**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 Hybrid R‑Function,
Section 2.2.1.4, all shifts are set to zero, and this is also an option
for the General R‑matrix, Section D.1.5.2.

![](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 *l*′*s*′*J*′ 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

**D.1.5.1. Calculation of the R‑matrix elements**

![](media/image43.wmf)The basic element in an R‑matrix calculation is
the contribution of a single resonance to the R‑matrix,

or in terms of the laboratory quantities defined in Section 2.2.1.4.2,

![](media/image44.wmf)The matrix itself is obtained by summing over the
appropriate λ‑values and adding in the background R‑matrix:

![](media/image45.wmf)If there are altogether N channels, then the
calculation will involve N(N + 1)/2 distinct R‑matrix elements, R~c,d~,
since the matrix is symmetric. Those elements with different J and π in
the two channels are zero, and the remaining ones can be rearranged in
diagonal blocks labeled by J and π. The information needed to compute
the elements of each block is contained in the spin‑group list, together
with the background R-matrix list, which specifies how the evaluation
deals with distant-level effects.

Since the format gives the channel indices for each spin‑group, i.e.,
each Jπ sub-matrix, explicitly in the spin‑group list, the using code is
relieved of the necessity of doing any angular momentum bookkeeping. The
evaluator needs to be concerned with the details of the multiple sums
over ***l***, ***l***′, s, s′, and J, but the user needs only to invert
the sub‑matrices specified by each spin-group, and then sum the partial
cross sections according to the MT-list. Unlike the SLBW, MLBW, and
Reich‑Moore formalisms, here the burden of specifying ***all*** the
necessary channels is on the evaluator, not the processing codes.

Following the usual procedure, the background R‑matrix is assumed to be
diagonal and non‑zero only in the incident neutron channels, i.e., those
for which PMT=SMT=2 in the spin‑group list.

*RBK~c,d~ = RBK~n~δ~c,n~δ~c,d~ ,*

> where n stands for an incident neutron channel.

Three different representations are allowed for the background R‑matrix:

a\. a tabulated complex function of the energy,

b\. the logarithmic parameterization used in SAMMY,

*RBK~n~ = R0~n~ + R1~n~ E + R2~n~ E^2^ - S1~n~ \[EU~n~ - ED~n~\]*

*- \[S0~n~ + S1~n~ E\] **ln**\[{EU~n~ - E}/{E - ED~n~}\]*

c\. a statistical parameterization using R~∞~ (RIN), a strength function
(SF), the energy interval (INT) over which the explicit resonances are
given, the interval midpoint (EBAR) and an average gamma width (AVGG)
over that interval (references 6 and 10):

*RBK~n~ = RIN~n~ + 2SF~n~ \[arctanh\[2 (E - EBAR)/INT\]*

*+ i \[AVGG INT/{(INT)^2^ - 4(E - EBAR)^2^}\]\]*

**D.1.5.2. Calculation of the U‑matrix elements**

The collision, or U‑matrix, is obtained by the usual inversion formula
from the R‑matrix:

*U = Ω P^1/2^ \[1 - R (L- B)\]^-1^ \[1 - R(L\*- B)\] P^-1/2^ Ω*

### Ω = exp (-iθ) , L = S + iP

It has the same diagonal block structure as the R‑matrix, but its
elements are complicated functions of all the R‑matrix elements involved
in the inversion. In addition to the R‑matrix elements, this step in the
calculation requires phase shifts, penetrabilities, shift factors,
nuclear radii, and boundary‑condition parameters. These are specified in
the following lists, or are calculated by the using code, depending on
the options specified by the evaluator.

As an option, the shift factor may be set to zero. In R‑matrix theory,
the shift factor occurs because the underlying complete set of nuclear
states on which the theory is built do not coincide with the actual
nuclear states, due to the artificial boundary condition imposed on the
former. In practice, only the ***form*** of the theory is used, and not
its basic theoretical content, so that for neutron cross section
evaluation in ENDF, the shift is an irrelevant artifact and can be
ignored. It is true that a shifted and a resonance that is not shifted
fitted to the same experimental data will have slight shape differences,
but these will have no technological significance. Eliminating the shift
also removes one barrier to the use of R‑matrix parameters in multigroup
slowing‑down and self‑shielding codes, which naively assume that a
resonance energy ER will be associated with a Lorentzian bump at that
energy. Anti‑resonances, and strange channel‑interference patterns will
still cause problems in such applications, but will at least occur in
the expected locations.

For the same reason, it is recommended that the boundary‑condition
parameter be set to zero. With S = B = 0, the above formula reduces to

*U = Ω P^1/2^ \[*1 *- iRP\]^-1^ \[*1 *+ iRP\] P^-1/2^ Ω*

which can often reproduce experimental data as well as the more general
form above.

It is assumed that the phase shifts in

*Ω = exp (-iθ)*

are either hard‑sphere values, or the ones defined in the phase‑shift
list. That is, hard‑sphere values are assumed to be ***included*** in
the tabulated values, and are not to be added by the user.

**D.1.5.3. Calculation of the channel cross section**

Each element of a U-sub-matrix, U~c,d~, defines a channel-to-channel
cross sections,

![](media/image46.wmf)These cross sections are not observable, but are
the ingredients for ones that are. The statistical factor depends on *c*
through *J*.

**D.1.5.4. Calculation of the PMT‑cross sections (Particular
MT-values)**

An observable PMT‑cross section is obtained by summing the
channel‑to‑channel cross section over all the incident channels, and
over those exit channels belonging to a particular MT‑value, PMT. Both
of these summations are defined by the MT‑list, which tells which
channels are incident, SMT=PMT=2, and which channels belong to each PMT.
The code used needs only to keep track of the indices, and is not
required to book keep the angular momentum quantum numbers.

*σ~PMT\ =~ Σ~c\ in\ n,\ d\ in\ PMT~ σ~c,d~*

**D.1.5.5. Calculation of the SMT‑cross sections (Summed MT-values)**

The complete reaction cross sections, corresponding to the summed‑MT\'s,
are then obtained by summing over the appropriate PMT\'s. The MT‑list
specifies which PMT\'s belong to each SMT.

*σ~SMT~ = Σ~PMT\ in\ SMT~ σ~PMT~*

The elastic cross section, SMT=2, is the same as its only component,
σ(PMT=2). The same is true of the fission cross section, SMT=PMT=18.

**D.1.5.6. Calculation of the total cross section**

The total cross section is obtained from the diagonal elements
corresponding to the incident channels. It cannot be obtained by summing
partial cross sections because radiative capture is not calculated
according to the above scheme, but is treated as an eliminated channel.
Hence its cross section is not available for summing.

![](media/image47.wmf)The summation here is over all channels belonging
to SMT=PMT=2.

**D.1.5.7. Calculation of the capture cross section**

This is obtained as the total cross section minus the sum of all the
SMT‑cross sections:

*σ~CAP~ = σ~TOT~ - Σ~SMT~ σ~SMT~*

If the eliminated width were read in as the sum of the capture width and
some other reaction width(s), the above difference cross section would
correspond to the sum of those reactions. This fact is exploited in the
Hybrid R‑function formalism to simplify the treatment of competitive
reactions, which often do not require the complication of a full
R‑matrix treatment.

**D.1.5.8. Calculation of the absorption cross section**

*σ~ABS~ = σ~TOT~ - σ~SMT=2~*

If the inelastic cross section, σ~SMT=4~ , is not zero, it is necessary
also to subtract it.

**D.1.5.9. Calculation of the angular distributions**

These may be found from the Blatt and Biedenharn formalism. However, the
summations required do not lend themselves to expression in terms of
channel indices, the way the integrated cross sections do. If the
angular distributions are calculated directly from the scattering
amplitudes, one has to deal with magnetic quantum numbers in addition to
α, *l*, s, and J. If one eliminates the magnetic quantum numbers ala
Blatt and Biedenharn, the sums over *l*, s, and J are each treated
differently and do not stay together in neat channel packages. To
utilize these formulas, it is necessary to pick apart the c\'s and d\'s
and extract the quantum numbers. The format has not been optimized for
that, although the information is contained in the MT‑ and
spin‑group‑lists. To use the angular distribution formula, the code
needs to perform the indicated 9‑fold summation, and to locate a given
U‑matrix element from its seven arguments α~1~, α~2~, *l*~1~, *l*~2~,
s~1~, s~2~, and J. To facilitate the summation, the limits are given
explicitly in the following paragraph.

Using a normalization in which B~0~ is also the angle‑integrated cross
section, the Blatt and Biedenharn formula is[^4]

![](media/image48.wmf)where P~L~(μ) is a Legendre polynomial in cos
θ~αα′~, and

*(l~1~l~2~00;L0)* and *W* are the Clebsch-Gordan and Racah coefficients.

## From outside to inside, the sum limits are

1\) 0 ≤ L ≤ 2\*(NLSC-1), where NLSC-1 is a cutoff value chosen by the
evaluator to converge the angular distributions. (See Sections 2.4.23
and 2.4.24).

2\) I - i ≤ s ≤ I + i (incident spins)

3\) I′ - i′ ≤ s′ ≤ I′ + i′ (exit spins)

4\) 0 ≤ *l*~1~ ≤ NLSC-1

5\) 0 ≤ *l*′~1~ ≤ NLSC-1

6\) L - *l*~1~ ≤ *l*~2~ ≤ min{NLSC-1, L+*l*~1~}

7\) L - *l*′~1~ ≤ *l*′~2~ ≤ min{NLSC-1, L+*l*′~1~}

8\) max{*l*~1~ - s, *l*′~1~ - s′} ≤ J~1~ ≤ min{*l*~1~ + s, *l*′~1~ +
s′}

9\) max{*l*~2~ - s, *l*′~2~ - s′, L - J~1~} ≤ J~2~ ≤ min{*l*~2~ +
s, *l*′~2~ + s′, L + J~1~}

**D.1.6 Hybrid R‑function (HRF):** LRU=1, LRF=6

The formulae are from standard R‑matrix theory, References 1, 5, and 6,
augmented by the single‑level Breit‑Wigner formalism for the calculation
of competitive reactions. The mixing of the two methods is indicated by
the term \"hybrid.\" As in the previous section, D.1.5, R, S, and U are
center‑of‑mass quantities, written in terms of laboratory widths and
energies.

**D.1.6.1. Elastic Scattering Cross Section**

> ![](media/image50.wmf)where S~*l*sJ~(E) ≡ 1 - U~*l*sJ~(E)

![](media/image51.wmf)The eliminated width, Γ~er~, in the denominator of
the R‑function is a constant, evaluated at the resonance energy. It is
the sum of the partial widths for capture and competitive reactions (see
Section 2.2.1‑4). Its omitted energy‑dependence has a negligible effect
on the shape of the elastic resonances. If the flag LPS is 0, the phase
shift δ~*l*sJ~ will be calculated from the hard‑sphere formulas. If
LPS=1, tabulated phase shifts will be supplied in every channel. This
permits parameterization of optical‑model evaluations. Hard‑sphere
values depend on the channel‑ dependent radius AC. The formulas of
section D.1.1 are used with ρ *=* ![](media/image52.wmf) being dependent
on *l*, s and J. Note that the above equations are all scalar. No matrix
inversion is required. The background R‑function ![](media/image53.wmf),
if supplied, must be tabulated, although the analytic formulae of
Section 2.2.1.4.3 may be used as a guide.

As in the footnote on page D.1, the processing codes must sum over all
channels, whether they contain resonances or not. The evaluator must
specify suitable values of NLS and NLSC to converge the cross‑section
and angular‑distribution calculations.

**D.1.6.2. Partial Reaction Cross Sections**

The following formulas are the same as SLBW, except that an explicit sum
on channel spin has been introduced, so that the same J‑value can occur
in two different channels, *l*s~1~J and *l*s~2~J:

![](media/image54.wmf)x can represent any of four reaction types:

a\) [Radiative capture]{.underline}

*Γ~xr~(E) = Γ~γr~*, a constant in energy

b\) [Fission]{.underline}

![](media/image55.wmf)Γ~xr~(E) = Γ~fr~, a constant in energy. This
convention is the same as that used in the SLBW and MLBW formalisms.

c\) [Inelastic scattering]{.underline}

![](media/image56.wmf)![](media/image57.wmf)is the laboratory threshold
energy for exciting the p^th^ level in the target nucleus. It is
determined from the Q‑value in File 2 (QRE~n~) by

P*~l′~* is to be calculated using the AC values for the first channel in
the file. The prime on the subscript *l* denotes the fact that an
inelastically‑scattered neutron may have a different *l*‑value from the
incident neutron. *l*′ is the library quantity ALRE~n~. AWRI is the same
as for the incident neutron, because the reduced mass in an inelastic
channel is (essentially) the same as in the incident channel. P~*l*′~ is
zero if its argument is a negative incident energy. If its argument is a
negative resonance energy, P~*l*′~ is evaluated at the absolute value of
its argument.

d\) [Charged‑particle scattering]{.underline}

The charged‑particle penetrability will be supplied by the evaluator as
a tabulated function of the incident neutron\'s laboratory energy. The
conversion from the charged‑particle

channel energy to the neutron\'s energy is discussed in Section
D.3.1(c). This choice of energy eliminates the need for the user to do
any conversion, and the energy‑dependent charged‑particle exit width is
simply

![](media/image58.wmf)The choice also eliminates the need to specify an
exit mass ratio, AWRIC, an exit Q‑value, or an exit channel radius, as
these are incorporated by the evaluator into the tabulated
penetrability.

**D.1.6.3. Total Reaction Cross Section**

![](media/image59.wmf)This is calculated as the sum of the above partial
reaction cross sections,

![](media/image60.wmf)In the important case that radiative capture is
the only reaction specified (NFRE = NIRE = NCRE = 0), the total reaction
cross section is also the radiative capture cross section, and is to be
calculated from the R‑function formula

> ![](media/image61.wmf)where

This formula is applicable also in the \"trivial\" case of pure elastic
scattering (NGRE = NFRE = NIRE = NCRE = 0). This formula, when
applicable, eliminates all reference to the SLBW formalism, and reduces
the hybrid R-function to a \"pure\" R-function.

**D.1.6.4. Total Cross section**

The total cross section is calculated as the sum of the elastic
scattering and total reaction cross section defined above,

*σ~T~(E) = σ~n,n~(E) + σ~R~(E)*

In the two special cases noted in D.1.6.3, the total cross section may
also be calculated from the mathematically‑equivalent form:

![](media/image62.wmf)These special cases are standard R‑function
theory, as no use is made of the single‑level Breit‑Wigner formalism
(see Section D.3.3.).

**D.1.6.5. The Angular Distribution of Elastically‑Scattered Particles**

![](media/image63.wmf)Scattering‑matrix theory predicts an angular
distribution which varies across a resonance. For some applications this
variation might be worth calculating. The general formula of Blatt and
Biedenharn, Reference 7, reduced to the present case of an R‑function
that is diagonal in *l*, s, and J is (center of mass):

> ![](media/image64.wmf)where
>
> and *i* is the neutron spin (1/2). Here (*l*~1~*l*~2~00; L0) and
> W(*l*~1~J~1~*l*~2~J~2~l;sL) are Clebsch‑Gordan and Racah coefficients,
> respectively.
>
> The limits on the summation variables are as follows, from outside to
> inside:

1\) 0 ≤ L ≤ 2 (NLSC‑1), where NLSC‑1 is the highest partial wave
required to specify the angular distribution. It may be necessary to
calculate U~*l*sJ~ for non‑resonant channels which contribute just
hard‑sphere or optical‑model scattering, U~*l*sJ~ = exp(-2iδ~*l*sJ~).

2\) I - ½ ≤ s ≤ I + ½

3\) 0 ≤ *l*~1~ ≤ NLSC-1

4\) L - *l*~1~ ≤ *l*~2~ ≤ min{NLSC-1, L+ *l*~1~}

5\)  *l*~1~ - s ≤ J~1~ ≤ *l*~1~ + s

6\) max{ *l*~2~ - s, J~1~ - L} ≤ J~2~ ≤ min{*l*~2~ + s, J~1~ + L}

**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/image65.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/image66.wmf)The asymmetric term in EE´~r~ is assumed to
average to zero under the energy-averaging denoted by \< \>.

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

![](media/image67.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/image69.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/image70.wmf) = GX, the average competitive reaction width

![](media/image71.wmf) = GN0, the average reduced neutron width

![](media/image72.wmf) = GG, the average radiation width

![](media/image73.wmf) = GF, the average fission width

![](media/image74.wmf) = D, the average level spacing

The average neutron widths are defined in Section D.2.2.2, Equation
(31c), where ![](media/image75.wmf). Degrees of freedom are discussed in
Section 2.4.20.

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

![](media/image76.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/image77.wmf)

and their level spacing is

![](media/image78.wmf)

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

subset x is

![](media/image79.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/image80.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/image81.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/image82.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,[^5]

![](media/image83.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/image84.wmf)

which may again restrict μ~*l*,J~ to the value one.[^6]

The other new notation is the line through the quantum number s, meaning
that the quantity ![](media/image85.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/image86.wmf)

> where the bar over ![](media/image87.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/image88.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/image89.wmf)

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

![](media/image90.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/image91.wmf)

where we use the assumed independence of ![](media/image92.wmf) on s to
get the same result on the left‑hand‑side.

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

![](media/image93.wmf)

The corresponding sum and average forms are

![](media/image94.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/image95.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/image96.wmf)

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

![](media/image97.wmf)

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

![](media/image98.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 I≥½ , 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/image99.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\":[^7]

![](media/image100.wmf)

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

![](media/image101.wmf)

or, since g is independent of s,

![](media/image102.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/image103.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/image104.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/image105.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/image106.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/image107.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/image108.wmf)

which, together with the assumption

![](media/image109.wmf) ,

leads to

![](media/image110.wmf),

> where

![](media/image111.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/image112.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/image113.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 μ≥20. 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/image114.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     |
|        | (medi | ](media | over all resonances with given *l* and   |
|        | a/ima | /image1 | J.                                       |
|        | ge115 | 16.wmf) |                                          |
|        | .wmf) |         |                                          |
+--------+-------+---------+------------------------------------------+
|        | ![]   | ![      | The average neutron width. In practice,  |
|        | (medi | ](media | the energy‑ dependence of this quantity  |
|        | a/ima | /image1 | is not averaged, but extracted before    |
|        | ge117 | 18.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/image119.wmf)

(Here and in the following, ![](media/image120.wmf)and ***v~l~*** are
average values appropriate to the energy interval.)

## We know that the correct relationship is

![](media/image121.wmf)

If we assume that ![](media/image122.wmf) in Equation (6) is to agree
with experiment, ![](media/image123.wmf) then must be

![](media/image124.wmf)

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

![](media/image125.wmf)and now

![](media/image126.wmf)which is correct. Thus an ENDF reduced width
![](media/image127.wmf) will sometimes be half what an experimentalist
would measure.

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

![](media/image128.wmf)

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

For p-waves,

![](media/image129.wmf)and Equation (5) would read:

![](media/image130.wmf)

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

page D-9 ![](media/image131.wmf)

> page D-11 ![](media/image132.wmf)
>
> page D-12 ![](media/image133.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/image134.wmf)The quantity 〈gΓ*^l^*~n~〉
is ![](media/image135.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/image136.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/image137.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/image138.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/image139.wmf)

> where E^\*^~1~ is (AWRI+1)/AWRI times the excitation energy of the
> first excited state, ![](media/image140.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 General
R-matrix and Hybrid R-function formalisms allow for this possibility
(see Sections D.1.5 and D.1.6).

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/image141.wmf)with appropriate modification below each
threshold. Note that the exit *l*-values are independent of the
incident-neutron *l*-value.

![](media/image142.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} formalisms allow
multiple reaction channels, in addition to the elastic channel. The
above discussion of penetrabilities pertains here also. For purposes of
coding the penetrability factor as a subroutine, one possibility is to
supply six arguments, an energy E, a Q-value Q, a mass ratio AWR, a
radius R, an *l*-value L, and a sentinel K. Inside the subroutine, one
calculates the threshold T= -(AWR +1.0) Q/AWR and then branches on K. If
K=0, signaling an \"incident\" energy, the penetrability will be set to
0 if E is below threshold, but if K=1, signaling a \"resonance\" energy,
the penetrability will be calculated at the absolute value of E-T. Thus
an inelastic width would be calculated as

![](media/image143.wmf)where P is a subroutine or function statement.

![](media/image144.png){width="5.5in" height="2.842361111111111in"}For
both the R-formalisms, the evaluator must supply any Coulomb
penetrabilities as tabulated functions of the incident neutron\'s
laboratory energy. This choice keeps the energy consistent with that
used in all other ENDF quantities and simplifies the problem of
interpolating in the tables.

The evaluator will most likely calculate the Coulomb penetrabilities as
functions of the center-of-mass energy in the exit channel, E~cm,exit~.
This is related to the center-of-mass energy in the incident channel by

*E~cm,inc~ = E~cm,exit~ - Q*

and the incident laboratory energy is

![](media/image145.wmf)

In case the evaluator obtains his Coulomb penetrabilities as functions
of the charged-particle ![](media/image146.wmf)laboratory energy,
E~lab,exit~, one more transformation is required:

![](media/image147.wmf)where AWRIC is the mass ratio in the exit
channel.[^8] In this case,

These formulas enable the evaluator to supply the charged-particle
penetrabilities as functions of the incident neutron\'s laboratory
energy. This leaves to the user only the task of incorporating the
dependence on the individual resonance energies, Section D.3.1.

**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/image148.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/image149.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 General R-matrix and Hybrid R-function do 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. Depending on what
non-elastic channels are specified, the total cross section may be
calculated either as the sum of the partial cross sections, or from
{1-ReU(*l*sJ)}, as described in Sections D.1.5.6 and D.1.6.4. 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]: Sum variables in curly brackets. This formula is not applicable to
    either capture or fission. Both dσ/dΩ and B~L~ are center-of-mass
    quantities, in terms of laboratory widths and energies.

[^5]: This is not true for the individual resonances

[^6]: For example, if I = 1/2, μ~1,2~ = 1.

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

[^8]: ^8^ For an n,p reaction AWRIC = M~daughter~/m~p~ ≈ AWRI; but for
    an n,α reaction, AWRIC = M~daughter~/m~α~ ≈ (AWRI-3)/4.