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Page 1: Divakth.diva-portal.org/smash/get/diva2:11732/FULLTEXT01.pdf · "!$#% $&' ()+* ,.-/*102- 0 , -43.* -5* 6 798;:9@?A8B798;C1DE 6AF -G0; IH HJ 79KLGE MNKA>@O PQ8SR8BC E

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3N?�6.:\<@6C�N>W8SDf<WCT7\MN8�P)6�?A8Bcd_NC1L 7\<WK�CYcdKA:\OI_N>=6�7\<WK�CI7\K79MN8�79MN:\8B8�D9F16�7\<@6A>1L2K�K�:\R�<@C16.798;D;m >QK�C1L28;F�-79_16A>W>@wAo�7\MN<=Df<@Df6'OT6 64KA: D�<@OTFN>W< %VL;6�79<@KACqm u !f :\8B>@<W8SDqK�C%74P�K�cd_NC1R16.OT8BC�7\6A>�79MN8;KA:\8BOJD;m$)MN8%1:GD47�7\MN8BK�:98;O R�_N8�7\K[��K�MN8BC-�y8B:\X[6.C1Rs�'KAMNC U j k V D�7\6.798SD)79M16.7�cdKA: 6AC�w D9w�D�798;O KAc <@C�798B:9-6AL279<@CNX[8B>@8;L279:\KAC1D'<@C�6.C"8Bi�798B:\C16A>qFyK.798;C�79<=6.>

Vext(r)oy79MN8YFyK.798;C�79<=6.>

Vext(r)<=D�_NCN< ��_18B>@w

R�827\8B:\OT<WCN8SR`o`82iNLB8BF�7�cdKA:6.C 6ARNR�<W79<@?A8TL2K�C1D�7\6.C�7So ��w^7\MN8TXA:\KA_NC1R�-/D�7\6.798%8;>W8SL 79:\KAC R�8BCVD�<W74wn0(r)

mY)MN8T<WOTFN>@<@L;6�7\<WK�C KAcl79MN<=DD�7\6�7\8BOT8BC�7<=D�79M16.76A>W> 7\MN8TFN:9K�FV8;:�7\<W8SD�K.c�6^D9w�D�798;O K.cC�_1L2>@8B<�6.CVR�8;>W8SL 79:\KACVD'6A:98TL2K�OYF1>W8B798B>@w"R�8B798B:\OT<WC18;RE��ws7\MN8TXA:\KA_NCVR�-/D�7\6.798a8B>@8;L 7\:9K�C R�8;C�-D�<W74wAm�)MN8�:\8;6�D�K�CN<WC1X �y8BMN<@C1R 79M18�>=6�79798;:�L2KACVL2>@_1D�<@KAC <@D�79M16.7)E�CNK�P�>@8;RNXA8�K.cq7\MN8FyK.798;C�79<=6.>Vext(r)

<=D�8���_1<W?�6.>@8BC�7'79KUE�CNK�P�<@CNXT7\MN8Icd_1>W> M16.OT<@> 7\KACN<=6.C 6.C1RrLBKAC1D98 ��_N8;C�7\>Ww 79M18aOJ6.C�w�-�VK�R�w P�6�?�82- cd_NC1L 7\<WK�C <=DJL2KAOTFN>@827\8B>@w R�827\8B:\OT<WCN8SR`o�P�MN<@LGM <@C FN:\<@C1L2<@FN>@8 X�<W?�8;DT6AL;L28;D\Da79K6.C�wUF1:9K�FV8;:�74w^K.cf79M18IOJ6�7\8B:\<@6A> m�)MN<=D�6.FNFN>@<@8;D�7\K �yK.79M�XA:\KA_NC1R�-/D�7\6.798%6.CVRs82iNL2<W798SR�-/D�7\6.798FN:\KAFy8B:979<@8;D;m[�5C F16.:979<=L2_1>@6A:�79MN8�%1:GD�7a��K�MN8BC-�y8B:\X.-5� K�MNC"7\MN8BK�:98;O <@DI?�6.>@<@R cdKA:%79MN8J7\K.7G6.>8BCN8;:9X�wAo�N_�7;ohR�8SD�FN<W798aFN:\K�?�<@CNXT7\MN8%82i�<=D47\8BC1LB8IKAc�6J79K.7G6.>W- 8;CN8B:\XAwUcd_NC1L279<@KAC16A> oy< 7�R�K�8;D'CNK.7XA<@?A8�_1D�6AC�w[:98SL2<@FV8�MNK�P 79K[L;6.>=L2_N>=6�7\8�< 7Sm )MN8%D98;L2K�C1RU79MN8;KA:\8BO RN_N8�79K[� KAMN8;C:�y8B:\XJ6.C1R� K�MNC"FN:\K�?�<@RN8;D 6 ?�6.:\<@6.79<@KAC16A>qFN:\<@C1L2<@FN>@8�U j k V/mI)MN8;:98a<@D�6 _NCN<@?A8;:\D\6.>$cd_NC1L279<@KAC16A>$cdK�: 7\MN88BCN8;:9X�w <@C"7\8B:\OJD KAcl79MN8TR�8;C1D�<W74w

n(r)oE[n(r)]

on6.CVRs79MN8TOT<@CN<WOa_NO K.c 79MN<=D�cd_1C1L 7\<WK�C16.><@DY7\MN8sXA:\KA_NCVR�-/D�7\6.798^8BCN8;:9X�wAm �rKA:\8BK�?�8B:I79MN8"R�8;C1D9< 74w 7\M16�7JOT<@CN<WOT<;.B8SD

E[n(r)]<@DY7\MN8

XA:\KA_NCVR�-/D�7\6.798TR�8BCVD�<W74wn0(r)

m[(�w�RN8 %1CN<W79<@KACqoq7\MN8[D�8SL2K�C1R"7\MN8BK�:98;O R�K�8;D�C1K.7%6.FNF1>Ww�79K82iNL2<W798SR D�7\6.798FN:\KAFy8B:979<@8;D)6AC1R 79MN8SD�8�OI_VD47��y8%R�8B:\<@?A8;R <@C^6ACNK.7\MN8B:�P)6�wAm

)MN8 X�:9K�_NC1R�-5D�7\6�7\8[8B>@8;L279:\KAC R�8;C1D9< 74w 6AC1R 79M18U7\K.7\6A>Q8;CN8B:\XAw 6.:\8 L;6.>=L2_N>=6�7\8;R P�<W79M1<WC79MN8 �'KAMNC�-4��M16AO cdK�:9Oa_N>=6�79<@KAC K.c u !f U j�j V/o'P�MN<=LGM 6�D9D9_NOT8;DU79M16.7 7\MN8 R�8BCVD�<W74w K.ca6D�w�D�798;O KAc1<@C�7\8B:G6AL279<@CNX 8;>W8SL 79:\KACVD$L;6.C �y8�KC��7G6.<@CN8;Ra6ADq79MN8)R�8BCVD�<W74wK.cV6ACI6._Ni�<@>@<@6A:9wD9w�D�798;OK.c�CNK�CN<WC�7\8B:G6AL 7\<WC1Xr8B>@8;L 7\:9K�C1D�7\M16�7IOTK�?�8J<WC 6AC 82gh8;L 7\<W?�8 R�8;C1D9< 74w�-5R�8BFy8BC1RN8BC�7IFyK.798;C�79<=6.>V (r)

m�)MN8 KA:\<WX�<WCV6.>lOJ6.C�w�- �VK�R�w FN:9K��N>@8BO M16AD%79M�_1D �V8;8BC 7\:\6AC1D4cdK�:9OT8SR <@C�79K�6�D�8B7aK.cKACN8B- 8;>W8SL 7\:9K�CUFN:\KC�N>@8BOJD;ml)MN8�79KA7\6A>n8BCN8;:9X�wJK.c 79MN<=D�D9w�D47\8BO KAc C1KACN<@C�798B:G6AL279<@CNXT8B>@8;L279:\KAC1D

Page 16: Divakth.diva-portal.org/smash/get/diva2:11732/FULLTEXT01.pdf · "!$#% $&' ()+* ,.-/*102- 0 , -43.* -5* 6 798;:9@?A8B798;C1DE 6AF -G0; IH HJ 79KLGE MNKA>@O PQ8SR8BC E

, ������� ����� � ��� � ��� ��������� � � � � ����� � � � ����������

<@D�X�<W?�8BC ��wJ79MN8%�'KAM1C�-5��M16.OZ8;CN8B:\XAwTcd_NCVL 79<@KACV6.> U j ,YV

E[{φj(r)}] = 2∑

j

φj(r)(− ~2

2m)∇2φj(r)dr +

n(r)Vext(r)dr ++ kNm H-,

+e2

2

n(r)n(r′)

|r − r′| drdr′ +EXC [n(r)] +Eion(RI),

P�MN8B:\8l7\MN8φj(r)

x Df6.:\8�79M18QK�CN82-/8B>@8;L279:\KACP)6�?A8�cd_NCVL 79<@KACVDBmf)MN8K%1:\D�7q7\8B:\O <WCa�%�ymfkNm H':98;F�-:98SD�8;C�7GDf79M18�79KA7\6A>1E�<WC18279<=L�8;CN8B:\XAwaK.ch79MN8'CNKAC1<WC�798;:\6�L 7\<WCNX�8;>W8SL 79:\KACVDl6.C1RY7\MN8'D98;LBKAC1Ra7\8B:\O:98;FN:98SD�8;C�7\DQ79MN88;CN8B:\XAw[RN_N8�79KY79MN8<@C�798;:\6�L 79<@KAC��V8B74PQ8;8BC 79MN8<@KACsL2K�:98SD)6.C1RU79MN8?�6A>W8;C1L288B>@8;L 7\:9K�C1D;m

Vext(r)<@D�7\MN8JL2K�:9:\8;D9FVK�C1R�<@CNXUFyK.7\8BC�79<=6.> mT)MN8Y79MN<@:\R�798;:9O <=D�79M18Y8;>W8SL 7\:9K�C�-

8B>@8;L 7\:9K�C9>QKA_N>@KAO �Y8;CN8B:\XAwa6.C1REion(RI)

<@Df7\MN8 >QK�_N>WK�O �T8BC18B:\XAwcd:\KAO 79MN8�<@C�798;:\6�L 79<@KAC6.OTKACNXU79MN8TC�_1LB>W8;< m! <WCV6.>@>Ww�o

EXC [n(r)]<=D'79MN8T82iNLGM16ACNXA8B-/LBKA:\:98;>@6.79<@KAC^8BCN8;:9X�w^6AC1Rr7\MN<=D

<@DY7\MN8sKACN>@w ��_16.C�79<W74w <@C �%�ym k�m H�7\M16�7[LB6ACNCNKA73�y8s8B?�6.>@_16�7\8;R 82iN6AL279>@wAm )MN8s8;>W8SL 79:\KACR�8BC1D9<W74wU<=D�:\8B>=6�798SR[7\KT79MN8IKACN8B- 8;>W8SL 79:\KAC^P)6�?A8'cd_NC1L 7\<WK�C1D���w

n(r) = 2∑

j |φj(r)|2 m�)MN8KACN8B- 8;>W8SL 79:\KAC P)6�?A8Ycd_NC1L279<@KAC1D;oφj(r)

of6A:98JL;6.>=L2_N>=6�7\8;R cd:\KAO 7\MN8UD9K.-5LB6A>W>@8;R �'KAMNC�-4��M16AO8 ��_16�7\<WK�C1D U jAj o j ,YV/o$P�M1<@LGM 6.:\8JR�8B:\<@?A8;RE��w 6AFNFN>@w�<@CNXs7\MN8J?�6.:\<@6.79<@KAC16A>�FN:9<@C1LB<WFN>@8T79Ks7\MN8� K�MNC�-4��M16.O+8;CN8B:\XAw�cd_1C1L 7\<WK�C16.> + �%�ymlk�m H:,$6.C1Ra<WOTFyK�D9<WC1X'7\MN8�KA:979M1KAXAK�C16.>@<W74w�LBKAC1D�79:G6.<@C�7\DKACT79M18 KAC182-/8B>@8;L 7\:9K�CaP)6�?A8)cd_1C1L 7\<WK�C1DBm�)MN8'� K�MNC�-4��M16.O 8 ��_16.79<@KAC1D�MV6�?A8)79M18 D96AOT8�cdKA:\O6AD�79M18KACN8B- 8;>W8SL 79:\KACs��LGMN:\b�R�<WC1XA8B:)8���_V6�79<@KAC

(−~2

2m∇2 + V (r)

)

φj(r) = εjφj(r),+ k�m 3C,

P�MN8B:\8T79MN8εjx D%6A:98T7\MN8UD9<WCNX�>W8JFV6.:979<=L2>@8[� K�MNC�-4��M16.O 8BCN8;:9X�w"8B<@XA8;C�?�6A>W_N8SDBm

V (r)<@Da6.C

82gh8;L 7\<W?�8FVKA798;C�7\<@6A>`FN:\K�RN_1L28SR���wJ7\MN8K.7\MN8B:�8;>W8SL 7\:9K�C1D)6.C1R 79M18C�_VL2>@8B<

V (r) = Vext(r) + e2∫

n(r)n(r′)

|r − r′| dr

′ +δEXC [n(r)]

δn(r).

+ k�m ��,

)MN8 D9827lK.cn�'KAMNCN-5��MV6.O�8���_V6�79<@KACVDf<WCJ�%�ymlk�m 3�6.:\8�L2K�_NFN>@8;Ra79MN:\KA_1XAMY79MN8'R�8BC1D9<W74waR�8;FV8;C�-R�8BC1LB8'KAc

V (r)6.C1R[79MN8;<W:�D9KA>@_�79<@KAC[<=D�_1D9_16.>@>WwJD9KA_NX�M�7QP�<W79Ms6ID98B>Wc -/LBKAC1D9<@D�798;C�7QFN:\K�L28;RN_N:98�o

P�MN<@LGMs<@C�?AKA>@?A8SDQ79MN8�cdK�>W>@K�P�<WC1XYD�798;F1DBx0Am�� 8%D47G6.:97�P�< 7\MsD9KAOT8�79:\<=6.>nR�8;C1D�<W74w

n(r)m

k�mA>�6A>@LB_N>@6.798V (r)

cd:\KAO �%�ym kNm �1mj m���K�>W?�8

φj(r)cd:\KAO �%�ym kNm 3Nm

H1mA>�6A>@LB_N>@6.7986YCN8BP R�8;C1D�<W74wJcd:\KAOn(r) = 2

i |φj(r)|2 m3�m)�/c

n(r)MV6AD�L2K�C�?A8B:\XA8SRJP�8�MV6�?A8'cdKA_NC1R 7\MN8%D�K�>W_N79<@KACno�KA79MN8;:9P�<=D�8�X�Ka7\KJD47\8BF"k�m

�5CYF1:\6�L 79<=L28Q79M18�CN8BP R�8BC1D9<W74w%<WCTFyKA<@C�7fH�<=D�KC�N7\6.<@CN8SR ��wI6.F1FN>Ww�<@CNX6�OY<Wi�<WC1X�D\LGMN8;OY8)P�<W79MR�8BC1D9<W79<@8;Dfcd:\KAO�F1:98;?�<@KA_VDf< 7\8B:G6�79<@KACTD�798BFVDEU H���o.H10WV mf)MN<=D�FN:\K�LB8;R�_1:98�<@DlCN8;LB8;D\D96A:9w<@CTKA:GR�8B:79Ks6AD\D�_N:\8ILBKAC�?A8;:9X�8BC1LB8Am')M18%79KA7\6A> 8;CN8B:\XAw^<=DA%VC16.>@>Wws8;?�6A>W_V6�798SRscd:\KAO 79M18Y�'KAMNCN-5��MV6.O

Page 17: Divakth.diva-portal.org/smash/get/diva2:11732/FULLTEXT01.pdf · "!$#% $&' ()+* ,.-/*102- 0 , -43.* -5* 6 798;:9@?A8B798;C1DE 6AF -G0; IH HJ 79KLGE MNKA>@O PQ8SR8BC E

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EXC [n(r)]onLBKAC�7\6A<WCVD�79M18aOJ6.C�w�- �VK�R�ws82gh8;L27\D 79M16.7

6.:\8 OT<=D9D9<WC1X�<@Ca79MN8)<@C1R�8BFy8BCVR�8BC�7�-/8B>@8;L279:\KACIcdKA:\OI_N>=6�7\<WK�CIK.c u !f 6.C1Ra< 7�<=D 79MN8)K�CN>Ww���_16AC�-79<W74w'<@C%�%�ymfk�m H�79M16.7$LB6ACNCNKA7 �V8Q8B?�6.>@_16.798;R�8BiN6AL 7\>Ww�mq)MN8�D�<@OTFN>W8SD47 6.FNFN:\K�i�<WOJ6.79<@KAC'cdKA:n7\MN882iNLGM16ACNXA8B-/LBKA:\:98;>@6.79<@KACJ8BC18B:\XAwJ<=D)79MN8>@K�LB6A>qR�8BCVD�<W74wU6.F1FN:9K�i�<@OJ6�79<@KACG+ p u �5, U jAj o j ��o1H�k V/m�5CI7\MN8�p u � KACN8�6AD\D9_NOT8;D$79M16.7�79MN8)8Bi�LGMV6.CNX�82-5L2KA:\:\8B>=6�79<@KAC8;CN8B:\XAwFy8B:�8;>W8SL 7\:9K�Cno

εXC(r)o

<WC"6JD9w�D47\8BO K.c�8B>@8;L279:\KAC1D�P�<W79M"RN8BC1D9< 74wn(r)

<@D�8���_16.>q79KT79M18%82iNLGM16ACNXA8B-/LBKA:\:98;>@6.79<@KAC[8;C�-8B:\XAw K.c 6JMNK�OTKAXA8;CN8BK�_1D�8;>W8SL 79:\KACrX�6AD�P�<W79M"79MN8YD96AOY8aR�8BC1D9<W74wAo

εhomXC (n(r))

ohP�MN<=LGMr<@D�6E�CNK�P�C���_16AC�7\< 74w R�8B:\<W?�8;RUcd:9K�O ��_16.C�7\_NO �rK�C�7\8 >�6.:\>WKTD�<@OI_1>@6.79<@KAC1D U H j V/ml��LBLBKA:GR�<@CNXA>@wAoELDA

XC [n(r)]<@D�XA<@?A8BC���w

ELDAXC [n(r)] =

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N =∑

i nio�<@D

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E = ELDA − UN(N − 1)/2 +1

2U∑

i 6=j

ninj .+ k�m ,�,

u <Wgy8;:98;C�79<=6�79<@KACaK.c1� �Vm�k�m ,�P�<W79Ma:98SD�Fy8;L27$7\K�7\MN8QK�:/�1< 7G6.>�K�L;L2_NFV6.C1LB<W8SDniXA<@?A8SDn7\MN8QK�:/�1< 7G6.>

8BCN8;:9X�<W8SDεi

εi = εLDA + U

(

1

2− ni

)

.+ k�m *�,

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2

ofP�MN<@>W8U_NCNK�LBLB_NFN<@8;R D�7\6�7\8;D +ni = 0

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ELDA+U [nσ(r), {nIσm }] = ELDA[nσ(r)] +EHub[{nlσ

m}] −EDC [{nlσm}]. + kNmW0S��,

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ELDA+U = ELDA +U − J

2

σ

[Trρσ − Tr(ρσρσ)] ,+ kNmW0�0 ,

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� MBJL4GJ JT=S8TJ�Q�2,?�8�� = ��M��/= Q � 6 R!8L4G256 J�5C K�:\RN8B:a79K D�K�>W?�8U7\MN8r� K�MNC�-4��M16AO 8���_16�7\<WK�C1Da79MN8rP)6�?A8Ucd_NC1L 7\<WK�C1DJ6.:\8^82i�F16.CVR�8;R <WCD�K�OT8�LBKAC�?A8;CN<@8BC�7 �V6AD9<@D�D9827

ψjk(r) =∑

l

ajk,lαjk,l,+ kNmW0 j ,

P�MN8B:\8αjk,l

6.:\8T79MN89�16AD9<@Dcd_NC1L 7\<WK�C1D;oajk,l

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D�_VLGM[6P�6�wI79M16.7 7\MN8'D9_NO <WC[� �VmlkNmW0 j L2KAC�?�8B:\XA8;D ��_N<=LGE�>Ww�o.7\M�_VD >@<WOT<W79<@CNX%7\MN8�C�_NO �V8;:QKAc798;:9OJD 79M16.7�CN8;8;R"79K9�y8Y<WC1LB>W_VR�8;R`m�5C�79MN<=D'79MN8SD�<=D'PQ8YM16�?A8IOT6A<WC1>Wwr8;OYF1>WK�w�8;RsOT827\MNK�RND�16AD98;R FN>=6.CN8TP)6�?A8Y82i�F16AC1D�<@KACVD�U �AH V o��N_�7IOT8B79MNK�RND�7\M16�7%:\8B>@w"K�C >@<WC18;6.:\<;.B8;R OI_SXTCN- 7\<WCKA:!�N<W7\6.>h8Bi�FV6.C1D9<WK�C1D U ��3 VnM16�?�86.>=D�K3�V8;8BCs_1D98;R`m

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ujk(r) =∑

G

cjk,GeiG·r,

+ kNmW0;H:,

P�MN8B:\8G6.:\8J79M18U:\8;L2<@FN:\K�LB6.>Q>@6.7�7\<@LB8[?A8SL 79K�:\Da6.C1R

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ψjk(r) =∑

G

cjk,Gei(k+G)·r.

+ kNmW0�3C,

)MN8UFN:\KC�1>W8;O KAc7%1C1R�<@CNXψjk(r)

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VMT (r) =

{

V (r) , r ≤ S.VMTZ , r ≥ S.

+ kNmW0S��,

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{

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αjk,lm(r, E) =∑

R

eik·Rχlm(r − R, E),+ k�m@0;,:,

P�MN8B:\8χlm(r − R, E)

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◦Gend −∑

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ED(T, V ) =9

8NkBθD + 3NkBTD(θD/T )

+ H1m ,�,

SD(T, V ) = 3NkB

(

4

3D (θD/T )− ln (1 − exp (−θD/T ))

)

.+ H1m *�,

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3

P

V

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k(ν) =

(

2

3

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+1

3

(

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)32

)− 13

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u <Wgh_1D�<@KAC <@C D9KA>@<@R1D':\82cd8;:\D 79K^7\MN8T:98SR�<@D�79:\<;�N_�79<@KAC K.cQ6.79K�OTD�K�:�<@KACVD�6�D�6U:\8;D9FVK�C1D98%79Kr6CNKAC18 ��_N<@>W<;�N:\<W_NO D�7\6.798IKAc�79MN8YOJ6�7\8B:\<@6A> m�)MN8aRN:9<@?�<WCNXUcdKA:GL28IcdKA:'RN< gh_1D9<WK�C�<=D'LBKAC1LB8BC�79:G6�-79<@KACJX�:\6�R�<W8;C�7\D;oAKA: 79K �V8'FN:98SL2<=D�8 XA:G6AR�<@8BC�7\Dl<WCULGM18BOT<@L;6.>VFVKA798BC�7\<@6A> m�!NK�:QLGM16A:9X�8;RTD9FV8SL2<@8;D+d<@KAC1D ,�R�<Wgh_1D�<@KAC LB6.CJ6A>@D9K �y8 R�:\<W?�8BC ��wY6ACJ8B>@8;L279:\<@L %V8B>=R`m u <Wgy_VD�<@KAC[F1>@6�w�Dl6ACJ<WOTFyKA:97\6.C�7:9K�>W8Q<WCa79M18�E�<@CN827\<@L;DfK.c1F1M16AD98 7\:\6AC1D�cdKA:\OT6.79<@KAC1D;o�P�MN<=LGMI<=D�6'D�79:\KACNX�R�827\8B:\OT<WC16AC�7 KAcVOJ6�7\82-:9<=6.>`OT<=L2:\K�D�79:\_1L 7\_N:\8;D)6.C1R 7\MN8B:\8���w[OJ6.C�w[<@OTFVK�:�7G6.C�7�OJ6�7\8B:\<@6A>@D�FN:\KAFy8B:979<@8;D;m ��KA>@<=R^D�7\6.798R�<Wgy_1D9<@KACUKAcq<@KACN<=L'D9Fy8;L2<@8;D�<=D�_�79<@>W<;.B8SRU<@C[cdK�:)82iN6.OTFN>@8�cd_N8;>nL28;>W>=DBmf�5CU79MN8�FN:\8;D98BC�7�LGMV6.F�7\8B:PQ83%1:GD�7#�1:9<@8 1w"RN<@D\L2_1D\D�79M18[L2KAC�7\<WC�_N_NO 79MN8;KA:\wrK.c�RN< gh_1D9<WK�CnoncdK�>W>@K�PQ8SR ��w�6^:\8B?�<W8;P KAc6�7\KAOT<@D�79<=LR�<Wgy_VD�<@KAC"OT8SLGM16.CN<=D9OTD;oV6AC1R %1C16A>W>@wUP�8%8Bi�7\8BC1R"KA_N: RN<@D\L2_1D\D9<WK�C 7\K[<@KACN<=L%L2K�C�-R�_1L279<@?�<W74wJ6.C1R[< 7GDQ6AFNFN>@<@L;6�79<@KACU<WCJcd_18B>`L28;>W>V798SLGMNCNKA>@KAX�wAm � 8�F16�wYF16.:979<=L2_N>=6.:�6�79798BC�7\<WK�CJ79KMNK�P+7\MN8YRN< gh_1D9<WK�C FN:\KAFy8B:979<@8;D�L;6.C �y8JD47\_1R�<@8;R _1D9<WCNX�%1:GD479- F1:9<@C1L2<@FN>@8;D 798SLGMNCN< ��_18;D;mY��8SL.m3�m k^<@D%OT6A<WC1>Ww :98;>@6.798SR�7\K ���J6.CVR <WC ��8SL.mr3�m j PQ8 R�<@D\L2_VD9D<=D\D�_N8SD�79MV6�7a6A:98T7\:98S6�798SR <WC��� "�m

�E��� � � K#�� �� � (*," . ��� �#�� � � + ( ��� � "�� � ��7�79MN8TOJ6ALB:9K�D9LBKAFN<=LILBKAC�79<@C�_N_NO >@8B?�8B>f79M18 V_�i�K.c)6�7\KAOJD�K.cQ74w�Fy8J� <WC 6 �N<@C16.:\w"��-5(D�w�D�798;O <@D)F1:9K�FVK�:�7\<WK�C16.>y79KT7\MN8%L2K�C1L28;C�79:G6�79<@KAC XA:G6AR�<@8BC�7

JA = −DA∇cA,+ 3�m@0 ,

P�MN8B:\8cA<=Dl7\MN8'LBKAC1LB8BC�79:G6�7\<WK�CJK.cn� 6.C1R

DA<@D 79M18'R�<Wgy_VD�<@KACULBK�8WX[L2<@8BC�7QK�:QR�<Wgh_1D�<@?�< 74w�m

�5C79M1:98;8QR�<@OT8BC1D9<WK�C1DDA

<=D$6'D�w�OTOT8279:\<=LlD�8SL2K�C1R:G6.CNE'798;C1D�K�:;o�P�MN<=LGM:\8;R�_1LB8;Dq79K�6�D9L;6.>=6.:< c)7\MN8UOJ6�7\8B:\<@6A>l<=DI<=D�KA79:\KAFN<=L.ms)MN8^R�< gh_1D9<WK�C L2K�8WX[LB<W8;C�7aKC�y8Bw�DI6AC ��:\:9MN8;CN<@_1D%:\8B>=6�7\<WK�CDA = D0

A exp(

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) o1P�MN8B:\8D0

A

<@D 6Icd:\8 ��_N8;C1L2w[c 6AL 7\KA:SoR<=D)79MN8X�6AD�LBKAC1D�7\6AC�7So

T<=D

79MN8�6C�1D�K�>W_N798�798;OYFy8B:G6�7\_N:\8)6AC1RQ<=Df79MN8�6�L 79<@?�6�7\<WK�Ca8;CN8B:\XAw�mq�%�yml3�m@0Q<=D�E�CNK�P�CY6AD�! <@LGE�� D

%1:GD47)>@6�P UW0�0;�YV/m��5C X�8BCN8;:\6A>y6 1_NiU<@D)L;6._1D98;R���w[6YRN:9<@?�<WCNXacdKA:GL28�o�P�MN<@LGM^<=D�6IX�:\6�R�<W8;C�7�KAc6TFVKA798;C�7\<@6A> m���7\:9<=L 7\>Ww^D�Fy8;6AE�<@CNXVo1L2K�C1L28;C�7\:\6.79<@KAC^<@D CNK.7 6[FVKA798BC�7\<@6A>q6.C1R 7\M�_VD ∇cA

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,a6.CVR ∇cA<@C �%�ym 3�m@0 D9MNK�_N>@R �V8^:98;FN>=6AL28SR ��w ∇µA

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JA = −DA∇cA = −LA∇µA = −MAcA∂µA

∂cA∇cA.

+ 3�m kC,

��8B:\8LA

<=D�6'E�<@CN8B79<=L ��_16AC�79<W74w79M16.7lLB6.C3�y8�<=R�8;C�7\< %V8;RT6AD$79M18�F1:9K�R�_1L27�K.cy6�OTKC�1<W>@< 74wMA6.C1RI79MN8�LBKAC1LB8BC�79:G6�7\<WK�C%K.cV�Io�P�MN<=LGM%79K�XA8B79MN8;: P�<W79Ma79M18�:\8B>=6�7\<WK�C ∇µA = ∂µA

∂cA∇cA

w�<@8B>=RND79MN8s>=6AD�7Y8���_V6.>@< 74w�m

MAKC�y8Bw�DT6.C ��:\:9M18BCN<@_1DY:98;>@6.79<@KACno

MA = M0A exp

(

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) m (�wL2KAOTF16A:9<@CNXa�%��DBm 3NmW0�6.C1R^3�m kP�8�<@R�8;C�79<Wcdw

DA = MAcA∂µA

∂cA

ml!$<=LGE�� DK%1:GD�7�>=6�P 6AD�D�7\6�7\8;R6 �yK�?A8%6.F1FN>W<@8;D�79K �1<WC16A:9wsD9w�D�798;OJD�6.C1Rs8Bi�798BCVD�<@KACs7\K[OI_N>W79<=L2K�OTFVK�CN8BC�7'D9w�D47\8BOJD�:\8;D9_N>W7\D<WC 79MN8^!$<=LGE�- 4'CVD96AXA8B:>=6�P UW0�0�� V/m )M18 R�<Wgy_VD�<@KAC F1:9K�L28SD9D%OI_1D�7YD\6�7\<@D�cdw 79MN8^L2KAC�7\<WC�_N<W74w8 ��_16�7\<WK�Cno ∂cA

∂t + ∇ · JA = 0o�P�MN<=LGMU7\KAXA8B79MN8;:�P�< 7\Ms�%�ym 3�m@0�w�<@8B>=RND�! <@LGE�� D�D�8SL2K�C1RU>=6�P

∂cA∂t

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a) b) c)

! <WX�_N:98�3�m@0Ax$6�,$��iNLGM16.C1XA8�OY8SLGM16.C1<@D9O +E → F

, 6.C1Ra:\<WCNX�OT8;LGM16ACN<=D�O +A → D → C →

B → A)mQ)MN8:\<@CNXTOY8SLGM16.C1<@D9OsoN<WCsP�MN<=LGM^79MN:\8B8K�:�OTKA:\8�6.79K�OTD�6A:98�<@C�?AKA>@?A8SR +dcdKA_N:�<@C

79MN<=D�FN<=L 79_1:98$, o`<=D�6 ?�6.:\<@6.79<@KACrKAcl79MN8JR�<@:\8;L 7�82iNLGM16ACNXA8aOY8SLGM16.C1<@D9Osm �=,')M18a<@C�798B:GD�79<W79<=6.>OT8;LGM16ACN<@D9O +

A,I6.C1R 79MN8 <WC�798;:\D�79<W79<=6.>=L2w OT8;LGM16ACN<=D�O +

BC,2m L ,&l6AL;6.C1LBw RN< gh_1D9<WK�C ��w

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B, m

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i, j = 1K�:

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jij = 4niAc

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!$<@XA_1:98a3Nm kNx���LGMN8BOJ6.79<=LFN<=L 79_1:98aK.clCN8S6.:\8;D�7�CN8B<@XAM-�yKA:'RN< gh_1D9<WK�C"<WC"79MN8Ic LBL%>=6�7979<=L28�m )MN8XA:\8Bw�6.79KAO M16ADT0�k CN8;6A:98SD47CN8;<WX�M:�yKA:>=6�7979<=L28[D9< 7\8;D;oq>=6 �y8B>@8;R 0T79K 0�k�m[)MN8JX�:98;w�6�7\KAOsoD�<W798J3�o`D9< 7\8Y�1ohD9<W798T�[6.C1R D�<W798Y,U>@<@8I<@C"79M18YD\6.OT8IF1>@6ACN8�+ FN>@6ACN8^0 , m%��<W798SDI0�onk�o j 6AC1R"HK�L;L2_NF�wU6ACNK.7\MN8B:�F1>@6ACN8 +dFN>=6.CN8Ik�, m

cdKA:GL28wij = Γ

6

oy6.C1R^OTKA:\8BK�?�8B:Soc1v = c2v = cv

mQ� 8B:\8Γ:\8BFN:\8;D98BC�7\D�7\MN8 64_NOTFr:G6�798<@C"6.>@>

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6(n1

A − n2A) = −cv

Γ

6a20

∂cA∂x

.+ 3�m ��,

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6.CVRw∗

j

6.:\8s79MN8 CNKA:\OJ6.>�OTK�R�8"cd:\8 ��_N8;C1L2<@8;D[P�MN8BC 7\MN8�6.79KAO K�LBLB_NFN<@8;DJ7\MN8>@6.7�7\<@LB8%D�<W798FyK�D9< 7\<WK�Cr6.C1R 79M18%D96�RNR�>@8FVK�<WC�7�FyK�D9< 7\<WK�CnoN:\8;D9FV8SL 79<@?A8;>Ww�m )MN8%6�L 79<@?�6�7\<WK�C^8;C�-8B:\XAw�o

E∗ oQL2KA:\:\8;D9FVK�C1RND�7\K�7\MN8Ucd_1>W>@w :98;>@6.i�8;R D�7\6.798 6.7a7\MN8sD96�RNR�>@8UFyKA<@C�7Sm )MN8[c 6�L 7\KA:Γ0 exp

(

SMv /kB

) <@Cs�%�ym 3�m ,Y<@D)<=R�8BC�7\< %18SR^P�< 7\M 7\MN8FN:\82c 6�L 79K�:�<WC^�%�ymQ3NmW0S�Nm u 8;?�<=6�7\<WK�C1Dcd:9K�O

kharmTST

R�_18T79K":\8;LB:9K�D9D9<WC1X�D5�V8BcdKA:\8T79MN8[OT<@XA:G6�7\<WK�C D�798;F <@D%LBKAOTFN>@827\8;R LB6AC �V8[6�L -L2K�_NC�798;R�cdKA:#��w"<@C�79:\K�RN_1L2<@CNXr6 79:G6.C1D9OT<@D\D�<@KAC"c 6AL 7\KA:So

κoq6.C1R LBKA:\:98SL 79<@CNXU79MN8JM16A:9OTK�CN<@L

6.FNF1:9K�i�<@OT6.79<@KACU6�LBLBKA:GR�<WC1XI7\KkTST = κkharm

TST

m�/c �%�ym j m 3a<=D�6.FNF1>W<@8;RUcdKA:

cv6.C1R^�%�ym 3Nm ,acdKA:

Γo�79MN8%RN< gh_1D9<WK�CsLBK�8WX[L2<@8BC�7�<@D

DncA = a2

0

Γ0

6exp

(

−HFv +HM

v

kBT+SF

v + SMv

kB

)

=

= Dnc,0A exp

(

−HFv +HM

v

kBT

)

,+ 3NmW0�0 ,

P�MN8B:\8Dnc,0

A = a20

Γ0

6 exp(

SFv +SM

v

kB

) <=D[<@C1R�8;FV8;C1R�8BC�7UKAc�798BOTFy8B:G6�7\_N:98�m � �Vm+3�m@0A0r<=DFV8;:�cd8SL 7\>Wwr?�6.>@<=R�<Wc�D9_ �1D98 ��_N8;C�7 64_NOTF1D6A:98a_NC1L2K�:9:\8B>=6�7\8;R`oy7\MNKA_NX�M"7\MN<@DLBKAC1RN< 7\<WK�C�<=D�CNK.7cd_N> %1>W>@8;R cdKA:J?�6�LB6.CVL2w OT8;LGM16ACN<=D�OJD;m ��C 6�7\KAO 7\M16�7[M16�D564_NOTFV8SR <@C�79K 6 CN8B<@XAM-�yKA:\<WCNX?�6ALB6AC1L2w MV6ADJ6 >@6A:9X�8B:YFN:\KC�16C�N<W>@<W74w K.cK64_NOTFN<WC1X �16ALGE <@C�79K <W7\DTKA:\<@XA<@C16.>)FyK�D9< 7\<WK�C 7\M16.C79K 64_NOTF <@C�KA79MN8;:�R�<@:\8;L 7\<WK�C1D;oyP�M1<@LGM"<=D�R�_N8a79K 79MN8ac 6AL27'7\M16�76�c 7\8B:'79MN8 %1:GD47 64_NOTF"7\MN8?�6ALB6AC1L2w�P�<W>@>I�y8J<WC 79M18UL2K�:9:\8;L27�FyK�D9< 7\<WK�C cdKA:�79M18U6�7\KAO 7\K�64_NOTF �16ALGEhm[)MN8J8Bgy8SL 7IKAc

Page 50: Divakth.diva-portal.org/smash/get/diva2:11732/FULLTEXT01.pdf · "!$#% $&' ()+* ,.-/*102- 0 , -43.* -5* 6 798;:9@?A8B798;C1DE 6AF -G0; IH HJ 79KLGE MNKA>@O PQ8SR8BC E

H�k ������������� ����� � ��� � ����� ��� � � � ��� �

L2KA:\:\8B>=6�79<@KAC^LB6AC �y8I6�LBL2K�_NC�798SR[cdK�:7��w <WC�7\:9K�R�_1LB<WC1X[6YLBKA:\:98;>@6.79<@KACUc 6AL 7\KA:fc,v

+fc,v ≤ 1

,6ALBLBKA:GR�<@CNXs79K

DA = DncA fc,v

m�)M18U?�6.>@_N8UKAc�79MN8^L2K�:9:\8B>=6�7\<WK�C�c 6�L 79K�:aR�8;FV8;C1RNDaKAC 7\MN8R�< gh_1D9<WK�C OT8;LGM16ACN<@D9O 6AC1R 79MN8[LB:9w�D�7\6A>lD47\:9_1L279_N:\8JK.c�79M18[OJ6�798;:9<=6.> mU!1KA:OTKACNK�?�6�LB6.CVL2wR�< gh_1D9<WK�C <@C c L;L LB:9w�D�7\6.>=D

fc,1v ≈ 0.78U@0A0;, V%6AC1R 79M18 LBKA:\:98SD�FyKAC1RN<WCNX ?�6A>W_18"cdKA: 79M18

R�<W?�6�LB6.CVL2w OT8SLGM16.CN<=D9O <=Dfc,2v ≈ 0.47

U@0A0S,YV/mq�/c�D�8;?A8B:G6.>.RN< gh_1D9<WK�COY8SLGM16.C1<@D9OJDn6.:\8�6AL 7\<W?�86�7 7\MN8 D\6.OT8�7\<WOT8�7\MN8�7\K.7\6A>VR�<Wgh_1D�<@KAC L2K�8WX[L2<@8BC�7 <=DlX�<W?�8BC���wI7\MN8'D9_NO K.c`7\MN8�<@C1R�<@?�<=R�_16A>R�< gh_1D9<WK�CrL2K�8WX[L2<@8BC�7GDBo�FN:\K�?�<@RN8;RU79M16.7�79MN8%R�<Wgh8B:\8BC�7�OY8SLGM16.C1<@D9OJD�6.:\8�<@C1R�8;FV8;C1R�8;C�7Sm

� 4 � � JL4G256 �KMB?@MOF�=>8T=>?@J Q�?@2/F � ?@J 8 �K? 476 RI4��ENG=>J

HFv

6AC1RHM

v

<@C �%�ym 3�m@0A0^LB6AC �y8 K���7\6A<WC18;R cd:9K�O %1:\D�7�-/FN:\<WCVL2<@FN>W8SDaL;6.>=L2_N>=6�7\<WK�C1DY6.C1RP�< 7\M�7\MN<@D�<@C�cdK�:9OJ6�7\<WK�C�<W7<@D�FyK�D\D�<;�N>@8I7\K^6AC16.>@w-.B8a79MN8a798;OYFy8B:G6�7\_N:\8YR�8;FV8;C1R�8;C�7F16A:�7�K.c79MN8JR�<Wgy_VD�<@KAC L2K�8WX[L2<@8BC�7�<WC �%�yma3NmW0�0Am

HMv

<=D'8;?�6.>@_16�7\8;R�6�LBLBKA:GR�<WC1XJ79K 79MN8Y7\:\6AC1D�<W79<@KACD47G6�798T7\MN8BK�:9wE��w�FN>=6ALB<WCNX^7\MN8[OT<WX�:\6.79<@CNXr6�7\KAO <@C 79MN8[D\6AR1R�>W8JFyKA<@C�7%FVK�D�<W79<@KAC 6AC1R LB6.>W-L2_N>=6�7\<WCNX[79MN87\K.7G6.>q8;CN8B:\XAw[cdKA:�7\MN8%:\8B>=6�i�8;RsD�79:\_1L 7\_N:\8Am))MN<@D OT8279M1K�Rs:\8 ��_N<@:\8;D)7\M16�7�7\MN8FVK�D�<W79<@KAC K.cQ79MN8[D\6AR1R�>W8TFyKA<@C�7%<@DE�CNK�P�CqmT!1KA:%D9KAOT8JLB6�D�8SD'<W7%<=DFVK�D9D9<;�N>W8Y7\KsRN82798;:9OT<@CN8< 7GDQ>@K�LB6.79<@KACJcd:\KAOZD9<WOTFN>@8 X�8BKAOT8B79:\wY6A:9X�_NOT8BC�7\D;o��1_�7�<@CUX�8BCN8;:\6A>N7\MN<@DQ<=DQC1K.7 7\MN8�LB6AD98Am �/7<@D�FyK�D\D9< �N>@8�79K3%1C1R 7\MN8%D96�RNR�>@8'FyKA<@C�7�FVK�D�<W79<@KAC ��w[OJ6.FNF1<WCNXJ6�?�6.<@>=6 �N>@8'OT<@XA:G6�7\<WK�CUF16.79M1D;m�5C 7\MN<=D�FN:9K�LB8;R�_N:\8a79MN8TOT<WX�:\6.79<@CNXs6�79K�O OI_1D�7CNK.7��y8J6.>@>WK�P�8;R"79Ks:98;>@6.ir<WC 79M18YOT<@XA:G6�-79<@KAC R�<@:98SL 7\<WK�Cno D9<@C1L28[<W7IP�KA_N>=R 79MN8;C :\8B>=6�i �16�LGE�<@C�79Kr7\MN8[8 ��_N<@>@< �N:\<@_NO FyK�D9<W79<@KACnms)MN8C�_VR�XA8SRJ8B>=6AD�79<=L��16AC1R[OT8B79MNK�R U@0Sk�k V`<@D�6.C[8DX[L2<@8BC�7�6A>WX�KA:\< 7\MNO cdK�:�RN82798;:9OT<@CN<@CNXI:\8;6�L 79<@KAC�16.:\:9<@8B:GD + D96�RNR�>@8[FVK�<WC�7GD!,2m")MN<@DaOT827\MNK�R L2:\8;6�7\8;Da6sC�_1O �y8B:YK.c�L2K�FN<W8SDK.c�7\MN8 D9w�D47\8BOsoLB6.>@>@8;R �y8;6�RNDBo�79MV6�7QLBKACNCN8SL 7Q79MN8'<WCN<W79<=6.>h6.CVR3%1C16.>VFVK�D�<W79<@KACVDBmf� 6�LGM �V8S6AR[LBKA:\:98SD�FyKAC1R1D 79K6�D9Fy8;L2< %VL)X�8BKAOT8B79:\w�KAcV7\MN8�6�7\KAOJDfKACa79MN8;<W:�P)6�w�cd:\KAO+7\MN8)<WCN<W79<=6.>N79K�79MN8 %1CV6.>ND�7\6�7\8Amf)MN8�V8S6ARND'6.:\8�LBKACNC18;L 7\8;R ��w D9FN:\<WC1X�D�6AC1R 79M�_1D�F1:98;?A8BC�7\8;RU79K[:\8B>=6�i9�V6ALGEU<@C�79KT7\MN8ID�7\6.:979<@CNXKA: %1C16A>nFyK�D9< 7\<WK�C1DBm�)MN8%D�79:G6�7\8BX�wTcdKA:�LB6A>@LB_N>=6�79<@CNX

HFv

P�6�D�R�8;D\L2:\< �y8;R <@C >QMV6.F�7\8B: j m�5C K�:\R�8;:Y79K FN:\8;R�<=L 7J7\MN8rR�< gh_1D9<WK�C L2K�8WX[L2<@8BC�7T79M18^F1:98B- 8Bi�FyKAC18BC�79<=6.>�c 6AL279K�:Y<@C �%�ym

3�m@0A0OI_VD47A�y8%8SD47\<WOJ6�7\8;R`m�)MN<=D�OJ6�w��V8IRNKACN8%P�<W79M1<WCs7\MN8cd:G6.OT8BP�KA:\E[K.cf7\MN8%M16A:9OTKAC1<@L79:G6.C1D9< 7\<WK�C�D�7\6�7\8I7\MN8BK�:9w�m�)M18aCNK�:9OJ6A>$OTK�R�8;D

wj6AC1R

w∗j

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�E��� � � ��.�. � (� .0#�� ��� #T � MN8BCU6ACJ8B>@8;L279:\<@L %18;>@RT<=D 6AFNFN>@<W8SRY79KI6ID�K�>W<=Ra79MN8�LGM16.:\XA8SRIF16A:�7\<@LB>W8SD�P�<@>W>�V8�D�_ � 648;L 7 79KI6cdKA:GL28�79M16.7lP�<W>@>�7\:9wI79K�:98;KA:\<W8;C�7\6�7\8 7\MN8BO 79K%6A>W>@K�P 1K�P K.chL2_N:\:98;C�7 7\MN:\KA_NX�MI7\MN8�OJ6�798;:9<=6.> m�5C OT8B7\6.>=D%6AC1R D�8;OY<=L2K�C1R�_1L279K�:\D'79M18[L2_N:\:98;C�7�<=D%L;6.:\:9<@8;RE��w"8B>@8;L279:\KAC1D�K�:MNKA>@8;D;o �N_N7%<WCD�K�OY8r<@C1D9_N>=6�79<@CNX <@KACN<=LrL2K�OTFVK�_NC1RNDT7\MN8"<@KAC1DJ79MN8;OTD98B>@?A8SDJOJ6�w �y8;LBKAOT8rOTKA:\8^OTK��N<W>@879M16AC[79M18'8;>W8SL 79:\KACVDBm �5CJ79M18'>=6�79798B:)LB6�D�8 8B>@8;L 7\:9<=L LB_N:\:98;C�7 1K�P�D�7\MN:\KA_NX�MJ79MN8'OJ6�798;:9<=6.>=��wR�< gh_1D9<WK�CsK.c�LGM16.:\XA8SR[<@KAC1D;m

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cA, oh79M18B<@:'OTKC�1<W>@< 74w +

MA, 6AC1Rr7\MN8acdKA:GL28a6�L 7\<WCNX KAC�79MN8;O

Page 51: Divakth.diva-portal.org/smash/get/diva2:11732/FULLTEXT01.pdf · "!$#% $&' ()+* ,.-/*102- 0 , -43.* -5* 6 798;:9@?A8B798;C1DE 6AF -G0; IH HJ 79KLGE MNKA>@O PQ8SR8BC E

� ��� � ��������� ������� �������� ����� H j

+d�%�ym 3Nm k�, mfpn827)_1DQCNK�P L2K�C1D9<@R�8;: 7\MN8'L;6AD98 P�MN8;C � <@D�6.CU<WK�CN<@L�D9FV8SL2<@8;D;m$�/cn7\MN8B:\8�<@DK�VKA79M6YXA:G6AR�<@8BC�7�K.c 79MN88;>W8SL 7\:9<=L�FVKA798;C�7\<@6A>q6.CVR`oNK.c 79M18%LGMN8BOT<=LB6.>`FyK.7\8BC�79<=6.>$6.>@KACNXY79M18

x-56�i�<@D;o

79MN8 V_�iUK.c�� <=D)82i�FN:\8;D\D�8SR 6�D U kAk V

JA = −MAcA

(

∂µA

∂x+ qA

∂φ

∂x

)

.+ 3NmW0�kC,

��8;:98qA

<@DI79M18 LGM16A:9X�8JK.c�79M18^� D�Fy8;LB<W8SDBoφ<=D%7\MN8 8;>W8SL 7\:9<=LJFyK.798;C�79<=6.>)6.C1R

µA<=D%7\MN8

LGMN8BOT<=LB6A>�FyK.7\8BC�79<=6.> m !1KA:JR�<@>W_N798"D9KA>@_�79<@KAC1DYK�CN8 OJ6�w R�8;:9<@?A8 79M18^� <WC1D�798;<WC K�:T� 8B:\C1D�7�-�l<@C1D47\8B<@C 8 ��_16.79<@KACnof:\8B>=6�7\<WCNX"79MN8 OYK��N<@>W<W74w 6AC1R 7\MN8 R�<Wgh_1D�<@?�< 74w�o

DA = MARTm")MN<=D

:98;>@6.79<@KAC cdKA>@>WK�P�DYcd:\KAODA = MAcA

∂µA

∂cA

6.C1R ∂µA

∂cA= RT

cA

cdK�:J<@R�8S6.>�D9KA>@<@R D9KA>@_�79<@KAC1D;m� D\D�_1OY<@CNXT7\M16�7'79MN8ILBKAC1LB8BC�79:G6�7\<WK�C X�:\6�R�<@8BC�7�<@DA.B8;:9KVo�79MN8 1_�isK.clLGM16.:\XA8�o

I = JAqAoV<=D

XA<@?A8;C9��w

I =DAq

2AcA

RT

(

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)

.+ 3NmW0 j ,

! <WC16A>W>@wJ7\MN8<WK�CN<=L�LBKAC1RN_1L 7\<W?�<W74wU<=D�R�8 %1C18;Rs6ADσ = I/

(

−∂φ∂x

) oNw�<W8;>@R�<@CNX

σ =DAq

2AcA

RT=D0

Aq2A

RTexp

(

−HM +HF

kBT

)

=

=σ0

Texp

(

− Ha

kBT

)

.+ 3NmW0;H:,

�5C��%�yma3NmW0;HUP�8I_1D98;RDA = cνa2 = ν0a

2 exp(

−HF −TSF

kBT − HM−TSM

kBT

) o`P�MN8B:\8ν =

ν0 exp(

−HM−TSM

kBT

) <=D79M18#64_NOTF cd:98���_18BC1LBwAoν0<@D%79MN8[6.7�7\8BOTF�7%cd:\8 ��_N8;C1L2w�o

a<=D�7\MN8

64_NOTF�R�<=D47G6.C1LB8a6AC1Rc = exp

(

−HF −TSF

kBT

) <=D�7\MN8aLBKAC1LB8BC�79:G6�7\<WK�CsK.c�79M18aR�8Bcd8;L 7�79MV6�7'<=D<WC�?�KA>@?A8;R <@Cs79MN8YR�< gh_1D9<WK�CrK.clD9FV8SL2<@8;D �Im

HF +SF ,�<=D�7\MN8%8BC18B:\XAwE+ 8BC�79:\KAF�w ,)6AD\D9K�LB<@6.798SR

P�< 7\M�79MN8lcdKA:\OT6.79<@KAC�K.c�79M18 L2K�:9:\8;D9FVK�C1R�<@CNX�R�82cd8SL 7$6AC1RHM +

SM ,h<=D`79M18 R�82cd8SL 7qOT<@XA:G6�7\<WK�C�16.:\:\<W8;:�+d8;C�7\:9K�F�w�,2mq! <WCV6.>@>Ww�o�PQ8 RN8 %1CN8

D0A = ν0a

2 exp(

SF +SM

kB

) 6.C1Rσ0 =

D0Aq2

A

R

mf)MN8798;:9OJDQ<WCJ7\MN8'798BOTFy8B:G6�7\_N:98�R�8BFy8BCVR�8BC�7)82i�FVK�CN8BC�7\<@6A>VKAcn�%�ym 3�m@0BHI6A:98 LBKAO �N<WC18;RJ<@C�79K%7\MN879KA7\6.>n6�L 7\<W?�6�7\<WK�C 8;C�79M16A>WF�wJcdK�:�R�< gh_1D9<WK�Cno

Ha = HF +HM m

�:=>Q =>R!8 Q 2,? F�M 8L4G2/6 MO6 C MBJTJT2�R>4GM 8L4G2/6)MN8

HF F16A:\6AOT82798;:l<@CJ�%�ym�3NmW0;H%R�8B:\<@?A8;D�cd:9K�O 7\MN8�82i�FN:\8;D\D9<WK�CacdK�:l7\MN8 L2K�C1L28;C�79:G6�79<@KACTKAcR�82cd8SL 7GD 79M16.7 F16A:�7\<@LB<WFV6�798�<@CI7\MN8�R�< gh_1D9<WK�CaF1:9K�L28SD9D;m � 8�P�<W>@>�82i�FN>@KA:\8 7\MN8)OT8;6.C1<WCNX�K.c179MN<=DF16.:G6.OT8B798B: ��wI_VD�<@CNX�79MN8�LBKAC1RN_1L 7\<W?�<W74wIK.chK�i�w�XA8BCa<WK�C1Df<@CYF1_N:98�6AC1RTR�KAFy8;RYL28;:9<=6�+ >Q8D4

2,

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56 58 60 62 64 66 68Atomic number

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0.2

0.3

0.4

0.5

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V)

a)

La Pr Nd Pm Sm Gd DyCe Tb Ho ErY

Nearest neighbor M

3+

Nearest neighbor M 3+

(La/Sm, Pr/Sm, Nd/Sm or Pr/Gd)

Separated M +3

Separated M 3+

(La/Sm, Pr/Sm, Nd/Sm or Pr/Gd)

Effective average

Effective average(La/Sm, Pr/Sm, Nd/Sm or Pr/Gd)

Experiment (123)

Pr/GdNd/Sm

Pr/SmLa/Sm

56 58 60 62 64 66 680

0.05

0.1

0.15

0.2

0.25M

3+

M3+

mixed

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Nd/SmPr/SmLa/Sm

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Eb (

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