Fission — chain reaction, critical mass: Question bank

2.3.23 · D5 · Physics › Modern Physics › Fission — chain reaction, critical mass

Symbols jo tumhe dikhenge (sab parent note mein build kiye gaye hain): $k$ = multiplication factor (is generation ke neutrons ÷ pichhli generation ke neutrons); $ν$ = neutrons emitted per fission ( $\approx 2.5$ ); $R$ = fissile lump ki radius; $M_{c}$ = critical mass; $τ$ = generation time; $Q$ = energy released per fission.

True ya false — justify karo

True/false batao AUR reason bhi do. Sirf truth value se kuch nahi milega.

Ek akele

^{235} U

nucleus ko fission hone se pehle poori critical mass ki zaroorat hoti hai.

False — ek nucleus tab fission karta hai jab ek neutron absorb hota hai, kisi bhi mass par. Critical mass ek self-sustaining chain ke liye threshold hai, yaani generations mein neutron survival ke liye — ek split ke liye nahi.

Critical mass se neeche bilkul bhi fission events nahi hote.

False — individual fissions hote rehte hain; woh bas cascade nahi karte.

k < 1

ke saath har generation mein pichle se kam neutrons bante hain, toh chain fizzle ho jaati hai, haalaanki single splits hote rehte hain.

Bilkul

k = 1

par neutrons ki sankhya time mein constant rehti hai.

True —

N_{n} = N_{0} k^{n} = N_{0}

sabhi

n

ke liye jab

k = 1

ho, toh population steady rehti hai. Yeh ek reactor ki normal operating state hai.

Ek subcritical lump ki mass ko double karne se woh hamesha supercritical ho jaati hai.

Saamanya roop mein False — iska fayda hota hai (leakage fraction

\propto 1/ R

ghatti hai jaise

R

badhti hai), lekin kya

k

1 cross karega, yeh shape, density, purity aur reflectors par depend karta hai. Do guni mass ki ek flat sheet phir bhi itna leak kar sakti hai ki subcritical rahe.

Fission se energy isliye niklti hai kyunki woh strong nuclear bonds todta hai.

False — energy isliye niklti hai kyunki fragments zyada tightly bound hote hain (zyada BE/A,

\approx 8.5

7.6

MeV). Loosely bound se tightly bound hone par difference release hota hai; tum binding paate ho, kharach nahi karte. Dekho Binding Energy per Nucleon Curve.

Sphere optimal shape hai kyunki usme sabse zyada fuel aa sakta hai.

False — ek sphere ka unit volume par sabse kam surface area hota hai, jo leakage minimize karta hai (jo surface

R^{2}

ke scale par hai production

R^{3}

ke against). Yeh critical mass minimize karta hai, fuel maximize nahi.

Fissile material ko compress karne se uski critical mass ghatti hai.

True — zyada density

ρ

se same mass ke liye

R

chhota hota hai aur nuclei ek doosre ke itne paas aa jaate hain ki neutrons jaldi kisi ko hit karte hain;

M_{c} \propto 1/ ρ^{2}

. Yahi physics hai implosion assembly ke peeche.

Ek moderator chain reaction ko speed up karta hai neutrons ko zyada energy dekar.

False — ek moderator neutrons ko elastic collisions se slow karta hai. Slow (thermal) neutrons ka

^{235} U

mein fission cross-section kaafi bada hota hai, toh unhe slow karne se agli fission ki probability badhti hai. Dekho Neutron Cross-section.

Delayed neutrons ek reactor ko control karna mushkil banate hain.

False — woh effective generation time

τ

\sim 1 0^{- 8} s

se barhaakar

\sim 1 0^{- 1} s

tak kheench dete hain, jisse operators ko microseconds nahi balki seconds milte hain respond karne ke liye. Yahi cheez reactors ko steer karna possible banati hai.

Kyunki

Q = Δ m c^{2}

hai, fragments milaakar original nucleus se zyada wazni hote hain.

False — woh kam wazni hote hain; woh khoya hua mass

Δ m = m_{before} - m_{after}

released energy ban gaya. Dekho Mass-Energy Equivalence E=mc^2.

Error dhundho

Har statement mein ek kami hai. Use naam do aur theek karo.

"Fusion aur fission dono BE/A curve par iron ki taraf chadh kar energy release karte hain."

Sahi hai — aur internalize karne layak. Dono nuclei ko iron peak ki taraf le jaate hain: fission heavy nuclei ko daayein se neeche laata hai, fusion light nuclei ko baayein se upar. Dekho Nuclear Fusion.

k = 1.01

ke saath, neutron count har generation mein apni original value ka 1% badhta hai."

Error: growth multiplicative hai, additive nahi. Har generation

1.01

se multiply hoti hai, toh woh current count ka 1% badhti hai — yeh exponential hai (

N_{n} = N_{0} (1.01)^{n}

), linear nahi.

"Control rods neutrons ko core mein reflect karke kaam karte hain."

Error: control rods neutrons ko absorb karte hain (cadmium/boron) unhe remove karne aur

k = 1

rakhne ke liye. Neutrons ko wapas reflect karna ek reflector/tamper ka kaam hai, jo critical mass ko ghataata hai — bilkul ulta maqsad.

"Fuel ko 3%

^{235} U

tak enrich karne se ek reactor bomb ki tarah blast karne mein capable ho jaata hai."

Error: 3% enrichment prompt-supercritical nahi ho sakta. Bomb ko

> 90%

^{235} U

aur fast neutrons chahiye; reactor-grade fuel physically ek nuclear explosion mein assemble nahi ho sakta.

"Kyunki

ν \approx 2.5

neutrons har fission se niklte hain,

k

hamesha lagbhag

2.5

hota hai."

Error: $ν$ production hai; $k$ net survival hai. Un

2.5

neutrons mein se zyaadatar leak ho jaate hain, non-productively absorb ho jaate hain, ya fission cause karne mein fail ho jaate hain.

k

sirf unhe count karta hai jo agali generation trigger karte hain — ek working reactor mein

1

ke aaspaas.

"Leakage aur non-fission absorption dono surface area ke saath scale karte hain, toh dono chhote lumps mein dominate karte hain."

Error: absorption volume ( $R^{3}$ ) ke saath scale karta hai, production ki tarah. Sirf leakage surface (

R^{2}

) ke saath scale karti hai. Wahi surface/volume mismatch,

\propto 1/ R

, isiliye chhote lumps subcritical ho jaate hain.

Why questions

Mechanism explain karo, sirf fact nahi.

Critical mass ek finite number ke roop mein kyun exist karta hai, na ki "koi bhi itna bada amount trivially kaam karega"?

Kyunki leakage se neutron loss fraction

1/ R

ki tarah ghatti hai: sirf ek specific radius par production

\propto R^{3}

exactly losses ke saath balance hoti hai,

k = 1

deta hai. Woh balance point ek sachcha geometric threshold hai.

Ek neutron reflector (tamper) critical mass ko kyun ghataata hai?

Yeh escape karne wale neutrons ko core mein wapas bounce karta hai, leakage loss ghataata hai. Kam leakage ka matlab hai production chhoti radius par win karta hai, toh

k = 1

reach karne ke liye kam material chahiye.

Reactors jaanbujhkar

k

1

par kyun rakhte hain,

1

se comfortably upar nahi?

k > 1

ka matlab power ki exponential growth hai — runaway.

k = 1

rakhna constant, controllable output deta hai; upar/neeche chhoti deliberate excursions se operators power smoothly badha ya ghata sakte hain.

Generation time

τ

reactor safety ke liye itni crucial kyun hai, haalaanki yeh nahi badalti ki reaction sustain hogi ya nahi?

τ

change ki timescale set karta hai,

N (t) = N_{0} k^{t / τ}

. Chhota

τ

kisi bhi

k > 1

ko microseconds mein blast kar deta hai; bada

τ

(delayed neutrons ki wajah se) usi

k

ko ek slow, catchable drift mein badal deta hai.

Sirf kaafi natural uranium pile karke bomb kyun nahi bana sakte?

Natural uranium mein sirf

\sim 0.7%

^{235} U

hota hai; dominant

^{238} U

un energies par fission kiye bina neutrons absorb kar leta hai, toh

k

1 se neeche rehta hai. Fissile isotope ko concentrate karne ke liye enrich karna zaroori hai.

^{235} U

slow neutrons prefer kyun karta hai jabki fission emit karta hai fast neutrons?

Fission cross-section — woh effective target size jo ek neutron dekhta hai — slow (thermal) neutrons ke liye kaafi badi hoti hai. Fast neutrons itni tezi se guzar jaate hain ki efficiently capture ho sakein, toh pehle unhe moderate karna padta hai. Dekho Neutron Cross-section.

Edge cases

Boundary aur degenerate scenarios jo topic quietly assume karta hai.

R \to \infty

par

k

ka kya hoga (effectively infinite slab of fuel)?

Leakage fraction

\propto 1/ R \to 0

, toh koi neutron escape nahi karta aur

k

apni maximum "infinite-medium" value

k_{\infty}

ke aaspaas pahunch jaata hai. Yeh ideal upper limit hai; har real, finite lump ka

k < k_{\infty}

hota hai.

R \to 0

par

k

ka kya hoga (fissile dust ka ek tukda)?

Surface-to-volume

\to \infty

, toh almost har neutron fission cause karne se pehle leak ho jaata hai aur

k \to 0

. Isiliye fine powder ya thin foils material ki parwah kiye bina chain reactions se safe hote hain.

Agar

k

exactly

1

hai lekin ek fluctuation kuch neutrons hata de, toh kya reactor turant mar jaata hai?

Nahi —

k = 1

par population sirf marginally stable hai; ek dip se thodi giraahat hoti hai aur woh drift karta hai, lekin koi exponential collapse nahi hota. Delayed neutrons se bada

τ

operators ko

k

wapas 1 par nudge karne ka poora time deta hai.

Degenerate case

ν = 0

(fission par zero neutrons emitted) consider karo. Kya chain reaction possible hai?

Nahi — zero neutrons produce hone se, har fission ek dead end hai aur

k = 0

hamesha. Ek chain ke liye

ν > 1

chahiye taki saari losses ke baad kam se kam ek neutron survive kare continue karne ke liye.

Melting/compression limit par, kya akela density badhana kisi bhi subcritical lump ko critical bana sakta hai?

Sirf tab tak jab tak physical limits (solid-density, phir nuclear-matter density) reach ho jaayein, aur sirf tab agar kaafi total mass present ho;

M_{c} \propto 1/ ρ^{2}

ghatti hai lekin bahut chhoti mass phir bhi chain sustain nahi kar sakti chahe kitna bhi compress karo.

Recall Quick self-test

Woh single sentence jo yahan zyaadatar traps resolve karta hai ::: Critical mass neutron survival across generations ( $k$ ) ke baare mein hai, jo geometry aur leakage se govern hoti hai — na ki is baare mein ki kya ek nucleus split ho sakta hai ya kitni total energy store hai.