation in the framework of the classical Born-Mayer-model.

In order to obtain the core energy which usually is related to a plane perpendicular to the dislocation line, the interaction energy between ions in this plane and lattice rows parallel to the dislocation must be computed and then summed over all ions in the core region. The interaction energy of an ion with ions of it's own row must not be taken into account because it doesn't change when a dislocation is formed in an ideal crystal.

2 . 1 THE ELECTROSTATIC ENERGY. - Depending on the geometry of tlie dislocation' the lattice rows parallel t o the dislocation line may be occupied by ions with alternating signs or may be charged uni- formly these two possibilities being realized for

< ] l o > , (110) or <110>, (100) edge disloca- tions respectively corresponding to the glide system encountered predotninantly in crystals with rock salt structure.

2 . 1 . 1 T l ~ e < 1 10 > ^{, {} 1 I O)-glic/e system.

Refer- ring t o figure 5 the electrostatic energy uij between an ion i (with charge (li) in the xy-reference plane and

FIG. 5. - Geometry of alternatively charged [001]-rows parallel to the dislocation line.

an alternately charged row intersecting the reference plane at the ion j (with charge qj) at the distance r i j from the ion i is given by :

KO denotes the rapidly convergent McDonald func- tion, a is the distance of nearest neighbours and the summation is over all ions in the j ' t h row.

2 . 1 . 2 Tlie + < ] l o > , (100)-glide system. -

Contrary t o figure 5, where only two different types of ionic rows exist which may be transformed into one another by a simple interchange of charges, there exist now, depending on the signs of the charges and the point of intersection with the reference plane four different types (denoted by j, k, m, n in Fig. 6) of

~n the plonc

m the plane

FIG. ^6. - Geometry of uniformly charged [Olll-rows parallel to the dislocation line.

uniformly charged rows. The electrostatic energy bet- ween an ion with charge q i in the reference plane and a row k or

intersecting the reference plane a t a distance r,,, or rig,, from the ion i is now given by the equation :

The energy between the same ion i and rows j or m,

intersecting the reference plane exactly halfway bet-

ween ions at distances riVj or r i , , from the ion i is

given by :

C9-362 F. GRANZER, V. BELZNER, M. B ~ ~ C H E R , P. PETRASCH AND C. TEODOSIU Due to the slowly convergent logarithmic terms in (2b),

the environment of an ion in the reference plane, where interaction with uniformly charged rows must be taken into account, has to be enlarged considerably thus leading t o laborious numerical calculations.

2 . 2 THE REPULSION ENERGY. - AS demonstrated by Granzer et al. [loc. cit.] whatever type of the repulsion potential used by Born, Mayer, Huggins, Pauling (see review article of Tosi [7] !) to calculate cohesive energies in alkali and silver halides is only of little influence upon the core energy and Peierls stress of edge dislocations in ionic crystals. The results, listed in table 11, were obtained by the combined Born- Mayer-Huggins potential :

I'.

.

rp(ri,jl) = bi. bjl exp ( ^- ?) ^{. (3)}

The distances between the ion i in the reference plane and the Ith ions in rows of type j or k are given by :

ri,jl ⁼ Jr:j + (la)' (see Fig. 5 !) and

ri,k, ⁼ + (I. a 47)' (see Fig. 6 !) respectively for the two different glide systems.

The Pauling-factors

depending on the valence Z and the number N of the outer electrons of the two interacting ions i and j, are collected in table I, together with the repulsion para- meters b,, b- and p for LiF, NaCI, and AgCI.

2 . 3 THE VAN DER WAALS ENERGY.

In ionic crystals consisting of ions with large polarizabilities and in crystal regions where neighbouring ions approach each other up to very short distances as at the center of the dislocation, van der Waals-interaction must be taken into account. The van der Waals- energy of a pair of ions, separated by a distance ri,jl (where the above notations have been used) is given by :

different ion combinations as well as the ionic polari- zabilities m i are listed in table I.

2.4 THE POLARIZATION ENERGY. - From the expressions (1) to (4) it follows, that all potentials used so far were based on two-body central force interac- tions between ion pairs. The cohesion energies of ideal ionic crystals in the Born-Mayer-model were indeed obtained only by a straight forward summation over such pair-potentials. In real crystals however, supplementary contributions to the total energy must be taken into account. The electric field strength, vanishing by symmetry at a lattice site in an ideal crystal, may increase considerably at ionic positions near defects, where the original symmetry is heavily destroyed. The relaxation of the electric field by a polarization of the ions then results in a gain of energy, the so-called polarization energy.

Around an edge dislocation, where the symmetry is violated only in the xy-plane perpendicular to the dislocation line, this energy is given by :

The electric field strength at the position of the ion i with polarizability cci in the xy-reference plane may be expressed by the gradients of one of the electrostatic potentials of lattice rows already used in eq. (1) and (2), which in the case of the < 110> ( 1 10)-glide system yields :

while for the <110> (100)-glide system, depending on the type of lattice rows under consideration, one obtains in analogy to eq. (2a) and (2b) :

1 4 n

i , )

a J2.

and

The van der Waals coefficients ci,j and di,j of the respectively.

8++ 8-- 8 , - by"" pH

rlai cL- (1;-

r - LiF

NaCl

AgCl

ATOMISTIC CALCULATIONS ON EDGE DISLOCATIONS IN IONIC CRYSTALS C9-363

N , gives the number of rows which contribute to the field strength at the position of the reference ion i, K , denotes the first derivative of the McDonald- function KO.

Substituting the expressions (6) and (7) in eq. (5) it becomes evident that the polarization energy contrary t o the preceding energy contributions cannot be obtained by a simple summation over

pairs.

In crystals with small lattice parameters (e. g. LiF), or crystals consisting of ions with large polarizabilities the polarization energy in heavily distorted regions may exceed the other energy terms and some precau- tions are necessary to avoid the crpolarization catastrophe

2 . 5 ENERGY CONTRIBUTIONS RESULTING FROM M A N Y BODY FORCES. - While the appearance of many body interactions in the polarization energy was due to a violation of the symmetry around crystal defects, the failure of the Cauchy relations in crystals with the centrosymmetric rock salt structure clearly indicates that many body non-central forces must be present also in ideal crystals.

Therefore Sarkar and Sengupta [8], [9] in their calculations of elastic constants and phase transitions of various alkali halides completed the two body potentials used in the classical Born-Mayer-model by a term :

where the interaction between two ions i and j depends not on their distance ^I.,,, but on the positions and the species of their common nearest neighbour (cnn) ions ; their specific contributions to the many body energy being expressed by A(1i). The repulsive parameter p is the same as in eq. (3).

The incorporation of many body interaction poten- tials like those of eq. (8) into the total lattice energy of ionic crystals is inevitable if the parameters appearing in the different potential terms have to be fitted to elastic constants which d o not obey the Cauchy relations as observed in LiF and AgCI.

Recently C11owdhu1-y

(11. [lo] using additiollally an interaction potential of type (8) have calculated the formation energy of Scl~ottky defects in alkali halides.

3. The elastic boundary conditions. - I n order to improve the linear elastic (Volterra) PI-escl-iption of the displacement field around nn edge dislocation core 170111ii1ei11. tecllniques on the fundament of third order anisotropic elasticity theory have been applied to calculate more exactly the boundary conditions serving as basis for the tomis is tic treatment of the core region.

The rather complicated mathematical program was supplied by C. T e o d v s i ~ ~ [ I I]. By means of this quite voluminous proced~ire i t is possible to determine exact nonlinear fields relati\.ely near LIP to the dislo- cation linc. Such a ficlcl is composed of the ~vell- known volt err.:^ field. thc third-order-ficld ( a s nien- tioned above. consistirlg of' more tlxtn fifty term\). and

four hyperbolic functions arising from both, the linear as well as the nonlinear parts of the field. The four constants belonging to these functions finally have to be included into the variational procedure of the atomistic core calculations.

The very simplified way through the complicated problems to finally obtain the above mentioned for- mulas is scheduled in the following scheme :

Airy's Stresses Strains Displacements

fct

The formula for the displacements obtained in this way consists of a linear combination of functions of the following kinds :

1 I n r U ( r ) * In J-, - , - .

r r

Figure 7 schematically shows the results of the calcu- lations. From the figure it is clearly t o be recognized

FIG. 7. - The calculated displacements at a distance of about 16 .\ from the dislocation line. Dashed line denotes the Volterra positions of the ions, dotted line the nonlinear contribution on a scale about 5 times larger than the one used for the slightly

enlarged Volterra field.

that the nonlinear terms of the displacement field cause the density to decrease above as well as below the glide plane thus diminishing the compression in the former but fortifying the dilatation in the latter region. That is in good agreement with experimental results expecting the dilatation A V / V , after bringing a dislocation into the crystal, to become one atom volume per plan?.

4. Results.

4 . 1 Colu. lirtl<c;lrs 01. EI)C;E IIISLO-

Cr\l.lOKS O N I ) I I I

I:III_NT

T H I ( I R

ANISOTROPY

IONIC

I t is well known that ionic crystals

though having the same structure, niny dil'fer in the

C9-364 F. G R A N Z E R , V. BELZNER, M . B ~ ~ C H E R . P. PETRASCH A N D C. TEODOSIU glide systems activated under plastic deformation.

Correlating different physical properties to tlie preferential glide system observed in 14 different crystals with rock salt structure Gilman [I21 assumed that the glide plane in NaC1-type crystals seems to be determined solely by the structure of the res- pective dislocation core. To prove this assumption the structures (Fig. 8) and core energies of edge dis-

FIG. 8. - Ionic configuration of edge dislocations in crystals of rock salt structure :

< 1 l o > , ( I 10)-glide system ;

b) < 1 l o > , {loo)-glide system ; hatched circles

ions in

plane r

0 ; open circles

ions in plane z

^&/2.

locations of (100) and {I 10)-planes in LiF, NaCl and AgCl were calculated, the polarization energy (eq. ( 5 ) ) for the first time being included in the total interaction energy of the core. The contri- butions of the different interaction potentials used in these calculations, the total core energies and the ratios of the core energies of dislocations on the two different glide planes are collected in table 11. From tlie last column in table 11, a correlation may be deduced between the ratio of core energies on different glide planes, the observed plastic anisotropy being largest for LiF where, as it is well known, the {I 10)-glide plane clearly dominates and approaching unity for AgCl where pencil glide is observed.

4 . 2 THE INFLUENCE OF IONIC POTENTIALS ON CORE STRUCTURE AND CORE ENERGY. - TO demonstrate the influence of various sets of ionic potentials the structure and the energy of a < 1 l o > , { I 10) edge

Electrostatic (110) (100) - -

Li F 0.368 1.052 0.230 0.658 NaCl 0.270 0.774 0.169 0.484 AgCl 0.365 0.914 0.228 0.571

dislocation were calculated for NaCl and AgCl using for the minimization procedure :

I ) the potentials 2.1-2.4 with parameters given in table I ;

2) the potentials 2.1-2.4 and additionally the many body potential 2.5, the parameters being fitted to the second order elastic constants and to the equilibrium lattice constant.

In both cases the initial ionic configuration has been derived from linear anisotropic elastic theory.

The displacements of some core ions to their final positions after the minimization procedure are indi- cated in figure 9 by dotted arrows when only two body

FIG. 9. - Displacements of ions (enlarged by a factor of 10) during the mi~iimization process for NaCl

- - -

without

I many body potential.

--+

with

central potentials were used and by full arrows when these potentials were supplemented by many body terms.

The incorporation of the latter into the interaction energy of the ions give rise t o a further dilatation of the inner core region.

Lpper line : erg/cm lower Ii~te : lo8 eV/cm Repulsion

(110) (100)

-

0.652 0.609 0.408 0.381 0.726 0.609 0.454 0.381 1.586 1.050 0.913 0.656

Van der Waals (110) (100)

- -

-

0.077 - 0.077

- 0.048

0.048

- 0.065 - 0.080

- 0.041

0.050

- 0.266 - 0.181

- 0.166 - 0.113

Polarization (110) (100)

-

- - 0.187 - 0.058 -0.117 -0.036

- 0.168 - 0.057

- 0.105 - 0.036

- 0.233 - 0.153

- 0.146 - 0.096

Total energy E ( ,

,,,

(110) (100) E ( , , o ,

-

- - 0.756 1.526

0.473 0.954 2.02 0.776 1.245

0.485 0.778 1.60 1.451 1.630

0.907 1.019 1.12

ATOMISTIC CALCULATIONS ON EDGE DISLOCATIONS IN IONIC CRYSTALS C9-365

MAD REP VDW POL MBE Total

- - - - - -

a NaCl - 0.132 1.424 - 0.070 ^- 0.309 - 0.913

NaCl - 0.052 1.512 - 0.064 - 0.265 - 0.065 1.066

AgCI 0.583 1.391 - 0.196 ^- 0.271 - 1.507

AgCl 0.168 1.458 - 1.095 - 0.782 0.849 0.598

b ) NaCl - 0.000 2 1.465 - 0.102 - 0.253 - 0.124 0.986

The core energy (Table llla in eV per plane) is increased by 15 "/ for NaCl and decreased by a factor of 2.5 for AgCI when the many body potential was used and the parameters were fitted to the elastic constants. That the influence of many body terms turns out to be considerably larger in AgCl than in NaCl is in good agreement with the fact that the Cauchy relation, being heavily violated in silver halides, is nearly fulfilled in NaCI. Furthermore the relatively low core energy of AgCl may be related to the fact that the concentt-ation of dislocations in as grown AgCl crystals is observed to be two or three orders of magnitude larger than in NaCl crystals grown under comparable conditions.

4 . 3 T H E COMI3INATION OF T H I R D O R D t R ELASTIC B O U N D A R Y

IMPROVLD IONIC INTER-

ACTION POTLNTIALS.

AS a matter of fact Itnea~.

elasticity theol-y cannot adequately descrlbe the realistic response of a crystal to a dlslocat~on as may easily be recognized from the average dilatation becoming zero (see Chapter I ). Nonluiear calculations, however, overwheln~ this discrepancy to experimental results for it is almost exactly the one-atom-volume- per-plane dilatation that is obtained by the help of nonlinear elasticity theory.

The next step to a more realistic model calculation is to combine the nonlinear elastic theory with the improved ionic potentials. Therefore the potentials

(Chap. 2.1-2.5) with the parameters fitted to second and also third order elastic constants were used in the minimization procedure starting with the confi- guration given by nonlinear elastic calculations.

The core energy thus obtained for NaCl is given in table IlIb in eV per plane. Compared to table lIla this improvement of the elastic boundary condition leads to a lower core energy.

PROBLEMS TO COME.

Being in possession of a realistic core configuration a number of interesting problems should be expected to be solved such as the exact calculation of Peierls stresses on different glide systems for different crystals, the interaction between point defects and dilatations, the dissocia- tion of a complete dislocation into partials and related problems.

Acknowledgments.

The authors appreciate very helpful and stimulating discussions with Prof.

Dr. A. Seeger and Dr. R. Bauer, Max Planck-Institut fiir Metallforschung, Stuttgart, Dr. J. Eisenblgtter, Battelle Institut, FrankfurtIMain, and G. Wagner, AEG-Telefunken, Konstanz.

This work has been supported by the Deutsche Forschungsgemeinschaft. Numerical calculations have been performed by the help of the Univac 1108 of the ZRI, University of FrankfurtIMain.

References

[I] HUNTINGTOS, H . B., PIIJS. Re\'. 59 (1941) 942. T / ? ~ o I . . I

p. 1332 (Nat. Bur. Stand. (US), Spec. Publ.

[ 2 ] HUNTINGTON, H . B., DICKEY, J . E. and THOMSON, R., 317, Vol.

Dec. 1970).

P / ~ J ~ s . Rcl.. 100 (1955) 1 117. ^[7] T o s ~ , M. P., Solicl State P11j.s. 16 (1964) 1.

[8] SARKAK, A. K. and SENGUPTA, S., Solid Statr Corntnlm.

[3] KUROSAWA, T., J. P11)js. SOC. Japan 19 (1964) 2096. 7 (1969) 135.

[4] GRANZER, F., WAGNER, G . and E I S E N R L A ~ ~ ~ ~ ,

P h ~ s . [g] SARKAR, A. K. and SENGUPTA, S., Phys. Slat. Sol. 36

Star. Sol. 30 (1968) 587. (1969) 359. .

[5] SEEGER, A , , in

Interatomic Potentials and Simulatio~l of [lo] CHOWDHURY, S., SEN, S. K. and ROY, D., Phys. Stat.

Lattice Dcfecfs n, p. 566, 764-65 (Battelle Institute Sol. (b) 56 (1973) 403.

Material Science Colloquia, June 1971). [11] TEODOSIU, C., to be published.

[6] SEEGER, A., in

F~rnciatnental Aspects of Dislocatiot~ [I21 GILMAN, J. J., Acta Met. 7 (1959) 608.