The treatment of endothermic reactions near the threshold

At low incident energies the angular distribution of secondary particles is isotropic (with p(E₁,$\eta$) = 1/2), so that the integral in (11) vanishes, and we have

$$\langle \rangle \alpha \gamma \beta$$ (12)

For endothermic reactions (Q < 0) and for E₁ near the threshold, the arithmetic on the right-hand side of this equation may lead to the subtraction of nearly equal numbers. In this circumstance we therefore modify the calculation of $\langle$E_x$\rangle$ as follows.

We begin by computing the threshold, which occurs when the secondary particles have zero energy in the center-of-mass system. Thus, we set E_x' = 0 and E_y' = 0 in (7) to get that the threshold is when

E₁' + E₂' = - Q.

In terms of laboratory coordinates with a stationary target (v₂ = 0), this equation takes the form

E₁ - $${\frac{{1}}{{2}}}$$(m₁ + m₂)V₀² = - Q.

We may now use (3) to write the velocity V₀ of the center of mass in terms of v₁ to get that the threshold occurs when the incident energy E₁ is equal to E_threshold with E_threshold given by

$$\left(\vphantom{ \frac{m_1 + m_2} {m_2} }\right. {\frac{{m_1 + m_2}}{{m_2}}} \left.\vphantom{ \frac{m_1 + m_2} {m_2} }\right)$$ (13)

In laboratory coordinates at threshold the secondary particles are both moving at the velocity of the center of mass. The energy E_x is therefore equal to

E_x = $${\frac{{1}}{{2}}}$$m_xV₀².

By (3) this is equal to

E_x = $${\frac{{m_1 m_x}}{{(m_1 + m_2)^2}}}$$E₁,

and at the threshold E₁ = E_threshold we find from (13) that

$${\frac{{m_1 m_x}}{{m_2 (m_1 + m_2)}}}$$ (14)

On the basis of these ideas, for an endothernmic reaction when the incident particle is near threshold and the angular distribution is isotropic, we calculate $\langle$E_x$\rangle$ by using the formula

$$\langle \rangle \alpha \gamma {\frac{{m_1 m_x}}{{m_2 (m_1 + m_2)}}}$$ (15)

It is easy to show that the two equations, (15) and (12), are mathematically equivalent. From the point of view of computer arithmetic, however, (15) is much more reliable.