Yilda kristallografiya , a kasr koordinatalar tizimi a koordinatalar tizimi unda qirralarning birlik hujayrasi asosiy sifatida ishlatiladi vektorlar atom yadrolarining pozitsiyalarini tavsiflash. Birlik hujayrasi a parallelepiped uning qirralarining uzunligi bilan belgilanadi a , b , v {displaystyle a, b, c} a , β , γ {displaystyle alfa, eta, gamma}
Umumiy ish
Kosmosdagi davriy tuzilish tizimini va ishlatilishini ko'rib chiqing a {displaystyle {mathbf {a}}} b {displaystyle mathbf {b}} v {displaystyle mathbf {c}} r {displaystyle mathbf {r}}
r = siz a + v b + w v . {displaystyle {mathbf {r}} = u {mathbf {a}} + v {mathbf {b}} + w {mathbf {c}}.} Bizning vazifamiz kasr koordinatalari deb nomlanuvchi skalar koeffitsientlarini hisoblash siz {displaystyle u} v {displaystyle v} w {displaystyle w} r {displaystyle mathbf {r}} a {displaystyle mathbf {a}} b {displaystyle mathbf {b}} v {displaystyle mathbf {c}}
Shu maqsadda quyidagi hujayra yuzasi vektorini hisoblab chiqamiz
σ a = b × v , {displaystyle mathbf {sigma} _ {mathbf {a}} = {mathbf {b}} imes {mathbf {c}},} keyin
b ⋅ σ a = 0 , v ⋅ σ a = 0 , {displaystyle {mathbf {b}} cdot mathbf {sigma} _ {mathbf {a}} = 0, {mathbf {c}} cdot mathbf {sigma} _ {mathbf {a}} = 0,} va hujayraning hajmi
Ω = a ⋅ σ a . {displaystyle Omega = {mathbf {a}} cdot mathbf {sigma} _ {mathbf {a}}.} Agar biz vektor ichki (nuqta) mahsulotni quyidagicha bajaradigan bo'lsak
r ⋅ σ a = siz a ⋅ σ a + v b ⋅ σ a + w v ⋅ σ a = siz a ⋅ σ a = siz Ω , {displaystyle {egin {aligned} {mathbf {r}} cdot mathbf {sigma} _ {mathbf {a}} & = u {mathbf {a}} cdot mathbf {sigma} _ {mathbf {a}} + v {mathbf {b}} cdot mathbf {sigma} _ {mathbf {a}} + w {mathbf {c}} cdot mathbf {sigma} _ {mathbf {a}} & = u {mathbf {a}} cdot mathbf {sigma } _ {mathbf {a}} & = uOmega, oxiri {hizalanmış}}} keyin olamiz
siz = 1 Ω r ⋅ σ a . {displaystyle u = {frac {1} {Omega}} {{mathbf {r}} cdot mathbf {sigma} _ {mathbf {a}}}.} Xuddi shunday,
σ b = v × a , v ⋅ σ b = 0 , a ⋅ σ b = 0 , b ⋅ σ b = Ω , {displaystyle mathbf {sigma} _ {mathbf {b}} = {mathbf {c}} imes {mathbf {a}}, {mathbf {c}} cdot mathbf {sigma} _ {mathbf {b}} = 0, { mathbf {a}} cdot mathbf {sigma} _ {mathbf {b}} = 0, {mathbf {b}} cdot mathbf {sigma} _ {mathbf {b}} = Omega,} r ⋅ σ b = siz a ⋅ σ b + v b ⋅ σ b + w v ⋅ σ b = v b ⋅ σ b = v Ω , {displaystyle {mathbf {r}} cdot mathbf {sigma} _ {mathbf {b}} = u {mathbf {a}} cdot mathbf {sigma} _ {mathbf {b}} + v {mathbf {b}} cdot mathbf {sigma} _ {mathbf {b}} + w {mathbf {c}} cdot mathbf {sigma} _ {mathbf {b}} = v {mathbf {b}} cdot mathbf {sigma} _ {mathbf {b}} = vOmega,} biz etib boramiz
v = 1 Ω r ⋅ σ b , {displaystyle v = {frac {1} {Omega}} {{mathbf {r}} cdot mathbf {sigma} _ {mathbf {b}}},} va
σ v = a × b , a ⋅ σ v = 0 , b ⋅ σ v = 0 , v ⋅ σ v = Ω , {displaystyle mathbf {sigma} _ {mathbf {c}} = {mathbf {a}} imes {mathbf {b}}, {mathbf {a}} cdot mathbf {sigma} _ {mathbf {c}} = 0, { mathbf {b}} cdot mathbf {sigma} _ {mathbf {c}} = 0, {mathbf {c}} cdot mathbf {sigma} _ {mathbf {c}} = Omega,} r ⋅ σ v = siz a ⋅ σ v + v b ⋅ σ v + w v ⋅ σ v = w v ⋅ σ v = w Ω , {displaystyle {mathbf {r}} cdot mathbf {sigma} _ {mathbf {c}} = u {mathbf {a}} cdot mathbf {sigma} _ {mathbf {c}} + v {mathbf {b}} cdot mathbf {sigma} _ {mathbf {c}} + w {mathbf {c}} cdot mathbf {sigma} _ {mathbf {c}} = w {mathbf {c}} cdot mathbf {sigma} _ {mathbf {c}} = wOmega,} w = 1 Ω r ⋅ σ v . {displaystyle w = {frac {1} {Omega}} {{mathbf {r}} cdot mathbf {sigma} _ {mathbf {c}}}.} Agar ko'p bo'lsa r {displaystyle mathbf {r}}
siz = r ⋅ σ a ′ , v = r ⋅ σ b ′ , w = r ⋅ σ v ′ , {displaystyle {egin {aligned} u & = {{mathbf {r}} cdot mathbf {sigma} _ {mathbf {a}} ^ {prime}}, v & = {{mathbf {r}} cdot mathbf {sigma} _ {mathbf {b}} ^ {prime}}, w & = {{mathbf {r}} cdot mathbf {sigma} _ {mathbf {c}} ^ {prime}}, oxiri {hizalangan}}} qayerda
σ a ′ = 1 Ω σ a , σ b ′ = 1 Ω σ b , σ v ′ = 1 Ω σ v . {displaystyle {egin {aligned} mathbf {sigma} _ {mathbf {a}} ^ {prime} = {frac {1} {Omega}} {mathbf {sigma} _ {mathbf {a}}}, mathbf {sigma } _ {mathbf {b}} ^ {prime} = {frac {1} {Omega}} {mathbf {sigma} _ {mathbf {b}}}, mathbf {sigma} _ {mathbf {c}} ^ { prime} = {frac {1} {Omega}} {mathbf {sigma} _ {mathbf {c}}}. oxiri {hizalanmış}}} Kristalografiyada
Yilda kristallografiya , uzunliklar ( a {displaystyle a} b {displaystyle b} v {displaystyle c} a {displaystyle alfa} β {displaystyle eta} γ {displaystyle gamma} a {displaystyle mathbf {a}} b {displaystyle mathbf {b}} v {displaystyle mathbf {c}} parallelepiped birlik hujayrasi ma'lum. Oddiylik uchun chekka vektor tanlanadi a {displaystyle mathbf {a}} x {displaystyle x} b {displaystyle mathbf {b}} x − y {displaystyle x-y} y {displaystyle y} v {displaystyle mathbf {c}} z {displaystyle z}
Uzunliklarga ega parallelepiped yordamida birlik hujayraning ta'rifi
a {displaystyle a} ,
b {displaystyle b} ,
v {displaystyle c} va tomonlari orasidagi burchaklar
a {displaystyle alfa} ,
β {displaystyle eta} va
γ {displaystyle gamma} [1] Keyin chekka vektorlarni quyidagicha yozish mumkin
a = ( a , 0 , 0 ) , b = ( b cos ( γ ) , b gunoh ( γ ) , 0 ) , v = ( v x , v y , v z ) , {displaystyle {egin {aligned} {mathbf {a}} & = (a, 0,0), {mathbf {b}} & = (bcos (gamma), bsin (gamma), 0), {mathbf { c}} & = (c_ {x}, c_ {y}, c_ {z}), oxiri {hizalanmış}}} hamma qayerda a {displaystyle a} b {displaystyle b} v {displaystyle c} gunoh ( γ ) {displaystyle sin (gamma)} v z {displaystyle c_ {z}} v {displaystyle mathbf {c}}
v ⋅ a = a v cos ( β ) = v x a , v ⋅ b = b v cos ( a ) = v x b cos ( γ ) + v y b gunoh ( γ ) , v ⋅ v = v 2 = v x 2 + v y 2 + v z 2 . {displaystyle {egin {aligned} {mathbf {c}} cdot {mathbf {a}} & = accos (eta) = c_ {x} a, {mathbf {c}} cdot {mathbf {b}} & = bccos (alfa) = c_ {x} bcos (gamma) + c_ {y} bsin (gamma), {mathbf {c}} cdot {mathbf {c}} & = c ^ {2} = c_ {x} ^ { 2} + c_ {y} ^ {2} + c_ {z} ^ {2} .end {aligned}}} Keyin
v x = v cos ( β ) , v y = v cos ( a ) − cos ( γ ) cos ( β ) gunoh ( γ ) , v z 2 = v 2 − v x 2 − v y 2 = v 2 { 1 − cos 2 ( β ) − [ cos ( a ) − cos ( γ ) cos ( β ) ] 2 gunoh 2 ( γ ) } . {displaystyle {egin {hizalanmış} c_ {x} & = ccos (eta), c_ {y} & = c {frac {cos (alfa) -cos (gamma) cos (eta)} {sin (gamma)}} , c_ {z} ^ {2} & = c ^ {2} -c_ {x} ^ {2} -c_ {y} ^ {2} = c ^ {2} chap {1-cos ^ {2} (eta) - {frac {[cos (alfa) -cos (gamma) cos (eta)] ^ {2}} {sin ^ {2} (gamma)}} ight} .end {aligned}}} Oxirgisi davom etmoqda
v z 2 = v 2 gunoh 2 ( γ ) − gunoh 2 ( γ ) cos 2 ( β ) − [ cos ( a ) − cos ( γ ) cos ( β ) ] 2 gunoh 2 ( γ ) = v 2 gunoh 2 ( γ ) { gunoh 2 ( γ ) − gunoh 2 ( γ ) cos 2 ( β ) − [ cos ( a ) − cos ( γ ) cos ( β ) ] 2 } {displaystyle {egin {aligned} c_ {z} ^ {2} & = c ^ {2} {frac {sin ^ {2} (gamma) -sin ^ {2} (gamma) cos ^ {2} (eta) - [cos (alfa) -cos (gamma) cos (eta)] ^ {2}} {sin ^ {2} (gamma)}} & = {frac {c ^ {2}} {sin ^ {2} (gamma)}} chap {sin ^ {2} (gamma) -sin ^ {2} (gamma) cos ^ {2} (eta) - [cos (alfa) -cos (gamma) cos (eta)] ^ { 2} ight} oxiri {hizalanmış}}} qayerda
gunoh 2 ( γ ) − gunoh 2 ( γ ) cos 2 ( β ) − [ cos ( a ) − cos ( γ ) cos ( β ) ] 2 = gunoh 2 ( γ ) − gunoh 2 ( γ ) cos 2 ( β ) − cos 2 ( a ) − cos 2 ( γ ) cos 2 ( β ) + 2 cos ( a ) cos ( γ ) cos ( β ) = gunoh 2 ( γ ) − cos 2 ( a ) − gunoh 2 ( γ ) cos 2 ( β ) − cos 2 ( γ ) cos 2 ( β ) + 2 cos ( a ) cos ( β ) cos ( γ ) = gunoh 2 ( γ ) − cos 2 ( a ) − [ gunoh 2 ( γ ) + cos 2 ( γ ) ] cos 2 ( β ) + 2 cos ( a ) cos ( β ) cos ( γ ) = gunoh 2 ( γ ) − cos 2 ( a ) − cos 2 ( β ) + 2 cos ( a ) cos ( β ) cos ( γ ) = 1 − cos 2 ( a ) − cos 2 ( β ) − cos 2 ( γ ) + 2 cos ( a ) cos ( β ) cos ( γ ) . {displaystyle {egin {hizalanmış} va sin ^ {2} (gamma) -sin ^ {2} (gamma) cos ^ {2} (eta) - [cos (alfa) -cos (gamma) cos (eta)] ^ { 2} & = sin ^ {2} (gamma) -sin ^ {2} (gamma) cos ^ {2} (eta) -cos ^ {2} (alfa) -cos ^ {2} (gamma) cos ^ {2} (eta) + 2cos (alfa) cos (gamma) cos (eta) & = sin ^ {2} (gamma) -cos ^ {2} (alfa) -sin ^ {2} (gamma) cos ^ {2} (eta) -cos ^ {2} (gamma) cos ^ {2} (eta) + 2cos (alfa) cos (eta) cos (gamma) & = sin ^ {2} (gamma) -cos ^ {2} (alfa) - [sin ^ {2} (gamma) + cos ^ {2} (gamma)] cos ^ {2} (eta) + 2cos (alfa) cos (eta) cos (gamma) & = sin ^ {2} (gamma) -cos ^ {2} (alfa) -cos ^ {2} (eta) + 2cos (alfa) cos (eta) cos (gamma) & = 1-cos ^ {2} ( alfa) -cos ^ {2} (eta) -cos ^ {2} (gamma) + 2cos (alfa) cos (eta) cos (gamma) .end {hizalanmış}}} Eslab qolish v z {displaystyle c_ {z}} v {displaystyle c} gunoh ( γ ) {displaystyle sin (gamma)}
v z = v gunoh ( γ ) 1 − cos 2 ( a ) − cos 2 ( β ) − cos 2 ( γ ) + 2 cos ( a ) cos ( β ) cos ( γ ) . {displaystyle c_ {z} = {frac {c} {sin (gamma)}} {sqrt {1-cos ^ {2} (alfa) -cos ^ {2} (eta) -cos ^ {2} (gamma) + 2cos (alfa) cos (eta) cos (gamma)}}.} Hujayraning pastki sirt maydonining absolyut qiymati bo'lgani uchun
| σ v | = a b gunoh ( γ ) , {displaystyle left | mathbf {sigma} _ {mathbf {c}} ight | = absin (gamma),} parallelepiped xujayrasining hajmi quyidagicha ifodalanishi mumkin
Ω = v z | σ v | = a b v 1 − cos 2 ( a ) − cos 2 ( β ) − cos 2 ( γ ) + 2 cos ( a ) cos ( β ) cos ( γ ) {displaystyle Omega = c_ {z} chap | mathbf {sigma} _ {mathbf {c}} ight | = abc {sqrt {1-cos ^ {2} (alfa) -cos ^ {2} (eta) -cos ^ {2} (gamma) + 2cos (alfa) cos (eta) cos (gamma)}}} [2] Tovush yuqoridagi kabi hisoblangandan so'ng, bitta bo'ladi
v z = Ω a b gunoh ( γ ) . {displaystyle c_ {z} = {frac {Omega} {absin (gamma)}}.} Endi chekka (davr) vektorlarning ifodasini umumlashtiramiz
a = ( a x , a y , a z ) = ( a , 0 , 0 ) , b = ( b x , b y , b z ) = ( b cos ( γ ) , b gunoh ( γ ) , 0 ) , v = ( v x , v y , v z ) = ( v cos ( β ) , v cos ( a ) − cos ( β ) cos ( γ ) gunoh ( γ ) , Ω a b gunoh ( γ ) ) . {displaystyle {egin {aligned} {mathbf {a}} & = ({a} _ {x}, {a} _ {y}, {a} _ {z}) = (a, 0,0), {mathbf {b}} & = ({b} _ {x}, {b} _ {y}, {b} _ {z}) = (bcos (gamma), bsin (gamma), 0), { mathbf {c}} & = ({c} _ {x}, {c} _ {y}, {c} _ {z}) = (ccos (eta)), c {frac {cos (alfa) -cos ( eta) cos (gamma)} {sin (gamma)}}, {frac {Omega} {absin (gamma)}}). end {hizalanmış}}} Dekart koordinatalaridan konversiya Avval hujayraning quyidagi sirt maydoni vektorini hisoblaymiz
σ a = ( σ a , x , σ a , y , σ a , z ) = b × v , {displaystyle mathbf {sigma} _ {mathbf {a}} = (mathbf {sigma} _ {mathbf {a}, x}, mathbf {sigma} _ {mathbf {a}, y}, mathbf {sigma} _ {mathbf {a}, z}) = {mathbf {b}} imes {mathbf {c}},} qayerda
σ a , x = b y v z − b z v y = b gunoh ( γ ) Ω a b gunoh ( γ ) = Ω a , σ a , y = b z v x − b x v z = − b cos ( γ ) Ω a b gunoh ( γ ) = − Ω cos ( γ ) a gunoh ( γ ) , σ a , z = b x v y − b y v x = b cos ( γ ) v cos ( a ) − cos ( β ) cos ( γ ) gunoh ( γ ) − b gunoh ( γ ) v cos ( β ) = b v { cos ( γ ) cos ( a ) − cos ( β ) cos ( γ ) gunoh ( γ ) − gunoh ( γ ) cos ( β ) } = b v gunoh ( γ ) { cos ( γ ) [ cos ( a ) − cos ( β ) cos ( γ ) ] − gunoh 2 ( γ ) cos ( β ) } = b v gunoh ( γ ) { cos ( γ ) cos ( a ) − cos ( β ) cos 2 ( γ ) − gunoh 2 ( γ ) cos ( β ) } = b v gunoh ( γ ) { cos ( a ) cos ( γ ) − cos ( β ) } . {displaystyle {egin {aligned} mathbf {sigma} _ {mathbf {a}, x} & = {b} _ {y} {c} _ {z} - {b} _ {z} {c} _ {y } = bsin (gamma) {frac {Omega} {absin (gamma)}} = {frac {Omega} {a}}, mathbf {sigma} _ {mathbf {a}, y} & = {b} _ { z} {c} _ {x} - {b} _ {x} {c} _ {z} = - bcos (gamma) {frac {Omega} {absin (gamma)}} = - {frac {Omega cos ( gamma)} {asin (gamma)}}, mathbf {sigma} _ {mathbf {a}, z} & = {b} _ {x} {c} _ {y} - {b} _ {y} { c} _ {x} = bcos (gamma) c {frac {cos (alfa) -cos (eta) cos (gamma)} {sin (gamma)}} - bsin (gamma) ccos (eta) & = bcleft { cos (gamma) {frac {cos (alfa) -cos (eta) cos (gamma)} {sin (gamma)}} - sin (gamma) cos (eta) ight} & = {frac {bc} {sin ( gamma)}} chap {cos (gamma) [cos (alfa) -cos (eta) cos (gamma)] - sin ^ {2} (gamma) cos (eta) ight} & = {frac {bc} {sin (gamma)}} chap {cos (gamma) cos (alfa) -cos (eta) cos ^ {2} (gamma) -sin ^ {2} (gamma) cos (eta) ight} & = {frac {bc } {sin (gamma)}} chap {cos (alfa) cos (gamma) -cos (eta) ight}. end {hizalanmış}}} Hujayraning yana bir sirt maydoni vektori
σ b = ( σ b , x , σ b , y , σ b , z ) = v × a , {displaystyle mathbf {sigma} _ {mathbf {b}} = (mathbf {sigma} _ {mathbf {b}, x}, mathbf {sigma} _ {mathbf {b}, y}, mathbf {sigma} _ {mathbf {b}, z}) = {mathbf {c}} imes {mathbf {a}},} qayerda
σ b , x = v y a z − v z a y = 0 , σ b , y = v z a x − v x a z = a Ω a b gunoh ( γ ) = Ω b gunoh ( γ ) , σ b , z = v x a y − v y a x = − a v cos ( a ) − cos ( β ) cos ( γ ) gunoh ( γ ) = a v gunoh ( γ ) { cos ( β ) cos ( γ ) − cos ( a ) } . {displaystyle {egin {aligned} mathbf {sigma} _ {mathbf {b}, x} & = {c} _ {y} {a} _ {z} - {c} _ {z} {a} _ {y } = 0, mathbf {sigma} _ {mathbf {b}, y} & = {c} _ {z} {a} _ {x} - {c} _ {x} {a} _ {z} = a {frac {Omega} {absin (gamma)}} = {frac {Omega} {bsin (gamma)}}, mathbf {sigma} _ {mathbf {b}, z} & = {c} _ {x} {a} _ {y} - {c} _ {y} {a} _ {x} = - ac {frac {cos (alfa) -cos (eta) cos (gamma)} {sin (gamma)}}) & = {frac {ac} {sin (gamma)}} chap {cos (eta) cos (gamma) -cos (alfa) ight} .end {aligned}}} Hujayraning oxirgi sirt maydoni vektori
σ v = ( σ v , x , σ v , y , σ v , z ) = a × b , {displaystyle mathbf {sigma} _ {mathbf {c}} = (mathbf {sigma} _ {mathbf {c}, x}, mathbf {sigma} _ {mathbf {c}, y}, mathbf {sigma} _ {mathbf {c}, z}) = {mathbf {a}} imes {mathbf {b}},} qayerda
σ v , x = a y b z − a z b y = 0 , σ v , y = a z b x − a x b z = 0 , σ v , z = a x b y − a y b x = a b gunoh ( γ ) . {displaystyle {egin {aligned} mathbf {sigma} _ {mathbf {c}, x} & = {a} _ {y} {b} _ {z} - {a} _ {z} {b} _ {y } = 0, mathbf {sigma} _ {mathbf {c}, y} & = {a} _ {z} {b} _ {x} - {a} _ {x} {b} _ {z} = 0, mathbf {sigma} _ {mathbf {c}, z} & = {a} _ {x} {b} _ {y} - {a} _ {y} {b} _ {x} = absin ( gamma) .end {hizalangan}}} Xulosa qiling
σ a ′ = 1 Ω σ a = ( 1 a , − cos ( γ ) a gunoh ( γ ) , b v cos ( a ) cos ( γ ) − cos ( β ) Ω gunoh ( γ ) ) , σ b ′ = 1 Ω σ b = ( 0 , 1 b gunoh ( γ ) , a v cos ( β ) cos ( γ ) − cos ( a ) Ω gunoh ( γ ) ) , σ v ′ = 1 Ω σ v = ( 0 , 0 , a b gunoh ( γ ) Ω ) . {displaystyle {egin {aligned} mathbf {sigma} _ {mathbf {a}} ^ {prime} & = {frac {1} {Omega}} {mathbf {sigma} _ {mathbf {a}}} = chap ({ frac {1} {a}}, - {frac {cos (gamma)} {asin (gamma)}}, bc {frac {cos (alfa) cos (gamma) -cos (eta)} {Omega sin (gamma)) }} ight), mathbf {sigma} _ {mathbf {b}} ^ {prime} & = {frac {1} {Omega}} {mathbf {sigma} _ {mathbf {b}}} = chap (0, {frac {1} {bsin (gamma)}}, ac {frac {cos (eta) cos (gamma) -cos (alfa)} {Omega sin (gamma)}} ight), mathbf {sigma} _ {mathbf {c}} ^ {prime} & = {frac {1} {Omega}} {mathbf {sigma} _ {mathbf {c}}} = chap (0,0, {frac {absin (gamma)} {Omega} } ight) .end {hizalangan}}} Natijada[3]
[ siz v w ] = [ 1 a − cos ( γ ) a gunoh ( γ ) b v cos ( a ) cos ( γ ) − cos ( β ) Ω gunoh ( γ ) 0 1 b gunoh ( γ ) a v cos ( β ) cos ( γ ) − cos ( a ) Ω gunoh ( γ ) 0 0 a b gunoh ( γ ) Ω ] [ x y z ] {displaystyle left [{egin {matrix} u v wend {matrix}} ight] = left [{egin {matrix} {frac {1} {a}} & - {frac {cos (gamma)} {asin ( gamma)}} & bc {frac {cos (alfa) cos (gamma) -cos (eta)} {Omega sin (gamma)}} 0 & {frac {1} {bsin (gamma)}} & ac {frac {cos ( eta) cos (gamma) -cos (alfa)} {Omega sin (gamma)}} 0 & 0 & {frac {absin (gamma)} {Omega}} end {matrix}} ight] left [{egin {matrix} x y zend {matrix}} ight]} qayerda ( x {displaystyle (x} y {displaystyle y} z ) {displaystyle z)} r {displaystyle mathbf {r}}
Dekart koordinatalariga o'tish Ortogonal koordinatalarni qaytarish uchun angstromlar kasr koordinatalaridan tepada birinchi tenglama va chekka (davr) vektorlarning ifodasini ishlatish mumkin[4] [5]
[ x y z ] = [ a b cos ( γ ) v cos ( β ) 0 b gunoh ( γ ) v cos ( a ) − cos ( β ) cos ( γ ) gunoh ( γ ) 0 0 Ω a b gunoh ( γ ) ] [ siz v w ] . {displaystyle left [{egin {matrix} x y zend {matrix}} ight] = left [{egin {matrix} a & bcos (gamma) & ccos (eta) 0 & bsin (gamma) & c {frac {cos (alfa) -) cos (eta) cos (gamma)} {sin (gamma)}} 0 & 0 & {frac {Omega} {absin (gamma)}} end {matrix}} ight] left [{egin {matrix} u v wend { matritsa}} yopiq].} A uchun maxsus holat monoklinik hujayra (umumiy holat) qaerda a = γ = 90 ∘ {displaystyle alfa = gamma = 90 ^ {circ}} β > 90 ∘ {displaystyle eta> 90 ^ {circ}}
x = a siz + v w cos ( β ) , y = b v , z = Ω a b w = v w gunoh ( β ) . {displaystyle {egin {aligned} x & = au + cwcos (eta), y & = bv, z & = {frac {Omega} {ab}} w = cwsin (eta) .end {aligned}}} Fayl formatlarini qo'llab-quvvatlash
Adabiyotlar
^ Parallelepiped yordamida uzunlikdagi birlik katakchasini aniqlash a , b , v va tomonidan berilgan qirralarning orasidagi burchaklar a , β , γ " . Ccdc.cam.ac.uk . Arxivlandi asl nusxasi 2008-10-04 kunlari. Olingan 2016-08-17 .^ "Muvofiqlashtiruvchi tizim o'zgarishi" . www.ruppweb.org . Olingan 2016-10-19 .^ "Muvofiqlashtiruvchi tizim o'zgarishi" . Ruppweb.org . Olingan 2016-10-19 .^ Sussman, J .; Xolbruk, S .; Cherch, G.; Kim, S (1977). "Cheklangan va cheklangan parametrlardan foydalangan holda makromolekulyar tuzilmalar uchun struktura-omil eng kam kvadratlarni takomillashtirish tartibi". Acta Crystallogr. A . 33 (5): 800–804. Bibcode :1977AcCrA..33..800S . CiteSeerX 10.1.1.70.8631 doi :10.1107 / S0567739477001958 . ^ Rossmann, M.; Blow, D. (1962). "Kristalografik assimetrik birlik ichida kichik birliklarni aniqlash". Acta Crystallogr . 15 : 24–31. CiteSeerX 10.1.1.319.3019 doi :10.1107 / S0365110X62000067 . 
![{displaystyle {egin {hizalanmış} c_ {x} & = ccos (eta), c_ {y} & = c {frac {cos (alfa) -cos (gamma) cos (eta)} {sin (gamma)}} , c_ {z} ^ {2} & = c ^ {2} -c_ {x} ^ {2} -c_ {y} ^ {2} = c ^ {2} chap {1-cos ^ {2} (eta) - {frac {[cos (alfa) -cos (gamma) cos (eta)] ^ {2}} {sin ^ {2} (gamma)}} ight} .end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68f4b2b245ad322aa239174a1c015a928cde0a45)
![{displaystyle {egin {aligned} c_ {z} ^ {2} & = c ^ {2} {frac {sin ^ {2} (gamma) -sin ^ {2} (gamma) cos ^ {2} (eta) - [cos (alfa) -cos (gamma) cos (eta)] ^ {2}} {sin ^ {2} (gamma)}} & = {frac {c ^ {2}} {sin ^ {2} (gamma)}} chap {sin ^ {2} (gamma) -sin ^ {2} (gamma) cos ^ {2} (eta) - [cos (alfa) -cos (gamma) cos (eta)] ^ { 2} ight} oxiri {hizalanmış}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/935e638c35a037b78b7b2bd379351fb7cbaf8691)
![{displaystyle {egin {hizalanmış} va sin ^ {2} (gamma) -sin ^ {2} (gamma) cos ^ {2} (eta) - [cos (alfa) -cos (gamma) cos (eta)] ^ { 2} & = sin ^ {2} (gamma) -sin ^ {2} (gamma) cos ^ {2} (eta) -cos ^ {2} (alfa) -cos ^ {2} (gamma) cos ^ {2} (eta) + 2cos (alfa) cos (gamma) cos (eta) & = sin ^ {2} (gamma) -cos ^ {2} (alfa) -sin ^ {2} (gamma) cos ^ {2} (eta) -cos ^ {2} (gamma) cos ^ {2} (eta) + 2cos (alfa) cos (eta) cos (gamma) & = sin ^ {2} (gamma) -cos ^ {2} (alfa) - [sin ^ {2} (gamma) + cos ^ {2} (gamma)] cos ^ {2} (eta) + 2cos (alfa) cos (eta) cos (gamma) & = sin ^ {2} (gamma) -cos ^ {2} (alfa) -cos ^ {2} (eta) + 2cos (alfa) cos (eta) cos (gamma) & = 1-cos ^ {2} ( alfa) -cos ^ {2} (eta) -cos ^ {2} (gamma) + 2cos (alfa) cos (eta) cos (gamma) .end {hizalanmış}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84639c5dd72fa894c1b432deb4a37048ec16689c)
![{displaystyle {egin {aligned} mathbf {sigma} _ {mathbf {a}, x} & = {b} _ {y} {c} _ {z} - {b} _ {z} {c} _ {y } = bsin (gamma) {frac {Omega} {absin (gamma)}} = {frac {Omega} {a}}, mathbf {sigma} _ {mathbf {a}, y} & = {b} _ { z} {c} _ {x} - {b} _ {x} {c} _ {z} = - bcos (gamma) {frac {Omega} {absin (gamma)}} = - {frac {Omega cos ( gamma)} {asin (gamma)}}, mathbf {sigma} _ {mathbf {a}, z} & = {b} _ {x} {c} _ {y} - {b} _ {y} { c} _ {x} = bcos (gamma) c {frac {cos (alfa) -cos (eta) cos (gamma)} {sin (gamma)}} - bsin (gamma) ccos (eta) & = bcleft { cos (gamma) {frac {cos (alfa) -cos (eta) cos (gamma)} {sin (gamma)}} - sin (gamma) cos (eta) ight} & = {frac {bc} {sin ( gamma)}} chap {cos (gamma) [cos (alfa) -cos (eta) cos (gamma)] - sin ^ {2} (gamma) cos (eta) ight} & = {frac {bc} {sin (gamma)}} chap {cos (gamma) cos (alfa) -cos (eta) cos ^ {2} (gamma) -sin ^ {2} (gamma) cos (eta) ight} & = {frac {bc } {sin (gamma)}} chap {cos (alfa) cos (gamma) -cos (eta) ight}. end {hizalanmış}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eadbb81a1bf8feb74e2a08963ebe9b631ef39bcf)
![{displaystyle left [{egin {matrix} u v wend {matrix}} ight] = left [{egin {matrix} {frac {1} {a}} & - {frac {cos (gamma)} {asin ( gamma)}} & bc {frac {cos (alfa) cos (gamma) -cos (eta)} {Omega sin (gamma)}} 0 & {frac {1} {bsin (gamma)}} & ac {frac {cos ( eta) cos (gamma) -cos (alfa)} {Omega sin (gamma)}} 0 & 0 & {frac {absin (gamma)} {Omega}} end {matrix}} ight] left [{egin {matrix} x y zend {matrix}} ight]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccbf2aa3407bf0ee76088a2a068ba2fcd067d7cd)
![{displaystyle left [{egin {matrix} x y zend {matrix}} ight] = left [{egin {matrix} a & bcos (gamma) & ccos (eta) 0 & bsin (gamma) & c {frac {cos (alfa) -) cos (eta) cos (gamma)} {sin (gamma)}} 0 & 0 & {frac {Omega} {absin (gamma)}} end {matrix}} ight] left [{egin {matrix} u v wend { matritsa}} yopiq].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e44ae0b4bf199429eb0763ac90ea96872c37690)
