The diffraction pattern produced by a turbostratically disordered structure can be divided into two parts, the a b-plane (hkO) structure and the c-axis (001) series. Simulations of these are carried out separately: the (hkO) simulations assume a very small crystallite size N3 (usually 1), while the (001) simulations use a much larger N3, because the c-axis coherence length is the significant factor. The c-axis simulations can be simplified considerably by ignoring the ab-plane structure entirely and only including the interplane spacing in the calculations planes. When dealing with the diffraction data produced by a pure powder sample the two simulated patterns are combined and compared with the data, but given the technique used here produced separate ab-plane
and c-axis diffractograms the simulations can be compared independently. The method of Reynolds (1989) was used to produce these simulations.
F igure 2.10: The reciprocal space construction used f o r Equation 24. The calculated hemisphere o f radius 1/d in reciprocal space is shown interacting with the reciprocal space rods produced by a turbostratically disordered compound. After Reynolds
In producing the simulation of the (hkO) diffraction patterns, the technique involves calculating the value of Equation (25) at each increment of 20 for all values of (hkO) that fall on the (hemi-)sphere l/d=2sin0/^ in reciprocal space. This summation is shown in Figure 2.10, and proceeds as follows: at an angle 20, the reciprocal distance 1/d is calculated; the hemi-sphere is enumerated as a series of circles of radius R i=cos(yi)/d, where the angle yi is the angle a radius of the (1/d) sphere makes with the hk-plane. At each point of the circle described by y2, the values of the continuous functions of h, k
and 1 shown in Equations (27), (28) and (29) are determined, and these values used to calculate the structure factor F(hkO) and interference functions O j, d>2 and O3 at that
point.
2 . 7 . 1 Simulation of the a b-plane Pattern
12
-k/b
y ,= 90 y 2 =180
X 2;tÄi X |/r(M/)|:0 lO,<I>2A y1Ay2
where
sin2(7r/iA^j) _ sin: (/rkN2) _ sin2(7r//V3)
sin~(/r/z) ~ sin ~(Kk) ancj sin~(/r/) (26)
and h, k and 1 are replaced by the continuous functions:
h = asinß(Ricosy2 - sinyi/dtanß*) (27)
k = Rißsiny2 (28)
1 = (c/d)siny2 (29)
where Ri = cos(yi)/d, ß* = 180-ß, and d = A,/2sin0.
Figure 2.11 and 2.12 show a number of simulated diffraction patterns. These all use the same crystal structure, a P6 hexagonal lattice o f rubidium ions with the ab-plane dimensions a=4.93 Ä and the c-axis spacing c=9.0Ä between the layers, with the wavelength ^=1.54178 Ä. This structure is not chosen at random, it is the structure of a 2x2 hexagonal superlattice of rubidium ions in a stage-II binary GIC, but the diffraction pattern of the graphite is not included in the patterns. The primary difference between the structures used to produce these simulated patterns is the size of the crystallites. In Figure 2.11, the ab-plane crystallite dim ensions Ni= N 2=30, while the three patterns are calculated for N3= l, 5 and 30. At N ß=l, only the (hkO) peaks are produced by the simulation, and these now possess the characteristic band shape of a turbostratically disordered structure. In Figure 2.12, N3= l for all three simulated patterns, producing the same turbostratic band shapes, while the size of the in-plane crystallite is varied, N 1,2= 10, 20 and 30. This shows the broadening of the peaks as N12 is decreased, but also that the intensity maximum of the peaks moves to a higher 20 as N p2 is decreased, a phenomena described by Warren( 1()), which is a result of the interaction of the sphere of reflection with the broadened (hkO) cylinders produced by a small in-plane crystallite.
20 25 30 35 40 20 (Degrees)
F ig u re 2.11: s im u la te d d iffr a c tio n p a tte rn s u sing c ry s ta llite s c o n s is tin g o f a hexagonal P6 la ttic e o f ru b id iu m ions, w ith a - b - 4 . 9 3 k , c= 9 .0 Ä , w ith a v a ry in g c- axis c ry s ta llite size Ny, w h ile N 1 2 = 3 0 f o r a ll three sim u la tio n s . The w avelength X - l.5 4 1 7 8 k .
---N
20 25 30 35 40
20 (Degrees)
Figure 2.12: s im u la te d d iffr a c tio n p a tte rn s u sin g c ry s ta llite s w ith the same stru c tu re as used in F ig u re 2.11, setting N j = l f o r a ll three p a tte rn s b u t the in -p la ne c ry s ta llite size N jy2 v a ry in g as shown. As the size o f the in -p la n e c ry s ta llite dom ain is reduced, the center o f the p eak shifts aw ay fr o m the true (hkO) p o s itio n to a h ig h e r
20.
W hile application of the formula for intensity provided by Equation (25) can reproduce the purely turbostratic band shape from an isotropically oriented powder sample, some of the samples studied also included a small degree of interplanar interaction, which caused a significant degree o f 3D ordering without producing a fixed c-axis crystallite size N3.
This was added to the simulation by replacing the interference function O3 with the Fourier series equivalent shown in Equation (30):
rt=yv3-i
<t>3 = yV3 + 2 (/V3 - n)cos(2mil)
«=1 (30)
where N3 is the same crystallite, and n is the number of interfaces separating any two layers in the crystal. Into this equation a negative exponential term is introduced:
n = N) - 1
<b3 = A3 + 2 (jV3 - rt)exp(-A2 / <5)cos(2/m/)
«=1 (31)
where d is the average defect-free distance along the c-axis in each crystallite, in units of number of coherent interfaces(2), and the term exp(-n/5) is the probability of a defect-free region of n interfaces in the crystallite.
5= 1.5
5= 0.5
20 25 30 35 40
20 (Degrees)
Figure 2.13: the effect o f the ordering parameter 5 on the simulated diffraction pattern. Here the same P6 cell as above is used, with N1 2=30 and the c-axis crystallite Nj=30, but the alternative interference function & 3 in Equation (31) is employed. At 5 -0 .0 there are no defect-free interfaces along the c-axis, and the pattern is purely turbostratic.
The effect of this ordering parameter 6 on the simulated diffraction pattern is shown in Figure 2.13, where even small values of 6 produce considerable changes to the simulated diffraction patterns. At 5=1.5, the probability that any two adjacent layers are 3D ordered is exp(-l/1.5)=0.51, that is, 49% of the layers are not 3D ordered at all and remain in a
purely turbostratic structure, but the diffraction patten has lost many of the characteristic features of the turbostratic pattern produced by 8=0.0.
An additional correction is needed, as Equation (25) assumes a perfect powder distribution, but the diffraction experiments used oriented samples, as described above, to drastically reduce the contribution of the c-axis structure to the data. To account for the ab-plane oriented samples used in the diffraction experiments, an empirical correction is used to reduce the structure factor F(hkl) as 1 increases, in the same manner as the temperature correction:
F(hkl) = F e x p (-C D /2) (3, }
CD is the c-axis damping factor, which was used as a refinable value in fitting the simulations to the diffraction data. Its effect on the turbostratic peak shape is to suppress the "tail" at high 20, as seen in Figure 2.14. It can be thought of as a disorder amplitude in the 1 direction of the graphene sheet.
--- CD=0.0 - - CD=0.5
20 (Degrees)
Figure 2.14: the effect o f the c-axis damping fa cto r CD on the turbostratic profile shape. The sim ulated crystallite has the same P6 unit cell used in the previous simulations, with N i 2=30 and N s= l.