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A diffracted x-ray beam is observed from an unknown cubic metal at angles 30.7301 degree, 35.6406...

A diffracted x-ray beam is observed from an unknown cubic metal at angles 30.7301 degree, 35.6406 degree, 51.1984 degree, 60.9706 degree, 64.0327 degree and 75.4344 degree when x-ray of 0.1248 nm wavelength is used.

Determine the indices of the planes (hkl) that produce each of the peaks.

Expert Solution

The formula to determine the indices of the planes using XRD data is,

where, = diffraction angle

= wavelength of X- rays in nm

a = lattice parameter of a crystal in nm

h,k,l = the indices of the peaks

procedure to calculate the indices of the peaks,

1. Calculate for different values of 2

Peak no.	2		1x /	2x/	3x/	hkl
1	30.7301	0.0702
2	35.6406	0.0937
3	51.1984	0.1867
4	60.9706	0.2574
5	64.0327	0.2811
6	75.4344	0.3743

2. Find out the minimum value of (here it is 0.0702) and divide all the values of using this minimum value.Multiply the final result by 1, 2 and 3 and tabulate the results. This is done to get a rounded-off figure for our values.

Peak no.	2			1x /	2x/	3x/	hkl
1	30.7301	0.0702	1	1	2	3
2	35.6406	0.0937	1.3348	1.3348	2.6696	4
3	51.1984	0.1867	2.6595	2.6595	5.319	8
4	60.9706	0.2574	3.6667	3.6667	7.3334	11
5	64.0327	0.2811	4.0043	4.0043	8.0086	12
6	75.4344	0.3743	5.3319	5.3319	10.6638	16

The following values of h,k,l are predetermined indies for specific unit cell systems;

Primitive = 1,2,3,4,5,6,8,9,10,11,12,13,14,16.....

Body-centered = 2,4,6,8,10,12,14,16,....

Face-centered = 3,4,8,11,12,16,19,20,24,27,32,.....

Diamond Cubic = 3,8,11,16,19,24,27,32,....

From the table, 3x/ shows the rounded-off values matching closely with the indices. The indices closely match with the values of Face-centered cubic crystal system. The corresponding h,k,l values of FCC system are listed that would yield the values of .

Peak no.	2			1x /	2x/	3x/		hkl
1	30.7301	0.0702	1	1	2	3	3	111
2	35.6406	0.0937	1.3348	1.3348	2.6696	4	4	200
3	51.1984	0.1867	2.6595	2.6595	5.319	8	8	220
4	60.9706	0.2574	3.6667	3.6667	7.3334	11	11	311
5	64.0327	0.2811	4.0043	4.0043	8.0086	12	12	222
6	75.4344	0.3743	5.3319	5.3319	10.6638	16	16	400

Amanda K answered 7 months ago

A diffracted x-ray beam is observed from an unknown cubic metal at angles 30.7301o, 35.6406o, 51.1984o,...

A diffracted x-ray beam is observed from an unknown cubic metal at angles 30.7301o, 35.6406o, 51.1984o, 60.9706o, 64.0327o and 75.4344o when x-ray of 0.1248 nm wavelength is used. Find the lattice parameter (in nm) of the metal for each of the peaks.

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