Question

The power density (Watts per kg) produced by a given nuclear reaction in a gas of density ρ between species a and b with mass fractions X, charge numbers Z and atomic mass numbers A varies as 1/6 1/3] where Q and So are constants for the given reaction, m is the reduced mass, and Eg is the Gamow energy: th (MKS units). (a) Show that the dependence of e at a temperature T near To can be approximated as a power law x T®, where 1/3 m(姦)"? (Hint: find In e using the expression above, then calculate d(n e)/d(ln T)lr,. This is the first order coefficient in a Taylor expansion of In ε as a function of In T (a pure power-law relation between € and T would obey ln € const. + β ln T) (b) Evaluate β at T。= 1.5 x 107 K, for (i) the reaction p +p → d+e+ + ve, and (ii) the reaction p + 12C → My + γ.

Answer 1

E infinity rho X_a X_b Q s_0/A_a A_b squareroot m E_c1^1/6/T^2/3 xp [-3(E_g/4 KT)^1/3] E_g = pi^2 m^bar z_1^2 z_2^2 e^4/2 elementof_0^2 h^2 (MKS) (a) Taking ln in both side of equation 1 ln (epsilon) = ln(w) - 2/3 ln(T) -3[E_g/4KT)^1/3 (where w = rho X_a X_b Q S_0/A_a A_b squareroot m^bar E_n^1/6 = constant) Now evaluating d ln(epsilon)/d (ln T) of above expression we get d ln(epsilon)/d ln (T) Vertical-bar _T = T_0 = (E_g/4KT_0)^1/3 - 2/3 {-3(E_g/4KT)^1/3 iscongruentto -3(E_g/4KT)^1/3 (-1/3 ln(T)) from above we can say that ln(epsilon) as a function of ln T on ln(epsilon) = a_0 + f^'(T_0) ln (T) + f^''(T_0)(ln (T))^2 + ... (Taylor series) That f^'(T_0) = (E_g/4KT_0)^1/3 - 2/3 ln(epsilon) = a_0 + (E_g/4KT_0)^1/3 - 2/3 ln(T) ln (epsilon) = a_0 + beta ln(T) In other way, we can rewrite, epsilon = aT^beta on epsilon alpha T^B where beta = (E_g/4KT_0)^1/3 = 2/3 (b) T_0 = 1.5 times 10^7 k E_g = pi^2 m^bar z_1^2 z_2^2 e^4/2 elementof_0^2