Современные методы исследования и описания кристаллических структур: Методические указания

1 FBGBKL?JKL...

Author: Самойлов А.М.

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1 FBGBKL?JKL?J:PBB
KH<J?F?GGU?F?LH>UBKKE?>H<:GBYBHIBK:GBY DJBKL:EEBQ?KDBOKLJMDLMJ F_lh^bq_kdb_mdZaZgby^eyklm^_glh\dmjkZ^g_\gh]hhl^_e_gby obfbq_kdh]hnZdmevl_lZ

KhklZ\bl_ev dZg^obfgZmd K:FHCEH<:F

<?GB?

2 < kh\j_f_gghc obfbb hkh[_ggh g_hj]Zgbq_kdhc ij_^klZ\e_gby h kljmdlmj_ b ijhkljZgkl\_gghfkljh_gbbj_Zevguo\_s_kl\ b]jZxl\Z`g_crmxjhevGbjZajZ[hldZ ijbgpbibZevguo nmg^Zf_glZevguo ijh[e_f h[s_c nbabq_kdhc b g_hj]Zgbq_kdhc obfbb gb j_r_gb_ [hevrbgkl\Z ijbdeZ^guo obfbq_kdbo aZ^Zq ijZdlbq_kdb g_fukebfu [_a m]em[e_ggh]h ihagZgby \k_o Zki_dlh\ kljh_gby \_s_kl\Z b ij_`^_ \k_]h_]h]_hf_ljbq_kdh]hkljh_gbyijhkljZgkl\_ggh]hjZkiheh`_gbyZlhfh\ Ihke_^gb_^_kylbe_lbyhlf_q_guklj_fbl_evgufijh]j_kkhf\jZa\blbblhqguo bgkljmf_glZevguo f_lh^h\ bamq_gby ijhkljZgkl\_ggh]h kljh_gby obfbq_kdbo kh_^bg_gbc - j_gl]_gh]jZnbq_kdh]h ZgZebaZ g_cljhgh]jZnbb we_dljhgh]jZnbb we_dljhgghc b lmgg_evghc fbdjhkdhibb
3 kh\f_klgh k ZgZebahf hkgh\guo nbabq_kdbo ijbgpbih\ e_`Zsbo \ hkgh\_ bgkljmf_glZevguof_lh^h\ijbf_gy_fuo^eywdki_jbf_glZevgh]hbkke_^h\Zgbywlbo \_s_kl\

<

gZklhys_f

ihkh[bb

ijb\_^_gh

djZldh_

hibkZgb_

hkgh\guo

wdki_jbf_glZevguof_lh^h\bkke_^h\ZgbydjbklZeebq_kdbokljmdlmjZlZd`_^Zxlky ij_^klZ\e_gby h l_o l_hj_lbq_kdbo fh^_eyo kljh_gby djbklZeebq_kdbo l_e dhlhju_ fh`ghihkljhblvgZhkgh\Zgbbihemq_gguohiulguo^Zgguo

=E:<:J?GL=?GH=J:NBQ?KDBC:G:EBA 1.1. F?LH>J?GL=?GH
kl_i_gv

mihjy^hq_gghklb

kljmdlmju

\_ebqbgm

djbklZeeblh\

\

ihebdjbklZeebq_kdhf fZl_jbZe_ fZ]gblgmx kljmdlmjm kh_^bg_gby b fgh]h_ ^jm]h_ H^gbfbaf_lh^h\iha\heyxsbo^hklZlhqghmki_rghj_rZlvih^h[gu_aZ^Zqby\ey_lky j_gl]_gh]jZnbq_kdbcZgZeba J_gl]_gh\kdbfbemqZfbgZau\Zxlkywe_dljhfZ]gblgu_dhe_[Zgbyk^ebghc\heguλ hl-3^hgf>eybamq_gbykljmdlmjudjbklZeeh\ijbf_gyxlj_gl]_gh\kdb_emqbk λ

≈ gfIhdZaZl_evij_ehfe_gbyj_gl]_gh\kdboemq_cf_gvr_ohlybfZehhlebqZ_lky hlg__

q = 1 − 1,3 ×10-5 ρλ2,

( 1.1 )

]^_ρ -iehlghklv\_s_kl\Z\]/kf3, λ -^ebgZ\heguj_gl]_gh\kdh]hbaemq_gby\gf <\b^m [ebahklb qd_^bgbp_nhdmkbjh\Zlvj_gl]_gh\kdb_emqbijbihfhsbdZdboeb[h ebga g_\hafh`gh < j_gl]_gh\kdhc hilbd_ imqdb nhjfbjmxlky qZs_ \k_]h k ihfhsvx^bZnjZ]f
hν = E1 − E2 ,

( 1.2 )

]^_E1 , E2 -wg_j]bywe_dljhgh\khhl\_lkl\_ggh^hbihke_khm^Zj_gbykZgh^hf

2

FZdkbfZevgZyqZklhlZνmax fbgbfZevgZy^ebgZ\hegu λmin khhl\_lkl\m_liheghc hklZgh\d_we_dljhgh\ ( E2 = 0 ).

hνmax = hc/λmin = E1 = eU,

( 1.3 )

]^_U -mkdhjyxs__gZijy`_gb_ Ihkdhevdm E2 fh`_lbf_lvex[mx\_ebqbgmf_gvrmx E1lhg_ij_ju\gucki_dlj kh klhjhgu ^ebgguo \heg h]jZgbq_g ebrv ih]ehs_gb_f fy]dbo emq_c \ fZl_jbZe_ hdhrdZljm[dbb\\ha^mo_ ?keb mkdhj_ggu_ we_dljhgu h[eZ^Zxl wg_j]b_c [hevr_c ihjh]Z \ha[m`^_gby l_ ^hklZlhqghcqlh[u\u[blvbaZlhfZwe_dljhggZoh^ysbckygZ\gmlj_gg_ch[hehqd_gZ nhg_ g_ij_ju\gh]h ki_dljZ \hagbdZ_l ebgby oZjZdl_jbklbq_kdh]h j_gl]_gh\kdh]h baemq_gbyWe_dljhguZlhfZi_j_oh^yk\g_rgboh[hehq_dbf_xsbo[hevrmxwg_j]bx

EgZ\ZdZglgh_f_klh\h\gmlj_gg_ch[hehqd_bkimkdZxlnhlhgu hν = E − E0

( 1.4 )

<_ebqbgZ hij_^_ey_lky oZjZdl_jghc ^ey dZ`^h]h we_f_glZ kbkl_fhc _]h wg_j]_lbq_kdbo mjh\g_c Ebgbb ki_dljZ jZa^_eyxlky gZ k_jbb K, L, M, N ...\aZ\bkbfhklbhllh]hk dZdhch[hehqdb[uem^Ze_gwe_dljhg
α2 ..., β1, β2 ... \ khhl\_lkl\bb k mjh\g_f k dhlhjh]h kh\_jrbeky wg_j]_lbq_kdbc i_j_oh^ GZb[he__ bgl_gkb\gufb y\eyxlky ebgbb Kα1 i_j_oh^ we_dljhgZ LIII - KI ) b [ebadZydg_cihbgl_gkb\ghklbgh[he__keZ[ZyebgbyKα2 i_j_oh^we_dljhgZLII - KI ). Bgl_gkb\ghklv ebgbb Kβ1 i_j_oh^ we_dljhgZ MIII - KI ) khklZ\ey_l - 25 % bgl_gkb\ghklb Kα1 >jm]b_ ebgbb bf_xl _s_ f_gvrmx bgl_gkb\ghklv QZklhlZ dZdhceb[hebgbbbaf_gy_lkyijbi_j_oh^_hlh^gh]hwe_f_glZd^jm]hfmihaZdhgmFhaeb

√ ν = (Z − σ ),

( 1.5 )

]^_Z -aZjy^y^jZcbσ -ihklhyggu_ < j_gl]_ghkljmdlmjghf ZgZeba_ \ dZq_kl\_ fhghojhfZlbq_kdh]h baemq_gby qZs_ \k_]hbkihevamxlkyebgbb Kα1b Kα2 f_lZeeh\hlojhfZ Z ^hfheb[^_gZZ = ^ebgu\hegdhlhjuoe_`Zl\bgl_j\Ze_hl^hgf Wdki_jbf_glZevgu_ b l_hj_lbq_kdb_ hkgh\u j_gl]_ghkljmdlmjgh]h b nZah\h]h ZgZebaZ l\_j^uo l_e [ueb aZeh`_gu \ -191 ]] \ jZ[hlZo F EZmw1 b _]h

3 khljm^gbdh\ AgZy qlh fhev NaCl ] aZgbfZ_l \ l\_j^hf khklhygbb h[t_f jZ\guc kf3 b qlh \ wlhf h[t_f_ kh^_j`blky qbkeh :\h]Z^jh NA= 6,023×1023 +

−

dZ`^h]h ba bhgh\ Na b Cl Z \k_]h ×1024 bhgh\ e_]dh jZkkqblZlv kj_^gb_ agZq_gby +

−

f_`Zlhfguo jZkklhygbc r0 = a/2, ijbgbfZy qlh bhgu Na b Cl ihhq_j_^gh aZgbfZxl maeu dm[bq_kdhc yq_cdb \ djbklZeebq_kdhc kljmdlmj_ NaCl GZ hkgh\Zgbb wlh]h ^hims_gbyihemqZ_lky

r0 =

2 6 ,9 = 2 ,8 × 1 0 − 8 ñì = 0 ,2 8 í ì 24 1 ,2 × 1 0

3

( 1.6 )

LZdbf h[jZahf ihemq_ggZy \_ebqbgZ ^ey r0 khbaf_jbfZ k ^ebghc \hegu oZjZdl_jbklbq_kdh]hj_gl]_gh\kdh]hbaemq_gbyD- k_jbbBkoh^ybamkeh\by^bnjZdpbb

1 ≤

U0 ≤ 10 λ

( 1.7 )

FEZmwij_^iheh`beqlh\khhl\_lkl\bbkaZdhgZfb^bnjZdpbbk\_lZ

1

FZdkL_h^hjN_ebdknhgEZmw - 1960 ) -g_f_pdbcnbabdeZmj_ZlGh[_e_\kdhc

ij_fbb H[eZklb bkke_^h\Zgby l_hjby bgl_jn_j_gpbb b ^bnjZdpbb k\_lZ Hkgh\gh_ ^hklb`_gb_ EZmw - ij_^kdZaZgb_ ^bnjZd-pbhgghc dZjlbgu ijb ijhoh`^_gbb kd\hav djbklZeebq_kdmx j_r_ldm j_gl]_gh\kdbo emq_c Wdki_jbf_gl ih^l\_j^b\rbc ]bihl_am EZmw [ue hkms_kl\e_g < Njb^jbohf b I Dgbiibg]hf \ Fxgo_gkdhf mgb\_jkbl_l_ >bnjZdpbhggu_ dZjlbgu hl Zlhfh\ djbklZeeZ k l_o ihj gZau\Zxl eZmw]jZffZfb

djbklZee\qZklghklboehjb^ZgZljbyfh`_lkem`blv\dZq_kl\_^bnjZdpbhgghcj_r_ldb ^ey j_gl]_gh\kdbo emq_c Wdki_jbf_gl ih^l\_j^b\rbc ]bihl_am EZmw [ue [e_klys_ hkms_kl\e_g<Njb^jbohfbI Dgbiibg]hf \ Fxgo_gkdhf mgb\_jkbl_l_ J_amevlZlu wlh]h hiulZ iha\hebeb jZkdjulv kljmdlmjmdZf_gghckheb MJ:
4 HljZ`_gb_\b^bfuoemq_chla_jdZeZdZdba\_klghfh`_lbf_lvf_klhih^ex[uf m]ehfijbq_fm]heiZ^_gbyjZ\_gm]emhljZ`_gby
>Zggh_ mjZ\g_gb_ \u\_^_gh \ ] Zg]ebckdbfb mq_gufb Mbevyfhf =_gjb

;jw]]hf hlphf b Mbevyfhf Ehj_gphf ;jw]]hf kughf G_aZ\bkbfh hl gbo d ZgZeh]bqghfm mjZ\g_gbx ijbr_e ijhn_kkhj Fhkdh\kdh]h mgb\_jkbl_lZ Xjbc
5

JbkKo_fZihykgyxsZy\u\h^mjZ\g_gby<mevnZ-;jw]]Z

iZ^_gbyjZ\_gm]emhljZ`_gby j_gl]_gh\kdb_emqbijhgbdZxl\]em[v djbklZeeZb©hljZ`_gb_ªijhbkoh^blg_lhevdhhlih\_joghklbghbhlgb`_e_`Zsbo iehkdhkl_c jbk Z [ A^_kv njhgl iZ^Zxsbo emq_c h[hagZq_g HH’ njhgl

6 ©hljZ`_gguoª - HH’’. Ba jbkmgdZ \b^gh qlh dZ`^uc ke_^mxsbc emq gZijbf_j 2, ^he`_gijhclbimlvgZ2x[hevrbcih kjZ\g_gbx k ij_^u^msbf \ gZr_f kemqZ_ emqhf 1. Ijb wlhf hq_\b^gh qlh x = d

sinΘ]^_Θ -m]heiZ^_gbybhljZ`_gbyemq_cGhg_\kydbcm]heα y\ey_lky[jw]]h\kdbf m]ehf Θ^eydhlhjh]h\hafh`ghh[jZah\Zgb_©hljZ`_ggh]hªj_ne_dkZLZddZd\k_emqb f_`^m njhglhf iZ^_gby HH’ b njhglhf hljZ`_gby ijhoh^yl jZagu_ imlb b m njhglZ hljZ`_gby hgb ^he`gu bgl_jn_jbjh\Zlv lh h[jZah\Zlvky hljZ`_gguc j_ne_dk fh`_l lhevdh\lhfkemqZ__kebj_amevlbjmxsZyZfieblm^Z\k_oemq_c[m^_lhlebqgZhlgmeyZ ijZdlbq_kdb - ^hklZlhqgh agZqbl_evgZ qlh[u [ulv aZj_]bkljbjh\Zgghc Ijb hljZ`_gbb ijbfblb\ghc j_r_ldhc h[gZjm`b\Z_lky qlh ^ey wlh]h 2x ^he`gh [ulv jZ\gh p_ehfm qbkem^ebg\heg, beb nλ. Lh]^Z

n λ = 2 dhkl sinΘ

( 1.8 )

LZdbfh[jZahfihemqZ_lkymjZ\g_gb_<mevnZ-;jw]]Z DZd\b^ghgZjbkd, ijbwlhf\k_emqbkh\iZ^ZxlihnZa_nZah\ucm]hejZ\_gb kha^Zxlky hilbfZevgu_ mkeh\by ^ey ©hljZ`_gbyª ?keb m\_ebqblv m]he iZ^_gby lZd qlh[u 2x klZeh > λ gZijbf_j λ. Ba jbk k hq_\b^gh qlh \hagbdZxl emqb \ ijhlb\hiheh`guonZaZobj_amevlbjmxsZyZfieblm^ZdZ`^uoq_luj_oemq_cjZ\gZgmex Bf__lf_klhih]ZkZgb_j_ne_dkZ Fgh`bl_ev n gZau\Zxl ihjy^dhf ki_dljZ LZd dZd sinΘ = nλ/2dhkl lh ijb nbdkbjh\ZgguoagZq_gbyo λ b dhkl \aZ\bkbfhklbhlagZq_gbcn kbgmkm]eZ^bnjZdpbb

sinΘ ijbgbfZ_l jZagu_ agZq_gby ijhihjpbhgZevgu_ wlbf qbkeZf ?keb ^ey n = 1 sinΘ = 0,300 ( Θ = 17h 27’ lh^eyn = 2 sinΘ = 0,600 ( Θ = 36h 52’ ^eyn = 3 sinΘ = 0,900 ( Θ = 76h 10’ >ey n = 4 sinΘ > j_ne_dk g_\hafh`_g >Z`_ _keb ijb n = 1 sinΘ hq_gv fZe dZd \ ijb\_^_gghfijbf_j_ijZdlbq_kdb\hagbdZxlki_dljuebrvi_j\uo lj_oihjy^dh\?keb`_sinΘ i_j\h]hj_ne_dkZ[hevr_l_Θ > 76h 10’ ), lhhljZ`_gby^Z`_\lhjh]hihjy^dZhl^Zggh]hdhkl g_\hafh`gusinΘ > 1 ). Ijb bamq_gbb ki_pbZebabjh\Zgguo fhgh]jZnbc beb hjb]bgZevguo klZl_c \ i_jbh^bq_kdhc i_qZlb ihk\ys_gguo ijh[e_fZf bkke_^h\Zgby djbklZeebq_kdhc kljmdlmjuf_lh^hfj_gl]_gh\kdh]hZgZebaZfh`_l

7 \hagbdgmlv \iheg_ mf_klguc \hijhk ihq_fm ijb jZkq_lZo \hafh`guo j_ne_dkh\ ih mjZ\g_gbx ijbgbfZ_lkyihjy^hdki_dljZ n b\dZdhcf_j_baf_gylky\u\h^u ijb n b l ^ G_h[oh^bfh hl\_lblv qlh \u\h^u g_ baf_gylky Ijb aZibkb mjZ\g_gby \\b^_

λ 2 sin Θ

d hkl = n

( 1.9 )

,

]^_n - ihjy^hdki_dljZp_eh_qbkehjZ\gh_bl^ Ihkdhevdm

d = d n hkl

nhnknl

= d

( 1.10 )

h * k * l *

gZijbf_j

d 2

111

= d

222

;

d 3

111

= d

333

Ih^klZ\eyy \ fh`ghihemqblv

dh *

k * l*

=

λ 2 sin Θ

( 1.11 )

>jm]bfb keh\Zfb hl iehkdhklb ( h*k*l*) \hagbdZ_l j_ne_dk ih^qbgyxsbcky mjZ\g_gbx<mevnZ-;jw]]Zb\ua\Zggucki_dljZfb-]h-]hbl^ihjy^dZ 1.3. MKLJHCKLEYJ?GL=?GH=J:NBQ?KDBO BKKE?>H<:GBC IjbgpbibZevgZyko_fZijb[hjZ^eyijh\_^_gbyj_gl]_gh]jZnbq_kdbobkke_^h\Zgbc djbklZeebq_kdhckljmdlmjuijb\_^_gZgZjbkHkgh\guf

8

JbkIjbgpbibZevgZyko_fZmkljhckl\ZZiiZjZlZ^eyijh\_^_gby j_gl]_gh\kdbobkke_^h\Zgbc 1 -j_gl]_gh\kdZyljm[dZ-bklhqgbd\g_rg_]hgZijy`_gby 3 -j_gl]_gh\kdh_baemq_gb_jZkk_yggh_bkke_^m_fufh[jZaphf 4 -bkke_^m_fuch[jZa_p-nbevlj-^bZnjZ]fZ

we_f_glhf ex[h]h j_gl]_gh\kdh]h ZiiZjZlZ y\ey_lky bklhqgbd fhghojhfZlbq_kdh]h baemq_gby j_gl]_gh\kdZy ljm[dZ 1 gZ dhlhjmx ih^Zxlky \g_rg__ we_dljbq_kdh_ gZijy`_gb_ 2 Kh\j_f_ggu_ j_gl]_gh\kdb_ ZiiZjZlu ^ey kljmdlmjgh]h ZgZebaZ

9 jZkkqblZgu gZ gZijy`_gb_ ^h d< b ^Zxl klZ[bevghklv baemq_gby - 0,3 %. Ih^Z\eyxsZy qZklv wg_j]bb iZ^Zxs_]h we_dljhggh]h imqdZ ljZlblky gZ gZ]j_\Zgb_ Zgh^Z \ lh \j_fy dZd \ j_gl]_gh\kdh_ baemq_gb_ ij_h[jZam_lky f_g__ wg_j]bb
I = I0 exp (− µ l ) ,

( 1.12 )

]^_ µ -kmffZjgucdhwnnbpb_glhkeZ[e_gby Dhwnnbpb_gl ih]ehs_gby ijhihjpbhgZe_g ijb[ebabl_evgh λ Z lZd`_ Z . 3

3

Ijbf_gyynbevlju 5 ( ðèñ. 2 )bawe_f_glZkih^oh^ysbf Zbihevamykv_]hk_e_dlb\guf ih]ehs_gb_f\[ebabdjZyD -k_jbbfh`ghgZj_gl]_gh]jZffZomf_gvrblvdhjhldh\hegh\mxkhklZ\eyxsmxnhgZbbkdexqblvβ -ebgbb

10 Baf_j_gb_bgl_gkb\ghklbdZjlbgj_amevlZlh\jZkk_b\Zgbyj_gl]_gh\kdboemq_c 3 ( jbk hljZ`_gguo hl bkke_^m_fh]h h[jZapZ 4 ijb\h^blky eb[h nhlh]jZnbq_kdbf f_lh^hfeb[hkihfhsvxkq_lqbdh\nhlhgh\Ijbnhlh]jZnbq_kdhcj_]bkljZpbbie_gdZ fh`_l h^gh\j_f_ggh beb ih hq_j_^b nbdkbjh\Zlv j_gl]_gh\kdb_ imqdb jZkk_yggu_ h[jZaphf ih^ jZagufb m]eZfb l_ hgZ y\ey_lky ^\mf_jguf ^_l_dlhjhf >_ckl\b_ j_gl]_gh\kdbo emq_c gZ nhlhie_gdm lZdh_ `_ dZd b \b^bfh]h k\_lZ b aZdexqZ_lky \ nhlhobfbq_kdhfjZaeh`_gbb[jhfb^Zk_j_[jZIhq_jg_gb_Dhij_^_ey_lkydZdeh]Zjbnf hlghr_gby bgl_gkb\ghkl_c imqdZ k\_lZ ijhr_^r_]h q_j_a ^Zgguc mqZklhd ie_gdb I b imqdZijhr_^r_]hq_j_ag_h[emq_ggucmqZklhdie_gdbI0:

D = lg ( I / I0 )

( 1.13 )

Ïî÷åðíåíèå ïëåíêè â èíòåðâàëå îò 0,3 äî 1,2 ïðîïîðöèîíàëüíî qbkem j_gl]_gh\kdbo nhlhgh\ miZ\rbo gZ _^bgbpm iehsZ^b >ey lhqgh]h hij_^_e_gby ihq_jg_gbc kem`Zl hilbq_kdb_ fbdjhnhlhf_lju beb fbdjh^_gkblhf_lju k hlkq_lhf ihq_jg_gby ih rdZe_ beb aZibkb ijhnbey ibdh\ \ \b^_ djb\hc b Z\lhfZlbq_kdb_ fbdjh^_gkblhf_lju kh_^bg_ggu_ g_ihkj_^kl\_gghkWjm]hc hkgh\ghc f_lh^ j_]bkljZpbb j_gl]_gh\kdbo emq_c - ^bnjZdlhf_ljbq_kdbc hkgh\Zgguc gZ bkihevah\Zgbb jZaebqgh]h jh^Z kq_lqbdh\ j_gl]_gh\kdbo d\Zglh\ bhgbaZpbhgguo kpbglbeeypbhgguo ihemijh\h^gbdh\uo b ^j Kq_lqbd dZd ijZ\beh fh`_l j_]bkljbjh\Zlv madbc imqhd emq_c \ dhg_qghf fZehf m]eh\hf bgl_j\Ze_ l_ y\ey_lkylhq_qguf^_l_dlhjhf GZb[he__

mihlj_[bl_evgu

]ZahjZajy^gu_

ijhihjpbhgZevgu_

kq_lqbdb

b

kpbglbeeypbhggu_kq_lqbdbdhlhju_ij_^klZ\eyxlkh[hckhq_lZgb_

exfbg_kp_glgh]h djbklZeeZ k\_lys_]hky ih^ ^_ckl\b_f j_gl]_gh\kdbo d\Zglh\ b nhlhwe_dljhggh]h mfgh`bl_ey Ijbf_gyxlky lZd`_ ihemijh\h^gbdh\u_ ^_l_dlhju ba ]_jfZgbybdj_fgby Wnn_dlb\ghklv lh]h beb bgh]h ^_l_dlhjZ hij_^_ey_lky hlghr_gb_f qbkeZ aZj_]bkljbjh\Zgguo nhlhgh\ d qbkem nhlhgh\ ihiZ\rbo gZ \oh^gh_ hdgh ^_l_dlhjZ b khklZ\ey_lhl^h G:IJ:<E?GBYJ?GL=?GH=J:NBQ?KDH=H:G:EBA:

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wdki_jbf_glZevgufiml_fgZdhie_guk\_^_gbyhkljh_gbb^_kyldh\lukyqjZaghh[jZaguo \_s_kl\ b wlZ [ZaZ ^Zgguo [ukljh ihihegy_lky H^gZdh ij_`^_ \k_]h g_h[oh^bfh ihfgblvhlhfqlhdhgdj_lgh_hibkZgb_dZdhc-eb[h kljmdlmju fh`gh ^Zlv ebrv \ nhjf_ fh^_eb [he__ beb f_g__ hljZ`Zxs_c gZrb ij_^klZ\e_gby h[ h[t_dl_ bkke_^h\Zgby D kh`Ze_gbx wlZ g_ba[_`gh ijb[eb`_ggZy fh^_ev m`_ aZeh`_gZ \ ljZdlh\dm wdki_jbf_glZ b qlh [he__ \Z`gh - \ khagZgb_ bkke_^h\Zl_eyijbq_fgZklhevdh]em[hdhqlhbgh]^Z\hagbdZ_lhiZkghklvg_hijZ\^Zgghc Z[khexlbaZpbb kljmdlmjguo ^Zgguo KijZ\_^eb\h iheZ]Zy mklZgh\e_ggmx kljmdlmjm wdki_jbf_glZevgufnZdlhf l_f g_ f_g__ ke_^m_l ihfgblv qlh wlhl nZdl ihemq_g gZ hkgh\_ hij_^_e_gguo l_hj_lbq_kdboij_^klZ\e_gbcKh\j_f_g_fwlbij_^klZ\e_gbyf_gyxlkyke_^h\Zl_evgh ^he`gu baf_gylvky b wdki_jbf_glZevgu_ nZdlu Ihwlhfm b k ijbgpbibZevghc b k ijZdlbq_kdhclhq_daj_gby\Z`gh]em[hdhihgbfZlv kmsghklv ijb[eb`_gbc dhlhju_ e_`Zl \ hkgh\_ bkihevam_fhc l_hj_lbq_kdhc fh^_eb Ijb\_^_f djZldbc ZgZeblbq_kdbc h[ahj hkgh\guo fh^_e_c kljmdlmj bkihevam_fuo \ djbklZeehobfbb 2.2. KL:LBKLBQ?KDB?FH>?EBKLJMDLMJ K ihfhsvx kh\j_f_gguo wdki_jbf_glZevguo f_lh^bd j_gl]_gh-kljmdlmjgh]h ZgZebaZ fh`gh h^gh\j_f_ggh ihemqZlv k\_^_gby h kj_^g_f \h \j_f_gb jZkiheh`_gbb Zlhfh\ \ h[t_f_ djbklZeebq_kdh]h \_s_kl\Z Z lZd`_ h ^bgZfbd_ bo l_ieh\uo dhe_[ZgbcH^gZdh\[hevrbgkl\_jZ[hl ^bgZfbq_kdb_ oZjZdl_jbklbdb l_ieh\uo dhe_[Zgbc Zlhfh\ g_ h[km`^Zxlky NZdlbq_kdb wlhhagZqZ_lqlhjZkkfZljb\Z_fu_fh^_ebhlghkylkydjZajy^m klZlbq_kdbo < klZlbq_kdhc fh^_eb djbklZeebq_kdhc kljmdlmju bf__lky g_kdhevdh mjh\g_clZ[ebpZ LZ[ebpZKlZlbklbq_kdZyfh^_evdjbklZeebq_dhckljmdlmju Lbifh^_eb

r - fh^_ev

I_j\bqgZyljZdlh\dZ ---

HibkZgb_

{ R } + { rj } + N

23 lhq_qgZy

j = 1,2 ... N/k

r’-fh^_ev lhq_qgh-rljboh\Zy ]jZn

\Ze_glgZyk\yav

rZjh\Zy

djbklZeehobfbq_kdb_

rZjh\u_ miZdh\db b

jZ^bmku

deZ^db

+ { ij }

r, ρ-fh^_ev ---

we_dljhggZyiehlghklv

r’, δρ - fh^_ev ^_nhjfZpbhggZy iehlghklv

+ρ(r)

\Ze_glgu_bg_ih^_e_gwe_dljhggZy gu_ we_dljhggu_ iZju i_j_ghk aZjy^Z iheyjb-aZpby

+ δρ ( r )

Zlhfh\bbhgh\

R } -j_i_jj_r_ldb jZ^bmk-\_dlhju rj dhhj^bgZlu Zlhfh\1 bijhkljZgkl\_ggmx n_^hjh\kdmx ]jmiimkbff_ljbb N, dhlhjZyiha\hey_lkqblZlvg_aZ\bkbfufblhevdh

N/k Zlhfh\, ãäå N - qbkehZlhfh\\

1

Iheh`_gb_ dhhj^bgZlu ex[h]h maeZ \ ijhkljZgkl\_gghc j_r_ld_ hij_^_eyxlky

jZ^bmkhf-\_dlhjhfrjijh\_^_gguf bagZqZeZdhhj^bgZlijbq_frj = ma + nb + pc, ]^_

a, b, c

-

kbkl_fZ [Zabkguo \_dlhjh\ ih dhhj^bgZlguf hkyf OX, OY b OZ

khhl\_lkl\_ggh m, n, p -ljbqbkeZgZau\Z_fuobg^_dkZfbmaeZ

1

we_f_glZjghcyq_cd_ k - nZdlhjkbff_ljbb . < lZdhf \b^_ h[uqgh [u\Zxl ij_^klZ\e_gu hkgh\gu_ wdki_jbf_glZevgu_ j_amevlZlu j_gl]_ghkljmdlmjgh]h ZgZebaZ Ih wlbf ^Zgguf fh`gh hij_^_eylv agZq_gby f_`iehkdhklguo b f_`Zlhfguo jZkklhygbc NZdlbq_kdb lZdZyfh^_evhdZau\Z_lkyfZeh bgnhjfZlb\ghc < kemqZ_ [he__ keh`ghc kljmdlmju hgZ ij_^klZ_l dZd oZhlbq_kdh_ gZ]jhfh`^_gb_ lhq_d ba dhlhjh]h ^hklZlhqgh ljm^gh ba\e_qv ihe_agmx bgnhjfZpbx >ey jZkrbnjh\db lZdbo kljmdlmj g_h[oh^bfh ijbf_gylv hkgh\gu_ iheh`_gby mq_gby h obfbq_kdhck\yabqlhijb\h^bldihy\e_gbxr’ -fh^_eb

24 Ijb gZebqbb gZijZ\e_gguo dh\Ze_glguo k\ya_c i_j\bqgZy ljZdlh\dZ khklhbl \ lhf qlh Zlhfu kh_^bgyxl \Ze_glgufb rljboZfb \ ^mo_ deZkkbq_kdhc l_hjbb kljh_gby LZd ihemqZ_lky lhq_qgh-rljboh\Zy fh^_ev b __ hibkZgb_ ^hihegy_lky fZkkb\hf dhhj^bgZljZkiheh`_gbydh\Ze_glguok\ya_c<_ebqbguf_`ZlhfguojZkklhygbch[uqgh mdZau\Zxl ijZ\bevguc kihkh[ ihkljh_gby \Ze_glguo rljboh\ dZd ijZ\beh jZkklhygb_:-< khhl\_lkl\mxs__dh\Ze_glghck\yabihdjZcg_cf_j_\jZaZdhjhq_q_fdjZlqZcr__ jZkklhygb_ f_`^m \Ze_glgh g_k\yaZggufb ZlhfZfb : b < b wlhl kihkh[ m^h\e_l\hjbl_evghkh]eZkm_lkykdeZkkbq_kdhcl_hjb_cobfbq_kdhck\yab M`_kkZfh]haZjh`^_gbygZqZeh-o]h^h\OO\_dZ j_gl]_ghkljmdlmjgucZgZeba \ dZq_kl\_ hkgh\gh]h j_amevlZlZ ^Z\Ze bgnhjfZpbx h jZkij_^_e_gbb we_dljhgghc iehlghklb ρ(r) ih yq_cd_ djbklZeeZ H^gZdh gbadZy lhqghklv baf_j_gbc hlkmlkl\b_ l_hj_lbq_kdbo jZajZ[hlhd dhlhju_ iha\heyxl mq_klv kbkl_fZlbq_kdb_ hrb[db ijb\h^beb d agZqbl_evghfm h]jm[e_gbx wlhc nmgdpbb qlh ^hklh\_jgufb hdZau\Zebkv ebrvdhhj^bgZlufZdkbfmfh\we_dljhgghciehlghklbdhlhju_ijbgbfZebkv bijbgbfZxlkyihgug_ aZdhhj^bgZluZlhfh\H[uqghdhhj^bgZluwlbofZdkbfmfh\ aZbkdexq_gb_fZlhfh\\h^hjh^Z fZehq_fhlebqZxlkyhlkj_^gbo\h\j_f_gbdhhj^bgZl y^_jZlhfh\^Zggu_hdhlhjuoihemqZxlijb 1

Kh\hdmighklv\k_owe_f_glh\kbff_ljbb^ZgghcdjbklZeebq_kdhcj_r_ldb

ihfhsbg_cljhgh]jZnbb JZa\blb_ j_gl]_ghkljmdlmjgh]h ZgZebaZ k^_eZeh [he__ lhqgufb ihemqZ_fu_ ba ^bnjZdpbhgguo ^Zgguo dZjlbgu jZkij_^_e_gby we_dljhgghc iehlghklb ρ(r). < blh]_ klZeZ^hklmighc r,ρ -fh^_evmqblu\ZxsZyZ^_d\Zlgu_j_Zevghklbhkh[_gghklbnmgdpbb

ρ(r). >ey lh]h qlh[u ij_^klZ\blv bgnhjfZpbx \ [he__ y\ghf \b^_ bkihevamxl ihkljh_gb_ lZd gZau\Z_fhc ©^_nhjfZpbhgghcª we_dljhgghc iehlghklb δρ(r), dhlhjZy ij_^klZ\ey_lkh[hcjZaghklv

δρ(r) = ρ(r) - ρ0(r) ,

( 2.1 )

]^_ρ0 -we_dljhggZyiehlghklv\Ze_glghg_k\yaZgguoZlhfh\

25 LZdbf h[jZahf nmgdpby δρ(r) ihdZau\Z_l dZdb_ baf_g_gby ijhbkoh^yl \ we_dljhgghc iehlghklbZlhfh\\j_amevlZl_h[jZah\Zgbyobfbq_kdbok\ya_c jbk >ZggZynmgdpbyiha\hey_lmklZgh\blv\Ze_glgu_bg_ih^_e_ggu_we_dljhggu_ iZju i_j_ghk aZjy^Z iheyjbaZpbx Zlhfh\ b bhgh\ qlh iha\hey_l km^blv h ki_pbnbq_kdbo hkh[_gghklyo dZd kljmdlmj k dh\Ze_glgufb k\yayfb lZd b kljmdlmj \ dhlhjuo gZijZ\e_ggh_ f_`Zlhfgh_ \aZbfh^_ckl\b_ hlkmlkl\m_l < khq_lZgbb k r’ fh^_evx©^_nhjfZpbhggZyªwe_dljhggZy iehlghklv ^Z_l r’, δρ - fh^_ev dhlhjZy gZ k_]h^gyrgbc ^_gv ih \k_c \b^bfhklb ij_^klZ\ey_l kh[hc gZb\ukrbc ^hklmiguc mjh\_gv klZlbq_kdh]h \ZjbZglZ bah[jZ`_gby djbklZeebq_kdhckljmdlmju 2.3. >BG:FBQ?KDB?FH>?EBKLJMDLMJ J_amevlZlu

j_gl]_ghkljmdlmjguo

bkke_^h\Zgbc

gZjy^m

k

ihabpbhggufb

iZjZf_ljZfb kh^_j`Zl b ^bgZfbq_kdb_ oZjZdl_jbklbdb Mq_l wlbo oZjZdl_jbklbd ijb\h^bldihy\e_gbx^bgZfbq_kdbofh^_e_ckljmdlmj
26

Jbk JZkij_^_e_gb_ ^_nhjfZpbhgghc we_dljhgghc iehlghklb \ f_`fhe_dmeyjghc \h^hjh^ghc k\yab ijbkmlkl\mxs_c \ djbklZeeZo α-]ebpbgZ NH2-CH2-COOH IhdZaZgZ qZklvk_q_gbyijhoh^ys_]hq_j_aebgbx\h^hjh^ghck\yab

lhq_dZ\\b^_weebikhb^h\bogZau\Zxl©l_ieh\ufbweebikhb^Zfbª jbk IZjZf_lju Uj

kl

ij_^klZ\eyxsb_ g_aZ\bkbfuo dhfihg_glh\ kbff_ljbqgh]h l_gahjZ

27 \lhjh]hjZg]ZoZjZdl_jbamxlnhjfmbhjb_glZpbxwlboweebikhb^h\bbf_xlke_^mxsbc nbabq_kdbckfukekg_dhlhjhcnbdkbjh\Zgghc\_jhylghklvxj\ex[hcfhf_gl\j_f_gb Zlhf gZoh^blky \gmljb beb gZ ih\_joghklb lZdh]h weebikhb^Z Hq_\b^gh qlh gZijZ\e_gb_gZb[hevr_c\ulygmlhklbweebikhb^Zkhhl\_lkl\m_lgZijZ\e_gbx\dhlhjhf Zlhf kh\_jrZ_l fZdkbfZevgu_ ih Zfieblm^_ dhe_[Zgby < [hevrbgkl\_ kemqZ_\ weebikhb^ughjfbjh\ZgugZ\_jhylghklv j = Ijbnbdkbjh\ZgghfagZq_gbb jjZaf_j weebikhb^ZdZd\b^ghgZjbkaZ\bkblhll_fi_jZlmju Mlhqg_gb_ kljmdlmju \ mkeh\byo Zg]Zjfhgbq_kdh]h ijb[eb`_gby ijb\h^bl d kms_kl\_gghfmmkeh`g_gbxfZl_fZlbq_kdh]hZiiZjZlZ-ijboh^blkyijb\e_dZlvl_gahju lj_lv_]hbq_l\_jlh]hjZg]h\LZdb_jZkq_lufh`ghhkms_kl\eylvlhevdh^eyijhkl_crbo kljmdlmj\dhlhjuoZlhfuaZgbfZxlqZklgu_ihabpbb GZ mjh\g_ ]Zjfhgbq_kdh]h ijb[eb`_gby fh^_ev fh`gh h[h]Zlblv ^h[Z\b\ d g_c dZjlmjZkij_^_e_gbywe_dljhgghciehlghklbqlhijb\_^_ld

r, U, ρ - fh^_eb EYK:FHDHGLJHEYD=E:
28

JbkWeebikhb^ul_ieh\uodhe_[ZgbcZlhfh\ Z - h[sbc \b^ b hjb_glZpby weebikhb^Z [ - l_ieh\u_ weebikhb^u \ kljmdlmj_ µZp_lbe_gZ-[bk-pbdehi_g^b_gbeZgbd_eyijbD\-lh`_ijbD

29 Hkgh\gZyebl_jZlmjZ ;hdbc=;DjbklZeehobfby-FGZmdZ RZkdhevkdZyFIDjbklZeeh]jZnby-F
>hihegbl_evgZyebl_jZlmjZ Kh\j_f_ggZydjbklZeeh]jZnby/ Ih^j_^;D
KhklZ\bl_evKZfhceh\:e_dkZg^jFboZceh\bq J_^Zdlhj;mgbgZL>

Современные методы исследования и описания кристаллических структур: Методические указания

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