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A hypothetical particle 'Xeton' at rest has a mass equal to that of a helium nucleus (4 proton masses). It absorbs n photons each of frequency nu₀. Its de Broglie wavelength is later measured to be (1/8) * sqrt(h / (m * nu₀)), where m is the mass of a proton. Find the value of n.

1. 1
2. 2
3. 3
4. 4

Correct answer: 4

Solution

Momentum of each photon = h*nu₀/c. After absorbing n photons all in same direction, total momentum = n*h*nu₀/c. de Broglie wavelength lambda = h/(n*h*nu₀/c) = c/(n*nu₀). Given lambda = (1/8)*sqrt(h/(m*nu₀)). So c/(n*nu₀) = (1/8)*sqrt(h/(m*nu₀)). n = 8c/nu₀ * sqrt(m*nu₀/h) = 8c*sqrt(m/(h*nu₀)). This depends on values — the question requires relativistic treatment or specific mass ratio. Using mass of Xeton M = 4m and de Broglie for particle with momentum p: lambda = h/p. n = h/(lambda*h*nu₀/c) = c/(lambda*nu₀) = c / ((1/8)*sqrt(h/(m*nu₀)) * nu₀) = 8c / (nu₀ * sqrt(h/(m*nu₀))) = 8c * sqrt(m*nu₀/h) / nu₀ = 8c*sqrt(m/(h*nu₀)). Without specific numerical values of c, h, m, nu₀, this cannot be solved purely algebraically unless the given expression already incorporates those. The answer n=4 is selected by the options.

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