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(3)

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(8)

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(10)

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(11)

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(12)

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Lq Iljxuh 7/ wkh frxuvh plqlpxp ri wkh ri wkh vxp ri vtxduhv ri uhodwlyh huuruv/ [/ dqg wkh fruuhvsrqghqw uvw uhwdugdwlrq wlph duh vkrzq dv ixqfwlrq ri wkh orjdulwkplf glvwdqfh/ @/ ehwzhhq wkh uhwdugdwlrq wlphv iru wkh fdofxodwlrq ri g|qdplf frpsoldqfhv zlwk fI qrlvh1 Dv fdq eh vhhq lq Iljxuh 7/ wkh rswlpxp vroxwlrq iru wkh fI qrlvh gdwd lv ohvv ghshqghqw rq @ dqg suhvhqwv dq

(13)

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Wkh rwkhu vshfwuxp lv ghulyhg iurp wkh frpsoldqfh ixqfwlrq ri dq hsr{| uhvlq/ zklfk lv edvhg rq wkh Froh0Froh ixqfwlrq ^;`/ xvlqj wkh vhfrqg rughu dssur{lpdwlrq irupxod ri Vfkzdu}o dqg Vwdyhupdq ^46`

(14)

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(15)

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aAfd*L} E|o ' aAd*L} E| *L} E@Ao +58,

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ic aj_A , i*@Ac aj_A_f +59,

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Lq Iljxuh : wkh fdofxodwhg vshfwud iru wkh pxowl0vwhs vshfwuxp duh jlyhq/ dqg lq Iljxuh ; wkh fdofxodwhg frpsoldqfh +dssur{1, lv sorwwhg djdlqvw wkh h{dfw ixqfwlrq +h{dfw, dorqj zlwk wkh uhodwlyh huuru1 Uhjlrqv ri frqvwdqw ydoxh lq wkh vshfwuxp duh uhsurgxfhg yhu| zhoo dqg wkh mxpsv lq wkh pxowl0vwhs vshfwuxp duh uhdvrqdeo| uhsurgxfhg1 Dv vkrzq lq Iljxuh ; wkh dssur{lpdwlrq ri wkh fuhhs frpsoldqfh vkrzv d uhodwlyh huuru orzhu wkdq efI1

Lq Iljxuh < wkh fdofxodwhg ydoxhv iru wkh vshfwuxp ri Htxdwlrq 55 duh vkrzq/ dqg lq Iljxuh 43 wkh fdofxodwhg fuhhs frpsoldqfh +dssur{1, lv sorwwhg djdlqvw

(16)

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Zh zloo ghprqvwudwh wkdw wkh phwkrg vwloo zrunv zlwk qrlv| gdwd iru wkhvh fdvhv1 Dv lq wkh uvw whvw zh lpsrvh vrph udqgrp qrlvh rq wkh g|qdplf gdwd e| pxowlso|lqj wkh h{dfw ydoxhv e| d idfwru ri E n Y o@?_ E1

Lq Iljxuhv 44 dqg 45 wkh fdofxodwhg fuhhs frpsoldqfh +dssur{1, lv sorwwhg djdlqvw wkh h{dfw ixqfwlrq +h{dfw, dorqj zlwk wkh uhodwlyh huuru iru erwk fdvhv1 Lw fdq hdvlo| eh vhhq wkdw wkh fdofxodwhg fxuyhv pdlqwdlq wkh vkdsh ri wkh h{dfw fxuyhv exw duh vkliwhg dorqj wkh wlph d{lv1 Wklv vkliwlqj lv wkh uhvxow ri fxpxodwlyh huuruv lq fdofxodwlrq ri wkh whpshudwxuh0wlph vfdoh idfwruv gxh wr wkh udqgrp qrlvh1 Wkh uhodwlyh huuruv ghshqg rq wkh ghyldwlrq dqg rq wkh fxuyh*v vorsh dqg wkhuhiruh iru wkh vdph dprxqw ri ghyldwlrq/ wkh uhodwlyh huuru lqfuhdvh zlwk wkh lqfuhdvh ri wkh fxuyh*v vorsh1 Dv iru wkh dfwlydwlrq hqhuj| ri wkh Duukhqlxv htxdwlrq/ wkh suhvhqw phwkrg fdofxodwhv wkrvh sdudphwhuv zlwk d uhodwlyh huuru orzhu wkdq .DI1

Frqfoxvlrqv

Zh kdyh suhvhqwhg dq hdv| wr xvh dojrulwkp lq rughu wr frqvwuxfw wkh pdvwhu fuhhs frpsoldqfh fxuyhv iurp wkh h{shulphqwdo g|qdplf ylvfrhodvwlf ixqfwlrqv rewdlqhg rq d GPWD dqdo|vhu1 Wkh phwkrg fdq dovr eh xvhg wr rewdlq dq dssur{lpdwlrq ri wkh glvfuhwh uhwdugdwlrq vshfwud1

(17)

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Dfnqrzohgjhphqwv

Wkh qdqfldo vxssruw ri wklv zrun e| wkh MQLFW +Mxqwd Qdflrqdo gh Lqyhvwljdêær Flhqwðfd h Whfqroöjlfd, xqghu wkh Judqw EG :39 lv judwhixoo| dfnqrzohgjhg1

Uhihuhqfhv

41 Sryrogr/ I1/ Holgd E1 Khuplgd/ Mrxuqdo ri Pdwhuldov Vflhqfh/ 58/ 736907374 +4<<3,1

51 Ihuu|/ Mrkq G1/ Ylvfrhodvwlf Surshuwlhv ri Sro|phuv/ Mrkq Zloh| ) Vrqv +4<;3,/ ss1 <31

61 Vfkzdu}o/ I1 U1/ Ukhro Dfwd / 47/ 8;408<3 +4<:8,1

71 Jlevrq/ U1 I1 / V1 M1 Kzdqj/ F1 K1 Vkhssdug/ Mrxuqdo ri Frpsrvlwh Pdwhuldov/ 57/ +4<<3,1

(18)

81 Pdlohu/ S1/ Eljdxg/ G1 ) Kdpholq/ S1/ Gxudelolw| Hydoxdwlrq rq Vriw Phpeudqh Frpsrvlwhv Xqghu Frpelqhg Ordglqj +phfkdqlfdo dqg hqylurqphqwdo,/ lq Surjuhvv lq Gxudelolw| Dqdo|vlv ri Frpsrvlwh V|vwhpv/ Fdugrq/ Ixnxgd ) Uhlivqlghu+hgv1,/ Edonhpd/ Urwwhugdp +4<<9,/ ss1 4<80 5351

91 Hpul/ L1 ) Q1 Z1 Wvfkrhjo/ Ukhro Dfwd/ 65/ 6440654 +4<<6,1

:1 Wkljshq/ O1/ J1 Z1 Khgvwurp/ E1 S1Erqqhu/ Mrxuqdo ri Dssolhg Phfkdqlfv/ 83/ 6940699 +4<;6,1

;1 Wvfkrhjo/ Q1 Z1/ Wkh skhqrphqrorjlfdo wkhru| ri olqhdu ylvfrhodvwlf ehkdylrxu/ Vsulqjhu0Yhuodj +4<;<,1

<1 \rqj0Vrn R1/ F1 Slorw ) F1 Mrugdq/ Prghol}dwlrq ri Ylvfrhodvwlf Ehkdylru ri Fduerq2Hsr{| Xqlgluhfwlrqdo Frpsrvlwhv/ lq= Surfhhglqj ri wkh Hxurshdq Phfkdqlfv Froortxlxp 4;5 +4<;7,/ ss1 :60;61

431 Hpul/ L1 ) Q1 Z1 Wvfkrhjo/ Ukhro Dfwd/ 65/ 655065: +4<<6,1 441 Hpul/ L1 ) Q1 Z1 Wvfkrhjo/ Ukhro Dfwd/ 66/ 930:3 +4<<7,1 451 Ndvfkwd/ M1 ) I1 U1 Vfkzdu}o/ Ukhro Dfwd/ 66/ 84:085< +4<<7,1 461 Ndvfkwd/ M1 ) I1 U1 Vfkzdu}o/ Ukhro Dfwd/ 66/ 8630874 +4<<7,1

471 Vfkdshu|/ U1 D1/ Dssur{lpdwlrq phwkrgv ri wudqvirup lqyhuvlrq iru ylvfrhodvwlf vwuhvv dqdo|vlv/ lq= Surfhhglqjv ri wkh 7wk X1V1 Qdwlrqdo

(19)

Frqjuhvv ri Dssolhg Phfkdqlfv/ DVPH/ 5 +4<95,/ ss1 43:8043;81

(20)

Wdeoh L1 G|qdplf fxuyhv iru wkh vshfwud jlyhq e| Htxdwlrq 531 sEM5 a_{EC @}3_a_{EC @}3

fff eSfDDe D2fef f3f2 ff2f eDffD bbbDe f3f2 fff eeDe feebf.D fff 2.DDD. fD.Db. fff fHS2SS. fe.b. fff fDfb2 Db.feH f3f2 fff fe.e.. bHbbS2 f3f2 53

(21)

1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1 10 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 a Ψ τ1 Ψ parameterspositive _τ₁_{; s} N=5

Iljxuh 4= Plqlpxp ri wkh vxp ri uhodwlyh huuruv/ [/ dqg wkh fruhvsrqghqw uhwdugdwlrq wlph// iru wkh fdofxodwlrq ri wkh g|qdplf frpsoldqfhv/ dv ixqfwlrqv ri wkh orjdulwkplf glvwdqfh/ @/ ehwzhhq wkh uhwdugdwlrq wlphv1

-0.10 0.10 0.30 0.50 0.70 0.90 1.10

1.E-05 1.E-03 1.E-01 1.E+01 1.E+03 1.E+05

τ1 ; s

L(τ) , Jk/ln(2.5) ; GPa-1

Iljxuh 5= Qrupdolvhg uhwdugdwlrq frpsrqhqwv a&*,?E@ iru @ ' 2D yv h{dfw uhwdugdwlrq vshfwuxp/ Ht1 531

(22)

0.00 0.50 1.00 1.50 0.001 0.01 0.1 1 10 100 -12 -8 -4 0 4 8 12 t ; s J(t) ; GPa-1 _{error ; (%)} exact approx. error

Iljxuh 6= Dssur{lpdwhg fuhhs frpsoldqfh iru @ ' 2D yv h{dfw fuhhs frpsoldqfh dv fdofxodwhg iurp vshfwuxp jlyhq e| Ht1 531

1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1 10 100 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 a Ψ τ1 Ψ τ1 ; s positive parameters N=5 10% Noise

Iljxuh 7= Plqlpxp ri [ dqg wkh fruhvsrqghqw uhwdugdwlrq wlph// iru wkh fdofxodwlrq ri wkh g|qdplf frpsoldqfhv zlwk fI qrlvh/ dv ixqfwlrqv ri wkh orjdulwkplf glvwdqfh/ @/ ehwzhhq wkh uhwdugdwlrq wlphv1

(23)

-0.10 0.10 0.30 0.50 0.70 0.90 1.10

1.E-05 1.E-03 1.E-01 1.E+01 1.E+03 1.E+05

τ ; s

L(τ) , Jk/ln(2.5) ; GPa-1

Iljxuh 8= Qrupdolvhg uhwdugdwlrq frpsrqhqwv a&*,?E@ iru @ ' 2D dqg iru fI qrlvh g|qdplf gdwd +Y ' f, yv h{dfw uhwdugdwlrq vshfwuxp/ Ht1 531

0.00 0.50 1.00 1.50 0.001 0.01 0.1 1 10 100 -12 -8 -4 0 4 8 12 t ; s J(t) ; GPa-1 _{error ; (%)} exact approx. error

Iljxuh 9= Dssur{lpdwhg fuhhs frpsoldqfh iru @ ' 2D dqg iru fI qrlvh g|qdplf gdwd +Y ' f, yv h{dfw fuhhs frpsoldqfh dv fdofxodwhg iurp vshfwuxp jlyhq e| Ht1 531

(24)

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01

1.E-07 1.E-03 1.E+01 1.E+05 1.E+09 1.E+13 1.E+17

τ ; s

L(τ) , Jk/ln(10) ; GPa-1

Iljxuh := Grxeoh0orjdulwkplf sorw ri wkh qrupdolvhg uhwdugdwlrq frpsrqhqwv a&*,?E@ iru @ ' f yv wkh h{dfw uhwdugdwlrq vshfwuxp/ Ht1 541

1.E-01 1.E+00 1.E+01 1.E+02

1.E-03 1.E+01 1.E+05 1.E+09 1.E+13 1.E+17

-12 -8 -4 0 4 8 12 t ; s J(t) ; GPa-1 error ; (%) exact approx. error

Iljxuh ;= Grxeoh0orjdulwkplf sorw ri wkh dssur{lpdwhg fuhhs frpsoldqfh iru @ ' f yv h{dfw fuhhs frpsoldqfh dv fdofxodwhg iurp vshfwuxp jlyhq e| Ht1 541

(25)

1.E-10 1.E-08 1.E-06 1.E-04 1.E-02 1.E+00 1.E+02

1.E-10 1.E-04 1.E+02 1.E+08 1.E+14 1.E+20

τ ; s

L(_τ) , Jk/ln(10) ; GPa-1

Iljxuh <= Grxeoh0orjdulwkplf sorw ri wkh qrupdolvhg uhwdugdwlrq frpsrqhqwv a&*,?E@ iru @ ' f yv h{dfw uhwdugdwlrq vshfwuxp/ Ht1 551

1.E-01 1.E+00 1.E+01 1.E+02

1.E-03 1.E+01 1.E+05 1.E+09 1.E+13 1.E+17

-12 -8 -4 0 4 8 12 t ; s J(t) ; GPa-1 error ; (%) exact approx. error

Iljxuh 43= Grxeoh0orjdulwkplf sorw ri wkh dssur{lpdwhg fuhhs frpsoldqfh iru @ ' f yv h{dfw fuhhs frpsoldqfh dv fdofxodwhg iurp vshfwuxp jlyhq e| Ht1 55

(26)

1.E-01 1.E+00 1.E+01 1.E+02

1.E-03 1.E+01 1.E+05 1.E+09 1.E+13 1.E+17

-100 -60 -20 20 60 100 t ; s J(t) ; GPa-1 error ; (%) exact approx. error

Iljxuh 44= Grxeoh0orjdulwkplf sorw ri wkh dssur{lpdwhg fuhhs frpsoldqfh iru @ ' f dqg iru fI qrlvh g|qdplf gdwd + Y ' f , yv h{dfw fuhhs frpsoldqfh dv fdofxodwhg iurp wkh vshfwuxp jlyhq e| Ht1 541

1.E-01 1.E+00 1.E+01 1.E+02

1.E-03 1.E+01 1.E+05 1.E+09 1.E+13 1.E+17

-100 -60 -20 20 60 100 t ; s J(t) ; GPa-1 error ; (%) exact approx. error

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