content_image/338331.jpg content_image/338329.jpg content_image/338330.jpg content_image/338332.jpg content_image/338333.jpg content_image/338334.jpg
Figs. 3 and 4 can be expressed as 

to the accuracy mentioned above, where 

Relation of the B x i and R x i ( x = S$_{j}$ or n$_{j}$ ) are expressed as follows: 

<img src='content_image/338331.jpg'>

<img src='content_image/338329.jpg'>

<img src='content_image/338330.jpg'>

where σ $^{x$_{j}$}$( U, J $^{π}$) denotes the formation cross section of J π states in reaction x$_{j}$ including the factor of (2 J + 1). Equation (4) can be rewritten as 

<img src='content_image/338332.jpg'>

Then, the decay probability for reaction i in surrogate S$_{1}$ measurement, R S$_{1}$ i , can be written as 

<img src='content_image/338333.jpg'>

Since the 2 surrogate reactions S$_{1}$ and S$_{2}$ are assumed to be carried out for a pair of nuclei having similar mass and structure, the distribution of the formation cross section σ $^{S$_{1}$}$( U, J $^{π}$) will be fairly close to that of σ $^{S$_{2}$}$( U, J $^{π}$) if the nuclear structure and reaction mechanisms are similar to each other. We can write this similarity as σ $^{S$_{1}$}$( U, J $^{π}$) = ασ $^{S$_{2}$}$( U, J $^{π}$), where the symbol α denotes a constant such as the kinematical factor. If the dependence of α on J π is ignorable, Eq. (8) reads 

<img src='content_image/338334.jpg'>

by definition. This equation is equivalent to Eq. (2). Since we know R n$_{2}$ i ( U ), and we measure R S$_{1}$ i /R S$_{2}$ i in surrogate ratio method, we can obtain R n$_{1}$ i to the accuracy mentioned above. 

This gives an explanation of the reason why the surrogate ratio method works. 

