CHAPTER 4 EVALUATION OF FORCE-CONSTANT MATRIX 4.- AIM OF THE WORK As antcpated n the ntodcton the am of the pesent ok s to obtan the nmecal vale of the foce-constant matx fo tantalm. In a fst step expesson of the total enegy of the elemental system s taken nto accont. It conssts of pa nteacton potentals embedded potentals and angla dependent potentals. etals abot the expesson fo the potentals sch as vales of paametes and lattce popetes ae pesented n the paagaph 4.. Statng fom the expesson of the total enegy of the system t s possble to evalate the fst and second devatve of sch enegy. Examples and explcatons abot ho to evalate these devatves ae pesented n the paagaph 4.3. Once evalated the tems of the devatves t s possble to eoganze them to evalate the foce-constant matx both n the eal doman that n the complex doman va Foe tansfom. These evalatons ae obtaned thanks to the se of the comptatonal softae pogam "Wolfam Mathematca 9.0". Moe detals abot ho the foce-constant matx as evalated and ts nmecal vale ae pesented n paagaph 4.4. 4.- TOTAL POTENTIAL ENERGY OF TANTALUM Tantalm a chemcal element hch symbol s T a and atomc nmbe 73 s a body-contended cbc BBC tanston metal.
Fge 4. - Tantalm The mechancal and physcal popetes depend to a geat extent on the metal pty. Pty s manly elated to the pesence of metallods sch as cabon oxygen ntogen and hydogen of hch even lo concentaton case a consdeable changes n the popetes. One of the man popetes of tantalm s ts hgh esstance to cooson. The alloys podced th tantalm have hgh meltng ponts ae stong and have good dctlty. It s sed ethe n a pe fom o as a component n alloys n a nmbe of technologcal applcatons ncldng hgh-tempeate stctal mateals and dffson baes n ntegated ccts. In aeospace applcatons t s manly sed n the podcton of spealloys fo et engne components. In ecent yeas sgnfcant effects have been made to mpove the basc ndestandng of the mechancal behavo of tantalm by means of compte modellng.
3 In the pesent ok to evalate the foce-constant matx the statng pont s the expesson of the total potental enegy fo tantalm pesented n scvee femento and hee shong: E tot φ F µ λ 6 ν 4. hee ndces and enmeate atoms and the spescpts 3 efe to a Catesan components of vectos and tensos. As befly ntodced n chapte 3.4. bt th dffeent notaton the fst tem n eqaton 4. epesents pa nteactons beng the pa-nteacton potental beteen an atom located at postons and an atom at poston. The fncton F s the enegy of embeddng atom n the host electon densty ndced at ste by all othe atoms of the system. The host electon densty s gven by : 4. hee s the electon densty fncton assgned to an atom. The fst to tems n 4. epesent the egla EAM pesented n chapte 3.4. and have a cental foce chaacte. They depend only on nteatomc dstances bt not on bond angles. The non-cental component of bondng s ntodced thogh the dpole vectos µ 4.3 and qadpole tensos
4 λ 4.4 hee ν s the tace of λ ν λ 4.5 The angla tems n 4.3-4.5 penalze the total enegy fo devatons of local atomc envonments fom cbc symmety. The expesson of the fnctons shoed above ee gven n a nmecal fom. A table th to colmns as sed fo each fncton n hch n one colmn thee ae the vales of the ndependent vaable and n the second thee ae the coespondng vales of the fncton. These tables can be fond n the C-ROM of the ok as a "Mcosoft Excel" fle named "modelo Tanalo Mshn.xlsx" Mshn Y. Elasse C. Gmbsch P. Lozovo Y. Hashbon A. 008. The man popetes of the BCC Tantalm ae shon n the follong fge :
5 Fge 4. - Popetes of BCC Tantalm calclated th the angla dependent potental n compason th expemental fst-pncple data It s mpotant to emak that the angla dependent potental s an empcal potental sed to evalate the total enegy of the system. It ncldes also the EAM potentals ntodced n chapte 3.4. and t s one of the many potentals avalable to model the total potental enegy. Each model can be bette than the othes dependng by the specfc applcatons. The potental sed fo ths ok s the one pesented n Mshn Y. Lozovo A. Y. 006. 4.3- FIRST AN SECON ERIVATIVE OF THE TOTAL POTENTIAL ENERGY
6 As pevosly ntodced n chapte 4. the foce constant matx can be constcted statng fom the second devatve of the total potental enegy expessed by 4.. In fact developng n sees of Taylo ntl the second ode the expesson of the potental of the cystal t s possble to obtan an expesson n hch appea the elements of the foce constant matx elated to the second devatve of the total potental enegy thogh the elaton : Φ k l m E l ToT k m 0 4.6 l Whee s the dsplacement of the atom m decton k s the dsplacement of the atom decton and E ToT s the total potental enegy The elaton 4. s fo a geneal case. l n the th coodnates m n the k th coodnate Applyng the dscete lattce model pesented n chapte 3 to the total potental enegy expessed by 4. the nteatomc dstance ae consdeed -cell elements an elementay segment hle the lattce ste poston ae consdeed 0-cell elements o nteconnected vetces.
7 Once made these assmptons t s necessay to choose the nmbe of neghbos consdeed n the expesson of the potental. In the pesent model ae taken nto accont the fst and second neghbos hch mply 4 atoms of efeence as schematcally shon n the fge.9. It s mpotant to note that n the expesson of the total potental enegy thee ae tems hch depend explctly fom the nteatomc dstance o elementay segment e sch as the pa nteacton potentals; and tems that depends explctly by the ste poston bt mplctly by the nteatomc dstance. Consdeng that to evalate the foce-constant matx t s necessay to deve the total potental enegy th espect to the -cell elementay segment hen devng the tems hch depends fom the 0-cell lattce poston sch as the angla-dependent nteatomc potentals o the EAM potental the chan le has to be appled. In the pesent ok to bette manage all the tems geneated by the second devatve of the total potental enegy sch devatves ae evalated ndvdal by fo each of the fve tems hch consttte the total potental enegy. Ths fomal expesson takng nto accont the chan le ae shon belo: E φ φ 4.7
8 F E k k k k F F F ] [ 4.8 E 3 k k k k k k k k k k k k k k
9 [ ] µ µ 4.9 4 E σ σ σ σ σ σ k k k k k k k k k
0 λ λ σ σ σ σ σ σ } σ σ 4.0 E 6 5 k 6 ν ν 6
4. The notaton sed s the one ntodced n chapte 4. epesented the nteatomc dstance o -cell elementay segment denotes the Konecke delta and the symbol denotes the dyadc podct. Wth the symbol ae denoted the atoms that ae the second exteme of a -cell elementay segment hch has as fst exteme the atom. Some smplfcatons ee sed to smplfy these expessons sch as : 4. λ λ 4.3 Whee λ s a vale obtaned hen the 3x3 matx λ s constcted µ 0 4.4 4.5 4.6 ν 4.7
The fst and second devatve of the fnctons φ F and hch appea nto the expessons 4.7-4. ae evalated nmecally as the fomat as nmecal. The fomlas sed to evalate these nmecal devatves ae: f t 6 f t 3 f t f t f ' t 6h 4 5 4 f '' t [ f t.. f t f t f t f t h 3 3 The nmecal vales can be seen n C-ROM nde the name "Mshn Tantalo.xlsx". 4.4- EVALUATION OF THE FORCE CONSTANT MATRIX Once evalated the second devatve of the fve tems of the total potental enegy chapte 4.3 t s possble to constct the foce-constant matx B. Fo each of the fve tems deved t s possble to constct a foceconstant matx egopng coectly the tems of each devatve. The fomal expesson of the foce-constant matx B of each tem s: B e e B P e e 0 Q 4. 4.0
3 B e e P Q 4. B e e R k k 4.3 B 3 e e P 3 Q 3 4.4 B 3 k k k e R3 T3 M 3 3 k k e N 4.5 B 4 σ σ e e P4 σ Q4 S4 4 E4 σ σ F4 G4λ 4.6 B 4 σ σ σ k k k e R4 T4 M 4 4 k k e N σ L 4 σ k k 4.7 B 5 e e P 5 Q 5 λ 4.8
4 B 5 e e R5 k k 4.9 Whee the geneal 3x3 elements B e e epesent the elements of the B e e man dagonal and the 3x3 elements ae the othes elements. The tems P Q R P... ae nmbes that mltples the dffeent 3x3 elements of B and ae obtaned egopng all the tems that fo each second devatve mltply the 3x3 element of B consdeed. The fnal foce-constant matx s obtaned by addng the fve matces: B e e B e e B e e B3 e e B4 e e B5 e e 4.30 B 3 4 B 5 e e e B e e B e e B e e e 4.3 As an example the elements B and B ae shon : 0.649868 B 0.397763 0.397763 0.397763 0.649868 0.397763 0.397763 0.397763 0.649868 0.009089 B 0.0487 0.06 0.0487.009089 0.06 0.06 0.06 0.057455 Whee epesents the -cell elements and shon n fge 3.. The fll expesson of the matx B can be fond n the C-ROM nde the name "vefca del fle nmec secondo vecno.nb"
5 Usng the Foe tansfom t s possble to evalate the matx Ψˆ θ l Ψ l m e θl 4.3 At last sng the elaton 3. t s possble to evalate the matx Φˆ hose nmecal expesson s: 0.07907 ˆΦ 0.08899 0.00363806 0.08899 0.33309 0.0056 0.00363806 0.0056 0.0665869 hee the expesson of Φˆ s evalated fst applyng the le : θ4 θ θ θ3 ; θ5 θ θ3 ; θ 6 θ θ3 ; θ 7 θ θ and then gvng the vales : θ 0. ; θ 0. ; θ 3 0. othese the matx Φˆ depends by θ and has a vey lage expesson. All these evalatons ee made sng the softae Wolfam Mathematca 9.0 a comptatonal softae pogam.