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838.Строение вещества учебное пособие.

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ij_^klZ\eylv k ihfhsvx kljmdlmjguo nhjfme lh _klv mijhszggh]h ieh
kdhklgh]h bah[jZ`_gby j_Zevgh]h jZkiheh`_gby Zlhfh\ \ fhe_dme_ <h
fgh]bo kemqZyo bkihevah\Zgb_ kljmdlmjguo nhjfme iha\hey_l emqr_
ij_^klZ\blv b kljh_gb_ kZfbo fhe_dme b bo obfbq_kdb_ k\hckl\Z b ob
fbafijhp_kkh\\dhlhjuofhe_dmeZmqZkl\m_l
2. 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' ). Kj_^bq_lujzoZlhfguofhe_dme\klj_qZ
xlky b ebg_cgu_ H − C ≡ C − H ) b iehkdb_ dZd h[jZamxsb_ jZ\ghklh
jhggbclj_m]hevgbd– BCl3, AlBr3lZdbL-h[jZagu_– ClF3)ghqZs_\k_]h
– h[tzfgu_ D ihke_^gbf hlghkylky fhe_dmeu \ \b^_ lj_m]hevguo ibjZ
fb^ k jZaebqgufb m]eZfb \klj_qZ_lky lZd`_ ^jm]h_ kljh_gb_ gZijbf_j
Zlhfu i_j_dbkb \h^hjh^Z jZkiheZ]Zxlky \ ^\mo i_j_k_dZxsboky iehkdh
klyo Fhe_dmeu k [hevrbf qbkehf Zlhfh\ qZs_ \k_]h h[jZamxl fgh]h
]jZggu_ ibjZfb^u b [bibjZfb^u M]e_\h^hjh^u fh]ml h[jZah\u\Zlv p_
ihq_qgu_kljmdlmjuk\Ze_glgufbm]eZfbZlZd`_pbdeu<gZkus_gguo
pbdebq_kdbom]e_\h^hjh^Zo\Ze_glgu_m]eubkdZ`_guqlhijb\h^bldgZ
15
ijy`_gbyfpbdeh\bh[mkeh\eb\Z_lih\ur_ggmxobfbq_kdmxZdlb\ghklv
wlbo \_s_kl\ < fhe_dme_ pbdehi_glZgZ q_luj_ ZlhfZ m]e_jh^Z e_`Zl \
h^ghciehkdhklbZiylucijbih^gylgZiheZg]klj_fZjbk3ZFhe_dmeZ
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iehkdhklbkbff_ljbb
17
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p_glj kbff_ljbb beb p_glj bg\_jkbb – h[hagZqZ_lky i hi_
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18
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z ?keb fhe_dmeZ bf__l g_kdhevdh hk_c \ukr_]h ihjy^dZ lh aZ ]eZ\gmx
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20
ij_h[jZah\Zgby dhhj^bgZl ( x, y, z ) \ (− x, − y, z ) Hi_jZpbx C 2 fh`gh
ij_^klZ\blvdZd
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C 21 ⋅  y  = − y  .
   
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aZibkZgZke_^mxsbfh[jZahf
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σ  y  = − y  .
   
 z   z 
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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
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 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\hjhlgZŒn
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 σvijhba\h^bfuogZ^fhe_dmehckbff_ljbb
K2#gZijbf_jG2H– wlhσvgZihfgbfqlhhkvK2y\ey_lkyhkvxz,fhe_
dmeZe_`bl\iehkdhklbyz, Ziehkdhklvyz _klviehkdhklvσv):
K2· σv = 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_jZpbyσvhkms_kl\ey_lkyi_j\hcZaZl_fke_^m_lhi_
jZpbyK2qlh^Z_llhl`_kZfucj_amevlZlqlhbhi_jZpbyσv<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 σv = σv K2]h\hjylqlh^\Zwe_
f_glZkbff_ljbbK2bσvdhffmlbjmxl
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 ) σv].
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
σvŽ
?
?
K3
C32
σv
σv
σvŽ
K3
K3
C32
?
σvŽ
σv
σv
C32
C32
?
K3
σv
σv
σv
σv
σvŽ
?
K3
C32
σv
σv
σvŽ
σv
C32
?
K3
σvŽ
σvŽ
σv
σv
K3
C32
?
σvŽ
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. HILBQ?KD:Y:DLB<GHKLV
Hilbq_kdZyZdlb\ghklv\_s_kl\–\Z`gZykl_j_hobfbq_kdZyoZjZdl_
jbklbdZ>Zggu_h\jZs_gbbiehkdhklbiheyjbaZpbbhilbq_kdbZdlb\guo
\_s_kl\ihkem`bebhkgh\hc^eykha^ZgbyY<[email protected] E_-;_e_f
l_hjbbhijhkljZgkl\_gghfl_ljZw^jbq_kdhfkljh_gbbf_lZgZb_]hijhba
\h^guo Iha^g__ wlb ij_^klZ\e_gby gZreb ijbf_g_gb_ ^ey fgh]bo deZk
kh\\_s_kl\\dexqZydhfie_dkgu_kh_^bg_gbyZlZd`_[bheh]bq_kdbZd
lb\gu_\_s_kl\Z
H^ghba\Z`g_crbok\hckl\fhe_dmehkh[_gghijbjh^guokh_^bg_
gbc–objZevghklvbebhilbq_kdZyZdlb\ghklvHgZh[mkeh\e_gZkms_kl
\h\Zgb_f a_jdZevghih^h[guo bahf_jh\ – wgZglbhf_jh\ dhlhju_ _s_ gZ
au\Zxlkyhilbq_kdbfbZglbih^ZfbH^bgbagbo\jZsZ_liehkdhklvihey
jbah\Zggh]hemqZ\e_\h>l beb–)[email protected]^jm]hc–gZlZdhc`_m]he\ijZ
\h>dbeb-nhjfZ]. <hlebqb_hl]_hf_ljbq_kdbobahf_jh\wgZglbhf_ju
wd\b\Ze_glguihk\hbfnbabq_kdbfbobfbq_kdbfk\hckl\ZfMgboh^b
gZdh\u_l_fi_jZlmjuieZ\e_gbybdbi_gby^Z\e_gbyiZjZiehlghklvih
dZaZl_ev ij_ehfe_gby H^gZdh wgZglbhf_ju ih-jZaghfm \jZsZxl iehk
dhklviheyjbaZpbbebg_cghiheyjbah\Zggh]hk\_lZ
Hkgh\gh_ mkeh\b_ objZevghklb fhe_dme – hlkmlkl\b_ p_gljZ kbf
f_ljbbiehkdhklbkbff_ljbba_jdZevgh-ih\hjhlghchkbkbff_ljbb S n \
fhe_dme_Hilbq_kdbZdlb\gu_fhe_dmeufh]mlbf_lvkbff_ljbx C n b D n .
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
nn 
µ S = 2µ B
 + 1 = µ B n(n + 2 ) .
22 
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
(4Œ0 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_fm4Œr2dr 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
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