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

Подождите немного. Документ загружается.

Chapter

CeRhGe,, neutrons,

1.494

8l,,,,,,#C8#,,#L

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Bragg angle,

(deg.)

Figure

6.18.

The observed and calculated powder diffraction patterns of CeRhGe3 after

refinement of all parameters. The region

35.2

39.1

(outlined in the inset) was

excluded from the refinement because it contains three additional peaks due to instrumental

contribution. The powder diMaction data were collected with a step

0.05'

20.

Table

6.22.

Figures of merit obtained at different stages during the full pattern decomposition

of the powder diffraction pattern of CeRhGe3 collected using thermal neutrons with the

wavelength

1.494

Re,,

3.09

The temperature of the experiment was

200

Refined parameters Illustration RD RwD

Background (linear)

27.11 34.38 124.0

20.57 27.18 77.52

rlo,

asymmetry

3.87 5.33 2.99

All, background

(4'h

order).

Figure

6.18

3.54 4.78 2.40

Table

6.23.

The list of Bragg reflections and observed structure factors squared determined

after Le Bail's full pattern deEomposition of the neutron powder diffraction data of CeRhGe3.

IF"^"^

(rlFObS12

IF^^^^^

OIF~~~~~

Crystal structure solution

543

Ignoring the dissimilarities in the values of both the scattered intensity

and structure factors for x-rays (Figure

6.16

and Table

6.10,

respectively)

and neutrons, which are due to the differences in the scattering factors,' the

Coherent neutron scattering lengths (b) are 4.84, 5.88, and 8.185 fm for Ce,

and Ge,

respectively. Since for these three elements the neutron scattering ability accidentally

increases with the decreasing atomic number, Ge is determined from neutron data with

better accuracy than Ce. On the contrary, the x-ray scattering factors are proportional to

the atomic number, and therefore, Ce is determined from x-ray diffraction with a better

precision than Ge.

544

Chapter

most significant difference between the x-ray and neutron diffraction profiles

is in the widths of Bragg peaks and the

mixing

parameter of the pseudo-

Voigt function as illustrated in

Figure

6.19. The Bragg peaks in the x-ray

experiment are narrow and they are well described by a nearly pure

Lorentzian. On the other hand, Bragg peaks are much broader and they are

closer to a pure Gaussian distribution in the neutron diffraction experiment.

Since we established in the previous section that the correct space group

symmetry for this material is

I4mm, we will not

try

any other space groups.

The distribution of peaks in the Patterson function calculated using the

observed structure factors obtained in the neutron powder diffraction

experiment is shown in

Table

6.24. The UOW cross-section of the three-

dimensional Patterson function using the same powder diffraction data is

illustrated in Chapter 2,

Figure

2.61, right. The height of an interatomic

vector between two atoms of type

and

is proportional to the product of

their scattering power, which in this case is proportional to the product of the

corresponding coherent scattering lengths, bibj. For example, Ce-Ce vectors

should be proportional to b2C, (-23~10-~' fm2), while Ge-Ge vectors should

be almost three times stronger: b2G,

69x10-~' fm2.

0 20 40 60 80 100 120 140 160

Bragg angle,

(deg.)

1 .o

0.9

0.8

0.7

0.6

z0.5

0.4

0.3

0.2

Figure 6.19.

Full widths at half maximum

(FWHM)

and mixing parameters

(q)

of the pseudo-

Voigt function used to approximate peak shapes in the x-ray

(Figure 6.16)

and neutron

(Figure 6.18)

powder diffraction patterns collected from the same CeRhGe3 powder.

CeRhGe3

neutrons

./---

zH?.

neutrws

---

I/'

/--------

,/'

0.1

FwHM, M0

Crystal structure solution

545

Table

6.24.

The three-dimensional distribution of the interatomic vectors in the symmetrically

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

determined from Le Bail's extraction employing neutron diffraction data

(Table

6.23).

Patterson map peak number

Peak height

0 0 0 6000

0 0 0.5 2675

3 0.5 0 0.355 1500

0.5 0 0.145 1500

0.5 0 0.250 1500

0 0 0.237 1124

0 0 0.346 687

0 0 0.404 489

0.125 0.125 0.088 266

0.218 0.21

0.457 260

Only the eight strongest independent peaks in

Table

6.24 (highlighted in

bold) have meaningful lengths between -2.4 and 5

The differences in the

heights of Patterson peaks between

Table

6.24 and

Table

6.11 are expected

because now Ge atoms are the strongest scattering species (see previous

paragraph). Between the two sites (see

Table

6.16) that may accommodate

six Ge atoms, the 2(a) site results in the 0,0,0 vector, and the 4(b) site yields

one additional 1/2,1/2,0 vector. Indeed, the second strongest vector found in

Table

6.24 is identical to the latter (0,0,1/2

1/2,1/2,1/2

112,112,O). Thus,

the Patterson function points to a strongly scattering atom in the 4(b) site of

the space group I4mm. Since the origin of coordinates here is not fixed along

the Z-axis, we may choose any z-coordinate for the Ge atom in this site.

Assume that four Ge atoms are located in

4(b) with z

0.000. Let's

try

use the Patterson function

(Table

6.24) and locate a second Ge, which should

be in a two-fold site because there is a total of six Ge atoms in the unit cell.

The coordinates of a point in 2(a) are 0,0,z. Hence, the corresponding vector

between the two independent Ge atoms should be

1/2,0,0

0,Oq

1/2,0,-z

112,Oq due to a mirror plane perpendicular to

at z

112, which is present in

the space group I4lmmm that describes Patterson symmetry.

second vector

can be found from 1/2,0,0

1/2,1/2,1/2

0,0,z

0,112,112-z. Given the

presence of a four-fold axis parallel to

this vector is identical to 1/2,0,1/2-

z. These are vectors No. 3 and 4 in

Table

6.24 assuming z

0.355 (or

0.145). Without other atoms in the model of the crystal structure, the two

choices are equivalent, and any of the two z-coordinates may be selected to

represent Ge atom in the 2(a) site. Thus, the two independent Ge atoms,

according to the Patterson function, are as follows: 4Gel in 4(b):

112,0,0.000, and 2Ge2 in 2(a): 0,0,0.355. The corresponding residual

29.0

The coordinates of peaks found on the three-dimensional Fourier

map calculated using phase angles determined from this partial model of the

crystal structure are listed in

Table

6.25.

546

Chapter

Table 6.25. The three-dimensional nuclear 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 6.23) and phase angles determined by Ge in

4(b)

with

0.000

and Ge in 2(a) with

0.355

in the space group

I4mm

(RF

29.0

%).

Fourier map peak number

Peak height

1 0

0.355 73

0.5 0 0

3 0 0 0.754 18

4 0

0.587 15

5 0 0 0.1

0 0 0.921 10

The next strongest peak (No. 3) also belongs to the 2(a) site with the

coordinate

0.754. Its shortest interatomic distances are

3.30 and

3.36

and

63-Ge2

3.27

All three correspond to Ce rather than to Rh,

even though Ce is a light scattering atom when compared to

Rh.

Thus, after

placing 2Ce in 2(a) with

0.754, the RF

19.8

and the subsequently

calculated Fourier map is illustrated in

Table

6.26.

Table 6.26. The three-dimensional nuclear 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 6.23) and phase angles determined by Ge in

4(b)

with

0.000,

Ge in 2(a) with

0.355,

and Ce in 2(a) with

0.754

in space group I4mm

(RF

19.8

%).

Fourier map peak number

Peak height

0.353

0.5 0 0 5 5

0.758

0 0 0.580 13

0 0 0.113 10

0.5 0 0.258 7

So far three atoms have been confirmed, and the fourth strongest peak

points to an atom that also belongs to the 2(a) site with

0.580. The only

atom, which is missing fiom our model, is

Rh.

When 2

atoms are placed

in this site, the interatomic distance

8Rh-Ce

1.75

which is too short. The

fourth peak is therefore, a false maximum and it should be discarded.' After

the next strongest peak (No. 5) is tested, all distances are normal. The RF

16.3

and the subsequent Fourier map is listed in

Table

6.27, which now

displays a sharp drop in peak heights after peak No. 4. The cross-section of

False peaks may sometimes be stronger than the real peaks on the Fourier map, especially

when a structural model is incomplete and/or structure factors accuracy is relatively low,

which is the case here. One atom, which is still missing from the model, is

Rh;

it is the

second strongest scattering atom and, therefore, phase angles are relatively imprecise thus

causing the appearance of a strong false peak.

Crystal structure solution

547

the nuclear density distribution in the

XOZ

plane calculated using the same

powder diffraction data is shown in

Figure

2.59,

right, in Chapter

It is

worth noting that

Figure

2.59

represents a different origin of coordinates

since its selection along the Z-axis is arbitrary in this space group symmetry.

Table 6.27. The three-dimensional nuclear 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 6.23) and phase angles determined by Ge in

4(b)

with

0.000, Ge

2(a) with

0.355, Ce in 2(a) with

0.754, and

in 2(a) with

0.1 13 in the space group I4mm (RF

16.3

%).

Fourier map peak number

Peak height

0 0 0.349 48

0.5 0 0 47

0 0 0.108 3 3

0 0 0.759 24

0 0 0.233

0 0 0.000 3

Since we did not change the coordinates of atoms to those refined fiom

the previous Fourier map computations, we will do it now and assume the

distribution of atoms in the unit cell of

CeRhGe3 as shown in

Table

6.28.

All

atoms were assigned identical isotropic displacement parameters

0.5

A2.

The resulting RF

9.8%,

which is quite close to that obtained using x-ray

diffraction data (see previous section). The coordinates of all atoms may be

further refined by calculating another Fourier map. This is, however,

unnecessary since the model of the crystal structure

(Figure

6.20)

appears

complete and all relevant structural parameters should and will be refined

using the Rietveld technique based on the available neutron powder

diffraction data (see Chapter

7).

Table

6.28.

Coordinates of atoms in the unit cell of CeRhGe3 as determined from neutron

powder diffraction data in the space group symmetry I4mm. RF

9.8%.

Atom Site

Ge 1 4(b) 0.5 0 0.000

Ge2 2(a) 0 0 0.349

2(a) 0 0 0.759

Rh 2(a) 0 0 0.108

comparison of

Table

6.21

and

Table

6.28

indicates that the coordinates

of the atoms are different in the two models. Nonetheless, the models are

identical: the two crystallographic bases are related to one another by the

center of inversion and by different origins of coordinates, which is easily

seen in

Figure

6.20.

548 Chapter

Figure

6.20.

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

(left) and neutron (right) powder

diffraction

data. The two models are identical except for the

inversion of the coordinate system and differently chosen origin of the unit cell.

6.12

Crystal structure

Nd5Si41

The next example is also an intermetallic compound, Nd5Si4, which is the

most complex among those considered so far. The powder diffraction pattern

(Figure 6.21, data files

Ch6ExO4-CuKa.dat

and

Ch6Ex04-CuKa.xy

are

found on the

CD)

has been indexed in a tetragonal crystal system with a

7.871 and

14.812

Analysis of the systematic absences indicates that

reflections hOO with h

2n and 001 with 1

4n are extinct and points to two

possible space groups: P41212 or P43212. The complexity of this example

arises from a large primitive unit cell of the material, which yields over 440

Bragg reflections possible between 18 and

120' 28 when using Cu

radiation. One of the major difficulties in the full pattern decomposition here

is in the uncertainty of the background at high Bragg angles, where multiple

reflections heavily overlap. Therefore, the background baseline should be

monitored at all times during the refinement.

The progression of the Le Bail full pattern decomposition is illustrated in

Table

6.29

and the results are shown in Figure 6.21. Bragg peaks were

represented by the pseudo-Voigt function with Howard's asymmetry

correction.

A.O.

Pecharsky,

K.A.

Gschneidner, Jr., and

V.K.

Pecharsky, unpublished. The alloy was

prepared by arc-melting a stoichiometric mixture of pure components.

Crystal structure solution

549

20 40 60 80 100 120

Bragg angle,

(deg.)

Figure

6.21.

The observed and calculated powder diffraction patterns of Nd5Si4 after

refinement of all parameters. The powder diffraction data were collected from

ground

sample of Nd& using Cu

radiation on a Rigaku TTRAX rotating anode diffractometer.

The divergence slit was 0.75' and the receiving slit was 0.03'. The experiment was carried

out in a continuous scanning mode with a sampling step 0.02" and a scan rate of 0.5

deglmin.

The inset shows the low intensity region of the powder diffraction pattern with a properly

determined background (solid line).

Table

6.29.

Figures of merit obtained at different stages during the full pattern decomposition

of the powder diffraction pattern of Nd5Si4. Wavelengths used:

hKal

1.54059

hKa2

1 .5444l

Re,,

4.64

Refined parameters Illustration R, RwD

Background (linear) 13.70 19.06 16.86

Background (5th order) 10.42 14.15 9.29

qo,

asymmetry 5.14 7.03 2.30

All

Figure

6.21

4.70 6.54 1.99

The measured gravimetric density of the alloy is 5.96 g/cm3 and one unit

cell contains

3.95

formula units of Nd5Si4. The two possible space

groups symmetry are enantiomorphous and they cannot be distinguished

using powder diffraction data. The results shown in

Table

6.29

and

Figure

550 Chapter 6

6.21 are obtained in the space group P41212. Due to a large number of Bragg

reflections, the list of individual structure factors squared is found on the CD

in the data file

Ch6Ex04-F2.dat.

The complexity of this crystal structure precludes easy interpretation of

the Patterson function because there is a total of 20 Nd atoms in the unit cell.

Therefore, we will try to solve the crystal structure of this alloy using direct

methods. The space group symmetry

P41212 contains only two possible sites,

which are listed in Table 6.30.

Table

6.30.

Sites available in space group symmetry

P4!2!2.

Site Coordinates of symmetrically equivalent points

- -

4(a)

x,X,o; x, x, 112; 112-x, 1/2+x, 114; 1 /2+x, 112-x,3/4

The array of the individual structure factors determined from Le Bail's

full pattern decomposition was processed using WinCSD (see footnote No. 4

on page 515) and, according to a combined figure of merit, one of the

possible solutions was notably better than the others. The subsequently

calculated E-map is listed in Table

6.31.

The first two peaks in Table 6.31 correspond to atoms in the general site

position

8(b) of the space group P41212, and the third peak indicates an atom

in the 4(a) site. Interatomic distances among peaks No. 1 to 3 vary from -3.4

to -3.7 A, which match well the atomic radius of Nd (-1.85 A). Thus, it

appears that the direct phase angles determination results in finding all

atoms, which are expected to be in one unit cell of the compound (8

20). Recalling that Nd is the strongest scattering atom in this crystal

structure, it makes little, if any, sense to further analyze the E-map: the

locations of Si atoms should be easily revealed on a subsequent Fourier map

(Table 6.32) after calculating reflection phases using the coordinates of the

first three peaks as three independent Nd atoms. At this point

1.1

Table

6.31.

The three-dimensional E-map in the symmetrically independent part of the unit

cell of Nd5Si4 calculated using the observed structure factors determined from Le Bail's

extraction and directly determined phase angles in the space group

P4,2,2.

E-map peak number

Peak height

1 0.3698 0.0021 0.4543 36

2 0.1268 0.9925 0.8768 28

0.81 18 0.1882 0.7500 19

4 0.2089 0.1725 0.6881 8

5 0.8661 0.9856 0.2168 7

6 0.1218 0.0046 0.0224

7 0.6805 0.1999 0.43 11 7

Crystal structure solution

55 1

Table

6.32.

The three-dimensional Fourier map in the symmetrically independent part of the

unit cell of Nd5Si4 calculated using the observed structure factors determined from Le Bail's

extraction and phase angles determined by three Nd atoms: 8Ndl in 8@) with

0.3698,

0.0021,

0.4543; 8Nd2 in 8(b) with

0.1268,

0.9925,

0.8768 and 4Nd3 in 4(a)

with

0.31 18 in the space group P41212 (RF

31.1

%).

Fourier map peak number

Peak height

1 0.3713 0.0053 0.4564 75

2 0.1276 0.9908 0.8780 72

3 0.8107 0.1893 0.7500 63

4 0.4238 0.1989 0.8100 13

5 0.1799 0.2036 0.31 14 12

6 0.2322 0.0429 0.4980 10

7 0.1282 0.0793 0.1500 9

8 0.745

1 0.0777 0.4221 9

All three Nd atoms were confirmed on the Fourier map. Even though

there is no sharp reduction of peak heights among peaks 4 through

calculation of the interatomic distances indicates that only peaks No. 4 and

can be used as positions of silicon atoms. Both coordinate triplets correspond

to the general site in the space group P41212. Assuming that these are the

missing Si atoms, the Nd-Si distances vary between -2.9 and -3.3

A, and

Si-Si distances are 2.50

(rsi

1.10 A). The calculated

27.3

when

all five atoms are placed as determined from the E-map (Nd) and the Fourier

map (Si), and the coordinates of the electron density peaks on the subsequent

Fourier map are shown in

Table

6.33.

The entire model of the crystal structure has been confirmed

the last

Fourier map, which in addition has a two-fold reduction in the heights of the

observed electron density maxima after peak No. 5. After the coordinates of

all atoms have been changed as determined

the latest electron density

distribution, the model of the crystal structure, which is shown in

Table

6.34,

results in

27.1

Table

6.33.

The three-dimensional Fourier map in the symmetrically independent part of the

unit cell of Nd5Si4 calculated using the observed structure factors determined from Le Bail's

extraction and phase angles determined by three Nd and two Si atoms: 8Ndl in 8(b) with

0.3698,

0.0021,

0.4543; 8Nd2 in 8(b) with

0.1268, y

0.9925,

0.8768; 4Nd3 in

4(a) with

0.31 18; 8Sil in 8(b) with

0.4238,

0.1989,

0.8100 and 8Si2 in 8(b)

with

0.1799,

0.2036,

0.3 114 in the space group P4J12 (RF

27.3

%).

Fourier map peak number

Peak height

1 0.3714 0.0057 0.4561 6 1

2 0.1280 0.9903 0.8779 5 9

3 0.8103 0.1897 0.7500 52

4 0.4220 0.2004 0.8095 17

5 0.1778 0.2070 0.3 108 15

6 0.2371 0.0395 0.5003 7

7 0.8345 0.1294 0.3169 7

8 0.1239 0.0444 0.1529 7