**Gamma-nurlarining kesishishi** - gamma nurlarining materiya bilan o'zaro ta'sir qilish ehtimolli o'lchovidir. Gamma nurlarining o'zaro ta'sirining umumiy kesimi bir nechta mustaqil jarayonlardan iborat: fotoelektrik effekt, Komptonning tarqalishi, yadro maydonida elektron-pozitron juftligi va elektron maydonida elektron-pozitron juftligi hosil bo'lishi (uchlik ishlab chiqarish). Yuqorida sanab o'tilgan bitta jarayon uchun kesma umumiy gamma nurlari kesimining bir qismidir.

Fotoyadroviy yutilish, Tomson yoki Rayleigh (kogerent) tarqalishi kabi boshqa effektlar energiyaning gamma-nurlari diapazoniga ahamiyatsiz hissa qo'shgani uchun olib tashlanishi mumkin.

Gamma-nurlarining materiya bilan o'zaro ta'siri bilan bog'liq bo'lgan barcha qayd etilgan effektlarning ko'ndalang kesimlari ( barn / atom) uchun batafsil tenglamalar quyida keltirilgan.

  

Ushbu maqola Mirzo Ulug'bek nomidagi O'zbekiston Milliy Universiteti Fizika fakulteti talabasi Do'stmuhamedova Shahzoda tomonidan wikita'lim loyihasi doirasida ingliz tilidan tarjima qilindi.

Fotoelektrik effekt ko‘ndalang kesimi
-------------------------------------

Ushbu hodisa gamma fotonning atom tuzilishida joylashgan elektron bilan o'zaro ta'sir qilish holatini tavsiflaydi. Bu elektronning atomdan chiqarilishiga olib keladi. Fotoelektrik effekt 50 keV dan past energiyaga ega rentgen va gamma-nurli fotonlar uchun dominant energiya uzatish mexanizmidir, lekin yuqori energiyalarda u kamroq ahamiyatga ega, ammo baribir e'tiborga olish kerak.

Odatda fotoeffektning kesmasini  ning soddalashtirilgan tenglamasi bilan taxmin qilish mumkin.

{\displaystyle \sigma \_{ph}={\frac {16}{2}}{\sqrt {2}}\pi r\_{e}^{2}\alpha ^{4}{\frac {Z^{5}}{k^{3.5}}}\approx 3\cdot 10^{12}{\frac {Z^{4}}{E\_{\gamma }^{3.5}}}}

Bu yerda *k = E g / E e*, va bu erda *E g = hn -* eV da berilgan foton energiyasi va *E e = m e c 2* ≈ 5,11∙10 5 eV - elektronning tinch massa energiyasi, *Z* - atom raqami absorber elementining *a = e 2 /(ħc)* ≈ 1/137 nozik struktur konstantasi, *r e 2 = e 4 /E e 2* ≈ 0,07941 b klassik elektron radiusining kvadrati.

Yuqori aniqlik uchun Sauter tenglamasi  juda mos keladi:

{\displaystyle \sigma \_{ph}={\frac {3}{2}}\phi \_{0}\alpha ^{4}{\biggl (}Z{\frac {E\_{e}}{E\_{\gamma }}}{\biggr )}^{5}(\gamma ^{2}-1)^{3/2}{\Biggl [}{\frac {4}{3}}+{\frac {\gamma (\gamma -2)}{\gamma +1}}{\Biggl (}1-{\frac {1}{2\gamma (\gamma ^{2}-1)^{1/2}}}\ln {\frac {\gamma +(\gamma ^{2}-1)^{1/2}}{\gamma -(\gamma ^{2}-1)^{1/2}}}{\Biggr )}{\Biggr ]}}

bu erda barcha kirish parametrlari quyidagi jadvalda keltirilgan.

{\displaystyle \gamma ={\frac {E\_{\gamma }-E\_{B}+E\_{e}}{E\_{e}}}}

va *E B* - elektronning bog'lanish energiyasi va ϕ *0* - Tomson kesimi (ϕ *0* = 8 {\displaystyle \pi }*e 4 /(3E e 2 )* ≈ 0,66526 barn ).

Yuqori energiyalar (>0,5 MeV ) uchun fotoelektr effektining kesimi juda kichik, chunki boshqa effektlar (ayniqsa, Komptonning tarqalishi ) ustunlik qiladi. Biroq, yuqori energiya diapazonida fotoeffekt kesimini aniq hisoblash uchun Sauter tenglamasi Pratt-Skofild tenglamasi bilan almashtirilishi kerak 

{\displaystyle \sigma \_{ph}=Z^{5}{\Biggl (}\sum \_{n=1}^{4}{\frac {a\_{n}+b\_{n}Z}{1+c\_{n}Z}}k^{-p\_{n}}{\Biggr )}}

|  |  |  |  |  |
| --- | --- | --- | --- | --- |
| n | a n | b n | c n | p n |
| 1 | 1,6268∙10 −9 | -2,683∙10 −12 | 4,173∙10 −2 | 1 |
| 2 | 1,5274∙10 −9 | -5,110∙10 −13 | 1,027∙10 −2 | 2 |
| 3 | 1,1330∙10 −9 | -2,177∙10 −12 | 2,013∙10 −2 | 3.5 |
| 4 | -9,12∙10 −11 | 0 | 0 | 4 |

Komptonning sochilish kesimi
----------------------------

  

Komptonning tarqalishi (yoki Kompton effekti) - bu o'zaro ta'sir bo'lib, unda tushgan gamma foton atom elektroni bilan o'zaro ta'sirlanib, uning chiqishi va asl fotonning kamroq energiya bilan tarqalishiga olib keladi. Foton energiyasining ortishi bilan Komptonning tarqalishi ehtimoli kamayadi. Komptonning tarqalishi 100 keV dan 10 MeV gacha bo'lgan oraliq energiya diapazonidagi gamma nurlari uchun asosiy yutilish mexanizmi hisoblanadi.

{\displaystyle \sigma \_{C}=Z2\pi r\_{e}^{2}{\Biggl \{}{\frac {1+k}{k^{2}}}{\Biggl [}{\frac {2(1+k)}{1+2k}}-{\frac {\ln {(1+2k)}}{k}}{\Biggr ]}+{\frac {\ln {(1+2k)}}{2k}}-{\frac {1+3k}{(1+2k)^{2}}}{\Biggr \}}}

100 keV dan yuqori energiyalar uchun (k>0,2). Pastroq energiyalar uchun esa bu tenglama quyidagi bilan almashtiriladi: 

{\displaystyle \sigma \_{C}=Z{\frac {8}{3}}\pi r\_{e}^{2}{\frac {1}{(1+2k)^{2}}}{\biggl (}1+2k+{\frac {6}{5}}k^{2}-{\frac {1}{2}}k^{3}+{\frac {2}{7}}k^{4}-{\frac {6}{35}}k^{5}+{\frac {8}{105}}k^{6}+{\frac {4}{105}}k^{7}{\biggr )}}

bu absorberning atom raqamiga mutanosib *Z* .

Kompton effekti bilan bog'liq bo'lgan qo'shimcha kesma faqat energiya uzatish koeffitsienti uchun hisoblanishi mumkin - foton energiyasining elektron tomonidan yutilishi : 

{\displaystyle \sigma \_{C,abs}=Z2\pi r\_{e}^{2}{\biggl [}{\frac {2(1+k)^{2}}{k^{2}(1+2k)}}-{\frac {1+3k}{(1+2k)^{2}}}-{\frac {(1+k)(2k^{2}-2k-1)}{k^{2}(1+2k)^{2}}}-{\frac {4k^{2}}{3(1+2k)^{3}}}-{\Bigl (}{\frac {1+k}{k^{3}}}-{\frac {1}{2k}}+{\frac {1}{2k^{3}}}{\Bigr )}\ln {(1+2k)}{\biggr ]}}

ko'pincha radiatsiyaviy himoya hisob-kitoblarida qo'llaniladi.

Juft ishlab chiqarish (yadro maydonida) kesma
---------------------------------------------

Yadroning elektr maydoni bilan o'zaro ta'sir qilish natijasida tushayotgan fotonning energiyasi elektron - pozitron (e - e + ) juftligi massasiga aylanadi. Juft ishlab chiqarish effekti uchun kesma odatda Maximon tenglamasi bilan tavsiflanadi: 

{\displaystyle \sigma \_{pair}=Z^{2}\alpha r\_{e}^{2}{\frac {2\pi }{3}}{\biggl (}{\frac {k-2}{k}}{\biggr )}^{3}{\biggl (}1+{\frac {1}{2}}\rho +{\frac {23}{40}}\rho ^{2}+{\frac {11}{60}}\rho ^{3}+{\frac {29}{960}}\rho ^{4}{\biggr )}} past energiya uchun ( *k* <4),

Bu yerda z(3)≈1,2020569 Riemann zeta funksiyasi. Juft ishlab chiqarish effekti uchun energiya chegarasi *k* =2 ( pozitron va elektronning dam olish massasi energiyasi ).

{\displaystyle \rho ={\frac {2k-4}{2+k+2{\sqrt {2k}}}}} .

Biroq, yuqori energiyalar uchun ( *k* >4) Maximon tenglamasi quyidagi shaklga ega

{\displaystyle \sigma \_{pair}=Z^{2}\alpha r\_{e}^{2}{\Biggl \{}{\frac {28}{9}}\ln {2k}-{\frac {218}{27}}+({\frac {2}{k}})^{2}{\biggl [}6\ln {2k}-{\frac {7}{2}}+{\frac {2}{3}}\ln ^{3}{2k}-\ln ^{2}{2k}-{\frac {1}{3}}\pi ^{2}\ln {2k}+2\zeta (3)+{\frac {\pi ^{2}}{6}}{\biggr ]}-({\frac {2}{k}})^{4}{\biggl [}{\frac {3}{16}}\ln {2k}+{\frac {1}{8}}{\biggr ]}-({\frac {2}{k}})^{6}{\biggl [}{\frac {29}{9\cdot 256}}\ln {2k}-{\frac {77}{27\cdot 512}}{\biggr ]}{\Biggr \}}}

Triplet ishlab chiqarish kesimi
-------------------------------

Pozitron va elektron boshqa elektronlar sohasida hosil bo'ladigan triplet ishlab chiqarish effekti, *k* = 4 bo'sag'asi bilan juft ishlab chiqarishga o'xshaydi. Biroq, bu ta'sir yadro sohasida juft ishlab chiqarishga qaraganda ancha kam. Uchlik kesmaning eng mashhur shakli Borsellino-Gizzetti tenglamasi sifatida tuzilgan 

{\displaystyle {\begin{aligned}\sigma \_{trip}=Z\alpha r\_{e}^{2}{\Biggl [}{\frac {28}{9}}\ln {2k}-{\frac {218}{27}}+{\frac {1}{k}}{\biggl (}-{\frac {4}{3}}\ln ^{3}{2k}+3\ln ^{2}{2k}-{\frac {60+16a}{3}}\ln {2k}&+{\frac {123+12a+16b}{3}}{\biggr )}+{\frac {1}{k^{2}}}{\biggl (}{\frac {8}{3}}\ln ^{3}{2k}-4\ln ^{2}{2k}+{\frac {51+32a}{3}}\ln {2k}-{\frac {123+32a+64b}{6}}{\biggr )}+{\frac {1}{k^{3}}}{\biggl (}\ln ^{2}{2k}-{\frac {53}{9}}\ln {2k}-{\frac {2915-288a}{216}}{\biggr )}\\&\qquad +{\frac {1}{k^{4}}}{\biggl (}-{\frac {49}{18}}\ln {2k}-{\frac {115}{432}}{\biggr )}+{\frac {1}{k^{5}}}{\biggl (}-{\frac {77}{36}}\ln {2k}-{\frac {10831}{8640}}{\biggr )}+{\frac {1}{k^{6}}}{\biggl (}-{\frac {641}{300}}\ln {2k}-{\frac {64573}{36000}}{\biggr )}+{\frac {1}{k^{7}}}{\biggl (}-{\frac {4423}{1800}}\ln {2k}-{\frac {394979}{216000}}{\biggr )}{\Biggl ]}\end{aligned}}}

  

Ushbu maqola Mirzo Ulug'bek nomidagi O'zbekisto Milliy Universiteti Fizika fakulteti talabasi Do'stmuhamedova Shahzoda tomonidan wikita'lim loyihasi doirasida ingliz tilidan tarjima qilindi.

{\displaystyle \sigma \_{trip,H}=Z\alpha r\_{e}^{2}[5.6+20.4(k-4)-10.9(k-4)^{2}-3.6(k-4)^{3}+7.4(k-4)^{4}]10^{-3}(k-4)^{2}}
{\displaystyle \sigma \_{trip,H}=Z\alpha r\_{e}^{2}(0.582814-0.29842k+0.04354k^{2}-0.0012977k^{3})}
{\displaystyle \sigma \_{trip,H}=Z\alpha r\_{e}^{2}{\biggl (}{\frac {3.1247-1.3394k+0.14612k^{2}}{1+0.4648k+0.016683k^{2}}}{\biggr )}}
*k* >14 uchun Haug Borsellino tenglamasining qisqaroq shaklini qo'llashni taklif qildi: 

{\displaystyle \sigma \_{trip,H}=Z\alpha r\_{e}^{2}{\Biggl [}{\frac {28}{9}}\ln {2k}-{\frac {218}{27}}+{\frac {1}{k}}{\biggl (}-{\frac {4}{3}}\ln ^{3}{2k}+3.863\ln ^{2}{2k}-11\ln {2k}+27.9{\biggr )}{\Biggr ]}}

Jami kesma
----------

Bir atomga to'g'ri keladigan umumiy tasavvurlar har bir ta'sirning oddiy yig'indisi sifatida taqdim etilishi mumkin: 

{\displaystyle \sigma \_{total}=\sigma \_{ph}+\sigma \_{C}+\sigma \_{pair}+\sigma \_{trip}}

Keyinchalik, Beer-Lambert-Bouger qonunidan foydalanib, *N* atom zichligidagi absorber bilan fotonlarning o'zaro ta'siri uchun chiziqli zaiflashuv koeffitsientini hisoblash mumkin:

{\displaystyle \mu =\sigma \_{total}N}

yoki massa zaiflashuv koeffitsienti :

{\displaystyle \mu \_{d}={\frac {\mu }{\rho }}={\frac {\sigma \_{total}}{uA}}}

Bu erda *r* - massa zichligi, *u* - atom massa birligi, *A* - absorberning atom massasi .

Bu to'g'ridan-to'g'ri amalda qo'llanilishi mumkin, masalan, radiatsiyadan himoya qilishda.

Har bir aniq hodisaning kesimini analitik hisoblash ancha qiyin, chunki tegishli tenglamalar uzoq va murakkab. Shunday qilib, gamma o'zaro ta'sirining umumiy kesimi Fornalski tomonidan tuzilgan bitta fenomenologik tenglamada taqdim  mumkin, buning o'rniga foydalanish mumkin:

{\displaystyle \sigma \_{total}(k,Z)=\sum \_{i=0}^{6}{\biggl [}(\ln {k})^{i}\sum \_{j=0}^{4}a\_{i,j}Z^{j}{\biggr ]}}

a i,j parametrlari quyidagi jadvalda keltirilgan. Bu formula turli energiyalar (1 MeV dan 10 GeV gacha, yaʼni 2< *k* <20.000) va absorberning atom raqamlari ( *Z* =1 dan 100 gacha) uchun gamma nurlarining materiya bilan oʻzaro taʼsirining umumiy koʻndalang kesimining taxminiy koʻrsatkichidir.

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| **a i, j** | ***i=0*** | ***i=1*** | ***i=2*** | ***i=3*** | ***i=4*** | ***i=5*** | ***i=6*** |
| ***j=0*** | 0.0830899 | -0,08717743 | 0.02610534 | -2,74655∙10 −3 | 4,39504∙10 −5 | 9.05605∙10 −6 | -3,97621∙10 −7 |
| ***j=1*** | 0.265283 | -0,10167009 | 0,00701793 | 2,371288∙10 −3 | -5,020251∙10 −4 | 3,6531∙10 −5 | -9,46044∙10 −7 |
| ***j=2*** | 2.18838∙10 −3 | -2,914205∙10 −3 | 1,26639∙10 −3 | -7,6598∙10 −5 | -1,58882∙10 −5 | 2.18716∙10 −6 | -7,49728∙10 −8 |
| ***j=3*** | -4,48746∙10 −5 | 4,75329∙10 −5 | -1,43471∙10 −5 | 1,19661∙10 −6 | 5,7891∙10 −8 | -1,2617∙10 −8 | 4,633∙10 −10 |
| ***j=4*** | 6,29882∙10 −7 | -6,72311∙10 −7 | 2,61963∙10 −7 | -5,1862∙10 −8 | 5,692∙10 −9 | -3,29∙10 −10 | 7,7∙10 −12 |

Pastroq energiya mintaqasi uchun (<1 MeV) Fornalski tenglamasi turli elementlarning katta funktsiyalari o'zgaruvchanligi tufayli murakkabroq. Shuning uchun o'zgartirilgan tenglama 

{\displaystyle \sigma \_{total}(E,Z)=\exp \sum \_{i=0}^{6}{\biggl [}(\ln {E})^{i}\sum \_{j=0}^{6}a\_{i,j}Z^{j}{\biggr ]}}

150 keV dan 10 MeV gacha bo'lgan foton energiyasi uchun yaxshi yaqinlikdir, bu erda foton energiyasi *E* MeVda berilgan va a **i, j** parametrlari quyidagi jadvalda ancha yaxshi aniqlik bilan berilgan. Analogik ravishda tenglama 1 dan 100 gacha bo'lgan barcha *Z* uchun amal qiladi.

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| **a i, j** | ***j=0*** | ***j=1*** | ***j=2*** | ***j=3*** | ***j=4*** | ***j=5*** | ***j=6*** |
| ***i=0*** | -1,539137959563277 | 0,3722271606115605 | -0,018918894979230043 | 5.304673816064956∙10 −4 | -7,901251450214221∙10 −6 | 5,9124040925689876∙10 −8 | -1,7450439521037788∙10 −10 |
| ***i=1*** | -0,49013771295901015 | 7.366301806437177∙10 −4 | -8,898417420107425∙10 −5 | 3,294237085781055∙10 −6 | -8,450746169984143∙10 −8 | 7,640266479340313∙10 −10 | -2,282367050913894∙10 −12 |
| ***i=2*** | -0,05705460622256227 | 0,001957234615764126 | -6,187107799669643∙10 −5 | 2.1901234933548505∙10 −6 | 1,9412437622425253∙10 −8 | -5,851534943255455∙10 −10 | 2,7073481839614158∙10 −12 |
| ***i=3*** | 0,001395861376531693 | -7,137867469026608∙10 −4 | 2,462958782088413∙10 −4 | -9,660290609660555∙10 −6 | 1,295493742164346∙10 −7 | -6,538025860945927∙10 −10 | 8,763097742806648∙10 −13 |
| ***i=4*** | 5.105805426257604∙10 −5 | 0,0011420827759804927 | -8,177273886356552∙10 −5 | 4,564725445290536∙10 −6 | -9,707786695822055∙10 −8 | 8,351662725636947∙10 −10 | -2,545941852995417∙10 −12 |
| ***i=5*** | -5,416099245465933∙10 −4 | 5,65398317844477∙10 −4 | -5,294089702089374∙10 −5 | 5.437298837558547∙10 −7 | 1,4824427385312707∙10 −8 | -2,8079293400520423∙10 −10 | 1,247192025425616∙10 −12 |
| ***i=6*** | 3,6322794450615036∙10 −4 | -2,186723664102979∙10 −4 | 1,739236692381265∙10 −5 | -3,7341071277534563∙10 −7 | 1,1585158108088033∙10 −9 | 3,1805366711255584∙10 −11 | -2,0806866173605604∙10 −13 |

XCOM kesmalarning ma'lumotlar bazasi
------------------------------------

AQSh Milliy Standartlar va Texnologiyalar Instituti turli energiyadagi turli materiallar bilan rentgen va gamma-nurlarining o'zaro ta'sirining kesma qiymatlarining to'liq va batafsil ma'lumotlar bazasini onlayn  nashr etdi. XCOM deb ataladigan ma'lumotlar bazasi, shuningdek, amaliy ilovalar uchun foydali bo'lgan chiziqli va ommaviy zaiflashuv koeffitsientlarini o'z ichiga oladi.

Shuningdek qarang
-----------------

Havolalar
---------

* XCOM ma'lumotlar bazasi

Ma'lumotnomalar
---------------

1. ↑  Davisson, C.M.*. Interaction of gamma-radiation with matter. In: Alpha-, Beta- and Gamma-ray Spectroscopy: Volume 1. Edited by Kai Siegbahn.*. Amsterdam: North-Holland Publishing Company, 1965.
2. 1 2  Fornalski, Krzysztof W (2018-03-22). "Simple empirical correction functions to cross sections of the photoelectric effect, Compton scattering, pair and triplet production for carbon radiation shields for intermediate and high photon energies". *Journal of Physics Communications* **2** (3): 035038. doi:10.1088/2399-6528/aab408. ISSN 2399-6528.
3. ↑  Davisson, Charlotte Meaker; Evans, Robley D. (1952-04-01). "Gamma-Ray Absorption Coefficients" (en). *Reviews of Modern Physics* **24** (2): 79–107. doi:10.1103/RevModPhys.24.79. ISSN 0034-6861. .
4. ↑  Pratt, R. H. (1960-02-15). "Atomic Photoelectric Effect at High Energies" (en). *Physical Review* **117** (4): 1017–1028. doi:10.1103/PhysRev.117.1017. ISSN 0031-899X. .
5. ↑  Scofield J.H. 1973. *Theoretical photoionization cross sections from 1 to 1500 keV*. Technical Report no. UCRL—51326, California Univ. Livermore. Lawrence Livermore Lab.
6. 1 2 3 4  Hubbell, J. H.; Gimm, H. A.; O/verbo/, I. (1980). "Pair, Triplet, and Total Atomic Cross Sections (and Mass Attenuation Coefficients) for 1 MeV‐100 GeV Photons in Elements Z =1 to 100" (en). *Journal of Physical and Chemical Reference Data* **9** (4): 1023–1148. doi:10.1063/1.555629. ISSN 0047-2689. .
7. ↑  Attix F.H. 1986. *Introduction to radiological physics and radiation dosimetry*. John Wiley & Sons
8. ↑  Maximon L.C. 1968. *Simple analytic expressions for the total Born approximation cross section for pair production in a Coulomb field*. J. Res. Nat. Bur. Stand. vol. 72B (Math. Sci.), no. 1, pp. 79-88
9. ↑  Haug E. 1981. *Simple analytic expressions for the total cross section for γ-e pair production*. Zeitschrift für Naturforschung, vol. 36a, pp. 413-414
10. ↑  Haug E. 1975. *Bremsstrahlung and pair production in the field of free electrons*. Zeitschrift für Naturforschung, vol. 30a, pp. 1099-1113
11. 1 2  Fornalski, Krzysztof Wojciech (2021-01-01). "Total Cross Section Phenomenological Formulas for X-Ray and Gamma Radiation Interaction With Matter for Different Energies and Absorber Types" (en). *Journal of Nuclear Engineering and Radiation Science* **7** (1). doi:10.1115/1.4045806. ISSN 2332-8983. .
12. ↑  Berger, M.J. Hubbell, J.H. Seltzer, S.M. Chang, J. Coursey, J.S. Sukumar, R. Zucker, D.S. and Olsen, K. 2010. XCOM: Photon Cross Section Database (version 1.5), National Institute of Standards and Technology, Gaithersburg, MD, USA, DOI: 10.18434/T48G6X