**APPENDIX D**

# Resonance Region Formulae

**D.1. The resolved resonance region**

The following resonance formalisms are given for a particular isotope in
the laboratory system, without Doppler broadening.

**D.1.1.** **Single-Level Breit-Wigner (SLBW):** LRU=1, LRF=1

**D.1.1.1.** **Elastic Scattering Cross Sections**[^1]

![](media/image1.wmf)where

![](media/image2.wmf)The hard‑sphere phase shifts φ*~l~*, the wave
number k, the primed resonance energy E´~r~, the neutron width Γ~nr~,
and through it the total width Γ~r~, are all functions of energy,
φ*~l~*(E), k(E), E~r~(E), ~nr~(E), and Γ~r~(E), but this dependence is
not shown explicitly. Also, each resonance parameter carries the
implicit quantum numbers *l* and J, determined by the appropriate
entries in the ENDF/B file. In case a given pair (*l*,J) is compatible
with two different values of the channel spin, s, the width is a sum
over the two partial channel spin widths. This allows one to omit an
explicit sum over channel spin when defining the cross sections.

**D.1.1.2.** **Radiative Capture Cross Section**

> ![](media/image3.wmf)where
>
> ![](media/image4.wmf)and ***Γ~γr~***, the radiative capture width, is
> constant in energy

**D.1.1.3. Fission Cross Section**

> ![](media/image5.wmf)where
>
> ![](media/image6.wmf)and ***Γ~fr~***, the fission width, is constant
> in energy.

**D.1.1.4.** **The Competitive Reaction Cross Section**.

The competitive reaction cross section, σ~n,x~(E), is given in terms of
analogous formulas involving Γ~xr~, the competitive width. By
convention, the cross section for the competitive reaction is given
entirely in File 3, and is not to be computed from the resonance
parameters. The reason for this is that the latter calculation can be
done correctly only for a single competitive channel, since the file can
define only one competitive width.

The statistical factor g~J~ = (2J+1)/2(2I+1) is obtained from the target
spin I and the resonance spin J given in File 2 as SPI and AJ,
respectively.

The sum on *l* extends over all *l*‑values for which resonance
parameters are supplied. There will be NLS terms in the sum. NLS is
given in File 2 for each isotope. In general, ENDF/B resonance files are
limited to *l*=0, 1, and 2, so that the potential‑scattering
contribution will be represented by hard‑sphere scattering up to the
energy where f‑wave (*l*=3) potential scattering starts. At that point,
the evaluator may have to supply File 3 scattering to simulate the
higher *l*‑values. He may also require a File 3 contribution at lower
energies to represent any differences between hard‑sphere scattering and
experiment.

The sum on J extends over all possible J‑values for a particular
*l*‑value. NR~J~ is the number of resonances for a given pair of *l* and
J values and may be zero. NRS is the total number of resonances for a
given *l*‑value and is given in File 2 for each *l*‑value.

> ![](media/image7.wmf)where
>
> ![](media/image8.wmf)and

![](media/image9.wmf)

Γ~nr~(\|E~r~\|) GN~r~ is the neutron width, for the r^th^ resonance for
a particular value of *l* and J, evaluated at the resonance energy E~r~.
For bound levels, the absolute value \|E~r~\| is used.

![](media/image10.wmf)Γ~r~ = Γ~nr~(E)+Γ~γr~+Γ~fr~+Γ~xr~ is the total
width, a function of energy through Γ~nr~ and Γ~xr~, since Γ~γr~ and
Γ~fr~ are constant with respect to energy. The \"competitive\" width,
Γ~xr~ is not entered explicitly in File 2. It is calculated from the
equation:

![](media/image11.wmf)The following quantities are given in File 2 for
each resonance:

E~r~ = ER, the resonance energy

J = AJ, the angular momentum (\"spin\") of the resonance state

I = SPI, the angular momentum (\"spin\") of the target nucleus

g~J~ = statistical factor (2J+1)/2(2I+1)

Γ~nr~(\|E~r~\|) = GN, the neutron width

Γ~γr~ = GG, the radiation width

Γ~fr~ = GF, the fission width and

> Γ~r~(\|E~r~\|) = GT, the total width evaluated at the resonance
> energy.

![](media/image12.wmf)Since the competitive width Γ~xr~, is not given,
Γ~r~ should be obtained from File 2 directly, and ***not*** by summing
partial widths.

For p‑, d‑ and higher *l*‑values, the primed resonance energy E´~r~ is
energy‑dependent:

![](media/image13.wmf)The fact that the shift is zero at ***each*** E~r~
is an artifact of the SLBW formalism, and implies a different R‑matrix
boundary condition for each resonance.

The neutron wave number in the center-of-mass system is given as:

> where

AWRI = ratio of the mass of a particular isotope to that of the neutron.

E = laboratory energy in eV.

The energy is written with absolute value signs so that the same formula
can be used for positive incident neutron energies and for negative
(bound state) resonance energies. (When inelastic scattering can occur,
resonances below the level threshold are at \"negative energy\" in the
inelastic channel.)

S*~l~* is the shift factor,

![](media/image14.wmf)

(the quantity ρ is defined below).

For higher *l*-values, S*~l~* is defined by Equation (2.9) in
Reference 1. In conventional R‑matrix theory, the shift factors are
defined differently for negative energies (Reference 1, Equations
2.11a‑c). In ENDF, the positive‑energy formulas are used, but the
absolute value of E is used in SLBW and MLBW. For the R-Matrix Limited
format, Section D.1.7, a flag indicates whether shifts are to be
calculated or assumed to be zero for each particle-pair.

![](media/image15.wmf)P*~l~* is the penetration factor,

For higher *l*-values, the P*~l~* are defined by Equation (2.9) in
Reference 1. In conventional R-matrix theory, the penetrabilities are
zero for negative energies. The theory uses the \"theoretical\"
definition of a reduced width, Γ(E)=2P*~l~*(E)γ^2^, where E is a channel
energy (center‑of‑mass), and it suffices to say that P*~l~*(E) = 0 if E
\< 0.

In ENDF, the \"experimental\" definition is used, Γ(E)=
Γ(\|E~r~\|)P*~l~*(E)/P*~l~*(\|E~r~\|), and it is necessary to make the
convention that a penetrability for a negative ***resonance*** energy is
evaluated at its absolute value. A negative kinetic energy can occur in
an exit channel if the reaction is exothermic, and in this case P*~l~*(E
\< 0) is zero.

φ~*l*~ is the (negative of a) hard‑sphere phase shift,

![](media/image16.wmf)

For higher *l*‑values, the φ*~l~* are defined by Equation 2.12 in
Reference 1. It is not necessary to evaluate a phase shift at negative
energies.

ρ and ρˆ are defined as k RADIUS, where RADIUS is defined as follows:

Let *a* = channel radius in units of 10^-12^ cm

= 0.123 AWRI^1/3^ + 0.08 [^2] (D.0)

AP = energy‑independent scattering radius, which determines the
low‑energy scattering cross section. It is given in File 2 following
SPI.

AP(E) = energy‑dependent scattering radius, given as a TAB1 card
preceding the \"SPI AP\.... NLS\...\" card.

## If NRO = 0 (AP energy‑independent)

NAPS = 0 ρ = k*a*; ρˆ = k AP

NAPS = 1 ρ = ρˆ = k AP

If NRO = 1 (AP energy‑dependent)

NAPS = 0 ρ = k*a*; ρˆ = k AP(E)

NAPS = 1 ρ = ρˆ = k AP(E)

NAPS= 2 ρ = k AP; ρˆ = k AP(E)

**D.1.2.** **Multilevel Breit-Wigner (MLBW):** LRU=1, LRF=2

The equations are the same as **SLBW**,[^3] except that a
resonance‑resonance interference term is included in the equation for
elastic scattering of *l*-wave neutrons, σ~n,n~*^l^*(E):

![](media/image17.wmf)

This form, which has ![](media/image18.wmf) energy‑dependent terms and
can involve a great deal of computer time, may be written in the
following form with only NR~J~ terms: (See Section 2.4.14)

![](media/image19.wmf)

where

![](media/image20.wmf)

![](media/image21.wmf)

For the user who does not require ψ‑ and χ‑broadening, the following
equations, which are mathematically identical to the MLBW equations,
require less computing time: (See Section 2.4.19)

![](media/image22.wmf)

![](media/image23.wmf)

![](media/image24.wmf)

**D.1.3 Reich-Moore (R-M):** LRU = 1, LRF = 3

This description of the ENDF Reich-Moore formalism differs from previous
versions by using notation in closer agreement with References 1 and 5.
The dependence of all quantities on channel spin has been made explicit,
to support a format extension which permits specifying the individual
channel-spin components of the neutron width.

Partial cross sections may be obtained from a collision matrix
![](media/image25.wmf), which connects entrance channels *a* with exit
channels *b*. In ENDF, the formalism is applied to neutron reactions, *a
= n*:

![](media/image26.wmf)

These partial cross sections are not observable, but must be summed over
the appropriate entrance and exit channels to yield observable cross
sections. The statistical factor *g~n~* is a result of prior averaging
over channels with different magnetic sub-states, since the ENDF
formulae apply to unpolarized particles.

In the Reich-Moore formalism, the only reactions requiring explicit
channel definitions are elastic scattering and fission; capture is
obtained by subtraction (although it is possible to obtain it directly
from the collision matrix elements). Neutron channels are labeled by
three quantum numbers, *l*, *s*, and *J*. In the ENDF format, *l* runs
from zero to NLS-1, the highest *l*-value that contributes to the cross
section in the energy range of interest. The channel spin *s* is the
vector sum of the target spin *I* and the neutron spin *i* (1/2 ), and
takes on the range of values \| *I* - 1/2\| to *I* + 1/2. The total
angular momentum *J* is the vector sum of *l* and *s*, and runs from \|
*l - s* \| to *l + s*. The fission channels do not correspond to
individual two-body fission product breakup, but to Bohr-channels in
deformation space, which is why two are adequate for describing many
neutron-induced fission cross sections. It is not necessary to specify
the quantum numbers associated with the two "ENDF-allowed" fission
channels, and they can simply be labeled f1 and f2.

If one sums over all incident channels *n* and exit channels *b*, and
invokes unitarity, the resulting total cross section can be expressed in
terms of the diagonal matrix elements as

![](media/image27.wmf)

The elastic cross section is obtained by summing the incident neutron
channels over all possible *lsJ* values and the exit neutron channels
over those quantities *lsJ* that have the same ranges as *lsJ*.
Conservation of total angular momentum requires that *J* = *J*; the ENDF
format imposes additional "conservation rules" *l* = *l* and *s* = *s*
which are actually just simplifying assumptions, with some basis in
theory and experiment. The six-fold summation then reduces to the
familiar form

![](media/image28.wmf)

The absorption (non-elastic) cross section is obtained by subtraction:

![](media/image29.wmf)

Fission is obtained from the collision matrix by summing Eq. (D.8) over
all incident *lsJ* values and over the two exit fission channels, *b* =
f1 and *b* = f2,

![](media/image30.wmf)

The Reich-Moore formalism is described in Reference 2. Here we repeat
the level-matrix form of the collision matrix as given in the earlier
versions of this manual:

![](media/image31.wmf)

where

![](media/image32.wmf)

Here ![](media/image33.wmf) is zero for fission, ![](media/image34.wmf)
(defined previously), and the summation is over those resonances *r*
which have partial widths in both of the channels *n* and *b*; *E~r~* is
the resonance energy; *Γ~γr~* is the "eliminated" radiation width;
*Γ~nr~* and *Γ~br~* are the partial widths for the *r^th^* resonance in
channels *n* and *b*.

If we define a matrix ρ by the equation

![](media/image35.wmf)

then the various cross sections take the following forms:

Total:

![](media/image36.wmf)

Elastic:

![](media/image37.wmf)

Absorption (fission plus capture):

![](media/image38.wmf)

Fission:

![](media/image39.wmf)

The phase shifts and penetrabilities are evaluated in terms of *a* and
AP as described earlier. The shift factor has been set equal to zero in
the above equations (*E~r~E~r~*); hence they are strictly correct only
for s-wave resonances. Originally, the ENDF Reich-Moore format was used
for low-energy resonances in fissile materials, which **are** s-waves.
However, it is believed that the "no-shift" formulae can be safely
applied to higher *l*-values also, since the difference in shape between
a shifted resonance and one that is not shifted at the same energy has
no practical significance.

Footnote 1 applies to the Reich-Moore formalism also. Until this
revision, the format did not permit the specification of channel spin;
therefore, if an evaluation includes *l* \> 0 resonances for *I* \> 0
nucleus, it was necessary for the processing codes to include the
potential-scattering contributions from the "missing" channels. (It is
adequate to arbitrarily assume that the supplied values are for the *s*
= *I* 1/2 channels, and to use the same potential-scattering radius in
the missing *I* +1/2 channels. See Sections 2.4.23 and 2.4.24.) Having
the ability to specify which channel spin is intended does not solve
this problem, unless the evaluator actually supplies resonances for both
channels. In cases where the data can be fit with all the resonances in
the same *s*-channel, the "other one" will still be absent from the ENDF
file, since the format stipulates nothing about avoiding missing
channels. This is why it is reasonable for the processing codes to run
over the triple *lsJ* loop, inserting potential scattering in every
channel, and resonances whenever they are supplied.

Note: When both positive and negative AJ values are given in the file,
negative AJ implies *s* = *I*-1/2 and positive AJ implies *s* = *I*
+1/2. When AJ = 0, one and only one of *I*1/2 or *I* +1/2 is possible,
so the possible ambiguity of ±0 does not arise. In this case *s = l*,
parity conservation prevents the occurrence, for a given *J*, of two
*s*-values differing by one unit.

**D.1.4. Adler-Adler (AA):** LRU=1, LRF=4

The formulae, taken from References 3 and 4, are given for the total,
radiative capture, and fission cross sections. They have been slightly
re‑cast to make them conform to the definitions used earlier in this
Appendix. Furthermore, only the ***l***=0 terms are given, consistent
with current usage of this formalism. Procedures are discussed in
Section 2.4.15. Since only s‑waves are considered, higher ***l***‑wave
contributions to the potential scattering must be put into File 3 by the
evaluator.

1\. Total Cross Section:

![](media/image40.wmf)

2\. Radiative Capture Cross Section:

![](media/image41.wmf)

3\. Fission Cross Section:

![](media/image42.wmf)

Although the format uses different names for μ and v for each reaction,
they are equal:

DET~r~ = DEF~r~ = DEC~r~ = μ~r~

DWT~r~ = DWF~r~ = DWG~r~ = ν~r~

**D.1.5. General R‑Matrix (GRM):** LRU = 1, LRF = 5 no longer available

**D.1.6 Hybrid R‑function (HRF):** LRU = 1, LRF = 6 no longer available

**D.1.7 R-Matrix Limited Format (LRF) :** LRU =1, LRF = 7

In R-Matrix theory, a channel may be defined by *c* = (α, *l, s, J*),
where

-   α represents the two particles making up channel; α includes mass
    (*m~a~* and *m~b~* with subscript *a* indicating the incident
    particle for an entrance channel), charge (Z~a~ and Z~b~), spin
    (i~a~ and i~b~) and parity (π~a~ and π~b~) and all other quantum
    numbers for each of the two particles, plus the Q-value.

-   *l* is the orbital angular momentum; the associated parity is
    (-1)*^l^*.

-   *s* represents the channel spin (including the associated parity);
    that is, *s* is the vector sum of the spins of the two particles of
    the pair.

-   *J* is the total angular momentum (and associated parity); *J* is
    the vector sum of *l* and *s*.

Only *J* and its associated parity are conserved for any given
interaction. The other quantum numbers may differ from channel to
channel, so long as the sum rules for spin and parity are obeyed.

In the Reich-Moore approximation to R-matrix theory, the radiation width
is treated separately and differently from widths for other channels
(which are hereafter referred to as "particle channels"). In this LRF=7
format, there is assumed to be an "eliminated channel" which, for the
strict interpretation of the Reich-Moore approximation, contains all the
radiation width; in this format, it is possible for some portion of the
radiation width to be treated in the same fashion as the particle
widths. In the equations below, the eliminated width appears only in the
denominator of the R-matrix.

In all formulae given below, spin quantum numbers (e.g., *J* ) are
implicitly assumed to include the associated parity. Vector sum rules
are implicitly assumed to be obeyed; readers unfamiliar with these sum
rules are referred to Sect. D.1.7.6 for details.

Let the angle-integrated cross sections from entrance channel *c* to
exit channel *c*\' with total angular momentum *J* be represented by
σ~cc\'~. This cross section is given in terms of the scattering matrix
*U* ~cc\'~ as

$\sigma_{\text{cc}'} = \ \frac{\pi}{k_{a}^{2}}\ g_{J\alpha}\ |\ e^{\text{iw}_{c}}\delta_{\text{cc}'} - U_{\text{cc}'}|{}^{2}\ \delta_{\text{JJ}'}$
**,** (1)

where (l) *k~α~* is the center-of mass momentum associated with incident
particle-pair α, (2) is *g~Jα~* is the spin statistical factor, and (3)
w~c~ is zero for non-Coulomb channels. (Details for the charged-particle
case are presented later.) The spin statistical factor is given by

$g_{J\alpha} = \frac{2J + 1}{(2i_{a} + 1)(2i_{b} + 1)}$ , (2)

and center-of mass momentum *k*~α~ by

$k_{\alpha}^{2} = \frac{2m_{a}m_{b}^{2}}{\left( m_{a} + m_{b} \right)^{2}}E$
. (3)

The scattering matrix *U* can be written in terms of the matrix *W* as

$U_{\text{cc}'} = \Omega_{c}W_{\text{cc}'}\Omega_{c'}$ , (4)

where Ω is given by

$)$ (5)

Here again, *w*~c~ is zero for non-Coulomb channels, and the potential
scattering phase shifts for non-Coulomb interactions φ~c~ are defined in
many references (e.g., Ref. 1). The matrix *W* in Eq. (4) is related to
the R-matrix (in matrix notation with indices suppressed) via

$W = P^{\frac{1}{2}}(I - \text{RL})^{- 1}(I - \text{RL}^{}\ )P^{\frac{- 1}{2}}$
(6)

The quantity *I* in this equation represents the identity matrix. The
quantity *L* in Eq. (6) is given by

$L = (S - B) + \text{iP}$ , (7)

with *P* the penetration factor, *S* the shift factor, and *B* the
arbitrary boundary constant at the channel radius *a~c~*. Formulae for
*P* and *S* are likewise found in many references (see, e.g., Eq. (2.9)
in Ref. 1); for non-Coulomb interactions see Table D.1.7.1 for the
appropriate formulae. For fission, the penetrability is unity. For
non-eliminated capture channels, the penetrability is unity. For two
charged particles, formulae for the penetrabilities are given in Sect.
D.1.7.4.

In the eliminated-channel approximation, the R-matrix of Eq. (6) \[for
the spin group defined by total spin *J* and implicit parity π\] has the
form

$R_{\text{cc}'} = \left\lbrack \sum_{\lambda}^{}\frac{\gamma_{\lambda c}\ \gamma_{\lambda c'}}{E_{\lambda} - E - \frac{{i\Gamma}_{\text{λγ}}}{2}} + R_{c}^{\text{bkg}}\delta_{\text{cc}'} \right\rbrack\ \delta_{\text{JJ}'}$
, (8)

where all levels (resonances) of that spin group are included in the
sum. Subscripts λ designate the particular level; subscripts *c* and
*c\'* designate channels (including particle-pairs and all the relevant
quantum numbers). Again, the width Γ~λγ~ occurring in the denominator
corresponds to the "eliminated" non-interfering capture channels of the
Reich-Moore approximation.

The "background R-matrix" *R~c~^bkg^* of Eq. (8) will be discussed in
Sect. D.1.7.7.

The channel width Γ~λc~ is given in terms of the reduced width amplitude
γ~λc~ by

$\Gamma_{\lambda c} = 2\ \gamma_{\lambda c}^{2}\ P_{c}(E)$ , (9)

where *P~c~* is the penetrability, whose value is a function of the type
of particles in the channel, of the orbital angulat momentum *l*, and of
the energy *E*. Note that the reduced width amplitude *γ~λc~* is always
independent of energy, but the width Γ~*λc*~ may depend on energy via
the penetration factor.

Cross sections may be calculated by using the above expressions for *R*
and *L* to calculate *W*, and from there calculating *U* and,
ultimately, σ. However, while Eq. (6) for *W* is correct, an equivalent
form which is computationally more stable is

$W = I + 2\ i\ X$ , (10)

where *X* is given in matrix notation by

$X = P^{\frac{1}{2}}L^{- 1}\ \left( L^{- 1} - R \right)^{- 1}R\ P^{\frac{1}{2}}$
(11)

When the suppressed indices and implied summations are inserted, the
expression of *X* becomes

$X_{\text{cc}'} = P_{c}^{\frac{1}{2}}L_{c}^{- 1}\ \sum_{c\text{\} \}  \{` left [ \textbackslash( L rSup \{ size 8\{ - 1\} \}  - R \textbackslash)  rSup \{ size 8\{ - 1\} \}  right ] rSub \{ size 8\{ ital }cc}^{}R_{c\text{c'\} \} `P rSub \{ size 8\{c'\} \}  rSup \{ size 8\{1/2\} \} δ rSub \{ size 8\{ ital }JJ\text{'\} \} \} \} \{}}$
. (12)

The various cross sections are then written in terms of *X*.

**Table D.1.7.1. Hard sphere penetrability (penetration factor) *P*,
level shift factor *S*, and potential scattering phase shift φ for
orbital angular momentum *l*, center of mass momentum *k*, and channel
radius *a~c~*~,~ with ρ = *ka~c~*.**

+---+----------------+-----------------------------+-------------------+
| * | ***P~l~***     | ***S~l~***                  | $$\varphi_{l}$$   |
| * |                |                             |                   |
| * |                |                             |                   |
| l |                |                             |                   |
| * |                |                             |                   |
| * |                |                             |                   |
| * |                |                             |                   |
+---+----------------+-----------------------------+-------------------+
| 0 | ρ              | 0                           | ρ                 |
+---+----------------+-----------------------------+-------------------+
| 1 | ρ^3^/(1 +      | -1 / (1 + ρ^2^)             | ρ-tan^-1^ ρ       |
|   | ρ^2^)          |                             |                   |
+---+----------------+-----------------------------+-------------------+
| 2 | ρ^5^ / (9 + 3  | -(18 + 3 ρ^2^) / (9 + 3     | ρ-tan^-1^\[3ρ /   |
|   | ρ^2^ + ρ^4^)   | ρ^2^ + ρ^4^)                | (3 - ρ^2^)\]      |
+---+----------------+-----------------------------+-------------------+
| 3 | ρ^7^ / (225 +  | -(675 + 90 ρ^2^ + 6 ρ^4^) / | ρ-ta              |
|   | 45 ρ^2^) +     |                             | n^-1^\[ρ(15-ρ^2^) |
|   |                | (225 + 45 ρ^2^ + 6 ρ^4^ +   | / (15-6 ρ^2^)\]   |
|   | 6ρ^4^ + ρ^6^)  | ρ^6^)                       |                   |
+---+----------------+-----------------------------+-------------------+
| 4 | ρ^9^ /         | -(44100 + 4725 ρ^2^ + 270   | ρ                 |
|   | (11025 + 1575  | ρ^4^ + 10 ρ^6^) /           | -tan^-1^\[ρ(105 - |
|   | ρ^2^ +         |                             | 10 ρ^2^) /        |
|   |                | (11025 + 1575 ρ^2^ + 135    |                   |
|   | 135ρ^4^ +      | ρ^4^ + 10 ρ^6^ + ρ^8^)      | (105 -- 45 ρ^2^ + |
|   | 10ρ^6^ + ρ^8^  |                             | ρ^4^)\]           |
+---+----------------+-----------------------------+-------------------+
| * | $$\b           | $$\frac{\rho^{2}(l - S_{l   | $\                |
| l | egin{array}{r} |  - 1})}{(l - S_{l - 1})^{2} | varphi_{l - 1} -  |
| * | \frac{\        |  + P_{l - 1}^{2}}\  - \ l$$ | \text{tan}^{- 1}\ |
|   | rho^{2}P_{l -  |                             | left( (\frac{P_{l |
|   | 1}}{\left( l - |                             |  - 1}}{(}l - S_{l |
|   |  S_{l - 1} \ri |                             |  - 1}) \right)$or |
|   | ght)^{2} + P_{ |                             |                   |
|   | l - 1}^{2}} \\ |                             | $$B_{l} = (B_{    |
|   |                |                             | l - 1} + X_{l})$$ |
|   | \end{array}$$  |                             |                   |
|   |                |                             | $(1 -             |
|   |                |                             |  B_{l - 1}X_{l})$ |
|   |                |                             |                   |
|   |                |                             | with              |
|   |                |                             |                   |
|   |                |                             | $$B_{l}           |
|   |                |                             | = \text{tan}(\rho |
|   |                |                             |  - \varphi_{l})$$ |
|   |                |                             |                   |
|   |                |                             | and               |
|   |                |                             |                   |
|   |                |                             | $$X_{l} = (P_     |
|   |                |                             | {l - 1}\frac{)}{( |
|   |                |                             | }l - S_{l - 1})$$ |
+---+----------------+-----------------------------+-------------------+

**D.1.7.1 Energy-Differential (Angle-Integrated) Cross Sections
(Non-Coulomb Channels)**

The observable cross sections are found in terms of *X* by first
substituting Eq. (4, 5, and 10) into Eq. (1), summing over spin groups
(i.e., over *J*^π^), and then summing over all channels corresponding to
those particle pairs and spin groups. If *X^r^* represents the real part
and *X^i^* the imaginary part of *X*, then the angle-integrated (but
energy-differential) cross section for the interaction which leads from
particle-pair α to particle-pair α^′^ has the form

$\sigma_{\alpha,\alpha'}\ (E)\  = \ \frac{4\pi}{k_{\alpha}^{2}}\ \sum_{J}^{}{\ g_{J\alpha}\ \sum_{c}^{}{\ \left\lbrack \left( \text{sin}^{2}\phi_{c}\ (1 - 2X_{\text{cc}}^{\ i})\  - X_{\text{cc}}^{\ i}\text{sin}\ (\ 2\phi_{c})\  \right)\ \delta_{\alpha,\alpha'}\  + \ \sum_{c'}^{}\left\{ \ X_{\text{cc}'}^{{\ i}^{2}} + X_{\text{cc}'}^{{\ r}^{2}}\  \right\}\  \right\rbrack\ }}\ \text{.}$
(13)

\[This formula is accurate only for cases in which one of particles in α
is a neutron; however, both particles in α^′^ may be charged.\]

In Eq. (13) the summations are over those channels *c* and *c′* {of the
spin group defined by *J*^π^} for which the particle-pairs are
respectively α and α^′^*.* More than one "incident channel" *c* = (α,
*l, s, J* ) can contribute to this cross section, e.g., when both *l =*
0 and *l* = 2 are possible, or when, in the case of incident neutrons
and non-zero spin target nuclei, both channel spins are allowed.
Similarly, there may be several "exit channels" *c*^′^ *=* (α^′^, *l*
^′^*, s*^′^*, J* ^′^ *)*, depending on the particular reaction being
calculated (elastic, inelastic, fission, etc.).

The total cross section (for non-Coulomb initial states) is the sum of
Eq. (13) over all possible final-state particle-pairs α^′^*,* assuming
the scattering matrix is unitary (that is, assuming that the sum over
*c*′ of \|*U~cc~′\|^2^ =* 1). Written in terms of the *X* matrix, the
total cross section has the form

$\sigma_{\alpha,\ \text{total}}\ (E\ )\  = \ \frac{4\pi}{k_{\alpha}^{2}}\ \sum_{J}^{}{\ g_{J\alpha}}\ \sum_{c}^{}{\ \left\lbrack \ \frac{1}{2}\ \text{sin}^{2}\phi_{c}\  + \ X_{\text{cc}}^{i}\text{cos}\ (2\ \phi_{c})\  - X_{\text{cc}}^{r}\text{sin}\ (2\ \phi_{c}) \right\rbrack}$
, (14)

where again the sum over *c* includes only those channels of the *J^π^*
spin group for which the particle-pair is α.

The angle integrated elastic cross section is given by

$\sigma_{\alpha,\ \alpha}\ (E)\  = \ \frac{4\pi}{k_{\alpha}^{2}}\ \sum_{J}^{}g_{J\alpha}\ \sum_{c}^{}\left\lbrack \text{sin}^{2}\phi_{c}\ (\ 1 - 2X_{\text{cc}}^{i})\  - X_{\text{cc}}^{r}\text{sin}\ (\ 2\ \phi_{c}\ )\  + \sum_{c'}^{}\left\{ X_{\text{cc}'}^{i} + X_{\text{cc}'}^{r^{2}} \right\} \right\rbrack$
. (15)

In this case, both *c* and *c*′ are limited to those channels of the
*J*^π^ spin group for which the particle-pair is α; again, there may be
more than one such channel for a given spin group.

Similarly, the reaction cross section from particle-pair α to
particle-pair α′ (where α′ is not equal to α) is

$\sigma_{\alpha,\ \alpha}(\ E\ )\  = \ \frac{4\pi}{k_{\alpha}^{2}}\ \sum_{J}^{}g_{J\alpha}\ \sum_{c}^{}{\ \sum_{c'}^{}{\ \left\lbrack X_{\text{cc}'}^{i^{2}} + X_{\text{cc}'}^{r^{2}} \right\rbrack}}$
. (16)

Here *c* is restricted to those channels of the *J^π^* spin group from
which the particle-pair is α, and *c′* to those channels for which the
particle-pair is α′.

The absorption cross section has the form

$\sigma_{\alpha\ \text{absorption}}\ (\ E\ )\  = \ \frac{4\pi}{k_{\alpha}^{2}}\ \sum_{J}^{}g_{J\alpha}\ \sum_{c}^{}{\ \left\lbrack X_{\text{cc}}^{i} - \ \sum_{c'}^{}{\ \left\{ \ X_{\text{cc}'}^{i^{2}} + X_{\text{cc}'}^{r^{2}}\  \right\}} \right\rbrack}$
. (17)

Here both the sum over *c* and the sum over *c′* include
[all]{.underline} [incident]{.underline} particle channels (i.e.,
particle-pair α only) for the *J^π^* spin group.

The capture cross section for the eliminated radiation channels can be
calculated directly as

$\sigma_{\alpha\ \gamma}\ (\ E\ )\  = \ \frac{4\pi}{k_{\alpha}^{2}}\ \sum_{J}^{}g_{J\alpha}\ \sum_{\text{inc}\ c}^{}{\ \left\lbrack X_{\text{cc}}^{i} - \sum_{\text{all}\ c'}^{}\left\{ X_{\text{cc}'}^{i^{2}} + X_{\text{cc}'}^{r^{2}} \right\} \right\rbrack}$
, (18)

or may be found by subtracting the sum of all reaction cross sections
from the absorption cross section. In Eq. (18), the sum over *c*
includes all incident particle channels for the *J^π^* spin group, and
the sum over *c′* includes [all]{.underline} particle channels, both
incident and exit, for that spin group.

**D.1.7.2 Angular Distributions**

Angular distributions (elastic, inelastic, or other reaction) cross
sections for incident neutrons can be calculated from Reich-Moore
resonance parameters. Following Blatt and Biedenharn \[Ref. 7\] with
some notational changes, the angular distribution cross section in the
center-of-mass system may be written

$\frac{d\ \sigma_{\text{ss}'}}{d\ \Omega_{\text{CM}}}\  = \ \sum_{L}^{}{\ C_{L\text{αα}'}\ (\ E\ )\ P_{L}\ (\ \text{cos}\beta\ )}$
, (19)

in which the subscript αα′ indicates which type of cross section is
being considered *P~L~* is the Legendre polynomial of degree *L*, and β
is the angle of the outgoing neutron (or other particle) relative to the
incoming neutron in the center-of-mass system. The coefficients *C~Lαα′~
(E)* are given by

$\begin{array}{r}
C_{L\alpha\ \alpha'}(\ E\ )\  = \ \frac{1}{4k_{\alpha}^{2}}\ \sum_{J_{1}}^{}{\ \sum_{J_{2}}^{}{\ \sum_{c_{1}\  = \ (\alpha\ l_{1}\ s_{1}\ J_{1})}^{}{\ \sum_{c_{1}'\  = \ (\alpha'\ l_{1}'\ s_{1}'\ J_{1})}^{}{\ \sum_{c_{2}\  = \ (\alpha\ l_{2}\ s_{2}\ J_{2})}^{}{\ \sum_{c_{2}'\  = \ (\alpha'\ l_{2}'\ s_{2}'\ J_{2})}^{}}}}}} \\
 \\
\  \times \ B_{\left\{ l_{1}\ s_{1}\ l_{1}'\ s_{1}'\ J_{1} \right\}\ \left\{ l_{2}\ s_{2}\ l_{2}'\ s_{2}'\ J_{2}\  \right\}\ L}\ \frac{1}{(2i_{a} + 1)\ (2i_{b} + 1)}\ \text{Re}\ \left\lbrack (\ \delta_{c_{1}c_{1}'} - U_{c_{2}c_{2}'}\ )\ (\ \delta_{c_{1}c_{1}'} - U_{c_{2}c_{2}'}^{}\ ) \right\rbrack\ ,
\end{array}$ (20)

where the various summations are to be interpreted as follows:

\(1\) sum over all spin groups defined by spin *J*~1~ and the implicit
associated parity.

\(2\) sum over all spin groups defined by spin *J~2~* and implicit
associated parity.

\(3\) sum over all those channels c~1~ belonging to the *J~1~* spin
group and having particle-pair α

\[*c~1~* = (α, *l~1~, s~1~, J~1~*)\].

\(4\) sum over those channels *c~1~*′ *J~1\ ~*spin group with
particle-pair α′ \[*c~1~*′ = (α′, *l′~1~, s′~1~, J~1~*)\].

\(5\) sum over those channels *c~2~* *J~2\ ~*spin group with
particle-pair α \[*c~2~* = (α, *l~2~, s~2~, J~2~*)\].

\(6\) sum over those channels *c~2~*′ *J~2\ ~*spin group with
particle-pair α′ \[*c~2~*′ = (α′, *l′~2~, s′~2~, J~2~*)\].

Also note that *i~a~* and *i~b~* are spins of the two particles in
particle-pair α.

The geometric factor *B* can be exactly evaluated as a product of terms

$$
$B_{\left\{ \ l_{1}\ s_{1}\ l_{1}^{'}\ s_{1}^{'}\ J_{1}\  \right\}\ \left\{ \ l_{2}\ s_{2}\ l_{2}^{'}\ s_{2}^{'}\ J_{2}\  \right\}\ L}\  = \ A_{\ l_{1}\ s_{1}\ l_{1}^{'}\ s_{1}^{'}\ ;\ J_{1}}\ A_{\ l_{2}\ s_{2}\ l_{2}^{'}\ s_{2}^{'}\ ;\ J_{2}}\ D_{l_{1}\ s_{1}\ l_{1}^{'}\ s_{1}^{'}\ l_{2}\ s_{2}\ l_{2}^{'}\ s_{2}^{'}\ ;\ L\ J_{1}\ J_{2}}$
, (21)

where the factor $A_{\ l_{1}\ s_{1}\ l_{1}\ 's_{1}'\ ;\ J_{1}}$is of the
form

$A_{\ l_{1}\ s_{1}\ l_{1}^{'}\ s_{1}^{'}\ ;\ J_{1}}\  = \ \sqrt{(\ 2\ l_{1} + \ 1\ )\ (\ 2\ l_{1}^{'} + \ 1\ )}\ (\ 2\ J_{1} + \ 1\ )\ \Delta\ (\ l_{1}\ J_{1}\ s_{1})\ \Delta\ (\ l_{1}^{'}\ J_{1}\ s_{1}^{'}\ )$
. (22)

The expression for *D* is

$\begin{array}{r}
D_{l_{1}\ s_{1}\ l_{1}^{'}\ s_{1}^{'}\ l_{2}\ s_{2}\ l_{2}^{'}\ s_{2}^{'};\ L\ J_{1}\ J_{2}} = \ (2L + 1\ )\ \Delta^{2}(J_{1}\ J_{2}\ L\ )\ \Delta^{2}(l_{1}\ l_{2}\ L\ )\ \Delta^{2}(l_{{}_{1}}^{'}\ l_{{}_{2}}^{'}\ L) \\
\ w\ (\ l_{1}\ J_{1}\ l_{2}\ J_{2},\ s\ L\ )\ w(\ l_{{}_{1}}^{'}\ J_{1}\ l_{{}_{2}}^{'}\ J_{2},\ s_{{}_{1}}^{'}\ L\ )\ \delta_{s_{1}\ s_{2}}\ \delta_{s_{1}^{'}\ s_{2}^{'}}\ ( - 1\ )^{s_{1} - \ s_{1}^{'}} \\
\ \frac{n\ !\ ( - 1\ )^{n}}{(n - 1_{1}\ )!\ (n - l_{2}\ )!\ (n - L\ )!}\ \frac{n'\ !\ ( - 1)^{n'}}{(n' - l_{1}^{'}\ )!\ (n^{'} - l_{2}^{'}\ )!\ (n^{'} - L\ )!}\ ,
\end{array}$ (23)

in which *n* is defined by

$2\ n\  = \ l_{1} + l_{2} + L\ ;$ (24)

*D* is zero if *l*~1~*+l*~2~*+L* is an odd number. A similar expression
defines *n′*. The Δ^2^ term is given by

$\Delta^{2}(\text{abc}) = \frac{(a + b - c)!\ (a + b + c)!\ (a + b + c)!}{(a + b + c + 1\ )!}\ ,$
(25)

for which the arguments *a, b,* and *c* are to be replaced by the
appropriate values given in Eqs. (22) and (23). The expression for Δ^2^
(*a b c*) implicitly includes a selection rule for the arguments; that
is, the vector sum must hold:

$\overrightarrow{a} + b = \overrightarrow{c}$ . (26)

The quantity *w* in Eq. (23) is defined as

$\begin{array}{r}
w\ (l_{1}\ J_{1}\ l_{2}\ J_{2},\ s\ L\ )\  = \ \underset{k = k\text{min}}{\overset{k\text{max}}{\Sigma}}\ \frac{( - 1\ )^{k + l_{1} + J_{1} + l_{2} + J_{2}}\ (k + 1)!}{\left( k - (\ l_{1} + J_{1} + s) \right)\ !\ \left( k - (\ l_{2} + J_{2} + s) \right)\ !} \\
\ \frac{1}{\left( k - (l_{1} + l_{2} + L) \right)\ !\ \left( k - (J_{1} + J_{2} + L) \right)\ !} \\
\ \frac{1}{(\ l_{1} + J_{1} + l_{2} + J_{2} - k\ )\ !\ (\ l_{1} + J_{2} + s + L - k\ )\ !\ (\ l_{2} + J_{1} + s + L - k\ )\ !}
\end{array}$ (27)

(and similarly for the primed expression), where *kmin* and *kmax* are
chosen such that none of the arguments of the factorials are negative.
That is,

$\begin{array}{r}
\text{kmin}\  = \ \text{max}\ \left\{ \ (\ l_{1} + J_{1} + s),\ (\ l_{2} + J_{2} + s\ ),\ (\ l_{1} + l_{2} + L\ ),\ (\ J_{1} + J_{2} + L\ )\  \right\} \\
\text{kmax}\  = \ \text{min}\ \left\{ \ (\ l_{1} + J_{1} + l_{2} + J_{2}\ ),\ (\ l_{1} + J_{2} + s + L\ ),\ (\ l_{2} + J_{1} + s + L\ )\  \right\}
\end{array}$ . (28)

[Single-channel case]{.underline}

For the single-channel case, the coefficients *C~Lαα'~* (E) reduce to

$\begin{array}{r}
C_{L\ \alpha\ \alpha}(E)\  = \ \frac{1}{4\ k_{\alpha}^{2}}\ \sum_{J_{1}}^{}{\ \sum_{J_{2}}^{}{\ \sum_{c_{1} = \ (\alpha\ l_{1}\ s_{1}\ J_{1}\ )}^{}{\ \sum_{c_{2} = \ (\alpha\ l_{2}\ s_{2}\ J_{2}\ )}^{}{\ B_{\left\{ \ l_{1}\ s_{1}\ l_{1}\ s_{1}\ J_{1}\  \right\}\ \left\{ \ l_{2}\ s_{2}\ l_{2}\ s_{2}\ J_{2}\  \right\}\ L}}}}} \\
\ \frac{1}{(\ 2\ i_{a} + 1\ )\ (\ 2\ i_{b} + 1\ )}\ \text{Re}\ \left\lbrack \ (\ 1 - U_{c_{1}\ c_{1}})\ (\ 1 - U_{c_{2}\ c_{2}}^{})\  \right\rbrack
\end{array}$ . (29)

where the existence of only one channel requires that the primed
quantities of Eq. (25) be equal to the unprimed (e.g., α = α'). The
geometric factor *B* becomes

$B_{\left\{ \ l_{1}\ s_{1}\ l_{1}\ s_{1}\ J_{1}\  \right\}\ \left\{ \ l_{2}\ s_{2}\ l_{2}\ s_{2}\ J_{2}\  \right\}\ L}\  = \ A_{\ l_{1}\ s_{1}\ l_{1}\ s_{1};\ J_{1}}\ A_{\ l_{2}\ s_{2}\ l_{2}\ s_{2};\ J_{2}}\ D_{\ l_{1}\ s_{1}\ l_{1}\ s_{1}\ l_{2}\ s_{2}\ l_{2}\ s_{2};\ L\ J_{1}\ J_{2}}$
, (30)

where the factor *A* reduces to the simple form

$A_{\ l_{1}\ s_{1}\ l_{1}\ s_{1};\ J_{1}}\  = \ (\ 2\ l_{1} + 1\ )\ (\ 2\ J_{1} + 1\ )\ \Delta^{2}(\ l_{1}\ J_{1}\ s_{1}\ )$
, (31)

and the expression for *D* reduces to

$\begin{array}{r}
D_{\ l_{1}\ s_{1}\ l_{1}\ s_{1}\ l_{2}\ s_{2}\ l_{2}\ s_{2};\ L\ J_{1}\ J_{2}}\  = \ (\ 2\ L + 1\ )\ \Delta^{2}(\ J_{1}\ J_{2}\ L\ )\ \Delta^{4}(\ l_{1}\ l_{2}\ L\ )\ w^{2}(\ l_{1}\ J_{1}\ l_{2}\ J_{2},\ s_{1}\ L\ )\ \delta_{s_{1}s_{2}} \\
\ \left\lbrack \frac{n\ !}{(n - l_{1}\ )\ !\ (n - l_{2}\ )\ !\ (n - L\ )\ !\ } \right\rbrack^{\ 2}
\end{array}$ . (32)

in which *n* is again defined as in Eq. (24).

**D.1.7.3 Kinematics for angular distributions of elastic scattering**

If *E* represents the laboratory kinetic energy of the incident neutron,
*E′* the lab kinetic energy of the outgoing particle, θ the laboratory
angle of the outgoing neutron, and *Q* the Q-value for the reaction, the
*E′* may be expressed in terms of *E*, θ, and *Q* as

$E\ '\  = \ E\ \left\lbrack \ \frac{m_{a}}{m_{a} + m_{b}}\ \text{cos}\theta\  + \ \sqrt{\ \left( \frac{m}{m_{a} + m_{b}} \right)^{\ 2} - \ \text{sin}^{2}\theta\ \left( \frac{m_{a}}{m_{a} + m_{b}} \right)^{\ 2}}\  \right\rbrack^{\ 2}$$$
. (33)

where *m~a~* represents the mass of the incident particle (neutron) and
*m~b~*, the mass of the sample (target) nucleus. Similarly, the
center-of-mass angle β between outgoing and incoming neutron is found
from

$\text{cos}\beta\  = \  \pm \ \frac{m_{a}}{m_{b}}\ \left\{ \ \text{cos}\theta\ \sqrt{\ \frac{m_{b}^{2}}{m_{a}^{2}}\  - \ \text{sin}^{2}\theta}\  - \ \text{sin}^{2}\theta\  \right\}$
, (34)

and the Jacobian of transformation from center-of-mass to laboratory
system is

$\frac{d(\text{cos}\beta)}{d(\text{cos}\theta)}\  = \ 2\text{cos}\theta\ \frac{m_{a}}{m_{b}}\  + \ \frac{1 + (\ 2\text{cos}^{2}\theta - 1\ )\ \frac{m_{a}^{2}}{m_{b}^{2}}}{\sqrt{\ 1 - \text{sin}^{2}\theta\ \frac{m_{a}^{2}}{m_{b}^{2}}}}$
. (35)

The elastic angular distribution cross section in the laboratory system
is then found by combining Eq. (15 or 29) with (35), using the
relationship in Eq. (34), to give

$\frac{d\ \sigma}{d\ \Omega_{\text{lab}}}(\theta)\  = \ \frac{d\ \sigma}{d\ \Omega_{\text{CM}}}\ \frac{d(\text{cos}\beta)}{d(\text{cos}\theta)}$
. (36)

Note that the lowest energy into which a neutron may scatter (i.e., the
energy of a neutron after 180-degree scattering) is

$E\ '\ (\text{cos}\theta = - 1)\  = \ E\ \left\lbrack \frac{m_{b} - m_{a}}{m_{b} + m_{a}} \right\rbrack^{\ 2}$
, (37)

and the energy of 90-degree scattering is

$E\ '(\text{cos}\theta = 0)\  = \ E\ \left\lbrack \frac{m_{b} - m_{a}}{m_{b} + m_{a}} \right\rbrack$
. (38)

**D.1.7.6 Spin and Angular Momentum Conventions**

The spin and angular momentum conventions used in the Classic
Reich-Moore Format are described in Table D.1.7.6. Note that the word
"channel" refers to the physical configuration as well as to the quantum
numbers given here. For example, for an incident neutron (intrinsic spin
*i* = ½) impinging on a target (sample) whose spin is *I*, the channel
spin is s, where $\overrightarrow{s}\  = \ i\  + \ \overrightarrow{I}$.
The relative orbital angular momentum of this channel (neutron + target)
is *l*, and total spin is *J*, where
$\overrightarrow{J}\  = \ \overrightarrow{s}\  + \overrightarrow{l}$.
The exit channel might be the same as the entrance channel, or it might
include, for example, two particles whose individual spins are *i*′ and
*I′* and whose channel spin is *s*′, where
$\overrightarrow{s}'\  = \ \overrightarrow{i}'\  + \overrightarrow{I}'$.
The relative angular momentum of the two particles is *l*′, and the
total *J* must satisfy
$\overrightarrow{J}\  = \ \overrightarrow{s}'\  + \overrightarrow{l}'$.

For readers unfamiliar with vector summation, the rules are as follows:
All quantum numbers are either integer (0, 1, 2, ...) or half-integer
(1/2, 3/2, 5/2, ...). If vectors of magnitude *a* and *b* are to be
added, then the sum *c* has magnitude in the range \| *a-b* \| ≤ *c* ≤
*a* + *b*; *c* takes on only integer values if *a* + *b* is integer, and
half-integer values if *a* + *b* is half-integer. The parity associated
with *c* is the product of the parities associated with *a* and *b*.
Note also that parity associated with orbital angular momentum *l* is
rarely expressed explicitly, as it is always (-1)*^1^.*

**Table D.1.7.6 Spin and angular momentum conventions**

+---------+------------------------------------+-----------------------+
| **S     | **Meaning**                        | **Value or range of   |
| ymbol** |                                    | values**              |
+---------+------------------------------------+-----------------------+
| *i* or  | Intrinsic spin of incident neutron | ½ for incident        |
| *i′*    | or outgoing particle               | neutron               |
+---------+------------------------------------+-----------------------+
| *I* or  | Spin of target or residual nuclei  | integer or            |
| *I ′*   |                                    | half-integer          |
+---------+------------------------------------+-----------------------+
| *l* or  | Orbital angular momentum of        | non-negative integer  |
| *l′*    | incident or outgoing particle      |                       |
+---------+------------------------------------+-----------------------+
| *s* or  | Incident or outgoing channel spin, | $\ove                 |
| *s′*    | equal to target spin plus incident | rrightarrow{s}\  = \  |
|         | particle spin                      | \overrightarrow{I}\   |
|         |                                    | + \overrightarrow{i}$ |
|         |                                    | or                    |
|         |                                    | $\overri              |
|         |                                    | ghtarrow{s}'\  = \ \o |
|         |                                    | verrightarrow{I}'\  + |
|         |                                    |  \overrightarrow{i}'$ |
+---------+------------------------------------+-----------------------+
| *J*     | \(1\) Spin of resonance            | $$\overrightarr       |
|         |                                    | ow{J}\  = \ \overrigh |
|         | \(2\) Spin of excited level in the | tarrow{l}\  + \overri |
|         | compound nucleus                   | ghtarrow{s}\  = \ \ov |
|         |                                    | errightarrow{l}'\  +  |
|         | \(3\) Total angular momentum       | \overrightarrow{s}'$$ |
|         | quantum number                     |                       |
+---------+------------------------------------+-----------------------+

**D.1.7.7 Extensions to R-matrix theory**

As stated in Eq. (6), the R-matrix has the form

$R_{\ c\ c'}\  = \ \left\lbrack \ \sum_{\lambda}^{}{\frac{\gamma_{\lambda\ c}\ \gamma_{\ \lambda c'}}{E_{\lambda} - E - i\ \frac{\Gamma_{\text{λγ}}}{2}}\  + \ R_{\ c}^{\ \text{bkg}}\ \delta_{\text{cc}'}}\  \right\rbrack\ \delta_{J\ J\ '}$
. (39)

The external or background R-matrix $R_{c}^{\text{bkg}}$can be written
in many different ways; four options are available in the RML format:

Option 0. $R_{c}^{\text{bkg}}$= 0 (in which case the background is
described by "dummy" resonances whose energies lie outside the range of
validity of this parameterization).

Option 1. $R_{c}^{\text{bkg}}$is a tabulated complex function of the
energy.

Option 2. $R_{c}^{\text{bkg}}$is a real statistical parameterization of
the form available in SAMMY \[ref.11\],

$R_{c}^{\text{ext}}(E) = {\overline{R}}_{\text{con},c} + {\overline{R}}_{\text{lin},c}\ E + {\overline{R}}_{q,c}\ E^{2} - s_{\text{lin},c}(E_{c}^{\text{up}} - E_{c}^{\text{down}})\  - \ (s_{\text{con},c} + s_{\text{lin},c}E)\ \text{ln}\left\lbrack \frac{E_{c}^{\text{up}} - E}{E - E_{c}^{\text{down}}} \right\rbrack\ \text{.}\ $(40)

Option 3. $R_{c}^{\text{bkg}}$is a complex statistical parameterization
of the forms described by Fröhner \[ref. 6 & 10\],

$R_{\ c}^{\ \text{ext}}(E)\  = \ {\overline{R}}_{c}\  + \ 2s_{c}\ \left\lbrack \text{tanh}^{\text{-1}\ }\left( \frac{E - \overline{E}}{\frac{Q}{2}} \right)\  + \ i\ \frac{\frac{{\overline{\Gamma}}_{\gamma}}{Q}}{1 - \left( \frac{(E - \overline{E})}{\frac{Q}{2}} \right)^{\ 2}} \right\rbrack$
. (41)

with

$Q = E_{c}^{\ \text{up}} - E_{c}^{\ \text{down}}\ \text{and}\ \overline{E} = \left( E_{c}^{\ \text{up}} + E_{c}^{\ \text{down}} \right)\frac{\ }{2}$
. (42)

**D.2. THE UNRESOLVED RESONANCE REGION:** LRU=2, LRF=1 or 2

Average resonance parameters are provided in File 2 for the unresolved
region. Parameters are given for possible *l*‑ and J‑values (up to
d‑wave, *l* = 2) and the following parameters may be energy
dependent:![](media/image43.wmf). The parameters are for the
single‑level Breit‑Wigner formalism. Each width is distributed according
to a chi‑squared distribution with a designated number of degrees of
freedom. The number of degrees of freedom may be different for neutron
and fission widths and for different (*l*,J) values. These formulae do
not consider Doppler broadening.

**D.2.1. Cross Sections in the Unresolved Region**

Definitions and amplifying comments on the following are given in
Section D.2.2.

a\. [Elastic Scattering Cross Section]{.underline}

![](media/image44.wmf)The asymmetric term in EE´~r~ is assumed to
average to zero under the energy-averaging denoted by \< \>.

b\. [Radiative Capture Cross Section]{.underline}

![](media/image45.wmf)c. [Fission Cross Section]{.underline}

The sum over *l* in the above equations extends up to *l* = 2 or NLS‑1
(the highest *l*‑value for which data are given). For each value of *l*,
the sum over J has NJS terms. The number of J‑states for a particular
*l*‑state will depend on the value of *l*. NLS and NJS are given in File
2.

## The averages are re‑written as

![](media/image47.wmf)where R~γ*l*,J~, R~f*l*,J~ and R~n*l*,J~ are
width‑fluctuation factors for capture, fission, and elastic scattering,
respectively. Associated with each factor is the number of degrees of
freedom for each of the average widths, and the integrals are to be
evaluated using the MC^2^‑II method.

**Data given in File 2 for each (*l*,J) state**

> μ~n*l*,J~ = AMUN, the number of degrees of freedom for neutron widths

μ~f*l*,J~ = AMUF, the number of degrees of freedom for fission widths

μ~x*l*,J~ = AMUX, the number of degrees of freedom for competitive
widths

μ~γ*l*,J~ = AMUG, the number of degrees of freedom for radiation widths

![](media/image48.wmf) = GX, the average competitive reaction width

![](media/image49.wmf) = GN0, the average reduced neutron width

![](media/image50.wmf) = GG, the average radiation width

![](media/image51.wmf) = GF, the average fission width

![](media/image52.wmf) = D, the average level spacing

The average neutron widths are defined in Section D.2.2.2, Equation
(31c), where ![](media/image53.wmf). Degrees of freedom are discussed in
Section 2.4.20.

## The average total width, at energy E, is

![](media/image54.wmf) ,

## and all widths are evaluated at energy E. J=AJ, I=SPI, and *l*=L are given in File 2. The penetration factors and phase shifts are functions of *a* or AP, as describe earlier.

**D.2.2. Definitions for the Unresolved Resonance Region**

Editions of ENDF‑102 prior to ENDF/B‑V have had some errors in the
\"Definitions\" section of Appendix D (previously Section D.2.1). To
clarify the points and facilitate parallel reading with Gyulassy and
Perkins, Reference 8, their parenthesized indices will be used. Section
D.2.3 contains a table of equivalences to the notation used in D.2.1 and
Section D.2.4 compares the present discussion with those previously
given.

**D.2.2.1. Sums and Averages**

In an energy interval Δε, let the resonances be identified by a
subscript λ = 1, 2, ... which goes over all the resonances. The present
discussion is concerned with the ***combinatorial*** aspects of level
sequences, hence λ enumerates ***all*** the resonances, whether their
widths are observably large or not. One purpose of this section is to
permit estimation of missed resonances by comparing observed level
densities or strength functions with the theoretically‑expected
relations. The latter are concerned with the set of all resonances, and
not just those that are observable in a particular experiment.

Let x denote a set of quantum numbers that label a subset of resonances
in the interval. If there are N(x) such resonances, their level density
is

![](media/image55.wmf)

and their level spacing is

![](media/image56.wmf)

If y~λ~ is some quantity associated with each resonance, λ, the sum of
the y‑values over the

subset x is

![](media/image57.wmf)

In this section, the summation index λ is written as a subscript, and
the range of the summation is indicated by the superscript x. Equation
(D.25) says \"sum the quantity y over every resonance in the interval Δε
which has the quantum numbers x.\" Usually, these resonances will
possess other quantum numbers too, but it is the set x which determines
whether they are included or not.

An ***average*** of the quantity y over the set x is

![](media/image58.wmf)

**D.2.2.2. Reduced Widths**

In this section, reduced widths follow the experimental definition
rather than the theoretical usage Γ = 2Pγ^2^. A partial width for the
decay of a resonance into a particular channel carries many quantum
numbers, but we need only three, the total and orbital angular momenta J
and *l*, and the channel spin s. The reduced neutron width,
Γ*^l^*~nλ~(J,s), is defined by:

![](media/image59.wmf)

> where v*~l~* = P*~l~*/ρ

v~0~ = 1,

v~1~ = ρ^2^/(1+ρ^2^)

v~2~ = ρ^4^/(9+3ρ^2^+ρ^4^)

> and ρ =k*a*, *a* being the channel radius.

Assuming additivity of partial widths,

![](media/image60.wmf)

> where Σ~s~ is a summation over the 1 or 2 possible channel‑spin
> values.

If we average over resonances, and assume that the average partial width
is independent of channel spin,[^4]

![](media/image61.wmf)

Equation (D.29) introduces the multiplicity μ~*l*,J~, which for neutrons
can have the value 1 or 2, depending on whether the channel spin has one
or two values. For *l*=0, or I=0, or J=0, μ~*l*,J~ = 1. In other cases,
s can take on the values I ±1/2 subject to the additional vector sum

![](media/image62.wmf)

which may again restrict μ~*l*,J~ to the value one.[^5]

The other new notation is the line through the quantum number s, meaning
that the quantity ![](media/image63.wmf) does not depend on the value of
s. This is ***not*** the same as omitting s from the parentheses, since
***that*** defines the left‑hand side quantity. This is the primary
source of confusion in previous discussions. Since v*~l~* only depends
on *l*,

![](media/image64.wmf)

> where the bar over ![](media/image65.wmf)denotes some average value
> appropriate to the interval.

**D.2.2.3. Strength Function**

The pole‑strength function was originally introduced as an average over
the R‑matrix reduced widths for a given channel, γ~c~^2^. Using the
experimental convention.

![](media/image66.wmf)

Since the channel spin values are uniquely determined by J and *l*,
together with the target spin I which is common to all the resonances, s
is superfluous in defining the subset over which the average is taken,
and

![](media/image67.wmf)

If the parity π were used as an explicit quantum number, *l* could be
dropped,

![](media/image68.wmf)

because *l* and π are equivalent for labeling resonances. That is, every
resonance with a given J and π will have channels labeled by the same
set of *l*‑values, whether their partial widths are observably large or
not. Some authors go one step further and drop π, so that J ***means***
J,π, but that is an invitation to confusion.

Expressing S(*l*,J,s) as a sum over reduced widths gives

![](media/image69.wmf)

where we use the assumed independence of ![](media/image70.wmf) on s to
get the same result on the left‑hand‑side.

The strength function S(*l*,J) is defined as

![](media/image71.wmf)

The corresponding sum and average forms are

![](media/image72.wmf)

The next \"natural\" summation would be to collect the different
*l*-contributions to the total width, to form S(J), but this is not what
is observable. Instead one defines S(*l*) as a weighted sum of the
S(*l*,J,s):

![](media/image73.wmf)

This equation occurs in Lynn, Reference 1, as 6.126, with a confusing
typographical error, namely the index s is missing from S(*l*,J,s).

Actually, the strength function was introduced first in the \"s‑wave\"
form

![](media/image74.wmf)

and later generalized by Saplakoglu et al. Reference 9, to the p‑wave
form

![](media/image75.wmf)

For expository purposes, it is clearer to start from (31). The sum on J
and s is for fixed *l*:

![](media/image76.wmf)

It is important to note that the outer sum on channel spin is correct as
written. It goes over the values I±½ if I½ , and over the single value
½, if I=0. It is ***not*** further constrained by Equation (D.30)
because now ***it*** is the \"independent variable.\" The inner sum on J
enumerates some J‑values once, and some twice, the latter occurring when
both s‑values can produce that J‑value. The number of times J occurs is
the same μ~*l*,J~ that appeared previously.

If we are summing a quantity that is independent of s, then (D.41) can
be rewritten:

![](media/image77.wmf)

The multiplicity μ~*l*,J~ takes care of the sum on s, and the tilde over
the sum on J, as emphasized by Gyulassy and Perkins, Reference 8,
reminds us that J goes over its full range, \"once‑only\":[^6]

![](media/image78.wmf)

####### The denominator in Equation (D.38) can be shown to be

![](media/image79.wmf)

or, since g is independent of s,

![](media/image80.wmf)

G‑P assume, and later approximately justify by comparison to experiment,
that S(*l*,J,s/) is also independent of J. With this, Equation (D.38)
becomes

![](media/image81.wmf)

using Equation (D.36.b).

Note the peculiar fact that S(*l*) and S(*l*,J,s) are independent of J,
but S(*l*,J) is not. This is a consequence of the fact that more than
one channel spin value can contribute to S(*l*,J), inducing a
\"J‑dependence\" in the form of a possible factor of two.

As a sum over resonances,

![](media/image82.wmf)

The right hand side of Equation (D.47c) says to sum Γ*^l^*~n~(J) over
all possible values of J, which is what is meant by Equations (D.39) and
(D.40). We can suppress the explicit J\'s and write, as in Equation
(D.40),

![](media/image83.wmf)

but we have to remember that Γ*^l^*~n~ is still Γ*^l^*~n~(J), and not a
new quantity.

As an average, using the same convention,

![](media/image84.wmf)

Otherwise, all the notation is correct: D(*l*) is the spacing of
*l*‑wave resonances without regard to their J‑values, and the average \<
\>*^l^* goes over all resonances possessing the quantum number *l*,
again without regard for their J-values. It is worth noting explicitly
that although S(*l*,J) is \"almost\" independent of J, this is not true
of ![](media/image85.wmf). As Equation (37b) shows, its J-dependence is
canceled by the J-dependence of D(*l*,J), up to the factor μ~*l*,J~.
This property is what makes strength functions useful.

**D.2.2.4. Level Spacings**

G‑P emphasize that

![](media/image86.wmf)

which, together with the assumption

![](media/image87.wmf) ,

leads to

![](media/image88.wmf),

> where

![](media/image89.wmf)

and is unity if *l*=0 or I=0.

The reader is referred to Reference 8 for a fuller discussion but here
we can point out that, for a given parity, ρ(*l*,J) is independent of
*l*, by definition. As noted, every resonance with a given J and π has
the same set of associated *l*‑channels, whether it has an observable
width or not. Hence

![](media/image90.wmf)

The further assumption of parity‑independence makes ρ(*l*,J) totally
independent of *l*. As a result, G‑P\'s K(*l*) from Equation (D.51) is
independent of *l*, and

![](media/image91.wmf) ,

where C depends on the nuclear species by not on any quantum numbers.

**D.2.2.5. Gamma Widths**

In the limited energy range of a few keV usually covered by the
unresolved resonance region, the gamma width may be assumed to be
constant and equal to that obtained from an analysis of the resolved
resonances. If, however, the energy range is rather wide, an energy
dependence as given by some of the well‑known theoretical models,
Reference 1, may be built in. Since the observed gamma width is the sum
of a large number of primary gamma transitions, each assumed to have a
chi‑squared distribution of μ=1, the sum is found to have a μ20. In
effect this implies that the gamma width is a constant, since a
chi‑squared distribution with a large number of degrees of freedom
approximates a δ‑function.

**D.2.2.6. Degrees of Freedom**

For the reasons enumerated in Chapter 2, Section 2.4.20, the following
values should be used:

1\. Neutron width, 1. AMUN 2., and specifically, AMUN= μ~*l*,J~ .

2\. Radiation width, AMUG=0.

3\. Fission width, 1. AMUF 4., to be determined by comparison with
experiment. Only integral values are permitted, although non‑integers
occur in some analyses.

4\. Competitive width, 1. AMUX 2., because only a single inelastic level
excitation is permitted as a competitive reaction. Specifically, AMUX=
μ~*l*,J~, where J is the spin of the resonance, and *l* is the orbital
angular momentum of the inelastically scattered neutron. Since the
daughter nucleus may have a spin ![](media/image92.wmf)different from
the target spin, *I*, *l* may be different from *l* and the number of
channel spin values μ~*l*,J~ may be different from μ~*l*,J~.

**D.2.3. Equivalent Quantities in Sections D.1 and D.2**

+--------+-------+---------+------------------------------------------+
| ####   |       |         | #### Definition                          |
| Symbol |       |         |                                          |
+--------+-------+---------+------------------------------------------+
| **in   | **in  | **in    |                                          |
| D.1**  | D.    | D.2.2** |                                          |
|        | 2.1** |         |                                          |
+--------+-------+---------+------------------------------------------+
| r      | \-    | λ       | This is a non‑equivalence. λ enumerates  |
|        |       |         | ***all*** resonances. r enumerates those |
|        |       |         | within a subset and hence ***implies***  |
|        |       |         | a set of quantum numbers.                |
+--------+-------+---------+------------------------------------------+
| Γ~nr~  |       | Γ~nλ~   | The neutron width, summed over channel   |
|        |       | (*l*,J) | spin.                                    |
+--------+-------+---------+------------------------------------------+
| Γ~γr   |       | ‑       | Not used in D.2.2, but the same          |
| ~,Γ~fr |       |         | implication of *l*, J holds.             |
| ~,\... |       |         |                                          |
+--------+-------+---------+------------------------------------------+
| P*~l~* |       | ρv*~l~* | Penetration factor.                      |
+--------+-------+---------+------------------------------------------+
|        | D~*   | D       | Average level spacing for a subset of    |
|        | l*,J~ | (*l*,J) | resonances with given *l* and J.         |
+--------+-------+---------+------------------------------------------+
|        | ![    | !       | The *l*-wave reduced width, averaged     |
|        | ](med | [](medi | over all resonances with given *l* and   |
|        | ia/im | a/image | J.                                       |
|        | age93 | 94.wmf) |                                          |
|        | .wmf) |         |                                          |
+--------+-------+---------+------------------------------------------+
|        | ![    | !       | The average neutron width. In practice,  |
|        | ](med | [](medi | the energy‑ dependence of this quantity  |
|        | ia/im | a/image | is not averaged, but extracted before    |
|        | age95 | 96.wmf) | averaging.                               |
|        | .wmf) |         |                                          |
+--------+-------+---------+------------------------------------------+

**D.2.4. Comparison with previous editions of ENDF‑102**

##### D.2.4.1. ENDF‑102, October 1970 edition

1\. Equation (1). D~observed~ is D(*l*).

2\. Line 9. *l* is not the angular momentum of \"the incident neutron.\"
The incident neutrons carry all angular momenta. *l* is the orbital
angular momentum of the resonance, or more precisely, of the channel or
channels which are involved. The resonant *l*^th^ phase shift will
interfere with the non‑resonant ones in angular distributions, but not
in angle‑integrated cross sections.

3\. Equation (2)

*ρ~J~* is *ρ(l,J); ρ~obs~* is *ρ(l)*; Σ is *Σ˜~J~*.

4\. Equation (3). D~J~ is D(*l*,J) and the right‑hand‑side should have a
factor ω~I,*l*~.

5\. Page D‑11, line 1. \"Level-spacing\" means D(*l*,J). Line 8. The
statement \"If we assume the s‑wave strength function is independent of
J\...\"presumably means assuming S(0,J) is independent of J, since the
s‑wave strength function itself, S(0), is a sum over J‑states and is
therefore \"independent\" of J by definition.

## Equation 4 means

*S(*0*) = S(0,J,s) = S(*0*,J)/μ*~0*,J*~

> which because μ~0,J~= 1 implies

*S(*0*) = S(*0*,J) = Γ*^0^*~n~(J)* ^0*,J*^*/D(*0*,J) .*

Equation 5 is trickier because μ~*l*,J~ is not identically equal to 1,
and the discussion appears to give the user the option of getting
D(*l*,J) from Equation (3) and \"the corresponding reduced neutron
width\" from Equation (5), or of using the ENDF/B convention, Equation
(6).

The problem lies in the failure to distinguish Γ*^l^*~n~(J) from
Γ*^l^*~n~(J,s). Equation (6) states the \"ENDF/B convention\":

![](media/image97.wmf)

(Here and in the following, ![](media/image98.wmf)and ***v~l~*** are
average values appropriate to the energy interval.)

## We know that the correct relationship is

![](media/image99.wmf)

If we assume that ![](media/image100.wmf) in Equation (6) is to agree
with experiment, ![](media/image101.wmf) then must be

![](media/image102.wmf)

If we use a subscript E to denote an ENDF-convention quantity,

![](media/image103.wmf)and now

![](media/image104.wmf)which is correct. Thus an ENDF reduced width
![](media/image105.wmf) will sometimes be half what an experimentalist
would measure.

In the notation of D.2.2., Equation (24) is

![](media/image106.wmf)

and Γ*^l^*~n~(J) is the reduced width determined by experiment.

For p-waves,

![](media/image107.wmf)and Equation (5) would read:

![](media/image108.wmf)

6\. The October 1970 edition uses three different symbols for the
reduced width. In a unified notation:

page D-9 ![](media/image109.wmf)

> page D-11 ![](media/image110.wmf)
>
> page D-12 ![](media/image111.wmf)

**D.2.4.2.** **Comments on ENDF-102, October 1975 edition, Section
D.2.1**

1\. Equation (1). D~*l*,observed~ is D(*l*).

2\. Same as comment 2 on the 1970 edition; above.

3\. Equation (2). D~obs~ is D(*l*); ρ~*l*,obs~ is ρ(*l*); Σ~J~ is Σ˜~J~.

4\. \"All allowed *l*-values label the same set of resonances\" means
ρ(*l*,J) does not depend on *l* (for given parity).

5\. Equation (3). (2J + 1)^-1^ is missing from the right hand side.

6\. Page D-12, second equation:

![](media/image112.wmf)The quantity gΓ*^l^*~n~
is![](media/image113.wmf). The bracket *^l^* means summed over all
J-values. The other two brackets are for particular J-values, *i.e.*,

7\. \"The strength functions for a given *l*-value but different
J-values\" means S(*l*,J). These are not all equal\--it is the ratio
S(*l*,J)μ~*l*,J~ which is independent of J.

8\. Equation (D.6) should read:

![](media/image114.wmf)where Γ*^l^*~n~(J) is the reduced width
determined by experiment. That is, the relation involving μ~*l*,J~ is
only valid for an ***average*** width, and hence ![](media/image115.wmf)
must also be some appropriate average value. The quantum number s should
be exhibited when μ is used.

###### D.3 The Competitive Width

**D.3.1 Penetrability Factor for the Competitive Width in the Resolved
Resonance Region**

A. [SLBW and MLBW]{.underline}

![](media/image116.wmf)For these formalisms, the only physical situation
which can be handled without approximation is that in which a
***single*** inelastic competitive process is possible, because the
formalism presently permits the definition of only ***one*** additional
quantity. The most common case will occur when inelastic scattering to
the first excited state of the target nucleus is energetically possible.
Ignoring, as in the case of elastic scattering, the possibility that the
partial widths depend on channel spin, the penetrability is identical to
that for elastic scattering, but the energy is reduced by the excitation
energy of the first excited state, corrected for recoil, so that

and

![](media/image117.wmf)

> where E^\*^~1~ is (AWRI+1)/AWRI times the excitation energy of the
> first excited state, ![](media/image118.wmf)in File 2).

This definition involves two conventions, both taken over from the
elastic case. One is the way in which an \"experimental\" reduced width
Γ^l^~n~ is defined in terms of the theoretical reduced width γ^2^, and
the other is the way in which negative energy levels are treated.
Neither of these problems arises in the theory, where Γ=2Pγ^2^ and all
quantities are defined in terms of the channel energy. Note that the
*l*-value to be used in the penetrability is not that of the incident
neutron, but of the \"exit\" inelastically scattered neutron.

It is conceivable that an (n,α) or (n,p) reaction to the ground state of
the daughter nucleus could be open, without inelastic competition, in
which case the formula for Γ~xr~ would be the same, but the P*~l~* would
be a Coulomb penetrability, and the excitation energy E~l~^\*^ would be
replaced by the approximate Q-value and reduced mass. The R-Matrix
Limited format allows for this possibility (see Section D.1.7).

If more than one competitive process is energetically possible, then the
SLBW and MLBW formats are inadequate to give the correct energy
dependence of the competitive width, since they supply only one number,
and a partial width is required for each process. *E.g.*, when two
inelastic levels can be reached,

![](media/image119.wmf)with appropriate modification below each
threshold. Note that the exit *l*-values are independent of the
incident-neutron *l*-value.

![](media/image120.wmf)For codes that presently approximate Γ as a
constant in the denominator, a possible procedure is to substitute a
step function

and then make some provision to handle the resultant discontinuity in
the cross section.

Users who are unable to handle this degree of complexity, and would like
to use GT~R~ from File 2 as the total width without regard for whether
the competitive process is energetically possible or not should at least
be aware of the problem.

B. When the [Adler-Adler and Reich-Moore]{.underline} formalisms are
used for low-energy fissile materials, no recommendation concerning the
treatment of Γ~xr~ need be given, and users can presume that it is zero.
When Reich-Moore is used above the thermal region, the same comments
apply as for the SLBW/MLBW formalism.

C. [General R-matrix and Hybrid R-function]{.underline} (deleted)

**D.3.2. Penetrability Factor for the Competitive Width in the
Unresolved Resonance Region**

Since many codes treat the average total width in the denominator of
expressions like \<Γ~n~Γ~γ~/Γ\> as an energy-independent constant, the
penetrability factor of the competitive width needs to be handled by
specifying energy-dependent unresolved resonance parameters.

The formalism, which is a simple average over SBLW line shapes, takes
account of the energy-dependence of the neutron widths in the numerator,
by extracting their penetrability factors before the averaging is done.
These then contribute to the energy-dependence of the average cross
section. The energy-dependence of the neutron width in the denominator,
i.e., in ![](media/image121.wmf), is neglected. No such fix is readily
available for the energy-dependence of the competitive width, whose
penetrability factor will involve the threshold dependence of an
inelastic cross section. The evaluator can circumvent this difficulty by
specifying energy-dependent parameters and setting \<Γ~x~\>=0 below its
threshold; then allowing it to build up according to the formulas given
in Section D.3

The degrees of freedom, AMUX, should be 1.0 or 2.0. (See Section
2.4.20.)

**D.3.3. Calculation of the Total Cross Section when a Competitive
Reaction is Specified**

When a competitive reaction is specified for SBLW or MLBW and Γ exceeds
Γ~n~ + Γ~γ~ + Γ~f~, the ENDF convention is that the scattering, capture,
and fission cross sections will be calculated from the sum of File 2 and
File 3 contributions, but the competitive reaction will be contained
entirely in File 3, and no File 2 contribution should be added to it.
The reason for this is that users can avoid problems in coding up
resonant competitive widths. In the File 2 calculations, the correct
total width Γ must be used in order to get the correct line shape.

This puts the ***total*** cross section in a special category. If it is
calculated as the sum of σ~n~, σ~γ~, σ~f~, σ~x~, then the above
prescription works satisfactorily. However, if it is calculated from the
SLBW formula,

![](media/image122.wmf)as it is in some applications, then it will
include the competitive reaction, and the user should not add the File 3
contribution to it. The ENDF convention presumes that σ~n,t~ will be
calculated by summing the partial reactions.

The R-matrix Limited format does not have this problem. The
\"competitive\" reactions are treated normally, and File 2 and File 3
are added together for all reactions. That is because the total width is
always the sum of the explicitly given partial widths. If a File 3
contribution were specified for the total cross section, then it would
be added to the {1-ReU} calculation, but not to the sum-of-parts
calculation, as the latter would already include the File 3 contribution
for each partial reaction. This assumes that the File 3 total is the sum
of the File 3 partials.

##### REFERENCES

1\. J.E. Lynn, **The Theory of Neutron Resonance Reactions**, Clarendon
Press, Oxford (1958)

2\. C.W. Reich and M.S. Moore, ***Phys. Rev.*** **111**, 929 (1958)

3\. F.T. Adler and D.B. Adler, *Interpretations of Neutron Cross
Sections of the Fissionable Materials for the Resolved Resonance
Region*, U.S. Atomic Energy Commission Report **CONF-660303**, ***Conf.
on Neutron Cross Section Technology*** (1966), Book 2, p. 873

4\. D.B. Adler and F.T. Adler, Argonne National Laboratory report
**ANL-6792**, 695 (1963)

5\. A.M. Lane and R.G. Thomas, ***Rev. Mod. Phys.*** **30**, 257 (1958)

6\. F.H. Fröhner, *Applied Neutron Resonance Theory*,
Kernforschungszentrum, Karlsruhe report **KFK 2669** (1978)

7\. J.M. Blatt and L.C. Biedenharn, ***Rev. Mod. Phys.*** **24**, 258
(1952)

8\. M. Gyulassy and S.T. Perkins, ***Nucl. Sci. Eng.*** **53**, 482
(1974)

9\. A. Saplakoglu, L. M. Bollinger, and R. E. Coté, ***Phys. Rev.***
**109**, 1258 (1958)

10. F.H. Fröhner, *New Techniques for Multi-Level Cross Section
    Calculation and Fitting*, Brookhaven National Laboratory report
    **BNL-NCS-51363**, ***Proceedings of the Conference on Nuclear Data
    Evaluation, Methods and Procedures*** (1981), Vol. I, P. 375

[^1]: ^1^ Processing codes should sum the cross section, as shown, from
    *l*=0 to *l*=NLS-1, including any \"empty\" or \"non-resonant\"
    channels, in order to get the potential-scattering contribution. If
    higher *l*-values contribute to the scattering in the resonance
    region, it is the responsibility of the evaluator to provide a
    suitable File 3 contribution. (See Sections 2.4.23 and 2.4.24.)

[^2]: ^2^ The channel radius, strictly speaking, involves A^1/3^ (the
    target mass in amu), and not (AWRI)^1/3^, but as long as the mass of
    the incident particle is approximately unity, as it is for neutrons,
    the difference is not important. AWRI = A/m~n~, where m~n~ is the
    neutron mass (see Appendix H).

[^3]: ^3^ Including footnote on page D.1.

[^4]: This is not true for the individual resonances

[^5]: For example, if I = 1/2, μ~1,2~ = 1.

[^6]: ^7^ Reference 8 has this written incorrectly.