Fbgbkl_jkl\hh[jZah\ZgbyJhkkbckdhcN_^_jZpbb <eZ^bfbjkdbc]hkm^Zjkl\_ggucmgb\_jkbl_l ;:Dmolbg<GEh[dh KLJH?GB?<?S?KL<: Mq_[gh_ihkh[b_ <eZ^bfbj M>D 541.5 D J_p_ga_glu >hdlhjobfbq_kdbogZmdijhn_kkhj aZ\.dZn_^jhcnbabq_kdhcbdheehb^ghcobfbb B\Zgh\kdh]h]hkm^Zjkl\_ggh]hobfbdh-l_ogheh]bq_kdh]hmgb\_jkbl_lZ F<Meblbg >hdlhjobfbq_kdbogZmdijhn_kkhj B\Zgh\kdh]h]hkm^Zjkl\_ggh]hobfbdh-l_ogheh]bq_kdh]hmgb\_jkbl_lZ H<E_n_^h\Z I_qZlZ_lkyihj_r_gbxj_^Zdpbhggh-ba^Zl_evkdh]hkh\_lZ <eZ^bfbjkdh]h]hkm^Zjkl\_ggh]hmgb\_jkbl_lZ Dmolbg;:Eh[dh<G. D Kljh_gb_ \_s_kl\Z: Mq_[ihkh[b_ /<eZ^bf ]hk mg-l<eZ^bfbj. 76k ISBN 5-89368-414-1 Baeh`_gu hkgh\gu_ ijbgpbiu kljh_gby \_s_kl\Z >Zgh hibkZgb_ kljmdlmju fhe_dme k bkihevah\Zgb_f l_hjbb kbff_ljbb dhlhjhc wlb fhe_dmeu h[eZ^Zxl JZah [jZgu^bwe_dljbq_kdb_k\hckl\Zfhe_dmebf_`fhe_dmeyjgh_\aZbfh^_ckl\b_ZlZd`_ ijb\_^_gug_dhlhju_k\_^_gbyhkljh_gbbdhg^_gkbjh\ZgguonZa. Ij_^gZagZq_gh^eyklm^_glh\obfbq_kdboki_pbZevghkl_c LZ[eBe. 38.;b[ebh]jgZa\. ISBN 5-89368-414-1 M>D 541.5 <eZ^bfbjkdbc]hkm^Zjkl\_gguc mgb\_jkbl_l H=E:<E?GB? <<?>?GB? .......................................................................................... 5 =eZ\Z1.HKGH<GU?IJ?>KL:<E?GBY DE:KKBQ?KDHC L?HJBBKLJH?GBY<?S?KL<: ........................................ 6 1.1. DjZldZybklhjbq_kdZykijZ\dZ ................................................... 6 1.2. L_hjbyobfbq_kdh]hkljh_gby:F;mle_jh\Z ........................... 8 1.3. KljmdlmjgZybahf_jby............................................................. 10 1.4. IjhkljZgkl\_ggZybahf_jby .................................................... 12 1.5 Hkgh\gu_oZjZdl_jbklbdb]_hf_ljbq_kdhcdhgnb]mjZpbb fhe_dme .................................................................................. 14 =eZ\Z 2. KBFF?LJBYFHE?DMEYJGUOKBKL?F...................... 17 2.1. We_f_glukbff_ljbbh[t_dlZbhi_jZpbbkbff_ljbb............... 17 2.2. Lhq_qgu_]jmiiu .................................................................... 27 2.3. Ihjy^hddeZkkbnbdZpbbfhe_dmeihlhq_qguf]jmiiZf ........... 31 2.4. G_dhlhju_ijhklu_k\hckl\Zkbff_ljbb.................................. 31 2.5. =jmiih\h_mfgh`_gb_k\hckl\Z]jmiiu .................................. 32 2.6. FZljbqgh_ij_^klZ\e_gb_ljZgknhjfZpbc OZjZdl_jhi_jZpbckbff_ljbb................................................. 34 2.7. G_ijb\h^bfu_ij_^klZ\e_gby ................................................. 38 =eZ\Z 3. WE?DLJBQ?KDB?BF:=GBLGU?K<HCKL<: FHE?DME ............................................................................. 39 3.1. >bihevgucfhf_gl ................................................................. 39 3.2. >bwe_dljbq_kdZyijhgbpZ_fhklvIheyjbaZpby ........................ 40 3.3. Hij_^_e_gb_^bihevguofhf_glh\fhe_dme ............................. 43 3.4. Iheyjbam_fhklvfhe_dme......................................................... 45 3.5. Hilbq_kdZyZdlb\ghklv............................................................ 47 3.6. FZ]gblgu_k\hckl\Z................................................................ 48 3 =eZ\Z4. [email protected]?DMEYJGH?<A:BFH>@?CKL<B? ............... 51 4.1. Hjb_glZpbhggu_bbg^mdpbhggu_\aZbfh^_ckl\by .................. 51 4.2. >bki_jkbhggh_\aZbfh^_ckl\b_wnn_dlEhg^hgZ ................... 56 4.3. HllZedb\Zgb_bkmffZjgh_\aZbfh^_ckl\b_ ............................. 58 4.4. <h^hjh^gZyk\yav .................................................................... 59 =eZ\Z5. G?DHLHJU?K<?>?GBYHKLJH?GBB DHG>?GKBJH<:GGUON:A ............................................. 61 5.1. KljmdlmjZ`b^dhkl_c .............................................................. 61 5.2. @b^db_f_lZeeu ..................................................................... 63 5.3. Kljh_gb_`b^dhc\h^u ............................................................ 65 5.4. @b^db_djbklZeeu .................................................................. 66 5.5. :fhjngh_khklhygb_ ............................................................... 68 DHGLJHEVGU?<HIJHKU ............................................................ 70 A:DEXQ?GB? .................................................................................. 73 ;B;EBH=J:NBQ?KDBCKIBKHD ................................................ 74 4 <<?>?GB? 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Kh_^bg_gb_ Zlhfh\ \ fhe_dme_ ijhbkoh^bl \ khhl\_lkl\bb k bo \Ze_glghklvxLh_klvhij_^_eyxsbfnZdlhjhfijbh[jZah\Zgbbfhe_dme y\ey_lky ijbjh^Z kZfbo Zlhfh\ khklZ\eyxsbo ^Zggmx fhe_dmem L_hjby ;mle_jh\Zg_j_rZeZ\hijhkhgZebqbbi_j_f_gghc\Ze_glghklbhl\_lgZ g_]h^ZeZ\ihke_^kl\bbi_jbh^bq_kdZykbkl_fZF_g^_e__\Zghhl\h^beh wlhfmihgylbxh^gmbahij_^_eyxsbojhe_c 3Obfbq_kdb_bnbabq_kdb_k\hckl\Z\_s_kl\ZaZ\bkylhlijbjh^u b dhebq_kl\Z Zlhfh\ khklZ\eyxsbo fhe_dmem ^Zggh]h \_s_kl\Z Z lZd`_ hlbo\aZbfgh]hjZkiheh`_gby\fhe_dme_ Wlhiheh`_gb_y\eyehkv[hev rbf ^hklb`_gb_f l_hjbb ;mle_jh\Z b hdZaZehkv hq_gv ieh^hl\hjguf <ebylv^jm]gZ^jm]Zfh]mlg_lhevdhZlhfu–[eb`Zcrb_khk_^bghb[h e__ m^Zezggu_ \ p_ihqd_ Zlhfh\ bg^mdpbhgguc wnn_dl <ebygb_ wlh kjZ\gbl_evghfZehgh\g_dhlhjuokemqZyofh`_l[ulv\_kvfZaZf_lguf gZijbf_j dh]^Z kbevgh we_dljhhljbpZl_evguc Zlhf hlly]b\Z_l gZ k_[y we_dljhggmx iehlghklv b fh`_l f_gylv obfbq_kdmx Zdlb\ghklv m^Zezg guoZlhfh\g_ihkj_^kl\_gghmqZkl\mxsbo\obfbq_kdboij_\jZs_gbyo MdZaZggZydhgp_ipbyiha\hebeZ\uy\eylv\keh`guofhe_dmeZonmgdpbh gZevgu_]jmiiu–]jmiiuZlhfh\hij_^_ezggh]hkljh_gby\klj_qZxsb_ky \ jZaebqguo fhe_dmeZo, \ agZqbl_evghc f_j_ hij_^_eyxsb_ obfbq_kdb_ k\hckl\Z \_s_kl\ b y\eyxsb_ky hkgh\hc ^ey bo deZkkbnbdZpbb GZijb f_jnmgdpbhgZevgZy]jmiiZ–OH ijbkmsZ hkgh\Zgbyf ]jmiiZ –COOH – hj]Zgbq_kdbfdbkehlZf LZdbfh[jZahfl_hjby;mle_jh\Zh[tykgyeZ\kzfgh]hh[jZab_hj]Z gbq_kdbo b g_hj]Zgbq_kdbo \_s_kl\ b y\eyeZkv \k_h[t_fexs_c H^gZdh ihgZqZemwlZl_hjby \ua\ZeZg_ihgbfZgb_khklhjhgujy^ZmqzguoH^bg g_f_pdbcobfbdbkoh^ybah^ghc[jmllh-nhjfmeubhkgh\u\ZykvgZl_h 9 jbb ;mle_jh\Z gZibkZe g_kdhevdh \hafh`guo kljmdlmjguo nhjfme kqb lZy wlh g_^hklZldhf l_hjbb Dmjvza khklhye \ lhf qlh \ihke_^kl\bb \k_ wlb\_s_kl\Z[uebhldjulu L_hjby^ZeZkbevguclheqhdjZa\blbxhj]Zgbq_kdhcobfbblZddZd [uemkljZgzgoZhkpZjb\rbc^hkboihj\bgl_jij_lZpbbh]jhfgh]hwdk i_jbf_glZevgh]hfZl_jbZeZ;uebjZajZ[hlZguobfbq_kdb_f_lh^ubkke_ ^h\Zgby kljh_gby fhe_dme Kmlv bo mijhszggh k\h^beZkv d ke_^mxs_fm Bkoh^ybaba\_klghc[jmllh-nhjfmeugZijbf_jC2H6O)aZibku\Zebkv\k_ \hafh`gu_kljmdlmjgu_nhjfmeu H H H H | | | | H − C − C − O − H b H − C − O − C − H ; | | | | H H H H ZaZl_fbamqZebkvobfbq_kdb_k\hckl\Z^Zggh]h\_s_kl\ZGZijbf_j_keb ijb^_ckl\bbf_lZeebq_kdh]hgZljbyaZf_sZekylhevdhh^bgZlhf\h^hjh ^Z \_s_kl\h fh]eh bf_lv lhevdh i_j\mx kljmdlmjgmx nhjfmem lh _klv y\eyehkvwlbeh\ufkibjlhfLZdbfh[jZahf,[uebmklZgh\e_gukljmdlmj gu_ nhjfmeu fgh]bolukyq\_s_kl\JZajZ[hlZggu_ ebrv \ XX\_d_nb abq_kdb_f_lh^ubkke_^h\Zgbykljh_gbyfhe_dmeiheghklvxih^l\_j^beb ijZ\bevghklv wlbonhjfme < ]h^m X Eb[bohf b N <ze_jhf [ueh hldjulh y\e_gb_ bah f_jbbaZdexqZxs__ky\lhfqlh\_s_kl\Zh^gh]hkhklZ\ZZlhfh\lh_klv bf_xsb_ h^gm b lm `_ [jmllh-nhjfmem h[eZ^Zeb jZaebqgufb obfbq_ kdbfb b nbabq_kdbfb k\hckl\Zfb >he]h_ \j_fy wlh g_ bf_eh m^h\e_l\h jbl_evgh]hh[tykg_gbybebrvl_hjby;mle_jh\Z^ZeZlZdh_h[tykg_gb_k ihabpbc kljh_gby fhe_dme JZkkfhljbf ^\Z ba\_klguo lbiZ bahf_jbb – kljmdlmjgmxbijhkljZgkl\_ggmx 1.KLJMDLMJG:YBAHF?JBY < [hevrbgkl\_ kemqZ_\ bahf_jby jZkkfZljb\Z_lky ih hlghr_gbx d hj]Zgbq_kdbf \_s_kl\Zf hkgh\m fhe_dme dhlhjuo khklZ\eyxl Zlhfu m] e_jh^Z Fu lZd`_ jZkkfhljbf ijbf_ju gZ hkgh\_ wlbo \_s_kl\ ohly \kz kdZaZggh_fh`gh[m^_lhlg_klbbddj_fgbchj]Zgbq_kdbf\_s_kl\Zfb\ g_dhlhjuokemqZyo,dg_hj]Zgbq_kdbf 10 Ijb jZkkfhlj_gbb hj]Zgbq_kdbo \_s_kl\ p_e_khh[jZagh aZ hkgh\m [jZlvZlhfum]e_jh^gh]hkd_e_lZlZddZdh[uqghZlhfu^jm]bowe_f_glh\ bnmgdpbhgZevgu_]jmiiu\ukljZb\Zxlky\hdjm]gbo. I_j\uc\b^kljmdlmjghcbahf_jbb–bahf_jbym]e_jh^gh]hkd_e_lZ dh]^Zihke_^gbcfh`_l[ulv\ukljh_gih-jZaghfmbah^gh]hblh]h`_dh ebq_kl\Zm]e_jh^guoZlhfh\ CH3–CH2–CH2–CH2–CH3 g-i_glZg b CH 3 − CH − CH 2 − CH 3 2-f_lbe[mlZg | CH 3 QbkehlZdbobahf_jh\[ukljh\hajZklZ_lkjhklhfqbkeZZlhfh\m] e_jh^gh]h kd_e_lZ ^ey ^\Z^pZlb Zlhfh\ m]e_jh^Z kms_kl\m_l m`_ [he__ ljzokiheh\bghckhl_gbahf_jh\;hevrb_jZaebqby\k\hckl\Zo^Zzlwlhl lbibahf_jbb^eypbdebq_kdbokh_^bg_gbc ?keb m]e_jh^guc kd_e_l h^bgZdh\ gh jZaebqgh iheh`_gb_ \ gzf nmgdpbhgZevguo]jmiibebhl^_evguoijbkh_^bgzgguoZlhfh\fubf__f \lhjhclbikljmdlmjghcbahf_jbb–bahf_jbxiheh`_gby CH3–CH2–CH2OH g-ijhibeh\uckibjl b CH 3 − CH − CH 3 bahijhiZghe | OH <g_dhlhjuokemqZyohj]Zgbq_kdb_\_s_kl\Zfh]ml[ulvihkljh_gu \hdjm] g_m]e_jh^guo Zlhfh\ _keb ijb wlhf jZ^bdZeu bf_xl jZaebqguc khklZ\ b kljh_gb_ Z [jmllh-nhjfmeu \_s_kl\ h^bgZdh\u gZ[ex^Z_lky lj_lbc\b^kljmdlmjghcbahf_jbb–f_lZf_jbyIjbf_jhffh]mlkem`blv f_lbeijhibeh\ucb^bwlbeh\ucwnbju CH3–O–CH2–CH2–CH3 b CH3–CH2–O–CH2–CH3. Hkh[mx ]jmiim bahf_jh\ khklZ\eyxl \_s_kl\Z hlebqZxsb_ky kljh_gb_fnmgdpbhgZevguo]jmii mdkmkgZydbkehlZ b CH3COOH CH 2 (OH) − C = O ]ebdhe_\ucZev^_]b^ | H 11 <g_dhlhjuokemqZyobahf_jufh]mle_]dhi_j_oh^blv^jm]\^jm]Z lh _klv bf__l f_klh h[jZlbfZy j_Zdpby ^ZxsZy jZ\gh\_kgmx kf_kv bah f_jh\ ijb wlhf gZ[ex^Z_lky lZmlhf_jby beb ^bgZfbq_kdZy bahf_jby DeZkkbq_kdbfijbf_jhf_zy\ey_lkyd_lh-_ghevgh_jZ\gh\_kb_ CH 3 − C = O = CH 3 − C = CH 2 | CH 3 | OH < XX \_d_ dh]^Z \ gZmqguc h[boh^ [ueb \\_^_gu jZ^bhZdlb\gu_ bahlhiuZlhfZm]e_jh^Z[uebihemq_gubahlhigu_bahf_ju 1.IJHKLJ:GKL<?GG:YBAHF?JBY <[hevrbgkl\_kemqZ_\fhe_dmeubf_xlh[tzfgh_kljh_gb_lh_klv khklZ\eyxsb_boZlhfug_e_`Zl\h^ghciehkdhklbbebgZh^ghcebgbb < qZklghklb, kljh_gb_ hj]Zgbq_kdbo fhe_dme \h fgh]hf hij_^_ey_lky kljh_gb_fZlhfZm]e_jh^Z\Ze_glghklbdhlhjh]hijbijhqbojZ\guomkeh \byo gZijZ\e_gu d \_jrbgZf ijZ\bevgh]h l_ljZw^jZ – ijZ\bevghc lj_ m]hevghc ibjZfb^u Ijb wlhf m]he f_`^m ^\mfy ex[ufb \Ze_glghklyfb khklZ\ey_l 109,5° ?kebkZlhfhfm]e_jh^Zk\yau\Zxlkyh^bgZdh\u_q_lu j_ ZlhfZ beb nmgdpbhgZevgu_ ]jmiiu wlhl m]he g_ baf_gy_lky gZebqb_ jZaguo Zlhfh\ beb ]jmii fh`_l _]h bkdZ`Zlv < fhe_dmeZo keh`gh]h kljh_gbywlhfh`_lijb\h^blvdijhkljZgkl\_gghcbahf_jbb^\mo\b^h\– ]_hf_ljbq_kdhcbhilbq_kdhc LZdbfh[jZahfijbbamq_gbbkljh_gbyfhe_dmeg_h[oh^bfhmqblu \Zlv bo ijhkljZgkl\_ggmx dhgnb]mjZpbx Wlh kms_kl\_ggh_ ^hiheg_gb_ l_hjbb;mle_jh\Z[uehk^_eZgh\]h^mY<[email protected]_;_ e_f JZa^_e obfbb bamqZxsbc ijhkljZgkl\_ggmx dhgnb]mjZpbx hj]Zgb q_kdbobg_hj]Zgbq_kdbofhe_dme,ihemqbegZa\Zgb_kl_j_hobfbb =_hf_ljbq_kdZy bahf_jby oZjZdl_jgZ ^ey g_ij_^_evguo b pbdebq_ kdbo hj]Zgbq_kdbo kh_^bg_gbc Z lZd`_ ^ey dhfie_dkguo kh_^bg_gbc. Ij_^_evgu_ m]e_\h^hjh^u b bo ijhba\h^gu_ g_ h[eZ^Zxl wlbf \b^hf bahf_jbb lZd dZd \hdjm] hkb k\yab C–C ijhbkoh^bl k\h[h^gh_ \jZs_gb_ khhl\_lkl\mxsbo ]jmii Zlhfh\ < ]_hf_ljbq_kdbo bahf_jZo ZlhfuaZf_klbl_ebbebnmgdpbhgZevgu_]jmiiuih-jZaghfmjZkiheZ]Zxlkyhlgh kbl_evgh^\hcghck\yabbebiehkdhklbpbdeZ?kebaZf_klbl_ebbebnmgd pbhgZevgu_ ]jmiiu m^Ze_gu ^jm] hl ^jm]Z fu bf__f ^_eh k ljZgk12 bahf_jhf _keb k[eb`_gu – bf__l f_klh pbk-bahf_j Fhe_dmeu ^boehjwlbe_gZ H − C = C − H | | Cl Cl pbk-nhjfZ Fhe_dmeu-^boehjpbdehi_glZgZ pbk-nhjfZ C3 − = C − Cl | | Cl H ljZgk-nhjfZ C ljZgk-nhjfZ Hilbq_kdZy beb a_jdZevgZy bahf_jby oZjZdl_jbam_lky l_f qlh ih obfbq_kdbf k\hckl\Zf lZdb_ bahf_ju g_jZaebqbfu DZd ijZ\beh \g_r gb_ijhy\e_gbylZdhcbahf_jbbk\h^ylkydjZaebqbx\hilbq_kdbok\hckl \Zo \_s_kl\ Z bf_ggh d \jZs_gbx iehkdhklb iheyjbah\Zggh]h k\_lZ ih jZagufgZijZ\e_gbyfih^jh[g__wlh[m^_ljZkkfhlj_gh\]e. III). GZebqb_ bahf_jbb wlh]h \b^Z h[mkeh\eb\Z_lky \hafh`ghklvx kms_kl\h\Zgby keh`guoh[tzfguofhe_dmey\eyxsbokydZd[ua_jdZevgufhljZ`_gb_f ^jm] ^jm]Z _keb bo hij_^_ezgguf h[jZahf jZkiheh`blv \ ijhkljZgkl\_ Ijbwlhf_kebh^gmbafhe_dmefuke_gghi_j_f_sZlv\ijhkljZgkl\_ex [ufbkihkh[Zfb_zg_\hafh`ghkh\f_klblvkhk\hbfa_jdZevgufhlh[jZ `_gb_fK\hckl\hwlhgZau\Z_lkyobjZevghklvxk\yaZghhghkkbff_ljb_c b[m^_lih^jh[g__jZkkfhlj_gh\]e. IIIjhkl_crbfijbf_jhfobjZevguo bahf_jh\ wgZglbhf_jh\ fh`_l kem`blv fhe_dmeZ f_lZgZ \ dhlhjhc ljb ZlhfZ\h^hjh^ZaZf_s_gujZaebqgufbaZf_klbl_eyfb (jbk JbkWgZglbhf_juZ[lj_oaZf_szgguofhe_dmef_lZgZ 13 <Ze_glgu_ m]eu lZdbo fhe_dme ijZdlbq_kdb jZ\gu qlh h[mkeh\eb \Z_lh^bgZdh\hklvobfbq_kdbok\hckl\ <l\zj^hfkhklhygbbhilbq_kdb_Zglbih^uwgZglbhf_jufh]mlh[ jZah\u\ZlvdjbklZeeudhlhju_fh`ghjZa^_eblv^Z`_\bamZevghjbk 2) JbkDjbklZeeua_jdZevguobahf_jh\gZljb_\h-Zffhgbcghckheb \bgghc dbkehlukjZ\gbl_]jZgbZ – Zb[–[) Hilbq_kdZy bahf_jby ijbkmsZ \k_f fhe_dmeZf k Zkbff_ljbqgufb m]e_jh^gufbZlhfZfbijbq_fqbkehlZdbobahf_jh\fh`_l[ulv^h\hevgh \_ebdh 1.HKGH<GU?O:J:DL?JBKLBDB=?HF?LJBQ?KDHC DHGNB=MJ:PBBFHE?DME <ex[uokhklhygbyo\dhlhjuofh`_lgZoh^blvkyfhe_dmeZ_zy^jZ fh]mlkh\_jrZlvljbjh^Z^\b`_gbc Y^jZfh]ml^\b]ZlvkyihklmiZl_evgh\f_kl_ki_j_f_s_gb_f\k_c qZklbpudZdp_eh]hijbwlhfhlghkbl_evgh_jZkiheh`_gb_y^_jg_f_gy _lky Y^jZ fh]ml \f_kl_ kh \k_c fhe_dmehc \jZsZlvky \ ijhkljZgkl\_ hiylv-lZdbg_f_gyyk\h_]hhlghkbl_evgh]hjZkiheh`_gby Y^jZqZklbpufh]mldhe_[Zlvkyhdhehg_dhlhjuoiheh`_gbcjZ\ gh\_kbyijbwlhfjZaf_jufhe_dmeubhlghkbl_evgh_jZkiheh`_gb_y^_j g_ij_ju\gh f_gy_lky \ g_dhlhjuo ij_^_eZo Iheh`_gby jZ\gh\_kby y^_j khklZ\eyxljZ\gh\_kgmxdhgnb]mjZpbxfhe_dmeu Djhf_ lh]h, \ g_dhlhjuo fhe_dmeZo ]jmiiu Zlhfh\ qZs_ \k_]h nmgdpbhgZevgu_ ]jmiiu fh]ml k\h[h^gh \jZsZlvky \hdjm] hkb hj^bgZj guok\ya_c 14 LZdbf h[jZahf j_Zevgh fhe_dmeZ g_ bf__l kljh]h ihklhygguo ]_h f_ljbq_kdbo jZaf_jh\ b aZ^ZqZ bamq_gby kljmdlmju fhe_dme k\h^blky d mklZgh\e_gbxbojZ\gh\_kghcdhgnb]mjZpbb Hkgh\gu_oZjZdl_jbklbdb]_hf_ljbbfhe_dme: Zf_`ty^_jgu_jZkklhygby^ebguk\ya_c–jZkklhygbyf_`^mjZ\ gh\_kgufbiheh`_gbyfbobfbq_kdbk\yaZgguoiZjZlhfh\ [\Ze_glgu_m]euf_`^mebgbyfbk\yau\ZxsbfbmaeujZ\gh\_kby y^_j; \ jZ\gh\_kgu_ m]eu ih\hjhlZ h^gbo ]jmii y^_j ih hlghr_gbx d ^jm]bf; ]wg_j]byk\ya_chij_^_eyxsZyboijhqghklv. =jm[u_ ijbdb^db f_`ty^_jguo jZkklhygbc fh]ml [ulv k^_eZgu gZ hkgh\_^Zgguoihiehlghklbbbkihevah\Zgby^eyjZkqzlZqbkeZ:\h]Z^jh LZdb_ jZkqzlu iha\heyxl \_jgh m]Z^Zlv ihjy^hd \_ebqbgu dhlhjuc kh klZ\ey_lg_kdhevdhZg]klj_fÅ). Nbabq_kdb_f_lh^ubkke_^h\Zgbyiha\h eyxlhij_^_eblv^ebguk\ya_c^h\hevghlhqghJ_amevlZlulZdbobaf_j_ gbc ohjhrh hl\_qZxl aZdhghf_jghklyf i_jbh^bq_kdhc kbkl_fu qlh rb jhdhba\_klgh:gZebahiulguo^ZgguoihdZau\Z_lqlh^eyh^gh]hblh]h `_\Ze_glgh]hkhklhygby^ebguk\ya_ch^gbobl_o`_Zlhfh\\jZaguokh _^bg_gbyo ijbf_jgh h^bgZdh\u ^ey ZebnZlbq_kdbo kh_^bg_gbc ^ebgZ k\yab d C −C = 1,54 ÷ 1,58 Å, ^ey ZjhfZlbq_kdbo d C −C = 1,39 ÷ 1,42 Å. ?keb ^eyhj^bgZjguok\ya_c d C −C ≈ 1,54 Å, lh^ey^\hcguo d C =C ≈ 1,34 Å, Z^ey ljhcghck\yab d C ≡C ≈ 1,20 Å; qlhhljZ`Z_lmijhqg_gb_wlbok\ya_c <_ebqbgZ\Ze_glguom]eh\aZ\bkblhlijbjh^uZlhfh\b\b^Zk\yab Ebrv^\moZlhfgu_fhe_dmeu\k_]^Zebg_cgu[hevr__dhebq_kl\hZlhfh\ ijb\h^bldjZaebqguf\ZjbZglZfLjzoZlhfgu_fhe_dmeufh]ml[ulveb g_cgu k jZ\guf beb g_jZ\guf f_`ty^_jguf jZkklhygb_f (BeCl2, ZnBr2, CdI2, CO2, CS2, HCN) bebg_ebg_cguH2O – \Ze_glgucm]he 104°28' ; H2S – 92° ; H2Se – 91° ; H2Te – 89°30' ). 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Z [ JbkKbff_ljbyfhe_dmeZffbZdZZb\h^u[ Hq_\b^gh qlh hkv ijhoh^ysZy q_j_a p_glj fhe_dmeu [_gaheZ i_ji_g^b dmeyjgh _z iehkdhklb jbk y\ey_lky hkvx r_klh]h ihjy^dZ K6 Ih kdhevdm n r_klv ih\hjhlh\ gZ h (360h/n ijb\h^bl dr_klb wd\b\Z e_glguf hjb_glZpbyf LZd`_ hq_\b^gh qlh fhe_dmeZ [_gaheZ bf__l r_klv hk_c \lhjh]h ihjy^dZ K2 e_`Zsbo \ iehkdhklb fhe_dmeu Ljb ba gboijhoh^ylq_j_aiZjuijhlb\hiheh`ghe_`ZsboZlhfh\m]e_jh^Zljb ^jm]b_q_j_ap_gljuk\ya_cK–KjbkGZebqb_hkb\jZs_gbyihjy^dZ n, dhlhjuc fh`_l ijbgbfZlv agZq_gby hl ^h ki_pbnbq_kdhc hkb K 19 khhl\_lkl\m_lhi_jZpbyih\hjhlZgZkdhevm]h^ghfZeucm]heex[Zy eb g_cgZyfhe_dmeZbf__lh^gmhkvK –jbk]h\hjblhkms_kl\h\Zgbbn hi_jZpbc Kn , C n2 , Cn3 ,…, C nn −1 , C nn ;he__ lh]h hi_jZpby C 42 wd\b\Z e_glgZhi_jZpbbK2hi_jZpby C62 –hi_jZpbbK3Zhi_jZpby C nn _klvhi_ jZpbylh`^_kl\_ggh]hij_h[jZah\Zgby K6bhk_cK2 JbkHkbkbff_ljbb[_gaheZK6G6 JbkKljh_gb_ fhe_dmeu HCl Hkv\jZs_gbyfhe_dmeu\ukr_]hihjy^dZgZau\Z_lky]eZ\ghchkvx fhe_dmeu?keb\k_hkbKnwd\b\Ze_glguex[mxbagbofh`gh\u[jZlv\ dZq_kl\_ ]eZ\ghc hkb >hklZlhqgh m^h[gh jZkkfZljb\Zlv fhe_dmem b _z we_f_glu kbff_ljbb \ ^_dZjlh\hc kbkl_f_ dhhj^bgZl Ijbq_f ^\b`_gb_ hlydoy\ey_lky\jZs_gb_fkiheh`bl_evgufagZdhfP_gljfZkkfhe_dm eu ijb wlhf ihf_sZxl \ gZqZeh dhhj^bgZl ?keb m fhe_dmeu bf__lky lhevdh h^gZ hkv kbff_ljbb lh hgZ kh\iZ^Z_l k hkvx z, _keb `_ hk_c g_ kdhevdhlh]eZ\ghchkvxfhe_dmeukqblZxllmdhlhjZykh\iZ^Z_lkhkvx z ?keb fhe_dmeZ bf__l g_kdhevdh hk_c \ukr_]h ihjy^dZ lh aZ ]eZ\gmx ijbgbfZxllmhkvdhlhjZyijhoh^blq_j_agZb[hevr__qbkehZlhfh\?keb fhe_dmeZ iehkdZy b hkv z e_`bl \ iehkdhklb fhe_dmeu dZd gZijbf_j \ fhe_dme_\h^uaZhkvo\u[bjZ_lkyhkvi_ji_g^bdmeyjgZyiehkdhklbLZ dbf h[jZahf _keb z – hkv \jZs_gby lh hi_jZpby \jZs_gby khklhbl ba 20 ij_h[jZah\Zgby dhhj^bgZl ( x, y, z ) \ (− x, − y, z ) Hi_jZpbx C 2 fh`gh ij_^klZ\blvdZd x − x C 21 ⋅ y = − y . z z A_jdZevgZyiehkdhklv σ ?kebkms_kl\m_liehkdhklvdhlhjZy^_ ebl fhe_dmem gZ ^\_ iheh\bgu y\eyxsb_ky a_jdZevgufb hljZ`_gbyfb h^gZ ^jm]hc lh fhe_dmeZ h[eZ^Z_l we_f_glhf kbff_ljbb dhlhjuc gZau \Z_lkya_jdZevghciehkdhklvxA_jdZevgZyiehkdhklvg_fh`_le_`Zlv\g_ fhe_dmeuhgZh[yaZgZijhoh^blvq_j_ag_zIjbhljZ`_gbb\a_jdZevghc iehkdhklb fhe_dmeZ h[t_dl g_hlebqbfZ hl i_j\hgZqZevghc nhjfu Ih \lhjgZy hi_jZpby σ ^Zzl i_j\hgZqZevgmx dhgnb]mjZpbx ihwlhfm a^_kv bf__lky lhevdh h^gZ hij_^_ezggZy hi_jZpby σ k = σ ( k – g_qzlgh_ ld σ k = E _keb k –qzlgh_?keba_jdZevgZyiehkdhklvijhoh^blq_j_a]eZ\ gmx hkv kbff_ljbb l_ y\ey_lky \_jlbdZevghc lh hgZ h[hagZqZ_lky σ v jbkZ[ ?keb hgZ i_ji_g^bdmeyjgZ ]eZ\ghc hkb l_ y\ey_lky ]hjbahglZev ghc lh h[hagZqZ_lky σ h jbk \ ?keb iehkdhklv \_jlbdZevgZ b ijb wlhf^_eblihiheZfm]hef_`^m^\mfyhkyfb C 2 dhlhju_kZfbi_ji_g^b dmeyjgu]eZ\ghchkblh_zh[hagZqZxl σ d l_^bZ]hgZevgZybeb^bw^jb q_kdZy<fhe_dme_H2Obf_xlky^\_\aZbfghi_ji_g^bdmeyjgu_hkb σ v b σ v ' jbk Z Fhe_dmeZ ZffbZdZ bf__l ljb σ v jbk \ ;_gahe bf__l ihfbfh\k_]hijhq_]hiehkdhklv σ h br_klviehkdhkl_c σv jbkb\ Ebg_cgu_ fhe_dmeu k C ∞ bf_xl [_kdhg_qgh_ fgh`_kl\h a_jdZevguo iehkdhkl_c σ v jbk ?keb xz a_jdZevgZy iehkdhklv lh hi_jZpby hljZ`_gby fh`_l [ulv aZibkZgZke_^mxsbfh[jZahf x x σ y = − y . z z P_glj kbff_ljbb i Bg\_jkby khklhbl \ kh_^bg_gbb ex[hc jZk kfZljb\Z_fhc lhqdb h[t_dlZ k p_gljhf kbff_ljbb b ijh^he`_gbb wlhc 21 ijyfhcaZp_gljkbff_ljbbgZlZdh_`_jZkklhygb_Ijbwlhf\ihemq_g ghclhqd_^he`_ggZoh^blvkywd\b\Ze_glgucwe_f_glh[t_dlZ [ Z \ JbkA_jdZevgu_iehkdhklbfhe_dmeuZffbZdZNH3 Zb\h^uH2H [ a_jdZevgZyiehkdhklv σ h fhe_dmeu[_gaheZ\ Fhe_dmeZbf__lp_gljkbff_ljbbbebp_gljbg\_jkbb_kebdZ`^uc _z Zlhf ijb ^\b`_gbb ih ijyfhc ebgbb ijhoh^ys_c q_j_a wlhl p_glj \klj_lblgZjZ\ghfjZkklhygbbhlp_gljZih^jm]mx_]hklhjhgmlZdhc`_ Zlhf ?keb Zlhf dbkehjh^Z a bhgZ ]bihgbljblZ jbk ^\b`_lky q_j_a p_gljbg\_jkbbhg\klj_lbl^jm]hcZlhfdbkehjh^ZbLh`_^he`ghijh bahclbkh[hbfbZlhfZfbZahlZ (Z) P_gljbg\_jkbb (b) JbkP_gljbg\_jkbbbhgZ]bihgbljblZ 22 P_glj kbff_ljbb bf_xl lZdb_ ]_hf_ljbq_kdb_ nb]mju dZd rZj dm[ ijZ\bevguc hdlZw^j jbk b lZdb_ fhe_dmeu dZd [_gahe kf jbk \ SF4 jbk CO2 G_ bf_xl p_gljZ kbff_ljbb fhe_dmeu H2O, NH3, CH4kfjbk b JbkP_gljbg\_jkbbijZ\bevgh]hhdlZw^jZ JbkKbff_ljbyfhe_dmeuSF4 ?kebfuihf_klbfp_gljkbff_ljbb\gZqZehkbkl_fudhhj^bgZllh khhl\_lkl\mxsZy wlhfm we_f_glm kbff_ljbb hi_jZpby [m^_l khklhylv \ baf_g_gbbdhhj^bgZldZ`^h]hZlhfZ ( x, y, z ) gZdhhj^bgZlu (− x, − y, − z ) . LZdbfh[jZahf^eyhi_jZpbbbg\_jkbbbf__f x − x x x − x i y = − y ^\Z`^u i ⋅ i ⋅ y = i ⋅ − y = y \h[s_f z − z z z − z i k = i ( k − q_lgh_ ), i k = E (k − g_q_lgh_) , l_ lhevdh h^gZ hi_jZpby y\ey_lky hij_^_ezgghc :lhfu ^he`gu gZoh ^blvky\p_gljhkbff_ljbqguoiZjZoaZbkdexq_gb_fp_gljZevgh]h Hkv g_kh[kl\_ggh]h \jZs_gby S WlZ hi_jZpby \dexqZ_l \jZs_ gb_\hdjm]hkbkihke_^mxsbf hljZ`_gb_f hl a_jdZevghciehkdhklbdh lhjZyi_ji_g^bdmeyjgZhkb\jZs_gbybebgZh[hjhll_\jZs_gb_–hljZ 23 `_gb_wd\b\Ze_glghhljZ`_gbx–\jZs_gbx?keb\j_amevlZl_wlbo^\mo hi_jZpbc ihy\ey_lky wd\b\Ze_glgZy kljmdlmjZ lh ]h\hjyl qlh fhe_dmeZ h[eZ^Z_l a_jdZevgh-ih\hjhlghc hkvx beb Zevl_jgZglghc Z hi_jZpby kh hl\_lkl\mxsZy wlhfm we_f_glm kbff_ljbb qZklh gZau\Z_lky g_kh[kl\_g guf\jZs_gb_fGb`gbcbg^_dk n \ S n mdZau\Z_lgZih\hjhlihqZkh\hc klj_ed_ gZ 2π n GZijbf_j S 2 mdZau\Z_l gZ ^\Z ih\hjhlZ ih qZkh\hc klj_ed_gZ180° Ihhij_^_e_gbxS2wd\b\Ze_glgZi. Hq_\b^gh_kebkms_kl\m_lhkvKnbi_ji_g^bdmeyjgZy_c1lhKnb _klvSnjbkZL_i_jvjZkkfhljbfkemqZcdh]^ZijbgZebqbbSngbKn, gb a_jdZevghc iehkdhklb i_ji_g^bdmeyjghc Kn, ih hl^_evghklb g_ kms_ kl\m_l<nhjf_wlZgZijb\_^_ggh]hgZjbk[k\yavK–Khij_^_ey_lhkv K3ghi_ji_g^bdmeyjgZy_ca_jdZevgZyiehkdhklvhlkmlkl\m_lH^gZdh_k ebih\_jgmlvfhe_dmem\hdjm]hkbK–KgZhZaZl_fhljZablv_z\iehk dhklbi_ji_g^bdmeyjghcwlhchkblhfuihemqbfwd\b\Ze_glgmxdhgnb ]mjZpbxKe_^h\Zl_evghkms_kl\m_lhkvS6dhlhjZy\lh`_\j_fyg_y\ ey_lkyhkvxK6. JZkkfhljbfq_fhlebqZxlkyg_kh[kl\_ggu_hkb\jZs_gbyq_lgh]hb g_q_lgh]hihjy^dh\?kebnq_lgh_ S nm ijb\h^bldgZ[hjm S 1n , S n2 , S n3 ,…, S nn dhlhjuc wd\b\Ze_gl_g C1n σ1 , Cn2 σ 2 , C nn σ 3 ,…, Cnn σ n hlf_lbf qlh aZibkv Cnn σ n h[hagZqZ_lqlh\gZqZe_ijhbkoh^blhljZ`_gb_ZaZl_f\jZ s_gb_ lh]^Z dZd aZibkv σ n C nn ]h\hjbl h lhf qlh \gZqZe_ ijhbkoh^bl \jZs_gb_ Z aZl_f hljZ`_gb_ ?keb m g_q_lgh_ fu bf__f khhlghr_gb_ σ m = σ Z _keb m q_lgh_ lh ihemqZ_fh_ khhlghr_gb_ bf__l \b^ σ m = E . Ihke_^g__ ijb\h^bl d lh`^_kl\Zf S nn = C nn E = E b S nm = C nm Hlf_lbf qlhbakms_kl\h\ZgbyhkbSnq_lgh]hihjy^dZ\k_]^Z\ul_dZ_lgZebqb_Kn/2, ihkdhevdm S n2 = Cn2 σ 2 = Cn / 2 Ijbf_ju fhe_dme bf_xsbo hkb \jZs_gby S2, S4, S6ijb\_^_gugZjbk[bbadhlhjuoke_^m_lqlhfhe_dmeZ SF4 bf__lljbhkb S 2 bq_luj_hkb S 6 kfjbkfhe_dmeZCH4bf__l 24 ljb hkb S 4 jbk g_kh[kl\_ggZy hkv \jZs_gby S6 bf__lky \ aZlhjfh `_gghc nhjf_ wlZgZ jbk [ LZdb_ fhe_dmeu dZd H2O, NH3 jbk j \hh[s_g_bf_xla_jdZevgh-ih\hjhlghchkb Z [ JbkDhgnhjfZpbbwlZgZ Z–aZl_gzggZy^bw^jZevgucm]he [–rZofZlgZyaZlhjfh`_ggZy^bw^jZevgucm]he 25 JZkkfhljbf gZ[hj S 61 , S 62 , S 63 , S 64 , S 65 , S 66 ^ey aZlhjfh`_gghc kljmdlmjuwlZgZjbk[ S 61 - g_evay aZibkZlv dZdbfeb[hbgufkihkh[hf S 62 = C62 σ 2 = C3 , S 63 = S2 = i hkvS2\k_]^ZjZ\ gZi), S 64 = C32 , S 65 –g_evayaZibkZlvdZdbfJbk<b^ukbff_ljbbfhe_dmeuKG4 eb[hbgufkihkh[hf S 66 = ? . IhegucgZ[hj^ey^Zggh]hkemqZyh[uqghaZibku\ZxllZd S6 C3 i C32 S 65 E. ?keb n g_q_lgh_ lh S nm ijb\h^bl d gZ[hjm S n , S n2 , S n3 , S n4 ,..., S n2n . Sn ijbg_q_lghfnkhhl\_lkl\m_lKnbi_ji_g^bdmeyjghc_c1Hlf_lbfqlh hi_jZpby S nn _kebng_q_lgh_wd\b\Ze_glgZ Cnn σ n = Cnn σ = σ . < dZq_kl\_ ijbf_jZ jZkkfhljbf S3 < wlhf kemqZ_ ij_`^_ \k_]h ^he`gh kms_kl\h\Zlv hljZ`_gb_ ijb\h^ys__ d dhgnb]mjZpbb wd\b\Z e_glghc bkoh^ghc ihkdhevdm ^ey ^Zgghc fhe_dmeu kms_kl\m_l 1 >Ze__ \jZs_gb_ \hdjm] K3 ^he`gh ^Z\Zlv dhgnb]mjZpbx khhl\_lkl\mxsmx S 31 . Ihkdhevdm S 31 –hi_jZpbykbff_ljbbfhe_dmeudhgnb]mjZpbyihke_\jZ s_gby\hdjm]K3^he`gZ[ulvwd\b\Ze_glghcbkoh^ghcblZdbfh[jZahf K3^_ckl\bl_evghkms_kl\m_l<hh[s_^eyex[hchi_jZpbbSnijbg_q_l ghfn) KnlZd`_y\ey_lkyhi_jZpb_ckbff_ljbbIjbf_jfhe_dmeubf_x s_chkv\jZs_gbyS3,ijb\_^_ggZjbkZK\_^_f\lZ[ehkgh\gu_ih gylbydhlhju_[uebbkihevah\ZguijbjZkkfhlj_gbbhi_jZpbcbwe_f_g lh\kbff_ljbb 26 We_f_glubhi_jZpbbkbff_ljbb Hi_jZpbykbff_ljbb Lh`^_kl\_ggh_ij_h[jZ ah\Zgb_ HljZ`_gb_ Bg\_jkby We_f_gl kbff_ljbb H[hagZq_gb_ Ijbf_j _ ? <k_fhe_dmeu Iehkdhklv G2H<F3 LhqdZp_glj kbff_ljbb I Cl2B–BCl2 Hkv Cn NH3, H2O Hkvbiehk dhklv Sn WlZg Kh[kl\_ggh_\jZs_gb_ G_kh[kl\_ggh_\jZs_gb_ ih\hjhlgZn kihke_^mxsbfhljZ`_ gb_f\iehkdhklbi_j i_g^bdmeyjghchkb LZ[ebpZ 2.2. LHQ?QGU?=JMIIU H[t_dlu\qZklghklbfhe_dmeudeZkkbnbpbjmxlkyihkbff_ljbb\ khhl\_lkl\bb k bo we_f_glZfb kbff_ljbb imlzf h[t_^bg_gby \ ]jmiiu <hafh`gu_ dhf[bgZpbb hi_jZpbc kbff_ljbb hklZ\eyxsbo [_a baf_g_ gby ohly [u h^gm lhqdm gZijbf_j p_glj fZkk gZau\Zxlky lhq_qgufb ]jmiiZfbkbff_ljbbKms_kl\mxl^\_kbkl_fukbf\heh\Lhq_qgu_]jmi iukbkl_fZRzgnebkZqZs_mihlj_[eyxlkyijbjZkkfhlj_gbbbg^b\b^m Zevguo fhe_dme Z ijhkljZgkl\_ggu_ ]jmiiu F_`^mgZjh^gZy kbkl_fZ =_jfZgZ-Fh]_gZ–ihqlbbkdexqbl_evghijbjZkkfhlj_gbbdjbklZeeh\ <^Zgghfihkh[bbdjbklZeeug_jZkkfZljb\Zxlkybke_^h\Zl_evgh j_qvihc^zllhevdhhkbkl_f_RzgnebkZ<lhq_qguo]jmiiZo\k_we_f_g lu kbff_ljbb ijhoh^yl q_j_a nbdkbjh\Zggmx lhqdm beb gZqZeh dhhj^b gZl < ijbgpbi_ kms_kl\m_l [hevrh_ fgh`_kl\h lhq_qguo ]jmii gh \k_ bofh`ghk\_klbdhkgh\guflbiZf =jmiiu C1 , C S , Ci h[hagZq_gb_]jmiiu C1 bwe_f_glZ C1 kh\iZ ^Zxl?kebfhe_dmeZg_kbff_ljbqgZhgZijbgZ^e_`bld]jmii_ C1 we_ f_gl E jbk?kebbf_xlkywe_f_glu E ba_jdZevgZyiehkdhklvkbf 27 f_ljbb σ fhe_dmeZijbgZ^e_`bldlhq_qghc]jmii_ C S jbk_keb E bbg\_jkby i –]jmiiZ Ci gZijbf_jf_ah\bggZyzdbkehlZjbk JbkIjbf_jg_kbff_ljbqghcJbk=jmiiZkbff_ljbb C S fhe_dmeu]jmiiZ C1 ) JbkKbff_ljbyfhe_dmeu f_ah\bgghcdbkehlu JbkFhe_dmeZi_j_dbkb\h^hjh^Z ]jmiiZK2) =jmiiu C n ( n > 1 h[hagZq_gb_wlbo]jmiikh\iZ^Z_lkh[hagZq_ gb_f we_f_glh\ Bo h[jZamxl we_f_glu kbff_ljbb E b C n Ijbf_j – i_j_dbkv\h^hjh^Zjbk =jmiiu C nv ( n > 1 ) –we_f_glu E , C n b n \_jlbdZevguoa_jdZev guoiehkdhkl_c σ v Fhe_dmeZ\h^ukh^_j`bllhevdhh^gmhkv\jZs_gbyb ^\_ iehkdhklb σ ( E , C , σ , σ ′ ), – ]jmiiZ C fhe_dmeZ ZffbZdZ v 2 v v ( E , C3 , σ v , σ v ′ , σ v ″ beb E , C3 , 3σ v ) –]jmiiZ C3v . 28 2v =jmiiu C nh ( n > 1 ) –we_f_glu E , C n b]hjbahglZevgZya_jdZev gZy iehkdhklv σ h i_ji_g^bdmeyjgZy hkb \jZs_gby C n jbk =jmiiZ C 2h Z\lhfZlbq_kdb bf__l we_f_gl i =_hf_ljbq_kdZy nb]mjZ dhgmk ijb gZ^e_`bl d ]jmii_ C ∞h ke_^h\Zl_evgh ^\moZlhfgu_ ]_l_jhy^_jgu_ fh e_dmeuijbgZ^e_`Zld]jmii_ C ∞h . =jmiiu Dn – we_f_glu E , C n b n hk_c C 2 i_ji_g^bdmeyjguo C n GZjbkihdZaZgh[t_dlk D6 –fhe_dmeZh^gh]hbaf_lZeehp_gh\ Jbk=jmiiu C 2h (C2H2Cl2 –^boehjwlZg Jbk=jmiiZkbff_ljbb D6 b C3h (H3BO3 –[hjgZydbkehlZ =jmiiu Dnh –wlh]jmiiZ Dn iexk σ h i_ji_g^bdmeyjgZyhkbKn jbk WlZ]jmiiZbf__lbn\_jlbdZevguoa_jdZevguoiehkdhkl_cIeh kdZy fhe_dmeZ BF3 – ]jmiiZ D3h [_gahe – D6h =_hf_ljbq_kdZy nb]mjZ pbebg^j \oh^bl \ ]jmiim D∞h ke_^h\Zl_evgh ^\moZlhfgu_ ]hfhy^_jgu_ fhe_dmeuijbgZ^e_`Zld]jmii_ D∞h . Z [ \ Jbk=jmiiukbff_ljbb Dnh : Z– D2h fhe_dmeuwlZgZ[– D3h fhe_dmeZ PCl5\– D4h fhe_dmeZAuCl4) 29 =jmiiu Dnd –wlh]jmiiZ Dn iexkiehkdhklb σ d \dhlhjuogZ oh^blky ]eZ\gZy Kn hkv gh g_l gb h^ghc ba i_ji_g^bdmeyjguo hk_c K2 jbkRZofZlgZyaZlhjfh`_ggZydhgnhjfZpbywlZgZkfjbk[– ]jmiiZ D3d . Dm[bq_kdb_]jmiiu T , O bboijhba\h^gu_jbkMh[t_dlh\ wlbo ]jmii bf__lky [he__ q_f h^gZ ]eZ\gZy hkv kbff_ljbb GZijbf_j \ fhe_dme_f_lZgZCH4bf_xlf_klhq_luj_jZ\ghp_ggu_hkb C3 Kj_^bdm [bq_kdbo\u^_eyxlkyl_ljZw^jbq_kdb_]jmiiu T , Td , Th bhdlZw^jbq_kdb_ ]jmiiu O, Oh =jmiiZ Td – ijZ\bevguc l_ljZw^j ]jmiiZ Oh – ijZ\bev guchdlZw^j?kebh[t_dlbf__l\jZsZl_evgmxkbff_ljbxl_ljZw^jZbeb hdlZw^jZghg_bf__lgbh^ghciehkdhklb σ lhhgijbgZ^e_`bl]jmiiZf T beb O khhl\_lkl\_ggh < ]jmiim Th \oh^yl h[t_dlu h[eZ^Zxsb_ oZ jZdl_jbklbdZfb]jmiiu T bp_gljhfbg\_jkbb i . JbkIjbf_j JbkFhe_dmeu]jmii Td (CCl4Zb Oh (SF6[ fhe_dmeu ]jmiiu Dnd =jmiiZihegh]h\jZs_gby R 3 –kn_jbq_kdbch[t_dl<wlm]jmiim \oh^ylZlhfugbh^gZfhe_dmeZg_ijbgZ^e_`bldwlhc]jmii_ =jmiiu S n bgh]^ZlZdb_]jmiiuhl^_evghg_\u^_eyxlkyH[t _dluwlbo]jmiioZjZdl_jbamxlkygZebqb_fwe_f_glh\kbff_ljbb S n . 30 2.3. IHJY>HDDE:KKBNBD:PBBFHE?DME IHLHQ?QGUF=JMII:F >ey h[e_]q_gby deZkkbnbdZpbb h[t_dlh\ ih ]jmiiZf fh`gh j_dh f_g^h\Zlv ke_^mxsbc ihjy^hd ijb_fe_fuc ^ey ih^Z\eyxs_]h qbkeZ h[t_dlh\ij_^iheZ]Z_lkyqlhh[t_dlg_h[eZ^Z_lkn_jbq_kdhckbff_ljb_c 1. Hij_^_eblvhlghkblkyebfhe_dmeZd]jmiiZf T beb O ihgZeb qbxq_lujzohk_c C3 \^hev^bZ]hgZe_cdm[Z 2. ?kebh[t_dlg_bf__lgbh^ghchkb Cn ghZbf__lkyiehkdhklv σ – ]jmiiZ C S [ bf__lky p_glj kbff_ljbb i – ]jmiiZ Ci \ gbq_]h g_ bf__lky–]jmiiZ C1 . 3. ?kebbf__lkyohly[uh^gZ C n hij_^_ey_lky]eZ\gZyhkv 4. ?kebbf__lkyhkv S 2n kh\iZ^ZxsZyk]eZ\ghcbebk^jm]hchkvx bijbwlhfg_l^jm]bodjhf_ i we_f_glh\kbff_ljbblhh[t_dlijbgZ^ e_`bld]jmii_ S m (m = 2n) . 5. ?kebij_^u^msbcimgdlg_ih^oh^blhij_^_ey_lkybf__lkyeb n hk_c C 2 i_ji_g^bdmeyjguo]eZ\ghcbeb^jm]hchkb?kebwlhlZdh[t _dlijbgZ^e_`bldh^ghcba]jmii D ?kebbf__lkyiehkdhklv σ h –]jmiiZ Dnh _keb_klviehkdhklb σ d –]jmiiZ D nd _kebiehkdhkl_c σ g_l\hh[ s_–]jmiiZ D n . 6. ?kebg_bf__lky n hk_c C 2 i_ji_g^bdmeyjguo]eZ\ghcbeb^jm ]hchkblhh[t_dlijbgZ^e_`bl]jmiiZf C ?kebbf__lkyiehkdhklv σ h – ]jmiiZ C nh _keb_klv n iehkdhkl_c σ v –]jmiiZ C nv _kebiehkdhkl_c σ g_l–]jmiiZ C n . 2.4. G?DHLHJU?IJHKLU?K<HCKL<:KBFF?LJBB >bihevgucfhf_gl_klv\_dlhjgZ\_ebqbgmbgZijZ\e_gb_dhlhjh]h g_ \ebyxl ex[u_ hi_jZpbb kbff_ljbb Kh[kl\_gguf ^bihevguf fhf_g lhf fh]ml h[eZ^Zlv lhevdh h[t_dlu ijbgZ^e_`Zsb_ d ]jmiiZf C n , C nv , C S , C1 < wlbo ]jmiiZo g_l i_j_\hjZqb\Zgby h[t_dlh\ \\_jo beb \gbadZdwlhbf__lf_klhkdZ`_f\]jmiiZo C3h , D b^jGZijbf_jCO2 b H2O fh]ml bf_lv kljh_gb_ khhl\_lkl\mxs__ kbff_ljbqghc ebg_cghc fhe_dme_bebg_kbff_ljbqghcebg_cghcfhe_dme_bebbah]gmlhcfhe_dm e_Ihkdhevdmwdki_jbf_glihdZau\Z_lqlh^bihevgucfhf_glCO2jZ\_g 31 gmexlh_]hfhe_dmeZ–kbff_ljbqgZyebg_cgZymfhe_dmeu\h^ubf__l kyy\gh\ujZ`_gguc^bihevgucfhf_glke_^h\Zl_evghhgZbah]gmlZld g_ fh`_l [ulv g_kbff_ljbqghc ebg_cghc Fhe_dmeZ [_gaheZ g_ fh`_l bf_lv^bihevgucfhf_glZgZijZ\e_gb_^bihevgh]hfhf_glZoehj[_gaheZ ijhoh^blq_j_aZlhfClbp_gljfhe_dmeu Hilbq_kdZyZdlb\ghklvFhe_dmeZfh`_l\jZsZlviehkdhklviheyjb ah\Zggh]hk\_lZ_keblhevdhhgZh[eZ^Z_lk\hckl\hfobjZevghklbl_jfbg ³objZevghklv´ ijhbkoh^bl hl ]j_q_kdh]h ³jmdZ´ ObjZevgu_ nb]mju km s_kl\mxl\\b^_^\moa_jdZevghjZ\guonhjf–wgZglbhf_jh\h^bggZau \Z_lky³ijZ\uf´^jm]hc³e_\uf´DZ`^uc wgZglbhf_jy\ey_lkya_jdZev guf hlh[jZ`_gb_f k\h_c iZju _keb bo jZaf_klblv \ ijhkljZgkl\_ khhl \_lkl\mxsbfh[jZahfh^gZdh_]hg_evaykh\f_klblvkg_cimlzfex[uo i_j_f_s_gbc \ ijhkljZgkl\_Wlh ohjhrh \b^gh gZ ijbf_j_ jmd _kebbo jZkiheh`blv eZ^hgyfb ^jm] d ^jm]m hgb [m^ml a_jdZevgufb hlh[jZ`_ gbyfbghbo g_\hafh`gh kh\f_klblv ^jm] k ^jm]hfObjZevgu_ kh_^bg_ gby\`b^dhfkhklhygbbih\hjZqb\ZxliehkdhklviheyjbaZpbbgZm]he α , ijhihjpbhgZevguclhesbg_ij_iZjZlZIjbwlhfbf__lky\\b^mqlh^Zg guc ij_iZjZl kh^_j`bl iheghklvx beb ij_bfms_kl\_ggh h^bg ba wgZg lbhf_jh\?kebkms_kl\m_lhkv S n lhfhe_dmeZZobjZevgZbg_bf__lhi lbq_kdhc Zdlb\ghklb gZijbf_j ]jmiiZ C nh \dexqZ_l \ k_[y S n g_y\gh ?kebbf__lkyp_gljkbff_ljbb i lhh[t_dlg_fh`_l[ulvhilbq_kdbZd lb\guf < fhe_dmeZo k \gmlj_ggbf k\h[h^guf \jZs_gb_f wlZg H2O2) \hafh`guhilbq_kdbZdlb\gu_dhgnhjfZpbbghhgbijbkmlkl\mxl\`b^ dhklb\jZ\guodhebq_kl\Zob]ZabebjZkl\hj\p_ehfhilbq_kdbg_Zdlb\gu Kbff_ljbqgu_ k\hckl\Z hj[blZe_c Ijb jZkkfhlj_gbb kbff_ljbb hj[blZe_c g_h[oh^bfh mqblu\Zlv agZd ψ-nmgdpbb S-hj[blZeb ijbgZ^e_ `Zl d ]jmii_ R3 , p -hj[blZeb Zglbkbff_ljbqgu \ i_ji_g^bdmeyjghc iehkdhklbbg_bf_xlp_gljZbg\_jkbb i , d -hj[blZebbf_xl i . 2.5. =JMIIH<[email protected]?GB?K<HCKL<:=JMIIU Ijhba\_^_gb_ ex[uo ^\mo hi_jZpbc kbff_ljbb hij_^_ey_fh_ dZd bo ihke_^h\Zl_evgh_ ijbf_g_gb_ [m^_l hi_jZpb_c kbff_ljbb Ihwlhfm ijhba\_^_gb_hi_jZpbcK2b σvijhba\h^bfuogZ^fhe_dmehckbff_ljbb K2#gZijbf_jG2H– wlhσvgZihfgbfqlhhkvK2y\ey_lkyhkvxz,fhe_ dmeZe_`bl\iehkdhklbyz, Ziehkdhklvyz _klviehkdhklvσv): K2· σv = σv. 32 <f_klh l_jfbgZ ©ijhba\_^_gb_ª hibkZggmx hi_jZpbx h[uqgh oZ jZdl_jbam_l l_jfbg ©dhf[bgZpbyª Ihjy^hd aZibkb hi_jZpbb l_ ke_\Z gZijZ\hijhlb\hiheh`_gihjy^dm\dhlhjhfhgb\uihegyxlky<ijb\_ ^_gghfijbf_j_hi_jZpbyσvhkms_kl\ey_lkyi_j\hcZaZl_fke_^m_lhi_ jZpbyK2qlh^Z_llhl`_kZfucj_amevlZlqlhbhi_jZpbyσv<hh[s_dh g_qguc j_amevlZl aZ\bkbl hl ihjy^dZ \uiheg_gby hi_jZpbc h^gZdh \ g_ dhlhjuo kemqZyo wlh g_ lZd ?keb j_amevlZl ^\mo hi_jZpbc g_ aZ\bkbl hl ihjy^dZbo\uiheg_gbygZijbf_jK2 σv = σv K2]h\hjylqlh^\Zwe_ f_glZkbff_ljbbK2bσvdhffmlbjmxl Hi_jZpbb kbff_ljbb fh]ml [ulv jZkkfhlj_gu k ihfhsvx l_hjbb ]jmii>eylh]hqlh[ukh\hdmighklvwe_f_glh\h[jZah\u\ZeZfZl_fZlbq_ kdmx]jmiim^he`gu\uihegylvkyke_^mxsb_mkeh\by 1. Dhf[bgZpby ex[uo ^\mo we_f_glh\ b d\Z^jZl ex[h]h we_f_glZ ^he`gu^Z\Zlvwe_f_glwlhc`_]jmiiu?keb\ha\_klb\d\Z^jZlwe_f_gl lhq_qghc]jmiiuK2vbeb\aylv\k_\hafh`gu_dhf[bgZpbbwe_f_glh\wlhc ]jmiiu lh hiylv `_ ihemqblky we_f_gl ijbgZ^e_`Zsbc d ]jmii_ K2v. <hlebqb_hlmfgh`_gby\h[uqghcZjbnf_lbd_g_gm`ghaZ[u\Zlvhih ke_^h\Zl_evghklb\uiheg_gbyhi_jZpbc\dhf[bgZpbb 2. H^gZbahi_jZpbc]jmiiu^he`gZdhffmlbjh\Zlvkh\k_fb^jm]b fb b hklZ\eylv bo [_a baf_g_gby LZdbf we_f_glhf y\ey_lky lh`^_kl\_g gh_ij_h[jZah\Zgb_ ?σ 2 = σ2? = σ2. 3. >he`_g\uihegylvkyZkkhpbZlb\gucaZdhgmfgh`_gbyl_ (O<)Z = X(YZ). J_amevlZl dhf[bgZpbb lj_o we_f_glh\ ^he`_g [ulv l_f `_ kZfuf _keb i_j\uc we_f_gl dhf[bgbjm_lky k ijhba\_^_gb_f ^\mo ^jm]bo >l_ σv(K2 σ[email protected]_kebijhba\_^_gb_^\moi_j\uodhf[bgbjm_lkykihke_^gbf we_f_glhf>l_σvK2 ) σv]. 4. DZ`^ucwe_f_gl^he`_gbf_lvh[jZlgucwe_f_gldhlhjuclZd`_ y\ey_lkywe_f_glhf]jmiiuLZdbfh[jZahf^eydZ`^hchi_jZpbbkbff_l jbb^he`gZkms_kl\h\Zlvhi_jZpbymkljZgyxsZyj_amevlZl^_ckl\byi_j \hc hi_jZpbb >ey ex[hc a_jdZevghc iehkdhklb h[jZlguf we_f_glhf ^he`gZ[ulvb^_glbqgZya_jdZevgZyiehkdhklvl_σ σ = ?>eykh[kl\_g gh]h\jZs_gby Cnm h[jZlgufwe_f_glhfy\ey_lky Cnn−m ; Cnm Cnn−m = ??keb :bf__lh[jZlgucwe_f_glBlh]^Z:< <: ??keb<ijhlb\hiheh`_g :lhb:ijhlb\hiheh`_g<<hh[s_we_f_glh[jZlgucwe_f_glm:fh` ghaZibkZlvdZd:-1. 33 We_f_glhfh[jZlgufwe_f_glmK3y\ey_lkywe_f_gl C32 : K3 C32 = C32 K3 = ? LZ[ebp_c]jmiih\h]hmfgh`_gbygZau\Z_lkylZ[ebpZ\k_odhf[bgZpbc Ihkdhevdm ijhklh_ mfgh`_gb_ dZd ijZ\beh g_ y\ey_lky dhffmlZ lb\guf fu ^he`gu ijb^_j`b\Zlvky hij_^_e_ggh]h ihjy^dZ mfgh`_gby H[uqgh ijbgbfZxl qlh hi_jZpby hkms_kl\ey_lky \ ke_^mxs_f ihjy^d_ we_f_gl klhe[pZ mfgh`Z_lky gZ we_f_gl kljhdb l_ _keb fu aZibku\Z_f ijhba\_^_gb_dZdKR]^_Khlghkblkydklhe[pmZR –dkljhd_lhfu[_ j_fkgZqZeZwe_f_glklhe[pZZaZl_fwe_f_glkljhdb Gb`_ ijb\_^_gZ ^ey ijbf_jZ lZ[e ]jmiih\h]h mfgh`_gby ^ey ]jmiiu C3v . LZ[ebpZ LZ[ebpZmfgh`_gby^eylhq_qghc]jmiiuK3# R C ? K3 C32 σv σv σv ? ? K3 C32 σv σv σv K3 K3 C32 ? σv σv σv C32 C32 ? K3 σv σv σv σv σv ? K3 C32 σv σv σv σv C32 ? K3 σv σv σv σv K3 C32 ? σv σv 2.6. F:LJBQGH?IJ?>KL:<E?GB?LJ:GKNHJF:PBC O:J:DL?JHI?J:PBCKBFF?LJBB FZljbpZ ij_^klZ\ey_l kh[hc jZkiheh`_gb_ qbk_e beb kbf\heh\ \ \b^_ijyfhm]hevgbdZke_^mxs_]hh[s_]h\b^Z a11 a12 a a22 21 a31 a32 34 a13 a23 . a33 D\Z^jZlgu_kdh[dbmdZau\ZxlgZlhqlhwlhfZljbpZZg_^_l_jfb gZgl?kebqbkehkljhdfZljbpujZ\ghqbkemklhe[ph\fZljbpZgZau\Z_l ky d\Z^jZlghc We_f_gl d\Z^jZlghc fZljbpu \b^Z Z11 Z22 Z33 gZau\Z_lky ^bZ]hgZevgufwe_f_glhfZ^jm]b_we_f_glughkylgZa\Zgb_g_^bZ]hgZev guo?keb\k_g_^bZ]hgZevgu_we_f_glujZ\gugmexfZljbpZgZau\Z_lky ^bZ]hgZebah\Zgghc beb ^bZ]hgZevghc ?keb dZ`^uc ^bZ]hgZevguc we_ f_gld\Z^jZlghcfZljbpujZ\_g_^bgbp_Zg_^bZ]hgZevgu_we_f_glujZ\ gugmexfZljbpZgZau\Z_lky_^bgbqghc<Z`gZyoZjZdl_jbklbdZd\Z^jZl ghcfZljbpuke_^beboZjZdl_jij_^klZ\ey_lkh[hcijhklhkmffm^bZ]h gZevguo we_f_glh\ FZljbpu fh`gh kdeZ^u\Zlv \uqblZlv ^_eblv b mf gh`Zlvbkihevamykhhl\_lkl\mxsb_ijZ\beZfZljbqghcZe]_[ju DZdjZg__[uehhlf_q_gh?, 1, i, Cn, Snhibku\Zxlkbff_ljbxh[t _dlh\DZ`^mxbawlbohi_jZpbcfh`ghhibkZlvfZljbp_c JZkkfhljbf lhqdm J k dhhj^bgZlZfb o y b z jZ\gufb jbk b khhl\_lkl\mxsbfbijh_dpbyfblhqdbgZwlbhkbLh`^_kl\_ggh_ij_h[jZ ah\Zgb_ gZ^ wlhc lhqdhc ijb\h^bl d gh\hfm gZ[hjm dhhj^bgZl dhlhjuc gbq_fg_hlebqZ_lkyhlklZjh]h >Zggmxhi_jZpbxhibku\Z_lfZljbpZ ' 1 0 0 X X 0 1 0 Y = Y ' . 0 0 1 Z Z ' Mfgh`_gb_fZljbp^Z_l X = X Y = Y . Z = Z JbkLhqdZ\^_dZjlh\hc kbkl_f_dhhj^bgZl =h\hjylqlhwlZ_^bgbqgZyfZljbpZy\ey_lkyij_^klZ\e_gb_flh`^_ kl\_ggh]hij_h[jZah\Zgby JZkkfhljbffhe_dmem C3v kp_gljZevgufZlhfhf S N blj_fyZlhfZfb S A , S B , SC jbkIjbhi_jZpbb σ v (S N , S A , SC , S B ) ← (S N , S A , S B , SC ) . 35 LZdmx ljZgknhjfZpbx fh`gh \ujZablv bkihevamy fZljbqgh_ mf gh`_gb_l_fh`ghgZclbfZljbpm D(σ v ) lZdmxqlh 1 0 (S N , S A , SC , S B ) ← (S N , S A , S B , SC ) 0 0 0 0 0 1 0 0 . 0 1 0 0 0 1 FZljbpZ D(σ v ) gZau\Z_lkyij_^klZ\e_gb_fhi_jZpbb σ v . Ij_^klZ\e_gb_hi_jZpbb C3− –fZljbpZ D (C3− ) : 1 0 0 0 0 1 ( S N , S C , S A , S B ) ← ( S N , S A , S B , S C ) 0 0 0 0 1 0 Ij_^klZ\e_gb_hi_jZpbb C3+ –fZljbpZ D(C3+ ) : 1 0 ( S N , S B , S C , S A ) ← ( S N , S A , S B , S C ) 0 0 0 0 . 1 0 0 0 0 0 0 1 . 1 0 0 0 1 0 Ij_^klZ\e_gb_ E : 1 0 0 0 0 1 0 0 D(E ) = . 0 0 1 0 0 0 0 1 JbkFhe_dmeZ]jmiiu kbff_ljbb C3v 36 >ey D (σ v ′ ) : (SN , SC , SB , S A) ← 1 0 0 0 0 0 0 1 ← ( S N , S A , S B , S C ) . 0 0 1 0 0 1 0 0 ″ >ey D(σ v ) : 1 0 ( S N , S B , S A , S C ) ← ( S N , S A , S B , S C ) 0 0 0 0 0 0 1 0 . 1 0 0 0 0 1 >eyijhba\_^_gby D(σ v ) D(C3+ ) bf__f 1 0 + D (σ v ) D (C3 ) = 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 = 1 0 0 0 0 0 1 0 0 1 l_ D(σ′v ) = D(σ v ) D(C3+ ) qlhkh\iZ^Z_lk σ v ′ = σ v C3+ . 0 0 0 1 = D(σ′v ) , 1 0 0 0 IhegZy lZ[ebpZ ]jmiih\h]h mfgh`_gby \hkijhba\h^blky Ze]_[jZb q_kdbfmfgh`_gb_ffZljbqguoij_^klZ\e_gbc GZ[hjbar_klbfZljbpy\ey_lkyfZljbqgufij_^klZ\e_gb_f]jmiiu C3v blZ[efh`ghihkljhblvZe]_[jZbq_kdbWe_f_glu S N , S A , S B , S C gZau\Zxlky[Zabkhf\^ZgghfkemqZ_hgq_luj_of_jgucqlhijb\h^bld q_luj_of_jguf fZljbpZf ^ey f_lZgZ ]jmiiZ Td [Zabk lZd`_ q_luj_o f_jguc ?keb^eygZr_]hijbf_jZkeh`blv^bZ]hgZevgu_we_f_glugZ]eZ\ ghc^bZ]hgZebfZljbqguoij_^klZ\e_gbclhihemqblky D( E ) D(C + ) D(C − ) D(σ ) D(σ ′ ) D(σ ″ ) 3 3 v v v χ= 4 1 1 2 2 2 KmffZ ^bZ]hgZevguo we_f_glh\ fZljbqgh]h ij_^klZ\e_gby hi_jZ pbb kbff_ljbb gZau\Z_lky oZjZdl_jhf χ ^Zgghc hi_jZpbb Hi_jZpbb k h^bgZdh\uf χ hlghkylkydh^ghfmdeZkkmOZjZdl_jhi_jZpbb χ( g i ) aZ\b kblhl[ZabkZOZjZdl_jujZaguodeZkkh\fh]ml[ulvh^bgZdh\ufb < dZq_kl\_ [ZabkZ fh`gh \aylv _^bgkl\_gguc Zlhf S N ex[Zy hi_ jZpbykbff_ljbb[m^_lk\h^blvkyd S N ← S N ij_^klZ\e_gb_ex[hchi_ jZpbb S N = S N ⋅ 1 ]^_ fh`gh jZkkfZljb\Zlv dZd fZljbpm oZjZdl_ju \k_oij_^klZ\e_gbc χ = 1Wlhij_^klZ\e_gb_ijZdlbq_kdbg_bgnhjfZlb\ gh b gZau\Z_lky g_^h\_jbl_evguf ij_^klZ\e_gb_f ]jmiiu \k_ we_f_glu \ujZ`_gu_^bgbp_c 37 2.7. G?IJB<H>BFU?IJ?>KL:<E?GBY <k_ fZljbqgu_ ij_^klZ\e_gby hi_jZpbc kbff_ljbb ]jmiiu C3v bf_xl[ehd-^bZ]hgZevgmxnhjfmlh_klv\kyfZljbpZfh`_l[ulvjZa[blZ gZ[ehdbf_gvrbofZljbp[_ag_^bZ]hgZevguowe_f_glh\f_`^m[ehdZfb 1 0 0 0 0 , 0 0 bhi_jZpbbkbff_ljbbgbdh]^Zg_i_j_f_rb\Zxl S N k^jm]bfbnmgdpby fb [ZabkZ S A , S B , S C l_ p_gljZevguc Zlhf b]jZ_l hkh[mx jhev \ kbf f_ljbb \ ^Zgghf [Zabk_ Q_luj_of_jgh_ ij_^klZ\e_gb_ fh`_l [ulv ih^ jZa^_e_ghgZ^\Zg_aZ\bkbfuo[ZabkZh^bg–kh^gbfqe_ghf^jm]hc–k lj_fyHgbh[hagZqZxlky D (1) b D (3) baZf_gyxl D ( 4) G_^h\_jbl_evgh_ ij_^klZ\e_gb_ D (1) ^Zevr_ih^h[gufh[jZahfg_fh`_l[ulvijb\_^_ghb y\ey_lkyg_ijb\h^bfufij_^klZ\e_gb_fLj_of_jgh_ij_^klZ\e_gb_ D (E ) χ= 1 0 0 0 1 0 3 0 0 1 D ( C 3+ ) 0 1 0 0 0 1 0 D ( C 3− ) 1 0 0 0 0 1 1 0 0 0 D (σ v ′ ) D (σ v ) 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 D (σ v ″ ) 1 0 0 0 1 0 1 0 0 1 0 0 1 G_ijb\h^bfu_ ij_^klZ\e_gby h[hagZqZxlky $ – h^ghf_jgu_ ( – ^\mf_jgu_g_imlZlvkb^_glbqghklvx7–lj_of_jgu_%–h^ghf_jgu_ ij_^klZ\e_gbyoZjZdl_jdhlhjuojZ\_g–ijbq_foZjZdl_jwlhlkhhl\_l kl\m_lhi_jZpbb C n ]^_ C n –]eZ\gZyhkvkbff_ljbb Kibkdb oZjZdl_jh\ \k_o \hafh`guo g_ijb\h^bfuo ij_^klZ\e_gbc jZaebqguo]jmiigZau\ZxlkylZ[ebpZfboZjZdl_jh\Hgbm`_khklZ\e_gu ^ey fgh`_kl\Z ]jmii b bfb gZ^h lhevdh mf_lv ihevah\Zlvky ld bf_ggh hgbij_^klZ\eyxlkh[hcgZb[he__bgl_j_kgucbklhqgbdbgnhjfZpbbDZ `^Zy pbnjZ \ lZ[ebp_ oZjZdl_jh\ ij_^klZ\ey_l kh[hc oZjZdl_j fZljbpu ^eyhi_jZpbbqvbfij_^klZ\e_gb_fhgZy\ey_lky>eyfZljbpu1× 1 oZjZd l_jbfZljbpZkh\iZ^Zxl>eydZ`^h]hg_ijb\h^bfh]hij_^klZ\e_gbybf_ _lkyk\hykljhdZ Ijb\_^_fl_hj_fml_hjbb]jmii[_a^hdZaZl_evkl\Z³Qbkehg_ijb \h^bfuoij_^klZ\e_gbcjZ\ghqbkemdeZkkh\´. Qbkehg_ijb\h^bfuoij_^klZ\e_gbckjZ\gbl_evghfZeh<lZ[ebpu oZjZdl_jh\ fh]ml \dexqZlvky g_ijb\h^bfu_ ij_^klZ\e_gby ij_h[jZah\Z gby dhhj^bgZl b bo d\Z^jZlbqguo nhjf Z lZd`_ \jZs_gb_ hlghkbl_evgh hk_c x, y, z dZd^hihegbl_evgZybgnhjfZpby 38 =eZ\Z3. WE?DLJBQ?KDB?BF:=GBLGU?K<HCKL<: FHE?DME 3.>BIHEVGUCFHF?GL H[sb_ ij_^klZ\e_gby h kljh_gbb fhe_dme dZd h kbkl_fZo iheh`b l_evgh aZjy`_gguo y^_j b we_dljhgguo h[eZdh\ jZaebqghc iehlghklb Z lZd`_ aZdhgu kbff_ljbb fhe_dme ijb\h^yl d \u\h^m h \hafh`ghklb g_ jZ\ghf_jgh]h jZkij_^_e_gby we_dljbq_kdh]h aZjy^Z \ fhe_dme_ \ p_ehf we_dljbq_kdbg_cljZevghc<g_rgb_ijhy\e_gbylZdhcg_jZ\ghf_jghklbb h[mkeh\eb\Zxl we_dljbq_kdb_ k\hckl\Z fhe_dme Ijb wlhf dhgdj_lgh_ jZkij_^_e_gb_ we_dljbq_kdh]h aZjy^Z ih h[tzfm fhe_dmeu [hevrh]h agZ q_gbyg_bf__l\Z`ghebrviheh`_gb_kbeh\h]hp_gljZiheh`bl_evguob hljbpZl_evguo aZjy^h\ Wlb p_glju fh]ml kh\iZ^Zlv ^jm] k ^jm]hf gZ ijbf_jm\ukhdhkbff_ljbqguofhe_dmebebg_kh\iZ^Zlv<khhl\_lkl\bb k wlbf \ k\h[h^ghf khklhygbb fhe_dmeu fh]ml [ulv iheyjgufb b g_ih eyjgufb<kzwlhih^l\_j`^Z_lkyfgh]hqbke_ggufbwdki_jbf_glZevgufb ^Zggufb < h^ghc qZklb fhe_dmeu fh`_l gZoh^blvky ba[ulhd we_dljhggh]h aZjy^Z iehlghklb \ ^jm]hc __ qZklb – ba[ulhd iheh`bl_evgh]h aZjy^Z <wlhfkemqZ_fhe_dmeZy\ey_lkyiheyjghcij_^klZ\ey_lkh[hcwe_dljbq_ & kdbc^bihevbh[eZ^Z_l^bihevguffhf_glhf µ ?keb^\ZaZjy^Z q b − q jZa^_e_gujZkklhygb_f r lh & µ = qr . & >bihevgucfhf_gl µ –wlh\_dlhjgZijZ\e_gguchlhljbpZl_evgh]h aZjy^Zdiheh`bl_evghfmjbkBaf_jy_lky^bihevgucfhf_gl\dmeh gZo gZ f_lj De ⋅ f ) \g_kbkl_fgZy _^bgbpZ – ^_[Zc ( > ; gZa\ZgZ \ q_klv ]heeZg^kdh]hmqzgh]hI>_[Zy 1 > = 3,336 ⋅10 − 30 De ⋅ f . & >bihevgucfhf_gl µ fh`ghjZkkqblZlv^ey ex[h]h jZkij_^_e_gby lhq_qguo aZjy^h\ qi , ihe gucaZjy^dhlhjuojZ\_ggmex & & & µ = ∑ (qi ri ) ]^_ ri –jZ^bmk-\_dlhjbagZqZ i eZdhhj^bgZldaZjy^m qi . µ x = ∑ (qi xi ); µ y = ∑ (qi yi ); µ z = ∑ (qi zi ) , i i ]^_ xi , yi b zi –dhhj^bgZluaZjy^Z qi . i Jbk. 24Dhij_^_e_gbx ^bihevgh]hfhf_glZ 39 >bihevgu_ fhf_glu ^Zxl bgnhjfZpbx h kljmdlmj_ fhe_dmeu ih gbf fh`gh ijh\_jblv \uqbke_ggu_ \hegh\u_ nmgdpbb GZ ijZdlbd_ hgb ijbf_gyxlky ^ey gZoh`^_gby ih^oh^ys_]h jZkl\hjbl_ey ^ey lh]h beb bgh]h\_s_kl\Z>bihebjZkl\hjbl_eyfh]mljZajmrZlvbhggu_djbklZeeu b^jm]b_kljmdlmjuldfh]mlkhev\Zlbjh\Zlvbhgubmf_gvrZlvkbemdm ehgh\kdh]hijbly`_gbyGZijbf_j\kemqZ_\h^udmehgh\kdbcihl_gpbZe mf_gvrZ_lkyijbf_jgh\jZa 3.>BWE?DLJBQ?KD:YIJHGBP:?FHKLVIHEYJBA:PBY Fhe_dmeufh]ml[ulvihf_s_gu\we_dljbq_kdh_ihe_dZd\ihklh yggh_ lZd b i_j_f_ggh_ jZaebqghc qZklhlu Ijb wlhf fhe_dmeu dZd ih eyjgu_lZdbg_iheyjgu_[m^ml\aZbfh^_ckl\h\Zlvkwlbfbiheyfb ?keb\_s_kl\hihf_klblv\h\g_rg__we_dljbq_kdh_ihe_kha^Z\Z_ fh_dhg^_gkZlhjhfjbklh_fdhklv_]h\hajZklZ_l\ ε jZal_ C ε= , C0 ]^_ C0 b C –zfdhklbdhg^_gkZlhjZ\\Zdmmf_bk\_s_kl\hf ε –^bwe_d ljbq_kdZyijhgbpZ_fhklv\_s_kl\Z <hajZklZgb_ zfdhklb \ j_amevlZl_ mf_gvr_gby kb eu we_dljbq_kdh]h ihey \u au\Z_lky dZd gZebqb_f ihklh yggh]h ^bihevgh]h fhf_glZ µ ijbkms_]h fhe_dmeZf ^Zg gh]h \_s_kl\Z lZd b bo ^_ nhjfZpb_c ih^ ^_ckl\b_f ih eyBgufbkeh\Zfbih^\eby gb_f we_dljbq_kdh]h ihey ijhbkoh^bl g_ lhevdh hjb_g lZpby fhe_dme iheyjgh]h \_ s_kl\Z ih gZijZ\e_gbx ih Jbk 5 >bihevgu_ fhe_dmeu \ h^ghjh^ ghf we_dljbq_kdhf ihe_ l – jZkklhygb_ ey gh b \hagbdgh\_gb_ ^h f_`^mieZklbgZfbS – iehsZ^vieZklbgu ihegbl_evgh]h gZ\_^_ggh]h bg^mpbjh\Zggh]h ^bihevgh ]hfhf_glZ µ bg^ aZkq_lkf_s_gbywe_dljhgh\hlqZklbby^_j 40 ?keb ih\_joghklgmx iehlghklv aZjy^Z ieZklbg dhg^_gkZlhjZ h[h agZqblv σ lh^ey\ZdmmfZgZijy`_gghklvwe_dljbq_kdh]hihey[m^_ljZ\ gZ E = σ ε 0 ]^_ ε 0 –^bwe_dljbq_kdZyijhgbpZ_fhklv \ZdmmfZ < nbabd_ qZs_bkihevamxll_jfbg©we_dljbq_kdZyihklhyggZyªqbke_ggh_agZq_gb_ dhlhjhc jZ\gh · 10-11 N· f-1 Hlf_lbf qlh \ =Zmkkh\hc kbkl_f_ wlZ nhjfmeZ bf__l \b^ E = 4πσ < ^Zgghc kbkl_f_ ^bwe_dljbq_kdZy ijhgb pZ_fhklvy\ey_lky[_ajZaf_jghc\_ebqbghcjZ\ghc^ey\ZdmmfZ_^bgbp_). Ihkdhevdm dhg_qgu_ jZkq_lgu_ nhjfmeu ijb\h^bfu_ \ [hevrbgkl\_ mq_[gbdh\h[uqgh^Zxlky\]Zmkkh\hckbkl_f_[m^_fbkihevah\Zlv_z^ey gZrbo^Zevg_crbojZkkm`^_gbc Ijb ihf_s_gbb ^bwe_dljbdZ f_`^m ieZklbgZfb ih^ ^_ckl\b_f \g_rg_]h we_dljbq_kdh]h ihey ijhbkoh^bl _]h iheyjbaZpby l_ hij_^_ & e_ggZyhjb_glZpbyfhe_dmeijb\h^ysZydhkeZ[e_gbxgZijy`_gghklb E , \wlhfkemqZ_ 4πσ E= . (1) ε < hlkmlkl\bb \g_rg_]h we_dljbq_kdh]h ihey ^bihevgu_ fhf_glu ^bwe_dljbdZ beb jZ\gu gmex g_iheyjgu_ fhe_dmeu beb jZkij_^_e_gu ihgZijZ\e_gbyf\ijhkljZgkl\_oZhlbq_kdbfh[jZahfiheyjgu_fhe_dm eu<h[hbokemqZyokmffZjgucwe_dljbq_kdbcfhf_gl^bwe_dljbdZjZ\_g gmexIh^^_ckl\b_f\g_rg_]hihey^bwe_dljbdiheyjbam_lkyl_\gmljb _]h \hagbdZ_l kh[kl\_ggh_ we_dljbq_kdh_ ihe_ Wlh hagZqZ_l qlh j_amev lbjmxsbc we_dljbq_kdbc fhf_gl ^bwe_dljbdZ klZgh\blky hlebqguf hl gmey < dZq_kl\_ \_ebqbgu oZjZdl_jbamxs_c kl_i_gv iheyjbaZpbb ^b we_dljbdZ[_jmlwe_dljbq_kdbcfhf_gl_^bgbpuh[t_fZ & µ& P= , Sl ]^_ S –iehsZ^vdhg^_gkZlhjZl –jZkklhygb_f_`^mieZklbgZfb AZf_gyy_]hagZq_gb_fbf__f ql q P= = , Sl S l_iheyjbaZpbyjZ\gZih\_joghklghciehlghklb aZjy^Z σ' ijb gZebqbb ^bwe_dljbdZ\dhg^_gkZlhj_ >ey bahljhigh]h \_s_kl\Z k\hckl\Z dhlhjh]h g_ aZ\bkyl hl hjb_g lZpbb \_dlhj iheyjbaZpbb k\yaZg k \_dlhjhf gZijy`_gghklb we_dljbq_ kdh]hihey\lhc`_lhqd_ijhklufkhhlghr_gb_f P = χE , (2) ]^_ χ –g_aZ\bkysZyhl E \_ebqbgZgZau\Z_fZy^bwe_dljbq_kdhc\hkijb 41 bfqb\hklvx ^bwe_dljbdZ JZaf_jghklv P b E h^bgZdh\Z ke_^h\Zl_evgh χ –[_ajZaf_jgZy\_ebqbgZ >ey ^bwe_dljbdh\ khklhysbo ba g_iheyjguo fhe_dme nhjfmeZ \ul_dZ_lbake_^mxsboijhkluokhh[jZ`_gbc<ij_^_euh[t_fZ ∆V ih iZ^Z_ldhebq_kl\hfhe_dmejZ\gh_ n ∆V ]^_ n –qbkehfhe_dme\_^bgbp_ h[t_fZDZ`^Zybafhe_dme\j_amevlZl_^_ckl\by\g_rg_]hiheyijbh[j_ lZ_lwe_dljbq_kdbcfhf_gl\_ebqbgZdhlhjh]hhij_^_ey_lkynhjfmehc µ = αE , ]^_ α –\_ebqbgZgZau\Z_fZyiheyjbam_fhklvxfhe_dmeubeb^_nhjfZ pbhgghciheyjbam_fhklvxhgZl_f[hevr_q_f[he__ih^^Z_lkyfhe_dm eZ^_nhjfZpbbl_q_ff_g__`_kldb__we_dljhggu_h[hehqdbLZdbfh[ jZahf ∑ µ i = n∆VαE . ∆V JZa^_eb\wlh\ujZ`_gb_gZ ∆V ihemqbf\_dlhjiheyjbaZpbb P = nαE . (3) <\_^yh[hagZq_gb_ χ = nα ijboh^bfdnhjfme_ < kemqZ_ ^bwe_dljbdh\ ihkljh_gguo ba iheyjguo fhe_dme hjb_g lbjmxs_fm^_ckl\bx\g_rg_]hiheyij_iylkl\m_ll_ieh\h_^\b`_gb_fh e_dmeklj_fys__ky©jZa[jhkZlvªbo^bihevgu_fhf_gluih\k_fgZijZ\ e_gbyf < j_amevlZl_ mklZgZ\eb\Z_lky g_dhlhjZy ij_bfms_kl\_ggZy hjb _glZpby ^bihevguo fhf_glh\ fhe_dme \ gZijZ\e_gbb ihey Khhl\_lkl \mxsbc klZlbklbq_kdbc jZkq_l ihdZau\Z_l \ kh]eZkbb k hiulhf qlh ijb g_baf_gghc l_fi_jZlmj_ \_dlhj iheyjbaZpbb ijhihjpbhgZe_g gZijy`_g ghklb ihey l_ ijb\h^bl d nhjfme_ Ijb ihklhygghc gZijy`_gghklb ihey \_dlhj iheyjbaZpbb ^bwe_dljbdh\ khklhysbo ba iheyjguo fhe_dme mf_gvrZ_lky k ih\ur_gb_f l_fi_jZlmju >bwe_dljbq_kdZy \hkijbbfqb \hklvlZdbo^bwe_dljbdh\h[jZlghijhihjpbhgZevgZZ[khexlghcl_fi_jZlmj_ K\yav f_`^m ^bwe_dljbq_kdhc ijhgbpZ_fhklvx b ^bwe_dljbq_kdhc \hkijbbfqb\hklvx ^bwe_dljbdZ fh`gh ihemqblv ba ke_^mxs_]h jZkkm` ^_gbyIhkdhevdmih^^_ckl\b_fihey^bwe_dljbdiheyjbam_lkylhgZ_]h ih\_joghklyo ihy\ylky aZjy^u iehlghklb σ' dZd [ueh ihdZaZgh \ur_ P = σ' dhlhju_ kha^Z^ml \gmljb h^ghjh^gh_ ihe_ H[Z ihey gZijZ\e_gu gZ\klj_qm^jm]^jm]mke_^h\Zl_evgh\gmljb^bwe_dljbdZ E = σ − σ' . Ih^klZ\eyy\^Zggh_mjZ\g_gb_agZq_gb_ σ \aylh_bajZ\_gkl\Z 42 ihemqbf^ey\_ebqbgu P ke_^mxsmxaZ\bkbfhklv (ε − 1) P= E. 4π KjZ\gb\ZymjZ\g_gbybfh`ghk^_eZlv\u\h^qlh ε = 1 + 4πχ . <kbkl_f_KB^ZggZynhjfmeZ[m^_l\u]ey^_lvke_^mxsbfh[jZahf: ε =1+ χ. (4) 3.3. HIJ?>?E?GB?>BIHEVGUOFHF?GLH<FHE?DME Hl^_evgmxfhe_dmem\^bwe_dljbq_kdhckj_^_fh`ghjZkkfZljb\Zlv, dZd qZklbpm gZoh^ysmxky \ kn_jbq_kdhc ihehklb gZ ih\_joghklb dhlh jhcgZoh^blkyhij_^_ezggZyiehlghklvaZjy^Zjbk GZijy`_gghklv ihey \ p_glj_kn_jbq_kdhcihehklb 4 E * = E + πP , (5) 3 ]^_ E – gZijy`_gghklv ihey \ kiehrghf ^bwe_dljbd_ < kbkl_f_ KB wlZ nhjfmeZ 1 P bf__l\b^ E * = E + <u 3 ε0 Jbk26. Ko_fZihdZau\ZxsZygZijy`_gghklv \h^ nhjfmeu ^ey gZijy`_g ^hihegbl_evgh]hihey\ihehklbkn_jbq_kdhc ghklb ^hihegbl_evgh]h ihey nhjfu \ ihehklb kn_jbq_kdhc nhj fu^Z_lky\khhl\_lkl\mxs_fjZa^_e_nbabdbKe_^m_lh^gZdhaZf_lblv qlh jZ\_gkl\h kljh]h \uihegy_lky lhevdh \ kemqZ_ djbklZeebq_kdh]h ^bwe_dljbdZdm[bq_kdhckbkl_fu>ey`b^dbob]Zahh[jZaguo^bwe_dljb dh\ gZijy`_gghklv ihey ^_ckl\mxs_]h gZ hl^_evgmx fhe_dmem hij_^_ ey_lkyagZq_gb_febrvijb[eb`_ggh LZdbfh[jZahf^eyh[s_]hiheyhdhehjZkkfZljb\Z_fhcfhe_dmeu mqblu\ZyjZ\_gkl\hfh`ghaZibkZlv 4 4πP 4 1 4πP (ε + 2) 1 E * = E + πP = + πP = 4πP + = , (ε − 1) 3 3 3(ε − 1) ε −1 3 hldm^Z 3(ε − 1) * P= E . (6) 4π(ε + 2) 43 BakjZ\g_gbymjZ\g_gbcbke_^m_lqlh nα = 3(ε − 1) . 4π(ε + 2) <ujZabf\_ebqbgm n q_j_aiehlghklv ρ fheyjgmxfZkkm M bqbk eh:\h]Z^jh N A : ρ n= N A. (7) M Lh]^Zfh`ghaZibkZlv ε −1 M 4 = π N A. ε+2 ρ 3 Ihemq_ggh_mjZ\g_gb_gZau\Z_lkymjZ\g_gb_fDeZmabmkZ–Fhkkhl lbbijbf_gbfhdg_iheyjguf\_s_kl\Zf >ey]Zah\ZlZd`_^eyiheyjguo\_s_kl\jZkl\hj_gguo\g_iheyj guojZkl\hjbl_eyof_`^m ε b µ kms_kl\m_lk\yav\ujZ`Z_fZymjZ\g_gb _fEZg`_\_gZ–>_[Zy ε −1 M 4 µ 2 = πN A α + . (8) ε+2 ρ 3 kT 3 I_j\h_ keZ]Z_fh_ \ mjZ\g_gbb hl\_qZ_l ^_nhjfZpbhgghfm wnn_dlm \lhjh_ – hjb_glZpbhgghfm wnn_dlm Ihke_^gbc hq_\b^gh ^he`_g [ulv l_f agZqbl_evg__ q_f iheyjg__ fhe_dmeZ l_ q_f [hevr_ b q_f gb`_ l_fi_jZlmjZ lZd dZd gZ]j_\Zgb_ mkbeb\Zy l_ieh\h_ ^\b`_gb_ fhe_dme ij_iylkl\m_lbohjb_glZpbb<khhl\_lkl\bbkmjZ\g_gb_fijbgbadbo l_fi_jZlmjZoij_h[eZ^Z_lhjb_glZpbhggucwnn_dlijb\ukhdbo–^_nhj fZpbhggucwnn_dl <\_^yke_^mxsb_h[hagZq_gby 4παN A 4πµ 2 N A ε −1 M 1 = A; = B b = x , = y; ε+2 ρ T 3 9k mjZ\g_gb_fh`ghaZibkZlvlZd y = A + Bx . ?kebhldeZ^u\ZlvihhkbZ[kpbkkh[jZlgmxZ[khexlgmxl_fi_jZlm ε −1 M jmZihhkbhj^bgZl\_ebqbgm lhihemqblkyijyfZyihgZdehgm ε+2 ρ 44 dhlhjhc e_]dh gZclb ^bihevguc fhf_gl >ey lh]h qlh[u ihkljhblv wlm ijyfmxgZ^hbaf_jblv_fdhklvdhg^_gkZlhjZkbamqZ_fuf\_s_kl\hfoh ly[uijb^\mol_fi_jZlmjZoZlZd`_agZlviehlghklv\_s_kl\Zijbwlbo l_fi_jZlmjZo <hafh`gh lZd`_ hij_^_eblv iheyjbam_fhklv b ^bihevguc fhf_gl gZhkgh\_baf_j_gby^bwe_dljbq_kdhcihklhygghcijbh^ghcl_fi_jZlmj_ bihdZaZl_e_ij_ehfe_gbykj_^u n . NhjfmeZEhj_gpZ–Ehj_gpZ n2 −1 M 4 = παN A n2 + 2 ρ 3 iha\hey_lg_ihkj_^kl\_gghihbaf_j_gbx n hij_^_eblv α baZl_fihagZ q_gbx ε ijbh^ghcl_fi_jZlmj_ihnhjfme_jZkkqblZlv µ . 3.IHEYJBAM?FHKLVFHE?DME DZd m`_ [ueh hlf_q_gh iheyjbam_fhklv oZjZdl_jbam_l gZ\_^_gguc ^bihevguc fhf_gl \hagbdZxsbc \ fhe_dme_ ih^ ^_ckl\b_f \g_rg_]h iheybf_xs_]hgZijy`_gghklv E : µ = αE . (9) JZaf_jghklviheyjbam_fhklbjZ\gZL3qlhg_ihkj_^kl\_gghke_^m_l baijb\_^_ggh]hmjZ\g_gby [α] = [µ] = [e][L]2 = [L]3 . [E ] [e]/[L] Iheyjbam_fhklv[ebadZih\_ebqbg_dh[t_fmfhe_dmeuWlhke_^m _l\qZklghklbba ke_^mxs_c ]jm[hcfh^_eb ;m^_f hibku\Zlv ^_ckl\b_ we_dljbq_kdh]h ihey gZ Zlhf dZd k^\b] djm]h\hc hj[blu jZ^bmkZ R gZ jZkklhygb_ ∆ jbkGZjbkmgd_imgdlbjhfihdZaZgZijh_dpbyhj[b lu\i_j\hgZqZevghfiheh`_gbbbkiehrghcebgb_cihke_\dexq_gbyihey JZkkfhljbfjZ\gh\_kb_kbeihhkb x .Ijh_dpbyp_gljh[_`ghckbeu ihwlhchkbjZ\gZgmexKbeZhlly]b\ZxsZywe_dljhghly^jZjZ\gZ eE . e KbeZ ijbly`_gby we_dljhgZ d y^jm jZ\gZ 2 aZjy^ y^jZ ijbgyl jZ\guf R 45 aZjy^m we_dljhgZ Ijh_dpby wlhc kbeu ih hkb x jZ\gZ h[hagZq_gbc R b α \b^_gbajbk, cos α = e2 R 2 cos α Kfuke ∆ LZdbfh[jZahf R e2 ∆ eE = , R 2 R hlkx^Z µ = e∆ = R 3 E . Ke_^h\Zl_evgh \ khhl\_lkl\bb k nhjfmehc 9), α = R 3 _keb ∆ fZeh ih kjZ\g_gbx k R LZd dZd R ≈ 10 −8 kf lh α ≈ 10 − 24 kf3. Ihkdhevdm iheyjbam_fhklv dZdhceb[h Zlhfghc kbkl_fu ZlhfZ ]jmiiu Zlhfh\k\yabbl^[ebadZd_zh[t_fmfh`ghh`b^Zlvqlhwlb\_ebqbgu 4 bhij_^_ey_fZy_xfhevgZyj_njZdpby παN A ^he`guZ^^blb\ghkdeZ 3 ^u\Zlvky >_ckl\bl_evgh j_njZdpby g_hj]Zgbq_kdbo kh_^bg_gbc kdeZ^u \Z_lkybaj_njZdpbcbhgh\j_njZdpbyhj]Zgbq_kdbofhe_dme–baj_njZd pbck\ya_cZlhfh\beb]jmii DZ`^hfmZgbhgmfh`ghijbibkZlvk\hxj_njZdpbxijZ\^Zwlb\_ ebqbgu[m^mlijb[eb`zggufbHiuluihdZau\Zxlqlhj_njZdpbyZgbh gZaZ\bkblbhldZlbhgZkdhlhjufhggZoh^blky\kh_^bg_gbb?keb`_\ i_j\hf ijb[eb`_gbb kqblZlv j_njZdpbx bhgh\ ihklhygghc lh fh`gh ijhke_^blvdZd\eby_laZjy^y^jZgZj_njZdpbxbhgh\ <havfzf bhgu k h^bgZdh\uf kljh_gb_f we_dljhgghc h[hehqdb lZ[e 3). BalZ[ebpu\b^ghqlhkm\_ebq_gb_faZjy^Zy^jZ\_ebqbgZj_n jZdpbb Z ke_^h\Zl_evgh b iheyjbaZpby bhgh\ mf_gvrZ_lky LZdbf h[jZ ahf q_f kbevg__ y^jh m^_j`b\Z_l hdheh k_[y we_dljhgu l_f hgb keZ[__ kf_sZxlky\h\g_rg_fwe_dljbq_kdhfihe_ LZ[ebpZ <ebygb_aZjy^Zy^jZgZj_njZdpbx R bhgh\ JbkKo_fZiheyjbaZpbb we_dljhgZZlhfZ QZklbpu AZjy^y^jZ Z J_njZdpby R,kf3 O28 7 F9 2,5 Ne 10 1,0 We_dljhggZyh[hehqdZs22s22p6 ^ey\k_oqZklbp. 46 Na+ 11 0,5 Mg2+ 12 0,2 <ebygb_dZlbhgZgZj_njZdpbxZgbhgZihdZaZgh \lZ[e LZ[ebpZ J_njZdpbybhgZoehjZ\aZ\bkbfhklbhlkhk_^g_]hdZlbhgZ DZlbhg* RCl− , kf3 H+ 6,7 Li+ 7,4 Na+ 8,0 K+ 8,6 Cs+ 8,97 * hlghr_gb_ aZjy^Z d jZ^bmkm bhgZ e r ke_\Z gZijZ\h mf_gvrZ_lky Q_f kbevg__ dZlbhg l_f dj_iq_ hg m^_j`b\Z_l we_dljhgu b l_f f_gvr_ hgb kf_sZxlkyih^^_ckl\b_f\g_rg_]hwe_dljbq_kdh]hiheyDZlbhgumdj_i eyxlwe_dljhggmxh[hehqdmZgbhgh\ 3.5. 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GZ]ey^ghc fh^_evx objZevguo fhe_dmefh`_l kem`blvijZ\Zybeb e_\Zy kibjZeb Djbl_jbchkgh\ZggucgZkbff_ljbb^eymklZgh\e_gbykms_kl\h\Z gby hilbq_kdb Zdlb\guo bahf_jh\ k\h^blky d ke_^mxs_fm ex[Zy fhe_ dmeZ dhlhjZy g_ bf__l g_kh[kl\_gghc hkb \jZs_gby gZau\Z_lky ^bkkbf f_ljbqghcZhilbq_kdbZdlb\gu_fhe_dmeu^he`gu[ulv^bkkbff_ljbqgufb 47 Ijbf_jhffh]mlkem`blvkibjZevgu_fhe_dmeugZijbf_jfhe_dm eu d\ZjpZ khklhyl ba kibjZevguo p_i_c SiO2 b hg h[eZ^Z_l hilbq_kdhc Zdlb\ghklvx Baf_g_gb_hilbq_kdhcZdlb\ghklbkqZklhlhck\_lZgZau\Z_lky^bk i_jkb_c hilbq_kdh]h \jZs_gby Ijbf_gy_lky hgZ ^ey bkke_^h\Zgby kl_ j_hobfbq_kdhcdhgnb]mjZpbb[hevrbofhe_dme F_jhchilbq_kdhcZdlb\ghklby\ey_lkyhilbq_kdh_\jZs_gb_ α dh lhjh_ baf_jyxl k ihfhsvx iheyjbf_ljh\ GZ \_ebqbgm α \ebyxl ^ebgZ kehyhilbq_kdbZdlb\gh]h\_s_kl\Zdhgp_gljZpbybebiehlghklv\kemqZ_ qbklh]h`b^dh]h\_s_kl\ZjZkl\hjbl_evl_fi_jZlmjZb^ebgZ\hegu λ . 3.6F:=GBLGU?K<HCKL<: :lhfgu_ y^jZ b we_dljhgu bf_y hij_^_e_gguc we_dljbq_kdbc aZ jy^fh]mlh[eZ^Zlvbg_dhlhjuffZ]gblguffhf_glhfijbq_fmy^jZhg ijbf_jghgZljbihjy^dZf_gvr_q_fmwe_dljhgZFhe_dmeZdZdkbkl_fZ khklhysZy ba wlbo aZjy`_gguo qZklbp lZd`_ fh`_l oZjZdl_jbah\Zlvky \_dlhjhf fZ]gblgh]h fhf_glZ dhlhjuc k\yaZg ]eZ\guf h[jZahf k hj[b lZevgufbkibgh\uf^\b`_gbyfbwe_dljhgh\<hlkmlkl\b_\g_rg_]hih ey\kbemoZhlbq_kdhchjb_glZpbbfZ]gblguofhf_glh\hl^_evguofhe_ dme kmffZjguc fZ]gblguc fhf_gl l_eZ jZ\_g gmex Ih^ ^_ckl\b_f ihey fZ]gblgu_ fhf_glu fhe_dme ijbh[j_lZxl ij_bfms_kl\_ggmx hjb_glZ pbx \ h^ghf gZijZ\e_gbb \ke_^kl\b_ q_]h \_s_kl\h gZfZ]gbqb\Z_lky – _]hkmffZjgucfZ]gblgucfhf_glklZgh\blkyhlebqgufhlgmey GZfZ]gbqb\Zgb_ \_s_kl\Z _kl_kl\_ggh oZjZdl_jbah\Zlv fZ]gblguf fhf_glhf_^bgbpuh[t_fZWlm\_ebqbgmgZau\Zxl\_dlhjhfgZfZ]gbqb \Zgbybh[hagZqZxl J DZdihdZau\Z_lhiul\_dlhjgZfZ]gbqb\Zgbyk\y aZgkgZijy`_gghklvxihey H \lhc`_lhqd_khhlghr_gb_f J = χf H , (10) ]^_ χ f – \_ebqbgZ gZau\Z_fZy fZ]gblghc \hkijbbfqb\hklvx \_s_kl\Z IhkdhevdmjZaf_jghklb J b H h^bgZdh\u χ f hdZau\Z_lky[_ajZaf_jghc \_ebqbghcQZklh\f_klh\hkijbbfqb\hklb_^bgbpuh[t_fZ χ f ihevamxl ky hlg_k_gghc d h^ghfm dbehfhex \_s_kl\Z dbehfheyjghc ^ey obfbq_ kdb ijhkluo \_s_kl\ – dbehZlhfghc \hkijbbfqb\hklvx χ df F_`^m agZq_gbyfbwlbo\hkijbbfqb\hkl_cbf__lkykhhlghr_gb_ χdf = χfVdf ,]^_ 48 Vdf –h[t_fdbehfhey\_s_kl\Zf3dfhev<aZ\bkbfhklbhlagZdZb\_eb qbgu fZ]gblghc \hkijbbfqb\hklb \k_ \_s_kl\Z ih^jZa^_eyxlky gZ ljb ]jmiiu 1) ^bZfZ]g_lbdbmdhlhjuo χ f hljbpZl_evgZbfZeZihZ[khexlghc \_ebqbg_ χ df ~ 10-8 ÷ 10-7 f3dfhev); 2) iZjZfZ]g_lbdb m dhlhjuo χ f lh`_ g_\_ebdZ gh iheh`bl_evgZ ( χ df ~ 10-7 ÷ 10-6 f3dfhev); 3) n_jjhfZ]g_lbdbmdhlhjuo χ f iheh`bl_evgZb^hklb]Z_lhq_gv [hevrbo agZq_gbc χ df ~ 103 f3dfhev). Djhf_ lh]h \ hlebqb_ hl ^bZ- b iZjZfZ]g_lbdh\ ^ey dhlhjuo χ f ihklhyggZ fZ]gblgZy \hkijbbfqb\hklv n_jjhfZ]g_lbdh\y\ey_lkynmgdpb_cgZijy`_gghklbfZ]gblgh]hihey LZdbf h[jZahf \_dlhj gZfZ]gbqb\Zgby J fh`_l dZd kh\iZ^Zlv ih gZijZ\e_gbxk H miZjZ-bn_jjhfZ]g_lbdh\lZdb[ulvgZijZ\e_gguf \ijhlb\hiheh`gmxklhjhgmm^bZfZ]g_lbdh\ Qlh[u hibkZlv ih\_^_gb_ \_s_kl\Z \ fZ]gblghf ihe_ m^h[gh \hk ihevah\ZlvkyiZjZf_ljhfgZau\Z_fuffZ]gblghcbg^mdpb_c B WlZ\_eb qbgZ oZjZdl_jbam_l kbeh\h_ ^_ckl\b_ fZ]gblgh]h ihey gZ lhd b ke_^h\Z l_evghy\ey_lkyZgZeh]hfgZijy`_gghklbwe_dljbq_kdh]hihey E dhlhjZy oZjZdl_jbam_lkbeh\h_^_ckl\b_we_dljbq_kdh]hiheygZaZjy^FZ]gblgZy bg^mdpby k\yaZgZ k gZijy`_gghklvx fZ]gblgh]h ihey \ \Zdmmf_ ke_^mx sbfkhhlghr_gb_f B H= , µ0 ]^_ µ 0 –fZ]gblgZyihklhyggZy_zqbke_ggh_agZq_gb_jZ\ghπ· 10-7=gf >eyhibkZgbyihey\\_s_kl\_kem`blke_^mxs__mjZ\g_gb_ B H= −J. (11) µ0 Ih^klZ\eyy\nhjfmemagZq_gb_ J bajZ\_gkl\Zihemqbf B − χf H , H= µ0 hldm^Z B H= . µ o (1 + χ f ) ;_ajZaf_jgZy\_ebqbgZ µ = 1 + χ f gZau\Z_lkyhlghkbl_evghcfZ]gbl ghc ijhgbpZ_fhklvx \_s_kl\Z Hlf_lbf qlh \ =Zmkkh\hc kbkl_f_ µ = 1 + 4πχf . 49 BlZd_kebfZ]gblgucfhf_glZlhfh\ µ f hlebq_ghlgmey\_s_kl\h hdZau\Z_lky iZjZfZ]g_lbdhf FZ]gblgZy \hkijbbfqb\hklv χ f iZjZfZ] gblguo \_s_kl\ ijb µ f B << kT \ujZ`Z_lky ih l_hjbb EZg`_\_gZ nhj fmehc χf = µo n µ 2F , (12) 3kT ]^_ n – dhgp_gljZpby qZklbp k – ihklhyggZy ;hevpfZgZ Mqblu\Zy aZ\b kbfhklvbamjZ\g_gbyihemqbf ρ N Aµ f χf = µo . (13) M 3kT IDxjbwdki_jbf_glZevghmklZgh\be aZdhgkh]eZkghdhlhjhfmiZ jZfZ]gblgZy\hkijbbfqb\hklv\_s_kl\Z χf = C T , (14) ]^_ C – ihklhyggZy Dxjb aZ\bkysZy hl jh^Z \_s_kl\Z KhihklZ\e_gb_ nhjfmeb^Z_l^eyihklhygghcDxjbke_^mxs__\ujZ`_gb_ 2 ρ N Aµ f C = µo . M 3k GZb[hevrbc \deZ^ \ iZjZfZ]gblguc wnn_dl \ghkbl kibgh\uc fZ] gblgucfhf_glwe_dljhgh\ µ S dhlhjuch[mkeh\e_g^\b`_gb_fk\h[h^gh ]hwe_dljhgZ\hdjm]kh[kl\_gghchkb<_ebqbgZ µ S aZ\bkblhlkibgh\h]h d\Zglh\h]hqbkeZ S : e! µ S = 2µ B s ( s + 1) ]^_ µ B –fZ]g_lhg ;hjZ µ B = . 2me >eyZlhfZbebbhgZbf_xs_]hng_kiZj_gguowe_dljhgh\khkibghf S = 1 kmffZjgh_ kibgh\h_ d\Zglh\h_ qbkeh jZ\gh S = n / 2 b kibgh\uc 2 fZ]gblgucfhf_glkhklZ\ey_l nn µ S = 2µ B + 1 = µ B n(n + 2 ) . 22 Ke_^m_lh[jZlblv\gbfZgb_gZlhqlhfZ]gblgucfhf_glg_baf_gy _lkyijyfhijhihjpbhgZevghqbkemg_kiZj_gguowe_dljhgh\ Ijbgbadbol_fi_jZlmjZo\iZjZfZ]gblghfl\_j^hf\_s_kl\_fh`_l ijhbahclbnZah\uci_j_oh^\n_jjhfZ]gblgh_khklhygb_ijbl_fi_jZlmj_ Dxjb – dh]^Z kibgu gZ [hevrbo mqZkldZo hjb_glbjmxlky iZjZee_evgh beb \ Zglbn_jjhfZ]gblgh_ khklhygb_ ijb l_fi_jZlmj_ G__ey – dh]^Z kibguq_j_^mxlky 50 2 =eZ\Z [email protected]?DMEYJGH?<A:BFH>?CKL<B? < jZaebqguo nbabq_kdbo ijhp_kkZo dhg^_gkZpby Z^khj[pby b ^j kms_kl\_ggmx jhev b]jZxl fhe_dmeyjgu_ kbeu ijb\h^ysb_ gZ [hevrbo jZkklhygbyo d ijbly`_gbx f_`^m fhe_dmeZfb JZaebqby f_`^m ^Zggufb kbeZfb b kbeZfb hij_^_eyxsbfb obfbq_kdh_ \aZbfh^_ckl\b_, aZdexqZ xlkyij_`^_\k_]h\ke_^mxs_f 1. Wg_j]by fhe_dmeyjgh]h \aZbfh^_ckl\by f_gvr_ wg_j]bb obfbq_ kdh]hijb[ebabl_evghgZihjy^hd\_ebqbgu 2. Kbeu obfbq_kdh]h kjh^kl\Z ki_pbnbqgu Z kbeu fhe_dmeyjgh]h ijbly`_gby mgb\_jkZevgu Ex[u_ Zlhfgu_ kbkl_fu ijbly]b\Zxlky gZ [hevrbo jZkklhygbyo Wlh h[klhyl_evkl\h ke_^m_l ba lh]h qlh \k_ ]Zau dhg^_gkbjmxlkyijb^hklZlhqghgbadbol_fi_jZlmjZo 3. < hlebqb_ hl obfbq_kdbo kbe h[eZ^Zxsbo gZkusZ_fhklvx hk gh\ghc lbi fhe_dmeyjguo kbe oZjZdl_jbam_lky Z^^blb\ghklvx Wlh hagZ qZ_lqlhwg_j]by\aZbfh^_ckl\by^\mofhe_dmeg_aZ\bkblhlijbkmlkl\by lj_lv_c Ih^h[gh^_e_gbxkbeobfbq_kdh]hkjh^kl\ZgZdh\Ze_glgu_bbhg gu_fhe_dmeyjgu_kbeu^_eylkygZljb\b^Z • hjb_glZpbhggu_ kbeu \hagbdZxl f_`^m fhe_dmeZfb h[eZ^Zx sbfb Zkbff_ljb_c aZ kq_l we_dljhklZlbq_kdh]h \aZbfh^_ckl\by aZjy^h\ bf_xsboky\fhe_dmeZo • bg^mdpbhggu_kbeu^_ckl\mxlf_`^m^bihevghcfhe_dmehcbg_ ^bihevghc \ j_amevlZl_ iheyjbaZpbb ihke_^g_c \ ihe_ kha^Z\Z_fhf ^b ihevghcfhe_dmehc • ^bki_jkbhggu_ kbeu h[mkeh\e_gu ki_pbnbq_kdbf \aZbfh^_ckl \b_fwe_dljhgh\fhe_dme HJB?GL:PBHGGU?BBG>MDPBHGGU? <A:BFH>?CKL<BY Hjb_glZpbhggu_ kbeu ihemqbeb k\hz gZa\Zgb_ \ k\yab k l_f qlh fhe_dmeu ^he`gu hjb_glbjh\Zlvky \ ijhkljZgkl\_ wg_j]_lbq_kdb \u]h^ guf h[jZahf Ij_^klZ\bf ^\_ ^bihevgu_ fhe_dmeu GZ jbk ihdZaZgZ wg_j]_lbq_kdb[he__\u]h^gZybodhgnb]mjZpby Jbk>\_hjb_glbjh\Zggu_^bihevgu_fhe_dmeu 51 Hq_\b^ghqlh_kebl_fi_jZlmjZ^hklZlhqghgbadZlhhkms_kl\blky hjb_glZpby b ke_^h\Zl_evgh ijbly`_gb_ fhe_dme Ijb hq_gv \ukhdhc l_fi_jZlmj_ wg_j]by \jZs_gby [m^_l [hevr_ wg_j]bb \aZbfh^_ckl\by hjb_glZpbybkq_ag_lbfhe_dmeug_[m^mlijbly]b\Zlvky^jm]d^jm]m LZdbf h[jZahf hjb_glZpbhggu_ fhe_dmeyjgu_ kbeu aZ\bkyl hl l_fi_jZlmjubmf_gvrZxlkyijb_z\hajZklZgbbJZkkfhljbf^\Zij_^_ev guokemqZyhq_gvgbadbobhq_gv\ukhdbol_fi_jZlmj Ijbhq_gvgbadbol_fi_jZlmjZodh]^Zwg_j]byijbly`_gbykms_kl \_ggh [hevr_ kL ijhbahc^_l ihegZy hjb_glZpby fhe_dme Lh]^Z gZijb f_j\aZbfh^_ckl\b_fhe_dmegZoh^ysbokygZh^ghcebgbbgZjZkklhygbb R^jm]hl^jm]Zhibr_lkydZd\aZbfh^_ckl\b_aZjy^h\kf jbk 28): e2 e2 e2 e2 . U =− + + − 4πε o R 4πε o ( R + ∆ 2 ) 4πε o ( R + ∆1 ) 4πε o ( R + ∆1 + ∆ 2 ) A^_kvû1bû2 –jZkklhygbyf_`^mp_gljZfbly`_klbaZjy^h\jZ\guo ih\_ebqbg__KqblZyqlhwlbjZkklhygbyfZeuihkjZ\g_gbxkR, ihemqbf U =− 2 R∆1∆ 2 e 2 4πε o R 4 . 1 – dhwnnbpb_gl ijhihjpbhgZevghklb ]^_ 4πε o ε 0 –we_dljbq_kdZy ihklhyggZy Ihhij_^_e_gbx µ1 = ∆1e b µ 2 = ∆ 2 e ]^_ µ1 b µ 2 –^bihevgu_fhf_glufhe_dmeLh]^Z 2µ1µ 2 U =− . (15) 3 4πε o R Ba ihemq_ggh]h mjZ\g_gby \b^bf qlh ijb gbadbo l_fi_jZlmjZo wg_j]by ijbly`_gby ^bihevguo fhe_dme h[jZlgh ijhihjpbhgZevgZ dm[m jZkklhygby JZkkfhljbfl_i_jvbg^mdpbhggu_kbeuZaZl_f\_jg_fkyd\ujZ`_ gbx hjb_glZpbhgguo kbe ijb \ukhdbo l_fi_jZlmjZo >ey ijhklhlu ijh ZgZebabjm_fkemqZcdh]^Zg_^bihevgZyfhe_dmeZgZoh^blkygZebgbbgZ dhlhjhcjZkiheh`_g\_dlhjoZjZdl_jbamxsbc^bihevgucfhf_gljbk 29). <_ebqbgZ jZ\gZy Jbk 29Ko_fZ^eyjZkq_lZgZijy`_gghklbihey^bihey 52 Ih^ \ebygb_f we_dljbq_kdh]h ihey kha^Z\Z_fh]h ^bihevghc fhe_ dmehc \ g_^bihevghc fhe_dme_ \hagbdZ_l \ke_^kl\b_ iheyjbaZpbb we_d ljhgh\gZ\_^_ggucfhf_glghij_^_ey_fucnhjfmehc µ g = αE , (16) ]^_ α –iheyjbam_fhklvg_^bihevghcfhe_dmeu E –gZijy`_gghklvihey kha^Z\Z_fh]h^bihevghcfhe_dmehcgZjZkklhygbbRBajbk 29\b^ghqlh e e 2∆e 2µ − = = E= , (17) 4πε o R 2 4πε o (R + ∆ )2 4πε o R 3 4πε o R 3 ]^_ µ –^bihevgucfhf_glfhe_dmeuKe_^h\Zl_evgh µg = 2αµ 4πε o R 3 . LZddZdgZijZ\e_gb_gZ\_^_ggh]hfhf_glZ µ g ijhlb\hiheh`ghgZijZ\e_ gbx µ lhwg_j]bybo\aZbfh^_ckl\byfh`_l[ulvjZkkqblZgZihnhjfme_ (15bkhklZ\bl U =− 4αµ 2 (4πε o ) 2 6 . (18) R G_h[oh^bfh h^gZdh mq_klv qlh iheyjbaZpby we_dljhgZ lj_[m_l aZljZlu g_dhlhjhcwg_j]bb>eyjZkq_lZwlhcwg_j]bb[m^_fg_ij_ju\ghbaf_gylv gZijy`_gghklv ihey hl gmey ^h dhg_qgh]h agZq_gby E Wg_j]by \aZbfh ^_ckl\byhjb_glbjh\Zggh]h^bihey µ kihe_fjZ\gZ µE . >_ckl\bl_evgh wg_j]bx \aZbfh^_ckl\by ^\mo aZjy^h\ ^bihey k ih e_ffh`ghhij_^_eblvbake_^mxs_]hmjZ\g_gby U i = eV1 − eV2 = e(V1 − V2 ), ]^_ V1 b V 2 – agZq_gby ihl_gpbZeh\ ihey \ lhqdZo ]^_ gZoh^ylky p_glju ly`_klbiheh`bl_evgh]hbhljbpZl_evgh]haZjy^h\H^gZdh V1 − V2 = − E∆ , lZdbfh[jZahf: U i = − Ee∆ = −µE . (19) Baf_g_gb_ E gZ dE ihlj_[m_laZljZluwg_j]bb µdE H[sZyaZljZlZ wg_j]bbgZiheyjbaZpbxfhe_dmeu\ujZablkyke_^mxsbfh[jZahf αE 2 . ∫ µdE = ∫ αEdE = 2 0 0 E E 53 Ih^klZ\b\agZq_gb_ E banhjfmeu17ihemqbf^eyjZkkfZljb\Z_ fhcwg_j]bb Ui = 2αµ 2 (4πε h ) 2 R 6 . LZdbfh[jZahfiheh\bgZwg_j]bb\u^_eyxs_ckyijb\aZbfh^_ckl \bb jZkkfZljb\Z_fuo ^bihe_c aZljZlblky gZ iheyjbaZpbx we_dljhgZ HdhgqZl_evgh\f_klhmjZ\g_gby18ihemqbf U =− 2αµ 2 (4πεo ) (20) 2 6 R BlZd wg_j]by bg^mdpbhggh]h \aZbfh^_ckl\by g_ aZ\bkbl hl l_fi_ jZlmjubh[jZlghijhihjpbhgZevgZr_klhckl_i_gbjZkklhygbyf_`^mfh e_dmeZfb>eyihemq_gbykj_^g_cwg_j]bbbg^mdpbhggh]h\aZbfh^_ckl\by fhe_dme ijhba\hevgh jZkiheh`_gguo \ ijhkljZgkl\_ g_h[oh^bfh gZclb gZijy`_gghklvihey E mkj_^g_ggh]hihjZaebqgufm]eZff_`^mebgb_c, kh_^bgyxs_cfhe_dmeubgZijZ\e_gb_f^bihevgh]hfhf_glZLZdhcmq_l iha\hey_lijb\_klbnhjfmem20dke_^mxs_fm\b^m U =− αµ 2 (4πε o )2 R 6 . (21) Ijb\_^_gguc jZkq_l wg_j]bb bg^mdpbhggh]h \aZbfh^_ckl\by ^Z_l \hafh`ghklv \uqbkeblv hjb_glZpbhggh_ \aZbfh^_ckl\b_ gZ hkgh\_ ke_ ^mxs_cZgZeh]bb>bihevgZyfhe_dmeZhjb_glbjmykvih^\ebygb_fihey E h[jZam_l g_dhlhjh_ kj_^g__ aZ\bkys__ hl l_fi_jZlmju agZq_gb_ ijh _dpbbfhf_glZijhlb\gZijZ\e_gbyiheyIhdZ`_fqlhwlZijh_dpbyijh ihjpbhgZevgZ E l_\hagbdZ_lg_dhlhjucZgZeh]iheyjbam_fhklb >ey gZoh`^_gby wlhc aZ\bkys_c hl l_fi_jZlmju wnn_dlb\ghc ih eyjbam_fhklbhij_^_ebfkj_^g__agZq_gb_ijh_dpbbfhf_glZfhe_dmeugZ hkvkh\iZ^ZxsmxkgZijZ\e_gb_fihey>eyijh_dpbbfhf_glZmfh`gh aZibkZlvmjZ\g_gb_ m = µ cos θ , (22) ]^_ θ – m]he f_`^m fhf_glhf ^bihey b gZijZ\e_gb_f ihey <uqbke_gb_ kj_^g_]h agZq_gby m k\yaZgh k hij_^_e_gb_f \_jhylghklb h[jZah\Zgby m]eZe_`Zs_]hf_`^magZq_gbyfb θ b dθ <_jhylghklvh[uqgh_zgZau \Zxl]_hf_ljbq_kdhc dw]bf_lvlZdhcm]hehij_^_ey_lkyhlghr_gb_fih 54 \_joghklbrZjh\h]hihykZhl\_qZxs_]hwlhfmm]emdh\k_cih\_joghklb rZjZ_^bgbqgh]hjZ^bmkZjbk0): 2π sin θ dθ sin θ dθ = . 4π 2 GZjy^m k ^Zgghc \_jhylghklvx g_h[oh^b fh mq_klv qlh wg_j]by \aZbfh^_ckl\by U ^bihevgh]hfhf_glZ µ kihe_faZ\bkblhl dw] = m]eZ θ >_ckl\bl_evgh kh]eZkgh mjZ\g_ gbx(19) U = − Em Zmqblu\Zy\ujZ`_gb_ (22 ihemqbf U = − Eµ cos θ LZdbf h[jZ ahfh[smx\_jhylghklvlh]hqlhm]he[m ^_l gZoh^blvky f_`^m θ b dθ bkihevamy Jbk0Ko_fZ^eyjZkq_lZ]_h aZdhg ;hevpfZgZ fh`gh aZibkZlv \ ke_ f_ljbq_kdhc\_jhylghklbjZkih eh`_gbym]eZf_`^mbd ^mxs_f\b^_ 1 Eµ cos θ dW = sin θ dθ exp . 2 kT <_ebqbgm m gZc^_fbajZ\_gkl\Z π m= 1 ∫ µ cos θ sin θ ⋅ e 2 0 Eµ cos θ kT d θ π1 . ∫ sin θ d θ 02 IheZ]Zyqlh kT >> Eµ cos θ jZaeh`bfwdkihg_glm\jy^b,h]jZgbqb\Zykv ^\mfyi_j\ufbqe_gZfbihemqbf m=− µ2E . 3kT LZdbf h[jZahf kh]eZkgh mjZ\g_gbx 16) agZq_gb_ wnn_dlb\ghc iheyjb am_fhklb α hij_^_eblkymjZ\g_gb_f µ2 α= . 3kT (23) 55 Ih^klZ\b\wlh\ujZ`_gb_\mjZ\g_gb_21^eybg^mdpbhggh]h\aZbfh^_c kl\by ihemqbf ijb[eb`_ggh_ \ujZ`_gb_ ^ey hjb_glZpbhggh]h \aZbfh ^_ckl\byijb\ukhdbol_fi_jZlmjZo U hj ≈ − µ12 µ 22 (40 o ) 2 1 6 . 3kT R Ijb[eb`_gghklv ijb\_^_ggh]h jZkq_lZ hij_^_ey_lky l_f qlh [ueh \aylh kj_^g__ agZq_gb_ ihey kha^Z\Z_fh]h h^ghc fhe_dmehc Z g_ [ueh k^_eZgh^_lZevgh]hjZkq_lZ\_jhylghklbjZaebqguo\aZbfguohjb_glZpbc ^bihevguofhf_glh\LZdhcjZkq_l^Zklke_^mxs__mjZ\g_gb_ µ12µ 22 2 1 . (24) U hj = − 3 (4πε h )2 kT R 6 LZdbf h[jZahf hjb_glZpbhggh_ \aZbfh^_ckl\b_ ijb \ukhdbo l_f i_jZlmjZo h[jZlgh ijhihjpbhgZevgh l_fi_jZlmj_ b r_klhc kl_i_gb jZk klhygbyf_`^mfhe_dmeZfb 4.>BKI?JKBHGGH?<A:BFH>?CKL<B? WNN?DL EHG>HG: JZkkfhljbfgZb[he__ijhklhc kemqZc \aZbfh^_ckl\by^ey ^\mo Zlh fh\\h^hjh^ZgZoh^ysbokygZklhev[hevrhfjZkklhygbbqlhwe_dljhgu hklZxlky\Zlhfguokhklhygbyobg_i_j_oh^ylgZfhe_dmeyjgu_hj[blu <dZ`^hfZlhf_p_gljly`_klbhljbpZl_evgh]haZjy^Zkh\iZ^Z_lkp_gljhf ly`_klb iheh`bl_evgh]h y^jhf LZdbf h[jZahf Zlhfu g_ bf_xl ^b ihevguo fhf_glh\ H^gZdh \ dZ`^uc ^Zgguc fhf_gl \j_f_gb Zlhfu h[ eZ^Zxl ³f]gh\_gguf´ ^bihevguf fhf_glhf Kbkl_fZ [m^_l h[eZ^Zlv [h e__gbadhcwg_j]b_c_kebwlbfhf_glu[m^mlhjb_glbjh\Zgu\ijhkljZg kl\_hij_^_e_ggufh[jZahfLZd_keb\h^ghfZlhf_we_dljhggZoh^blky ke_\Zhly^jZlhb\^jm]hf^eymf_gvr_gbyhllZedb\Zgbywe_dljhgm\u ]h^g__lh`_[ulvke_\Z LZdbfh[jZahfwe_dljhgujZaguoZlhfh\\dZdhc-lhkl_i_gb^he` gu^\b]Zlvky\lZdlWlhhagZqZ_lebrvqlh\jZaguoZlhfZoqZs_^he`gu [ulv ij_^klZ\e_gu dhgnb]mjZpbb we_dljhgh\ ijb\h^ysb_ d f_gvr_fm hllZedb\Zgbx >bki_jkbhggu_ kbeu fh`gh jZkkfZljb\Zlv dZd g_dhlhjh_ ih^h[b_ hjb_glZpbhgguo kbe b ihwlhfm fh`gh ijb[eb`_ggh ihemqblv fZl_fZlbq_kdh_ \ujZ`_gb_ ^ey wlbo kbe gZ hkgh\_ nhjfmeu K wlhc p_evx g_h[oh^bfh hij_^_eblv ZgZeh]b \_ebqbg kT b µ \oh^ysbo \ wlh mjZ\g_gb_ 56 H]jZgbqbfky jZkkfhlj_gb_f \aZbfh^_ckl\by h^bgZdh\uo fhe_dme Wg_j]bx l_ieh\h]h ^\b`_gby ij_iylkl\mxs_]h hjb_glZpbb ijb \aZbfh ^_ckl\bb^bihevguofhe_dmeoZjZdl_jbam_l kTF]gh\_ggufwe_dljhgguf fhf_glZf f_rZ_l hjb_glbjh\Zlvky bo dbg_lbq_kdZy wg_j]by f_jhc dhlh jhcdZdke_^m_lbad\Zglh\hcf_oZgbdbfh`_l[ulvihl_gpbZebhgbaZpbb V Ihwlhfm\f_klhkT\mjZ\g_gb_24ih^klZ\bf\_ebqbgm V F]gh\_g guc^bihevgucfhf_glp_e_khh[jZaghaZf_gblvwdi_jbf_glZevghhij_^_ ey_fhcaZ\bkys_chlg_]h\_ebqbghcdhlhjhc\jZfdZofh^_eb³f]gh\_g guofhf_glh\´[m^_liheyjbam_fhklvfhe_dmeuKh]eZkghmjZ\g_gbx23) ^bihevgZy fhe_dmeZ ijb l_fi_jZlmj_ L bf__l wnn_dlb\gmx iheyjbam_ µ2 . Ke_^h\Zl_evgh fhe_dmeZ k f]gh\_gguf ^bihevguf fh fhklv α = 3kT µ2 . <f_klh kT kh]eZkgh \u 3V r_baeh`_gghfm\\_^_gihl_gpbZebhgbaZpbbV . f_glhf ^he`gZ bf_lv iheyjbam_fhklv α = LZdbfh[jZahfih^klZ\eyy\mjZ\g_gb_24\ujZ`_gby^eykTb µ 2 , ihemqbfhimklb\qbke_ggucfgh`bl_ev U ≈− α 2V (4πε o )2 R 6 . LhqgZynhjfmeZ^eyh^bgZdh\uoZlhfh\bf__l\b^ U =− 3α 2V (4πεo ) 2 6 . (25) 4R <aZbfh^_ckl\b_ jZaguo Zlhfh\ hibku\Z_lky ke_^mxs_c nhjfmehc Ehg^hgZ V1V2 3 α1α 2 1 U =− . 2 (4πε o )2 V1 + V2 R 6 >bki_jkbhggu_kbeug_aZ\bkylhll_fi_jZlmjuHgbmgb\_jkZevgu lZddZdkms_kl\mxl\ex[uokbkl_fZo]^_bf_xlkywe_dljhgu>\b`_gb_ we_dljhgh\³\lZdl´ij_^klZ\e_ghebrv\keZ[hckl_i_gbl_bf_y\ukh dmx dbg_lbq_kdmx wg_j]bx we_dljhgu h^ghc fhe_dmeu \ fZehc kl_i_gb mqblu\Zxlf]gh\_ggh_iheh`_gb_we_dljhgh\^jm]hcfhe_dmeuJ_amevlZ lhf wlh]h y\ey_lky Z^^blb\ghklv ^bki_jkbhgguo kbe We_dljhgu h^ghc fhe_dmeu fh]ml mqblu\Zlv iheh`_gb_ we_dljhgh\ g_kdhevdbo khk_^gbo fhe_dme qlh [ueh [u dhg_qgh g_\hafh`gh ijb ^\b`_gbb we_dljhgh\ \ lZdl 57 JZkq_luihdZau\Zxlqlhbg^mdpbhggu_kbeufZeuihkjZ\g_gbxk hjb_glZpbhggufbb^bki_jkbhggufbbbfbfh`ghij_g_[j_qv>eyfhe_ dme g_ bf_xsbo ^bihevguo fhf_glh\ _^bgkl\_gguf \b^hf fhe_dmeyj guokbey\eyxlky^bki_jkbhggu_M^bihevguofhe_dmehjb_glZpbhggu_ kbeu kjZ\gbfu k ^bki_jkbhggufb Hlghkbl_evgZy jhev ^bki_jkbhgguo kbehij_^_eylkyh[uqghqbkehfwe_dljhgh\ HLL:EDB<:GB?BKMFF:JGH?<A:BFH>?CKL<B? Cmffbjmywg_j]bxhjb_glZpbhggh]hbg^mdpbhggh]hb^bki_jkbhg gh]h \aZbfh^_ckl\bc ^ey h^bgZdh\uo fhe_dme b h[t_^bgyy \k_ ihklhyg gu_\khhl\_lkl\bbkmjZ\g_gbyfb21bihemqbfwg_j]bxf_` fhe_dmeyjgh]hijbly`_gby K U ij = − , (26) r6 2 2µ 4 2 3α V + αµ + ]^_ C = . 4 (4πεo )2 3kT 1 Ijb fZeuo jZkklhygbyo f_`^m fhe_dmeZfb dh]^Z bo we_dljhggu_ h[hehqdbkbevghi_j_dju\Zxlkyl_dh]^Zwe_dljhklZlbq_kdh_hllZedb\Z gb_y^_jbwe_dljhgh\klZgh\blky[hevr_bo\aZbfgh]hijbly`_gbyijh y\ey_lky^_ckl\b_kbehllZedb\ZgbyGZkms_kl\h\Zgb_wlbokbemdZau\Z xl fgh]b_ nZdlu \ qZklghklb fZeZy k`bfZ_fhklv `b^dhkl_c b l\_j^uo l_e < i_j\hf ijb[eb`_gbb wg_j]by hllZedb\Zgby fhe_dme fh`_l [ulv \ujZ`_gZmjZ\g_gb_f < U hl = , (27) 12 r ]^_ B – ihklhyggZy dhgklZglZ hllZedb\Zgby Ba mjZ\g_gby \b^gh qlhkbeuhllZedb\ZgbygZqbgZxlijhy\eylvkygZhq_gvfZeuojZkklhygbyo b\_kvfZ[ukljhjZklmlkmf_gvr_gb_fr. IhegZywg_j]by\aZbfh^_ckl\byf_`^mfhe_dmeZfb U = U ij + U hl beb\khhl\_lkl\bbkb U =− 58 C B + 12 . 6 r r Wlh mjZ\g_gb_ gZau\Z_lky nhjfmehc E_ggZj^Z – >`hgkZ =jZnbq_ kdb ^ZggZy aZ\bkbfhklv khhl\_lkl\m_l djb\uf ijb\_^_gguf gZ jbk 31. FbgbfmfgZkmffZjghcdjb\hcoZjZdl_jbam_lwg_j]bxf_`fhe_dmeyjgh]h \aZbfh^_ckl\by U 0 bjZ\gh\_kgh_jZkklhygb_ r0 f_`^mfhe_dmeZfb Jbk31Djb\u_ihl_gpbZevghcwg_j]bb f_`fhe_dmeyjgh]h\aZbfh^_ckl\by <H>HJH>G:YK<YAV Ijhf_`mlhqghc f_`^m fhe_dmeyjgufb b obfbq_kdbfb kbeZfb y\ ey_lky lZd gZau\Z_fZy \h^hjh^gZy k\yav < jy^_ kemqZ_\ Zlhf \h^hjh^Z k\yau\Z_lky g_kh^gbfZlhfhf dZd wlh hl\_qZeh [u _]h \Ze_glghklb Z k ^\mfyWlZk\yavh[mkeh\e_gZl_fqlhkf_s_gb_we_dljhgZhlZlhfZ\h^h jh^Z ij_\jZsZ_l _]h \ qZklbpm k mgbdZevgufb k\hckl\Zfb dhlhjZy _keb jZkkfZljb\Zlv_zdZddZlbhgZg_bf__lwe_dljhgh\bihwlhfm\hlebqb_ hlhklZevguodZlbhgh\g_hllZedb\Z_lkywe_dljhggufbh[hehqdZfb^jm]bo qZklbp Z bkiulu\Z_l lhevdh ijbly`_gb_ [ h[eZ^Z_l gbqlh`gh fZeuf jZaf_jhfijhlhg\lukyqbjZaf_gvr_hklZevguobhgh\LZdbfh[jZahf fZe_gvdbciheh`bl_evghaZjy`_ggucbhg\h^hjh^Zkihkh[_g^ZlvagZqb l_evgmxwg_j]bxk\yabkkhk_^gbfbhljbpZl_evgufbbhgZfbZdhhj^bgZ 59 pbhggh_qbkeh^\Zfh`_lhij_^_eylvky]_hf_ljbq_kdbfbh[klhyl_evkl\Zfb [hevr_^\moZlhfh\g_fh]mlijb[ebablvkydfZehfmdZlbhgm\h^hjh^Z <h^hjh^gZy k\yav ijhy\ey_lky l_f kbevg__ q_f [hevr_ we_dljhhl jbpZl_evghklvZlhfZ-iZjlg_jZbq_ff_gvr__]hjZaf_juihwlhfmhgZoZ jZdl_jgZij_`^_\k_]h^eykh_^bg_gbcnlhjZZlZd`_dbkehjh^Z\f_gv r_c kl_i_gb ZahlZ \ _s_ f_gvr_c kl_i_gb ^ey oehjZ b k_ju ;eZ]h^Zjy \h^hjh^guf k\yayf fhe_dmeu h[t_^bgyxlky \ ^bf_ju b ihebf_ju LZd fmjZ\vbgZy dbkehlZ dZd \ `b^dhc lZd b \ ]Zahh[jZaghc nZa_ kms_kl\m_l ]eZ\gufh[jZahf\\b^_^bf_jZjbk32). Kihkh[ghklvdZkkhpbZpbbhlebqZ_l\h^mZffbZdkibjlubfgh]b_ ^jm]b_ `b^dhklb hl g_Zkkhpbbjh\Zgguo `b^dhkl_c gZijbf_j hl m]e_\h ^hjh^h\ Ijhqghklv^bf_jZjbk3hij_^_ey_lky^\mfy\h^hjh^gufbk\yay fb hkms_kl\ey_fufb dZ`^uf Zlhfhf \h^hjh^Z k dbkehjh^hf ³qm`hc´ fhe_dmeuLZdb_k\yabhlghkylkydkemqZxf_`fhe_dmeyjghc\h^hjh^ghc k\yab < lh `_ \j_fy \klj_qZ_lky \h^hjh^gZy k\yav dhlhjZy h[t_^bgy_l qZklb h^ghc b lhc `_ fhe_dmeu l_ y\ey_lky \gmljbfhe_dmeyjghc GZ jbk 3ihdZaZglbibqguckemqZclZdhck\yab\fhe_dme_hjlhgbljhn_gh eZ\ke_^kl\b_q_]h h[jZam_lky\lhjh_dhevphqlhoZjZdl_jgh^eyfgh]bo hj]Zgbq_kdbo\_s_kl\ Jbk3KljmdlmjZ^bf_jZ fmjZ\vbghcdbkehlu Jbk33<gmlj_ggyy\h^hjh^gZyk\yav \fhe_dme_hjlhgbljhn_gheZ ;hevrmxjhevb]jZ_l\h^hjh^gZyk\yavb\ijhqghklbjy^ZdjbklZe ebq_kdbo l_e LZd l\_j^Zy [hjgZy dbkehlZ khklhbl ba keh_\ fhe_dme <HG3dhlhju_k\yaZguf_`^mkh[hc\h^hjh^gufbk\yayfb 60 =eZ\Z 5.G?DHLHJU?K<?>?GBYHKLJH?GBB DHG>?GKBJH<:GGUON:A 5.KLJMDLMJ:@B>DHKL?C HlijZ\ghc lhqdhc ijb jZkkfhlj_gbb l\_j^uo \_s_kl\ y\ey_lky kljh]hmihjy^hq_ggZykljmdlmjZkh\_jr_gguodjbklZeeh\Zijbbamq_gbb ]Zah\ – iheghklvx oZhlbq_kdh_ jZkij_^_e_gb_ qZklbp \ b^_Zevghf ]Za_ Ijbbamq_gbb`b^dhkl_cbkoh^ylbaboijhf_`mlhqgh]hkljh_gbyIh^h[ gh ]ZaZf `b^dhklb ijbgbfZxl nhjfm lh]h khkm^Z \ dhlhjhf gZoh^ylky l_h[eZ^Zxll_dmq_klvxH^gZdhihk\hbfk\hckl\Zfbkljh_gbx`b^dh klb]hjZa^h[eb`_klhyldl\_j^ufl_eZfq_fd]ZaZfhkh[_gghijbl_f i_jZlmjZo [ebadbo d l_fi_jZlmj_ ieZ\e_gby LZdh_ aZdexq_gb_ \ul_dZ_l bakjZ\g_gbybok\hckl\ 1. Ijhp_kkieZ\e_gbykhijh\h`^Z_lkyg_[hevrbfm\_ebq_gb_fh[t _fZhlf_lbfqlhbf_xlkybkdexq_gby^eyg_dhlhjuo\_s_kl\gZijbf_j G2H\nhjf_ev^Zdhlhjh_h[uqghg_ij_\urZ_lWlhhagZqZ_lqlh jZkklhygb_f_`^mkhk_^gbfbqZklbpZfb\`b^dhklbfZehbaf_gy_lkyijb ieZ\e_gbb–g_[he__q_fgZIjbi_j_oh^_`_`b^dhklb\]Zahh[jZa gh_ khklhygb_ h[t_f m\_ebqb\Z_lky agZqbl_evgh gZijbf_j ijb i_j_oh^_ \h^u\iZj–\jZa 2. Wgljhibyijbi_j_oh^_l_eZbal\_j^h]hkhklhygby\`b^dh_m\_ ebqb\Z_lky gZ agZqbl_evgh f_gvrmx \_ebqbgm ijbf_jgh \ jZa ih kjZ\g_gbxkbaf_g_gb_fwgljhibbijbbkiZj_gbb 3. Dhwnnbpb_glu k`bfZ_fhklb `b^dhklb b l\_j^h]h l_eZ [ebadb b \hfgh]hjZaf_gvr_q_fm]ZaZ 4. @b^dhklbbl\_j^u_l_eZbf_xlih\_joghklgh_gZly`_gb_Z]Zag_l 5. <hij_^_e_gguomkeh\byo`b^dhklbdZdbl\_j^u_l_eZh[eZ^Zxl ojmidhklvx LZd ijb m^Zj_ e_lys_c imeb h kljmx \h^u ihke_^gyy dZd nbdkbjm_ldbghkt_fdZjZae_lZ_lky\\b^_hkdhedh\lbiZdmkhqdh\ev^Zgh [email protected]^dhklvklZgh\blkyojmidhcijbhq_gv[ukljuo^_nhjfZpbyo gZijbf_jijb\aju\Zo\`b^dhklbfh]ml\hagbdZlvdjZldh\j_f_ggu_lj_ sbgu ;ebahklv k\hckl\ l\_j^h]h b `b^dh]h khklhygbc iha\hey_l h[t_^b gylvwlb ^\Z khklhygbybgZau\Zlv boh[sbf l_jfbghf – dhg^_gkbjh\Zg gh_khklhygb_\_s_kl\Z 61 Ihegh_ hibkZgb_ jZaebqguo k\hckl\ `b^dhkl_c ^he`gh hibjZlvky gZg_dhlhju_[Zah\u_nmgdpbbi_j_^Zxsb_gZb[he__\Z`gu__zoZjZdl_ jbklbdb D lZdbf nmgdpbyf hlghkblky [bgZjgZy nmgdpby jZkij_^_e_gby hibku\ZxsZy [eb`gbc ihjy^hd JZkkfhljbf `b^dhklv khklhysmx ba Zlhfh\ <u[_j_f \gmljb lZdhc `b^dhklb rZjh\hc kehc k \gmlj_ggbf jZ ^bmkhf r b gZjm`guf r + dr k p_gljhf \ y^j_ ijhba\hevghc qZklbpu Kj_^g__ qbkeh Zlhfh\ dn(r ihiZ\rbo \ kehc [m^_l ijhihjpbhgZevgh h[t_fm4r2dr bg_dhlhjhfmfgh`bl_exR(r) aZ\bkys_fm hl r bgZau\Z_ fhfmiZjghcnmgdpb_cjZ^bZevgh]hjZkij_^_e_gbyZlhfh\hlf_lbfqlh\ g_dhlhjuomq_[gbdZo^Zggmxnmgdpbxh[hagZqZxlg(r)): dn(r) = 4πr 2 drR (r ) . (28) Nmgdpby R(r) ih k\h_fm kfukem y\ey_lky ehdZevghc iehlghklvx qbkeZqZklbpihhlghr_gbxdijhba\hevghfmZlhfm\u[jZgghfmaZgZqZ ehdhhj^bgZlGZ^hklZlhqgh[hevrhfjZkklhygbbrihkjZ\g_gbxkf_` Zlhfguf jZkklhygb_f Zlhfu g_ bkiulu\Zxl \ebygby p_gljZevghc qZklb pu b ihwlhfm jZkiheZ]Zxlky g_aZ\bkbfh hl g_z < wlhf kemqZ_ \_ebqbgZ R(r) = Robij_^klZ\ey_lkj_^gxxiehlghklvqbkeZqZklbpGZjZkklhygbb ro f_gvr_f q_f ^bZf_lj ZlhfZ \ rZjh\hc kehc g_ ihiZ^Z_l gb h^gh]h y^jZZlhfZbihwlhfma^_kvR(r) = <ijhf_`mlhqghch[eZklbnmgdpby R(r) hkpbeebjm_lhdhehagZq_gbyRo bbf__lg_kdhevdhfbgbfmfh\bfZd kbfmfh\G_dhlhju_f_`Zlhfgu_jZkklhygby\[ebabp_gljZevghcqZklbpu j_Zebamxlky[he__qZklhGZebqb_wlbowe_f_glh\mihjy^hq_gby\`b^dh klbbihemqbehgZa\Zgb_³[eb`g_]hihjy^dZ´BamjZ\g_gby\ul_dZ_l aZ\bkbfhklv dn(r) = 4πr 2 R(r ) , dr ij_^klZ\eyxsZy kh[hc kj_^g__ qbkeh qZklbp ijboh^ysboky gZ _^bgbq gmx lhesbgm rZjh\h]h kehy QZklh \f_klh nmgdpbb R(r) bkihevam_lky nmgdpby jZ^bZevgh]h jZkij_^_e_gby Zlhfghc iehlghklb !(r) wlb aZ\bkb fhklbk\yaZguf_`^mkh[hcke_^mxsbfkhhlghr_gb_f g(r) = !(r)/!o , ]^_!o –kj_^g__qbkehqZklbp\_^bgbp_h[t_fZ ρ 0 = N A / V ; N A –qbkeh :\h]Z^jh V – Zlhfguc h[t_f AZ\bkbfhklv nmgdpbb 4r2R(r) hl r beb 4r2!(r) hl r ij_^klZ\eyxl ]jZnbq_kdb GZ jbk 34 bah[jZ`_gh Zlhfgh_ jZkij_^_e_gb_ ^ey `b^dh]h pbgdZ IehsZ^v ibdZ khhl\_lkl\mxsZy i_j \hfmfZdkbfmfmhl\_qZ_lkj_^g_fmdhhj^bgZpbhgghfmqbkem\`b^dhklb 62 G_h[oh^bfu_ ^Zggu_ ^ey ihkljh_gby djb\hc jZ^bZevgh]h jZkij_^_e_gby ihemqZxlihdjb\ufbgl_gkb\ghklbjZkk_ygbyj_gl]_gh\kdboemq_c Jbk:lhfgh_jZkij_^_e_gb_\`b^dhfpbgd_hlex[hcqZklbpu \aylhcaZp_gljZevgmx: 1 –]Za–`b^dhklv–l\_j^h_ @B>DB?F?L:EEU Qbklu_`b^db_f_lZeeuhlghkyldlZdgZau\Z_fufijhkluf`b^dh klyf dhlhjufb y\eyxlky k`b`_ggu_ bg_jlgu_ ]Zau k <Zg-^_j<ZZevkh\ufb kbeZfb \aZbfh^_ckl\by Ijhklu_ `b^dhklb khklhyl ba kn_ jbq_kdb kbff_ljbqguo qZklbp f_`^m dhlhjufb ^_ckl\mxl g_gZijZ\e_g gu_ b g_gZkus_ggu_ kbeu k\yab Fhe_dmeu lZdbo `b^dhkl_c h[eZ^Zxl kn_jbq_kdbf ihl_gpbZehf \aZbfh^_ckl\by b g_ h[gZjm`b\Zxl ki_pbnb q_kdbok\ya_c^_ckl\mxsbo\dZdhf-eb[hhij_^_e_gghfgZijZ\e_gbbdZd \kemqZ_]hf_hiheyjguobeb\h^hjh^guok\ya_c <`b^dboqbkluof_lZeeZof_`qZklbqgZyk\yavhkms_kl\ey_lkydhe e_dlb\babjh\Zggufb \Ze_glgufb we_dljhgZfb WlZ k\yav dZd b <Zg-^_j\ZZevkh\Z k\yav g_gZijZ\e_gZ b g_gZkus_gZ F_lZeebq_kdb_ bhgu kh klZ\eyxsb_ jZkieZ\ kn_jbq_kdb kbff_ljbqgu lZd dZd bf_xl kljmdlmjm 63 \g_rgbowe_dljhgguoh[hehq_dZgZeh]bqgmxkljmdlmj_h[hehq_dZlhfh\ [eZ]hjh^guo ]Zah\ Ih^h[gh_ hlh`^_kl\e_gb_ kljmdlmju k`b`_gguo bg_jlguo]Zah\b`b^dbof_lZeeh\fh`_l[ulvhijZ\^Zghebrvijbhlgh kbl_evgh [hevrbo i_j_]j_\Zo f_lZeeh\ gZ^ lhqdhc ieZ\e_gby Ijb g_ [hevrbo`_i_j_]j_\Zofh`ghij_^iheh`blvqlh`b^db_f_lZeeu[eb`_ ihk\h_ckljmdlmj_ddjbklZeebq_kdhfmkhklhygbxgh\lh`_\j_fybf_ xlbk\hboZjZdl_jgu_hkh[_gghklb >ey bamq_gby kljmdlmju `b^dbo f_lZeeh\ bkihevamxl mijm]hjZk k_b\Z_fu_qZklbpu–j_gl]_gh\kdb_nhlhguwe_dljhgubebg_cljhgu<k_ hgbbf_xl^ebgm\heguihjy^dZf_`ZlhfguojZkklhygbcIjbbkke_^h\Z gbbgZh[t_dlgZijZ\eyxldhgp_gljbjh\ZggucimqhdwlboqZklbpj_]bkl jbjmxlihemqZ_fmx^bnjZdpbhggmxdZjlbgmgZhkgh\Zgbbdhlhjhckm^yl hkljh_gbb\_s_kl\ZGZb[hevr__jZkijhkljZg_gb_\kljmdlmjghfZgZeba_ `b^dbof_lZeeh\ihemqbebj_gl]_gh\kdb_emqb < `b^dbo f_lZeeZo j_gl]_gh\kdb_ emqb jZkk_yggu_ h^gbf Zlhfhf bgl_jn_jbjmxl k emqZfb jZkk_yggufb _]h [eb`Zcrbfb khk_^yfb Ijb q_f[eb`Zcrb_khk_^bbjZkklhygb_f_`^mgbfbg_ij_ju\ghbaf_gyxlky \ke_^kl\b_\ukhdhcih^\b`ghklbZlhfh\\`b^dbof_lZeeZoihwlhfm\ha gbdZxsbc^bnjZdpbhggucwnn_dly\ey_lkyj_amevlZlhfmkj_^g_ggh]h\h \j_f_gb jZkij_^_e_gbyZlhfh\Ijbwlhf kj_^g__ jZkij_^_e_gb_ g_ aZ\b kbl hl ZlhfZ \u[jZggh]h aZ gZqZeh hlkq_lZ GZ hkgh\_ djb\uo bgl_gkb\ ghklb jZkk_yggh]h baemq_gby dZd m`_ ]h\hjbehkv \ur_ jZkkqblu\Zxl b kljhyldjb\u_jZ^bZevgh]hjZkij_^_e_gbyZlhfghciehlghklb!(r). <k_ f_lZeeu g_aZ\bkbfh hl bo kljmdlmju \ l\_j^hf khklhygbb \ `b^dhf khklhygbb bf_xl koh^gh_ gh g_ ZgZeh]bqgh_ kljh_gb_ \ujZ `Zxs__ky\ijbf_jghh^bgZdh\hfoZjZdl_j_djb\uojZ^bZevgh]hjZkij_ ^_e_gbyZlhfghciehlghklb4r2!(r) dZdihdZaZghgZjbkJZ^bmki_j \hcdhhj^bgZpbhgghckn_ju\`b^dbo 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