Диссертация (1149591), страница 14
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Êîëè÷åñòâî èíòåðâàëîâ ñåòêèïî ïåðåìåííîé θ áûëî âûáðàíî ðàâíûì 499, ÷òî îïðåäåëÿåò õàðàêòåðíûé ðàçìåð îäíîãî áëîêà áëî÷íî-òð¼õäèàãîíàëüíîé ìàòðèöû ðàâíûì 998 (äëÿ êâàðòåòà). Êîëè÷åñòâî áëîêîâ íà äèàãîíàëè îïðåäåëÿåòñÿ øàãîì ∆ρ ðàâíîìåðíîé ñåòêè ïî ïåðåìåííîé ρ, êîòîðûé áûë âûáðàí ðàâíûì 0.01 ôì.  ðåçóëüòàòå, áûëè ïîëó÷åíû çíà÷åíèÿ àìïëèòóäíîé ôóíêöèè áèíàðíîãî ðàññåÿíèÿaJ0 (q, ρmax) è êîýôôèöèåíòîâ ðàçëîæåíèÿ aJi,k (E, ρmax) àìïëèòóäíîé ôóíêöèèêàíàëà ðàçâàëà äëÿ ðàçíûõ çíà÷åíèé ρmax .Áûëè âû÷èñëåíû íàáîðû aJi,k (E, ρmax ),k = 1, . . . , Nφ äëÿ ðàçíûõ Nφ ,÷òî ïîçâîëèëî íàéòè îïòèìàëüíîå ÷èñëî Nφ äëÿ äîñòàòî÷íî òî÷íîãî âîññòàíîâëåíèÿ äîïðåäåëüíîé è ïðåäåëüíîé àìïëèòóäíîé ôóíêöèè êàíàëà ðàçâàëà. Ïåðâûå êîýôôèöèåíòû ðàçëîæåíèÿ ñîîòâåòñòâóþò íèçøèì ãàðìîíèêàìè îïðåäåëÿþò îáùèé âèä àìïëèòóäíîé ôóíêöèè.
Îíè ÿâëÿþòñÿ íàèáîëüøèìè ïî àáñîëþòíîé âåëè÷èíå ÷ëåíàìè ðÿäà. Èõ çíà÷åíèÿ, ïî ìåíüøåé ìåðå,íà äâà ïîðÿäêà ïðåâûøàþò ïîãðåøíîñòü âû÷èñëåíèé è ïîñòåïåííî ñòàáèëèçèðóþòñÿ ñ óâåëè÷åíèåì ρmax . Èçìåíåíèå çíà÷åíèé ïåðâûõ êîýôôèöèåíòîâíà èíòåðâàëå 300 ôì < ρmax < 1500 ôì êàê ìèíèìóì íà ïîðÿäîê ìåíüøå èõ âåëè÷èíû. Ñîîòâåòñòâåííî, ïðîôèëü àìïëèòóäû ðàçâàëà îñòàåòñÿ ïîñòîÿííûì, íî åãî ôîðìà ñãëàæèâàåòñÿ çà ñ÷åò óìåíüøåíèÿ âêëàäà âûñîêèõãàðìîíèê ïðè óâåëè÷åíèè ρmax .
Ïðè ρmax < 500 ôì âñå ïîëó÷àåìûå êîýôôè99öèåíòû äëÿ âûñîêèõ ãàðìîíèê ïðåâûøàþò ïîãðåøíîñòü âû÷èñëåíèé íà ïîðÿäîê. Ýòî ïðîèñõîäèò äëÿ ðàçíûõ çíà÷åíèé Nφ è íå ïîçâîëÿåò îïðåäåëèòüîïòèìàëüíîå çíà÷åíèå Nφ äëÿ äàííîé îáëàñòè ρmax . Ïðè óâåëè÷åíèè ρmax ,íà÷èíàÿ ñ ρmax ∼ 500 ôì, çíà÷åíèÿ êîýôôèöèåíòîâ äëÿ âûñîêèõ ãàðìîíèêïîñòåïåííî âûñòðàèâàþòñÿ ïî àáñîëþòíîé âåëè÷èíå òàê, ÷òî ñ óâåëè÷åíèåìk èõ âåëè÷èíà óìåíüøàåòñÿ è äîñòèãàåò âåëè÷åíû ïîãðåøíîñòè ïðè k ∼ 30äëÿ âñåõ ρmax > 700 ôì. Òàêèì îáðàçîì, íà÷èíàÿ ñ ρmax ∼ 700 ôì âûñîêèå ãàðìîíèêè íå âíîñÿò ñóùåñòâåííîãî âêëàäà, è ïîýòîìó äîïðåäåëüíàÿ èïðåäåëüíàÿ àìïëèòóäíûå ôóíêöèè êàíàëà ðàçâàëà ìîãóò áûòü äîñòàòî÷íîòî÷íî âîññòàíîâëåíû ñ èñïîëüçîâàíèåì Nφ = 30 êîýôôèöèåíòîâ ðàçëîæåíèÿaJi,k (E, ρmax).
Ýòîãî êîëè÷åñòâà êîýôôèöèåíòîâ äîñòàòî÷íî äëÿ õîðîøåãî âîññòàíîâëåíèÿ äîïðåäåëüíîé àìïëèòóäíîé ôóíêöèè êàíàëà ðàçâàëà ïðè âñåõâîçìîæíûõ çíà÷åíèÿõ ρ. Ïåðåõîä ê ïðåäåëüíîé àìïëèòóäíîé ôóíêöèè êàíàëà ðàçâàëà ïðè ρ → ∞ â äàííîì ñëó÷àå îñóùåñòâëÿåòñÿ èñêëþ÷èòåëüíîïðåäåëüíûìè ñâîéñòâàìè áàçèñíûõ ôóíêöèé φk (ρ|θ). Âîçìóùåíèå àìïëèòóäíîé ôóíêöèè êàíàëà ðàçâàëà â îêðåñòíîñòè 90◦ , àíàëîãè÷íî ñëó÷àþ ìîäåëüíîé çàäà÷è (ñì. Ðèñ. 6.11), óìåíüøàåòñÿ ñ ðîñòîì ρ è â ïðåäåëå èñ÷åçàåòñîâñåì. Äîñòàòî÷íàÿ òî÷íîñòü ïîëó÷àåìîé àìïëèòóäíîé ôóíêöèè äîñòèãàåòñÿ ïðè ρmax ∼ 1400 ôì. Íà Ðèñ. 6.12, 6.13, 6.14, 6.15, 6.16, 6.17 äëÿ çíà÷åíèÿ ρ ∼ 1400 ôì ïðåäñòàâëåíû äîïðåäåëüíûå àìïëèòóäíûå ôóíêöèè êàíàëàðàçâàëà AJ (θ, E, ρmax , ρ), êîýôôèöèåíòû ðàçëîæåíèÿ êîòîðûõ áûëè âû÷èñëåíû ïðè ρmax = 1300 − 1550 ôì, à òàêæå àíàëîãè÷íûå ãðàôèêè ïðåäåëüíûõàìïëèòóäíûõ ôóíêöèé. Äàííûå ïðåäåëüíûå àìïëèòóäíûå ôóíêöèè êàíàëàðàçâàëà õàðàêòåðèçóþòñÿ ïðåäåëüíûì çíà÷åíèåì ρ = ∞.
Ëåãêî âèäåòü, ÷òîó ïðåäåëüíûõ àìïëèòóäíûõ ôóíêöèé îòñóòñòâóåò âîçìóùåíèå â îêðåñòíîñòè 90◦ . Ïðåäåëüíàÿ àìïëèòóäíàÿ ôóíêöèÿ â äàííîì ñëó÷àå àíàëèòè÷åñêèâîññòàíàâëèâàåòñÿ ÷åðåç ñâîè êîýôôèöèåíòû ðàçëîæåíèÿ ïî ôîðìóëå (4.2).Äàííàÿ ðåãóëÿðèçàöèÿ ïîâåäåíèÿ àìïëèòóäû ðàçâàëà è ñîîòâåòñòâóþùåéàìïëèòóäíîé ôóíêöèè â îêðåñòíîñòè 90◦ äîñòèãàåòñÿ èñêëþ÷èòåëüíî ïðåäåëü1003/2Òàáëèöà 6.1. Çíà÷åíèÿ àìïëèòóäû ðàññåÿíèÿ a3/2, ôàçî0 , êîýôôèöèåíòà íåóïðóãîñòè ηâîãî ñäâèãà δ3/2 è ïàðàìåòðà Λ3/2, õàðàêòåðèçóþùåãî òî÷íîñòü âûïîëíåíèÿ îïòè÷åñêîéòåîðåìû, â çàâèñèìîñòè îò ðàññòîÿíèÿ ρmax, ïðè êîòîðîì çàäàþòñÿ ãðàíè÷íûå óñëîâèÿ(çíà÷åíèÿ ïîëó÷åíû äëÿ ýíåðãèè Elab = 14.1 ÌýÂ)3/23/21000.32570.87100.9873 69.36 1.03072000.32830.86030.9749 68.83 0.99563000.32820.85810.9716 68.75 0.98795000.32890.85680.9706 68.67 0.98547000.32980.85770.9732 68.66 0.99069000.32990.85860.9746 68.69 0.993011000.33030.85950.9763 68.71 0.996513000.33020.86060.9778 68.76 0.999414000.33000.86090.9781 68.78 0.999815000.32970.86140.9784 68.81 1.0014ρmaxRe(a0 ) Im(a0 )η 3/2δ 3/2Λ3/2íûìè ñâîéñòâàìè áàçèñíûõ ôóíêöèé, èñïîëüçóåìûõ â íîâîì ïðåäñòàâëåíèèäëÿ êîìïîíåíòû Ôàääååâà âîëíîâîé ôóíêöèè.
Ïðåäëîæåííàÿ â äèññåðòàöèèàñèìïòîòèêà è ïðåäñòàâëåíèå äëÿ àìïëèòóäû ðàçâàëà ïîçâîëÿåò ïîëó÷èòüêîððåêòíîå ïîâåäåíèå äàííîé àìïëèòóäû â îêðåñòíîñòè 90◦ , ñîîòâåòñòâóþùåé ðàçâàëó, ïðè êîòîðîì äâå ÷àñòèöû ðàçëåòàþòñÿ íà íåáîëüøîì ðàññòîÿíèè äðóã îò äðóãà.Êàê ôóíêöèè ρmax , çíà÷åíèÿ àìïëèòóäíûõ ôóíêöèé êàíàëà ðàçâàëà ïðèôèêñèðîâàííûõ óãëàõ θ = 30◦ , 60◦ , 85◦ ïîêàçàíû íà Ðèñ. 6.18. Íà ðèñóíêàõâèäíî, ÷òî çíà÷åíèÿ àìïëèòóä ðàçâàëà ñòàáèëèçèðóþòñÿ ïðè ρmax > 1000 ôì.Çíà÷åíèÿ àìïëèòóäíîé ôóíêöèè áèíàðíîãî ðàññåÿíèÿ aJ0 (q, ρmax ) äëÿJ = 3/2, ïîëó÷åííûå ïðè íåêîòîðûõ çíà÷åíèÿõ ρmaxè ýíåðãèè â ëàáîðàòîðíîé ñèñòåìå îòñ÷åòà Elab = 14.1 ÌýÂ, ïðåäñòàâëåíû â Òàáë.
6.1. Äëÿ ïðîâåð101Òàáëèöà 6.2. Çíà÷åíèÿ êîýôôèöèåíòà íåóïðóãîñòè η, ôàçîâîãî ñäâèãà δ è ïàðàìåòðà Λ,õàðàêòåðèçóþùåãî òî÷íîñòü âûïîëíåíèÿ îïòè÷åñêîé òåîðåìû, äëÿ ðàçëè÷íûõ ýíåðãèéE , ÌýÂlab4.014.142.0êâàðòåò äóáëåò êâàðòåò äóáëåò êâàðòåò äóáëåòη0.99980.96460.97810.46480.90310.5021δ101.48143.5968.78105.4037.6641.21Λ0.99961.00020.99980.99970.99951.0002êè òî÷íîñòè ïîëó÷åííûõ çíà÷åíèé áèíàðíîé àìïëèòóäíîé ôóíêöèè, â òàáëèöå òàêæå äàþòñÿ âåëè÷èíû Λ3/2 , êîòîðûå îïðåäåëÿþò òî÷íîñòü âûïîëíåíèÿîïòè÷åñêîé òåîðåìû. Çíà÷åíèÿ ρmax , ïðè êîòîðûõ â àñèìïòîòè÷åñêîì ðåãèîíå äîñòèãàåòñÿ íàèáîëåå òî÷íîå âûïîëíåíèå îïòè÷åñêîé òåîðåìû, ïîçâîëÿþòïðåäïîëîæèòü, ÷òî ïðè äàííûõ ρmax àìïëèòóäû óïðóãîãî ðàññåÿíèÿ âû÷èñëÿþòñÿ òî÷íåå âñåãî. Êàê ñëåäóåò èç Òàáë.
6.1, íàèáîëåå óñòîé÷èâûå è áëèçêèåê 1 çíà÷åíèÿ Λ3/2 äîñòèãàþòñÿ ïðè ρmax = 1400 ôì. Äëÿ ñëó÷àÿ J = 1/2 íàèáîëåå òî÷íûå çíà÷åíèÿ Λ1/2 äîñòèãàþòñÿ òàêæå ïðè ρmax ∼ 1400 ôì. Âàæíîîòìåòèòü, ÷òî ïðèâåäåííûå âû÷èñëåííûå çíà÷åíèÿ ìîãóò ââåñòè â íåêîòîðîå çàáëóæäåíèå, ò.ê. íå ïîçâîëÿþò ñäåëàòü âûâîä îá îñöèëëÿöèÿõ áèíàðíîéàìïëèòóäíîé ôóíêöèè êàê ôóíêöèè ρmax . Áîëåå äåòàëüíîå îáñóæäåíèå ïîâåäåíèÿ áèíàðíîé àìïëèòóäíîé ôóíêöèè áóäåò äàíî íà ãðàôèêàõ â ñëåäóþùåìðàçäåëå.Çíà÷åíèÿ àìïëèòóäíîé ôóíêöèè áèíàðíîãî ðàññåÿíèÿ aJ0 (q, ρmax ) äëÿJ = 3/2 è J = 1/2 ïðè ðàçëè÷íûõ ýíåðãèÿõ ïðèâîäÿòñÿ â Òàáë.
6.2.  òàáëèöå, äàííûå âåëè÷èíû ïðåäñòàâëåíû â òåðìèíàõ êîýôôèöèåíòà íåóïðóãîñòèη è ôàçîâîãî ñäâèãà δ [ñì. ôîðìóëó (2.7)]. Çíà÷åíèÿ âû÷èñëÿëèñü âìåñòå ññîîòâåòñòâóþùèìè êîýôôèöèåíòàìè ðàçëîæåíèÿ aJi,k (E, ρmax ) àìïëèòóäíûõôóíêöèé êàíàëà ðàçâàëà, ïðåäñòàâëåííûõ íà Ðèñ. 6.126.17, ïðè îäíèõ è òåõæå çíà÷åíèÿõ ρmax ∼ 1400 ôì. Òàêèì îáðàçîì, äîñòèãàåòñÿ íàèáîëüøàÿ òî÷102íîñòü ïîëó÷àåìûõ çíà÷åíèé àìïëèòóäíûõ ôóíêöèé. Ïðåäñòàâëåííûå çíà÷åíèÿ àìïëèòóä aJ0 (q, ρmax ) ñîâïàäàþò ñ õîðîøåé òî÷íîñòüþ ñî çíà÷åíèÿìè, ïðèâåäåííûìè â ðàáîòàõ [37, 38].0.3Re(A3/2)Im(A3/2)0.20.020.1A(θ,E,ρmax,ρ)00−0.028990−0.1−0.2−0.3−0.401020304050θ, degree607080907080900.3Re(A3/2)Im(A3/2)0.20.020.1A(θ,E,ρmax)00-0.028990-0.1-0.2-0.3-0.401020304050θ, degree60Ðèñ.
6.12. Êâàðòåò: äîïðåäåëüíàÿ (ââåðõó) è ïðåäåëüíàÿ (âíèçó) àìïëèòóäíûå ôóíêöèèêàíàëà ðàçâàëà A3/2 äëÿ Elab = 4.0 ÌýÂ è ρ = ρmax = 1400 ôì.1030.25Re(A11/2Im(A11/2Re(A21/2Im(A21/20.2))))A(θ,E,ρmax,ρ)0.150.10.050.0200−0.028990−0.0501020304050θ, degree6070809080900.25Re(A11/2Im(A11/2Re(A21/2Im(A21/20.2))))A(θ,E,ρmax)0.150.10.050.0200−0.028990−0.0501020304050θ, degree6070Ðèñ. 6.13. Äóáëåò: äîïðåäåëüíûå (ââåðõó) è ïðåäåëüíûå (âíèçó) àìïëèòóäíûå ôóíêöèèêàíàëà ðàçâàëà A1/21,2 äëÿ Elab = 4.0 ÌýÂ è ρ = ρmax = 1400 ôì.1041.5Re(A3/2)Im(A3/2)10.06A(θ,E,ρmax,ρ)00.5−0.0689900−0.5−101020304050θ, degree607080907080901.5Re(A3/2)Im(A3/2)10.060A(θ,E,ρmax)0.5-0.0689900-0.5-101020304050θ, degree60Ðèñ.
6.14. Êâàðòåò: äîïðåäåëüíàÿ (ââåðõó) è ïðåäåëüíàÿ (âíèçó) àìïëèòóäíûå ôóíêöèè êàíàëà ðàçâàëà A3/2 äëÿ Elab = 14.1 Ìý è ρ = ρmax = 1400 ôì. Êâàäðàòàìè è êðóæî÷êàìèïîêàçàíû ðåçóëüòàòû ðàáîòû [37].1050.7Re(A10.60.5A(θ,E,ρmax,ρ)0.41/2)Im(A11/2Re(A21/2Im(A21/2)))0.30.20.100.02−0.10−0.2−0.028990−0.301020304050θ, degree607080907080900.7Re(A10.60.50.41/2)Im(A11/2Re(A21/2Im(A21/2)))A(θ,E,ρmax)0.30.20.100.02−0.10−0.2−0.028990−0.301020304050θ, degree60Ðèñ. 6.15.
Äóáëåò: äîïðåäåëüíûå (ââåðõó) è ïðåäåëüíûå (âíèçó) àìïëèòóäíûå ôóíêöèè êàíàëà ðàçâàëà A1/21,2 äëÿ Elab = 14.1 Ìý è ρ = ρmax = 1550 ôì. Êâàäðàòàìè è êðóæî÷êàìèïîêàçàíû ðåçóëüòàòû ðàáîòû [37].1061.4Re(A3/2)Im(A3/2)1.210.060A(θ,E,ρmax,ρ)0.8−0.06890.6900.40.20−0.201020304050θ, degree607080907080901.4Re(A3/2)Im(A3/2)1.210.060A(θ,E,ρmax)0.8-0.06890.6900.40.20-0.201020304050θ, degree60Ðèñ.
6.16. Êâàðòåò: äîïðåäåëüíàÿ (ââåðõó) è ïðåäåëüíàÿ (âíèçó) àìïëèòóäíûå ôóíêöèè êàíàëà ðàçâàëà A3/2 äëÿ Elab = 42.0 Ìý è ρ = ρmax = 1350 ôì. Êâàäðàòàìè è êðóæî÷êàìèïîêàçàíû ðåçóëüòàòû ðàáîòû [37].1070.7Re(A10.60.5A(θ,E,ρmax,ρ)0.41/2)Im(A11/2Re(A21/2Im(A21/2)))0.30.20.100.02−0.10−0.2−0.028990−0.301020304050θ, degree60708090607080900.7Re(A10.60.50.41/2)Im(A11/2Re(A21/2Im(A21/2)))A(θ,E,ρmax)0.30.20.100.02−0.10−0.2−0.028990−0.301020304050θ, degreeÐèñ. 6.17. Äóáëåò: äîïðåäåëüíûå (ââåðõó) è ïðåäåëüíûå (âíèçó) àìïëèòóäíûå ôóíêöèè êàíàëà ðàçâàëà A1/21,2 äëÿ Elab = 42.0 Ìý è ρ = ρmax = 1400 ôì.
Êâàäðàòàìè è êðóæî÷êàìèïîêàçàíû ðåçóëüòàòû ðàáîòû [37].108θ = 30oθ = 30o-0.52-0.54-0.56Im(A)Re(A)-0.58-0.6-0.62-0.64-0.6605001000ρmax, fm15001.281.271.261.251.241.231.221.211.21.191.1802000500-0.381.46-0.41.44200015002000150020001.42-0.421.4-0.44Im(A)Re(A)1500θ = 60oθ = 60o-0.461.381.361.34-0.481.32-0.51.3-0.521.2805001000ρmax, fm1500020005001000ρmax, fmθ = 85oθ = 85o-0.020.48-0.040.460.44-0.060.42-0.08Im(A)Re(A)1000ρmax, fm-0.10.40.380.36-0.120.34-0.140.320.3-0.1605001000ρmax, fm1500200005001000ρmax, fmÐèñ.
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