Homework 7 Due 5 December 2017 The numbers following each question give the approximate percentage of marks allocated to that question.

Transcription

1 Name: Homework 7 Due 5 December 2017 The umbers followig each questio give the approximate percetage of marks allocated to that questio. 1. Use the reciprocal metric tesor agai to calculate the agle betwee (112) ad (431) plaes i a moocliic uit cell with lattice costats a = 1 Å, b = 2 Å, c = 4 Å, β = 95. You ca use dspace to check your aswer: 10

2 2. Isertig a sigle atom ito the 1a site i space group P3 (attached) results i a poit symmetry which is ot 3 as listed. a. What is the resultat poit symmetry (remember to thik i three dimesios)? Cojecture why it is ot the same as what is listed. Use a diagram to help explai if ecessary. 5 b. Briefly explai why oe would ot describe a crystal i which oly the 1a site is occupied i this space group. What space group would be more correct? 5

3 3. a. Draw a represetatio of objects with 2/m poit symmetry, icludig all the ecessary symmetry elemets. Usig this diagram, costruct a multiplicatio table for this poit group. HINT: The order of the group is four. 8 b. Draw a represetatio of objects with 4mm poit symmetry, icludig all the ecessary symmetry elemets ad uiquely labelig all of the mirror symmetry elemets preset. Usig this diagram, costruct a multiplicatio table for this poit group usig the same labels. HINT: The order of the group is eight. 8 c. Which, if either, of these two groups is/are abelia? 2

4 4. Calculate the structure factor F for the followig reflectios of a ideal cubic perovskite (ABO3) i terms of fa, fb, fo. Your aswers should have o imagiary parts. a) b) c) d) Which is likely to be the weakest of the four? 2 The atomic positios are: A B ½ ½ ½ O ½ ½ 0 ½ 0 ½ 0 ½ ½ F ( hkl) = f 1 e 2πi ( hu + kv + lw )

5 5. For the electro diffractio patter show below: a. What are the idices of the reflectio marked hkl? 3 b. What is the zoe axis? 3 c. What ca you coclude about the structure based o the systematic abseces ad the differet spot itesities? hkl 420

6 6. Briefly explai the Curie Priciple A object with poit symmetry m3 m (Oh) distorts uder uiaxial stress alog [001]. What is the poit group of the distorted object? What is it whe the stress is alog [111]? If a applied electric field removes the cetre of symmetry i each case, to which poit groups would the distorted objects ow belog? May space group otatios are depedet upo the orietatio of the axes which oe chooses to describe a structure. I such cases there is a stadard settig, but potetially may other o-stadard oes as well. I the example of Pba, if the axes were re-orieted such that abc cab, use what you kow about symmetry operatios ad space-group otatio to determie what the ew otatio would be, ad explai your aswer. 10

7 9. Carbo atoms i graphite are boded withi (001) via strog triagular (120 ) sp 2 hybrid σ bodig. The remaiig uboded 2p electros form delocalized π bods, ad the adjacet sheets are held together by weak Va der Waals forces. For this reaso, both the electrical ad thermal coductivities of graphite are higher i the (001) tha ormal to it. The thermal coductivity kij is a tesor that relates the heat flux h to the temperature gradiet dt/dx accordig to: h = k i ij dt dx j The thermal coductivity tesor of graphite is: k = Wm -1 K -1 ij a. If a gradiet of 100 K/m is applied alog the y directio, what is the resultig heat flow (i kw/m 2 )? 5 b. If a gradiet of 100 K/m is applied alog the z directio, what is the resultig heat flow (i kw/m 2 )? 5

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