Download Atomic disordering in YB56 detected by high

PHILOSOPHICAL MAGAZINE A, 1999, VOL. 79, N O. 4, 821± 834
Atomic disordering in YB56 detected by high-resolution
electron microscopy with residual indices
Takeo Oku
Institute of Scienti® c and Industrial Research, Osaka University, Mihogaoka 8-1,
Ibaraki, Osaka 567-0047, Japan
and Jan-Olov Bovin
National Center for HREM, Inorganic Chemistry 2, Chemical Center,
Lund University, PO Box 124, S-221 00 Lund, Sweden
[Received 9 March 1998 and accepted in revised form 20 May 1998]
A bstract
Disordering of both yttrium and boron atoms in YB56 was observed, in three
dimensions, from di€ erence images of high-resolution electron microscopy
(HREM) and simulated images from an X-ray determined structural model.
Averaged HREM images recorded using a slow-scan charge-coupled device
camera from thin regions (less than 5 nm) along the [100], [110] and [111]
directions of the YB56 were compared with calculated images based on the Xray data. A residual index RHREM was used for image analysis to determine the
crystal thickness and defocus value of the observed images. Based on the
experimental conditions which minimized the RHREM values, di€ erence images
between observed and simulated images were calculated. These di€ erence images
observed along the three directions of the YB56 showed the atomic disorder,
which are yttrium positions and boron clusters around the yttrium atoms. This
approach is useful for the evaluation of disordered atomic positions in the crystal.
§ 1. Introduction
High-resolution radiation-resistant monochromators are required for soft X-ray
synchrotrons in the 1± 2 keV energy region. Gadolinium gallium garnet and beryl are
used for monochromators because of the d spacings of these materials. However, the
crystal quality of these materials is low, and they su€ er from synchrotron radiation
damage. Recently, high-quality YB56 single crystals, with the YB66 structure (Seybolt
1960, Richards and Kasper 1969, Slack et al. 1977) , have been synthesized for use as
high-resolution and synchrotron-radiation-resistant monochromators (Tanaka et al.
1994) . After the discovery of the application of YB56 as a soft-X -ray monochromator, the crystal structure of YB56 a 2.346 00 nm has been determined
using single-crystal X-ray di€ ractometry (Higashi et al. 1997) . The boron framework
of YB56 is basically made of eight super-icosahedra B12 13 and eight non-icosahedral B80 clusters (total, 1584 boron atoms) . The YB56 has yttrium site occupancies of
0.575, which suggests that the peanut-shaped `Y hole’ (a pair of yttrium atoms ) in the
boron clusters should be occupied in most cases by only one yttrium atom (Richards
and Kasper 1969, Slack et al. 1977, Higashi et al. 1997) . The structure of YB56- 66 has
been investigated by transmission electron microscopy (Slack et al. 1977) , scanning
tunnelling microscopy (Perkins et al. 1996) , and atomic force microscopy
(Kuhlmann et al. 1996) . The scanning tunnelling microscopy showed a periodicity
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1999 Taylor & Francis Ltd.
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T. Oku and J.-O. Bovin
of 1.2 nm, which indicates that there is no surface reconstruction at the YB66 surface
(Perkins et al. 1996) . To our knowledge, no direct imaging of single yttrium atomic
positions in the boron clusters has yet been done.
In our previous work (Oku et al. 1998) , the Y holes in the YB56 were directly
detected in the boron clusters by digital high-resolution electron microscopy
(HREM) using a slow-scan charge-coupled device (CCD) camera, and the `local’
structure model was proposed for yttrium atom arrangements. In addition, digital
HREM images recorded along [100], [110] and [111] directions of the YB56 crystals
showed `averaged’ yttrium atom arrangements inside the non-icosahedral boron
clusters. However, the dark contrast of yttrium atom positions in crystal regions
thinner than 5 nm was smeared compared with those of calculated images based on
X-ray data, which indicated non-unif ormity of yttrium atom positions. When the Y
holes are formed, the cubic system should be destroyed locally around them. It is
di cult to determine the atomic arrangement of boron around the Y holes because
of the low atomic number of boron. However, it is believed that the `averaged’
images should have `averaged’ information for the disordering of the yttrium and
boron atoms around the Y holes.
The purpose of the present work is to obtain the information for disordering of
atomic arrangements in YB56 from digital HREM images recorded along three
di€ erent directions. Imaging plates (Shindo et al. 1991, 1994, Oku et al. 1995) and
a slow-scan CCD camera (Pan and Crozier 1993, Sasaki et al. 1995, Ohnishi and
Hiraga 1996, Oku et al. 1998) , which have high linearity and electron sensitivity, are
suitable tools for recording HREM images digitally. In the present work, a slow-scan
CCD camera was used to compare the observed images with calculated images
because of the ease of handling digital data. In addition, a residual index
RHREM
Iobs Ical /
Iobs was used for image analysis because of its simple
form and usefulness for determining the crystal thickness and defocus value of
observed images (Shindo et al. 1994). Based on the crystal thickness and defocus
value with the minimum RHREM , di€ erence images between observed and simulated
images have been calculated. These di€ erence images were expected to show directly
disordered atomic positions in the YB56 crystal.
§ 2. Experimental procedures
High-quality single crystals of YB56 were grown by an indirect heating ¯ oatingzone method (Tanaka et al. 1994) . The molten zone was heated by radiation from an
inductively powered tungsten ring, placed between the work coil and the molten
zone. Growth conditions for high quality YB56 single crystals were as follows:
1
1
growth direction, [100]; growth rate, 10 mm h ; rotation rate, 6 rev min (about
the growth axis) ; atmosphere, 0.3 MPa (He) ; crystal composition ratio [B]/[Y] 56;
molten zone composition ratio [B]/[Y] 40.
Samples for HREM observations were prepared by dispersing crushed material
on a holey carbon grid. HREM observations were performed with 400 kV electron
microscope (JEOL JEM-4000EX) having a point-to-point resolution of 0.16 nm. The
electron microscope was equipped with a slow-scan CCD camera (Gatan SSC model
694) . The area of detection of the CCD camera was 1024 1024 pixels with a pixel size
of 24 24 m m. For image processing of the observed HREM images, Digital
Micrograph (Gatan, Inc. ) , Adobe Photoshop and Semper software (Synoptics Ltd)
were used. As a ® rst step, the digital images were masked and fast Fourier transformed.
The reciprocal lattice was indexed, and the lattice parameters were determined using
Atomic disordering in Y B56
823
the positions of the strongest peaks in the transform. The local background was
subtracted, and the amplitudes and phases of the peaks were re® ned using symmetrization (HovmoÈ ller et al. 1984, Carlsson 1992) . Before correcting the phases, the
phase origin was determined by investigating the origin shift which gave the best
accordance with the phase conditions for the two-dimensional space group.
Averaged symmetrized images were reconstructed from the corrected Fourier transform. To compare observed images with calculated images, HREM images were
produced from the multislice method (Cowley and Moodie 1957, Cowley 1981)
using the MacTempas software (Total Resolution ). The parameters used in the
image calculations were as follows: accelerating voltage, 400 kV; radius of the objec1
tive aperture, 6.3 nm ; spherical aberration Cs 1.0 mm; focus spread ¢ 8 nm;
convergence semi-angle a
0.55 mrad.
§ 3. R esults
3.1. Preliminary observation of Y B56
Three directions of the YB56 crystals were selected in order to obtain the structural image of the YB56 . Figures 1 (a) , (b) and (c) show HREM images of the YB56
recorded along the [100], [110] and [111] directions respectively using the slow-scan
CCD camera. The left-hand side of each image is the thinnest region of the YB56
crystals. To get an optimal resolution (less than 0.016 nm per pixel) , the digital
6
images were recorded at microscope magni® cations in the range 1.0- 1.2
10 .
The images were recorded close to the Scherzer defocus. In order to observe the
atomic arrangements more clearly, image processing was carried out using Fourier
® ltering, lattice averaging, symmetrization and convolution. The processed images of
the thinnest parts (less than 5 nm ) of ® gures 1 (a) ± (c) are shown in ® gures 1 (d) ± ( f )
respectively.
3.2. High-resolution electron microscopy along [100] of Y B56
To investigate this `averaged’ YB56 structure, HREM images were calculated on
the basis of the structure model of YB56 determined by X-ray di€ raction as listed in
table 1. HREM images calculated along the [100] direction are shown in ® gure 2. The
image calculations were carried out for various under-def ocus values and crystal
thicknesses to determine the imaging condition of the observed image. Since the
unit cell parameter of YB56 is large a 2.346 nm; 1608 atoms per unit cell) , the
unit cell was divided into four subslices. As observed in ® gure 2, contrast change of
an image is more sensitive to the defocus value the crystal thickness. In the images
with a thickness of one slice, asymmetry contrast due to the structure in one subslice
is observed.
To compare the observed image of YB56 recorded along the [100] with the
calculated images, RHREM values were calculated as shown in ® gure 3. The image
calculations were carried out for the image size of 512 512 pixels and an image
depth of 0± 255 grey scales. Several times calculations were carried out for minimization of RHREM values. The cumulative distribution functions were also used to get
the minimum RHREM values (Smith and Eyring 1982) . RHREM values of the HREM
image of [100] incidence are shown as a function of defocus value (® gure 3 (a)) and
crystal thickness (® gure 3 (b)) . In ® gure 3 (a), the crystal thickness is ® xed at four
slices (2.346 nm) , and the defocus value is changed. RHREM values were reduced to
0.3 in the range 30± 60 nm, and the minimum RHREM value of 0.142 was obtained at a
defocus of 50 nm. In ® gure 3 (b) , the defocus value is ® xed at 50 nm, and the crystal
824
T. Oku and J.-O. Bovin
Figure 1. HREM images of YB56 crystals recorded along the (a) [100], (b) [110] and (c) [111]
directions using a slow-scan CCD camera. Processed images of thin parts of the
crystals along the (d) [100], (e) [110] and ( f ) [111] directions.
thickness is changed. RHREM values were reduced to 0.2 in the range of two to six
slices, and the minimum RHREM value of 0.142 was obtained at a crystal thickness of
four slices (2.346 nm) .
HREM images of one unit cell of YB56 along the [100] direction and the projected model are shown in ® gure 4. Figure 4 (a) is one unit cell of the observed
HREM image taken from ® gure 1 (d) . A calculated image at a defocus of 50 nm
825
Atomic disordering in Y B56
Table 1.
Structural parameters used for image calculations of YB56 (Higashi et al. 1997)
(space group, Fm3c; a 2.346 nm).
Atom
x
y
z
Occupancy
Ba
nm2
Y
B(1)
B(2)
B(3)
B(4)
B(5)
B(6)
B(7)
B(8)
B(9)
B(10)
B(11)
B(12)
B(13)
0.05789
0
0
0
0
0
0.0389
0.0397
0.0631
0.0635
0.1318
0.2329
0.1734
0.2362
0.25
0.0376
0.0763
0.0387
0.1482
0.1856
0.1400
0.0812
0.0773
0.1457
0.1745
0.1586
0.1279
0.2362
0.25
0.0599
0.1169
0.1815
0.2419
0.1716
0.1217
0.2296
0.1590
0.1947
0.1960
0.3004
0.2588
0.2362
0.575
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.71
0.64
0.33
0.22
0.0209
0.008
0.009
0.007
0.007
0.008
0.010
0.007
0.008
0.009
0.027
0.058
0.033
0.017
a
Isotropic temperature factor.
and a crystal thickness of four slices (2.346 nm) , which showed the minimum RHREM
values in ® gure 3, is shown in ® gure 4 (b) . The yttrium atom positions appear as
black dots in ® gure 4 (a) and (b) , and there are two kinds of darkness at the yttrium
positions in both images. One set of two sites yttrium sites lies along [100] (single
asterisk) and one set of two sites is perpendicular to [100] (double asterisk) , thus
giving a yttrium atom density along [100] which is twice that of the former set.
However, the darkness of both yttrium atoms in the experimental image in ® gure
4 (a) is fairly weak compared with the calculated image in ® gure 4 (b) . Double rings
with dark contrast are observed at the centre and the corners of the unit cell in both
images, which correspond to the super-icosahedral B12 13 clusters as shown in the
projected structure model in ® gure 4 (d) . Square-like rings with dark contrast are also
observed around the darker yttrium atom positions in both images, which correspond to the non-icosahedral B80 clusters. A di€ erence image of ® gures 4 (a) and (b)
is shown in ® gure 4 (c) . Yttrium atom positions (indicated by arrows) and nonicosahedral B80 clusters around the Y holes show white contrast, which indicates
that the di€ erence between the observed image (® gure 4 (a)) and the calculated image
(® gure 4 (b)) is large at these positions. The super-icosahedra B12 13 do not show a
large di€ erence.
3.3. High-resolution electron microscopy along [110] and [111] of Y B56
To investigate the YB56 structure three-dimensionally, HREM image calculations along the [110] and [111] directions were carried out. Calculated HREM images
for various under defocus values and crystal thicknesses are shown in ® gure 5 (a) .
Since the unit cell parameter of YB56 is large along the [110] direction (3.318 nm) , the
unit cell was divided into four subslices along the [110] direction. As observed in
® gure 5 (a) , contrast changes in an image are more sensitive to the defocus value than
the crystal thickness. In the images with a thickness of one and three slices, asymmetry contrast due to the structure in one subslice is observed.
826
Figure 2.
T. Oku and J.-O. Bovin
HREM images of YB56 calculated along the [100] direction. One slice is 0.586 nm.
To compare the observed image of YB56 recorded along [110] with the calculated
images, RHREM values of the HREM image of [110] incidence are calculated as a
function of defocus value and crystal thickness, as shown in ® gures 5 (b) and (c)
respectively. Image calculations were carried out for an image size of 509 359
pixels and an image depth of 0± 255 grey scale. In ® gure 5 (b) , the crystal thickness
is ® xed at four slices (3.318 nm) , and the defocus value is changed. Low RHREM
values of about 0.2 are observed in the range 30± 60 nm, and a minimum RHREM
value of 0.193 was obtained at a defocus of 39 nm. In ® gure 5 (c) , the defocus value is
® xed at 39 nm, and the crystal thickness is changed. RHREM values are reduced to
0.25 in the range of two to seven slices, and the minimum RHREM value of 0.193 was
obtained at a crystal thickness of four slices (3.318 nm) .
Figure 6 shows HREM images and a projected model of one unit cell of YB56
along the [110] direction. A HREM image of one unit cell taken from ® gure 1 (e) is
shown in ® gure 6 (a) . Figure 6 (b) is a calculated image at a defocus of 39 nm and a
crystal thickness of four slices (3.318 nm) , which corresponds to the minimum
RHREM values in ® gures 5 (b) and (c). The yttrium atom positions appear as black
Atomic disordering in Y B56
827
Figure 3.
RHREM values of [100] image as a function of (a) defocus value and (b) crystal
thickness.
Figure 4.
(a) Observed image, (b) calculated image, (c) di€ erence image of (a) and (b), and (d)
structure model of YB56 projected along the [100] direction.
828
T. Oku and J.-O. Bovin
(a)
Figure 5. (a) HREM images of YB56 calculated along the [110] direction. One slice is
0.829 nm. RHREM values of [110] image as a function of (b) defocus value and (c)
crystal thickness.
Atomic disordering in Y B56
Figure 6.
829
(a) Observed image, (b) calculated image, (c) di€ erence image of (a) and (b), and (d)
structure model of YB56 projected along the [110] directions.
dots in ® gures 6 (a) and (b) as indicated by asterisks. For the [110] direction, two
types of yttrium atom position are observed in the unit cell as shown in the projected
model of ® gure 6 (d) . A di€ erence between the yttrium atom distances of the projected model (® gure 6 (d)) and the calculated image (® gure 6 (b)) is observed, which is
due to imaging conditions and the boron clusters around the Y holes. In the
observed image in ® gure 6 (a) , only one type of yttrium position is separated as
indicated by the double asterisks. The darkness of yttrium atoms in the observed
image in ® gure 6 (a) is weak compared with the calculated image in ® gure 6 (b) , which
indicates a smearing of the yttrium atom positions. This will be explained later. The
boron clusters show dark contrast in the observed (® gure 6 (a)) and calculated (® gure
6 (b)) images. The di€ erence image of ® gures 6 (a) and (b) is shown in ® gure 6 (c) . To
observe the di€ erence more clearly, the image contrast was enhanced. Yttrium atom
positions and non-icosahedral B80 clusters around the Y holes show white contrast.
This indicates that the di€ erence between the observed image (® gure 6(a)) and the
calculated image (® gure 6 (b )) is large at these positions.
Figure 7 (a) shows HREM images calculated along the [111] direction for various
defocus values (10± 70 nm) and crystal thicknesses. The unit cell was divided into six
subslices along the [111] direction because of the large unit-cell parameter of YB56
along the [111] direction (4.063 nm) . The contrast change of the images is sensitive to
both the defocus value and the crystal thickness as observed in ® gure 7 (a) .
Asymmetry contrast due to the structure is clearly observed in the images of thickness less than six slices.
The RHREM values of the observed HREM image of [111] incidence are shown as
a function of defocus value and crystal thickness in ® gures 7 (b) and (c) respectively.
830
T. Oku and J.-O. Bovin
(a)
Figure 7. (a) HREM images of YB56 calculated along the [111] direction. One slice is
0.677 nm. RHREM values of [111] image as a function of (b) defocus value and (c)
crystal thickness.
Atomic disordering in Y B56
831
The image calculations were carried out for the image size of 512 443 pixels with
0± 255 grey scale. In ® gure 7 (b) , the crystal thickness is ® xed at six slices (4.063 nm) ,
and the defocus value is changed. Low RHREM values of about 0.4 are observed in
the range 30± 60 nm, and the minimum RHREM value of 0.251 was obtained at a
defocus of 45 nm. In ® gure 7 (c) , the defocus value is ® xed at 45 nm, and the crystal
thickness is changed. RHREM values are reduced to 0.3 in the range of ® ve to ten
slices, and the minimum RHREM value of 0.251 was obtained at a crystal thickness of
six slices (4.063 nm) .
HREM images and a projected model of one unit cell of YB56 along the [111]
direction are shown in ® gure 8. A HREM image of one unit cell cut from ® gure 1 ( f )
is shown in ® gure 8 (a) . Figure 8 (b) is a calculated image at a defocus of 45 nm and a
crystal thickness of six slices (4.063 nm ), which corresponds to the minimum
RHREM values in ® gure 7 (b) and (c). For the [111] direction, yttrium atom positions
appear as dark spots in both the observed (® gure 8 (a)) and the calculated (® gure 8 (b))
images as indicated by arrows. The shortest distance of yttrium atoms in the projected
plane is 0.192 nm, which is clearly separated in both images. However, the darkness of
the yttrium atom positions in the observed image of ® gure 8 (a) is weaker than those in
the calculated image in ® gure 8 (b) , just as in the [100] and [110] directions. Boron
clusters appear dark in both ® gure 8 (a) and ® gure 8 (b). Figure 8 (c) is a di€ erence
image of ® gures 8 (a) and (b). Yttrium atom positions and non-icosahedral B80 clusters
around the Y holes show white contrast. The super-icosahedral B12 13 clusters also
Figure 8.
(a) Observed image, (b) calculated image, (c) di€ erence image of (a) and (b), and (d)
structure model of YB56 projected along the [111] directions.
832
T. Oku and J.-O. Bovin
show dark contrast. This indicates that the di€ erence between the observed image
(® gure 8 (a)) and the calculated image (® gure 8 (b)) is large at these positions.
§ 4. Discussion
4.1. Disordering of yttrium and boron atoms
Dark spots corresponding to yttrium atom positions in the `averaged images’
recorded at thin regions from the three directions are weak compared with those of
the calculated images. These results suggest that the yttrium atom is not ® xed at only
one position. The yttrium site occupancy of 0.575, which was determined by X-ray
di€ raction, is merely a statistical value, and this implies that the `Y hole’ (the Y± Y
pair with the shortest distance) should be occupied by only one yttrium atom. These
results indicate alternating yttrium atom positions in the boron cluster from cell to
cell. Based on this procedure, a local structural model for atomic arrangement of
yttrium atoms in YB56 have been proposed in our previous paper (Oku et al. 1998) .
The local yttrium atom arrangement was directly determined by digital HREM
imaging. The boron atom, which has a fairly low atomic number ( Z 5 compared
with the yttrium atom Z 39 , still shows dark contrast in both the observed and
the calculated HREM images.
The regions which show white contrast in the di€ erence images (® gures 4 (c) , 6 (c)
and 8 (c)) indicate a large di€ erence between the observed and calculated images
based on the X-ray data. Structure models of Y± B80 clusters (Y holes with nonicosahedral B80 clusters ) and super-icosahedral B12 13 clusters projected along the
[100], [110], and [111] directions are shown in ® gure 9. Comparing ® gure 9 with the
di€ erence images (® gures 4 (c) , 6 (c) and 8 (c)) , the positions with the large di€ erence
of image intensity are Y holes and B80 clusters. This result indicates that the atomic
arrangements of the Y holes and non-icosahedral B80 clusters are disordered compared with the X-ray data. When the Y holes are formed, the cubic system would be
destroyed locally around them. Although the B80 clusters has fourfold symmetry in
the previous papers, another boron cluster should be proposed for a local structure
model. However, it is di cult to determine directly the atomic arrangement of boron
around the Y holes because of the low atomic number of boron. Higashi et al. (1997)
proposed B80 clusters around the Y holes, which corresponds to B(10) , B(11) , B(12)
and B(13) in table 1. They have low occupancies (0.22± 0.71) of boron atoms and
large thermal parameters (0.017± 0.058 nm2 ) as listed in table 1, which makes the
determination more uncertain. In the present work, disordering of the boron atom
positions was detected three-dimensionally from the di€ erence images. Since there
are various types of higher borides and fullerene compounds, this kind of disordering
detection would be very useful for evaluation of disordering of light elements such as
boron and carbon atoms.
4.2. E€ ectiveness of RHREM values
In the present work, a residual index between observed and calculated images
was used because of the simple form and usefulness for determination of the defocus
value and crystal thickness. The crystal structure of high- Tc superconductor of
Tl2 Ba2 CuO6 had been investigated using these RHREM values, and the thallium
occupancy was determined after the determination of the crystal thickness and defocus value of the observed images (Shindo et al. 1994) . In the present work, a threedirection analysis was carried out for observation of atomic disordering in the YB56
crystal, and three-dimensional information of the disordering was obtained. In order
Atomic disordering in Y B56
833
Figure 9. Structure models of Y± B80 clusters (Y holes with non-icosahedral B80 clusters) and
super-icosahedral B 12 13 clusters projected along the [100], [110] and [111] directions.
to analyze more clearly, through-f ocus imaging would be useful. In addition to the
present RHREM values, various methods for estimation of image agreement between
experimental and simulated images are known such as the fractional mean absolute
di€ erence (Smith and Eyring 1982) , cross-correlation function (MoÈ bus and RuÈ hle
1994) , mean relative di€ erence (MoÈ bus and RuÈ hle 1994) , normalized Euclidean distance (Hof mann and Ernst 1994) and nonlinear least-square methods (King and
Campbell 1994) . These are useful methods for structure evaluation by image matching. In the present work, the observed image was ® xed after the processing in order
to get information on the thickness e€ ect in the calculated image. It is believed that
more accurate atomic positions would be determined (HovmoÈ ller et al. 1984, Oku
et al. 1995) by combining the RHREM values with observed HREM images recorded
under the optimum experimental conditions with quantitative devices.
§ 5. Conclusion
Atomic disordering in YB56 was detected three-dimensionally by digital HREM
imaging. Averaged HREM images recorded from thin regions (less than 5 nm) along
834
Atomic disordering in Y B56
the [100], [110] and [111] directions of the YB56 were compared with the calculated
images. the defocus values and crystal thicknesses of the observed HREM images
were determined as (50 nm and 2.346 nm) , (39 nm and 3.318 nm) , and (45 nm and
4.063 nm) with RHREM values of 0.142, 0.193 and 0.251 along the [100], [110] and
[111] directions respectively. Di€ erence images between observed and simulated
images were calculated for the experimental conditions which reduced the RHREM
values. All the di€ erence images showed disordered atomic positions, which are
yttrium positions and non-icosahedral B80 clusters around the yttrium atom positions. If the disordering of B80 clusters was reduced, the resolution of monochromators could be higher. This method is useful for evaluation of disordered structure in
crystals with a light element such as boron and carbon.
A CKNOWLEDGEMENTS
The authors would like to acknowledge I. Higashi (Chiba Institute of
Technology, Japan) , T. Tanaka (National Institute for Research in Inorganic
Materials, Japan) and Y. Ishizawa (Iwaki Meisai University ) for providing the
sample. The authors also would like to thank L. R. Wallenberg, J.-O. Malm, A.
Carlsson and K. Suganuma for warm encouragement. The work was supported by
the Swedish Natural Science Research Council.
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