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Experiment 9: Structure of solids. Inter-atomic distance from Bragg reflection and from theory
September 22th, 2019
TA: Aleksander Pikalov
In this experiment, two procedures were performed. First, ab-initio strategies were utilized to get structural parameters of NaCl. additionally, the second a part of the experiment concerned utilizing Bragg’s law of reflection to work out the ionic radii of atoms within the NaCl lattice victimization diffraction (XRD). it absolutely was all over that the optimized NaCl structure obtained a bond distance of 2.82A ? for the Na-Cl bonds severally. additionally, these bond distances were compared to the theoretical values that XRD would reveal of 28201A ? for the Na-Cl bonds severally. A % error of 0.0035% was tabulated for the Na-Cl bonds consequently. Moreover, it absolutely was found that NaCl obtained a lattice constant of 563.28 pm. additionally, the theoretical lattice constant of NaCl was found to be 564.02 pm that gave a % error of 0.13120 %. However, a spacing of 281.64 pm was for the NaCl monocrystal.
Theoretical background.
The item of this test is to decide the crystal structure of a solid substance from x-ray powder diffraction styles. This includes willpower of the symmetry class that’s the cubic for this experiment, the dimensions of the unit cellular, the quantity of atoms or ions of each kind in the unit mobile, and the location of every atom or ion within the unit mobile. Owing to inherent limitations of the powder approach, simplest materials within the cubic device can be easily characterized on this way, and a cubic material could be studied inside the experiment.1
Bragg Reflections decide the geometric situations beneath which diffraction of X rays by means of a crystal shape might take vicinity. These are (except for depth considerations) the same as the situations for diffraction from the crystal lattice if a “point scattering center” is located at every point of the lattice. We shall therefore look at the geometry of diffraction through this kind of crystal lattice. This produces radiation within the X-ray area, of the electromagnetic spectrum. The principle by that X rays would be diffracted by such a lattice is constant as that by which lightweight is diffracted by a dominated grating. once a plane wave impinges on a point-scattering center, the scattering center radiates a spherical wave. If there are 2 or additional such scattering centers, the spherical waves can in bound regions tend to strengthen each other and in certain other regions tend to cancel one another out, that is, interfere. 1
It is not necessary, however, as a condition for optimum reinforcement that everyone the trail lengths be equal, providing people who don’t seem to be equal, differ by a wavelength ? or by associate integral range of wavelengths n? , that the condition for complete formula reinforcement is
n?=dsin? (1)
where n is associate whole number, usually referred to as the order of the diffraction; D is that the interplanar spacing; and ? is the angle that the incident ray and therefore the mirrored ray create with the reflective planes. Equation (1) is named the general equation, and ? is commonly referred to as the general angle, indicating the path of the incident and the meditated waves for the set of community planes and is frequently referred to as The Bragg’s legal guidelines of reflection can be used to determine lattice constants ao, in cubic crystals, by way of a distance d similar to half the lattice consistent.1
d=ao2 (2)
Materials and Methodology
This lab was using the Instruments which Leybold’s 554 800 X-ray apparatus, computer and the NaCl crystal was using as the reagents. X-ray apparatus, GaussView program were the software which worked through this experiment.
In order to execute this, experiment the pdf “Structure of solids. Inter-atomic distance from Bragg reflection and from theory” lab procedure was attained for detailed directions. The first part of the experiment was obtained structural parameters of NaCl crystal by using the ab-intio calculation, utilizing a computer. First, the GuassView program was opened on the computer, then created the new molecule by click on Create New Molecule Group which found in File. PBC editor was opened by click on Tool then PBC. In the first tab Symmetry specifies the dimensionality of the unit cell, the lattice system and space group constraints. For NaCl, three was used for the Lattice Dimensions, the “Enable Space Group Symmetry” box and the “Always Track Spaces Group Symmetry” box were checked. Current Space Group was set 225 [F m-3m] and for “All changes”. Clicked on the tab Cell in PBC editor to specify the dimensions of the unit cell, a= 5.64, b=5.64, c=5.64. Snap on the tab Contents in the PBC manager. This tab enables you to add molecules to the cell. In Row 1 snap the “?” in the segment Symbol and select Cl, the sections a, b and c ought to have an estimation of zero (0) in every section. These columns show partial directions of particles in the unit cell. Next snap on the content Add in column Highlight. Four Cl iotas ought to be added to the rundown with the fragmentary coordinates decided dependent on the chose Space Gathering. Next include sodium molecules by tapping the “? “in the Row 5 in the segment Symbol and choosing Na. The partial arranges in sections a, b and c ought to be 0.5. Next snap on the content Add in segment Highlight in Row 5. Four Na molecules ought to be added to the rundown. Open Gaussian Calculation Setup from the Calculate menu. Select Optimization hands on Type tab, Select the Method tab, Use the Link 0 tab to determine the Memory Limit: 48 GB and the quantity of Mutual Processors: 24, at that point snap submit to run the compute Check just the *.log record. GaussView will neglect to open the *chk record. You should change over it physically to organized checkpoint document *fch later.Select Summary from the Results menu. You can reproduce the unit cell to show a bigger part of the. Open the PBC manager, menu Tools – > PBC… what’s more, select the tab View (Error! Reference source not found.). In the Cell Replication segment modify the estimations of a, b, c, – a, – b, – c to recreate the unit cell in any of the three. Open a terminal or Windows PowerShell and change the present index to the catalog where you spared your info record *gjf. The *log and *chk yield documents ought to be in a similar registry. To open PowerShell opens the PowerShell.In PowerShell type thecommand underneath supplanting “your_folder” with the name of the organizer you made and press enter.cduserspchemdesktopyour_folder.The checkpoint record *.chk is a twofold document that contains data important to produce atomic orbital surfaces. Before it tends to be utilized it should be changed over to a normal book record, designed checkpoint *fch.To convert the checkpoint document *.chk to the organized checkpoint record type the order be low in PowerShell C : gl6wformchk exe your_output. chk. The order above will make a designed checkpoint record with the expansion *.fch. Create the most elevated involved sub-atomic orbital (HOMO) and least vacant sub-atomic orbital (LUMO) surfaces. To make Homo orbital sort the accompanying direction in the terminal c:gl6wcubegen.exe 0 MO=HOMO *.fch homo.cube – 80 h. So as to make the LUMO orbital supplant Homo with Lumo in the above direction. To see the HOMO and LUMO orbitals first it must to stack the *cube records. Select or open the window with the *log document at that point open menu Results — > Surfaces/Contours. Snap the Cube Actions catch and Load Cube. Burden both 3D square records homo.cube and lumo.cube. The sheet Cubes Available should list two MO. Select the primary MO and snap Surface Actions catch and New Surface. The MO ought to be noticeable in the principle windows with the *log document and in the sheet Surfaces Available in the Surfaces and Contours. Select the subsequent MO in Cubes Available sheet and snap Surface Actions catch and New Surface. Both orbitals are currently appearing in the fundamental window. To conceal one of the MOs, select it in the Surfaces Available sheet and snap Surface Actions catch and Hide Surface. To show a shrouded MO initially select the surface and afterward snap Surface Actions catch and Show Surface.To make the sub-atomic orbital surface straightforward open menu View – + Display Format… select the tab Surface and change the Format to Transparent. Save the MO pictures. Select the primary window and ensure it is demonstrating the MO to spare or print. To spare the picture open menu File — > Save Image pick a name and a goal and snap Save.2
The second a part of this experiment was analysis of NaCl crystal exploitation XRD. The crystal was already placed within the slot, therefore instrument standardization was nonheritable. First, the ability button was switched on the lower left facet. Following, the ZERO key was accessed. The U button was ironed so as to line the thermionic tube high voltage and also the change was turned till 35.0 kV appeared within the readout space. The emission current was set by pressing the I button and rotating the change dial till a readout of one.0 mA read. Next, the COUPLED key was accessed to activate the coupled scanning mode and set the target to 7.2 exploitation the change knob. The HV on/off button was turned on. The device button was activated and exploitation the change apprehends the utmost enumeration rate for the primary reflection maximum of the K? line. Next, the TARGET button was ironed and the change button followed so as to search out the enumeration rate most. Following, shift occurred between the adult male and target scanning modes so as to examine if the enumeration rate most had been nonheritable. Next, within the COUPLED the target was enraptured back by 7.2 exploitation the change knob. The positions were saved because the “zero position of the system” by pressing TARGET, COUPLED, and ? LIMITS at the same time. Next, the X-ray equipment software system was opened on the pc desktop. Next, F4 was ironed so as to clear any previous work. The ?t button was ironed and with the change dial it had been set to ten s. Following, the ?? button was ironed and exploitation the change knob it had been set to zero.1o. The COUPLED key was activated then the ?limits was ironed and with the change knob the lower limit was set to 4o. Next, the ?limits key was used once more however now the higher limit was set to 30o exploitation the change knob. The measure was running by pressing the SCAN key on the instrument. Once the information had been complete within the spectrum provided the command “Calculate Peak Center” was ironed on to guage the optical phenomenon spectra. in addition, exploitation the left push the breadth of every peak of interest was clicked on so as to show the height center, and glancing angle ?, that were displayed at the left-hand corner of screen. The spectrum was saved and written. After, the instrument was reset by pressing COUPLED, then exploitation change button it had been revolved till the surface was horizontal. Finally, TARGET, COUPLED, and ? LIMITS were ironed at the same time.2
Data, Calculations and Error Analysis
Table 1. Data of bond angles of the structure NaCl from the ab-initio calculations.
Bond Angle
Na-Cl 90?
Na-Cl-Na 180?
Figure 1: The spectrum obtained for the NaCl crystal utilizing XRD.
Table 2
? 6.4 7.2 12.9 14.6 22.2
Sin ? 0.1115 0.1253 0.2233 0.2521 0.3778
Line K? K? K? K? K?
n 1 1 2 2 3
n? (pm) 63.09 71.08 126.18 142.16 213.24
1?= 100 pm
Sample calculation for ? = 6.40 for the NaCl crystal:
K? = 71.08 pm, K? = 63.09 pm
Sin ? = Sin (6.40) = 0.1115
n? = (1 * 63.09) = 63.09 pm
Figure 2. The plot of Braggs Law for the NaCl crystal
Calculation for lattice constant, ao, and percent error of NaCl:
Experiment Lattice Constant (ao) = slope(m)from Figure 2 = 563.28 pm
Calculation Lattice Constant (ao)
ao= d X 2=2.82 X 2= 5.64 ?= 564 pm
Using Eq: d=ao2 to calculate the distant
d=ao2 = 563.282 = 281.64 pm
Percent error for lattice constant:
1?= 100 pm
Thereotical lattice constant =5.6402? = 564.02 pm
Calculattion percent error between theoritical and experiment
Percent error=Actual-experimentalactual ? 100%
Percent error= 564.02-563.28564.02? 100%=0.13120%
Calculation percent error between Experiment vs Calculation
Percent error=Actual-experimentalactual ? 100%
Percent error= 564.00-563.28564.00? 100%=0.1277%
Calculation percent error between Calculation vs Theoretical
Percent error=Actual-experimentalactual ? 100%
Percent error= 564.02-564.0/564.02? 100%=0.0035%
Table 3. Shows the lattice cosntnat, spacing, and percent error obtained for the NaCl crystal
Experiment Lattice Constant (ao) Calculation Lattice Constant (ao) Theoretical Lattice Constant (ao)
563.28 pm 564.00 564.02 pm
Experiment vs Theoretical Experiment vs Calculation Calculation vs Theoretical
Percent Error 0.1312% 0.1277% 0.0035%
In the first part of this experiment theoretical methods were acquired to obtained structural parameters of the compound NaCl. Furthermore, a bond angle of 90? and 180 were obtained for both the bonds Na-Cl and the bond of Na-Cl-Na respectively, utilizing ab-initio calculations, as seen in Table.1. It was concluded that the optimized NaCl structure obtained a bond distance of 2.82 A Na-Cl bonds respectively, as seen in Table.3. In addition, these bond distances were compared to the theoretical values that XRD would reveal of 5.6402A of Na-Cl bond respectively. A percent error of 0.003% NaCl bonds, accordingly, as observed in Table.3. Such small percent error concluded that XRD and ab-initio methods are close reliable and close accurate in conducting the bond distance and bond angles of a sample being analyzed. All the Na-Cl distances in the optimized structure and corresponding Na-Cl-Na angles were the same. The optimized distances and angles were support cubic lattice system of Halite. Since the HOMO and the LUMO were failed to analyze to localized so it cannot tell HOMO or LUMO are composed of specific atomic orbital, and its localized om certain atoms. It is also the major reason to make this calculation have error percentage with the theoretical values. The application of ab-initio techniques for deriving structure are the try and remedy the digital Schrodinger equation given the positions of the nuclei and the number of electrons which will yield useful data such as electron densities, energies, and extraordinary residences of the system. ab-initio strategies are also useful in figuring out the bond lengths and bond angles as seen on this look at. Even though, XRD can derive bond lengths a few examples in which it’d now not be able to include no presence of a crystal. Otherwise, the Calculate type of this calculation was FOPT which mean full option, and the Calculate Method was Ground State DFT PBEPBE which mean Density Functional Theory and PBEPBE were the correction and exchange function3. The Basic Set for this calculation was 6.31G Fitting Set: Auto Charge 0 spin Singlet.
In the second a part of the science lab Bragg’s regulation to make your mind up the ionic radii of atoms within the NaCl lattice the usage of XRD. what is more, from the ? and ? angles of the spectrum established in Figure one, a graph returns to be computed plotting n? as a feature of sin? for the NaCl monocrystal as visible in figure.2. use the linear formula to calculate the slope of the graph lands up obtained. The slope corresponds to the lattice consistent of the monocrystal. It had been determined that NaCl experiment lattice consistent of 563.28 pm as seen in table three. Further, the theoretical lattice regular of NaCl became situated to be 564.02 pm that gave a proportion error of 0.13120 %. Such low proportion errors terminated that XRD find yourself a robust and correct tool in activity the lattice regular of a monocrystal. However, A spacing of 281.64 pm turned into computed for the NaCl monocrystal
Usual, this experiment become not successfully, and also the effects obtained showed by mistake from pc computer for the HOMO and the LUMO were did not analyze to localized therefore it cannot tell HOMO or LUMO are composed of specific atomic orbital, and its localized om sure atoms. It became terminated that the optimized NaCl form nonheritable a bond distance of 2.82 A ? with Na-Cl bonds severally. Further, a blunder of 0.0035% became tabulated for the Na-Cl bonds so, in comparison to the theoretical values that XRD may screen. experiment lattice steady for the NaCl crystal became 563.28 pm. It turned into compared to its theoretical lattice consistent value of 564.02 pm that discovered a proportion error of zero.13120%. what is more, a spacing of 281.64 pm turned into computed for the NaCl monocrystal.
1Carl Garland, Joseph Nibler, David Shoemaker – Experiments in Physical Chemistry-McGraw-Hill Science_Engineering_Math (2008)
2Lab Manual Hand Out
3Density Functional (DFT) Methods:

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