Using the notation T i = T ( α i ) {\displaystyle T_{i}=T(\alpha _{i})} for compactness, the DWBA equation can be written as:
I d ( q z ) = | T i T f F ( + Q z 1 ) + T i R f F ( − Q z 2 ) + R i T f F ( + Q z 2 ) + R i R f F ( − Q z 1 ) | 2 = T i 2 T f 2 | F ( + Q z 1 ) | 2 + T i 2 T f R f F ( + Q z 1 ) F ( − Q z 2 ) + T i R i T f R f F ( + Q z 1 ) F ( + Q z 2 ) + T i R i T f R f F ( + Q z 1 ) F ( − Q z 1 ) + T i 2 R f 2 | F ( − Q z 2 ) | 2 + T i 2 T f R f F ( + Q z 1 ) F ( − Q z 2 ) + T i R i T f R f F ( + Q z 2 ) F ( − Q z 2 ) + T i R i R f 2 F ( − Q z 1 ) F ( − Q z 2 ) + R i 2 T f 2 | F ( + Q z 2 ) | 2 + T i R i T f 2 F ( + Q z 1 ) F ( + Q z 2 ) + T i R i T f R f F ( + Q z 2 ) F ( − Q z 2 ) + R i 2 T f R f F ( − Q z 1 ) F ( + Q z 2 ) + R i 2 R f 2 | F ( − Q z 1 ) | 2 + T i R i T f R f F ( + Q z 1 ) F ( − Q z 1 ) + T i R i R f 2 F ( − Q z 1 ) F ( − Q z 2 ) + R i 2 T f R f F ( − Q z 1 ) F ( + Q z 2 ) {\displaystyle {\begin{aligned}I_{d}(q_{z})&=|T_{i}T_{f}F(+Q_{z1})+T_{i}R_{f}F(-Q_{z2})+R_{i}T_{f}F(+Q_{z2})+R_{i}R_{f}F(-Q_{z1})|^{2}\\&{\begin{aligned}=\,\,&T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}&&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&&&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(+Q_{z2})+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}&&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&&&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})\\&+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}&&+T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})\\&&&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\&+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}&&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&&&+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\\end{aligned}}\\\end{aligned}}} The DWBA equation is: I d ( q z ) = T i 2 T f 2 | F ( + Q z 1 ) | 2 + T i 2 R f 2 | F ( − Q z 2 ) | 2 + R i 2 T f 2 | F ( + Q z 2 ) | 2 + R i 2 R f 2 | F ( − Q z 1 ) | 2 {\displaystyle I_{d}(q_{z})=T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}}
Taking advantage of the fact that | F ( Q ) | 2 = I ( Q ) {\displaystyle |F(Q)|^{2}=I(Q)} and I ( + Q ) = I ( − Q ) {\displaystyle I(+Q)=I(-Q)} , we can rearrange to:
I d ( q z ) = T i 2 T f 2 | F ( + Q z 1 ) | 2 + T i 2 R f 2 | F ( − Q z 2 ) | 2 + R i 2 T f 2 | F ( + Q z 2 ) | 2 + R i 2 R f 2 | F ( − Q z 1 ) | 2 + T i 2 T f R f F ( + Q z 1 ) F ( − Q z 2 ) + T i R i T f R f F ( + Q z 1 ) F ( + Q z 2 ) + T i R i T f R f F ( + Q z 1 ) F ( − Q z 1 ) + T i 2 T f R f F ( + Q z 1 ) F ( − Q z 2 ) + T i R i T f R f F ( + Q z 2 ) F ( − Q z 2 ) + T i R i R f 2 F ( − Q z 1 ) F ( − Q z 2 ) + T i R i T f 2 F ( + Q z 1 ) F ( + Q z 2 ) + T i R i T f R f F ( + Q z 2 ) F ( − Q z 2 ) + R i 2 T f R f F ( − Q z 1 ) F ( + Q z 2 ) + T i R i T f R f F ( + Q z 1 ) F ( − Q z 1 ) + T i R i R f 2 F ( − Q z 1 ) F ( − Q z 2 ) + R i 2 T f R f F ( − Q z 1 ) F ( + Q z 2 ) = T i 2 T f 2 | F ( + Q z 1 ) | 2 + T i 2 R f 2 | F ( − Q z 2 ) | 2 + R i 2 T f 2 | F ( + Q z 2 ) | 2 + R i 2 R f 2 | F ( − Q z 1 ) | 2 + 2 × T i 2 T f R f F ( + Q z 1 ) F ( − Q z 2 ) + T i R i T f R f [ 2 F ( + Q z 1 ) F ( − Q z 1 ) + F ( + Q z 1 ) F ( + Q z 2 ) + 2 F ( + Q z 2 ) F ( − Q z 2 ) ] + 2 × T i R i R f 2 F ( − Q z 1 ) F ( − Q z 2 ) + T i R i T f 2 F ( + Q z 1 ) F ( + Q z 2 ) + 2 × R i 2 T f R f F ( − Q z 1 ) F ( + Q z 2 ) {\displaystyle {\begin{aligned}I_{d}(q_{z})=\,\,&T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}\\&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(+Q_{z2})+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\=\,\,&T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}\\&+2\times T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}[2F(+Q_{z1})F(-Q_{z1})+F(+Q_{z1})F(+Q_{z2})+2F(+Q_{z2})F(-Q_{z2})]\\&+2\times T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})\\&+2\times R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\\end{aligned}}}
for compactness, the DWBA equation can be written as:
The DWBA equation is:
and 
, we can rearrange to:
![{\displaystyle {\begin{aligned}I_{d}(q_{z})=\,\,&T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}\\&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(+Q_{z2})+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\=\,\,&T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}\\&+2\times T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}[2F(+Q_{z1})F(-Q_{z1})+F(+Q_{z1})F(+Q_{z2})+2F(+Q_{z2})F(-Q_{z2})]\\&+2\times T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})\\&+2\times R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc7ce8eb06c1820f96123b1e63533c795cdc783f)
