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

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Chapter

Bragg angle,

(deg.)

Figure

6.15.

The observed and calculated powder diffraction patterns of CeRhGe3 after all

parameters were refined in the same approximation as in the previous example. The powder

diffraction data were collected from a ground sample of CeRhGe3 using Mo

radiation on a

Rigaku

TTRAX

rotating anode diffractometer. The divergence slit was 0.38'; the receiving

slit was 0.03'. The experiment was carried out in a step scan mode with a step 0.01' and

counting time

sec per step. The inset illustrates an inadequate asymmetry approximation.

improvement of the peak shape function

(Figure

6.16)

was obtained

by choosing a different asymmetry correction, i.e. that suggested by Finger,

Cox and Jephcoat (FCJ, see the footnote on page 531 and section 2.9.1),

where the effects of axial divergence are treated by introducing a two-

parameter asymmetry function. The residuals show some improvement (see

Table

6.9), and this modification has an effect on the lattice parameters: the

unit cell dimensions and zero shift refined using Howard's asymmetry

approximation are

4.3986,

10.0331

628

0.044", while they

become

4.3981,

10.0320

628

0.038" in the FCJ approximation.

Given the fact that the least squares standard deviations of lattice parameters

are smaller than

this difference is statistically significant and the

result shown in

Figure

6.16

should be considered as a better outcome of the

full pattern decomposition.

Crystal structure solution

533

Bragg angle,

(deg.)

Figure

6.16.

The observed and calculated powder diffraction patterns of CeRhGe3 after

refinement of all parameters with an asymmetry correction in the Finger, Cox and Jephcoat

approximation. The inset illustrates an adequately treated asymmetry (compare with the inset

Figure

6.15).

Table

6.10.

The list of Bragg reflections with their corresponding observed structure factors

squared determined after

Bail's full pattern decomposition of the x-ray powder diffraction

pattern of CeRhGe,.

~l~~~~l~

IF^^^^^

~l~~~~l~

00 2 13

146

534

Chapter

0 15

101 1

Some of the errors are listed as zeros because of automatic rounding to integers.

Just as in the previous example, CeRhGes is an intermetallic compound

and it makes sense to assume the highest symmetry space group

I4lmmm

as a first attempt to solve the crystal structure. Once again, we are looking

for a total of

atoms, 2 Ce, 2

and

Ge; Ce is the strongest scattering

atom

(58

electrons), while Ge is the weakest (32 electrons). Considering the

simplicity of the unit cell and the presence of only two heavy atoms (Ce), the

Crystal structure solution 535

Patterson technique should be an adequate tool to solve this crystal

structure.

The file with the observed structure factors (Table 6.10) was processed

accordingly, and the resultant distribution of the interatomic vectors in the

asymmetric part of the unit cell in the space group I4lmmm is listed in Table

6.11. A cross section of the three-dimensional distribution of the interatomic

vectors in the unit cell of

CeRhGe3 at

0 is also illustrated in Chapter 2,

Figure 2.61, left. It only makes sense to take into consideration those

vectors, which have reasonable lengths, i.e. which are approximately equal

or longer than the expected shortest interatomic distance. In the case of

CeRhGe3 they should exceed -2.5 A, and these are the six strongest peaks,

in the Patterson function which are highlighted in bold in Table 6.11

(rCe

1.85, r~,

1.35, andrGe

1.25 A).

Table 6.11. The three-dimensional distribution of the interatomic vectors (the Patterson

function) in the symmetrically independent part of the unit cell of CeRhGe3 calculated using

the observed structure factors determined from Le Bail's extraction (Table 6.10).

Peak number

Peak height

1 0 0 0 6000

2 0.5 0 0.253 1673

3 0 0 0.341 1666

0 0.5 1424

5 0 0 0.239 926

6 0.5 0 0.390 829

7 0.5 0.271 0.474 112

8 0.266 0 0.301 62

0.268 0 0.433 41

10 0.176 0.176 0.322 20

After consulting the International Tables for Crystallography (see Table

6.12) there are two sites in the unit cell with I4/mmm symmetry that may

accommodate 2 Ce atoms: 2(a) with the coordinates 0,0,0; and 2(b) with the

coordinates

0,0,1/2. Both sites differ only by a (0,0,1/2) shift of the whole

structure and produce no additional peaks on the Patterson map except

0,0,0

and its symmetrical equivalent 1/2,1/2,1/2. We note that since there are only

2 Rh atoms in the unit cell, they can be accommodated in the second two-

fold position. The corresponding Ce-Rh interatomic vector is

0,0,1/2

[(O,O,O)ce

(0,0,1/2)Rh

(0,0,1/2)ce.Rh], and it is peak No. 4 in Table 6.11.

Thus, it appears that we have to test only one possible model: 2 Ce in 2(a)

and 2 Rh in 2(b) because switching the occupancy of these two sites is the

Patterson method has

high success rate when applied to intermetallic structures mainly

because of the high resolution of the resultant three-dimensional distribution of

interatomic vectors. The latter is due to noticeably greater minimal interatomic distances

and lower atomic displacement parameters when compared to other classes of compounds.

536

Chapter

result of shifting the entire structure by 0,0,1/2.

The interatomic distance

Ce-Rh (3.11

is slightly shorter than the sum of the corresponding atomic

radii, but it remains within normal limits.'

Table

6.12.

Low multiplicity sites available in the space group

14lrnrnn-1.

Site

Coordinates of symmetrically equivalent points

Phase angles were calculated using the model where Ce is in the 2(a) and

is in the 2(b) sites. The

55.6%

is nearly identical to the previous

example, and the resultant electron density distribution is listed in

Table

6.13.

Analysis of

Table

6.13

immediately indicates that the

atom, which

was placed in the 2(b) site with coordinates 0,0,1/2, was not confirmed on

the Fourier map because there is no peak with the coordinates 0,0,1/2 or

1/2,1/2,0, which are symmetrically equivalent due to the presence of the

body-centered translation. Thus, our reasoning may have been flawed, and

and Ge could be distributed statistically between two four-fold sites or in

a single eight-fold site in this space group symmetry instead of being

ordered. The second strongest peak in the Fourier map has coordinates

1/2,0,1/4, which corresponds to the 4(d) site in space group I4/mmm. After

placing

25%

and

75%

Ge in this site and removing

from the 2(b) site,

the corresponding

44.0% and the subsequent three-dimensional Fourier

map is shown in

Table

6.1

4.2

When compared to sums of empirical atomic radii, 5 to 10% reduction of the interatomic

distances is common in intermetallic phases.

Another possibility would be to place

and

Ge (i.e. a 50:50 mixture of

and Ge in

this site) assuming that the remaining four-fold site is fully occupied by Ge, and we leave

this exercise to the reader.

Crystal structure solution

537

Both atoms have been confirmed on the Fourier map and it appears that

the next missing atom is located in the 4(e) site with coordinates 0,0,0.368.

All

interatomic distances are normal and after including this atom into the

computation of structure factors and phase angles, the corresponding

32.2%. This value is quite high, but it is still worthwhile to calculate a third

Fourier map, which is shown in

Table

6.15.

Table

6.13.

The three-dimensional electron density distribution in the symmetrically

independent part of the unit cell of CeRhGe3 calculated using the observed structure factors

determined from Le Bail's extraction

(Table

6.10)

and phase angles determined by Ce atoms

placed in the

2(a),

and

atoms placed in the

2(b)

sites of the space group

I4/mmm

(RF

55.6%).

Peak number

Peak height

0 0 0 72

0.5 0 0.250 13

0 0.3424 11

4 0 0 0.434

5 0 0 0.243 7

6 0.5

0.372 5

0.5 0.25 0.472

Table

6.14.

The three-dimensional electron density distribution in the symmetrically

independent part of the unit cell of CeRhGe3 calculated using the observed structure factors

determined from Le Bail's extraction

(Table

6.10)

and phase angles determined

Ce atoms

placed in the 2(a), and

0.25Rh+0.75Ge

atoms placed in the

4(d)

sites of the space group

I4/mmm

(RF

44.0%).

Peak number

Peak height

0 58

0.5 0 0.250 43

0 0 0.368 15

0.5 0 0

All atoms present in the model have been confirmed (peaks No.

to 3 in

Table

6.15). The fourth peak, however, is only -113 of peak No. 3 and,

therefore, it corresponds to more than 10 electrons. Assuming that an atom

or a fraction of atom is located in this position, it has prohibitively short

distances (6) with all other atoms, already located in the unit cell:

6i4

2.12

62-4

1.57

and 63-4

2.22

This result combined with a high

(compare

32.2% obtained in the last iteration with

6.9%

when all

atoms were found in the crystal structure of LaNi4,s5Sno.15) is usually

strong

indicator that the symmetry of the material has been overestimated.

538

Chapter

Table

6.15.

The three-dimensional electron density distribution in the symmetrically

independent part of the unit cell of CeRhGe3 calculated using the observed structure factors

determined from Le Bail's extraction (Table

6.10)

and phase angles determined by Ce atoms

placed in the 2(a), and 0.25Rh+0.75Ge atoms placed in the 4(d) and

4(c)

0.3681 sites of

the space group I4lmrnm

(RF

32.2%).

Peak number

z Peak height

1 0 0 0 47

2 0 0 0.363 25

3 0.5 0 0.250 23

4 0 0 0.21 1 8

0 0 0.100

It is possible to use this model of the crystal structure and attempt

Rietveld refinement (as will be illustrated in Chapter

7),

but we will proceed

with testing other space groups from the list of

possible (14/mmm, 14rn2,

142m, I4mm, 1422, I4/m,

and 14). Analysis of space groups 14m2 and

142m indicates that available low-multiplicity sites are essentially identical

to those of the space group I4/mmm. When these two groups are tested as

described above, the resultant models are also quite suspicious.

The next space group on the list is I4mm (Table

6.16).

This group has no

fixed origin along the Z-axis: one available two fold site 2(a) has coordinates

0,0,z, and the only available 4-fold site has coordinates 1/2,0,z. We note that

there is no reason to recalculate the Patterson function, because its symmetry

remains

I41mmm. To ensure that we do not place any of the atoms

incorrectly we now position only 2 Ce atoms in the 2(a) site in this space

group. Because the origin along the Z-axis can be chosen arbitrarily in this

space group, it does not matter which

z-coordinate is chosen for Ce. After

placing 2 Ce in

2(a) with z

0.000, the RF is 42.6

and the resultant

electron density distribution is shown in Table

6.17.

Table

6.16.

Low multiplicity sites available in the space group

I4mm.

Site

Coordinates of symmetrically equivalent points

2(a) o,o,z; 1/2,1/2,1/2+z

4(b) 0,112,z;

1/2,o,z; 1/2,0,1/2+z; 0,1/2,1/2+z

8(c)

XJA

xJ,z;

1/2+x, 112+x, 1/2+z; 112-x, 1/2+x, 1/2+z;

x, x,z; x, 42; 112-x, 112-x, 1 12+z; 1 12+x, 1 /2+x,

/2+z

8(d) x,o,z;

OAz;

1/2+x, 1/2,1/2+z; 112,1/2+x, 1 /2+z;

x,o,z;

0, x,z;

112-x, 1/2,1/2+z; 112,112-x, 1 /2+z;

There is no sharp reduction of peak heights between any pair of peaks in

Table

6.1 7

except after the first, and therefore, we proceed by adding just

one two fold-site for the next iteration. Choosing peak No.2 and assuming

that it is the next strongest scattering atom, i.e.

Rh,

the distance between this

peak and Ce atom in 2(a) with

0.000 is normal. The residual did not

Crystal structure solution 539

change after

in 2(a) with

0.662 has been added to the model (RF

42.3

%).

The subsequent Fourier map is shown in Table 6.18.

Table

6.17. The three-dimensional electron density distribution in the symmetrically

independent part of the unit cell of CeRhGe3 calculated using the observed structure factors

determined from Le Bail's extraction

(Table

6.10) and phase angles determined by the Ce

atom placed in the 2(a) site of the space group I4mm with

0.000 (RF

42.6

%).

Peak number

Peak height

1 0 0 0 60

2 0 0 0.662 9

0.5 0 0.253 9

4 0 0 0.341 9

5 0 0 0.500

6 0 0 0.429

Table

6.18. The three-dimensional electron density distribution in the symmetrically

independent part of the unit cell of CeRhGe3 calculated using the observed structure factors

determined from Le Bail's extraction

(Table

6.10) and phase angles determined by the Ce and

Rh atoms placed in the 2(a) sites of the space group I4mm with

0.000 and 0.662,

respectively (RF

42.3

%).

Peak number

Peak height

1 0 0 0 5 5

2 0 0 0.665 27

0 0 0.427 9

4 0.5 0 0.259 9

5 0 0 0.242

0.495

Proceeding slowly, we add the third peak as Ge and change the

coordinate of

from 0.662 to 0.665 as had been determined from the latest

electron density map (Table 6.18). All distances remain normal and the

residual lowers to

32.1

the Fourier map calculated using phase

angles determined by all three independent atoms is shown in Table 6.19.

The fourth peak is now about twice as high as the fifth, and all atoms

already present in the unit cell have been confirmed. After changing

coordinates of

and the first Ge atom to 0.666 and 0.429, respectively, and

adding the coordinates of the fourth peak as Ge in 4(b) with

0.262 the

9.7

which is a much lower value when compared to that achieved in the

space group I4lmmm. The Fourier map calculated using phase angles

determined by all four independent atoms is shown in Table 6.20. As follows

from this Fourier map, all four atoms have been confirmed and in addition,

there is a sharp reduction in peak heights after the fourth maximum in Table

6.20. The latter is a clear indication that the structure solution is completed.

Therefore, the model (Table 6.21), built in the non-centrosymmetric space

group

I4mm, may be considered a correct solution of the problem and all

Chapter

structural parameters of CeRhGe3 should be finalized by Rietveld refinement

(see section

7.4).

Table 6.19. The three-dimensional electron density distribution in the symmetrically

independent part of the unit cell of CeRhGe, calculated using the observed structure factors

determined from Le Bail's extraction (Table 6.10) and phase angles determined by Ce in 2(a)

with

0.000,

in 2(a) with

0.665 and Ge in 2(a) with

0.427 in the space group

I4mm (RF

32.1

%).

Peak number

Peak height

0 0 0 47

2 0 0 0.666 33

3 0 0 0.429 23

0.5 0 0.262 9

5 0.5 0 0.400 5

Table 6.20. The three-dimensional electron density distribution in the symmetrically

independent part of the unit cell of

CeRhGe, calculated using the observed structure factors

determined from Le Bail's extraction (Table 6.10) and phase angles determined by Ce in 2(a)

with

0.000, Rh in 2(a) with

0.666, Ge in 2(a) with

0.429 and Ge in 4(b) with

0.262 in the space group I4mm (RF

9.7%). See also Figure

2.59,

left, in Chapter 2, where

this Fourier map is visualized, although with a different selection of the origin of coordinates.

Peak number

Peak height

0 0 0 42

2 0 0 0.665 32

0.428 22

4 0.5 0 0.265 20

5 0 0 0.173 2

6 0.25 0 0.700 2

Table 6.21. Coordinates of atoms in the unit cell of CeRhGe, as determined from x-ray

powder diffraction data in the space group symmetry

I4mm.

Atom

Site

2(a)

0 0

0.000

2(a)

0 0.665

Ge 1

2(a) 0 0 0.428

Ge2 4(b) 112

0 0.265

Pearson symbol of this crystal structure is

tIlO

and after consulting

Pearson's Handbook it easy to find that it belongs to the BaNiSn3-type

structure.' The model of this crystal structure is shown in

Figure

6.1

Doerrscheidt and

Schaefer, The structure of barium-platinum-tin (BaPtSn,), barium-

nickel-tin (BaNiSn,) and strontium-nickel-tin (SrNiSn,) and their relation to the thorium-

chromium-silicon (ThCr2Si2) structure type,

Less-Common Met. 58,209 (1978).

Crystal structure solution

Figure

6.17.

The model of the crystal structure of CeRhGe3 as determined from x-ray powder

diffraction data.

6.11

Crystal structure of CeRhGe3 from neutron data1

We now consider a neutron powder diffraction pattern of CeRhGe3. The

file with experimental data

(Ch6ExO3-Neut-dat)

is found on the

accompanying this book. The full pattern decomposition has been conducted

using LHPM-Rietica and peak shapes have been represented by means of the

pseudo-Voigt function with asymmetry in Howard's approximation (see the

footnote on page 5

18).

The progression of the Le Bail refinement is shown

by the corresponding figures of merit, which are listed in

Table

6.22.

The

obtained result is illustrated in

Figure

6.18

and the individual structure

factors are listed in

Table

6.23.

few additional intensity maxima appear

between

35.2

and

39.1"

due to instrumental contribution, and this region

of the powder diffraction pattern was excluded from calculations.

Zaharko, V.K. Pecharsky and

K.A.

Gschneidner, Jr., unpublished. Neutron diffraction

data were collected using the High Resolution Position Sensitive Detector Powder

Diffractometer for Thermal Neutrons at the Paul Scherrer Institute, Switzerland

(http://sinq.web.psi.ch/sinq/instr/hrpt.html).

The same alloy was used in the two

experiments (x-ray data in section

6.10,

and neutron data in this section).