Gamov nazariyasi - radioaktiv yadrolarning
-parchalanishini kvant tasavvurlar asosida tushuntirib beruvchi nazariya.
-parchalanish deganda, ogʻir yadrolarning oʻzidan geliy atomining yadrosini chiqarib, boshqa yadroga aylanishi tushuniladi.
-parchalanish uchun energetik shart quyidagi koʻrinishga ega:
![{\displaystyle M(A,\ Z)>M(A-4,\ Z-2)+M_{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/731309d4dc476c130c18e5264e677b096984c863)
Parchalanishda hosil boʻlgan energiya quyidagi ifoda orqali topiladi:
![{\displaystyle Q_{\alpha }=(M(A,\ Z)-M(A-4,\ Z-2)-M_{\alpha })c^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58a5b9424c2ffdf6749b4179b1f2a30e2493edc2)
Energiya va impulsning saqlanish qonunlaridan foydalanib,
-zarra yadroni tark etayotganida ega boʻladigan kinetik energiyasini aniqlash mumkin:
![{\displaystyle T_{\alpha }={\dfrac {M(A-4,\ Z-2)}{m_{\alpha }+M(A-4,\ Z-2)}}Q_{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b555f700f92b9a2795a056740cd7df449fad29e)
-parchalanishda hosil boʻlgan
-zarra kinetik energiyasi 2-8 MeV atrofida boʻladi,
-zarraning yadroga bogʻlanish energiyasi esa 28.3 MeV ga teng. U holda
-zarra yadroni qanday tark etadi, degan tabiiy savol tugʻiladi. Bunga G. A. Gamov kvant tasavvurlar asosida ishlab chiqqan nazariyasi orqali javob topish mumkin.
Gamov nazariyasi quyidagi prinsiplarga asoslanadi:
-zarra yadro ichida tayyor holda mavjud boʻladi.
-zarra tinimsiz harakatda boʻladi va potensial toʻsiq orqali yadro ichida tutib turiladi.
-zarra ushbu potensial toʻsiq orqali oʻtishi mumkin.
Vaqt birligidagi parchalanish ehtimolligi
ni quyidagi ifoda orqali aniqlash mumkin:
![{\displaystyle \lambda =nT}](https://wikimedia.org/api/rest_v1/media/math/render/svg/158a184d8562bd78fbe68c12542bc643893d44ee)
—
-zarraning yadro ichida potensial toʻsiq bilan toʻqnashishlari soni,
—
-zarraning toʻsiq orqali oʻtish ehtimolligi.
Aytaylik, ixtiyoriy vaqt momentida yadro ichida faqat bitta
-zarra mavjud boʻlsin va u yadro diametri boʻylab oldinga va orqaga harakatlanayotgan boʻlsin.
![{\displaystyle \nu ={\dfrac {v}{2R_{0}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbfb7e65cdcdaffb77d88b3cdb0eb03c198d29b4)
—
-zarraning yadroni tark etayotgandagi tezligi.
Kengligi
boʻlgan toʻsiqdan zarraning oʻtish ehtimolligi:
![{\displaystyle T=e^{-2k_{2}L},\ \ \ \ \ \ \ \ \ (1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a78dcc45f483e1f9c0c5c1276a5f374408646a7)
![{\displaystyle k_{2}={\dfrac {\sqrt {2m(U-E)}}{\hbar }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d449cf2c6a9ac56e8bb0409cd5492499bbdaed88)
(1) tenglama toʻgʻri burchakli potensial toʻsiqni ifodalaydi,
-zarra yadro ichida toʻsiq bilan koʻp marta toʻqnashadi.
![{\displaystyle \ln T=-2k_{2}L,\ \ \ \ \ \ \ \ (2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac927ff6578701d615940978b8c73163293f999)
![{\displaystyle \ln T=-2\int \limits _{0}^{L}k_{2}(r)\,dr=-2\int \limits _{R_{0}}^{R}k_{2}(r)\,dr,\ \ \ \ \ \ \ \ \ (3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c772f224c7b3e40b7158b6e691a902844dd7db2e)
— yadro radiusi,
—
boʻlganda, yadrogacha masofa
-zarraning
masofadagi elektr potensial energiyasi:
![{\displaystyle U(r)={\dfrac {2Ze^{2}}{4\pi \varepsilon _{0}r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f578d90d5fc03de65e2805641ae944af739f2912)
U holda,
![{\displaystyle k_{2}={\dfrac {\sqrt {2m(U-E)}}{\hbar }}=\left({\dfrac {2m}{\hbar ^{2}}}\right)^{1/2}\cdot \left({\dfrac {2Ze^{2}}{4\pi \varepsilon _{0}r}}-E\right)^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03a8b0b40849e0e719c243458c580f5fa7ac4a42)
boʻlganda
boʻlgani uchun,
![{\displaystyle E={\dfrac {2Ze^{2}}{4\pi \varepsilon _{0}R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18e852c93ed26158730146c31ca6d49de65364ee)
ni quyidagicha yozish mumkin:
![{\displaystyle k_{2}=\left({\dfrac {2m}{\hbar ^{2}}}\right)^{1/2}\cdot \left({\dfrac {R}{r}}-1\right)^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f8549db6d998039c48be20d5a43354315fabd7)
![{\displaystyle \ln T=-2\int \limits _{R_{0}}^{R}k_{2}(r)\,dr=-2\left({\dfrac {2mE}{\hbar ^{2}}}\right)^{1/2}\cdot \left({\dfrac {R}{r}}-1\right)^{1/2}\,dr\ \ \ \ \ \ \ \ (4)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2511625c956eb7c5d6b40433545d31c8b07c4e74)
![{\displaystyle \left[\int \left({\dfrac {R}{r}}-1\right)^{1/2}\,dr=-2R\int \sin ^{2}\theta \,d\theta \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/150713c7a41a4e7a15cb7e2c4ef71599a790db67)
![{\displaystyle \ln T=-2\left({\dfrac {2mE}{\hbar ^{2}}}\right)^{1/2}R\left[\cos ^{-1}\left({\dfrac {R_{0}}{R}}\right)^{1/2}-\left({\dfrac {R_{0}}{R}}\right)^{1/2}\left(1-{\dfrac {R_{0}}{R}}\right)^{1/2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73f23de79ef1fb2a9ad2735513b2a074cc292998)
Potensial toʻsiq yetarlicha keng boʻlgani uchun,
hamda
![{\displaystyle \cos \left({\dfrac {\pi }{2}}-\theta \right)=\sin \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4bad6df1291fe870d90891cae6930972d933cd)
![{\displaystyle \sin \left({\dfrac {R_{0}}{R}}\right)^{1/2}\approx \left({\dfrac {R_{0}}{R}}\right)^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d7842499ec59578a98bfa55c7e0d5a25a6e898)
![{\displaystyle \cos ^{-1}\left({\dfrac {R_{0}}{R}}\right)^{1/2}\approx {\dfrac {\pi }{2}}-\left({\dfrac {R_{0}}{R}}\right)^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32344c36d0c7a809ae6878a6c5a827aeb71c27aa)
![{\displaystyle 1-\left({\dfrac {R_{0}}{R}}\right)^{1/2}\approx 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/276face64cf3e37e54b50e3d1dafa986ebfb33b3)
![{\displaystyle \ln T=-2\left({\dfrac {2mE}{\hbar ^{2}}}\right)^{1/2}r\left[{\dfrac {\pi }{2}}-2\left({\dfrac {R_{0}}{R}}\right)^{1/2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec69c691af17aa7b6f40c59e17ddee9dc03e841)
![{\displaystyle \left[R={\dfrac {2Ze^{2}}{4\pi \varepsilon _{0}E}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3edc6e43c3f9d15a417e1760c461809e8adb4562)
Bundan kelib chiqadiki,
![{\displaystyle \ln T={\dfrac {4e}{\hbar }}\left({\dfrac {m}{\pi \varepsilon _{0}}}\right)^{1/2}Z^{1/2}R_{0}^{1/2}-{\dfrac {e^{2}}{\hbar \varepsilon _{0}}}\left({\dfrac {m}{2}}\right)^{1/2}ZE^{-1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71dcebd899b98992c83372abd6d2a6a432c1892b)
Tenglamadagi doimiylarning qiymatlarini oʻrniga qoʻyib hisoblasak:
![{\displaystyle \ln T=2,97Z^{1/2}R_{0}^{1/2}-3,95ZE^{-1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56b5819247a95281451c3af0c356519dd9b5ab5b)
— energiya,
— yadro radiusi,
— hosilaviy yadroning tartib raqami.
![{\displaystyle \log _{10}A=\left(\log _{10}e\right)(\ln A)=0,4343\ln A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/704565f128d39ee13822ff16ddadbdaeeb3309e5)
![{\displaystyle \log _{10}T=1,29Z^{1/2}R_{0}^{1/2}-1,72ZE^{-1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7626ce102281300f1406de2ca395a814d9cac33d)
Taʼrifga binoan, parchalanish doimiysi
![{\displaystyle \lambda =\nu T={\dfrac {v}{2R_{0}}}T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/385c23503f69c180525278a4e62d1729a9399dec)
Tenglamaning ikkala tomonidan
olamiz va
bilan almashtiramiz:
![{\displaystyle \log _{10}\lambda =\log _{10}\left({\dfrac {v}{2R_{0}}}\right)+1,92Z^{1/2}R_{0}^{1/2}-1,72ZE^{-1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35feecd0fdd30e70a3b2b135826f23f5dd81798b)
Hosil boʻlgan bu formulaga Geyger-Nettol qonuni deyiladi. Ushbu qonun
-parchalanish energiyasi va radioaktiv yadrolarning yarim yemirilish davrlari orasidagi bogʻliqlikni ifodalaydi va katta amaliy ahamiyatga ega.