A beam of x-rays whose wavelegth is 0.03875 nm impinges on acrushed crystal pack

Question

A beam of x-rays whose wavelegth is 0.03875 nm impinges on acrushed crystal packet with a single interplane separationof d nm. A sheet of film lies normal to the x-ray beam, adistance 50 cm from the crushed crystal. The first scatteringmaximum (Bragg peak) forms a circle of radius 20 cm as shown in thefigure below. Only the smallest radius circle is shown.

A view of the scattering at the atomic level is shownschematically in the figure below.

a) Calculate the distance between Bragg scattering planes.
d = nm    

HELP:  Considerany ray (for example, the topmost ray shown in the upper figure)emerging from the sample and striking the screen. What angle doesthis outgoing ray make with the incoming beam?

HELP:  The anglebetween the outgoing and incoming beam is NOT the Bragg angle. Fromthe lower figure, you should be able to see a relationship betweenthe incoming/outgoing beam angle and the Bragg angle. What is thatrelationship?

b) Find the maximum number of Bragg peaks which could appear onthe film plane even if it had infinite extent.

Number of Bragg Peaks =      *
3   OK

HELP:  If thescreen were of infinite extent, what is the minimum angle betweenthe incoming beam and outgoing beam that would miss thescreen?

HELP:  From thelower figure, what is the relationship between theincoming/outgoing beam angle and the Bragg angle?

c) Find the minimum radius of the film plane necessary tocontain the largest radius Bragg peak computed in part b.

rmin = cm    

HELP:  You need toworry about the accuracy of your answer. The tangent function is arapidly varying function, so you must keep at least foursignificant digits in your previous calculations in order tomaintain the required accuracy here.

d) Find the longest wavelength x-ray that will produce a Braggpeak on a sheet of film of the radius found in part c for theinterplane separation found in part a.

Explanation / Answer

a)According to Bragg's law we have 2d * sin = m * or d = (m * /2sin) -----------(1) m = 1, = 0.03875 nm = 0.03875 * 10-9 m and = 90o b)the condition for maxima in the interference pattern at theangle bright is d * sinbright = m * m =0,±1,±2,±3,................. or m = (d * sinbright/)-----------(2) the value of d is obtained from equation (1) andbright = 90o c)the minimum radius of the film plane necessary to containthe largest radius Bragg peak computed in part b = (m * * D/d) the value of m is obtained from equation (2) and D = 50 cm =50 * 10-2 m d)the general condition for destructive interference is sindark = m * (/d) m =±1,±2,±3,............... or = (sindark * d/m) dark = 90o and m = 3 b)the condition for maxima in the interference pattern at theangle bright is d * sinbright = m * m =0,±1,±2,±3,................. or m = (d * sinbright/)-----------(2) the value of d is obtained from equation (1) andbright = 90o c)the minimum radius of the film plane necessary to containthe largest radius Bragg peak computed in part b = (m * * D/d) the value of m is obtained from equation (2) and D = 50 cm =50 * 10-2 m d)the general condition for destructive interference is sindark = m * (/d) m =±1,±2,±3,............... or = (sindark * d/m) dark = 90o and m = 3

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