Theoretical treatment of the resonant hyper-Raman scattering of light in crystals
The work is aimed at discussion of different mechanisms of the resonant hyper-Raman scattering of light with participation of excitons in semiconductor crystals and analysis of their contributions to the scattering intensity.
Збережено в:
Дата: | 2004 |
---|---|
Автори: | , |
Формат: | Стаття |
Мова: | English |
Опубліковано: |
НТК «Інститут монокристалів» НАН України
2004
|
Назва видання: | Functional Materials |
Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/135242 |
Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
Цитувати: | Theoretical treatment of the resonant hyper-Raman scattering of light in crystals / L.E. Semenova, K.A. Prokhorov // Functional Materials. — 2004. — Т. 11, № 1. — С. 89-100. — Бібліогр.: 17 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraineid |
irk-123456789-135242 |
---|---|
record_format |
dspace |
spelling |
irk-123456789-1352422018-06-15T03:08:14Z Theoretical treatment of the resonant hyper-Raman scattering of light in crystals Semenova, L.E. Prokhorov, K.A. The work is aimed at discussion of different mechanisms of the resonant hyper-Raman scattering of light with participation of excitons in semiconductor crystals and analysis of their contributions to the scattering intensity. Работа посвящена обсуждению различных механизмов резонансного гиперкомбинационного рассеяния света с участием экситонов в полупроводниковых кристаллах и анализу их вкладов в интенсивность рассеяния. Роботу присвячено обговоренню різних механiзмiв резонансного гіперкомбінаційно-го розсіяння світла за участю екситонів у напівпровідникових кристалах та аналізу їх внесків в інтенсивність розсіяння. 2004 Article Theoretical treatment of the resonant hyper-Raman scattering of light in crystals / L.E. Semenova, K.A. Prokhorov // Functional Materials. — 2004. — Т. 11, № 1. — С. 89-100. — Бібліогр.: 17 назв. — англ. 1027-5495 http://dspace.nbuv.gov.ua/handle/123456789/135242 en Functional Materials НТК «Інститут монокристалів» НАН України |
institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
collection |
DSpace DC |
language |
English |
description |
The work is aimed at discussion of different mechanisms of the resonant hyper-Raman scattering of light with participation of excitons in semiconductor crystals and analysis of their contributions to the scattering intensity. |
format |
Article |
author |
Semenova, L.E. Prokhorov, K.A. |
spellingShingle |
Semenova, L.E. Prokhorov, K.A. Theoretical treatment of the resonant hyper-Raman scattering of light in crystals Functional Materials |
author_facet |
Semenova, L.E. Prokhorov, K.A. |
author_sort |
Semenova, L.E. |
title |
Theoretical treatment of the resonant hyper-Raman scattering of light in crystals |
title_short |
Theoretical treatment of the resonant hyper-Raman scattering of light in crystals |
title_full |
Theoretical treatment of the resonant hyper-Raman scattering of light in crystals |
title_fullStr |
Theoretical treatment of the resonant hyper-Raman scattering of light in crystals |
title_full_unstemmed |
Theoretical treatment of the resonant hyper-Raman scattering of light in crystals |
title_sort |
theoretical treatment of the resonant hyper-raman scattering of light in crystals |
publisher |
НТК «Інститут монокристалів» НАН України |
publishDate |
2004 |
url |
http://dspace.nbuv.gov.ua/handle/123456789/135242 |
citation_txt |
Theoretical treatment of the resonant hyper-Raman scattering of light in crystals / L.E. Semenova, K.A. Prokhorov // Functional Materials. — 2004. — Т. 11, № 1. — С. 89-100. — Бібліогр.: 17 назв. — англ. |
series |
Functional Materials |
work_keys_str_mv |
AT semenovale theoreticaltreatmentoftheresonanthyperramanscatteringoflightincrystals AT prokhorovka theoreticaltreatmentoftheresonanthyperramanscatteringoflightincrystals |
first_indexed |
2025-07-09T22:54:12Z |
last_indexed |
2025-07-09T22:54:12Z |
_version_ |
1837211748345577472 |
fulltext |
7KHRUHWLFDO�WUHDWPHQW�RI�WKH�UHVRQDQWK\SHU�5DPDQ�VFDWWHULQJ�RI�OLJKW�LQ�FU\VWDOV
/�(�6HPHQRYD��.�$�3URNKRURY
$�3URNKRURY�*HQHUDO�3K\VLFV�,QVWLWXWH��5XVVLDQ�$FDGHP\�RI�6FLHQFHV����9DYLORY�6W����������0RVFRZ��5XVVLD
7KH�ZRUN�LV�DLPHG�DW�GLVFXVVLRQ�RI�GLIIHUHQW�PHFKDQLVPV�RI�WKH�UHVRQDQW�K\SHU�5DPDQVFDWWHULQJ�RI�OLJKW�ZLWK�SDUWLFLSDWLRQ�RI�H[FLWRQV�LQ�VHPLFRQGXFWRU�FU\VWDOV�DQG�DQDO\VL V�RIWKHLU�FRQWULEXWLRQV�WR�WKH�VFDWWHULQJ�LQWHQVLW\��
cȺ�È�¹º°m«�ËÓÈ�º°�Î�ËÓÒ
¯ÈÏãÒ�Ó©²�ä˲ÈÓÒÏäºm�¯ËϺÓÈÓ°Óº�º��ҹ˯}ºäÒÓÈ��ÒºÓÓº�º� ¯È°°Ë«ÓÒ«� °mË�È� °� ��Ȱ�ÒËä� ª}°Ò�ºÓºm� m� ¹ºã�¹¯ºmº�ÓÒ}ºm©²� }¯Ò°�ÈããȲ� ÒÈÓÈãÒÏ��Ò²�m}ãÈ�ºm�m�ÒÓ�ËÓ°ÒmÓº°� ¯È°°Ë«ÓÒ«�
7KH� 5DPDQ� VFDWWHULQJ� �56�� RI� OLJKW� LVRQH� RI� LPSRUWDQW� LQYHVWLJDWLRQ� WHFKQLTXHVRI� VHPLFRQGXFWRU� FU\VWDOV� IRU� D� ORQJ� WLPH�2I� VSHFLDO� LQWHUHVW� LV� WKH� UHVRQDQW� 5DPDQVFDWWHULQJ��556��ZKHUH�WKH�LQFLGHQW�RU�VFDW �WHUHG� IUHTXHQF\� LV� FORVH� WR� WKH� UHVRQDQFHZLWK� IXQGDPHQWDO� HOHFWURQLF� WUDQVLWLRQV�VLQFH� WKH� 556� PDNHV� LW� SRVVLEOH� WR� REWDLQLQIRUPDWLRQ� RQ� H[FLWDWLRQV� LQ� VHPLFRQGXF �WRUV� DQG� WKHLU� LQWHUDFWLRQV� >�@�� 7KH� K\SHU�5DPDQ� VFDWWHULQJ� �+56�� RI� OLJKW� LV� D� QRQ �OLQHDU� RSWLFDO� SURFHVV� ZKHUH� WKH� VLPXOWDQH �RXV� DEVRUSWLRQ� RI� WZR� LQFLGHQW� SKRWRQV� LVDFFRPSDQLHG� E\� FUHDWLRQ� RI� SKRQRQ� DQGHPLVVLRQ� RI� VFDWWHUHG� SKRWRQ� >�@�� $ORQJZLWK�WKH�556��RI�D�SDUWLFXODU�LQWHUHVW�LV�WKHUHVRQDQW� K\SHU�5DPDQ� VFDWWHULQJ� �5+56�ZKHUH�WKH�GRXEOHG�HQHUJ\�RI�H[FLWLQJ�UDGLD �WLRQ� LV� FORVH� WR� WKH� HQHUJ\� JDS�� )RU� WKH5+56�� WZR�SKRWRQ� DOORZHG� WUDQVLWLRQV� FDQEH�LQ�UHVRQDQFH�ZLWK�H[FLWLQJ�UDGLDWLRQ��L�H�LQ� WKH� FDVH� RI� +56�� RWKHU� LQWHUPHGLDWHVWDWHV� DUH� LQYROYHG� LQ� WKH� VFDWWHULQJ� SURF �HVV�� 7KLV� UHVXOWV� LQ� QHZ� IHDWXUHV� LQ� K\SHU�5DPDQ�VSHFWUD��7KXV��WKH�+56�SURYLGHV�DQDGGLWLRQDO� LQIRUPDWLRQ� RQ� GLIIHUHQW� H[FLWD �WLRQV�LQ�VHPLFRQGXFWRUV�([SHULPHQWDO�LQYHVWLJDWLRQV�RI�WKH�5+56LQ�D�QXPEHU�RI�VHPLFRQGXFWRU�FU\VWDOV�ZHUHUHSRUWHG� LQ� >��@�� ,Q� WKH� ZRUN� E\� *DUVLD�&ULVWREDO�HW�DO��>��@��WKH�5+56�PHGLDWHG�E\GLSROH�DOORZHG� )U|KOLFK� LQWHUDFWLRQ� ZDVWKHRUHWLFDOO\� VWXGLHG� DQG� WKH� K\SHU�5DPDQ
HIILFLHQF\� ZDV� FDOFXODWHG� WDNLQJ� LQWR� DF �FRXQW�H[FLWRQLF�HIIHFWV��7KHRUHWLFDO�DQDO\VLVRI� VRPH� 5+56� PHFKDQLVPV� WKH� KDV� EHHQSHUIRUPHG� IRU� D� &G6 FU\VWDO� >��Ü��@�� 7KHGLIIHUHQW� VFDWWHULQJ� PHFKDQLVPV� IRU� 5+56ZHUH� FRQVLGHUHG� DQG� WKH� H[SUHVVLRQV� IRUWKHLU�FRQWULEXWLRQV�WR�WKH�FURVV�VHFWLRQ�ZHUHGHULYHG� LQ� >��@�� 7KH�SUHVHQW�ZRUN� LV� DLPHGDW� WKH� GLVFXVVLRQ� RI� IHDWXUHV� RI� WKH� 5+56RQ� RSWLFDO� SKRQRQV� XQGHU� LQYROYHPHQW� RIH[FLWRQV��$V�PHQWLRQHG�DERYH��LQ�WKH�+56�SURFHVV�D� SKRQRQ� DULVHV� DQG� VFDWWHUHG� SKRWRQ� LVHPLWWHG� GXH� WR� DEVRUSWLRQ� RI� WZR� LQFLGHQWSKRWRQV�� L�H�� WKH� +56� FKDQJHV� WKH� YLEUD �WLRQDO� VWDWH� RI� D� FU\VWDO�� EXW� WKH� HOHFWURQLFVWDWH� UHPDLQV�XQFKDQJHG��DOWKRXJK�WKH�YLU �WXDO�LQWHUPHGLDWH�VWDWHV�DUH�H[FLWHG��:H�DV �VXPH� WKDW� WKH� LQWHUPHGLDWH� VWDWHV� RI� HOHF �WURQLF� V\VWHP� DUH� WKH� HOHFWURQ�KROH� SDLUVERXQG� E\� WKH� &RXORPE� LQWHUDFWLRQ�� L�H�� WKH:DQQLHU�H[FLWRQV�,Q� WKLV� FDVH�� WKH�+56�HYHQW� LV�GHVFULEHGE\�WKH�IROORZLQJ�VHTXHQFH�RI�SURFHVVHV������DWUDQVLWLRQ� RI� HOHFWURQLF� V\VWHP� IURP� WKHJURXQG�VWDWH�WR�DQ�H[FLWRQLF�VWDWH�ZKHQ�WKHLQFLGHQW� SKRWRQ� ZLWK� IUHTXHQF\� ω/�� SRODUL�]DWLRQ� εε/ DQG� ZDYH� YHFWRU� T/ LV� DEVRUEHG����� D� WUDQVLWLRQ�EHWZHHQ� H[FLWRQV�XQGHU� DE �VRUSWLRQ� RI� DQRWKHU� SKRWRQ� RI� H[FLWLQJ� UD �GLDWLRQ�� ���� FUHDWLRQ� RI� D� SKRQRQ� ZLWK� IUH �TXHQF\�ω3 DQG�ZDYH�YHFWRU� T3 GXH� WR�H[FL�WRQ�ODWWLFH� LQWHUDFWLRQ�� ���� DQQLKLODWLRQ� RI
)XQFWLRQDO�0DWHULDOV�����1R��������� ��������,QVWLWXWH�IRU�6LQJOH�&U\VWDOV
�� )XQFWLRQDO�PDWHULDOV�������������
DQ�H[FLWRQ�DFFRPSDQLHG�E\� WKH�RI� VFDWWHUHGSKRWRQ��ω6��εε6��T6��HPLVVLRQ��7KH�ILUVW�RUGHU+56� �L�H�� RQH�SKRQRQ� VFDWWHULQJ�� LV� GH �VFULEHG�LQ�WKH�IRXUWK�RUGHU�RI�WKH�SHUWXUED �WLRQ�WKHRU\�DQG�LWV�GLIIHUHQWLDO�FURVV�VHFWLRQFDQ�EH�ZULWWHQ�DV�>��@GσGΩ
=
=
�πK−H�η6ω6�(Q3 + �)P�F�η/�ω/� 1/9_εγ
6εβ
/εα
/βαβγ(�T/ − T6)_��
���
ZKHUH� H DQG� P DUH� HOHFWURQ� FKDUJH� DQGPDVV�� UHVSHFWLYHO\�� F LV� WKH� OLJKW� VSHHG�� Q3LV� WKH� QXPEHU� RI� SKRQRQV�� 9 LV� WKH� FU\VWDOYROXPH��1/ �Q/�9 LV� WKH� SKRWRQ� GHQVLW\� RILQFLGHQW� UDGLDWLRQ�� η/�η6�� LV� WKH� UHIUDFWLYHLQGH[�IRU�WKH�IUHTXHQF\�ω/�ω6���7KH�+56�WHQ�VRU�βαβγ�T��KDV�WKH�IRUP�>��@�
βαβγ(T) =
= ∑
Λ�Λ�Λ�
Π�Λ�γ 3Λ�Λ�ΠΛ�Λ�β ΠΛ��α
((Λ� − K−ωV)((Λ� − �K−ω/)((Λ� − K−ω/)
�
���
ZKHUH� WKH� LQGH[� Λ ��Fνλ� FRUUHVSRQGV� WRWKH� H[FLWRQLF� VWDWH� IRUPHG� GXH� WR� WKH� &RX �ORPE� LQWHUDFWLRQ� RI� HOHFWURQ� IURP� WKH� FRQ �GXFWLRQ�EDQG�F�DQG�D�KROH�IURP�WKH�YDOHQFHEDQG� ν��λ ��ζOP�� GHQRWHV�D� VHW�RI�TXDQWXPQXPEHUV�DVVRFLDWHG�ZLWK�D�UHODWLYH�HOHFWURQ�KROH� PRWLRQ�� WKH� PDLQ� TXDQWXP� QXPEHU� ζFRLQFLGHV� ZLWK� Q IRU� WKH� GLVFUHWH� VSHFWUXPRU�ZLWK� N IRU� WKH� FRQWLQXRXV� VSHFWUXP�� (ΛLV�WKH�HQHUJ\�RI�FRUUHVSRQGLQJ�LQWHUPHGLDWHH[FLWRQLF�VWDWH�LQ�WKH�%ULOORXLQ�]RQH�FHQWHU��:H� DVVXPH� WKDW� WKH� VFDWWHULQJ� SURFHVV� RF �FXUV� LQ� WKH� %ULOORXLQ� ]RQH� FHQWHU� GXH� WRVPDOO� YDOXHV� RI� SKRWRQ� ZDYH� YHFWRUV��� ΠΛ�αGHVFULEHV� WKH� GLSROH� WUDQVLWLRQ� IURP� WKHJURXQG�VWDWH�WR�DQ�H[FLWRQLF�VWDWH�>��@
ΠΛ�α =
πFνα − 0Fνα′αSα′
χFνλ (U)∗
U=�� ���
ZKHUH� SAα �ÜLK−¶ ⁄ ¶ Uα�� πFν LV� WKH� LQWHUEDQGPDWUL[�HOHPHQW�RI� WKH�PRPHQWXP�RSHUDWRU�
χFν
λ �U�� LV�WKH�K\GURJHQ�OLNH�ZDYH�IXQFWLRQ�RIUHODWLYH�HOHFWURQ�KROH�PRWLRQ��+HUH�WKH�ILUVWWHUP�LQ�WKH�FXUO\�EUDFNHWV�GHVFULEHV�WKH�DO �ORZHG�GLSROH�WUDQVLWLRQ�WR�D�V�H[FLWRQLF�VWDWH�ZLWK� WKH� RUELWDO� TXDQWXP� QXPEHU� O ����7KH�VHFRQG�WHUP�FRUUHVSRQGV�WR�DQRWKHU�W\SHRI�GLSROH�WUDQVLWLRQV�ZKLFK�DUH�ZHDNO\�IRUELG �
GHQ� DQG� FDXVH� H[FLWDWLRQ� RI� S�H[FLWRQV� �O ����7KH�SDUDPHWHU�0Fν
α′α LV�GHILQHG�LQ�>��@��
ΠΛ′Λ
β GHVFULEHV� WKH� WUDQVLWLRQV� EHWZHHQH[FLWRQLF�VWDWHV�Λ′ DQG�Λ >��@
ΠΛ′Λ
β =
πF′Fβ δν′ν − πνν′
β δF′F<χF′ν′
λ′ _χFνλ > +��δν′νδF′F0F′Fβ′β − 0νν′
β′β
<χFνλ′ _ SAβ′_χFνλ >� ���
,Q� H[SUHVVLRQ� ����� WKH� ILUVW� WHUP� FRUUH �VSRQGV� WR� WUDQVLWLRQV� EHWZHHQ� H[FLWRQLFVWDWHV�EHORQJLQJ�WR�GLIIHUHQW�SDLUV�RI�EDQGV�LQWHUEDQG� WUDQVLWLRQV��� ZKHUHDV� WKH� VHFRQGWHUP� GHVFULEHV� WKH� LQWUDEDQG� GLSROH� WUDQVL �WLRQV�EHWZHHQ�H[FLWRQLF�VWDWHV�3Λ′Λ GHVFULEHV�WKH�WUDQVLWLRQ�EHWZHHQ�H[ �FLWRQLF� VWDWHV� Λ′ DQG� Λ GXH� WR� LQWHUDFWLRQZLWK� SKRQRQV�� ,Q� WKH� ]HUR� ZDYH� YHFWRU� DS �SUR[LPDWLRQ�ZKLFK� FRPPRQO\� LV� XVHG� IRU� DILUVW�RUGHU� VFDWWHULQJ� WKH� PDWUL[� HOHPHQW3Λ′Λ LV�JLYHQ�E\�>��@
3Λ′Λ ≈ γ'
ΞF′Fδνν′ − Ξνν′δFF′
<χF′ν′
λ′ _χFνλ > ���IRU�WKH�GHIRUPDWLRQ�SRWHQWLDO�LQWHUDFWLRQW\SLFDO�RI�DOO�RSWLFDO�SKRQRQV�RU3Λ′Λ ≈ γ)
TA ⋅ UF′′Fδνν′ − TA ⋅ Uνννν′′δFF′
<χF′ν′
λ′ _χFνλ > +
+ γ)δνν′δFF′<χFνλ′ _TA ⋅ U_χFνλ >
���
IRU� WKH� )U|KOLFK� LQWHUDFWLRQ� ZKLFK� FRQWULE �XWHV� WR� D� VFDWWHULQJ� E\� ORQJLWXGLQDO� RSWLFDO�/2��SKRQRQV�RQO\��+HUH�γ' LV�FRQVWDQW�DVVR�FLDWHG� ZLWK� WKH� UHODWLYH� GLVSODFHPHQW� RIDWRPV�LQ�WKH�XQLW�FHOO�DQG�ΞQ�Q LV�WKH�GHIRU�PDWLRQ� SRWHQWLDO�� γ) LV� WKH� FRQVWDQW� RI� WKH)U|KOLFK�LQWHUDFWLRQ�DQG�TA LV�WKH�XQLW�YHFWRULQ� WKH�GLUHFWLRQ� T3�� ,W� LV� VHHQ�IURP� WKH�H[�SUHVVLRQ� ���� WKDW� WKH� )URKOLFK� LQWHUDFWLRQLQGXFHV�ERWK� WKH� LQWHUEDQG� �ILUVW� WHUP��DQGLQWUDEDQG��VHFRQG�WHUP��WUDQVLWLRQV�EHWZHHQH[FLWRQLF�VWDWHV�1RZ� WKH�+56�SURFHVV�ZLOO�EH� FRQVLGHUHGIRU�WKH�FDVH�ZKHQ�WKH�GRXEOHG�IUHTXHQF\�RILQFLGHQW� UDGLDWLRQ� �RU� WKH� VFDWWHUHG� OLJKWIUHTXHQF\���LV�FORVH�WR�WKH�HQHUJ\�JDS�ZLGWK��K−ω/��K−ω6 a�(Fν���KHQFH��WKH�QRQ�UHVRQDQFHFRQWULEXWLRQV�ZLOO�EH�QHJOHFWHG�7KH� )U|KOLFK� VFDWWHULQJ� PHFKDQLVP�� WRZKLFK�D�VHTXHQFH�RI�LQWHUPHGLDWH�VWDWHV�� V�S�V�� FRUUHVSRQGV�� LV� NQRZQ� WR� SOD\� D� OHDGLQJUROH�LQ�WKH�5+56�>��@��$V�LV�VHHQ�IURP�WKHH[SUHVVLRQ� ����� DEVRUSWLRQ� RI� LQFLGHQW� SKR �WRQ�LV�DFFRPSDQLHG�E\�WUDQVLWLRQ�WR�DQ� V�H[�FLWRQ� VWDWH� XQGHU� FRQGLWLRQ� WKDW� WKH� GLSROHWUDQVLWLRQ�LV�DOORZHG��πFν ≠ ����7KHQ�WKH�VHF�RQG�SKRWRQ�FDQ�LQGXFH�WKH�LQWHUEDQG�WUDQVL �
/�(�6HPHQRYD��.�$�3URNKRURY���7KHRUHWLFDO�WUHDWPHQW����
)XQFWLRQDO�PDWHULDOV������������� ��
WLRQ� WR� V�H[FLWRQ� EHORQJLQJ� WR� DQRWKHU� SDLURI� EDQGV� RU� WKH� LQWUDEDQG�GLSROH� WUDQVLWLRQWR�D�S�H[FLWRQ��8QGHU�UHVRQDQFH�FRQGLWLRQV�LW� LV� MXVW� WKH� ODWWHU� WKDW� LV� RI� LQWHUHVW�� ,QWKLV�FDVH��WKH�GHIRUPDWLRQ�SRWHQWLDO�DQG�LQ �WHUEDQG�)U|KOLFK�LQWHUDFWLRQV�UHVXOW�LQ�WUDQ �VLWLRQV� WR� S�H[FLWRQ�VWDWHV��:KHUHDV� WKH� LQ�WUDEDQG� )U|KOLFK� FRXSOLQJ� FDXVHV� D� WUDQVL �WLRQ� WR�DQ� V�H[FLWRQ�EHORQJLQJ�WR� VDPH�SDLURI�EDQGV��%XW�WKH�WUDQVLWLRQ�IURP�WKH� S�H[�FLWRQLF�VWDWH�WR�WKH�JURXQG�RQH�LV�D�SURFHVVRI�KLJKHU�RUGHU�DV�FRPSDUHG�WR�WKH�RUGLQDU\GLSROH� WUDQVLWLRQ� DQG� LWV� RVFLOODWRU� IRUFH� LVVLJQLILFDQWO\� OHVV� WKDQ� WKDW� IRU� WKH� WUDQVL �WLRQ� WR� V�H[FLWRQLF� VWDWH�� 7KXV�� WKH� +56SURFHVV� YLD� WKH� LQWUDEDQG� )U|KOLFK� VFDWWHU�LQJ�LV�SUHGRPLQDQW�7R� HYDOXDWH� WKH� FRUUHVSRQGLQJ� K\SHU�5DPDQ� WHQVRU�� LW� LV� QHFHVVDU\� WR� FDOFXODWHWKH� H[FLWRQ�SKRWRQ� DQG� H[FLWRQ�SKRQRQ�PD �WUL[�HOHPHQWV�DQG�WR�VXP�RYHU�DOO�LQWHUPHGL �DWH�H[FLWRQLF�VWDWHV�RI�GLVFUHWH�DQG�FRQWLQX �RXV�VSHFWUD��7KH�+56�HIILFLHQF\�ZDV�HYDOX �DWHG� E\� WKLV� DSSURDFK� LQ� >��@�� 7KHH[SUHVVLRQ� REWDLQHG� LV�� KRZHYHU�� WRR� FXP �EHUVRPH��,Q�RXU�ZRUN�>��@�� WKH�+56�WHQVRUGHVFULELQJ� WKH� FRQWULEXWLRQ� RI� WKH� FRQVLG �HUHG� VFDWWHULQJ� PHFKDQLVP� ZDV� FDOFXODWHGE\� DSSO\LQJ� WKH� *UHHQ�V� IXQFWLRQ� PHWKRGDQG�KDV�D�IRUP�
βαβγ
(�) (T) = L �K−γ)πνFγ πFν
α >0FFβ′β − 0νν
β′β@TAβ′�π5�D� κ/�κ6� ×
×
∑Q� − �Q�Q=�
∞ -�(Q�κ6)-�(Q�κ/)
ξ(ω/) − Q−� − LγQ +
+ ∫�
∞
N(� + N�)� − H[S(−�π⁄ N)
-�(L⁄N�κ6)-�(L⁄N�κ/)
ξ(ω/) + N� − LγN GN
���
LQ� WKH� DSSUR[LPDWLRQ� RI� WKH� ]HUR� SKRQRQZDYH�YHFWRU��+HUH�5 DQG�D DUH� WKH�ELQGLQJHQHUJ\� DQG� WKH� UDGLXV� RI� H[FLWRQ�� UHVSHF �WLYHO\�� γQ LV� WKH� H[FLWRQ� GDPSLQJ�� 7KH� SD �UDPHWHUV� ξ�ω�� DQG� κ/ DUH� GHILQHG� DV� ξ(ω) �(Fν Ü��K−ω�5 DQG�κ/ �√ 5⁄((Fν − K−ω/��+HUHDI�WHU�� WKH� LQGLFHV� / DQG� 6 FRUUHVSRQG� WR� WKHLQFLGHQW�DQG�VFDWWHUHG�OLJKW��UHVSHFWLYHO\��7KHLQWHJUDOV� -��Q�κ�� DQG� -��Q�κ�� DUH� GHILQHG� E\WKH�H[SUHVVLRQV������DQG������LQ�>��@�7DNLQJ� LQWR� DFFRXQW� RQO\� WKH� GLSROH�DO �ORZHG� WUDQVLWLRQV�� DQRWKHU� VFDWWHULQJPHFKDQLVP� FDQ� EH� FRQVLGHUHG� ZKLFK� VKRZVWKH�UHVRQDQFH�LQFUHDVH�ZKHQ�WKH�SKRWRQ�HQ �HUJ\�RI�VFDWWHUHG�OLJKW�LV�FORVH�WR�WKH��6�H[�FLWRQ�HQHUJ\��7KLV�+56�SURFHVV�LV�GHVFULEHG
E\�WKH�IROORZLQJ�VHTXHQFH�RI�WUDQVLWLRQV�����WUDQVLWLRQ� IURP� WKH� JURXQG� VWDWHV� WR� DQ� V�H[FLWRQ�FRPSRVHG�RI�HOHFWURQ�IURP�WKH� ORZ �HVW�FRQGXFWLRQ�EDQG�F DQG�WKH�KROH�IURP�WKHKLJKHVW�YDOHQFH�EDQG�ν������WUDQVLWLRQ�RI�WKHHOHFWURQ�WR�WKH�KLJKHU�FRQGXFWLRQ�EDQG� F′ RUWKH� KROH� WR� WKH� ORZHU� YDOHQFH� EDQG� ν′�� ���WUDQVLWLRQ� RI� WKH� HOHFWURQ� EHWZHHQ� WKH� FRQ �GXFWLRQ� EDQGV� F′ DQG� F RU� WKH�KROH� EHWZHHQWKH� YDOHQFH� EDQGV� ν′ DQG� ν GXH� WR� H[FLWRQ�ODWWLFH�FRXSOLQJ������UHWXUQ�RI�HOHFWURQLF�V\V �WHP�WR�WKH�JURXQG�VWDWH�8QGHU� WKH� DVVXPSWLRQ� RI� WKH� VDPH� SD �UDPHWHUV� RI� H[FLWRQV� EHORQJLQJ� WR�GLIIHUHQWSDLUV� RI� EDQGV�� WKH� FRUUHVSRQGLQJ� WHQVRU� LVJLYHQ�DV
βαβγ
′ (T) =
γπνFγ πFνα
π5�D� κ/�κ6� ×
× ∑F′ν′
ΘFF′δνν′ − Θν′νδFF′ πF′Fβ δνν′ − πνν′
β δFF′ ×
×
∑Q=�
∞ �Q� -a(Q�κ6)-a(Q�κ/)(F′ν′ − 5⁄Q� − �K−ω/ +
+ ∫�
∞
N� − H[S−�π ⁄ N
-a(L ⁄ N�κ6)-a(L ⁄ N�κ/)(F′ν′ + 5N� − �K−ω/GN
��
���
ZKHUH�-a�Q�κ�� ����Ü� �κ�Q���Ü���SDUDPHWHUV� γDQG�ΘQQ′ FRLQFLGH� ZLWK� γ' DQG� ΞQQ′ IRU� WKHGHIRUPDWLRQ�SRWHQWLDO�LQWHUDFWLRQ�RU�ZLWK� γ)DQG� TAMUQQ′
M IRU� WKH� )U|KOLFK� VFDWWHULQJ�� 7KH+56� HYHQW� GHVFULEHG� E\� WKH� H[SUHVVLRQ� ���LQFOXGHV� WUDQVLWLRQV� WR�KLJKHU�O\LQJ�FRQGXF �WLRQ� EDQGV� DQG� WR� GHHSHU� YDOHQFH� EDQGV�7KHUHIRUH�� LWV� FRQWULEXWLRQ� LV� FRQVLGHUDEO\VPDOOHU�WKDQ�WKDW�RI�WKH�LQWUDEDQG�)U|KOLFKVFDWWHULQJ�PHFKDQLVP�ZKLFK� LV�GHVFULEHG� LQWKH�IUDPHZRUN�RI�WKH�WZR�EDQG�PRGHO��βαβγ
(�) T��$OWKRXJK�βαβγ
(�) �T��GHFUHDVHV�PRUH�IDVWO\�WKDQ
β′αβγ�T�� DV� WKH� LQFLGHQW� IUHTXHQF\� GHYLDWHVDZD\� IURP� WKH� WZR�SKRWRQ� UHVRQDQFH�� RXUHVWLPDWLRQV� VKRZ� WKDW� β′αβγ�T�� LV� QHJOLJLEOHLQ� FRPSDULVRQ� ZLWK� βαβγ
(�) �T�� IDU� DZD\� IURPWKH�UHVRQDQFH��WRR�,I� LQ�D� VHPLFRQGXFWRU� WKH�VHOHFWLRQ�UXOHVDOORZ�WKH�WZR�SKRWRQ�WUDQVLWLRQ�EHWZHHQ�WKHKLJKHVW�YDOHQFH�EDQG�DQG�WKH�ORZHVW�FRQGXF �WLRQ�EDQG�DORQJ�ZLWK� WKH� RQH�SKRWRQ�GLSROHWUDQVLWLRQ�� WKH�DGGLWLRQDO� VFDWWHULQJ�PHFKD �QLVPV�ZKLFK� LQFOXGH� WKH� WZR�SKRWRQ� WUDQVL �WLRQV�LQ�V�H[FLWRQLF�VWDWHV�FDQ�EH�LQYROYHG�LQWKH� +56� SURFHVV�� $V� LV� VHHQ� IURP� WKH� H[ �
/�(�6HPHQRYD��.�$�3URNKRURY���7KHRUHWLFDO�WUHDWPHQW����
�� )XQFWLRQDO�PDWHULDOV�������������
SUHVVLRQV�����DQG������WKH�LQWHUEDQG�)U|KOLFKDQG� GHIRUPDWLRQ� SRWHQWLDO� LQWHUDFWLRQVFDXVH� WKHQ� WUDQVLWLRQV� WR� V�H[FLWRQV�EHORQJ�LQJ� WR� RWKHU�SDLU� EDQG� �βαβγ
(�) �T��� DQG� WKH� LQ�WUDEDQG� )U|KOLFK� LQWHUDFWLRQ� UHVXOWV� LQ� WKHWUDQVLWLRQ�WR� S�H[FLWRQ��βαβγ
(�) �T����7KHVH�VFDW�WHULQJ�PHFKDQLVPV�ZHUH�FRQVLGHUHG�LQ�GHWDLOLQ� >��@��%\�XVLQJ� WKH�*UHHQ�V� IXQFWLRQ� IRU �PDOLVP��WKH�H[SUHVVLRQV�IRU�WKH�FRUUHVSRQG �LQJ� +56� WHQVRUV�� βαβγ
(�) �T�� DQG� βαβγ
(�) �T��� ZHUHDOVR� REWDLQHG� >��@�� %HVLGHV�� WKH�PHFKDQLVPRI� WKH� IRUELGGHQ� )U|KOLFK� +56� �βαβγ
I �T��� LVZRUWK� WR� WDNH� LQWR� DFFRXQW� ZKLFK� LV� GH �VFULEHG� DV� IROORZV�� WKH� WZR�SKRWRQ� WUDQVL �WLRQ� WR� DQ� V�H[FLWRQ� VWDWH� DQG� WKHQ� WKH� LQ�WUDEDQG�WUDQVLWLRQ�EHWZHHQ�V�H[FLWRQV�GXH�WRWKH� T�GHSHQGHQW� )U|KOLFK� LQWHUDFWLRQ�ZKLFKLV� IRUELGGHQ� LQ� WKH� ]HUR� ZDYH� YHFWRU� DS �SUR[LPDWLRQ�7KXV��YDULRXV�VFDWWHULQJ�PHFKDQLVPV�FRU �UHVSRQGLQJ� WR� GLIIHUHQW� VHTXHQFHV� RI� LQWHU �PHGLDWH� VWDWHV� FRQWULEXWH� WR� WKH� +56� LIERWK�WKH�RQH�SKRWRQ�DQG�WZR�SKRWRQ�WUDQVL �WLRQV�DUH�DOORZHG�LQ�D�VHPLFRQGXFWRU��%XW�LWLV�REYLRXV�WKDW�WKH�+56�GHVFULEHG�E\�βαβγ
(�) �T�LV� GRPLQDQW� EHFDXVH� LW� LV� GHVFULEHG� LQ� WKHIUDPHZRUN� RI� WKH� WZR�EDQG� PRGHO� DQG� LQ �FOXGHV� RQO\� WKH� GLSROH�DOORZHG� WUDQVLWLRQV�XQOLNH�RWKHU�VFDWWHULQJ�PHFKDQLVPV�+RZHYHU�� LW� VKRXOG� EH� QRWHG� WKDW� WKHUHVRQDQFH�FRQGLWLRQV�IRU� βαβγ
(�) �T���βαβγ
(�) �T��DQG
βαβγ
I �T��DUH�HVWDEOLVKHG�ZKHQ�WKH�GRXEOHG�HQ �HUJ\�RI�H[FLWLQJ�UDGLDWLRQ�TXDQWXP�LV�FORVHWR� WKH� HQHUJ\� OHYHO� RI� WKH� �6 H[FLWRQ��K−ω/ ≈ (�6��� :KHUHDV� WKH� UHVRQDQFH� IRU
βαβγ
(�) �T��WDNHV�SODFH�ZKHQ��K−ω/ ≈ (�3��7KHUH�IRUH�� RWKHU� VFDWWHULQJ� PHFKDQLVPV� FDQ� EHQRQ�QHJOLJLEOH��2XU�HVWLPDWLRQV�KDYH�VKRZQWKDW�FORVH�WR�WZR�SKRWRQ�UHVRQDQFH�ZLWK�WKH�6 H[FLWRQ�� WKH� FRQWULEXWLRQ� RI� WKH� +56FDXVHG�E\�WKH�IRUELGGHQ�)URKOLFK�VFDWWHULQJFDQ� EH� FRPSDUDEOH� ZLWK� WKDW� RI� WKH� +56GHVFULEHG� E\� βαβγ
(�) �T�� LI� WKH� HIIHFWLYH�PDVVHVRI� HOHFWURQ� DQG� KROH� GLIIHU� FRQVLGHUDEO\IURP�RQH�DQRWKHU�>��@��$V�D�UHVXOW��WKH�IUH �TXHQF\�GHSHQGHQFH�RI�WKH�+56�LQWHQVLW\�FDQFRQWDLQ�D�VLQJXODULW\�DW��K−ω/ ≈ (�6�0RUHRYHU��βαβγ
(�) �T��GHSHQGV�RQ�WKH�LQFLGHQWSRODUL]DWLRQ�� %\� XVLQJ� WKH� I�VXP� UXOH�� WKHGLIIHUHQFH�>0FFβ′β − 0νν
β′β@�FDQ�EH�H[SUHVVHG�YLDHIIHFWLYH�PDVVHV�RI�HOHFWURQ�PH∗ DQG�KROH�PK∗
0FFβ′β − 0νν
β′β
= P[� ⁄ PHβ′β
∗ + � ⁄ PKβ′β
∗ ]� ���
7KH� HIIHFWLYH�PDVV� LV� JHQHUDOO\� WKH� WHQ �VRU� YDOXH� EXW� LWV� QRQ�GLDJRQDO� FRPSRQHQWVDUH� ]HUR� IRU� D� QXPEHU� RI� FU\VWDOV��%HFDXVHRI� WKLV�� WKH� H[SUHVVLRQ� ���� FDQ� EH� FKDQJHGE\�>0FFβ′β − 0νν
β′β@ = P ⁄ µδβ′β ZKHUH�µ LV�WKH�UH�GXFHG� PDVV� RI� H[FLWRQ�� ,Q� WKLV� FDVH�� WKHK\SHU�5DPDQ� VFDWWHULQJ� PHFKDQLVP� GH �VFULEHG�E\�WKH�H[SUHVVLRQ� ����FRQWULEXWHV�WRWKH�+56�LQWHQVLW\�RQO\�XQGHU�FRQGLWLRQ�WKDWWKH� H[FLWLQJ� UDGLDWLRQ� SRODUL]DWLRQ� OLHV� LQ� DSODQH�IRUPHG�E\�SKRWRQ�ZDYH�YHFWRUV��Tβ ≠ ���8QIRUWXQDWHO\�� WKH� PHDVXUHPHQWV� RI� WKH+56� SRODUL]DWLRQ� GHSHQGHQFHV� FORVH� WR� WKHWZR�SKRWRQ� UHVRQDQFH� ZLWK� WKH� �S H[FLWRQVWDWH�� WR� RXU� NQRZOHGJH�� DUH� DEVHQW� H[FHSWIRU� WKH� LQYHVWLJDWLRQV� LQ� D� WKLQ� ILOP� =Q6HVDPSOH�RQ�*D$V VXEVWUDWH�>�@�7KHUHIRUH�� LQ� D� VFDWWHULQJ� JHRPHWU\ZKHUH� Tβ =��� RWKHU� +56� PHFKDQLVPV� FRQ�WULEXWH� WR� WKH� K\SHU�5DPDQ� LQWHQVLW\�� ,QWKLV� FDVH�� REYLRXVO\�� LW� LV� MXVW� WKH� 5+56GXH�WR�WKH�WZR�SKRWRQ�WUDQVLWLRQ�WR�D� V�H[FL�WRQLF�VWDWH�WKDW�GRPLQDWHV��6XFK�+56� VHHPVWR�EH�VLPLODU�WR�WKH�556��%XW�WKH�+56�VSHFWUDIRU� �K−ω/ a�(�6 FDQ� GLIIHU� FRQVLGHUDEO\� IURPWKH� 556� VSHFWUD� IRU� K−ω/ a�(�6� ,W� LV� NQRZQIURP� WKH� WKHRU\� RI� WZR�SKRWRQ� DEVRUSWLRQ>��@� WKDW� WKH� WZR�SKRWRQ� WUDQVLWLRQ� WR� D� V�H[FLWRQ� FDQ� EH� GHVFULEHG� LQ� WHUPV� RI� WKHWZR�EDQG�PRGHO�ZKHUH�WKH�DEVRUSWLRQ�RI�LQ �FLGHQW� SKRWRQ� LV� DFFRPSDQLHG� E\� H[FLWDWLRQRI�S�H[FLWRQ�DQG�DQRWKHU�SKRWRQ�LQGXFHV�WKHLQWUDEDQG� WUDQVLWLRQ� WR� DQ� V�H[FLWRQ�� 7KHWZR�SKRWRQ� H[FLWDWLRQ�RI�D� V�H[FLWRQ�FDQ�EHGXH� DOVR� WR� WKH� WUDQVLWLRQV� WR� WKH� KLJKHU�O\LQJ� FRQGXFWLRQ� EDQG� F′ RU� WKH� ORZHU� YD�OHQFH� EDQG� ν′ �WKUHH�EDQG� PRGHO��� ,Q� RWKHUZRUGV��LQLWLDOO\�WKH�WUDQVLWLRQ�WR�WKH� V�H[FL�WRQ�EHORQJLQJ�WR� F′ DQG�ν EDQGV�RU�WR�F DQG
ν′ RQHV�RFFXUV��WKHQ�WKH�LQWHUEDQG�WUDQVLWLRQRI� DQ� HOHFWURQ� RU� KROH� WDNHV� SODFH�� 7KXV�ERWK� WKH� WZR�EDQG� DQG� WKUHH�EDQG� PRGHOVFRQWULEXWH� WR� WKH� +56� PHFKDQLVPV� FRQVLG �HUHG��6LQFH� WKHLU� FRQWULEXWLRQV�DUH�RSSRVLWHLQ�VLJQ�>��@��WKHLU�LQWHUIHUHQFH�FDQ�UHVXOW�LQDGGLWLRQDO�VLQJXODULWLHV� LQ�+56�VSHFWUD�DQGWR�D�PRUH�FRPSOH[�UHVRQDQFH�SURILOH�,Q�RUGHU�WR�HYDOXDWH�WKH�UROH�RI�WKH�VFDW �WHULQJ�PHFKDQLVP�GHVFULEHG�E\� β′αβγ�T��� WKHWHQVRUV�βαβγ
(�) �T���βαβγ
(�) �T��DQG�β′αβγ�T��ZHUH�FDO�FXODWHG� LQ� D� ZLGH� IUHTXHQF\� UDQJH� IRU� WKH)U|KOLFK� LQWHUDFWLRQ� RQO\�� 7KH� UHVXOWV� DUHSUHVHQWHG� LQ� )LJ����� $V� LV� VHHQ� IURP� WKH)LJXUH�� βαβγ
(�) �T�� GRPLQDWHV� RYHU� βαβγ
(�) �T�� DQG
βαβγ
(�) �T�� QHDU� WZR�SKRWRQ� UHVRQDQFH�� EXW
/�(�6HPHQRYD��.�$�3URNKRURY���7KHRUHWLFDO�WUHDWPHQW����
)XQFWLRQDO�PDWHULDOV������������� ��
β′αβγ�T��EHFRPHV�FRPSDUDEOH�ZLWK� βαβγ
(�) �T��IDUIURP�WKH�UHVRQDQFH��7R� FRQFOXGH�� LQ� JHQHUDO� FDVH�� YDULRXVVFDWWHULQJ� PHFKDQLVPV� DUH� LQYROYHG� LQ� WKH+56� SURFHVV�� 7KHLU� FRQWULEXWLRQV� DUH� GH �ILQHG� E\� WKH� V\PPHWU\� DQG� SDUDPHWHUV� RIFU\VWDO� DV�ZHOO� DV� E\� WKH� VFDWWHULQJ� JHRPH �WU\�DQG�GHSHQG�RQ�LQFLGHQW�IUHTXHQF\��7KXV�LQYHVWLJDWLRQV� RI� WKH� K\SHU�5DPDQ� VFDWWHU �LQJ� FORVH� WR� WKH� WZR�SKRWRQ� UHVRQDQFH�ZLWKH[FLWRQLF� OHYHOV� DUH� RI� LQWHUHVW� EHFDXVH� WKHDQDO\VLV�RI�UHVRQDQFH�SURILOHV�DOORZV�RQH�WRREWDLQ� DGGLWLRQDO� LQIRUPDWLRQ� RQ� WKH� VHPL �FRQGXFWRU��$FNQRZOHGJHPHQW�� 7KLV� ZRUN� ZDV� VXS�SRUWHG�E\�WKH�5XVVLDQ�)RXQGDWLRQ�IRU�%DVLF5HVHDUFK��3URMHFW�1R����������������5HIHUHQFHV���5�0�0DUWLQ�� /�0�)DOLNRY�� LQ�� /LJKW� 6FDWWHU�LQJ� LQ� 6ROLGV�� HG�� E\� 0�&DUGRQD�� 6SULQJHU�%HUOLQ�����������'�$�/RQJ�� /�6WDQWRQ�� 3URF��5R\��6RF��$�� ������������������<X�1�3ROLYDQRY�� 5�6K�6D\DNKRY�� 3LV�PD� =K�(NVS��7HRU��)L]����������������������9�$�0DVORY�� .�.�2QGULDVK�� <X�1�3ROLYDQRYHW�DO���/DV��3K\V��������������������.�,QRXH��.�:DWDQDEH�� 3K\V��5HY��%�� ���� ��������������.�:DWDQDEH��.�,QRXH��)�0LQDPL��3K\V��5HY��%��������������������.�,QRXH�� )�0LQDPL�� <�.DWR�� .�<RVKLGD�� -�&U\VW��*URZWK����������������������6�.RQR��1�1DND��0�+DVXR�HW�DO���6ROLG�6WDWH&RPPXQ���������������������/�)LOLSKLFOLV��+�6LHJOH��$�+RIIPDQQ��&�7KRP�VHQ��3K\V��6WDW��6ROLGL�%�������5�������������$�*DUVLD�&ULVWREDO�� $�&DQWDUHUR�� &�7UDOOHUR�*LQQHU�� 0�&DUGRQD�� 3K\V��5HY��%�� ���� ����������������/�(�6HPHQRYD�� .�$�3URNKRURY�� LQ�� 3URF��RI;9,WK� ,QWHUQ��&RQI��5DPDQ� 6SHFWURVFRS\
�&DSH� 7RZQ�� ������� -RKQ� :LOH\� � 6RQV�&KLFKHVWHU���������S���������/�6HPHQRYD��.�3URNKRURY�� 3URF��63,(�� ��������������������/�6HPHQRYD��.�3URNKRURY�� 3URF��63,(�� ��������������������/�(�6HPHQRYD��.�$�3URNKRURY��=K��(NVS��7HRU�)L]�������������������������0�0�'HQLVRY�� 9�3�0DNDURY�� 3K\V��6WDW��6ROLGL%�������������������.�5XVWDJL�� )�3UDGHUH�� $�0\V\URZLF]�� 3K\V�5HY��%���������������������5�0�0DUWLQ��3K\V��5HY��%�����������������
�˺¯Ë�Ò�ÓË��¯È}��mÈÓÓ«�¯ËϺÓÈÓ°Óº�º�L¹Ë¯}ºäLÓÈ�L®Óº�º�¯ºÏ°L«ÓÓ«�°m��ãÈ���}¯Ò°�ÈãȲ
����vËäËÓºmÈ��z�2��¯º²º¯ºm
cºº���¹¯Ò°m«�ËÓº�º�ºmº¯ËÓÓ
¯LÏÓÒ²�ä˲ÈÓLÏäLm�¯ËϺÓÈÓ°Óº�º��L¹Ë¯}ºäLÓÈ�L®Óº��º�¯ºÏ°L«ÓÓ«�°mL�ãÈ�ÏÈ���Ȱ�
Ë}°Ò�ºÓLm���ÓȹLm¹¯ºmL�ÓÒ}ºmÒ²�}¯Ò°�ÈãȲ��È�ÈÓÈãLÏ���²mÓ˰}Lm�m�LÓ�ËÓ°ÒmÓL°� ¯ºÏ°�«ÓÓ«�
)LJ����� 7KH� FRQWULEXWLRQV� RI� WKUHH� SRVVLEOHVFDWWHULQJ� PHFKDQLVPV� GHVFULEHG� E\� WKHK\SHU�5DPDQ� WHQVRUV� _βαβγ
(�) �T�_� ����� _βαβγ
(�) �T�_������DQG�_β′αβγ�T�_� ����DV�D�IXQFWLRQ�RI�SDUDPH�WHU�ξ(ω/) ��(Fν Ü��K−ω/��5��:KHQ�FDOFXODWLQJ�WKH� W\SLFDO� SDUDPHWHUV� RI� VHPLFRQGXFWRUVFU\VWDOV� ZHUH� XVHG�� 5 �����(Fν�� K−ω3 ����5DQG� γ ������� %HVLGHV�� WKH� YLUWXDO� WUDQVLWLRQWR�WKH�RQO\�RQH�KLJKHU�O\LQJ�FRQGXFWLRQ�EDQGF′ �(F′F ����(Fν��ZDV�WDNHQ�LQWR�DFFRXQW�
/�(�6HPHQRYD��.�$�3URNKRURY���7KHRUHWLFDO�WUHDWPHQW����
��� )XQFWLRQDO�PDWHULDOV�������������
|