Pecharsky V.K., Zavalij P.Y. Fundamentals of Powder Diffraction and Structural Characterization of Materials

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582

Chapter

12 14 16 18

22 24 26

Bragg angle,

(deg.)

Figure

6.36.

Low Bragg angle regions of the powder diffraction patterns of GdsGe4,

Gd5SizGez, and Gd5Si4. The three patterns are representative for materials in the Gd5Ge4-

based solid solution, the intermediate intermetallic phase, and the Gd5Si4-based solid solution,

respectively. The experimental data were collected on a Rigaku TTRAX rotating anode

powder diffractometer using Mo

Kcc

radiation from 8 to 50•‹ 28 in a step scan mode with a

step A20

0.01" and counting time 15 seclpoint. Distinct clusters of Bragg peaks observed in

the regions -10

-18" and -21

-26" point to close relationships between the

three crystal structures. Gradual redistribution of peak intensities from one pattern to another,

however, cannot be solely associated with the expected change in lattice parameters due to the

substitution of smaller Si for larger Ge atoms. Therefore, we conclude that three crystal

structures are closely related, yet different.

Table

6.48.

Coordinates of atoms in the unit cell of Sm5Ge4 as determined from single crystal

diffraction data in the space group symmetry Pnma, after Smith

et al.,

Acta Cryst.

22,

269

(1 967).

Atom Site

Sml 4(c) 0.2880 1 14 0.0024

Sm2 8(d) -0.0283 0.1004 0.1781

Sm3 8(d) 0.3795 0.8843 0.1612

4(c) 0.1761 1 14 0.3667

Ge2 4(c) 0.9132 1 14 0.8885

Ge3 8(d) 0.2206 0.9551 0.4688

state of the x-ray powder diffraction analysis in 1960's, it was difficult, if at all possible, to

establish the details of the distribution of atoms in these complex crystal structures.

Crystal structure solution

583

6.18.1

Crystal structure

Gd5Ge4

As reported by

Smith

et al.

(see reference 3 on page

581),

pure

gadolinium germanide, GdSGe4, has the following unit cell dimensions:

7.69,

14.75,

7.76

From the similarity of the electronic structure,

chemical properties and atomic volumes of Sm and Gd, the assumption

about the identity of the crystal structures of Gd5Ge4 and SmSGe4, appears to

be quite reasonable. Furthermore, Si and Ge are electronic and

crystallographic twins, and it is likely that they may substitute for one

another in an extended range of concentrations in metallic alloys.

Nonetheless, assumptions like that usually require initial verification by

computing the distribution of intensities in the powder diffraction pattern of

the compound in question using proper or approximate unit cell dimensions

coupled with the distribution of atoms taken from a prototype crystal

structure. The result is illustrated in

Figure

6.3

I I I I

8 10 12 14

18 20 22

24 26

Bragg angle,

(deg.)

Figure

6.37.

The observed and calculated intensities in a fragment of a powder diffraction

pattern of Gd5Ge4. The calculated intensity has been scaled to match the observed profile. The

observed data are available in the data files

Ch6ExlO-MoKa.xy

and

Ch6ExlO-MoKa.dat

on the

CD.

Chapter 6

The unit cell dimensions for this sample were determined by employing

Le Bail's method applied to the fragment of data from 8.5 to 27O

7.6984,

14.8278,

7.7849

along with the relevant peak shape and

background parameters, the results of which are illustrated in

Figure

6.38.

All of them were used in the modeling of the calculated powder diffraction

pattern using coordinates of atoms listed in

Table

6.48

and assuming that Gd

atoms occupy Sm-sites. Individual displacement parameters of all atoms

were set at

0.5

A2.

Even though some discrepancies between the observed and calculated

profiles are obvious from

Figure

6.37,

the overall good agreement and low

residuals are solid indicators that the crystal structure of this material indeed

belongs to the

Sm5Ge4 type. All relevant parameters of individual atoms

should and will be refined using the Rietveld method applied to the entire

measured range of Bragg angles, as will be described in Chapter 7. The

experimental powder diffraction pattern is available in the data files

Ch6ExlO-MoKa.xy

and

Ch6ExlO-MoKa.dat

on the CD.

2.79%

R,,

4.95%

Re,,

1.92%

6.60

111 111111 I1111 111111I

10 12 14 16 18 20 22 24 26

Bragg angle,

(deg.)

Figure

6.38.

The results of Le Bail's full pattern decomposition of the powder diffraction

pattern of Gd5Ge4. The discrepancies between the observed and calcul~ted profiles are small

and all residuals are low, indicating that the unit cell dimensions are accurately determined

and that the chosen peak shape function (pseudo-Voigt) is a good choice for this experiment.

Crystal structure solution

585

6.18.2

Crystal structure

Gd5Si4

As reported by Smith

et al.

(see reference

on page 581), pure

gadolinium silicide, Gd5Si4, has the following unit cell dimensions:

7.45,

14.67,

7.73

The reduction of lattice parameters, when compared

to the germanide, is expected given the smaller effective radius of Si in

comparison with that of Ge.

Following the same approach, i.e. after determining the unit cell

dimensions

7.4863,

14.7465,

7.7503

A),

peak shape and

background parameters by means of Le Bail's full pattern decomposition

(Figure

6.39), the observed and calculated powder diffraction profiles of the

Gd5Si4 alloy are shown in

Figure

6.40. The coordinates of atoms listed in

Table

6.48 were used in the computation of the scattered intensity assuming

that Gd occupies Sm-sites and Si occupies the corresponding Ge-sites.

Unlike in the case of Gd5Ge4

(Figure

6-37), the match between the observed

and calculated intensities in

Figure

6.40 is quite poor.

8 10

12 14 16

20 22 24 26

Bragg angle,

(deg.)

Figure

6.39.

The results of Le Bail's full pattern decomposition of the powder diffraction

pattern of Gd5Si4. The discrepancies between the observed and calculated profiles are small

and all residuals are low, indicating that the unit cell dimensions are accurately determined

and that the chosen peak shape function (Pearson-VII) is a good choice for this experiment.

The observed data are available in the data files

Ch6Exll-MoKa.xy

and

Ch6Exll-MoKa.dat

on the

CD.

586

Chapter

The Fourier map, calculated using this model of the crystal structure,

confirms the locations of all Gd and two independent Si atoms

simultaneously indicating quite significant deviations from the coordinates

listed in

Table

6.48.'

Thus, at this point we may conclude that although the

structure of the Gd5Si4 alloy is closely related to that of the Gd5Ge4 alloy, the

distributions of atoms in their unit cells are not truly identical. It is feasible

that by using Rietveld refinement, we will be able to establish the necessary

structural details, and the structure completion process will be described in

Chapter

I I I I

8 10 12 14 16 18 20 22 24 26

Bragg angle,

(deg.)

Figure

6.40.

The observed (circles) and calculated (lines) intensities in a fragment of powder

diffraction pattern of Gd5Si4. The calculated intensity has been normalized to match the

observed profile. Peak shape, background and lattice parameters employed to compute the

calculated profile have been obtained by a full pattern decomposition of the observed data

using Le Bail's technique, as shown in

Figure

6.39.

Note, that the difference plot has been

compressed ten-fold for clarity.

We leave this as a self-exercise to the reader noting that several sequential Fourier maps,

each preceded by the re-determination of the individual intensities, may be required to

establish the coordinates of all atoms in the unit cell leading to an acceptable match

between the observed and calculated intensities.

Crystal structure solution

587

6.18.3

Crystal structure

GdSSizGez

Attempts to index the powder diffi-action pattern of Gd5Si2Ge2 in any unit

cell in the space group symmetry Pnma with dimensions intermediate

between those of the Gd5Ge4 and Gd5Si4 alloys, see the pattern in the middle

Figure

6.36, fail because not all peaks even at low Bragg angles can be

indexed. Thus, we have to assume that the formation of the intermediate

phase is accompanied by the change of the symmetry of the material. The

initio

indexing leads to a monoclinic unit cell with lattice parameters

7.581,

14.809,

7.784

93.19". The obtained unit cell

dimensions are obviously related to those observed in the orthorhombic

Gd5Ge4, see section 6.18.1 and it is quite reasonable to assume that the

intermediate intermetallic phase in the Gd5(Si,Gel-,)4 system is a monoclinic

distortion of the Sm5Ge4-type structure.

We begin with establishing the space group symmetry of the material.

this case, a straightforward analysis of the systematic absences is difficult

because of the overwhelming number of Bragg reflections, which are

possible in the examined range of Bragg angles: a total of -1500 reflections

could be observed up to 28

50" when Mo

radiation is employed.

Considering a monoclinic distortion of the orthorhombic lattice, the analysis

is relatively easy, as illustrated in

Figure

6.41. When the angle between the a

and b basis vectors is no longer 90•‹, both the mirror plane m, which is

perpendicular to b, and the glide plane n, perpendicular to a, are no longer

possible. However, the glide plane, a, which is perpendicular to c, the screw

axis

parallel to c, and the center of inversion remain unaffected by this

distortion.

Thus, the likely resulting space group symmetry is P1 121/a, which is a

subgroup of Pnma. As also shown in

Figure

6.41, any atom located in a

general site position of the space group Pnma (the multiplicity of this site is

8) breaks into two symmetrically independent atoms located in two general

four-fold sites in the monoclinic symmetry (the open circles and the hatched

circles in the right hand part of the figure). Furthermore, atoms located in

special sites on mirror planes in the orthorhombic crystal system, where the

value of the y-coordinate has been fixed at y

114 (see

Table

6.48) are no

longer special sites and their y-coordinates become free parameters.

Before proceeding with creating a model of the crystal structure of the

monoclinically distorted Gd5Si2Ge2 we will verify the correctness of both the

lattice parameters and space group symmetry (PI 12Ja) by performing Le

Bail's refinement of the experimental profile. The result is shown in

Figure

6.42, which indicates a high probability for this unit cell.

Chapter

-x

Pnma

Figure

6.41.

The disappearance of mirror planes, m, and glide planes, n, during a monoclinic

distortion of the orthorhombic unit cell corresponding to the space group symmetry Pnma

(left) when the angle between

and

unit cell edges deviates from

90"

(right). The resulting

space group symmetry is P112,Ia: centers of inversion, two-fold screw axes perpendicular to

the

XYplane, and glide planes, a, parallel to the XYplane remain unaffected by this distortion.

The open circles on the left represent symmetrically equivalent atoms located in a general site

position, 8(d), in the space group Pnma. The eight-fold site splits into two independent 4(e)

four-fold sites in the space group symmetry P112,/a, as indicated by both open and hatched

circles on the right, for which

Considering

Table

6.48 and

Figure

6.41, the approximate model of the

crystal structure of Gd5Si2Ge2 can be easily derived

splitting the

coordinates of atoms located in 8(d) sites in the space group symmetry Pnma

into two 4(e) sites in the space group P112,Ia. The coordinates of atoms in

the split positions are approximately related as

x,y,z and x,l/2-y,z, where

x,y,z are the coordinates of the symmetrically independent atoms in the 8(d)

sites. The coordinate triplets of atoms located in 4(c) sites in the space group

Pnma remain unchanged except for the fact that the values of their y-

coordinates are no longer fixed at

114. The positions of all of the

independent atoms obtained in this way and which represent the initial

model of the Gd5SizGe2 crystal structure are listed in

Table

6.49.

Crystal structure solution

589

l l l

1111

lIlW

lllll ll l

11111

IIIII

UI Ill

IIIU

III

IIIUIIIIYII

IIllllYUllll111111U11111111111111

IIN

I I

10 12 14 16 18 20 22 24 26

Bragg angle,

(deg.)

Figure

6.42.

The results of Le Bail's full pattern decomposition of the powder diffraction

pattern of Gd5Si2Ge2. The discrepancies between the observed and calculated profiles are

small and all residuals are low, indicating that the unit cell dimensions are accurately

determined and that the chosen peak shape function (pseudo-Voigt) is a good choice for this

experiment. The observed data are available in the data files

Ch6Exl2-MoKa.xy

and

Ch6Exl2-MoKa.dat

on the

CD.

Table

6.49.

Approximate coordinates of atoms in the unit cell of Gd5Si2Ge2 as obtained by

splitting general site positions from the space group Pnma into two sites in the space group

symmetry P112,Ia. As a first approximation, a completely random distribution of Si and Ge

atoms in the corresponding sites is assumed.

Atom Site (Pnma) Site (PI 112,a)

4(c) 4(e) 0.29 0.25 0.00

Gd2a 8(d) 4(e) -0.03 0.10 0.18

Gd2b 4(e) -0.03 0.40 0.18

Gd3 a 8(d) 4(e) 0.38 0.88 0.16

Gd3b 4(e) 0.38 0.62 0.16

(Si,Ge)l 4(c) 4(e) 0.18 0.25 0.37

(Si,Ge)2 4(c) 4(e) 0.91 0.25 0.89

(Si,Ge)3a 8(d) 4(e) 0.22 0.96 0.47

(Si,Ge)3 b 4(e) 0.22 0.54 0.47

590

Chapter

The feasibility of this model of the crystal structure has been verified by

computing scattered intensity using all relevant parameters as determined

from Le Bail's decomposition and coordinates of atoms from

Table

6.49. It

is easy to see from

Figure

6.43 that although the discrepancies are

considerable, the calculated and observed profiles match better than in the

case of Gd5Si4

(Figure

6.40). Thus, we conclude that not only the model

derived by a monoclinic distortion of the orthorhombic lattice is feasible, but

also the structure of the

Gd5Si2Ge2 alloy is likely to be an intermediate

between those of the gadolinium silicide and germanide. Obviously, this

structure, as well as two others considered in this section,

should be

completed and fully refined using the Rietveld technique, which will be

described in Chapter

The relationships between these three crystal

structures together with their influence on the magnetism of the alloys in the

Gd5(SixGelJ4 system, will be also briefly mentioned in Chapter

Bragg angle,

(deg.)

Figure

6.43.

The observed (circles) and calculated (lines) intensities in a fragment of powder

diffraction pattern of Gd5Si2Ge2. The calculated intensity has been normalized to match the

observed profile. Peak shape, background and lattice parameters employed to compute the

calculated profile have been obtained by a full pattern decomposition of the observed data

using Le Bail's technique, as shown in

Figure

6.42.

Note, that the difference plot has been

compressed five-fold for clarity (the differences are too large to be well seen).

Crystal structure solution

6.19

Additional reading

F. David and

S. Sivia, Extracting integrated intensities from

powder diffraction data, in: Structure determination from powder

diffraction data. IUCr monographs on crystallography 13. W.

David,

Shankland, L.B. McCusker, and Ch. Baerlocher, Eds., Oxford

University Press, Oxford, New York (2002).

Wessels, Ch. Baerlocher, L. B. McCusker, and W.I.F. David,

Experimental methods for estimating the relative intensities of

overlapping reflections, in: Structure determination from powder

diffraction data. IUCr monographs on crystallography 13. W. I.

David,

Shankland, L.B. McCusker, and Ch. Baerlocher, Eds., Oxford

University Press, Oxford, New York (2002).

Peschar, A. Etz, J. Jansen and

Schenk, Direct methods in powder

diffraction

basic concepts, in: Structure determination from powder

diffraction data. IUCr monographs on crystallography 13. W. I.

David,

Shankland, L.B. McCusker, and Ch. Baerlocher, Eds., Oxford

University Press, Oxford, New York (2002).

4. C. Giacovazzo, A. Altomare,

C. Burla, B. Carrozzini, G. L.

Cascarano, A. Guagliardi, A. G. G. Moliterni,

Polidori and

Rizzi,

Direct methods in powder diffraction

applications, in: Structure

determination from powder diffraction data.

KJCr

monographs on

crystallography 13.

David,

Shankland, L.B. McCusker, and

Ch. Baerlocher, Eds., Oxford University Press, Oxford, New York

(2002).

Estermann and W.

David, Patterson methods in powder

diffraction: maximum entropy and symmetry minimum function

techniques, in: Structure determination from powder diffraction data.

IUCr monographs on crystallography 13. W.

David,

Shankland,

L.B. McCusker, and Ch. Baerlocher, Eds., Oxford University Press,

Oxford, New York (2002).

Gilmore,

Shankland and W. Dong, A maximum entropy

approach to structure solution, in: Structure determination from powder

diffraction data. IUCr monographs on crystallography 13. W. I.

David,

Shankland, L.B. McCusker, and Ch. Baerlocher, Eds., Oxford

University Press, Oxford, New York (2002).