Ответы на вопросы по кристаллохимии для теста П. М. Зоркого

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Hij_^_e_gb_hibkZgb_bkbkl_fZlbaZpbydjbklZeebq_kdbokljmdlmjdZd?)
Bgl_jij_lZpbydjbklZeebq_kdbokljmdlmjihq_fm?)
Ij_^kdZaZgbydhjj_eypbbkljmdlmjZ±k\hckl\Zgmbqlh?)
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djbklZeeZhlebqZ_lkyhlb^_Zevghc?
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QlhlZdh_we_f_glZjgZyyq_cdZ?
WlhiZjZee_e_ibi_^ih\lhjy_fhklbihkljh_ggucgZdjbklZeeh]jZnbq_kdbohkyodhhj^bgZl
ijbq_f _]h j_[jZ ± djZlqZcrb_ ljZgkeypbb \^hev khhl\_lkl\mxsbf hkyf We_f_glZjgu_
yq_cdbh^bgZdh\hhjb_glbjh\Zggubp_ebdhfaZihegyxlijhkljZgkl\h
Q_f jZaebqZxlky ihgylby djbklZeebq_kdZy kljmdlmjZ b djbklZeebq_kdZy
j_r_ldZ?
DjbklZeebq_kdZyj_r_ldZ±kh\hdmighklvljZgkeypbcbf_xsboky\b^_ZevghcdjbklZeebq_kdhckljmdlmj_
DjbklZeebq_kdZy kljmdlmjZ ± dhgdj_lgh_ jZkiheh`_gb_ Zlhfh\ \ djbklZeebq_kdhf ijhkljZgkl\_\_s_kl\Z
Ijb\_^bl_ijbf_juZlhfghfhe_dmeyjguokbkl_fkh^ghf_jghcj_r_ldhc
P_ib\kljmdlmjZoPdCl2, BeCl2k_jh]hSe, hdj_aheZ
Ijb\_^bl_ijbf_juZlhfghfhe_dmeyjguokbkl_fk^\mf_jghcj_r_ldhc
Kehb\kljmdlmjZo]jZnblZBN, CuCl2; iZjd_lguckehc\djbklZeeZogZnlZebgZ
Ijb\_^bl_ijbf_jubkihevah\Zgbyihgylbylj_of_jghcj_r_ldb ^ey hibkZgbyiheghklvxmihjy^hq_gguobqZklbqghmihjy^hq_gguodjbklZeebq_kdbokljmdlmj
Mihjy^hq_ggu_kljmdlmju: NaCl, CaF2, CuAu (bgl_jf_lZeeb^
QZklbqghmihjy^hq_ggu_kljmdlmju: β-F2bγ-O2jhlZpbhgghdjbklZeebq_kdh_khklhygb_
CuAukieZ\±klZlbklbq_kdh_jZkiheh`_gb_jZaebqguoZlhfh\
MdZ`bl_hkgh\gu_kihkh[ufh^_ebjh\ZgbydjbklZeebq_kdhckljmdlmju
1) Rfh^_ev±ZlhfuijbgbfZxlkyaZlhqdbkj_^gb_ih\j_f_gbiheh`_gbyy^_j^eyhibkZgby bkihevamxl iZjZf_lju we_f_glZjghc yq_cdb dhhj^bgZlu Zlhfh\ ijhkljZgkl\_ggmx
]jmiimkbff_ljbbbqbkehnhjfmevguo_^bgbpgZyq_cdm
2) R¶fh^_ev ± i_j\bqgZy bgl_jij_lZpby djbklZeebq_kdhc kljmdlmju iheh`_gby obfbq_kdbok\ya_cbebhibkZgb_\l_jfbgZoIRMIRD>eyhibkZgbybkihevam_lky]jZn
3) RUfh^_ev ± Zlhfu ijbgbfZxlky aZ l_ieh\u_ weebikhb^u >ey hibkZgby bkihevam_lky
l_gahjdhhj^bgZlZlhfh\bl_ieh\uoiZjZf_ljh\
1
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5) Rδρfh^_ev ± jZkkfZljb\Z_lky dhglbgmZevgh_ jZkij_^_e_gb_ ^_nhjfZpbhgghc we_dljhgghc iehlghklb iheh`_gby fZdkbfmfh\ mdZau\Zxl gZ obfbq_kdb_ k\yab beb iheyjbaZpbx
we_dljhgguoh[hehq_dZlhfh\
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< ]Zjfhgbq_kdhf ijb[eb`_gbb bf__l ke_^mxsbc kfuke \ ex[hc fhf_gl \j_f_gb y^jh
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DZdhckfukebf__lnmgdpbyijhkljZgkl\_ggh]h jZkij_^_e_gbywe_dljhgghc
iehlghklb"Q_fmkhhl\_lkl\mxlfZdkbfmfuwlhcnmgdpbb"
Wlhiehlghklv\_jhylghklbgZoh`^_gbywe_dljhgZ\^Zgghclhqd_ijhkljZgkl\ZFZdkbfmfu
ijb[ebabl_evghkhhl\_lkl\mxlkj_^g_fmih\j_f_gbiheh`_gbxZlhfguoy^_jFh`_l[ulv
hij_^_e_gZkihfhsvxklZg^Zjlgh]hJK:
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WlhjZaghklv f_`^m j_Zevghc we_dljhgghc iehlghklvx b we_dljhgghc iehlghklvx jZkkqblZgghc\ij_^iheh`_gbbhlhfqlh\k_Zlhfubf_xlkn_jbq_kdmxnhjfmFZdkbfmfumdZau\ZxlgZiheh`_gbyg_ih^_e_gguowe_dljhgguoiZjbeb^_nhjfZpbxwe_dljhgguoh[hehq_dlh_klvgZiheh`_gbyobfbq_kdbok\ya_c
Q_f hij_^_ey_lky kms_kl\h\Zgb_ ^Zgghc djbklZeebq_kdhc kljmdlmju ijb
^Zgguomkeh\byob^Zgghfkihkh[_ihemq_gbydjbklZeeh\"
Fbgbfmfhfk\h[h^ghcwg_j]bbF = U + Edhe – TS (U±ihl_gpbZevgZywg_j]byj_r_ldbE b
S±wg_j]bybwgljhibydhe_[ZgbcZlhfh\F – fgh]hf_jgZynmgdpbygZebqb_h^bgZdh\uo
ehdZevguofbgbfmfh\oZjZdl_jbam_lh[jZah\Zgb_ihebfhjnguofh^bnbdZpbc]eh[Zevguc
fbgbfmfh[uqghkhhl\_lkl\m_lgZb[he__mklhcqb\ZyZeehljhigZyfh^bnbdZpby
QlhlZdh_bahkljmdlmjghklvdjbklZeebq_kdbo\_s_kl\"
WlhgZebqb_h^bgZdh\uodjbklZeebq_kdbokljmdlmjklhqghklvx^h]_hf_ljbq_kdh]hih^h[bymjZaguoobfbq_kdbokh_^bg_gbc
Ijb\_^bl_ijbf_ju[bgZjguoobfbq_kdbokh_^bg_gbcdjbklZeebamxsboky
kh[jZah\Zgb_fbahkljmdlmjguo\_s_kl\
NaCl – LiCl – MgO; WO3 – ReO3; CsCl – CsBr.
Ijb\_^bl_ijbf_jubahkljmdlmjguodjbklZeeh\l_jgZjguoobfbq_kdbokh_^bg_gbc
CaTiO3 – NaWO3 – PbZrO3; KMnO4 – BaSO4; MgAl2O4 – ZnNi2O4.
QlhlZdh_kh\_jr_ggucbahfhjnbaf"Ijb\_^bl_ijbf_ju
Bahfhjnbaf ± h[jZah\Zgb_ l\_j^uo jZkl\hjh\ aZf_s_gby ± kh\_jr_g_g _keb bahfhjngu_
\_s_kl\Z h[jZamxl g_ij_ju\guc jy^ l\_j^uo jZkl\hjh\ >ey kh\_jr_ggh]h bahfhjnbafZ
g_h[oh^bfZbahkljmdlmjghklvkh_^bg_gbcGZijbf_jKAl(SO4)2·12H2O – KCr(SO4)2·12H2O –
NH4Al(SO4)2·12H2O; NaCl – KClijb\ukhdbol_fi_jZlmjZo
2
QlhlZdh_g_kh\_jr_ggucbahfhjnbaf"Ijb\_^bl_ijbf_ju
Bahfhjnbaf±h[jZah\Zgb_l\_j^uojZkl\hjh\aZf_s_gby±g_kh\_jr_g_g_kebbahfhjngu_
\_s_kl\Zh[jZamxll\_j^u_jZkl\hju\hij_^_e_gguoij_^_eZoGZijbf_jBaSO4 – KMnO4;
ZnSknZe_jbl±FeS.
Ijb\_^bl_ijbf_jubah\Ze_glgh]hbahfhjnbafZ.
NaCl – KClijb\ukhdbol_fi_jZlmjZoKH2PO4 – KH2AsO4; ZnSO4·7H2O – FeSO4·7H2O.
Ijb\_^bl_ijbf_ju]_l_jh\Ze_glgh]hbahfhjnbafZ.
FeCO3 – ScBO3; CaTiO3 – KMgF3; BaSO4 – KMnO4.
QlhlZdh_l\_j^u_jZkl\hju\g_^j_gby"Ijb\_^bl_ijbf_ju
WlhfZljbpZbaZlhfh\h^gh]hbebg_kdhevdbowe_f_glh\\imklhlZodhlhjhcklZlbklbq_kdb
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Fe\hevnjZfh\u_[jhgau±NaxWO3.
QlhlZdh_l\_j^u_jZkl\hju\uqblZgby"Ijb\_^bl_ijbf_ju.
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wlhflZdb_imklhluihl_gpbZevghfh]ml[ulvaZiheg_guih^h[gu_kljmdlmjugZau\Zxlky
^_n_dlgufbGZijbf_jFe1-xS; Cu2-xS.
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Wlhkihkh[ghklvobfbq_kdh]hkh_^bg_gbylh_klvgZ[hjZwe_f_glh\oZjZdl_jbamxsbokyhij_^_e_ggufb k\yayfb b dhhj^bgZpb_c Zlhfh\ \klj_qZlvky \ \b^_ jZaguo \_s_kl\ lh _klv
]hfh]_gguo nZa oZjZdl_jbamxsboky hij_^_e_ggufb k\hckl\Zfb b kljmdlmjhc GZijbf_j
S8 jhf[bq_kdZy b fhghdebggZy fh^bnbdZpbb K ZefZa b ehgk^_cebl ZnS knZe_jbl b
\xjlpblCaCO3dZevpblbZjZ]hgbl
Q_fihebfhjnbafijhkluo\_s_kl\hlebqZ_lkyhlZeehljhibb"
:eehljhiby ± kihkh[ghklv ijhklh]h \_s_kl\Z ^Zggh]h we_f_glZ kms_kl\h\Zlv \ \b^_ g_kdhevdboobfbq_kdbo\_s_kl\ihebfhjnbaf±kms_kl\h\Zgb_g_kdhevdboijhkluo\_s_kl\
h[jZah\Zgguo^Zggufwe_f_glhf\\b^_h^gh]hblh]h`_obfbq_kdh]hkh_^bg_gbyGZijbf_j fhghdebggZy b jhf[bq_kdZy fh^bnbdZpbb k_ju ihebfhjngu fhghdebggZy jhf[bq_kdZy b ieZklbq_kdZy k_jZ ± Zeehljhigu_ fh^bnbdZpbb :efZa b ehgk^_cebl ]jZnbl b β]jZnbl±ihiZjghihebfhjnguZ\k_\f_kl_dZj[bgnmee_j_g±Zeehljhigu_fh^bnbdZpbb
QlhlZdh_iheblbiby"Ijb\_^bl_ijbf_ju
Iheblbiby±kihkh[ghklvh^gh]hblh]h`_\_s_kl\ZdjbklZeebah\Zlvky\g_kdhevdbokehbkluokljmdlmjguofh^bnbdZpbyojZaf_juwe_f_glZjguoyq__d\iehkdhklbkehyh^bgZdh\u
^ey \k_o iheblbiguo fh^bnbdZpbc dhlhju_ jZaebqZxlky lhevdh q_j_^h\Zgb_f beb ih\hjhlhfh^bgZdh\uokeh_\GZijbf_j]jZnblbβ]jZnblTiO2 – jmlbe^\mokehcgZymiZdh\dZ [jmdbl ljzokehcgZy ZgZlZa q_lujzokehcgZy CdI2 ± ^\mo q_lujzo b r_klbkehcgZy
fh^bnbdZpbb
QlhlZdh_fhjnhljhiby"Ijb\_^bl_ijbf_ju
Wlh baf_g_gb_ kljmdlmjgh]h lbiZ \ jy^m jh^kl\_gguo obfbq_kdbo kh_^bg_gbc GZijbf_j
MgCO3, CaCO3 ± dZevpbl ]_dkZ]hgZevgZy fh^bnbdZpby CaCO3, SrCO3, BaCO3 ± ZjZ]hgbl
jhf[bq_kdZyfh^bnbdZpbcF2±h^gZkljmdlmjZCl2, Br2, I2±^jm]ZyLiNO3, NaNO3, KNO3 –
kljmdlmjZdZevpblZKNO3, RbNO3, CsNO3±kljmdlmjZZjZ]hgblZ
3
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khklZ\ZoZjZdl_jbamxsZykyhij_^_ezggufbkljmdlmjhcbk\hckl\Zfb
Ijbf_ju: ZnSknZe_jblb\xjlpbl±h^ghkh_^bg_gb_ghjZagu_obfbq_kdb_\_s_kl\ZC
ZefZa b ]jZnbl ± jZagu_ obfbq_kdb_ kh_^bg_gby b jZagu_ \_s_kl\Z C ZefZa b ehgk^_cebl±h^ghkh_^bg_gb_ghjZagu_obfbq_kdb_\_s_kl\Z6ieZklbq_kdZybjhf[bq_kdZy±
jZagu_ obfbq_kdb_ kh_^bg_gby b jZagu_ \_s_kl\Z 6L22 (djbklh[Zebl b ljb^bfbl ± h^gh
kh_^bg_gb_d\Zjp±^jm]h_obfbq_kdh_kh_^bg_gb_
I_j_qbkebl_ hkgh\gu_ lbiu iheghklvx mihjy^hq_gguo dhg^_gkbjh\Zgguo
nZa
kkk ± b^_Zevguc djbklZee ccq ± i_jbh^bqghklv \ ^\mo baf_j_gbyo cqq ± i_jbh^bqghklv \
h^ghfbaf_j_gbb; qqq – d\ZabdjbklZee
I_j_qbkebl_ hkgh\gu_ lbiu qZklbqgh mihjy^hq_gguo dhg^_gkbjh\Zgguo
nZa
Mihjy^hq_gu \ ^\mo baf_j_gbyo ccd ± i_jbh^bq_g \ ^\mo baf_j_gbb cqd ± i_jbh^bq_g \
h^ghfbaf_j_gbbqqd±g_i_jbh^bq_g
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QlhlZdh_d\ZabdjbklZee?
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QlhlZdh_g_khjZaf_jgZy(g_khjZaf_jghfh^mebjh\ZggZy) kljmdlmjZ?
WlhkljmdlmjZ\dhlhjhcfh]ml[ulv\u^_e_gu^\_ih^kbkl_fukg_khjZaf_jgufbi_jbh^ZfbgZijbf_j^\_kbkl_fup_i_ckjZagufbi_jbh^ZfbGZijbf_j±kljmdlmjuMgSix.
GZah\bl_hkgh\gu_lbiu`b^dbodjbklZeeh\
@b^db_ djbklZeeu ± khklhygb_ \_s_kl\Z \ dhlhjhf hgh h[eZ^Z_l k\hckl\Zfb `b^dhklbl_dmq_klvxbg_dhlhjufbk\hckl\Zfbl\_j^uodjbklZeeh\Zgbahljhib_c
Lbiu`b^dbodjbklZeeh\: ccd – kf_dlbdbmihjy^hq_gu\^\mobaf_j_gbyogZijbf_jiZjZZahdkbZgbahecdd – g_fZlbdbmihjy^hq_gu\h^ghfbaf_j_gbbgZijbf_jwlbeh\ucwnbj
Zahdkb[_gahcghc dbkehlu LZd`_ \hafh`gZ deZkkbnbdZpby ih nhjf_ fhe_dme kl_j`g_h[jZagu_^bkdhb^gu_b^jfhe_dmeu?kebkl_j`g_h[jZagu_fhe_dmeumdeZ^u\Zxlky\kehb
lhh[jZamxlkykf_dlbdb_keb\k^\bgmlu_^jm]hlghkbl_evgh^jm]Zkljmgu±g_fZlbdb
Dh]^Zbd_f[ueZhldjulZ^bnjZdpbyj_gl]_gh\kdboemq_cgZdjbklZeeZo"
]FEZmwIDgbiibg]b<Njb^jbo
Dh]^Z b d_f [ueb bamq_gu i_j\u_ djbklZeebq_kdb_ kljmdlmju" DZdb_ wlh
[uebkljmdlmju"
]Hl_pbkug;jw]]bNaCl, ZnSknZe_jblb\xjlpblCuZefZa
];jw]]feZ^rbcCaCO3.
4
DhfmijbgZ^e_`Zli_j\u_djbklZeehobfbq_kdb_ h[h[s_gby"Dh]^Zhgb[uebk^_eZgub\qzfaZdexqZebkv"
_]];jw]]feZ^rbc=hev^rfb^l±djbklZeehobfbq_kdb_jZ^bmku[_amqzlZlbiZk\ya_cihgylbybahfhjnbafZbihebfhjnbafZ
_]]Ihebg]±lbik\ya_c\djbklZeeZolZ[ebpubhgguojZ^bmkh\
MdZ`bl_ kljmdlmjguc deZkk ^Zgghc fhe_dmeu ihykgbl_ kfuke wlh]h kljmdlmjgh]hdeZkkZAZibrbl_kbf\heihR_gnebkm
G2O – 2mm (m; 2mm); C2v.
;_gahe – 6/mmm (2mm; 2mm); D6h.
CO2 – ∞m (∞m; ∞m); D∞h.
S8 – 82m (m); D4d.
CH4 – 43m (43m; 3m); Td.
NhjfZev^_]b^ – 2mm (2mm; m; 2mm) C2v.
G2H2 – pbknhjfZmm (m; m) C2v; ljZgknhjfZm (m; m) C2h;
ijhba\hevgZynhjfZK2.
P4 – 43m (3m); Td.
SF6 – m3m (m3m; 4mm) Oh.
C3H6 – 6m 2 (2mm; m) D3h.
MdZ`bl_kljmdlmjgucdeZkkp_ibihykgbl_kfuke
β-Se – Pc312, Z = 3 (2).
Ihebwlbe_g – Pcmmm, Z = 1 (mmm; 2mm).
PdCl2 – Pcmmm, Z = 1 (mmm; 2mm).
SiS2 – Pc42/mmc, Z = 2 ( 4m 2; 2mm).
DZj[bg – Pc ∞m, Z = 1 (∞m).
MdZ`bl_kljmdlmjgucdeZkkkehyihykgbl_kfuke
=jZnbl – Pl6/mmm, Z = 2 ( 6m 2 ).
BN – Pl 6m 2, Z = 1 ( 6m 2; 6m 2 ).
As – Pl 3m1, Z = 2 (3m).
Zn Pl6/mmm, Z = 1 (6/mmm).
MoS2 – Pl 6m 2, Z = 1 ( 6m 2; 3m).
MdZ`bl_kljmdlmjgucdeZkkdjbklZeebq_kdh]h\_s_kl\Z;ihykgbl_kfuke
CaF2 – Fm3m, Z = 4 (m3m; 43m).
CuAu – P4/mmm, Z = 1 (4/mmm; 4/mmm).
NiAs – P63/mmc, Z = 2 (3m; 6m 2).
Cu3Au – Pm3m, Z = 1 (4/mmm; m3m).
TiO2 (jmlbe) – P42/mnm, Z = 2 (mmm; mm2).
CaTiO3 – Pm3m, Z = 1 (m3m; m3m; 4/mmm).
FeS2 (fZjdZabl) – Pnnm, Z = 2 (2/m; m).
ReO3 – Pm3m, Z = 1 (m3m; 4/mmm).
Hg – R3m, Z = 3 (3m).
5
Se (k_juc) – P312, Z = 6 (2).
DZdh_nbabq_kdh_y\e_gb_e_`bl\hkgh\_j_gl]_ghkljmdlmjgh]h ZgZebaZ"Qlh
ij_^klZ\ey_lkh[hcfZl_jbZevgZykm[klZgpbyjZkk_b\ZxsZyj_gl]_gh\kdb_emqb"
Nbabq_kdh_ y\e_gb_ ± ^bnjZdpby j_gl]_gh\kdbo emq_c j_r_ldhc djbklZeeZ FZl_jbZevgZy
km[klZgpby±dhglbgmZevgh_jZkij_^_e_gb_we_dljhgghciehlghklb
GZah\bl_ hkgh\gu_ kihkh[u ihemq_gby ^bnjZdpbhgghc dZjlbgu \ j_gl]_gh]jZnbb
1) F_lh^EZmwihebojhfZlbq_kdbcimqhddjbklZeeg_ih^\b`_g>eyg_dhlhjuo^ebg\heg
gZ[ex^Z_lky^bnjZdpby
2) F_lh^ \jZs_gby ± fhghojhfZlbq_kdh_ baemq_gb_ djbklZee ^\b`_lky \h \j_fy wdki_jbf_glZIjbg_dhlhjuom]eZohjb_glZpbyodjbklZeeZgZ[ex^Z_lky^bnjZdpby
3) F_lh^ ihjhrdZ (>_[ZyR_j_jZ ± fhghojhfZlbq_kdh_ baemq_gb_ gZ[ex^Z_lky ^bnjZdpbyhldjbklZeeh\kg_dhlhjhchjb_glZpb_cihhlghr_gbxdimqdm
DZdb_ oZjZdl_jbklbdb djbklZeebq_kdh]h \_s_kl\Z fh`gh ihemqblv k ihfhsvxmjZ\g_gbcEZmw"
a (cos ϕ − cos χ ) = pλ
1
1

MjZ\g_gbyEZmw b (cos ϕ 2 − cos χ 2 ) = qλ

c cos ϕ3 − cos χ3 = rλ
(
)
(a, b, c±iZjZf_ljuyq_cdbϕ1, ϕ2, ϕ3±m]euhij_^_eyxsb_gZijZ\e_gb_i_j\bqgh]himqdZ
ih hlghr_gbx d hkyf dhhj^bgZl djbklZeeZ χ1, χ2, χ3 ± m]eu hij_^_eyxsb_ gZijZ\e_gb_
^bnjZdpbhggh]hemqZihhlghr_gbxdhkyfdhhj^bgZlp, q, r ±bg^_dkuemqZLZdbfh[jZahfmjZ\g_gbyEZmwiha\heyxlhij_^_eblviZjZf_ljuwe_f_glZjghcyq_cdbdjbklZeeZ
DZdb_ oZjZdl_jbklbdb djbklZeebq_kdh]h \_s_kl\Z fh`gh ihemqblv k ihfhsvxmjZ\g_gby;jw]]Z<mevnZ"
2 d hkl sin θ = nλ – mjZ\g_gb_;jw]]Z<mevnZ
(d±f_`iehkdhklgh_jZkklhygb_θ±m]heiZ^_gbyemqZh, k, l±bg^_dkuiehkdhklbn±ihjy^hd hljZ`_gby MjZ\g_gby ;jw]]Z<mevnZ iha\heyxl hij_^_eblv f_`iehkdhklgu_ jZkklhygby \ djbklZee_ Z ijhbg^bpbjh\Z\ j_gl]_gh]jZffm lh _klv ijbibkZ\ iehkdhklyf bg^_dkuhklkbg]hgbxbiZjZf_ljuwe_f_glZjghcyq_cdb
DZdb_nhjfmeu\ujZ`Z_laZ\bkbfhklvbgl_gkb\ghklb^bnjZdpbhggh]hemqZ
hldjbklZeebq_kdhckljmdlmju"QlhlZdh_kljmdlmjgZyZfieblm^Z?
2π i hx j + ky j +lz j τ j
n
(\uq )
(\uq ) 2
(\uq )

I hkl
; Fhkl
.
= Fhkl
= ∑ f je 
j =1
F kljmdlmjgZy Zfieblm^Z ± dhfie_dkgZy \_ebqbgZ oZjZdl_jbam_fZy ijb\_^zgguf mjZ\g_( wdki )
gb_f Nbabq_kdbc kfuke bf__l Fhkl
± Zfieblm^Z ^bnjZdpbhggh]h emqZ hl k_f_ckl\Z
iehkdhkl_c k bg^_dkZfb hkl. (fj ± nmgdpby Zlhfgh]h jZkk_ygby j ± bg^_dk ZlhfZ n ± qbkeh
Zlhfh\\yq_cd_τj±l_ieh\hciZjZf_lj
QlhlZdh_RnZdlhjnZdlhjg_^hklh\_jghklb\JK:"
6
Wlh [_ajZaf_jgZy \_ebqbgZ ihdZau\ZxsZy gZkdhevdh lhqgh \uqbke_ggZy djbklZeebq_kdZy
kljmdlmjZkhhl\_lkl\m_lj_Zevghcq_fgb`_RnZdlhjl_flhqg__jZkkqblZggZykljmdlmjZ
KljmdlmjZkqblZ_lkygZ^_`ghmklZgh\e_gghc_kebR < 4%.
( wdki)
(\uq)
−F
∑ Fhkl
hkl
R = hkl
.
( wdki)
∑ F
hkl hkl
DZdZy nhjfmeZ jZkij_^_e_gby we_dljhgghc iehlghklb bkihevam_lky \ JK:"
DZdb_oZjZdl_jbklbdbdjbklZeebq_kdh]h\_s_kl\Zba\e_dZxlbawlhcnmgdpbb"
−2π i  hx+ky +lz 
1
ρ ( x, y , z ) =
, V0±h[t_fwe_f_glZjghcyq_cdb
∑ ∑ ∑ Fhkl e
V hk l
0
FZdkbfmfuρ(x, y, zijb[ebabl_evghkhhl\_lkl\mxlkj_^gbfih\j_f_gbiheh`_gbyfZlhfguoy^_jlh_klvnmgdpbyjZkij_^_e_gbywe_dljhgghciehlghklb^Z_l\hafh`ghklvhij_^_eblvdhhj^bgZlubl_ieh\u_iZjZf_ljuZlhfh\ZlZd`_kljmdlmjgucdeZkk
<q_fkmsghklvijyfh]hklZlbklbq_kdh]hf_lh^Zhij_^_e_gbygZqZevguonZa
kljmdlmjguoZfieblm^"
< ih^[hj_ gZqZevguo nZa f_lh^Zfb fZl_fZlbq_kdhc klZlbklbdb lZd qlh[u we_dljhggZy
iehlghklv\ex[hclhqd_ijhkljZgkl\Z[ueZiheh`bl_evgZGZqZevgZynZaZDhklhij_^_ey_l
khhlghr_gb_ f_`^m kljmdlmjghc Zfieblm^hc bkihevam_fhc ijb \uqbke_gbb nmgdpbb jZkij_^_e_gby we_dljhgghc iehlghklb b fh^me_f kljmdlmjghc Zfieblm^u ba\e_dZ_fhc ba
^Zgguoj_gl]_gh\kdh]hwdki_jbf_glZwlhd\Z^jZlgu_dhjgb ba agZq_gbc khhl\_lkl\mxsbo
iD
( wdki )
e hkl .
bgl_gkb\ghkl_c Fhkl = Fhkl
<q_fkhklhblijhp_^mjZmlhqg_gbydjbklZeebq_kdhckljmdlmju\JK:"
Wlh fbgbfbaZpby d\Z^jZlZ agZq_gby g_dhlhjh]h nmgdpbhgZeZ ihdZau\Zxs_]h g_^hklh\_jghklvhij_^_e_gbydjbklZeebq_kdhckljmdlmjukihfhsvx\Zjvbjh\Zgbydhhj^bgZlZlhfh\
b bo l_ieh\uo iZjZf_ljh\ < dZq_kl\_ lZdh]h nmgdpbhgZeZ qZklh bkihevamxl RnZdlhj <
ij_pbabhgghfJK:lZd`_mqblu\ZxlkykemqZcgu_kihfhsvxih\lhjguokt_fhdbkbkl_fZlbq_kdb_ih]ehs_gb_bgkljmf_glZevgZyih]j_rghklbdjhf_wlh]hmlhqgy_lky[hevr__
qbkehiZjZf_ljh\±\qZklghklbZlhfgu_nmgdpbbjZkk_ygby
Ihq_fm klZg^Zjlguc j_gl]_ghkljmdlmjguc ZgZeba g_ fh`_l ^Zlv Z^_d\ZlgmxbgnhjfZpbxhnmgdpbbjZkij_^_e_gbywe_dljhgghciehlghklb"
Ihlhfm qlh ijb jZkq_l_ ijhkljZgkl\_ggh]h jZkij_^_e_gby we_dljhgghc iehlghklb bkihevamxlkyklZg^Zjlgu_Zlhfgu_nmgdpbbjZkk_ygbylZ[mebjh\Zggu_^eyb^_Zevguokn_jbq_kdbokhklhygbcZlhfh\LZdbfh[jZahfg_mqblu\Z_lky\ebygb_\Ze_glguok\ya_cZlZd`_
iheyjbaZpbywe_dljhgguoh[hehq_dZlhfh\<ij_pbabhgghfJK:Zlhfgu_nmgdpbbjZkk_ygbymlhqgyxlky\f_kl_k^jm]bfbiZjZf_ljZfbdjbklZeebq_kdhckljmdlmjuihwlhfm^Zgguc
f_lh^^Z_lZ^_d\ZlgmxbgnhjfZpbxhjZkij_^_e_gbbwe_dljhgghciehlghklb
DZdmx^hihegbl_evgmxbgnhjfZpbx^Zzlij_pbabhggucJK:ihkjZ\g_gbx
kh klZg^Zjlguf \ZjbZglhf wlh]h f_lh^Z Qlh lZdh_ fmevlbihevgh_ jZaeh`_gb_ we_dljhgghciehlghklb?
Ij_pbabhggucJK:iha\hey_lhij_^_eblvg_lhevdhijbf_jgu_iheh`_gbyZlhfh\ghb\u^_eblv^_n_dluwe_dljhgghciehlghklbmdZau\Zxsb_gZgZebqb_obfbq_kdbok\ya_cg_ih^_ezgguowe_dljhgguoiZjbebiheyjbaZpbxwe_dljhgguoh[hehq_d
7
Fmevlbihevgh_ jZaeh`_gb_ we_dljhgghc iehlghklb ± ij_^klZ\e_gb_ dhglbgmZevgh]h jZkij_^_e_gbywe_dljhgghciehlghklb\\b^_kmi_jihabpbbihe_chl^_evguofmevlbihe_clh
_klvgZ[hjh\lhq_qguoaZjy^h\Bkihevam_lky\ij_pbabhgghfJK:
DZdh\hgZagZq_gb_bh[sb_ijbgpbiumkljhckl\ZZ\lhfZlbq_kdh]h^bnjZdlhf_ljZ"
GZagZq_gb_ ± ^bnjZdpbhgguc ijb[hj ^ey JK: Mkljhckl\h djbklZee h[emqZ_fuc fhghojhfZlbq_kdbfj_gl]_gh\kdbfbaemq_gb_f\jZsZ_lky\hdjm]\k_\hafh`guohk_c^eyihemq_gby iheghc ^bnjZdpbhgghc dZjlbgu Kq_lqbd j_]bkljbjm_l bgl_gkb\ghklv jZkk_yggh]h
baemq_gby \ jZaguo lhqdZo ijhkljZgkl\Z ijb jZaguo iheh`_gbyo djbklZeeZ hlghkbl_evgh
i_j\bqgh]himqdZ
>eydZdhcp_ebbkihevam_lkyj_gl]_ghnZah\ucZgZeba"<q_fkmsghklvwlh]hf_lh^Z"
P_evj_gl]_ghnZah\h]hZgZebaZ±hij_^_e_gb_nZah\h]hkhklZ\Zkf_kbdjbklZeebq_kdbo\_s_kl\ Kmsghklv f_lh^Z ± b^_glbnbdZpby nZau ih oZjZdl_jbklbq_kdhc ^bnjZdpbhgghc
dZjlbg_h[mkeh\e_gghcjZaebqgufgZ[hjhff_`iehkdhklguojZkklhygbc\djbklZeebq_kdbo
j_r_ldZojZaguo\_s_kl\
DZdb_ ^bnjZdpbhggu_ f_lh^u djhf_ j_gl]_gh]jZnbb bkihevamxlky ^ey
hij_^_e_gbykljmdlmjudjbklZeeh\"<q_fboij_bfms_kl\Zbg_^hklZldb"
G_cljhgh]jZnbybwe_dljhgh]jZnby
G_cljhgh]jZnby±^bnjZdpbyg_cljhgh\h[uqghg_cljhgh\hij_^_e_gghckdhjhklblh_klv
hij_^_e_gghc wg_j]bb gZ Zlhfguo y^jZo Ij_bfms_kl\Z \hafh`ghklv g_ihkj_^kl\_ggh]h
hij_^_e_gby kj_^gbo ih \j_f_gb dhhj^bgZl Zlhfguo y^_j Z g_ fZdkbfmfh\ we_dljhgghc
iehlghklb gZ^z`gh_ hij_^_e_gb_ iheh`_gbc ez]dbo Zlhfh\ ]eZ\gh_ ± \h^hjh^Z G_^hklZldbg_h[oh^bfhklvgZebqbydjmiguodjbklZeeh\^hjh]h\bagZbkeh`ghklv
We_dljhgh]jZnby±^bnjZdpbywe_dljhgh\h[uqghwe_dljhgh\khij_^_e_gghc^ebghc\hegugZihl_gpbZeZoy^jZbwe_dljhgh\Ij_bfms_kl\Z\hafh`ghklvbkke_^h\ZgbydjbklZeeh\fZeuojZaf_jh\bih\_joghklbG_^hklZldbiehob_j_amevlZlugZdjmiguodjbklZeeZo
^hjh]h\bagZ
DZdh_ nbabq_kdh_ y\e_gb_ e_`bl \ hkgh\_ g_cljhgh]jZnbb b we_dljhgh]jZnbb"GZdZdbofZl_jbZevguoh[t_dlZojZkk_b\Zxlkyg_cljhgubwe_dljhgu"
Nbabq_kdh_ y\e_gb_ ± ^bnjZdpby g_cljhgh\ beb we_dljhgh\ gZ khhl\_lkl\mxsbo fZl_jbZevguo km[klZgpbyo jZkij_^_e_gb_ ihl_gpbZeZ y^_jguo kbe lh _klv Zlhfgu_ y^jZ \ g_cljhgh]jZnbbwe_dljhklZlbq_kdbcihl_gpbZey^_jbwe_dljhgh\\we_dljhgh]jZnbb
Ijb\_^bl_ ijbf_ju f_lZeeh\ kljmdlmjZ dhlhjuo hibku\Z_lky ^\mokehcghc
IRMDZdjZkiheh`_guZlhfu\we_f_glZjguoyq_cdZowlbof_lZeeh\"DZdh\Zdhhj^bgZpbyZlhfh\"
Mg, Be, β-Ce, Tl, Ti. :lhfujZkiheh`_gu\\_jrbgZo]_dkZ]hgZevghcyq_cdbbp_glj_h^ghc
ba^\moljb]hgZevguoijbafgZdhlhju_yq_cdmjZa[b\Z_liehkdhklvijhoh^ysZyq_j_afZemx^bZ]hgZevhkgh\ZgbyDQ ]_dkZ]hgZevgucZgZeh]dm[hhdlZw^jZ
Ijb\_^bl_ ijbf_ju f_lZeeh\ kljmdlmjZ dhlhjuo hibku\Z_lky lj_okehcghc
IRMDZdjZkiheh`_guZlhfu\we_f_glZjguoyq_cdZowlbof_lZeeh\"DZdh\Zdhhj^bgZpbyZlhfh\"
Cu, Au, Al, Ni, Pb. :lhfujZkiheh`_gu\\_jrbgZodm[bq_kdhcyq_cdbbp_gljZo\k_o
]jZg_cDQ dm[hhdlZw^j
8
Ijb\_^bl_ijbf_juf_lZeeh\bahkljmdlmjguoα-FeDZdjZkiheh`_guZlhfu
\we_f_glZjguoyq_cdZowlbof_lZeeh\"DZdh\Zdhhj^bgZpbyZlhfh\"
K, Ba, V, W:lhfujZkiheh`_gu\\_jrbgZobp_glj_dm[bq_kdhcyq_cdbDQ dm[
Ijb\_^bl_ ijbf_ju ZghfZevguo djbklZeebq_kdbo kljmdlmj f_lZeeh\ Hibrbl_wlbkljmdlmju
1) Zn, Cd±kljmdlmjZMgjZklygmlZy\^hevhkblj_lv_]hihjy^dZ
2) In ± kljmdlmjZ Cu jZklygmlZy \^hev hkb q_l\zjlh]h ihjy^dZ Hg ± kljmdlmjZ Cu k`ZlZy
\^hevhkblj_lv_]hihjy^dZ
3) La±q_lujzokehcgZyIRMSm±^_\ylbkehcgZyIRM
4) α-Po±ijhklZydm[bq_kdZydeZ^dZZlhfh\Po.
DZdmxkljmdlmjmbf_xlhl\_j^_\rbobg_jlgu_]Zau
< khhl\_lkl\bb k ijZ\behf XfJha_jb qbkeh dh\Ze_glguo k\ya_c \ djbklZeeZo bg_jlguo
]Zah\ jZ\gh gmex Ihwlhfm \k_ k\yab \ wlbo djbklZeeZ \Zg^_j\ZZevkh\u lh _klv ± g_gZijZ\e_ggu_LZdbfh[jZahfkljmdlmjuhl\_j^_\rbobg_jlguo]Zah\bf_lZeeh\koh`bbfh]ml[ulvhibkZgu\l_jfbgZoIRM DjbklZeebq_kdbc He ± ^\mokehcgZy IRM ZgZeh] Mg;
hklZevgu_bg_jlgu_]Zau±ljzokehcgZyIRMkljmdlmjZCu.
Hibrbl_hkh[_gghklb\ukhdhl_fi_jZlmjguodjbklZeebq_kdbofh^bnbdZpbc
nlhjZbdbkehjh^Z
<wlbodjbklZeeZo^\moZlhfgu_fhe_dmeukhhl\_lkl\mxsbowe_f_glh\gZoh^ylky\ khklhygbbk\h[h^gh]h\jZs_gby\hdjm]\_jrbgdm[bq_kdhcyq_cdbJK:nbdkbjm_llZdb_fhe_dmeudZdkn_ju>Zggh_khklhygb_gZau\Z_lkyjhlZpbhgghdjbklZeebq_kdbf. β-F2, γ-O2: Pm3n,
Z = 8.
DZdh_ \b^hbaf_g_gb_ djbklZeebq_kdhc kljmdlmju gZ[ex^Z_lky \ jy^m Cl2,
Br2, I2?
< khhl\_lkl\bb k ijZ\behf XfZJha_jb qbkeh dh\Ze_glguo k\ya_c \ djbklZeeZo ]Zeh]_gh\
jZ\ghlh_klv\kljmdlmj_bf_xlky^\moZlhfgu_fhe_dmeuL_fg_f_g__\mdZaZgghfjy^m
kl_i_gv]_l_jh^_kfbqghklbkljmdlmjuihgb`Z_lky\ke_^kl\b_hkeZ[e_gby\gmljbfhe_dmeyjguok\ya_cbmdj_ie_gbyf_`fhe_dmeyjguok\ya_cbaaZjhklZjZaf_jh\Zlhfh\bwe_dljhgguoh[hehq_d
DZdb_dhhj^bgZpbhggu_qbkeZbf_xlZlhfu\kljmdlmjZoTe, $VbGe?
Te±bahkljmdlmj_gk_jhfmSep_ihq_qgh_kljh_gb_DQ As±bahkljmdlmj_gq_jghfmnhknhjm]hnjbjh\Zggu_kehbbar_klbqe_gguodhe_p±©dj_k_eªDQ Ge±bf__lZefZahih^h[gmxkljmdlmjmDQ DZdh\oZjZdl_jdjbklZeebq_kdbokljmdlmjk_jh]hbdjZkgh]hk_e_gZ"
DjZkguck_e_g (α-6H±bahkljmdlmj_gjhf[bq_kdhck_j_\hkvfbqe_ggu_dhevpZSe8k\yaZgu
keZ[ufb\Zg^_j\ZZevkh\ufbk\yayfboZjZdl_jkljmdlmjuhkljh\gZy]_l_jh^_kfbq_kdZy
K_juck_e_g (β-Se±kljmdlmjZknhjfbjh\ZgZp_iyfbPc31p_ibk\yaZgukeZ[ufb\Zg^_j
\ZZevkh\ufbk\yayfboZjZdl_jkljmdlmjup_ihq_qgZy]_l_jh^_kfbq_kdZy
Hibrbl_kljmdlmjgu_hkh[_gghklb^\modjbklZeebq_kdbofh^bnbdZpbcZah-
lZ
α-N2±dm[bq_kdZywe_f_glZjgZyyq_cdZPa3, Z = 4 (3).
β-N2±]_dkZ]hgZevgZywe_f_glZjgZyyq_cdZP63/mmc, Z = 2 (6m 2).
9
\h[hbokemqZyomdZaZgukljmdlmjgu_deZkku^eyfhe_dmeN2).
DZdh\oZjZdl_jkljmdlmju[_eh]hq_jgh]hbdjZkgh]hnhknhjZ"
Qzjgucnhknhj±]hnjbjh\Zggu_kehbkhklhysb_ ba r_klbqe_gguo dhe_p \ dhgnhjfZpbb
©dj_kehªf_`^mkehyfb±keZ[u_\Zg^_j\ZZevkh\uk\yaboZjZdl_jkljmdlmjukehbklZy]_l_jh^_kfbq_kdZy
;_euc nhknhj ± dm[bq_kdZy iehlg_crZy miZdh\dZ l_ljZw^jbq_kdbo ©fhe_dmeª J4 oZjZdl_j
kljmdlmjuhkljh\gZy]_l_jh^_kfbq_kdZy
DjZkgucnhknhj±Zfhjngh_\_s_kl\hkdhj__\k_]hkhklhblbap_i_cg_mihjy^hq_gghjZkiheh`_gguo\ijhkljZgkl\_
I_j_qbkebl_\Z`g_crb_djbklZeebq_kdb_nhjfum]e_jh^Z
:efZa b ehgk^_cebl sp3]b[jb^baZpby Zlhfh\ m]e_jh^Z DQ ± l_ljZw^j kbg]hgby ±
dm[bq_kdZyb]_dkZ]hgZevgZykhhl\_lkl\_ggh
2) αbβ]jZnblsp2]b[jb^baZpbyZlhfh\m]e_jh^ZDQ ±lj_m]hevgbdkljmdlmjZknhjfbjh\ZgZ kehyfb khklhysbo ba ijZ\bevguo r_klbqe_gguo dhe_p < α]jZnbl_ kehb q_j_^mxlkyq_j_ah^bg\β]jZnbl_±q_j_a^\Z
3) dZj[bg±[_kdhg_qgu_p_ib Pc ∞m.
DZdh\Zdhhj^bgZpbyZlhfZm]e_jh^Z\nmee_j_g_"
Fhe_dmeZ nmee_j_gZ bf__l nhjfm mk_qzggh]h bdhkZw^jZ b knhjfbjh\ZgZ iylb b r_klbqe_ggufbpbdeZfbZlhfh\m]e_jh^ZDQ Zlhfum]e_jh^Z\\_jrbgZobkdZ`zgguoljb]hgZevguoibjZfb^
DZdh\Zdhhj^bgZpbyZlhfh\[hjZ\djbklZeebq_kdhf[hj_"
DjbklZeebq_kdZy kljmdlmjZ [hjZ knhjfbjh\ZgZ bdhkZw^jZfb <12 kh_^bgzggufb q_j_a fhklbdh\u_ Zlhfu [hjZ Dhhj^bgZpby Zlhfh\ DQ \ \_jrbg_ l_ljZ]hgZevghc ibjZfb^u
DQ \\_jrbg_i_glZ]hgZevghcibjZfb^uDQ \p_glj_i_glZ]hgZevghcibjZfb^u
Ijb\_^bl_ijbf_jubahkljmdlmjguog_f_lZeeh\baVIIIbVII]jmii
Ne – Ar – Kr– Xe (kljmdlmjZ Cu); Cl2 – Br2 – I2.
Ijb\_^bl_ijbf_jubahkljmdlmjguog_f_lZeeh\baVIbV]jmii
Jhf[bq_kdZyk_jZ±α-SedjZkguck_e_gβ-Sek_juck_e_g±Te.
Qzjgucnhknhj±As – Sb – Bi.
Ijb\_^bl_ijbf_jubahkljmdlmjguog_f_lZeeh\IV]jmiiu
:efZa ± Si – Ge – α-Sn k_jh_ heh\h \ukhdhl_fi_jZlmjgu_ fh^bnbdZpbb Si b Ge – β-Sn
[_eh_heh\h
Ijb\_^bl_ijbf_jubgl_jf_lZeeb^h\khkljmdlmjZfbih^h[gufbhkgh\guf
kljmdlmjguflbiZff_lZeeh\QlhlZdh_k\_jokljmdlmjZ?
K\_jokljmdlmjZ±]_hf_ljbq_kdh_ih^h[b_\fhlb\_jZkiheh`_gbyZlhfh\\[he__keh`ghf
obfbq_kdhf \_s_kl\_ ih kjZ\g_gbx k [he__ ijhkluf GZijbf_j Cu3Au ± k\_jokljmdlmjZ
Cu; Fe3Al±k\_jokljmdlmjZα-Fe.
QlhlZdh_nZauEZ\_kZ"Ijb\_^bl_ijbf_ju
NZau EZ\_kZ ± bgl_jf_lZeebq_kdb_ kh_^bg_gby \ dhlhjuo Zlhfu f_gvr_]h jZaf_jZ h[jZamxldhfiZdlgu_]jmiibjh\dbh[uqgh±l_ljZw^jbq_kdb_kh_^bgyxsb_ky\dZjdZku:lhfu [hevr_]h jZ^bmkZ aZgbfZxl h[jZamxsb_ky imklhlu LZdb_ kljmdlmju oZjZdl_jgu ^ey
10
[bgZjguof_lZeebq_kdbokh_^bg_gbc:<\kemqZ__kebZlhfu:b<[ebadbobfbq_kdbgh
bobhggu_jZ^bmkukms_kl\_gghjZaebqZxlky
Ijbf_ju: MgCu2, MgZn2, MgNi2.
Q_f kljmdlmjZ bgl_jf_lZeeb^Z &X$X hlebqZ_lky hl kljmdlmju kieZ\Z lh]h
`_khklZ\ZDZdh\Zkbff_ljbydjbklZeebq_kdbokljmdlmjkieZ\Zbbgl_jf_lZeeb^Z"
KieZ\±klZlbklbq_kdh_jZkiheh`_gb_Zlhfh\f_^bbahehlZl_fg_f_g___kebijbgylvwlb
ZlhfuaZh^bgZdh\u_lhkms_kl\mxlmihjy^hq_gghklvbi_jbh^bqghklvbojZkiheh`_gby
Bgl_jf_lZeeb^±jZkiheh`_gb_Zlhfh\f_^bbahehlZmihjy^hq_ggh
Kbff_ljbykljmdlmju\h[hbokemqZyowlhkljmdlmjZf_^bjZklygmlZy\^hevhkb\kieZ\_ZlhfujZkiheh`_guklZlbklbq_kdb\bgl_jf_lZeeb^_±kehbf_^bbahehlZq_j_^mxlky
Hibrbl_ hkgh\gu_ kljmdlmjgu_ lbiu bhgguo djbklZeeh\ DZdh\u dhhj^bgZpbhggu_qbkeZbhgh\\wlbo\_s_kl\Zo"
1. NaCl±ZlhfugZljby\\_jrbgZodm[bq_kdhcyq_cdbb\p_gljZo\k_o]jZg_cZlhfuoehjZ\p_glj_yq_cdbbk_j_^bgZo\k_oj_[_jDQ hdlZw^j
2. KsCl±ZlhfuoehjZ\\_jrbgZodm[bq_kdhcyq_cdbZlhfp_aby\p_glj_DQ dm[
3. CaF2nexhjbl±ZlhfudZevpby\\_jrbgZodm[bq_kdhcyq_cdbb\p_gljZo\k_o__]jZg_cZlhfunlhjZ\p_gljZo\k_o\hkvfbhdlZglh\DQKZ dm[DQF l_ljZw^j
4. Li2O (Zglbnexhjbl±Zlhfudbkehjh^Z\\_jrbgZodm[bq_kdhcyq_cdbb\p_gljZo\k_o
__]jZg_cZlhfueblby\p_gljZo\k_o\hkvfbhdlZglh\DQLi l_ljZw^jDQ H dm[
Q_fjZaebqZxlkykljmdlmjZ\xjlpblZbknZe_jblZ"
KnZe_jbl±dm[bq_kdZyiehlg_crZymiZdh\dZZlhfh\k_juZlhfupbgdZaZgbfZxliheh\bgm
l_ljZw^jbq_kdboimklhlKljmdlmjgucdeZkk F43m, Z = 4 (43m; 43m).
<xjlpbl±]_dkZ]hgZevgZyiehlg_crZymiZdh\dZZlhfh\k_juZlhfupbgdZaZgbfZxliheh\bgml_ljZw^jbq_kdboimklhlKljmdlmjgucdeZkkP6m1, Z = 2 (6m, 3m; 6m, 3m).
DZdh\ oZjZdl_j djbklZeebq_kdhc kljmdlmju Cu2O" DZdh\Z dhhj^bgZpby
Zlhfh\\wlhf\_s_kl\_"
OZjZdl_j kljmdlmju dZjdZkgZy ]_l_jh^_kfbq_kdZy DQ Cu ^bw^j DQ O l_ljZw^jKljmdlmjZCu2O±ijbf_jdZl_gZgZ\djbklZeehobfbbj_r_ldbZlhfh\f_^bbdbkehjh^Z©ijh^_luªh^gZ\^jm]mx
GZah\bl_ba\_klgu_fh^bnbdZpbbSiO2mdZ`bl_bokbg]hgbxbkhhlghr_gb_k
fh^bnbdZpbyfbm]e_jh^Z
Djbklh[Zebl±dm[bq_kdZykbg]hgbyb^_Zebabjh\ZggZykljmdlmjZdjbklh[ZeblZfh`_l[ulv
ihemq_gZ^h[Z\e_gb_fZlhfh\dbkehjh^ZgZk_j_^bgu\k_ok\ya_c\ZefZa_bihke_^mxsbf
ih^h[guf m\_ebq_gb_f we_f_glZjghc yq_cdb < j_Zevguo djbklZeeZo djbklh[ZeblZ Zlhfu
dbkehjh^Z g_kdhevdh k^\bgmlu ba b^_Zevguo iheh`_gbc lZd qlh \Ze_glguc m]he Si-O-Si
klZgh\blkyf_gvr_0.
Ljb^bfbl ± ]_dkZ]hgZevgZy kbg]hgby b^_Zebabjh\ZggZy kljmdlmjZ ljb^bfblZ fh`_l [ulv
ihemq_gZ ^h[Z\e_gb_f Zlhfh\ dbkehjh^Z gZ k_j_^bgu \k_o k\ya_c \ ehgk^_cebl_ b ihke_^mxsbf ih^h[guf m\_ebq_gb_f we_f_glZjghc yq_cdb < j_Zevguo djbklZeeZo ljb^bfblZ
Zlhfudbkehjh^Zg_kdhevdhk^\bgmlubab^_Zevguoiheh`_gbclZdqlh\Ze_glgucm]heSiO-SiklZgh\blkyf_gvr_0.
D\Zjp±dm[bq_kdZykbg]hgbyl_ljZw^juSiO4kh_^bg_guq_j_a\_jrbgubgZoh^ylky\\_jrbgZobp_glj_yq_cdbKms_kl\mxl^\_hilbq_kdbZdlb\gu_fh^bnbdZpbbbaaZ\hafh`ghklbkms_kl\h\Zgbyhk_c1beb2.
11
Klbrh\bl±l_ljZ]hgZevgZykbg]hgbybahkljmdlmj_gjmlbemTiO2).
MdZ`bl_\Z`g_crb_jZaebqbykljmdlmjKH2bSiO2.
CO2 ± fhe_dmeyjgu_ djbklZeeu khklhysb_ ba ebg_cguo fhe_dme CO2 jZkiheh`_gguo \
\_jrbgZodm[bq_kdhcyq_cdbKljmdlmjZ]_l_jh^_kfbq_kdZyhkljh\gZy
SiO2 ± kms_kl\m_l \ g_kdhevdbo fh^bnbdZpbyo d\Zjp djbklh[Zebl ljb^bfbl Kljmdlmju
\k_owlbokh_^bg_gbckhklhylbal_ljZw^jh\SiO4bg_y\eyxlkyfhe_dmeyjgufb
DZdh_kljh_gb_bf_xldjbklZeeuev^Z"Kdhevdh\h^hjh^guok\ya_ch[jZam_l
fhe_dmeZ\h^u\wlbodjbklZeeZo"DZdwlbk\yabjZkiheh`_gu"
Kms_kl\m_l^\_fh^bnbdZpbbev^Zkljmdlmjudhlhjuofh`ghihemqblvbakljmdlmjdjbklh[ZeblZbljb^bfblZ:lhfudbkehjh^ZaZgbfZxlihabpbbZlhfh\dj_fgbyZlhfu\h^hjh^Z
g_kdhevdh[eb`_dh^ghfmbaZlhfh\dbkehjh^Zg_`_ebd^jm]hfmIhwlhfmkljmdlmjZklZgh\blkyhkljh\ghc]_l_jh^_kfbq_kdhc<h^hjh^gu_k\yab±q_luj_gZfhe_dmem±kh_^bgyxl
©hkljh\Zª\djbklZee_Wlbk\yabgZijZ\e_gu\^hevgZijZ\e_gbck\ya_cH±G\fhe_dmeZo
Q_fihoh`bb\q_fjZaebqZxlkykljmdlmjuBNb]jZnblZ"
Koh^kl\Zh[_kljmdlmju]_l_jh^_kfbq_kdb_kehbklu_kehbkhklhylbaijZ\bevguor_klbqe_gguodhe_p
JZaebqby\BNkehbjZkiheZ]ZxlkyiZjZee_evgh^jm]^jm]m\]jZnbl_bf__lky^\mokehcgZy
miZdh\dZkhk_^gb_kehbg_iZjZee_evgu
Ijb\_^bl_ijbf_juhkljh\guodjbklZeebq_kdbokljmdlmjbaqbkeZijhkluo
b[bgZjguo\_s_kl\
Ijhklu_\_s_kl\ZCl2, S8.
;bgZjgu_kh_^bg_gbyH2O, CO2m]e_\h^hjh^u
Ijb\_^bl_ijbf_jup_ihq_qguodjbklZeebq_kdbokljmdlmj[bgZjguokh_^bg_gbc:ObXY.
PdCl2, BeCl2, SiS2.
Ijb\_^bl_ijbf_jukehbkluodjbklZeebq_kdbokljmdlmj:ObOY.
CuCl2, BN, MoS2, CdI2.
Ijb\_^bl_ijbf_ju]hfh^_kfbq_kdbofhgheblguokljmdlmjAXbXYkdh\Ze_glgufbk\yayfb
ReO3, ZnSknZe_jblSiO2 (djbklh[Zeblljb^bfbld\Zjp
Hibrbl_fhlb\aZiheg_gbyimklhl\dhjmg^_Al2O3.
:lhfudbkehjh^Zh[jZamxl]_dkZ]hgZevgmxiehlg_crmxmiZdh\dmZlhfuZexfbgbykbkl_fZlbq_kdb aZgbfZxl hdlZw^jbq_kdbo imklhl \ gZijZ\e_gbb i_ji_g^bdmeyjghf d iehlg_crbfkehyf^\_aZgylu_imklhluq_j_^mxlkykh^ghcg_aZgylhc<iehkdhklbiZjZee_evghc iehkdhklyf keh_\ Zlhfu Zexfbgby h[jZamxl r_klbqe_ggu_ dhevpZ \ p_glj_ dhlhjuo
jZkiheZ]Z_lkyimklhlZ
Ijb\_^bl_ ijbf_ju djbklZeebq_kdbo kljmdlmj :O ]^_ f_lZee bf__l dhhj^bgZpbhggucfgh]h]jZggbd\\b^_
ZhdlZw^jZ– NaCl, Al2O3.
[l_ljZ]hgZevghc^bibjZfb^u– CuCl2, HgCl2.
\ljb]hgZevghcijbafu – MoS2, CdI2.
12
Ijb\_^bl_ijbf_jukljmdlmjAX]^_Zlhfuf_lZeeZdhhj^bgbjh\Zguih
Zl_ljZw^jm– ZnS, Li2O.
[d\Z^jZlm– PdCl2, CuO.
H[sb_ ijbgpbiu kbkl_fZlbdb l_jgZjguo djbklZeebq_kdbo kljmdlmj <Z`g_crb_hkh[_gghklbkljh_gbydjbklZeeh\:<K:<O:O<XYZ.
ABC±bgl_jf_lZeebq_kdb_kh_^bg_gby_keb:<bK±lbibqgu_f_lZeeubhggu_jZ^bmku
dhlhjuo [ebadb lh h[jZamxs__ky kh_^bg_gb_ y\ey_lky k\_jokljmdlmjhc ih hlghr_gbx d
lbibqguff_lZeebq_kdbfkljmdlmjZfCu, Mg, α-Fe_keb:<bKjZaebqZxlkyobfbq_kdb
bebihbhggufjZ^bmkZflhfh]mlh[jZah\u\ZlvkykZfu_jZaghh[jZagu_kljmdlmju
ABX±h[uqghhibku\Zxlky\l_jfbgZoIRMkeh`gu_hdkb^ubkmevnb^u
AXY±IRMg_oZjZdl_jgudZj[hgZlugbljZlu[hjZlukmevnZlu
XYZ±oZjZdl_jghh[jZah\Zgb_fhe_dmeyjguokljmdlmj
Hibrbl_kljh_gb_djbklZeeh\i_jh\kdblZCaTiO3.
:lhfudZevpbybdbkehjh^Zkh\f_klghh[jZamxllj_okehcgmxIRMZlhfulblZgZaZgbfZxl q_l\_jlv hdlZw^jbq_kdbo imklhl <lhjhc \ZjbZgl hibkZgby Zlhfu lblZgZ b dbkehjh^Z
h[jZamxldZjdZkZgZeh]bqguckljmdlmj_ReO3ZlhfdZevpbygZoh^blky\p_glj_dm[bq_kdhc
yq_cdbOZjZdl_jkljmdlmjudZjdZkgZy]_l_jh^_kfbq_kdZyKljmdlmjgucdeZkkPm3m, Z = 1
(4/mmm; m3m; mmmDQCa) = 12 (dm[hhdlZw^jDQTi hdlZw^jDQH ]Zgl_ev
DZdb_baf_g_gby\ kljmdlmj_ BaTiO3 lbi i_jh\kdblZ ijhbkoh^yl ijb i_j_oh^_\k_]g_lhwe_dljbq_kdh_khklhygb_"
K_]g_lhwe_dljbdb±\_s_kl\Zkhkh[ufoZjZdl_jhf]bkl_j_abkaZ\bkbfhklbiheyjbaZpbbhl
gZijy`_gghklbwe_dljbq_kdh]hihey
Ijbi_j_oh^_\k_]g_lhwe_dljbq_kdh_khklhygb_Zlhfu[ZjbyblblZgZg_kdhevdhkf_sZxlky
ba b^_Zevguo iheh`_gbc \ kljmdlmj_ \ j_amevlZl_ ijhbkoh^bl nZah\uc i_j_oh^ II jh^Z
djbklZeeul_jyxlp_gljkbff_ljbb
MdZ`bl_\Z`g_crb_hkh[_gghklbkljh_gbydjbklZeeh\ribg_ebMgAl2O4.
:lhfu dbkehjh^Z h[jZamxl dm[bq_kdmx iehlg_crmx miZdh\dm :lhfu fZ]gby aZgbfZxl
\hkvfmxqZklvl_ljZw^jbq_kdboimklhlZZlhfuZexfbgby± q_l\_jlv hdlZw^jbq_kdbo Ijb
wlhfh[jZam_lky]hfh^_kfbq_kdZykljmdlmjZ?_hkh[_gghklvxy\ey_lky\hafh`ghklv©h[jZs_gbyª ± aZgylb_ qZklb hdlZw^jbq_kdbo imklhl ZlhfZfb lbiZ : Z khhl\_lkl\mxs_]h qbkeZ
l_ljZw^jbq_kdboimklhl± ZlhfZfb lbiZ < JZaf_s_gb_ Zlhfh\ f_lZeeh\ ih imklhlZf hij_^_ey_lkymklhcqb\hklvx^eygbokhhl\_lkl\mxsbodhhj^bgZpbc
DZdhc \ZjbZgl kljmdlmjgh]h lbiZ ribg_eb ghjfZevgZy h[jZs_ggZy ijhf_`mlhqgZyke_^m_lh`b^Zlv ^ey ^\hcguo hdkb^h\ NiFe2O4, ZnCr2O4, NiCr2O4" Hl\_l
h[hkgh\Zlv
ZnNi2O4±ghjfZevgZyribg_evZn(IIg_klZ[bebabjm_lkydjbklZeebq_kdbfihe_fgb\l_ljZw^jbq_kdhc gb \ hdlZw^jbq_kdhc dhhj^bgZpbyo baaZ aZiheg_ggh]h dih^mjh\gy Ni(III)
bf__lwe_dljhggmxdhgnb]mjZpbxd7[he__mklhcqb\mx\hdZlw^jbq_kdhfihe_eb]Zg^h\
NiCr2O4 ±ijhf_`mlhqgZyribg_evNi(II) (3d8) bCr(III) (3d3klZ[bebabjmxlky\hdZlw^jbq_kdhfihe_eb]Zg^h\
NiFe2O4±h[jZs_ggZyribg_evNi(II) (3d8klZ[bebabjm_lky\hdlZw^jbq_kdhcdhhj^bgZpbb
Fe(III) (3d5g_klZ[bebabjm_lkygb\hdlZw^jbq_kdhcgb\l_ljZw^jbq_kdhcdhhj^bgZpbyo
Hibrbl_kljmdlmjmK2PtCl6.
>Zggmxkljmdlmjmfh`ghhibkZlv^\mfykihkh[ZfbZlhfudZebybieZlbguh[jZamxldm[bq_kdmx iehlg_crmx miZdh\dm Zlhfu oehjZ aZgbfZxl \k_ l_ljZw^jbq_kdb_ imklhlu Zlhfu
13
dZebyboehjZh[jZamxl dm[bq_kdmx iehlg_crmx miZdh\dm Zlhfu ieZlbgu aZgbfZxl h^gm
\hkvfmxqZklvhdlZw^jbq_kdboimklhl
<q_fkoh^kl\hb\q_fjZaebqb_kljmdlmjdZevpblZbZjZ]hgblZ
Koh^kl\Zh^bgZdh\uc\Zeh\hckhklZ\\h[_bokljmdlmjZofh`gh\u^_eblv]jmiiuCO3.
JZaebqby \ kljmdlmj_ dZevpblZ CO3]jmiiu g_ khijbdZkZxlky \ kljmdlmj_ ZjZ]hgblZ wlb
]jmiiukhijbdZkZxlkyDjhf_wlh]h\kljmdlmj_dZevpblZbf_xlkydhgmkugme_\h]hjZkrbj_gby ± dhgbq_kdb_ ih\_joghklb g_ baf_gyxsb_ky ijb gZ]j_\Zgbb beb hoeZ`^_gbb <
kljmdlmj_ZjZ]hgblZih^h[gu_dhgmkuhlkmlkl\mxl
<q_faZdexqZ_lkyhkgh\gh_jZaebqb_djbklZeebq_kdbokljmdlmjZexfhkbebdZlh\bkbebdZlh\Zexfbgby"Ijb\_klbijbf_ju
<ZexfhkbebdZlZokms_kl\mxlZlhfuZexfbgbykdhhj^bgZpbhggufqbkehflZdb_Zlhfu
dhhj^bgbjh\Zgu\l_ljZw^jZolh_klv\oh^yl\ khklZ\ dj_fgbcdbkehjh^gh]h fhlb\Z Ijbf_juNaAlSiO4p_heblu
< kbebdZlZo Zexfbgby \k_ Zlhfu Zexfbgby bf_xl dhhj^bgZpbhggh_ qbkeh lh _klv g_
\g_^j_gu \ dj_fgbcdbkehjh^guc fhlb\ Ijbf_ju dZhebgbl Al2[Si2O5](OH)4 lhiZa
(Al2SiO4(OH)F).
Bah[jZabl_oZjZdl_jgu_dj_fg_dbkehjh^gu_fhlb\uhkljh\guokbebdZlh\b
mdZ`bl_bokhklZ\u
6.10 Ijb\_^bl_ ijbf_j oZjZdl_jgh]h jZkiheh`_gby dj_fg_dbkehjh^guo l_ljZw^jh\\kehbkluokbebdZlZobmdZ`bl_khklZ\kehy
6.11Bah[jZabl_oZjZdl_jgu_dj_fg_dbkehjh^gu_fhlb\up_ihq_qguokbebdZlh\
bmdZ`bl_bokhklZ\u
Ijb\_^bl_ijbf_juaZ\bkbfhklbk\hckl\kbebdZlh\hlbokljmdlmju
Al2(Si2O5)(OH)4 – dZhebgbl]ebgZbf__lkehbklmxkljmdlmjmbe_]dhgZ[moZ_lih]ehsZ_l
\h^mihkdhevdmfhe_dmeu\h^ue_]dhijhgbdZxlf_`^mkehyfbh[jZamy\h^hjh^gu_k\yab
«Kblh:» (Na12Al12Si12O48·24H2O) – p_hebl\ kljmdlmj_ bf_xlky hdgZ ^bZf_ljhf ~ 4.25 Å;
ijhimkdZ_l q_j_a k_[y fZe_gvdb_ fhe_dmeu gZijbf_j \h^m b aZ^_j`b\Z_l [hevrb_ gZijbf_jjZa\_l\ezgguchdlZg
KiZcghklv ± [hevrbgkl\h kbebdZlh\ h[eZ^Z_l ohjhr_c kiZcghklvx kihkh[ghklvx jZkdZeu\Zlvky \^hev gZijZ\e_gbc hij_^_ezgguo iehkdhkl_c [eZ]h^Zjy gZebqbx hkljh\guo p_ihq_qguobebkehbkluokljmdlmj
14
[SiO3]’
[Si2O5]’
[Si4O11]’
[Si6O17]’
15
[Si2O5]2’
DZdb_kljmdlmjgu_hkh[_gghklboZjZdl_jgu^eyp_heblh\"
1) DZjdZkgu_kljmdlmju
2) Al \oh^bl \ dj_fgbcdbkehjh^guc fhlb\ lh _klv ih djZcg_c f_j_ g_dhlhju_ Zlhfu
Zexfbgbydhhj^bgbjh\Zgu\l_ljZw^jZo
3) Bf_xlkyihehklbkh_^bgzggu_q_j_ahdgZpbdeul_ljZw^jh\– kbkl_fZ dZgZeh\
< q_f khklhbl hkgh\gh_ jZaebqb_ djbklZeebq_kdbo kljmdlmj dZj[hgZlh\ b
kbebdZlh\"
Kljmdlmju dZj[hgZlh\ ± hkljh\gu_ d\Zab]hfh^_kfbq_kdb_ [ebadb_ d ]hfh^_kfbq_kdbf
Zlhfum]e_jh^ZqZklhbf_xlDQ baaZkhojZg_gby\kljmdlmj_]jmiiCO3. KljmdlmjukbebdZlh\ – ]_l_jh^_kfbq_kdb_ jZaebqguo lbih\ hkljh\gu_ p_ihq_qgu_ kehbklu_ dZjdZkgu_Zlhfudj_fgby\k_]^Zdhhj^bgbjh\Zgu\l_ljZw^jZo
Hkh[_gghklbkljh_gbydZhebgblZ.
DZhebgbl (Al2(Si2O5)(OH)4 ± hkgh\gZy khklZ\eyxsZy ijbjh^ghc ]ebgu Bf__l kehbklmx
kljmdlmjm khklZ\ kehz\ ± [Si2O5]2’; e_]dh ]b^jZlbjm_lky b gZ[moZ_l baaZ aZo\ZlZ \h^u \
f_`kehcgh_ijhkljZgkl\h
Hkh[_gghklbkljh_gbyrZ[ZablZbnh`ZablZ.
RZ[Zabl(Ca2Al4Si8O24·12H22bnh`Zabl(Na13Ca12Mg11[Al59Si133O384]·235H2O) – p_heblu
lh_klvdZjdZkgu_ZexfhkbebdZlu\kljmdlmjZodhlhjuobf_xlkyihehklbkh_^bgzggu_q_j_ahdgZ<rZ[Zabl_hdgZr_kl_jgu_lh_klvpbdekhklhblbar_klbl_ljZw^jh\\nh`Zabl_
±^\_gZ^pZl_jgu_^bZf_lj~7.5 Å).
16
Hkh[_gghklbkljh_gbydjbklZeeh\©kblZ:ªb©kblZO».
«Kblh:» (Na12Al12Si12O48·24H22b«kblhO» (Na11AlnSi102-nO384·260H2O) – p_heblulh_klv
dZjdZkgu_ZexfhkbebdZlu\kljmdlmjZodhlhjuobf_xlkyihehklbkh_^bgzggu_q_j_ahdgZ
<©kbl_:ª^bZf_ljhdhgkhklZ\ey_l~ 4.25 Å, \©kbl_Oª~ 7.25 Å. Bkihevamxl^eyjZa^_e_gbyfhe_dmejZaebqguojZaf_jh\gZijbf_jghdlZgZbjZa\_l\ezggh]hhdlZgZ
DZdh\uhkgh\gu_hkh[_gghklbdjbklZeebq_kdbokljmdlmji_glZkbeh\?
I_glZkbeu±dZjdZkgu_ZexfhkbebdZlubkdmkkl\_ggu_p_hebluKh^_j`Zlmadb_iyl_jgu_
hdgZbkbkl_fmi_j_k_dZxsbokydZgZeh\dZdijyfuolZdbab]aZ]hh[jZaguoIhiZ^Zy\wlb
dZgZeuZlhfubebfhe_dmeui_j_oh^yl\bffh[bebah\Zggh_khklhygb_ihwlhfmi_glZkbeu
bkihevamxdZddZlZebaZlhju
Ijb\_^bl_ ijbf_ju djbklZeebq_kdbo \_s_kl\ hlghkysboky d ]hfhfhe_dmeyjgufb]_l_jhfhe_dmeyguf.
=hfhfhe_dmeyjgu_khklhylbaobfbq_kdbb^_glbqguofhe_dmegZnlZebgn_ghelheZg
=_l_jhfhe_dmeyjgu_ khklhyl ba obfbq_kdb jZaebqguo fhe_dme COOH)2·2H2O,
K2COOH)2·2H2O, deZljZlu (kh_^bg_gby \ kljmdlmjZo dhlhjuo h^gb fhe_dmeu h[jZamxl
dZjdZkZ^jm]b_±aZgbfZxl_]himklhlu
Qlh lZdh_ fhgh b ihebkbkl_fgu_ fhe_dmeyjgu_ djbklZeeu" Ijb\_^bl_
ijbf_ju
< fhghkbkl_fguo djbklZeeZo \k_ fhe_dmeu aZgbfZxl h^gm hj[blm \ ihebkbkl_fguo ± g_kdhevdhhj[bl
Fhghkbkl_fgu_djbklZeeugZnlZebg]_dkZf_lbe[_gahef_lhdkbn_ghe
Ihebkbkl_fgu_ djbklZeeulheZghj[blun_ghehj[bluk_e_ghfhq_\bgZhj[bl
DZdh\uhkh[_gghklbfhe_dmeyjguodjbklZeeh\hlghkysbokydlbiZf:<b
K"Ijb\_^bl_ijbf_jukljmdlmj
Lbi: ±\k_fhe_dmeu\djbklZee_objZevgubwd\b\Ze_glgulh_klvgZoh^ylky\h^ghcba
hjb_glZpbcijZ\hcbebe_\hcLbibqgu_kljmdlmjgu_deZkku: P212121, Z = 4 (1); P21, Z = 2
(1). Ijbf_ju^bf_lben_ghe^bf_lben_ghe
Lbi < ± \ djbklZee_ ijbkmlkl\mxl dZd e_\u_ lZd b ijZ\u_ fhe_dmeu jZp_fZl Lbibqgu_
kljmdlmjgu_deZkku: P21/c, Z = 4(1); P1, Z = 2(1); Pbca, Z = 8(1). Ijbf_juf_lhdkbn_ghe
y[ehqgZydbkehlZ
Lbi K ± \k_ fhe_dmeu \ djbklZee_ ZobjZevgu Lbibqgu_ kljmdlmjgu_ deZkku:
P21 / c, Z = 2(1); P1, Z = 1(1); Pbca , Z = 4(1). Ijbf_ju gZnlZebg ]_dkZf_lbe[_gahe mjhljh-
ibg[_gahe
>ZggZydeZkkbnbdZpbyhlghkblkyd]hfhfhe_dmeyjguf, fhghkbkl_fgufdjbklZeeZf
GZah\bl_g_kdhevdhkljmdlmjguodeZkkh\baqbkeZgZb[he__rbjhdhjZkijhkljZgzgguo kj_^b ]hfhfhe_dmeyjguo hj]Zgbq_kdbo djbklZeeh\ Ijb\_^bl_ ijbf_ju
kljmdlmjhlghkysbokydwlbfdeZkkZf
Lbi: – P212121, Z ^bf_lben_ghe
Lbi< – P21/k, Z f_lhdkbn_ghe
LbiK – P21/k, Z = 2 (1); gZnlZebgZgljZp_gn-^boehj[_gaheαfh^bnbdZpby
DZdhc we_f_gl kbff_ljbb qZs_ \k_]h \klj_qZ_lky \ hj]Zgbq_kdbo djbklZeeZo"DZdZykbg]hgbygZb[he__rbjhdhjZkijhkljZg_gZkj_^bwlbo\_s_kl\"Ijb\_^bl_
ijbf_ju
17
GZb[he__ jZkijhkljZgzgguc we_f_gl kbff_ljbb ± hkv 1 gZb[he__ jZkijhkljZgzgguc
kljmdlmjgucdeZkk±P21/c, Z = 4(1) – fhghdebggZykbg]hgbyIjbf_j±f_lhdkbn_ghe
QlhlZdh_dhglZdlgZydhgnhjf_jby"DZdh\u_zlbibqgu_ijhy\e_gby"Ijb\_^bl_ijbf_ju.
DhglZdlgZydhgnhjf_jby±wlhkhkms_kl\h\Zgb_\h^ghfdjbklZee_h^bgZdh\uoobfbq_kdb
fhe_dmebf_xsbojZagmx]_hf_ljbq_kdmxnhjfm
Lbibqgu_ijhy\e_gby
1) GZebqb_ \ h^ghf djbklZee_ fhe_dme \ jZaguo dhgnhjfZpbyo GZijbf_j pbdeh]_dkZg \
dhgnhjfZpbyo©dj_kehªb©\ZggZªP2/c, Z = 4 (1; 2)); L-bahe_cpbg\©pbkªb©]hrªnhjfZoP2/c, Z = 4(1; 1)).
2) DhgnhjfZpbhggucbahfhjnbaf±bahfhjnbafdjbklZeeh\kh^_j`Zsbofhe_dmeu\jZaguodhgnhjfZpbyogZijbf_j\jZs_gb_f_lhdkb]jmiiu\hkgh\ZgbyoRbnnZ
Ijb\_^bl_ ijbf_ju ehdZevghc ik_\^hkbff_ljbb fhe_dmeyjguo ih^kbkl_f \
hj]Zgbq_kdbodjbklZeeZo
EhdZevgZyik_\^hkbff_ljby±kbff_ljbydhlhjZyfh`_l[ulv\u^_e_gZ\hl^_evguoqZklyo
djbklZeeZghhlkmlkl\m_l\djbklZee_\p_ehfIjbf_jun_ghe±ehdZevgZykbff_ljbyp_i_c Pc312, Z = 3(2); C19H23NO3 (ijhba\h^gh_ ^bn_gbehdkb^Z ± ehdZevgZy kbff_ljby kehz\
Pl21/c (kbff_ljbydjbklZeebq_kdhckljmdlmju± P1, Z = 2(1)).
Ijb\_^bl_ijbf_ju]bi_jkbff_ljbb\hj]Zgbq_kdbodjbklZeeZo
=bi_jkbff_ljby ± we_f_glu kbff_ljbb y\eyxsb_ky dhf[bgZpb_c aZdjuluo we_f_glh\
kbff_ljbbbk^\b]ZgZijhba\hevguc\_dlhjijbwlhf^Zggu_hi_jZpbblZd`_y\eyxlkyaZdjulufblh_klvfh]ml[ulvijbf_g_guebrvh^bgjZaIjbf_ju\kljmdlmj_lheZgZbf__lkyhkvq; \kljmdlmj_D,L-f_lbeyglZjghcdbkehlu±iehkdhklvmp.
DZdhc we_f_gl kh[kl\_gghc kbff_ljbb fhe_dmeu qZs_ \k_]h khojZgy_lky \
djbklZeebq_kdhc kljmdlmj_" Ijb\_^bl_ ijbf_ju khojZg_gby wlh]h we_f_glZ kbff_ljbb\djbklZee_
Kh]eZkgh ijZ\bem DblZc]hjh^kdh]h fhe_dmeu \ djbklZeeZo h[uqgh l_jyxl kh[kl\_ggu_
we_f_glu kbff_ljbb aZ bkdexq_gb_f p_gljZ kbff_ljbb Z _keb fhe_dmeZ h[eZ^ZeZ p_gljhf kbff_ljbb lh hgZ ihiZ^Z_l \ ihabpbx 1 ijhkljZgkl\_gghc ]jmiiu LZdbf h[jZahf
wlhlwe_f_glkbff_ljbb±p_gljbg\_jkbb. Ijbf_jugZnlZebgP21/k, Z = 2 (1)); ]_dkZf_lbe[_gahe (P1, Z = 1(1)); [_gahePbca, Z = 4 (1)).
7 GZah\bl_ kljmdlmjgu_ deZkku \ dhlhjuo Z \k_ fhe_dmeu hjb_glbjh\Zgu
h^bgZdh\h[bf_xlky^\_jZaebqgu_hjb_glZpbb.
Z P1, Z = 1(1).
[31/c, Z = 2 (1); P1, Z = 2(1); P21, Z = 2(1); Pbca, Z = 4 (1).
KdhevdhjZaebqguohjb_glZpbcfhe_dmeqZs_\k_]hgZ[ex^Z_lky\kljmdlmjZohj]Zgbq_kdbodjbklZeeh\"MdZ`bl_khhl\_lkl\mxsb_kljmdlmjgu_deZkku
QZs_ \k_]h gZ[ex^Zxlky jZaebqgu_ hjb_glZpbb Kljmdlmjgu_ deZkku P212121, Z = 4(1);
P21/c, Z = 4(1).
7.12. Ko_fZlbq_kdb bah[jZabl_ jZkiheh`_gb_ fhe_dme \ djbklZeeZo gZnlZebgZ
KdhevdhjZaebqguohjb_glZpbcfhe_dmegZ[ex^Z_lky\wlhf\_s_kl\_"
18
hjb_glZpbb
DZdb_ fhe_dmeyjgu_ Z]ehf_jZluqZs_\k_]h\klj_q\hj]Zgbq_kdbodjbklZeeZo"
Dhg_qgu_Z]ehf_jZlu±hkljh\Zbebh[ju\db p_i_c gZijbf_j ^bf_ju \ kljmdlmjZo
ijhibhgh\hcbfZkeyghcdbkehl
;_kdhg_qgu_ Z]ehf_jZlu ± kehb beb p_ib
P_ib ©dhgvdh\uc rZ]ª Pc21 – 2,5^bf_lben_ghe Pcg – f_lhdkb
l_ljZf_lben_ghe ©[mkuª Pc1 – ^bf_lbe]ebhdkbf
Kehb ©iZjd_lguc kehcª Pl21/c – ]ebhdkbf
gZnlZebg
< q_f kmsghklv l_hjbb iehlghcmiZdh\dbfhe_dme?
1) Kms_kl\m_l kbkl_fZ \Zg^_j\ZZevkh\uo
jZ^bmkh\ we_f_glh\hj]Zgh]_gh\ lZdbo qlh fhe_dmeu fh^_ebjm_fu_ \g_rg_c ih\_joghklvxi_j_k_dZxsbokyZlhfguokn_jklZdbfbjZ^bmkZfbdZkZxlkylh_klvg_ijhgbdZxl^jm]\^jm]Zbg_\bkyl\imklhl_\Zg^_j\ZZevkh\ujZ^bmku±Zlhfgu_jZ^bmku
^ey\Ze_glghk\yaZgguoZlhfh\
2) < ex[hf fhe_dmeyjghf djbklZee_ _klv dhglZdlu ©\uklmi\iZ^bgZª dhlhju_ hij_^_eyxlkljmdlmjmdjbklZeeZb__mklhcqb\hklv
3) MiZdh\dZ fhe_dme \ djbklZee_ g_miehlgy_fZ gh b g_ y\ey_lky gZbiehlg_cr_c lh _klv
g_evay fZeufb baf_g_gbyfb ^hklbqv kms_kl\_ggh]h baf_g_gby dhwnnbpb_glZ iehlghklbmiZdh\db
QlhlZdh_\Zg^_j\ZZevkh\ZlhfgucjZ^bmk?
WlhZlhfgucjZ^bmk^ey\Ze_glghk\yaZggh]hZlhfZhij_^_ey_lkyklZlbklbq_kdbfZgZebahf
^ebgk\ya_c\[hevrhfdhebq_kl\_kljmdlmj?keb d(A–A) = l, lh rA = l/2; _keb d(A–B) = l, lh
2 rA rB = l.
Ijb\_^bl_ijbf_juki_pbnbq_kdbof_`fhe_dmeyjguodhglZdlh\
1) >\moZlhfgu_ dhglZdlu Cl–Cl \ oehj[_gahe_ S–S, M–M \ ^bf_lbe]ebhdkbfZl_ gbd_ey
M–O \kZebpbeZebfbgZl_gbd_eyb^j
2) LjzoZlhfgu_ dhglZdlu ± \h^hjh^gu_ k\yab XH...Y ^bf_lben_ghe ^bf_lben_ghemjZpbe
3) IhebZlhfgu_dhglZdlu±dhglZdlu[_gahevguodhe_p\[_gahe_hdj_ahe_]_dkZf_lbe[_gahe_
F_`fhe_dmeyjguc dhglZdl ± kms_kl\_ggh_ i_j_dju\Zgb_ \Zg^_j\ZZevkh\uo kn_j g_kdhevdboZlhfh\
MdZ`bl_\Z`g_crb_lbiuki_pbnbq_kdbof_`fhe_dmeyjguodhglZdlh\[_gahevguodhe_p
1) ©Kwg^\bqª±]_dkZf_lbe[_gahe
2) Hjlh]hgZevgu_ [_gahevgu_ dhglZdlu Lh[jZagu_ dhglZdlu kh k^\b]hf ± [_gahe hdj_aheZfbghoehj[_gahe
19
3) Kdhr_ggu_ dhglZdlu ©iZjd_lguc kehcª ©dhgvdh\uc rZ]ª ± gZnlZebg ^bf_lben_ghef_lhdkbn_ghe
DZdh\uoZjZdl_jgu_agZq_gbyhkgh\guo]_hf_ljbq_kdbooZjZdl_jbklbd\h^hjh^ghck\yab
ϕg_fgh]h[hevr_0; R ≈ 2 rX ry ≥ rX + rY . LZdbfh[jZahf\hagbdZ_l
g_[hevrh_i_j_dju\Zgb_Zlhfguokn_jX bY.
Ijb\_^bl_ ijbf_ju GZkkhpbZlh\ k mdZaZgb_f kbf\heh\
khhl\_lkl\mxsbo]jZnh\
m
H[sbckbf\he]jZnZ: Gn ( k ), ]^_G±nhjfZZ]ehf_jZlZI±hkljh\K±p_ivL±kehcF –
dZjdZk m ± djZlghklv \_jrbgu qbkeh k\ya_c bkoh^ysbo ba ^Zgghc \_jrbgu n ± qbkeh
\_jrbgkdhlhjufb^ZggZy\_jrbgZk\yaZgZ\h^hjh^gufbk\yayfbk ±jZaf_jghklvpbdeZ
2
2
Ijbf_juGZkkhpbZlh\: C2 (0) ±fmjZ\vbgZybmdkmkgZydbkehlu I1 (2) ±\ukrb_dZj[hgh4
\u_dbkehluijhibhgh\ZyfZkeygZyb^j L3 (2; 6) – mjZpbe
QlhlZdh_khijy`zggu_dhhi_jZlb\gu_\h^hjh^gu_k\yab?
Wlh Z]j_]Zpby bg^b\b^mZevguo \h^hjh^guo k\ya_c k h[jZah\Zgb_f _^bgh]h Z]j_]ZlZ >ey
hibkZgbylZdbok\ya_cbkihevamxl]jZnudhlhju_fh]ml[ulvdZddhg_qgufblZdb[_kdhg_qgufbH[sbckbf\he]jZnZ±\\hijhk_
Qlhij_^klZ\ey_lkh[hc gZ[hj kljmdlmjguo iZjZf_ljh\ hibkZgby kljh_gby
fhe_dmeyjguodjbklZeeh\"
>eyfhghkbkl_fguodjbklZeeh\
1) IZjZf_ljuwe_f_glZjghcyq_cdbZ, b, c, α, β, γ.
2) Dhhj^bgZlubwce_jh\um]eufhe_dmeuxi, yi, zi, αi, βi, γi.
3) M]eudhgnhjfZpbhggh]hih\hjhlZϕ, θ, ψ.
DZd \uqbkey_lky wg_j]by f_`fhe_dmeyjgh]h \aZbfh^_ckl\by \ Zlhf
Zlhfghfijb[eb`_gbb"
(
)
U ij = ∑ ϕ ij + ψ ij , ]^_ϕij – ihl_gpbZe\aZbfh^_ckl\byZlhfh\\oh^ysbo\fhe_dmeuψij –
i, j
ihl_gpbZe\aZbfh^_ckl\byhklZlhqguoaZjy^h\Zlhfh\; i,j – ghf_jZ Zlhfh\ \oh^ysbo \h
\aZbfh^_ckl\mxsb_fhe_dmeu
−6
−12
Kms_kl\mxl ^\Z kihkh[Z jZkqzlZ ihl_gpbZeZ ϕij: ϕ ij = − Aij rij + Bij rij
– ihl_gpbZe E_g−6
gZj^Z>`hgkhgZ beb ϕ ij = − Aij rij + Bij exp −α rij – ihl_gpbZe ;Zdbg]_fZBhjg_jZ A^_kv
Aij, Bij – wfibjbq_kdb_ dhgklZglu ih^[bjZ_fu_ ^ey dZ`^hc iZju Zlhfh\ rij – jZkklhygb_
qi q j
f_`^mZlhfZfbψ ij =
.
rij
(
)
7DZd\uqbkey_lkyihl_gpbZevgZywg_j]byhj]Zgbq_kdh]hdjbklZeeZ"
Ihl_gpbZevgZy wg_j]by hj]Zgbq_kdh]h lh _klv fhe_dmeyjgh]h djbklZeeZ ± wg_j]by dhlhjmx g_h[oh^bfh aZljZlblv ^ey m^Ze_gby ba djbklZeeZ h^ghc fhe_dmeu LZdbf h[jZahf
U = ∑ U m , ]^_Um – wg_j]by\aZbfh^_ckl\byjZkkfZljb\Z_fhcfhe_dmeukh^ghcba^jm]bo
m
20
fhe_dmedjbklZeeZ<j_ZevghklbgZagZq_gb_wg_j]bbhdZau\ZxlagZq_gb_lhevdhl_fhe_dmeujZkklhygb_^hdhlhjuog_ij_\urZ_l±Å, ihwlhfmkmffbjh\Zgb_ijhba\h^blkyih
h]jZgbq_gghfmqbkemfhe_dme
Qlh ij_^klZ\ey_l kh[hc wg_j]by hj]Zgbq_kdh]h djbklZeeZ dZd nmgdpby
kljmdlmjguoiZjZf_ljh\"Qlhfh`ghba\e_qvbawlhcnmgdpbb"
Wg_j]byhj]Zgbq_kdh]hdjbklZeeZ±nmgdpbyfgh]boi_j_f_gguokljmdlmjguoiZjZf_ljh\
_z ]jZnbq_kdbf bah[jZ`_gb_f y\ey_lky ih\_joghklv \ fgh]hf_jghf ijhkljZgkl\_ < b^_Zevghf kemqZ_ ]eh[Zevguc fbgbfmf wlhc nmgdpbb ^he`_g khhl\_lkl\h\Zlv wdki_jbf_glZevghgZ[ex^Z_fufkljmdlmjgufiZjZf_ljZfGZkZfhf^_e_wdki_jbf_glZevgufnZdlZf
khhl\_lkl\m_lh^bgbaehdZevguofbgbfmfh\qlhk\yaZghkhij_^_ezggufbij_g_[j_`_gbyfb ijb jZkqzl_ ihl_gpbZeZ f_`fhe_dmeyjgh]h \aZbfh^_ckl\by Z lZd`_ l_f qlh kljmdlmjZ
hij_^_ey_lkyfbgbfmfhfk\h[h^ghcwg_j]bbdjbklZeeZZg_ihl_gpbZevghcwg_j]bb
Qlh lZdh_ kljmdlmjgu_ ih^deZkku hj]Zgbq_kdbo djbklZeeh\" Ijb\_^bl_
ijbf_ju
Kljmdlmjguc ih^deZkk ± kh\hdmighklv djbklZeeh\ hlghkysboky d ^Zgghfm kljmdlmjghfm
deZkkmbkh^_j`Zsboh^bgZdh\u_ihkbff_ljbbfhe_dmeyjgu_Z]ehf_jZluh^bgZdh\hhjb_glbjh\Zggu_ihhlghr_gbxdwe_f_glZfkbff_ljbb
Ijbf_ju: 1 − Pl (YZ ) 21 / c − P21 / c ± gZnlZebg 1 − Pc (Y ) 21 ↑↑ Pl (YZ ) 21 ↑↓ P212121, Z = 4 –
^bf_lben_ghe1 − Pc (Y ) 21 ↑↑ Pl (YZ ) 21 ↑↑ P21, Z = 2 ±^bf_lben_ghe
DZdmxdhgnhjfZpbxiehkdmxbebg_iehkdmxbf_xlfhe_dmeu[bn_gbeZ\
djbklZeeZo hlghkysboky d kljmdlmjghfm deZkkm 31/c, Z = 2(1)? Hl\_l h[hkgh\Zlv
Fhe_dmeZbf__liehkdmxdhgnhjfZpbx ihkdhevdm g_iehkdZy fhe_dmeZ [bn_gbeZ kbff_ljbyg_kh^_j`blp_gljZbg\_jkbb
MdZ`bl_kljmdlmjgucdeZkkbih^deZkkdjbklZeeh\gZnlZebgZbZgljZp_gZ
KljmdlmjgucdeZkk: P21/c, Z = 2(1); kljmdlmjgucih^deZkk: 1 – Pl21/c ↑↑P21/c, Z = 2Ijbkmlkl\mxsb_ fhe_dmeyjgu_ Z]ehf_jZlu ij_^klZ\eyxl kh[hc ©iZjd_lgu_ª kehb iZjZee_evgh_gZeh`_gb_dhlhjuoijb\h^bldh[jZah\ZgbxieZklbgqZluodjbklZeeh\
DZdh\uhkh[_gghklbkljh_gbydjbklZeebq_kdh]h[_gaheZ"
KljmdlmjgucdeZkkPbca, Z = 4(1). < djbklZeeZo ijbkmlkl\mxl ©hjlh]hgZevgu_ [_gahevgu_
kehbª h[jZah\Zggu_ ©hjlh]hgZevgufb [_gahevgufb dhglZdlZfbª Lh[jZagu_ dhglZdlu kh
k^\b]hf\j_amevlZl_k^\b]Zkbff_ljbydhglZdlZihgb`Z_lkyg_mmZm).
DZdbfb ]jZnZfb hibku\Z_lky kbkl_fZ \h^hjh^guo k\ya_c 2+2 \ djbklZeeZodZj[hgh\uodbkehl&nH2n+1&22+]^_&nH2n+1 – gZedbe
2
n = 0; 1 (fmjZ\vbgZybmdkmkgZydbkehlu± C2 (0).
2
n > 1 (ijhibhgh\ZyfZkeygZyb^jdbkehlu± I1 (2).
DZdh\uhkh[_gghklbdjbklZeebq_kdbokljmdlmjn_gheZbhdj_aheZ"
<h[_bokljmdlmjZoijbkmlkl\mxlp_ibkhijy`_gguo+k\ya_ck\bglh\hchkvx1dhlhjZy
y\ey_lky\kemqZ_hdj_aheZlhqghcdjbklZeeh]jZnbq_kdhchkvxZ\kemqZ_n_gheZ±ehdZevghc hkvx ik_\^hkbff_ljbb < djbklZeebq_kdhf hdj_ahe_ bf__lky djhf_ lh]h kibjZevgZy
21
kbkl_fZ khijy`_gguo ©hjlh]hgZevguo [_gahevguo dhglZdlh\ª gZ i_jbh^ lZdhc kibjZeb
ijboh^blky^_\ylv[_gahevguodhe_p
MdZ`bl_kljmdlmjgucdeZkkbhibrbl_hkh[_gghklbdjbklZeebq_kdhckljmdlmju]_dkZf_lbe[_gaheZ
J_^dbckljmdlmjgucdeZkk P1, Z = 1(1); fhe_dmeyjgZykljmdlmjZ;jZ\w±\k_fhe_dmeuljZgkeypbhgghk\yaZgubbf_xlh^bgZdh\mxhjb_glZpbx;_gahevgu_dhevpZh[jZamxl©kdhr_ggu_ªklhidb\dhlhjuo[_gahevgu_dhevpZkhijbdZkZxlkyh[jZamydhglZdlulbiZ©kwg^\bqª
kh k^\b]hf \ lh `_ \j_fy dZ`^Zy fhe_dmeZ e_`bl \ iehkdhklb ik_\^h]_dkZ]hgZevgh]h
©kehyª H^gZ ba i_j\uo hj]Zgbq_kdbo djbklZeebq_kdbo kljmdlmj bamq_gguo k ihfhsvx
j_gl]_ghkljmdlmjgh]hZgZebaZEhgk^_ce
DZdh\uhkh[_gghklbdjbklZeebq_kdbokljmdlmj^bf_lbe]ebhdkbfZlZgbd_ey
bkZebpbeZebfbgZlZgbd_ey"
< djbklZeeZo ^bf_lbe]ebhdkbfZlZ gbd_ey ijbkmlkl\mxl p_ib kh^_j`Zsb_ ki_pbnbq_kdb_
f_`fhe_dmeyjgu_dhglZdlu1L1LwlbdhglZdlukhijy`_gu±\hagbdZxl[_kdhg_qgu_p_ib
1L1L1L<kZebpbeZebfbgZl_gbd_eyfhe_dmeuh[jZamxliZjd_lgu_kehbf_`^mfhe_dmeZfbbf_xlkyki_pbnbq_kdb_f_`fhe_dmeyjgu_dhglZdlu1L2
MdZ`bl_kljmdlmjgucdeZkkbhibrbl_hkh[_gghklbdjbklZeebq_kdhckljmdlmjulheZgZ
< djbklZeeZo lheZgZ ^bn_gbeZp_lbe_gZ hlghkysboky d kljmdlmjghfm deZkkm 31/c,
2
Z = 4(1 ), kbff_ljbq_kdbg_aZ\bkbfu_fhe_dmeuij_h[jZamxlky^jm]\^jm]Z]bi_jkbff_ljbq_kdhchi_jZpb_cqih\hjhlgZƒkhk^\b]hfhkvqiZjZee_evgZiehkdhklbYZbh[jZam_lm]heω ≈ƒkiehkdhklvxXZ:gZeh]bqgufbhkh[_gghklyfbh[eZ^Zxlihqlb\k_djbklZeeuwlh]hkljmdlmjgh]hdeZkkZ
MdZ`bl_hkh[_gghklbdjbklZeebq_kdbokljmdlmjb^bf_lben_gheZ
f_lhdkbn_gheZ
<h \k_o gZa\Zgguo djbklZeeZo bf_xlky kbkl_fu khijy`_gguo k\ya_c 2+2 ihkj_^kl\hf
dhlhjuofhe_dmeuh[t_^bgyxlky\p_ib3c21, Z <djbklZeeZo^bf_lben_gheZp_ibgZdeZ^u\ZxlkyiZjZee_evghkh[jZah\Zgb_fkeh_\3l (YZ) 21, Z = 2(1); ZglbiZjZee_evgh_gZeh`_gb_keh_\^Z_ldjbklZeedeZkkZ312121, Z = <djbklZeeZo^bf_lben_gheZZgZeh]bqgu_ kehb kh^_j`Zsb_ ZgZeh]bqgu_ p_ib gZeZ]Zxlky iZjZee_evgh k h[jZah\Zgb_f
kljmdlmjuhlghkys_ckyddeZkkm31, Z = <djbklZeebq_kdhff_lhdkbn_ghe_bf_xlky
e_\u_bijZ\u_p_ib3c21, Z=2 dhlhju_h[jZamxlkehbkkbff_ljb_c3l21/cgZeZ]Zxsb_kyiZjZee_evghkh[jZah\Zgb_fjZp_fZlZ31/c, Z = 4(1).
MdZ`bl_ hkh[_gghklb djbklZeebq_kdhc kljmdlmju n-^boehj[_gaheZ αfh^bnbdZpby
KljmdlmjgucdeZkkgZnlZebgZ31/c, Z = 2(1).DZdb\gZnlZebg_fhe_dmeuh[jZamxl©iZjd_lgu_ kehbª GZ \g_rgbo ih\_joghklyo wlbo keh_\ k h[_bo klhjhg jZkiheZ]Zxlky Zlhfu
oehjZ Ijb gZeh`_gbb keh_\ h[jZamxlky f_`fhe_dmeyjgu_ dhglZdlu Cl...&O ^ebghc Å
l_gZÅ [he__dhjhldb_q_frClLZdbfh[jZahf\hagbdZ_l[_kdhg_qguc\^\mobaf_j_gbyo&OZ]j_]Zl
MdZ`bl_hkh[_gghklbdjbklZeebq_kdhckljmdlmjuoehj[_gaheZ
:lhfuoehjZkhk_^gbofhe_dmek[eb`_gudjZlqZcrb_dhglZdluCl...&ObÅ); \
blh]_\hagbdZ_l[_kdhg_qguc&OZ]j_]Zlbf_xsbc\b^kl_j`gy
22
MdZ`bl_hkh[_gghklbdjbklZeebq_kdhckljmdlmjuZfbghn_gheZbZfbgh
oehjn_gheZ
DjbklZeeuZfbghoehjn_gheZkhklhylbakeh_\bf_xsboke_^mxs__kljh_gb_[_gahevgu_ dhevpZ h[jZamxl hjlh]hgZevguc [_gahevguc kehc 2%]/ Z]j_]Zl hjlh]hgZevguo [_gahevguodhglZdlh\hq_gvihoh`bcgZlhldhlhjucgZ[ex^Z_lky\djbklZeebq_kdhf[_gahe_Zlhfu&OjZkiheZ]Zxlkyihh^gmklhjhgmwlh]hkehy2+b1+2]jmiiu±ih^jm]mxGZeh`_gb_keh_\ijhbkoh^bllZdqlhh[t_^bgyxlkyZlhfu&O^\mokeh_\h[jZamy[_kdhg_qguc
\^\mobaf_j_gbyo&OZ]j_]Zlb+kh^_j`Zsb_]jmiiuh[jZamxsb_f_`keh_\u_+k\yab<
djbklZee_ ihke_^h\Zl_evgh q_j_^mxlky kehc 2%]/ &OZ]j_]Zl kehc 2%]/ kehc k\ya_c
NH22b2+1kehc2%]/bl^
<Zfbghn_ghe_dhglZdluCl…Clhlkmlkl\mxlihwlhfmk\yabf_`^mkehyfbklZgh\ylkyf_g__ijhqgufb
MdZ`bl_hkh[_gghklbdjbklZeebq_kdhckljmdlmjumjZpbeZ
Fhe_dmeuihiZjghh[t_^bgyxlky^\mfyk\yayfb1+2\iehkdb_p_gljhkbff_ljbqgu_^bf_ju < k\hx hq_j_^v ^bf_ju k\yayfb 1+2 b &+2 h[t_^bg_gu \ iehkdb_ iZjd_lgu_
kehblZdqlhdZ`^uc^bf_jk\yaZgkq_lujvfy[eb`Zcrbfb^bf_jZfbf_`^m^\mfyk\yaZggufb ^bf_jZfb bf__lky h^gZ k\yav 1+2 b ^\_ k\yab &+2 KljmdlmjZ beexkljbjm_l
jhevkeZ[uok\ya_c&+2\bohlkmlkl\b_kehbg_[ueb[uiehkdbfb=jZnk\ya_c1+2
[_amq_lZk\ya_c&+2 – L43 (2, 6).
Hkh[_gghklb djbklZeebq_kdhc kljmdlmju ^\mo ihebfhjnguo fh^bnbdZpbc
f_lbemjZpbeZ
<h[_bofh^bnbdZpbyofhe_dmeuihiZjgh h[t_^bg_gu \ iehkdb_ p_gljhkbff_ljbqgu_ ^bf_ju ^\mfy k\yayfb 1+2 < αnhjf_ ^bf_ju h[t_^bg_u k\yayfb 1+2 \ kehb f_`^m
^\mfyk\yaZggufb^bf_jZfbbf__lkyh^gZk\yav1+2]jZnk\ya_c1+2± L43 (2, 6). <βnhjf_ ^bf_ju h[t_^bg_gu k\yayfb 1+2 \ e_glu f_`^m ^\mfy k\yaZggufb ^bf_jZfb
bf_xlky^\_k\yab1+2]jZnk\ya_c1+2± C24 (2). IjbjZkl\hj_gbbwlboihebfhjnh\\
jZkl\hj_ \ i_j\hf kemqZ_ ijbkmlkl\mxl hl^_evgu_ ^bf_ju Z \h \lhjhf kemqZ_ ± h[ju\db
e_glqlhh[tykgy_lkms_kl\_ggh_jZaebqb_e_dZjkl\_gguok\hckl\f_lbemjZpbebkihevam_lky^eyaZ`b\e_gbyjZgijbq_fβnhjfZij_^ihqlbl_evgZ
MdZ`bl_hkh[_gghklbdjbklZeebq_kdhckljmdlmjumjhljhibgZ
I_j\Zy bamq_ggZy k ihfhsvx j_gl]_ghkljmdlmjgh]h f_lh^Z \ ] kljmdlmjZ hj]Zgbq_kdh]hdjbklZeeZMgbdZevguc^eyhj]Zgbq_kdbo\_s_kl\kemqZcdm[bq_kdhckbff_ljbbFhe_dmeZ \ djbklZee_ iheghklvx khojZgy_l kh[kl\_ggmx kbff_ljbx 3m Kljmdlmjguc deZkk
I43m, Z = 2(43m).
>Zcl_hibkZgb_©keh\_kgucihjlj_lªmdZaZgghckljmdlmju
Cu:Zlhfu\\_jrbgZodm[bq_kdhcyq_cdbb\p_gljZo\k_o__]jZg_c
Mg: Zlhfu \ \_jrbgZo ]_dkZ]hgZevghc yq_cdb b \ p_glj_ h^ghc ba ^\mo ljb]hgZevguo
ijbaf gZ dhlhju_ ^_eblky ]_dkZ]hgZevguc iZjZee_e_ibi_^ iehkdhklvx ijhoh^ys_c q_j_a
fZeu_h[t_fgu_^bZ]hgZebyq_cdb
α-Fe:Zlhfu\\_jrbgZob\p_glj_dm[bq_kdhcyq_cdb
α-Po:Zlhfu\\_jrbgZodm[bq_kdhcyq_cdb
:efZaZlhfu&\\_jrbgZodm[bq_kdhcyq_cdb\p_gljZo__]jZg_cb\p_gljZoq_luj_oba
\hkvfbhdlZglh\\rZofZlghfihjy^d_
Li2O:ZlhfuH\\_jrbgZob\p_gljZo\k_o]jZg_cLi±\p_glj_\k_o\hkvfbhdlZglh\
23
Cu2O:Zlhfu2\\_jrbgZob\p_glj_dm[bq_kdhcyq_cdbZlhfu&X\p_gljZoq_luj_oba
\hkvfbhdlZglh\\rZofZlghfihjy^d_
ZnSknZe_jblZlhfu6\\_jrbgZodm[bq_kdhcyq_cdbb\p_gljZo__]jZg_cZlhfu=Q\
p_gljZoq_luj_oba\hkvfbhdlZglh\\rZofZlghfihjy^d_
CaF2:Zlhfu&D\\_jrbgZodm[bq_kdhcyq_cdbb\p_gljZo\k_o__]jZg_cZlhfu)\ p_gljZo\k_o\hkvfbhdlZglh\
NaCl:Zlhfu1D\\_jrbgZodm[bq_kdhcyq_cdbb\p_gljZo\k_o]jZg_cZlhfu&O\p_glj_
yq_cdbb\k_j_^bgZo\k_o__j_[_j
CsCl:Zlhfu&O\\_jrbgZodm[bq_kdhcyq_cdbZlhf&V\__p_glj_
CaTiO3:Zlhfu7L\\_jrbgZodm[bq_kdhcyq_cdbZlhf&D\__p_glj_Zlhfu2\k_j_^bgZo
\k_oj_[_jyq_cdb
Cu3Au:Zlhfu$X\\_jrbgZodm[bq_kdhc yq_cdb Zlhfu &X \ p_gljZo \k_o ]jZg_c yq_cdb
Hibrbl_mdZaZggmxkljmdlmjm\l_jfbgZoIRMIRD.
La:q_lujzokehcgZyIRM
Sm:^_\ylbkehcgZyIRM
CuAu:ljzokehcgZyIRMkg_kdhevdh©jZa^\bgmlufbªkehyfbh[jZah\ZggufbZlhfZfbCu
b$Xkh\f_klgh
Cu3Au:ljzokehcgZyIRMiehlgu_kehbh[jZah\ZguZlhfZfbCu bAukh\f_klgh
Mg3Cd:^\mokehcgZyIRMiehlgu_kehbh[jZah\ZguZlhfZfbMg bCd kh\f_klgh
Fe3Al:ZlhfuZexfbgbyh[jZamxlljzokehcgmxIRMZlhfu`_e_aZaZgbfZxl\k_l_ljZw^jbq_kdb_bhdlZw^jbq_kdb_imklhlu
NiAs:ZlhfufurvydZh[jZamxl^\mokehcgmxIRMZlhfugbd_eyaZgbfZxl\k_hdlZw^jbq_kdb_imklhlu
Li2O:Zlhfudbkehjh^Zh[jZamxlljzokehcgmxIRMZlhfueblbyaZgbfZxl\k_l_ljZw^jbq_kdb_imklhlu
CaF2:ZlhfudZevpbyh[jZamxlljzokehcgmxIRMZlhfunlhjZaZgbfZxl\k_l_ljZw^jbq_kdb_imklhlu
NaCl: Zlhfu oehjZ h[jZamxl ljzokehcgmx IRM Zlhfu gZljby aZgbfZxl \k_ hdlZw^jbq_kdb_imklhlu
CsCl:ZlhfuoehjZh[jZamxijhklmxdm[bq_kdmxdeZ^dmZlhfup_abyaZgbfZxlp_glju\k_o
h[jZamxsbokydm[h\
CdCl2:ZlhfuoehjZh[jZamxlljzokehcgmxIRMZlhfudZ^fbyihkehcghaZgbfZxliheh\bgmhdlZw^jbq_kdboimklhl
=Q6knZe_jblZlhfuk_juh[jZamxl ljzokehcgmx IRM Zlhfu pbgdZ jZ\ghkehcgh aZgbfZxliheh\bgml_ljZw^jbq_kdboimklhl
ZnS(\xjlpbl Zlhfu k_ju h[jZamxl ^\mokehcgmx IRM Zlhfu pbgdZ jZ\ghkehcgh aZgbfZxliheh\bgml_ljZw^jbq_kdboimklhl
CaTiO3:ZlhfudZevpbybdbkehjh^Zkh\f_klghh[jZamxlljzokehcgmxIRMZlhfulblZgZ
aZgbfZxlq_l\_jlvhdlZw^jbq_kdboimklhl
MgAl2O4:Zlhfudbkehjh^Zh[jZamxlljzokehcgmxIRMZlhfufZ]gbyaZgbfZxl\hkvfmx
qZklvl_ljZw^jbq_kdboimklhlZlhfuZexfbgby±iheh\bgmhdlZw^jbq_kdbo
Bah[jZabl_ijh_dpbxmdZaZgghckljmdlmju
Ku3Au, CaTiO3, CaF2, ZnS (knZe_jblCu2O – kfkeh\_kgu_ihjlj_lu
:efZaZlhfuK\\_jrbgZodm[bq_kdhcyq_cdb\p_gljZo\k_o_z]jZg_cb\p_gljZoq_lujzoba\hkvfbhdlZglh\\rZofZlghfihjy^d_
α]jZnbl Zlhfu K h[jZamx kehb khklhysb_ bo khijy`zgguo ijZ\bevguo r_klbm]hevgbdh\KehbgZeZ]ZxlkyihaZdhgm«:<:<:<«kehc<k^\bgmlhlghkbl_evghkehy:gZ\_ebqbgm\_dlhjZjZ\gh]hk\yabK±K
24
BN: ZlhfuB bN, q_j_^mykvZlhf1hdjm`zgZlhfZfb<Zlhf<hdjm`zgZlhfZfbN)h[jZamxlkehbkhklhysb_bakhijy`zgguoijZ\bevguor_klbm]hevgbdh\ Kehb gZeZ]Zxlky lZd qlh r_klbqe_ggu_
pbdeu hdZau\Zxlky ^jm] gZ^ ^jm]hf Zlhfu < gZ^ ZlhfZfbNZlhfuNgZ^ZlhfZfb<
NiAs:
ReO3: :lhfuRe \\_jrbgZodm[bq_kdhcyq_cdbZlhfu
H±\k_j_^bgZo\k_o_zjz[_j
ZnS (\xjlpbl
DZdh\u oZjZdl_j kljmdlmju b dhhj^bgZpbyZlhfh\\mdZaZgguodjbklZeeZo"
Cu:DQ dm[hhdlZw^j]hfh^_kfbq_kdZy
Mg: DQ ]_dkZ]hgZevguc ZgZeh] dm[hhdlZw^jZ
]hfh^_kfbq_kdZy
La:DQ dm[hhdlZw^jb]_dkZ]hgZevgucZgZeh]dm[hhdlZw^jZ]hfh^_kfbq_kdZy
Sm: DQ dm[hhdlZw^jb]_dkZ]hgZevgucZgZeh]dm[hhdlZw^jZ]hfh^_kfbq_kdZy
CuAu:DQKu) = DQ(Au) bkdZ`zggu_±\ulygmlu_±dm[hhdlZw^ju]hfh^_kfbq_kdZy
Cu3Au:DQKu DQAu) = 12 (dm[hhdlZw^ju]hfh^_kfbq_kdZy
Mg3Cd:DQMg) = DQCd) = 12 (]_dkZ]hgZevgu_ZgZeh]bdm[hhdlZw^jh\]hfh^_kfbq_kdZy
Fe3Al:DQFe) = DQAl) = dm[]hfh^_kfbq_kdZy
NiAs:DQNi) bkdZ`zgguchdlZw^jDQAs) ljb]hgZevgZyijbafZ]hfh^_kfbq_kdZy
Li2O:DQLi) = 4(l_ljZw^jDQO) = 8 (dm[]hfh^_kfbq_kdZy
WO3:DQW) hdlZw^jDQH ]Zgl_ev]hfh^_kfbq_kdZy
H2O ez^DQG DQH m]hehdhkljh\gZy]_l_jh^_kfbq_kdZy
CO2:DQK ]Zgl_evDQH hkljh\gZy]_l_jh^_kfbq_kdZy
CaF2:DQKZ dm[DQF) = l_ljZw^j]hfh^_kfbq_kdZy
BeCl2:DQBe) l_ljZw^jDQCl) = bah]gmlZy]Zgl_ev]_l_jh^_kfbq_kdZyp_ihq_qgZy
CdCl2:DQCd) = 6 (hdlZw^jDQCl) = lj_m]hevgbd]_l_jh^_kfbq_kdZykehbklZy
PdCl2:DQPd) = 4 (iehkdbcd\Z^jZlDQCl) = bah]gmlZy]Zgl_ev]_l_jh^_kfbq_kdZyp_ihq_qgZy
MoS2: DQ Fh ljb]hgZevgZy ijbafZ DQ S) lj_m]hevgbd ]_l_jh^_kfbq_kdZy
kehbklZy
CaTiO3: DQKZ dm[hhdlZw^jDQTi) hdlZw^jDQH ]Zgl_ev]_l_jh^_kfbq_kdZydZjdZkgZy
DZdb_dhhj^bgZpbhggu_qbkeZbfgh]h]jZggbdboZjZdl_jgu^eyZlhfh\g_f_lZeebq_kdbowe_f_glh\"
Kh]eZkghijZ\bemXfJha_jb\ijhkluo\_s_kl\ZoDQ ± N]^_N±ghf_j]jmiiu\dhlhjhcgZoh^blkywe_f_glg_f_lZee<wlhfkemqZ_j_ZebamxlkyDQ ^ey]Zeh]_gh\DQ ^ey k_ju β-Se, Te (bah]gmlZy ]Zgl_ev DQ lj_m]hevgbd beb \ \_jrbg_ l_ljZw^jZ ^ey
qzjgh]hnhknhjZDQ l_ljZw^j^eyZefZaZ<hafh`gulZd`_ZghfZevgu_dhhj^bgZpbb
DQ \\_jrbg_l_ljZ]hgZevghcibjZfb^uDQ Zlhf\\_jrbg_i_glZ]hgZevghcibjZfb^ubDQ i_glZ]hgZevgZyibjZfb^Z^ey[hjZ
<kljmdlmjZo[bgZjguob[he__keh`guokh_^bg_gbcZlhfug_f_lZeeZijhy\eyxljZaebqgu_
dhhj^bgZpbhggu_qbkeZhl^hbDQ \h^hjh^\kljmdlmj_ev^ZDQ ]Zgl_ev
dbkehjh^\kljmdlmj_CaTiO3), DQ \\_jrbg_ljb]hgZevghcibjZfb^uoehj\CdCl2), DQ
l_ljZw^jnlhj\CaF2), DQ hdlZw^joehj\NaCl), DQ = 8 (dm[dbkehjh^\/L2O).
25
10 DZdb_ dhhj^bgZpbhggu_ qbkeZ b dhhj^bgZpbhggu_ fgh]h]jZggbdb oZjZdl_jgu^eyZlhfh\\oh^ysbo\khklZ\f_lZeeh\bgl_jf_lZeeb^h\bkieZ\h\"
Hkgh\gu_kljmdlmjuf_lZeeh\&X±DQ dm[hhdlZw^jMg – DQ ]_dkZ]hgZevguc
ZgZeh]dm[hhdlZw^jZα-Fe – DQ dm[
Bgl_jf_lZeeb^uh[uqgheb[hhibku\Zxlky\l_jfbgZoIRMby\eyxlkyk\_jokljmdlmjZfb
ihhlghr_gbxdhkgh\gufkljmdlmjguflbiZff_lZeeh\DQ dm[DQ dm[hhdlZw^j
b_]h]_dkZ]hgZevgucZgZeh]eb[hy\eyxlkynZaZfbEZ\_kZoZjZdl_jgu \ukhdb_ dhhj^bgZpbhggu_ qbkeZ b keh`guf dhhj^bgZpbhggu_ fgh]h]jZggbdb ± bdhkZw^ju ^h^_dZw^ju b
boijhba\h^gu_
Ijb\_^bl_ ijbf_ju djbklZeebq_kdbo kljmdlmj \ dhlhjuo Zlhf beb bhg
oehjZbf__ldhhj^bgZpbhggh_qbkeh
DQ &O2.
DQ PdCl2, BeCl2.
DQ = 3: CdCl2, CuCl2.
DQ = 6: NaCl, KCl.
DQ CsCl.
Ijb\_^bl_ ijbf_ju djbklZeebq_kdbo kljmdlmj \ dhlhjuo Zlhf dbkehjh^Z
bf__ldhhj^bgZpbhggh_qbkeh
DQ H2, CO2.
DQ= 2: CaTiO3, ReO3, WO3.
DQ = 3: TiO2jmlbe, SiO2 (klbrh\bl
DQ = 4: Cu2O,
DQ = 6: MgO,
10 GZah\bl_ ljb g_hj]Zgbq_kdb_ djbklZeebq_kdb_ kljmdlmju \ dhlhjuo Zlhf
dbkehjh^Zbf__ldhhj^bgZpbhggh_qbkeh
CaTiO3, ReO3, H2O.
10DZdb_\b^udhhj^bgZpbbdhhj^bgZpbhggh_qbkehdhhj^bgZpbhggucihebw^j\klj_qZxlkymZlhfh\m]e_jh^Zbdj_fgby"
C:DQ ]Zgl_ev±dZj[bg&22.
DQ lj_m]hevgbd±]jZnbl\\_jrbg_ljb]hgZevghcibjZfb^u±nmee_j_g
DQ l_ljZw^j±ZefZabehgk^_cebl
Si:DQ l_ljZw^j±SiO2 (djbklh[Zeblbljb^bfbld\Zjp
DQ hdlZw^j±klbrh\bl
DZdb_\b^udhhj^bgZpbbdhhj^bgZpbhggh_qbkehdhhj^bgZpbhggucihebw^j\klj_qZxlkymZlhfh\ZexfbgbyblblZgZ"
Ti:DQ hdlZw^j±CaTiO3, TiO2 (jmlbe[jmdblbZgZlZa
DQ = 8 (dm[±Ti7Sb2.
DQ dm[hhdlZw^jb_]h]_dkZ]hgZevgucZgZeh]±f_lZeebq_kdbclblZg, TiNi3, TiAl3.
Al:DQ l_ljZw^j±NaAlSiO4, p_heblu
DQ hdlZw^j±MgAl2O4, Al2O3.
DQ dm[±Fe3Al, Cr2Al.
DQ dm[hhdlZw^j±f_lZeebq_kdbcAl, KuAl3, TiAl3, Cu2Mg.
10Ijb\_^bl_ijbf_ju]Zeh]_gb^h\$;2]^_; )Cl, %U,bf_xsbop_ihq_qgmxkehbklmxbfhgheblgmxdjbklZeebq_kdmxkljmdlmjm
26
P_ihq_qgZykljmdlmjZPdCl2, BeCl2, HgCl2.
KehbklZy kljmdlmjZCuCl2DQCl=3), CdCl2.
FhgheblgZy kljmdlmjZNaCl, CsCl.
10 < iZjZee_e_ibi_^_ ih\lhjy_fhklb bf_xs_f nhjfm dm[Z jZkiheZ]Zxlky
ZlhfOkdhhj^bgZlZfbbZlhfYkdhhj^bgZlZfb/6, 1/6, 1/6). Hij_^_eblvlbi
j_rzldbdhhj^bgZpbhggh_qbkehboZjZdl_jkljmdlmjuHl\_lh[hkgh\Zlv
Ijbfblb\gZy ]_dkZ]hgZevgZy j_rzldZ _klv hkv lj_lv_]h ihjy^dZ DQ O DQ Y) = 1;
kljmdlmjZhkljh\gZy]_l_jh^_kfbq_kdZy
10.10. < iZjZee_e_ibi_^_ ih\lhjy_fhklb bf_xs_f nhjfm dm[Z jZkiheZ]Zxlky
ZlhfOkdhhj^bgZlZfbbZlhfYkdhhj^bgZlZfb, 0, 1/2). Hij_^_eblvlbij_rzldbdhhj^bgZpbhggh_qbkehboZjZdl_jkljmdlmjuHl\_lh[hkgh\Zlv
Ijbfblb\gZy l_ljZ]hgZevgZy yq_cdZ _klv hkv q_l\zjlh]h ihjy^dZ DQ O DQ Y) = 2;
kljmdlmjZp_ihq_qgZy]_l_jh^_kfbq_kdZy
10.11. < iZjZee_e_ibi_^_ ih\lhjy_fhklb bf_xs_f nhjfm dm[Z jZkiheZ]Zxlky
ZlhfOkdhhj^bgZlZfbbZlhfYkdhhj^bgZlZfb /2, 0, 1/2). Hij_^_eblv lbi
j_rzldbdhhj^bgZpbhggh_qbkehboZjZdl_jkljmdlmjuHl\_lh[hkgh\Zlv
Ijbfblb\gZyl_ljZ]hgZevgZyj_rzldZhkvDQO DQY) kljmdlmjZkehbklZy]_l_jh^_kfbq_kdZy
27