Modelling the Structure

The effects of deviations from the ideal structure were discussed earlier in Section 2.4, and were described by introducing a displacement term $\Delta{\bf r}$ in equation 2.9. Following that same methodology here, the modulated structure may be described by placing $\Delta{\bf r}={\bf U}$, where U is a function describing the modulation. The position of the $n$th atom in the $m$th unit cell can then be written as

$${\bf R} = {\bf m} + {\bf r}_n + {\bf U}_{n,m}$$ (9.1)

where ${\bf m}=m_1{\bf a}+m_2{\bf b}+m_3{\bf c}$ is a lattice vector of the $m$th unit cell, and ${\bf r_n}=x{\bf a}+y{\bf b}+z{\bf c}$ is the position of the $n$th atom in the cell. The fractional atomic coordinates $(x,y,z)$ can be generated for all equivalent positions in the unit cell by applying the appropriate symmetry operations of the space group to the set of unique coordinates. The necessary starting information to apply this (space group, atomic coordinates of the fundamental structure, and atomic modulation function) can be obtained from any of the structure refinements listed in Table 3.1 of Chapter 3. In the work which follows, the results of Yamamoto [85], which is amongst the most extensive and detailed of the studies, will be drawn upon exclusively for this data. Listed in Table 7.1 are the starting coordinates for each atom position and the necessary symmetry operations for the Bbmb space group are given in Table 7.2. A similar approach to generating the structure, though based upon a different model, has been described by Heinrich [185] for the purposes of simulating HREM images.

Table 7.1: The atomic parameters of the fundamental structure for each unique atomic position (from Yamamoto [85]).

Atom		$x_0$		$y_0$		$z_0$		Wyckoff	Occupation
								site	$p$
Bi		0.25		0.5		0.0528		8l	1.0
Sr		0.25		0.0		0.1409		8l	1.0
Ca		0.25		0.0		0.25		4e	1.0
Cu		0.25		0.5		0.1963		8l	1.0
O(1)		0.0		0.25		0.1963		8g	1.0
O(2)		0.5		0.25		0.1963		8g	1.0
O(3)		0.25		0.5		0.1170		8l	1.0
O(4)		0.25		0.0		0.0528		8l	0.5
O(5)		0.5		0.25		0.0528		8g	0.5
O(6)		0.0		0.25		0.0528		8g	0.5

Table 7.2: The symmetry operations for each type of Wyckoff position in the structure (from the International Tables for Crystallography [186]).

site
	origins	(0,0,0)+		(1/2,1/2,0)+
8l	$x,y,0$	$-x,-y,0$	$-x,y,1/2$	$x,-y,1/2$
8g	$x,0,1/4$	$-x,0,1/4$	$-x,0,3/4$	$x,0,3/4$
4e	$1/4,1/4,0$	$3/4,1/4,1/2$

The modulation function is contained within the displacement vector ${\bf U}_n(t)$ and is a periodic function of $t={\bf q.(m+r_n)}$ where q is the wavevector of the modulation. The function can be expressed as a Fourier series

$${\bf U}_n(t) = \sum_j {\bf u}_n^j cos(2\pi j t) + {\bf v}_n^j sin(2\pi j t)$$ (9.2)

with ${\bf u}^j_n$ and ${\bf v}^j_n$ being the Fourier amplitudes associated with an atom $n$, and the sum is over the number of harmonic components, $j$, of the Fourier series. In the analysis of Yamamoto [85], and so in this work also, the function is approximated to only the first order, $j=1$, to reduce the number of variables. In this case, there are six amplitudes associated with each atom, and the values refined for these by Yamamoto are listed in Table 7.3. The zero components, such as $u_y$, in the modulation function are the result of restrictions imposed upon particular sites by the symmetry operations of the space group.

Table 7.3: The parameters of the atomic modulation function, U, for each site in the structure (from Yamamoto [85]).

Atom		$u_x$	$u_y$	$u_z$	$v_x$	$v_y$	$v_z$
Bi		-0.01		-0.0052		0.0745
Sr		0.0161		-0.0091		0.0365
Ca		0.0142		-0.0099
Cu		0.0142		-0.0102		0.0094
O(1)				-0.0074	0.0040	-0.02
O(2)				-0.0097	0.0216	0.0173
O(3)		-0.0545		-0.0058		-0.0945
O(4)		0.0198		-0.0103		0.1280
O(5)				-0.0311	-0.1620	-0.0312
O(6)				0.0281	-0.1324	0.2337