DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE
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1 Determintion RevAdvMterSci of mechnicl 0(009) -7 properties of nnostructures with complex crystl lttice using DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE Olg S Lobod nd Anton M Krivtsov Deprtment of Theoreticl Mechnics St Petersburg Stte Polytechnicl University Polyteknichesky St Petersburg Russi Received: Februry Abstrct The pper considers discrete nno-crystlline model with complex crystl lttices There re non close-pcked lttices typicl for some metls nd for solids with covlent bonds for exmple dimond grphite Trditionlly to describe this kind of lttices one used mnyprticle-interction potentils The lterntive pproch considers rottionl degree of freedom nd llows the moment contribution in intertomic interction The prticles interct using forces nd moments In this pper the chrcteristics of intertomic bonds re determined for crystls with dimond structure (crbon silicon nd germnium) It is shown tht reltion of bending stiffness of crbon covlent bonds to longitudinl one is between (in dimond crystls) Hence the bending stiffness is comprble with longitudinl nd it should be tken into ccount on clcultion for covlent crystls For crystls of silicon nd germnium this reltion is equl 0 The strength of covlent bonds decreses with the rise of intertomic distnce in sequence C-Si-Ge COMPLEX CRYSTAL LATTICES determine lttice geometry: here is the cell number A simple lttice is lttice with equivlent nodes is the vector connecting prticle of given cell with prticle β from cell there re two prticles in ech cell β nd For the simple lttice the trnsltion long the vector connecting ny two nodes is n identicl trnsformtion (for exmple simple cubic lttice) A The mcroscopic power stiffness tensor is given by: complex crystl lttice is lttice which does not possess this property (for exmple grphite dimond NCl lttice) A discrete mechnicl model ' A A A A for the complex crystl lttice ws developed in A V [] The prticles in the lttice hve trnsltionl ' T T nd rottionl degrees of freedom The prticles A b Agb Ag b Ag interct by mens of forces nd moments Mcroscopic chrcteristics of crystls depend on longi- A A () tudinl stiffness of intertomic bonds s well s on V trnsversl stiffness The equtions for mcroscopic stiffness tensor The component A is the stiffness tensor under cn be expressed in terms of intertomic bonds uniform deformtion A ' is correction component describing shift between two sublttices V stiffness tensor nd vectors These vectors is Corresponding uthor: Olg S Lobod e-mil: lobod_o@milru 009 Advnced Study Center Co Ltd
2 O S Lobod nd A M Krivtsov Fig Dimond lttice the unit cell volume The stiffness tensors of intertomic bonds we cn represent s: A A + D d d () + d d E d i The coefficients A in () describe longitudinl stiffness of n intertomic bond coefficients D trnsversl stiffness of intertomic bond In non-deformed stte the distnce between neighboring toms is Ech tom intercts only with the nerest neighbor toms In tht cse ll nonzero stiffness coefficients re equl nd we cn write: A A A D D D () ELASTIC PROPERTIES OF NANOCRYSTAL WITH D DIAMOND LATTICE Now we consider three-dimensionl crystl with dimond lttice (Fig ) lttice orienttion is {00} Let i j nd k be the unit vectors of lttice principl directions Then vectors will be equl: b g b g b g b g i + j + k i + j k i j + k i j k () Substituting () in formul () we obtin the expressions for stiffness coefficients of dimond: A A A V A + D 9 A A A A A A A D f f (5) For elsticity modulus (5) the expression for the correction component A ' is equl zero This llows simplify the formule As is known from the clssicl theory of elsticity cubic crystls hve three independent elstic constnts The coefficients A A nd sher modulus cn be chosen s these constnts The sher modulus in the moment theory is determined by expression (6)
3 Determintion of mechnicl properties of nnostructures with complex crystl lttice using 5 Tble Experimentl dt for dimond elstic constnts No K GP C GP C GP C GP E GP Ref [] [] The following nottions re used: A C A C G C Tble Clculted elstic constnts for dimond No A N/m D N/m D/A C GP Intertomic distnce in dimond crystls is 05 nm G A + A f (6) Using formule () nd () we obtin ' DA D A A + A D ' D D A A + A D f f (7) Returning to (6) we obtin for the sher modulus 8 AD G (8) D The unit cell volume cn be found using () s f V b b b b 6 V 9 (9) Substituting (9) in (5) nd (6) we finlly obtin the formule for the three independent elstic constnts They re expressed through stiffness of n intertomic bond nd intertomic distnce : f f A A + D ; A A D ; AD G 8 D (0) Expression () shows the connection between the coefficients of the stiffness tensor nd Young modules: D E D E D E def D A A A + A A A + A A A A () The bulk modulus for tensor A cn be determined by formul (): K E A E () d In our cse it hs the form: K A () As ws expected the bulk modulus depends on the coefficient A only which chrcterizes longitudinl stiffness of n intertomic bond nd does not depend on the coefficient D which chrcterizes trnsversl stiffness of n intertomic bond As is known from the clssicl theory of elsticity (momentless) there re only three indepen-
4 6 O S Lobod nd A M Krivtsov Tble Experimentl dt for silicon elstic constnts No K GP C GP C GP C GP E GP Ref [] Tble Clculted elstic constnts for silicon A N/m D N/m D/A C GP Intertomic distnce in silicon crystls is 05 nm Tble 5 Experimentl dt for germnium elstic constnts K GP C GP C GP C GP E GP Ref [] Tble 6 Clculted elstic constnts for germnium A N/m D N/m D/A C GP Intertomic distnce in germnium crystls is 05 nm Tble 7 Comprtive tble of clculted trnsversl nd longitudinl stiffness for elements with dimond structure A N/m D N/m D/A nm C C Si Ge dent elstic constnts for exmple (0) If the vlues of these elstic constnts re known for some mteril then expressions (0) llow find microscopic chrcteristics of intertomic bonds coefficients A nd D Thus momentless mcroscopic chrcteristics of mteril llow find power chrcteristic intertomic bond A nd coefficient D The coefficient D chrcterizes the trnsversl stiffness intertomic bond It should be noted tht this coefficient presents only in the models which tke into ccount moment interction on the intertomic level Experimentl dt for dimond elstic constnts [] re shown in Tble bulk modules K nd young modules E re clculted from () As one cn see these dt re not the sme for different experimentl methods Using experimentl dt [] for C nd C nd formuls (0) we obtin the vlue of stiffness of intertomic bonds in dimond crystls: A 578 N/m D 85 N/m () Hence the rtio trnsversl stiffness / longitudinl stiffness in dimond crystl is D/ A 09
5 Determintion of mechnicl properties of nnostructures with complex crystl lttice using 7 Using experimentl dt [] for nd C nd formule (0) we obtin the vlue of stiffness of intertomic bonds in dimond crystls: A 7 N/m D 0 N/m (5) Hence the rtio trnsversl stiffness / longitudinl stiffness in dimond crystl is D/A 07 It mens tht in both cses the trnsversl stiffness is of the sme order s longitudinl one nd thus hs to be tken into ccount when investigting covlent crystls Using coefficients () (5) nd formule (0) () we cn determine the vlue of elstic constnts (Tble ) As one cn see from Tbles nd the clculted dt re nerer to the experimentl ones in the second cse The mximum devition does not exceed 6% in the second cse wheres in the first cse it is equl to 66% Anlogously we consider silicon Silicon hs dimond structure too The experimentl dt for silicon elstic constnts [] re shown in Tble Using experimentl dt for nd C nd formule (0) we obtin the vlue of stiffness of intertomic bonds in silicon crystls: A 60 N/m D 55 N/m (6) so tht D/A 0 The substitution of coefficients (6) into formule (0) () gives the vlue of elstic constnts (Tble ) In this cse the mximum devition from the experimentl dt is % We performed similr clcultion for germnium since it hs dimond structure too Experimentl dt for germnium re shown in Tble 5 The vlue of stiffness of intertomic bonds in germnium crystls: A N/m D 6 N/m (7) with D/A 08 From (7) we clculted the vlue of elstic constnts for germnium (Tble 6) In this cse the mximum devition from experimentl dt is 5% Thus from the bove results it is seen tht the method considered gives good coincidence with the experimentl dt The mximum devition is 66% However it is necessry to tke into ccount the spred of experimentl vlues Tble 7 mkes comprison of the clculted trnsversl nd longitudinl stiffness for elements with dimond structure (crbon silicon germnium) From it follows tht in the C-Si-Ge series the stiffness of covlent bonds decreses with the increse of n intertomic distnce CONCLUSIONS For crystls with dimond structure such s crbon silicon germnium the equtions for mcroscopic tensors of stiffness re obtined They depend on tensors of stiffness of intertomic bonds nd vectors which define the geometry of the lttice The chrcteristics of the intertomic bonds re lso defined It is shown tht the rtio trnsversl stiffness / longitudinl stiffness of crbon s covlent bond is between 07 nd 09 It mens tht the trnsversl stiffness is of the sme order s longitudinl one nd thus hs to be tken into ccount when investigting covlent crystls In the C-Si-Ge series the stiffness of covlent bonds decreses with the increse of n intertomic distnce REFERENCES [] EA Ivnov AM Krivtsov nd NF Morozov // Applied Mthemtics nd Mechnics 7 (00) 595 [] HF Mrkhm IN: Ntionl Physicl Lbortory mesurements presented by Musgrve: Dimond Conf (Reding 965) [] HJ McSkimin nd P Andretch // J Appl Phys (97) 9 []
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