Nuclear structure — protons, neutrons, nuclear forces: Question bank

2.3.18 · D5 · Physics › Modern Physics › Nuclear structure — protons, neutrons, nuclear forces

Shuru karne se pehle, is page pe use hone wale har symbol ki definition yahan di gayi hai — neeche kuch bhi yaad rakhne pe depend nahi karta.

Is poore page pe do pictures kaam karti hain. Jawab dene se pehle inhe dekh lo — questions baar baar inhi pe point karte hain.

Figure 1 — nucleon–nucleon potential. Nuclear physics ka sabse useful graph yahi hai: ye dikhata hai do nucleons ke beech ki energy jab unhe door se slide karke pass laate hain. Neeche ki dip ka matlab "yahan rehna chahenge" (attraction); upar ki wall ka matlab "andar mat aao" (repulsion).

Figure — Nuclear structure — protons, neutrons, nuclear forces

Figure 2 — saturation vs. packing. Left cartoon galat picture hai (sabka sabse attraction); right sahi picture hai (har ball sirf unse velcro hoti hai jo usse touch kar rahe hain). Ye ek difference hi wajah hai kyun binding energy $A^{2}$ ki jagah $A$ ki tarah badhti hai.

True or false — justify

True or false: Ek metre door do protons strong nuclear force se ek dusre ko attract karte hain.

False. Figure 1 mein dip

\sim 3

fm tak flat hokar zero ho jaati hai; ek metre

1 0^{14}

fm hai, isliye sirf Coulomb repulsion kaam karta hai jo unhe door dhakelta hai.

True or false: Ek neutron aur ek proton strong force se ek dusre ko utni hi strongly attract karte hain jitna do protons karte hain.

True. Nuclear force charge independent hai —

p

–

p

n

–

n

, aur

p

–

n

strong attractions almost barabar hain, kyunki ye force electric charge ko bilkul ignore karta hai.

True or false: Kyunki helium nucleus bound hai, isliye wo apne 2 protons + 2 neutrons se zyada weighs karta hai jo alag alag the.

False. Binding energy release karti hai, aur wo lost energy mass le jaati hai (

Δ m = \sum m_{parts} - M_{nucleus} > 0

). Bound nucleus halka hota hai (ye sirf

E = m c^{2}

ulta hai).

True or false: Bada nucleus (bada

A

, yani zyada balls) chhote se zyada dense hota hai.

False. Mass (

\propto A

) aur volume (

\propto R^{3} = R_{0}^{3} A

) dono

A

mein linearly badhte hain, isliye ye cancel ho jaate hain: density har nucleus ke liye same hoti hai (

\approx 2.3 \times 1 0^{17}

kg/m³ — estimate neeche kiya gaya hai).

True or false: Strong nuclear force hamesha attractive hoti hai.

False. Figure 1 mein curve dip karta hai (attraction,

\sim 1

–

2

fm) lekin ek wall mein rise bhi karta hai ==below

\sim 0.5

fm== (ek repulsive "hard core") jo nucleus ko ek point pe collapse hone se rokta hai.

True or false: Nucleons ki ginti double karne se roughly total binding energy bhi double ho jaati hai.

True. Kyunki force saturated hai (Figure 2, right) — har nucleon sirf nearest neighbours se bond karta hai —

E_{B} \propto A

A^{2}

nahi.

True or false: Neutrons heavy nucleus ko kam stable banate hain kyunki wo mass add karte hain.

False. Neutrons strong-force attraction add karte hain bina kisi extra Coulomb repulsion ke, isliye extra neutrons heavy nuclei ko bound rehne mein help karte hain. Isliye stability line

N > Z

ki taraf bend karti hai.

True or false: Radius formula

R = R_{0} A^{1/3}

assume karta hai ki nucleons roughly constant density pe packed hain.

True. Constant density ka matlab volume

\propto

nucleons ki ginti (

V \propto A

), aur

R \propto V^{1/3}

seedha cube root deta hai;

R_{0}

ek ball ke share ka size hai.

True or false: Ek element ke isotopes protons (

Z

) ki ginti mein differ karte hain.

False. ==Isotopes

Z

share karte hain== (same proton count, same element) aur

N

mein differ karte hain, isliye

A

mein bhi. Isobars woh hote hain jo

Z

mein differ karte hue

A

share karte hain.

Spot the error

"Protons sab ek dusre ko repel karte hain, isliye bahut saare protons wala nucleus simply exist hi nahi kar sakta." — galti dhundho.

Is reasoning mein doosri, zyada powerful force bhool gayi. Coulomb repulsion real hai, lekin

\sim 2

fm ke andar strong force (Figure 1 mein deep dip,

\sim 100 \times

bada) jeet jaati hai. Repulsion gayab nahi hoti — bas haar jaati hai.

"Strong force har nucleon ko har dusre nucleon ki taraf khichti hai, jaise gravity karta hai." — galti dhundho.

Ye Figure 2 ka left cartoon hai. Agar ye sach hota, binding energy pairs ki ginti

\sim A^{2}

ki tarah badhti. Experiment dikhata hai

E_{B} \propto A

, jo right cartoon force karta hai: saturation, har nucleon sirf apne immediate neighbours se bond karta hai.

"Heavy nuclei ke liye density badhni chahiye kyunki tum zyada nucleons thoos rahe ho." — galti dhundho.

Tum zyada nucleons thoos rahe ho aur volume unhe fit karne ke liye badhta hai (

V \propto A

). Mass aur volume saath scale karte hain, isliye unka ratio (density) constant rehta hai.

"Yukawa ka pion mass wala hai, isliye strong force infinitely door tak pahunchta hai." — galti dhundho.

Ulta hai. Ek massive mediator sirf thodi der ke liye "borrow" kiya ja sakta hai (Heisenberg uncertainty principle,

Δ E Δ t \sim ℏ

), isliye ye sirf

r \sim ℏ/ (m_{π} c) \approx 1.4

fm tak travel karta hai — ek finite range, jo Figure 1 mein dip ki width hai. Massless mediator (photon) hai jo infinite range deta hai.

"Binding energy find karne ke liye, nucleus ki mass mein se parts ki mass ghata do:

E_{B} = (M_{nucleus} - \sum m_{parts}) c^{2}

." — galti dhundho.

Subtraction ulti hai. Nucleus halka hota hai, isliye

Δ m = \sum m_{parts} - M_{nucleus}

(ek positive number), aur

E_{B} = Δ m c^{2} > 0

_{1}^{3} H

aur

_{2}^{3} He

isotopes hain kyunki unka mass number same hai." — galti dhundho.

Same

A

lekin alag

Z

unhe isobars banata hai, isotopes nahi. Isotopes ka

Z

same hona chahiye (same element ho).

"Nuclear force do protons ke beech zyada strong hoti hai jitni ek proton aur neutron ke beech, kyunki protons electricity bhi feel karte hain." — galti dhundho.

Electric force ek alag force hai, nuclear force ka part nahi. Nuclear attraction charge-independent hai aur

p

–

p

n

–

n

p

–

n

ke liye almost equal hai; protons ki extra Coulomb repulsion actually net

p

–

p

situation ko kamzor banati hai.

Why questions

Do positive protons ek dusre ke paas kaise glued reh sakte hain, jabki like charges repel karte hain?

≲ 2

fm pe strong nuclear force (Figure 1 mein deep dip,

\sim 100 \times

EM se zyada strong) Coulomb repulsion ko overwhelm kar deti hai; protons itne close hain ki repulsion jeet hi nahi sakti.

Strong force ki range short kyun hai, jabki gravity aur electromagnetism hamesha ke liye pahunchte hain?

Iska mediator (pion) mass wala hai; massive particle banane ke liye energy borrow karna sirf thodi der ke liye ho sakta hai, uski travel

r \sim ℏ/ (m_{π} c) \approx 1.4

fm tak limit kar deta hai. Massless mediators (graviton, photon) aisi koi time limit impose nahi karte, isliye wo forces infinite-ranged hain.

Heavy stable nuclei mein protons se zyada neutrons kyun hote hain?

Coulomb repulsion

Z^{2}

ki tarah badhti hai (har proton har doosre proton ko push karta hai — ye "everyone-to-everyone" left cartoon hai), lekin saturated strong attraction sirf

\sim A

ki tarah badhti hai. Extra neutrons attraction add karte hain bina repulsion add kiye, balance restore karte hue.

Total binding energy roughly

A^{2}

ki bajaye

A

ke proportional kyun hai?

Kyunki saturation hai (Figure 2, right): har nucleon ek fixed chote number of neighbours se bond karta hai, isliye bonds ki ginti nucleons ki ginti (

A

) ki tarah badhti hai, na ki sabhi possible pairs (

\sim A^{2}

) ki tarah.

Nucleus apne free nucleons ke sum se halka kyun weighs karta hai?

Usse assemble karne se binding energy release hoti hai, aur

E = m c^{2}

ke zariye woh jaati energy mass le jaati hai. Missing mass

Δ m = E_{B} / c^{2}

ko mass defect kehte hain.

Strong force bahut short range pe repulsive kyun honi chahiye, sirf attractive nahi?

Agar ye purely attractive hoti, nucleons bina kisi lower size limit ke ek dusre pe collapse ho jaate. Figure 1 mein

\sim 0.5

fm ke neeche repulsive wall ek minimum spacing set karta hai, nuclei ko roughly constant density deta hai.

Edge cases

Do nucleons ke beech strong force exactly

10

fm pe separated kya hogi?

Essentially zero — Figure 1 ki dip ke flat tail (

2

–

3

fm) se kaafi aage. Sirf (kamzor, long-range) Coulomb push charged nucleons ke beech bachti hai.

Ek akele free neutron ke liye (koi doosra nucleon aas paas nahi), kya strong-force binding feel hogi?

Kuch nahi — strong force ko kuch fm ke andar ek neighbour chahiye. Akele neutron ke paas koi velcro karne wala nahi; consistently, free neutrons unstable hote hain aur

\sim 10

-minute half-life ke saath beta-decay karte hain.

Kya constant-density result

R = R_{0} A^{1/3}

A = 1

(ek single proton) ke liye hold karta hai?

Scaling law many-nucleon nuclei ke liye ek smooth approximation hai;

A = 1

ke saath koi neighbours nahi hain aur koi "packing" nahi, isliye akele proton ko

R_{0}

-radius sphere treat karna sirf ek rough limiting statement hai, precise fit nahi.

Jab do protons

3

fm se andar

0.3

fm tak laaye jaate hain, to net force ka sign kaise change hota hai describe karo.

Figure 1 right-to-left follow karo:

2

–

3

fm ke paas attraction switch on hoti hai aur curve dip karta hai (net attractive);

\sim 0.5

fm ke andar wall control le leti hai aur net force strongly repulsive ho jaati hai.

"Density

A

se independent hai" argument bahut light nucleus jaise

^{2} H

ke liye kya hoga?

Ye kamzor ho jaata hai: sirf do nucleons ke saath ek badi "surface" hai aur koi interior nahi, isliye packed-marbles picture (interior + saturation) ek poor approximation hai — constant-density law asymptotic hai, bade

A

ke liye best.

Agar pion massless hota, to nuclear force ki range ka kya hota?

Ye infinite-ranged ho jaati (jaise electromagnetism), kyunki

r \sim ℏ/ (m_{π} c) \to \infty

jab

m_{π} \to 0

— aur nuclear physics reality jaisi bilkul nahi lagti.

$2.3 \times 1 0^{17}$ kg/m³ kahan se aata hai? (ek 4-line estimate)

Recall Har trap ka ek-line summary

Strong force = short-range (Fig 1 dip), charge-independent, saturated (Fig 2 right), attractive-then-repulsive; binding mass hatati hai; density constant hai; extra neutrons $Z^{2}$ Coulomb growth ke against stability khareedte hain.