O DAVE CHAPPELLE: 3am 1, e chegg Study I Guided Soluti acebook e.com/u/0/c/NTA2N

Question

O DAVE CHAPPELLE: 3am 1, e chegg Study I Guided Soluti acebook e.com/u/0/c/NTA2NDlyNjUzM1pa 3. Waves in more than one dimension have periodicities (usually different) in each spatial direction: A) Consider first what waves look like in the x -y plane. At right is shown the spatial periodicity y of the waves in this plane. Use geometry to find the relation between y and the periodicities in each direction x and Ay B) Now find the relation between the periodicities in each direction ,Ay, and , and the total periodicity in 3 dimensions. [Hint: Repeat the argument in A) with Axy and ày as the orthogonal periodicities.] ) From the relationship you found in B), show that k (k2 +ky2 +k2)1/, where k is magnitude of the total wavevector and k,ky, and ky are the wavenumbers in each direction. 4. Once Loschmidt had established a reasonable estimate for Avagadro's [Loschmidt's] number, NA 6 x 1023, subsequent experiments could establish estimates for other fundamental constants: If I mol of hydrogen is 1g and 1 mol of carbon is 12g, what are the masses of a single hydrogen atom and a single carbon atom? A) B) Graphite (pure C) Assuming the volu occupied by one atom? ubical, what is the distance Page 2 2

Explanation / Answer

3. A. in the given figure

let the lambdaxy make angle theta with the x axis

then

lambdax = lamdbaxy/cos(theta)

lambday = lambdaxy/sin(theta)

now, in terms of wave number

lambdax = 2*pi/kx

lambday = 2*pi/ky

lambdaxy = 2*pi/kxy

2*pi/sin(theta)kxy = 2*pi/ky

2*pi/cos(theta)kxy = 2*pi/kx

=> kxy = ky/sin(theta)

=> kxy = kx/cos(theta)

hence

(ky/kxy)^2 + (kx/kxy)^2 = 1

kx^2 + ky^2 = kxy^2

kxy = (kx^2 + ky^2)^1/2

B. from A, let assume lambda is the 3D direction we are considering

then angle between lambda and lambdaxy = alpha

lambda/cos(alpha) = lambdaxy

lambda/sin(alpha) = lambdaz

in terms of wave number

kxy/cos(alpha) = k

kz/sin(alpha) = k

C. hence

kxy^2 + kz^2 = k^2

but form the last part

kxy^2 = kx^2 + ky^2

hence

k = (kx^2 + ky^2 + kz^2)^1/2

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