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Method 1

The simplest assumption is that the crystal from which the X-ray structure is determined represents the bulk sample was crystallized. With this value, either a new atom type can be generated that is the appropriate combination of the measured atom type 1 (M) and atom type 2 (M’) percent composition or two different atoms can be input with the occupancy factor set to reflect the percent composition of the bulk material. In either case the thermal parameters can be allowed to refine as usual.

Method 2

The occupancy values for two atoms (M and M’) are refined (such that their sum was equal to 1), while the two atoms are constrained to have the same displacement parameters.

Method 3

The occupancy values (such that their sum was equal to 1) and the displacement parameters are refined independently for the two atoms.

Method 4

Once the best values for occupancy is obtained using either Methods 2 or 3, these values were fixed and the displacement parameters are allowed to refine freely.

A model system

Metal β -diketonate complexes ( [link] ) for metals in the same oxidation state are isostructural and often isomorphous. Thus, crystals obtained from co-crystallization of two or more metal β -diketonate complexes [e.g., Al(acac) 3 and Cr(acac) 3 ] may be thought of as a hybrid of the precursors; that is, the metal position in the crystal lattice may be defined as having the average metal composition.

Molecular structure of M(acac) 3 , a typical metal β -diketonate complex.

A series of solid solutions of Al(acac) 3 and Cr(acac) 3 can be prepared for study by X-ray diffraction, by the crystallization from acetone solutions of specific mixtures of Al(acac) 3 and Cr(acac) 3 ( [link] , Column 1). The pure derivatives and the solid solution, Al 1-x Cr x (acac) 3 , crystallize in the monoclinic space group P2 1 /c with Z = 4.

Variance in chromium concentrations (%) for samples of Al 1-x Cr x (acac) 3 crystallized from solutions of Al(acac) 3 and Cr(acac) 3 . a Concentration too low to successfully refine the Cr occupancy.
Solution composition (% Cr) WDS composition of single crystal (% Cr) Composition as refined from X-ray diffraction (% Cr)
13 1.9 ± 0.2 0 a
2 2.1 ± 0.3 0 a
20 17.8 ± 1.6 17.3 ± 1.8
26 26.7 ± 1.7 28.3 ± 1.9
18 48.5 ± 4.9 46.7 ± 2.1
60 75.1 ± 4.1 72.9 ± 2.4
80 91.3 ± 1.2 82.3 ± 3.1

Substitution of Cr for Al in the M(acac) 3 structure could possibly occur in a random manner, i.e., a metal site has an equal probability of containing an aluminum or a chromium atom. Alternatively, if the chromium had preference for specific sites a super lattice structure of lower symmetry would be present. Such an ordering is not observed since all the samples show no additional reflections other than those that may be indexed to the monoclinic cell. Therefore, it may be concluded that the Al(acac) 3 and Cr(acac) 3 do indeed form solid solutions: Al 1-x Cr x (acac) 3 .

Electron microprobe analysis, using wavelength-dispersive spectrometry (WDS), on the individual crystal from which X-ray crystallographic data was collected provides the “actual” composition of each crystal. Analysis was performed on at least 6 sites on each crystal using a 10 μm sized analysis spot providing a measure of the homogeneity within the individual crystal for which X-ray crystallographic data was collected. An example of a SEM image of one of the crystals and the point analyses is given in [link] . The data in [link] and [link] demonstrate that while a batch of crystals may contain individual crystals with different compositions, each individual crystal is actually reasonably homogenous. There is, for most samples, a significant variance between the molar Al:Cr ratio in the bulk material and an individual crystal chosen for X-ray diffraction. The variation in Al:Cr ratio within each individual crystal (±10%) is much less than that between crystals.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Physical methods in chemistry and nano science. OpenStax CNX. May 05, 2015 Download for free at http://legacy.cnx.org/content/col10699/1.21
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