PC Chapter 12

PC Chapter 12 - 12 The Solid State 12-1 U Total =...

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187 12 The Solid State 12-1 α =+= - + 2 Total attractive repulsive m k eB U UU r r . At equilibrium, Total U reaches its minimum value. α + = = +- 2 Total 21 0 m dU ke mB dr rr . Calling the equilibrium separation 0 r , we may solve for B α α + - = = 2 12 00 2 1 0 . m m mB ke ke B mr Substituting into the expression for Total U we find ( 29 α αα -  = - + = --   22 0 0 0 1 1 m m k e mr ke ke U r rm r . 12-2 a a a a 2 a 3 2 θ a 3 2 Since one carbon is at the center of the cube, it is easy to show three carbons from an isosceles triangle with sides. Using this triangle θ = sin 2 32 a a and θ 54.75 2 .
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188 CHAPTER 12 THE SOLID STATE 12-3 ( 29 α  = --   2 0 1 1 e Uk rm ( 29 ( ( ( 29 - -  ×  = - ×- ×  2 19 9 22 9 1.6 10 C 1 1.747 6 9 10 Nm 1 8 0.281 10 m UC - =- × 18 1.25 10 J 7.84 eV U . The ionic cohesive energy is +- = 7.84 eV Na -Cl U pair. 12-4 (a) 199 kcal/mole - × = × 18 23 199 kcal 4 186 J kcal 1.384 10 J Na Cl 6.02 10 Na -Cl pairs . As α 2 0 0 1 1 ke U , α α - + =+ 00 2 2 1 1 1 1 rU m ke m ke or α - 1 2 1 m ke so ( ( 29 ( ( ( - - - - ×× = - =≈ 1 9 18 2 9 2 2 19 0.257 10 m 1.384 10 J Na Cl 1 8 . 69 1.747 6 9 10 Nm C 1.6 10 C m . (b) α = 2 0 0 7 8 ke U r for = 8 m , α = 2 0 0 8 9 ke U r for = 9 m . The % change in - = ×= 0 89 78 100 1.6% 78 U . The value of 0 U is not very sensitive to a change in m since a 13% change in m only results in a 1.6% change in 0 U . 12-5 = --++--++- = - - + - + KK 22222222 2 111 21 223344 2 3 4 ke ke ke ke ke ke ke ke e rr rrrrrr r but (29 + = -+-+ K 234 ln1 xxx xx so ( 2 2ln2 ke U r . 12-6 (a) α = -+ 2 Total m ke B U r r From Problem 1, note that α - = 2 1 0 m Br m , which came from using the condition = = 0 0 dU dr . Therefore,
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MODERN PHYSICS 189 (29 αα α - - - +  = -+    =- + -   22 1 0 Total 2 1 1 Total 0 0 2 12 1 m m m m m ke r U rm r d U rr Fm d r mr r (b) If the atom is displaced by a small distance x from 0 r , we can replace r with + 0 rx , and F becomes ( 29 ( ( ( ( 29 α α - - -   = - - <<   ++  - ≈- - -  +  + -- + K K 2 1 0 21 0 00 2 2 0 0 2 0 2 0 1 Note: 1 1 1 11 1 2 1 m m m r x F r x r m xr r r k e m x r m r Neglecting the term 0 x r in the denominator, ( 29 ( 29 α α - 2 3 0 0 1 1 k e r x r mx F r r . Therefore, the ion experiences a restoring force F Kx , where the effective force constant K is α 2 3 0 1 Km r . (c) For NaCl, α= 1.747 6 , - = 9 0 0.281 nm 0.281 10 m r , = 8 m so that ( ( ( ( - - ×× ≈≈ × 2 9 2 2 19 3 9 9 10 Nm C 1.747 6 1.6 10 C 7 127 N m 0.281 10 m K . The corresponding frequency of vibration is π = 1 2 K f m where we use the mass of Na to estimate f ; - = 27 23 1.67 10 kg m so π - = 12 27 126 Nm 1 9.15 10 Hz 2 23 1.67 10 f .
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190 CHAPTER 12 THE SOLID STATE (d) λ - - × = = × 8 54 1 21 3 10 ms 3.16 10 m 3.16 10 nm 800 nm 9.5 10 s c f . Therefore, absorption is in the infrared. 12-7 (a) ( 29 ( 29 ( 29( 29 ( 29 α - - - +- ××  = -=  ×  - = ×= 2 9 2 2 19 2 0 19 0 18 1.747 6 9.00 10 Nm C 1.60 10 1 1 0.314 10 1 1 1.14 10 J 7.12 eV K -Cl 9 ke U rm (b) Atomic cohesive energy = ionic cohesive energy + energy needed to remove an electron from - Cl – energy gained by adding the electron to + =+-= K 7.12 eV 3.61 eV 4.34 eV 6.39 eV KCl . 12-8 (a) There are 6 - Cl ions at 0 r : =- 2 0 6 C ke U r . There are 12 + Na ions at 0 2 r : =+ 2 0 12 2 C ke U r .
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This note was uploaded on 11/05/2009 for the course PHSX 313 taught by Professor Staff during the Fall '09 term at Kansas.

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PC Chapter 12 - 12 The Solid State 12-1 U Total =...

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