The Application of the Three Dimensional Patterson Method and the Crystal Structures of Proustite, Ag₃AsS₃, and Pyrargyrite, Ag₃SbS₃ Citation Harker, David (1936) The Application of the Three Dimensional Patterson Method and the Crystal Structures of Proustite, Ag₃AsS₃, and Pyrargyrite, Ag₃SbS₃. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7x7x-cv79. https://resolver.caltech.edu/CaltechTHESIS:03312025-223924929 Abstract It is shown that the three dimensional Patterson method can be so simplified by the use of the symmetry properties of the crystal under consideration that its use in determining the positions of atoms in crystals is practicable. This method is then used to determine the positions of the heavy atoms in proustite, Ag 3 AsS 3 . The positions of the sulfur atoms are found by the use of the covalent atomic radii and assumptions concerning bond directions. The structure so derived is found to be compatible with the observed data. Proustite is found to contain pyramidal AsS 3 groups and silver atoms forming two bonds to sulfur in almost opposed directions. Pyrargyrite, Ag 3 SbS 3 , is found to have almost the same structure as proustite. The analytical statement of the structures is as follows:-- The space group C 6 3v - R3c, is common to both proustite and pyrargyrite. The special positions of C 6 3v are (in hexagonal axes) 2a) 0,0,z; 0,0,½+z 6b) x,y,z; y,x-y,z; y-x,x,z; y,x,½+z; x,x-y,½+z; y-x,y,½+z; and positions derived from these by the operations of the rhombohedral lattice. The parameter values are:-- Proustite a o = 10.74 Å, c o = 8.64 Å 2As in 2a) Z As = 0.000 6Ag in 6B) x Ag = 0.246 y Ag = 0.298 z Ag = 0.235 6S in 6b) x S = 0.220 y S = 0.095 z S = 0.385 Pyrargyrite a o = 11.04 Å, c o = 8.71 Å 2Sb in 2a) Z Sb = 0.000 6Ag in 6B) x Ag = 0.250 y Ag = 0.305 z Ag = 0.210 x S = 0.220 y S = 0.105 z S = 0.355 Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemistry) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Pauling, Linus Thesis Committee: Unknown, Unknown Defense Date: 1 January 1936 Record Number: CaltechTHESIS:03312025-223924929 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03312025-223924929 DOI: 10.7907/7x7x-cv79 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17121 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 21 Apr 2025 18:04 Last Modified: 21 Apr 2025 18:13 Thesis Files PDF - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

THE APPLICATION OF THE THREE DIMENSIONAL PATTERSON METHOD AND TI-lli CRYSTAL STRUCTURES OF PROUSTITE, Ag~sS , 3 AND PYRARGYRITE, Ag SbS Thesis by David Harker In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California. 1936 3 • 3 1 Abstract It is shovm. that the three dimensional Patterson method can be so simplified by t he use of the symmetry properties of the cry stal unde r consideration that its use in determining the positions of atoms in crystals is practicable. This method is then used to determine the position s of the heavy atoms in proustite, Ag 3AsS 3 • The positions of the sulfur atoms are found by the use of the covalent atomic radii and assumptions concerning bond directions. The structure so derived is found to be compatib le with the observed data. Proustite is found to contain pyramidal AsS 3 groups and silver e.toms forming two bonds to sulfur in a l most opposed directions. Pyrargyrite, Ag 3 SbS 3 , is found to have almost the same structure as proustite. The analytical statement of the structures is as follows:-The spe.ce group c36 v - ~32_, is common to both proustite and pyrargyrite. The special positions of c6 -3v are (in hexagonal axes) 2a) 0,0,z; o,o,½rz 6b) x,y,z; y ,x-y, z; y-x,x,z; --i x,x-y,½tz; y-x,y,i+z; y,x,i§+z; and positions derived from these by the operations of the rhombohedra.l lattice. The parameter values are:-Prousti te Pyrargyrite 2As in 2a) zAs = 0.000 2Sb in 2a) Zsb = 0.000 6Ag in 6B) XAg = o. 246 6Ag in 6b) XAg = o. 250 YAg = o.298 YAg = 0.305 o.235 ZAg = o. 210 = 0.220 Ys = 0.095 ZS = 0.385 = 0.220 Ys = 0.105 ZS = 0.355 ZAg 6S in 6b) XS XS 2 Introduction The determination of the inner structure of a crystal by means of X-rays has been, in most cases, a process of trial and error. Once the space group of the crystal is known, the direct use of the X-ray data is abandoned. A structure which satisfies the symmetry requirements of the space group is then assumed and the intensities of the diffracted X-ray beams are calculated on this basis. If these do not agree with the exper- imente.lly determined intensities, another structure is assumed and the process is repeated. Systematic methods of carrying out this process have been devised which a.re remarkable for the rapidity and accuracy with which, in some cases, they lead to the elimination of every structure but the true one. In other cases, however, the systematic trial and error method breaks down and either fails completely or leads to the loss of much time. If a convenient direct method for calculating the structure of a cryste.l from the experimental data were available, these inefficiencies and failures would, of course, be eliminated. appeared. Such a method has not as yet However, a very close approximation to such a treatment is provided by the Patterson method when used as described in the next section. The determination of the structure of proustite by means of this method is described later in this paper and illustrates the advantages and limitations of the treatment. 3 The Use of the Three Dimensional Patterson Method = = ·= = === ======--===== ==== Patterson(l) has shovm that the three dimensional Fourier Series ?;"" ~ , .,~:-.J ""' = P(x,y,z) "" } cC 2 F(hke)\ cos 27( (hx + ky +f z) I --where~, l_, and~ are the coordinates of points in the crystal lattice measured in units of the primitive translations and !(hk() is the amplitude of the diffracted beam "reflected" from the crystal plane with indices (hke) -- represents a function having maxima at vector distances from the origin equal to the vector distances between pairs of atoms in the crystal. The evaluation of such a series at a sufficient number of points in the unit cell to shovr the true form of the function ~(~,l,,~) has never been carried out, the calculation involving a prohibitive amount of labor. Instead, the projection of P(~1 y,~) on a plane perpendicular to one of the crystal axes has been used. If the projection is upon the plane perpendicular to the ~-axis, the projected function £.(~,l) is calculated thus I p(x,y) = = J,P(x,y,z) dz ~-f-~ {:~ j / 2 F(hke)1 ,~ l 2 F (hkO)) C 21( (hx + ky +f z) dz J: cos 2 if (hx + ky) . £.(~,;'i), which is represented by a two dimensional Fourier series, has been evaluated completely by several authors in the course of various crystal structure investigations. Its use has proved very fruitful, in some cases making possible the detenninations of structures which would have been impracticably difficult otherwise. The use of £.(~,;t) has, however, two disadvantages. In the first place, there is a large probability that a given maximum is composed of the 4 superposition of two or more maxima of t(:::::.,.Y.,£.) which have projected onto nearly the same spot. Secondly, the resolution of maxima which are near together is not as good as can be obtained by the use of P(~•l.•~), since only a small fraction of the tenns in P(~•L•~) are used in .E.(~,i). The function f.(~,l.,~) would consequently be preferred if the labor involved in its calcu- lation could be reduced. The use of the symmetry of the crystal under consideration often leads to the desired simplification. For exrunple, let the crystal have a two-fold axis (which ·will be taken coincident with the E_-axis). Then, if there is an atom at the point (x,;z_,~), there will be another atom, crystallographically equivalent to the first, at (x,,;z,,z). The vector between these atoms has the components (2~,0,2~) and consequently there wiil be a maximum in P(~,y_,~) at the point (2x,0,2~_). This maximum lies in the plane y_ = o. There will be a maximum in this plane for each (crystallographically) different kind of atom in the crystal. The~ and y_ co8rdinates of all the atoms in the crystal can consequently be found by evaluating ~(x,y_,~) for y_ == 0 only. In this case the expression for P (2::,,y_,~) can be simplified as follows: oO o<> ~ ~ ~ L-., = cos 2 7f (hx + .{' z) h=-""', £a -.t:.· • '(; 2 ~ Now let L--=,. == k==-""' 0,:, an d we h ave P ( x, 0 , z ) == «) _F, ~h= - · .n~ ~ F - -~ ch" ~ cos 2" (hx + -\o z) f.(~,O,~) is a two dimensional Fourier series of the same general type as .:e_(x,y_) and consequently its calculation is practicable. The same infonnation about x and z co8rdina.tes of the a.toms is contained in t(x,O,~) as in 5 £,(~,~), but in much clearer form; for (a) the max ima in £_(~,0,~) will be sharper, in general, since t here a.re usually mo re terms in its Fouri er series than in t hat of E_(~,~) and (b) all the maxima corre sponding to interatomic vectors not parallel to the plane l.. = 0 appear in£.(~,~), but £..(2:,0,~) can be used to determine the x and z a.re eliminated in £..(2:,0,~). co8rdinates of atoms in case the £_-axis is parallel to a two-, three-, four-, or six-fold symmetry axis of the crystal, for in all these cases the vectors between equivalent atoms are para llel to the plane l. = 0. In case the b-axis is a two-fold screw a.xis t he tre atment of f.(2:,l,,~) is sli ghtly changed. The co8rdinates of equivalent a.toms are now (~,l,,~) and (x~l.+½,I) and the vectors between equivalent a toms have comp onents (2~,-½·, 2~). In this case £_(~,I,,~) must be evaluat ed for l.. = -½ or, since f (2:,;z.,~) has e. center of symmetry, for ;L = ½. 'lhe calculation simplifies as follows: oc: Z ( F l 2 cos 21T (hx + f + ..\'.z) P(x,½,z) = h,k,f (hk{') I = ""' = L Ch i<. cos 2n (hx + .Rz) h,R=-"" = ( - l \ F(hk~) \ 2 It is easily seen what change s i n the calculation must be ma.de for three-fold, four-fold, and six-fold screw axes. If the cryst a l contains a pl ane of symmetry , the form to wh i ch ) reduces 1.·s even simp ler t han in t he case of an a.xis of symmetry. -P (:'.:.'L'~ 6 Suppose the plane is perpendicular to th e b-axis. atoms at <~,z.~) and (~,l_,~). Then there are equivalent The vector between these atoms has comp onents (o,2L1 0) and the corresponding maximum in P(~,z,~) lies on the £_-a.xis. find this maxunum it is only necessary to evaluate _!:(O,z,o). is as follows: To The cal culation oc 2 = P(O,y,O) ~ , ~ _J F(hkf) I cos 2~ky = k= - ..c This is a one-dimensional Fourier series and can be comp l et ely evaluated in a few hours. simple. In the ca se of a gl ide plane the ca lculat ion is equally In t his case there are e quiva lent atoms at (~,;z..,~) and (~•Y•~+½) ( the glide is taken to be along the ~-a.xis), and there will be a maximum The simplification of ~(~•L•~) takes place thus, in ~(~,x_,~) at (0,2z,½). o<, P(O,y,½) = L Bk cos(2 7iky) k=--C oC = e<:: z z h= -..-0, x = - oe (-/< 2 F (hk Q) In Table I .;is given a swnmary of the fonns of ~(~ 1 .;z::,,!,) best suited to the various symmetry elements. It is t o be noted that at least one of these forms is available for t he study of any crystal that is not triclinic. There a re other case s i n wh ich t he valu(:: s of so::1e of t he atomic coBrdi nates in a crystal can be determined by the evaluation of t(~.z,~) throughout a plane or alo ng a line only. For instance, it might be knovm from synm1etry consid- erations that the position (O,O,O) is occupied by an atom and t hat t here are 7 TABLi I Forms of ~(~.;t_.~) best suited to the determination of atomic co8rdina.tes in crJstals having various symmetry elements . Symmet-rJ element a) Axes parallel to b-axis i) 2 J 4, 42 J 4, 6 J 62 * ii) 21, 4i, 43, 61, 6 P(x,o,z) P(x,1/2,z) 3 iii) 31, 62, 64 P(x,1/3,z) b) Planes perpendicular to b-axis i) reflection planes P(o,y,O) ii) glide plane, glide = ½a, 0 P(J./2,y,O) iii) II II II - iv) 11 II 11 = ½(ao + co) P (1/2, y , 1/2) v) " II 11 = l(a + co) 4 0 P(l/4,y, 1/4) v.i.) II 11 II = le 3a. + C ) 0 4 O P(3/4,y,1/4) ~l 2 0 P(O,y,1/2) In cases (e.) x and z are determined for each atom. In cases (b) l.. is determined for each atom . *The nomenclautre of symmetry axes is ta.ken from the "Internationale Ta.bellen zur Bestimmung von Kristallstrukt uren". 8 other atoms in the plane l.. =½with coordinates (say) (~,½,z;i_), (x2 ,½,~), etc. Then there will be maxima in !:_(~,_l,~) at just these points, for vectors from the origin to these atoms are also interatomic vectors. The evaluation of!:_(~,½,~) would detennine the x and z co8rclinate s of all these atoms. Or, for another example, suppose again that (0,0,0) is occupied by an a.tom and that other a.toms musv lie on the line x = ½, z = ½. They:_ co8rdinates of all these a.toms could be found by calculating P(½,y:_,½), for there will be a maximum in this function at ea.ch value of y:_ occupied by one of these a.toms. Such examples could be multiplied almost indefinitely. It is plain from the above that the function !:_(~,y,~) furnishes a powerful and simple method of attack on the structures of all crystals not belonging to the triclinic class. All that is required is a. complete set of X-ray diffraction data., for from these not only can the symmetry of the crystal be determined, but also the values of / F (hkf) [ 2 calculated. An illustration of the use of !:_(~,;z__,~) is furnished by the determination of the structure of the mineral proustite, AgaAsS 3 , which will be described below. 9 Determination of the Structure of Proustite, Agt;sS 3. The proustite used in this detennina.tion ca.me from Schneeberg in Saxony. It occurred in the form of tiny, brilliant red, hexagonal prisms measuring about 0.2 mm. by 1.0 mm. which were bounded by beautiful faces. Groth(Z) describes proustite as ditrigonal pyramidal (C the axial ratio c/a = 0.8038. from 5. 55 to 5. 62. 3v ) with The density varies among different specimens This assignment to a polar class is made on the basis of many observations of unsymmet rical face development and seems exceptionally well grounded. A Laue photograph taken perpendicular to a prism face showed a two-fold axis of symmetry, another Laue photograph taken a.long a line bisecting the angle between two prism faces and perpendicular to the prism axis showed a plane of s~runetry parallel to the prism axis, and a third Laue photograph taken along the prism axis showed a three-fold axis and three planes of symmetry. The Laue symmetry is therefore D3 d and the crystal must belong to one of the point groups c3v, D3 , or D3d on the basis of X-ray data alone. Measurements on oscillation photographs led to the dimensions ;:o 10.74 K, ~ = 8.64 K for the hexagonal unit cell. The ratio~/~= 0.804 is in excelle nt agreement with the r atio~~= 0.8038 obtained from crystallographic measurements. prism The two Laue photo g~ raphs ta.ken perpendicular to the axis were indexed by the aid of gnomonic projections and ' the wave length of the radiation pro ducing each reflection calculated. It was t hen found that no pl anes reflected in the first order unless t he indices satisfied the condition imposed by the rhombohedral lattice, 2H· + K + L = 0 mod 3, although many planes not satisfying t hi s conditi on were in position to give first order reflections. The l at t ice is consequently rhombohedral and its 10 unit cell ha s one -th i rd the vo lume of th e he xagonal ce ll for v,hich dimension s a.re recorded ab ove. It i s i mp os s i b le t o a ccount f or mu ch of the ob se rved data on the basis of a unit ce ll smaller th an this and no data requiring a large r one has been found, although several hundred r e flections on vari ous ph ot ograph s have been indexed. This r hornb ohe dra.l cell is t here fore t he true m it cell of p r ous ti te. 1 For c onvenience , however, t he hexagona l cell will be used throughout this paper, much of the ane.lytic geometry of t he structure appeari ng s imple r ·wh en referre d to these axes. The numb e r of f ormulas of Aga,AsS 3 in this unit wa s ca lcul ated from the density range given above to lie between 5.89 and 5.96, or 6 within the experimental error. Determination of the Space Group No r eflections of the type (OK•!:_), L odd, were obse rved in the fir s t order on any photo graph, although many pl anes of t hi s t ype were i n pos ition to r e flect on both t he Laue and oscill ati on ph ot og raph s. The re a r e t wo sp a ce groups based on t he rhombohedral lattice which requ ire t his absence: c6 6 R3c and D -R3c. - -3d - -:3v The hemihedral face development of proustite eliminat es t he l a t te r. Conse quently 6 the spa ce group of proustite is C - R3c. -:3v - Detenaination of the Structure 6 The space group c ov - R3c furnishes t he following sets of equivalent posi tionst and 2a) 0,0,u; o,o,u+½ 6b) X,Y,Z; Y,X-Y ,Z; -X+Y,X,Z; X,X-J,Z;-½; J Y,X,Z¥z-; -X+Y,Y,Z;-½; and positions derived from these by the operations of t he rhor.1bohedral lattice. There are 2 As, 6 Ag, and 6 S atom s t o distribute e.mong these positi ons. The ars enic atoms must be placed i n 2a) and the pa rameter ~ rnay be o.rbitra rily t aken to be zero. The a rse nic atom s a r e nmY fixe d at (O, O,O) and (O,O,½) in t he hexagonal ce ll arid the po sitions derived fro m t hese ·oy t he operations of the rhombohe dr a l l at t ice. The s ilve r and sulfur a toms must be in the gene r a l positi ons 6b) and there are consequently six pa r ameters to be detennined: 11 In ord,__, r to determine these parameters, the Patterson method described in the previous section was applied. The crystal contains a glide plane and so it is necessary to evaluate the Patterson function, ,!:.(x,;t:_,~) only for points along a line, in order to determine the distances of the atoms from the plane. By subtracting the co8rdinates of point s produced from each other by the action of the glide plane perpendicular to the I_ axis, it was found that for each kind of atom in the crystal. It was, therefore, only necessary to evaluate f(O,;z::,, ½) in order to determine X and Y for each atom. Now .,,c P(O,y,½) = ~ \ FHK•L \ 2 cos 2'11' (Ky + ~) H,K,L= -~ z °" OK cos 27TKy K= -.a with oa 2 The coefficients~ were calculated from the intensities of reflections on a series of oscillation photographs so taken that each possible reflection appeared near the equator on at least one of the series. The intensities were divided by the Lorentz and polarization factors in order to obtain the values of l FHK.L j 2. No corrections for extinction or absorption were made, as there was no way of estimating their effects. The curve of ,!:.(O,;t:_,½)plotted against l.. appears in Figure 1. The large maximum at l.. = O is due, of course, to the vector between the arsenic atoms at (0 , 0 , 0) and (0 , 0 , .1.) 2 • The maxi'n1a at 1/6, 1/3, and 1/2 were considered to be due mainly to the vectors bet,veen silver atoms, for the maxima due to vectors between sulfur atoms would be expect ed to be only about one tenth the size of these. 12 10 5 - 0 .1 0 0.6 0. 1 Y- Figure 1. P( o,j,Yi,) f or proustite plotted against l...• The large maximum at l... = 0 is due to As-As interaction, the maxima at 0.17, 0.33, and 0.50 are due to Ag-Ag interactions. There is a center of symmetry in P(O,½,.r) at l.. = ½, consequently only half the curve is plotted. 13 The axes can be so chosen that the inequalities &g + "!.Ag>2XJ...g - 2S/l.g > ~g - °IAg > 0 are satisfied. This choice and the positions of the maxima in f.(0,l_,½) lead to the equations XAg + YAg = 1/2 2YAg XAg = 1/3 2X.A.g YAg = 1/6 These are solved by ~g = 4/18 ~ 0.22 and ~g = 5/18 ~ 0.28. silver atoms are now approximately located in the ~/£ plane. The The maxima. in P(0,z,½) are fairly broad and consequently the values of 2S/l.g and °I..Ag determined from these are not to be considered as accurate. However, a silver atom must lie near the line~= 4/18, Y = 5/18. There is an arsenic atom at (0,0,0) and there must therefore be a maximum in P(~,z,~) at (!J..g,"!.J...g,!:J...g)• This maximum must lie near enough to the line!= 4/18, !_ = 5/18 so that f.(4/18,5/18,f) should show a maximum at f = !:J..g• f.(4/18,5/18,f) vras accordingly evaluated and values plotted in Figure 2 were obtained. The two sharp maxima at .f = 0.227 and f = 0 743 0 can be due only to the vectors from the arsenic atoms at (0,0,0) and (0,0,-½) to the silver atom. The values of ~g calculated from the positions of these two maxima are !:J..g = o. 227 and ; g = o. 243. with a probable error of about 0.008. The mean of these is ~g = o. 235 The relation between the arsenic and silver atoms in the proustite structure is now approximately determined. Hofmann( 3 ) has stated that the AsS 3 and SbS 3 groups are pyr~~idal with bond angles of about 95° and bond lengths about equal to the covalent radius sums calculated from the table of Pauling and Huggins< 4 ). Corey and Wyc- koff(5) in the study of (A.g(NH 3 )21 2 S0 4 and West( 6 ) in the study of AgCN have shovm that Ag I forms two covalent bonds in apposed directions with a covalent radius of 1.36 K. The asswnption was accordingly made that the AsS 3 groups in proustite were of approximately the form described by Hofmann and that 14 3 0 0.6 -I Figure 2 . f. r(3/18 , 5/18,f) for proustite plotted against The two large maxima are due to As-Ag interactions. 15 silver atoms fonne d t wo covalent bonds to sulfur in opposed direction s with a sil ver-sulfur distance of about 2.40 fl.., which is t he sum of the covalent radii. It was also assumed that sulfur would form bonds in ap proximately tetrahedral directions. It was found that these conditions could all be satisfied approximately by placing t he sulfur atoms in proustite in the general positions 6a) with !s =0.22, !s ~ O.ll, !s ~ 0.40. The complete structure of proustite was now roughly knov1m. The parameters &g and :!J,.g were next fixed accurately by means of the intensities of the reflections (HH•O) positions as above. !!_ = 1,2,3,4,5,6; assuming t he sulfur These intensities are a s follows: H Observed Intensity of (HH• 0) 1 2 3 4 5 6 o. 50 o. 40 0.45 0.10 0.15 0,20 Figure 3 shows the values of !Ag and !A_g ruled out by the various compa risons. The center of the a llowed r egion is at ~g = o. 246 , !Ag= o.29 s , and these values give calculated inten sities in good ag re ement with the quantitative data with the sulfur para.me ters anywhere near ~ = 2/9 , !s = 1/9. A change in ~g or !A_g of 0.005 in eithe r direction places the point !A_g, I..A,g at, or beyond, the e dge of the allowed region as shov1m in the fi gure. The limits of error in the determination of ~g and ~g are conseque ntly :t_ 0.005. If the Patter son method determination of .0,_g is accepted, the silver parameters are now completely determined . The s i l ve r po sitions we re accordi ng ly asswned to be accurate l y known and the sulfur atoms placed so as to be 2.25 fl.. from arsenic atoms and 2. 40 fl.. from silver atoms , t hese being the distances cal cul ated from t he covalent radii of these elements, now comp letely determined. ~g '!.AP' 0 ~g = The structure of proustite is The par ameters a.re as follows: 0.246 + o. 005 !s = 0.220 :t_ 0.10 o. 298 :!:_ o. 005 !s = 0.095 :t_ 0.10 0 . 235 :!:, 0.008 ~ = 0. 385 + 0.10 ~s = 0.000 16 B t 0.2 D ~g 0.2A-...,______________. 0.23 0.24 0.25 0.26 X-+~g Figure 3 . Determination of ~g and ~g for proustite using comparisons of intensities of reflections (HH• O) . Region A is ruled out by I( 55 • 0) c alc . / I(66•0) Region Bis ruled out oy I(22• 0) calc. ,> I(33•0) Region C is ruled out by I(44•0) ca.le .> I(22•0) Region D is ruled out oy 1(44• 0) cal c . > I(55•0) X marks the point giving the best qua.nti t e.ti ve agreement . 17 Intensities of all r eflections for which (sin e )/,A calculated and compa r ed with the observed data . appear in Table 2. ~ o. 075 were The r e sults of this calculation The agreement between the observed and calcula.ted values prove s t he structure of proustite described above to be correct. ~ Crystal Str ucture of Pyrargy ri te Ag 3SbS 3 The pyrargyrite used in this study was from Freibe rg in Saxony and occurred in the form of metallic appearing , hexagonal pri sms measuri ng about 0.5 mm. by 2.0 mm. Groth( 2 ) assigned pyrargyr ite to the ditri gonal pyramidal class (S3y) with the axial r at io ~!?:.. = O. 7892. The density r anges from 5. 75 to 5. 85. In t his case, as in the case of proustite , t he assignment to the polar class Sv is well grounded on obse rvations of u.nsyrrnnetrical face deve lopment . Mee.surements of reflections on oscillat i on photo graph s led to t he axial l engths !:.a= 11.04 K, 2JJ = 8. 71 K f or t he hexa~ona l unit cell. The axial rat io i s ~-~ = o. 78 9, in good agreement with the ratio ~~ = o. 7892 ca lculated from crystallograph ic measurements. The number of formul as of Ag 3 SbS 3 in this unit cell was calculated from the density range given above to lie between 5. 92 and 6. 03. There are t hu s 6 formulas of Ag 3 SbS 3 in the unit cell. Both t he Laue and oscillation photograph s of pyrargyrite show great similarity to the corres ponding photo graph s of proustite. In particular, t he conditions r equired by the r hombohedra l lat t ice and the space group c6 _ R3c were never violated. These observations , t ogethe r with the simi l arity bet·we en t he chemical comp ositions, crystallographic data , and unit cell dimensi ons of pyrargyri te and proustite, led to the asswnpti on that these 4.5--2,86 o.40--0.16 (41.0) (22,0) 0.40--0.10 o. 8--0. 77 2.0--1,79 1,5--1,51 0,40--0,22 (32.1) (13•1) (23•2) (21,1) (03•0) (24•4) ( 41 • 3) (33•3) ( 25• 3) (52•3) 1.2--1.22 ( 50• 2) ( 15· 2) (34•2) (61•2) ( 51• 1) ( 4.3• 1) (16•1) (06•0) (52•0) (44•0) o.,1s--o.24 0,60--1.20 o.40--0.54 0.10--0.04 (33•0) o,40--0,53 0.00--0.07 o,30--0,37 0,15--0.12 (24•1) o.45--0.62 o.40--0.33 0.10--0.14 0,15--0.14 (42•2) o.30--0.13 o.2s--o.15 o.25--o.os 0.15--0.37 0.15--0.29 (14• 3) o.30--0.20 1.3--1.40 (04-2) ( 05• 4) 0.10--0.22 0.10--0.42 0.15--0.49 ( 32• 4) 1.0--0.60 (31•2) o,9--1.48 ( 40• 4) 0.15--0. 45 ( 23• 5) 2.5--1.97 (12.2) 0,50--0.35 1,8--1.71 (11• 0) (20•2) (01•2) 0,25--0.17 1.2--1.55 4.0--4.39 1.3--0.71 (13• 4) ( 21 • 4) (22•3) 1.3--1.00 o,7--1.93 0.45--1.02 0.9--0.87 (11•3) (02•4) (10• 4) (31•5) (12• 5) figure is the calculated intensity. In each square the top rov-r gives the indices, and in the second row t he first figure is the observed, the second Observed and Calculated Frequencies for Proustite Table 2 9'.l I-' 19 two substances have nearly the same structure, the chief difference being the substitution of the arsenic atoms :i. n proustite by antimony atoms in pyrargyrite. That this assumption was justified is amply proved by the agreement between the calculated and observed intensities of reflection which resulted from its adoption. Gessner and .Mussgnug( 7 ) made layer line measurements on oscillation photographs of pyrargyrite from which they determined the lengths of the hexagonal axes to be~= 11.06 ! and~ results obtained in this research. 8.84 Kin good agreement with the They also proved the latt ice to be rhombohedre.l and determined the space group to be C6 - R3_c. - 3v They did not determine the structure, however. The assumption that proustite and pyrargyrite have almost t he same structure leads at once to the approximate structure for pyrargyrite:-c6 : 2 Sb in 2a), ZSb = 0.000; 3v 6 Ag in 6b), XAg YAg ZAg 6 S o. 25 , - 0.30, ~ o. 23; in 6b), XS - o. 22, Ys - 0.10, Zs - o. 40. The silver parameters ~g and °X.Ag were determined, as in the case of proustite, by use of t he inte nsities of the reflections (IDI~O), H = 1 to 6. These intensities were as follo,Ys: 0 H Observed Intensity of (HH•O) 1 2 3 4 5 6 o.oo 5.0 6.0 o.4o 1.5 o.so The region s of variation of ~g and "IJ,.g ruled out by the ve.rious comparisons are sho,m in Figure 4. Values of the parameters differinb froia ~.Ag = o. 250 20 0.32~--------- A t 0. D- ~g Q29- + - - - - - - - . . . . . . , ; ~ ~ 0.24 0.25 0.26 0.27 1 X-+Ag Figure 4. Determinations of ~g and ~g for pyrargyrite by comparison s of intens ities of reflections (HH•O). Region A is ruled out by I(66•0) calc • ..(_ ~(55•0) Region B is rule d out. by I( 22 •0) calc. ) I(3 3•0) 1 Region C is r uled out by I( 44• 0) calc. )' 4I(22 •0 ) Region D is ruled out by I ( 66• 0) calc. > · I(55•0) X mar k s the point giving t he best quantitative a greement. 21 .1i_g = O. ;105 by as much a s o. 0 05 gave p oor a 0 ree;1:ent with the obse rved data , and these vahws were the ref ore v.ceepte d as bei ng correct to + 0.0 05 . The p aramete r ~g was determi ned by compa risons, so,ne of wh ic h are sh own i n Fi gure 5, t o be within 0.005 cf ~g make t h e SbS 3 = 0.210. Th e sul fur atoms we r e the n p laced so as to g ro ups pyramidal with t;h e Sb -S dist an ce 2 . 45 K and t he Ag -S distan ce 2 . 40 K, i n a g reeme nt vri th t he covalent radii of these e l ements . The dete nni nat i on of the structu re of py r a r gyrite was t he n c omp l e te. The pa r a.meters are o. 250 + 0.00 5 ~ o. 220 + 0 . 010 '!.Ag 0. 305 + 0 .00 5 !s 0.105 + 0.010 ~g 0.210 + 0.005 ~ = 0.35 5 + 0.010 ~g ~b = 0.000 Intens iti es were calcul a t e d on the ba sis of this structure for a ll the reflectio ns for wh ich ca lcul ations had been made for p roustite. resul ts of these c0.lcu l at ion s appear in Tab l e 3 . The The agreement betvreen the observed and calcul ate d i ntensit i e s of reflection proves the above structure to be correct. Discussion 9f th e Structures of ProustHe and Pyra r.§yri t e The st r ucture of proustite is s h ovm in Fi gure ti , th e structure of py r a r gy rite being so similar th at t h is f i ~ure may b e used fo r the discus s i on of both . Both structures contain (AgS ).,., g roup s of almo st the sa'lle sh ape . In t hes,..: , the sulfur a t oms fonn the corn ers of a tr iangular s p ira l, wh il e t h e si l v u r a to,:1 s 1 ie a l most in th e centers of t he l eg s of the s p iral. The sulfur to silver distance is 2,40 ft., t he S-Ag-S b ond ang le is about 1 65 °, and the ' -'Ag-S-Ag b ond anc le is aoouv 8,z}.O v-2 • One conp lete tu r n of the s oira l is 22 1.0 0.5 0 .0-t--r----.----.--+---. 0 .25 0 .20 z.. Detenni nati on of Z Fi gure 5. -,\g isons of i ntensities . a gainst '!:.;.,_P- . for pyra.rgyri te by compa.r- brK•L p lotted on a n arbitra r y scal e The dotted li ne s hows the chosen value . 0 observed intensi.t i es satisfy t h e inequalities: The 6. 0-- 3. 60 (01•2) (12•2) ( 31 •2) ( 33 • 3 ) (2 5• 3) ( 51• 1) (52• 3) (06•0) 3.0--3. 62 (33•0) 3,4--1.75 ( 41• 0) 1. 5--0. 92 (2 2•0) (03•0) ( ol • 2) ( 43 • 1) 2.5--3.12 (52•0) 0,40--0.16 ( 44•0) 0.1--0.80 (16•1) 0.50--0,27 1.0--0.91 (34•2) 0.00--0.19 1.3--1. 29 1.s--1.82 2.0--1.61 5.5--6.66 5.0--4.73 5.5--5.23 ( 24• l) (1 5• 2 ) (32•1) (42• 2) 2. 5--2. 00 ( so. 2) 3.0--2. 49 ( 23 •2) 5.0- -5.79 0.00--0.1 3 8.0--10. 00 3.o--1.39 (11•0) ( 41• 3) o . 45-- o . 4o o.45--0.43 o.5o--o.44 0.60--0. 08 o.45--0. 69 (14• 3) (13·1) 1.5--1.09 (04•2) (24•4) (21•1) 1 2. o-1 2.2a 10 .0-10. 50 4.0-- 3.19 ( 20 • 2 ) 0. 9--1.08 1 2. 0-15. 37 (0 5• 4) 0.00--0.40 o.oo--o.42 2.0--2.82 4 . 5-- 5. 65 7.0--6.04 6.0- - 5. 56 (2 2.3) 3. 0--1 .09 10.0--12.50 ( 32 • 4) ( 40 • 4) (13•4) ( 21 • 4 ) o . 45- -1.00 1. 2- - 3.59 2.0--2.94 (11• 3) (02.4) (10• 4) ( 23 .5) ( 31• 5) (1 2• 5) figure is the calculated intensity. In each square the top ro1N gives the indices , and in the second row the first figure is the observed , the second 0bser·ved and Calculated Intensities fo~ Pyi:_argyrite Table 3 N: CN 24 equi valent to a translation of~ in the crystal. These spirals are fa stened together by th e As or Sb atoms to make up the s tructural fr e.meworks of proustite or pyrargyrite respectively. The sulfur to arsenic bond distance in prousti te is 2. 25 ! a.11d the S-As-S bond angle is 103°. The sulfur atom in proustite forms three bo r '.'l : , two to silver and one to arsenic ·with these angles: As-S-Ag = 101° and 105½ 0 , Ag-S-Ag = 83½ 0 • In pyrargyrite the antimony atoms form three bonds to sulfur, 2. 45 .K long at mutual angles of 95½ 0 • The sulfur atom in pyrargyrite forms three bonds with the ang les: Sb-S-Ag and 111°, Ag-S-Ag = 83½ = 97° 0 • An interesting feature of these structures is the fact that they are composed of two interpenetrating fra.mevrorks or the type described in the last paragraph which are apparently not connected by any bonds at all. The relation of the two frame-works can best be understood by a study of Fi gure 6. The As or Sb atoms at (0,0,0) belong to one framework and the crystallo 6 raphically equivalent As or Sb atoms at (o,o,½) belong to the other. The two frameworks are in t he relation t o each other required by the operation of the glide planes. One framework cont r,.in s only ri 6 ht handed AgS spirals, the other only left handed ones. Anoth er example of bNo interpenetrating frameworks apparently not interconnected by cher!'lical bonds is furnished by the structure of cuprite, Cu2 0. Aclmowledgement The author wishes to express his gratitude to Professor Linus Pe.uling, who sugg;e sted the study of prousti te and pyrargyri te, for sparing neither criticism nor encouragement throughout the course of t h is research. 0 Figure 6. The struct ure of proustite. Pyrami ds r epresent AsS 3 groups, balls r ep resent silver atoms, and lines re pr e sent covalent bonds. of pyra r gyrite is a l mo st the srune. The structure 26 References (1) A. L. Patterson, Zei t. Krist., 90, 517 (1935) (2) P. Groth, Chemische Kristallo g raphie (3) W. Hofuann, Zeit. Krist.,~, 161 (1935) (4) L. Pauling and M. L. Hug gins, Zeit. Krist., 87, 205 (1934) (5) R. B. Corey and R. W. (6) C. D. West, Zeit. Krist., 90, 555 (1935) (7) B. Gossner and F. Mussgnug, Centralbl. f. Min. (A) 1928, 65 c. Wyckoff, Zeit. Krist.,~, 264 (1934)