HW2_2009 solutions

# HW2_2009 solutions - MSE 2620 Spring 09 Homework#2...

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MSE 2620, Spring ‘09 Homework #2 Instructions: Due Tuesday, February 26 th , 2009. Turn in one solution per group and be sure to include the group letter, the names of the group members, and if any group members did not participate enough to deserve full credit along with the reasons why. Question 1: Conductivity and The Mean Free Path a) Sodium is a monovalent metal with a density of 0.97g/cm 3 and a BCC crystal structure. Estimate the mean separation (nm) between an electron and one metal ion, assuming that electrons spend the most time between a metal ion and its second nearest neighbor. Let x be the atoms in the unit cell. nm a a x M dN volume atoms d Density mol atoms m kg mol kg at A cm g 429 . 0 ) 10 022 . 6 )( 10 97 . 0 ( ) 10 99 . 22 ( 2 97 . 0 3 / 1 23 3 3 3 3 3 The average separation s between atom and ions is roughly a/2 or 0.214 nm. If you went to great extremes, you could show that the Na nearest neighbor separation is really 2 3 a , which will give you a separation of 0.866s, or 0.186 nm. As this is an estimation , the above approximation is fine. I’ll use the simple approximation. s ≈ a/2 = 0.214 nm b) Use the potential energy between this electron and metal ion pair to find the mean speed of an electron (m/s) (HINT: Virial Theorem). eV J nm e s e PE o o 71 . 6 10 07 . 1 ) 214 . 0 ( 4 4 18 2 2  , the potential energy of the ion- electron pair. The kinetic energy is related to the potential energy via the Virial Theorem. s m e e m J v v m J KE 6 2 / 1 19 2 10 09 . 1 ) 10 37 . 5 ( 2 2 1 10 37 . 5 2 1 c) Given that the mean free time between scattering events from thermal vibrations of the lattice is 4.3E-14s, find the drift mobility (m 2 /Vs), the conductivity (1/Ωm) and the mean free path of an electron (m). Vs m e e m s e m e 2 3 14 10 55 . 7 ) 10 3 . 4 (

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m Vs m mol kg A m kg N e en 1 7 3 3 3 10 07 . 3 ) 10 55 . 7 ( ) 10 99 . 22 ( ) 10 97 . 0 ( 2 3 , as sodium is monovalent, the concentration of free electrons n equals n a , the concentration of atoms. m s v l s m 8 14 6 10 67 . 4 ) 3 . 4 )( 09 . 1 ( d) When the distance between impurities is approximately equal to the mean free path between collisions, we will see a significant decrease in the resistivity. Estimate the impurity content (unitless) needed to substantially increase the resistivity of this metal. (Hint: if n I is the
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## This note was uploaded on 09/30/2009 for the course MSE 2620 taught by Professor Malliaras during the Spring '09 term at Cornell.

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HW2_2009 solutions - MSE 2620 Spring 09 Homework#2...

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