ERIS — A Program for MEM analysis from X-Ray
and Neutron Diffraction Data
Koichi MOMMA1
National Museum of Nature and Science,
4-1-1 Amakubo, Tsukuba, Ibaraki 305-0005, Japan

July 25, 2024

1

E-mail: vesta.dev@gmail.com

Contents
LICENSE AGREEMENT

iii

1 INTRODUCTION

1

2 THEORETICAL BACKGROUND
2.1 The Fundamental Principle of MEM in Crystallography . . . . . . . . . . . . . .
2.2 The Maximum Entropy Patterson Method . . . . . . . . . . . . . . . . . . . . . .
2.3 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 ZSPA algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 L-BFGS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Cambridge algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Fourier Transform Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
3
4
4
4
4
5
6

3 CONSTRAINTS
3.1 F Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 G Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Generalized Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Linear Combination of Generalized Constraints . . . . . . . . . . . . . . . . . . .

7
7
7
8
9

4 WEIGHTING
11
4.1 Why weighting of observation is necessary . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Weighting based on exponential of d-spacing . . . . . . . . . . . . . . . . . . . . . 11
4.3 Weighting based on n-th power of d-spacing . . . . . . . . . . . . . . . . . . . . . 11
4.4 Weighting based on multiplicity of reflections . . . . . . . . . . . . . . . . . . . . 12
5 ADJUSTMENT OF STANDARD UNCERTAINTIES
13
5.1 Powder Diffraction Data (*.fos) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Single-Crystal Data (*.mem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 INSTALLATION AND EXECUTION
14
6.1 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.2 Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.2.1 Interactive mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.2.2 Running ERIS using condition files . . . . . . . . . . . . . . . . . . . . . . 17
7 INPUT FILES
18
7.1 Condition Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7.2 Intensity Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7.3 File format of *.eris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

i

7.4
7.5
7.6
7.7

7.3.1 title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.3 beam_source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.4 cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.5 data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.6 delta_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.7 elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.8 epsilon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.9 lambda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.10 lambda_coef . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.11 max_cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.12 resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.13 restart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.14 scale_sigma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.15 space_group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.16 scio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.17 use_prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.18 weight_cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.19 weight_m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.20 weight_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
File format of *.prf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
File format of *.alb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
File format of *.mem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
File format of *.fos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18
18
19
19
19
19
19
20
20
20
20
20
20
21
21
21
21
21
21
22
22
22
22
22

8 OUTPUT FILES
23
8.1 Output Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8.2 File format of *.pgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8.2.1 Periodic grid vs. general grid . . . . . . . . . . . . . . . . . . . . . . . . . 23
8.2.2 Pseudocode to write *.pgrid . . . . . . . . . . . . . . . . . . . . . . . . . . 24

ii

LICENSE AGREEMENTERIS LICENSE
Version 0.99
Copyright © 2024, Koichi Momma
ERIS is distributed free of charge but currently copyrighted with its source code not open
to the public because we wish to control its development and future by ourselves.
Permission to use this software is hereby granted under the following conditions:
1. Whenever original results acquired with ERIS are published in journals, proceedings, andreview articles, the program name ”ERIS” should explicitly be stated. A paper about
ERIS, describing its original features, is currently under preparation. Once it is published,
it should be cited.
2. ERIS is provided “as is” without any expressed or implied warranty. We do not provide
any technical support for free.
3. You should not redistribute any copy of the distributed files unless you have a written
permission from us.
If you find a problem or a bug on use of ERIS, please let us know and help us improve it.

iiiChapter 1

INTRODUCTION
The maximum entropy method (MEM) [1,2] is very useful to modify structural models adopted
in Rietveld analysis and determine distributions of electron and nuclear (coherent-scattering
length, bc ) densities from X-ray and neutron diffraction data, respectively [3]. The termination
effect is less serious in MEM analysis than in Fourier/D synthesis because MEM is capable ofestimating structure factors of high-Q reflections that have not been measured experimentally.
In addition, after observed structure factors of overlapped reflections have been estimated on thebasis of results of Rietveld analysis, subsequent MEM analysis more or less improves structure
factors, which makes MEM very effective in the extraction of the maximum amount of structural
information from powder diffraction data.
Three-dimensional (3D) visualization of electron/nuclear densities resulting from diffraction
data achieves a better understanding of the highly disordered structure, chemical bonding (Xray diffraction), or anharmonic thermal motion. For example, diffusion paths of mobile chemical
species in ionic conductors have been visualized in three dimensions from X-ray and neutron
powder diffraction data by a sophisticated structure-refinement technique called MEM-based
pattern fitting (MPF) [3–6]. MPF may be able to reveal even anharmonic thermal vibrations,which are very diﬀicult to analyze from powder diffraction data because of many parameters to
be refined by a nonlinear least-squares method.
ERIS is a successor of MEM analysis programs Dysnomia [6–8], PRIMA, and ALBA [3, 9].
PRIMA was a fortran program for MEM analysis of electron and nuclear density from X-ray
and neutron diffraction data. ALBA was a fortran program for MEM analysis of the Patterson
function from X-ray and neutron diffraction data to estimate intensities of unobserved and/or
individual reflections of overlapped peaks in powder diffraction. PRIMA was first rewritten fromscratch in the C++ language and released as Dysnomia in 2011.
Dysnomia provides the following features which were not supported in PRIMA:
1. linear combination of the generalized F and G constraints,
2. weightings based on lattice-plane spacings.
3. new implementation of optimization algorithms (a variant of the Cambridge algorithm [10]and the limited-memory BFGS algorithm [11]) that converge to solutions close to the true
maximum-entropy conditions,
4. automatic selection of fast Fourier transform (FFT) and discrete Fourier transform (DFT),
5. parallel processing using an API (application programming interface), OpenMP, for multiplatform shared-memory parallel programming,

1

ERIS integrates the features of Dysnomia and ALBA, with a couple of new and uniquefeatures.
1. a new type of weighting function, proportional to exponential of d-spacing,
2. a feature to automatically determine the best weighting parameters,
3. implemented the maximum entropy Patterson method (MEP),
4. automatically impose constraints of extinction conditions in MEP analysis,
5. generate an input file of Superflip after convergence of MEP analysis to solve crystal
structure by the charge flipping method.
ERIS is the name of the Greek goddess of strife and discord. She is mother of Dysnomia,
who is also the goddess of discord. Since the lawlessness state stands for higher-entropy statecompared to ordered state, it is suitable for the name of the MEM analysis program, in which
the information entropy, S, is maximized under some constraints. The program name ERIS also
stands for Entropy-based Real-space Inference System.

2

Chapter 2

THEORETICAL BACKGROUND
2.1

The Fundamental Principle of MEM in Crystallography

The general principle of MEM analysis is to find the maximum of the information entropy, S,
under several constraints by iterative procedures. A few variations of the MEM formalism have
hitherto been adopted in the literature [1,12,13]. Here, we will follow the formalism of Collins [1]based on Jaynes’s expression of S [14].
In MEM analysis from X-ray and neutron diffraction data, electron and nuclear densities in
the unit cell are represented by those in voxels (parallelepipeds) whose numbers along a, b, andc axes are Na , Nb , and Nc , respectively; densities within a voxel is regarded as even. Let N (=Na Nb Nc ) be the total number of voxels in the unit cell, ρk the normalized density at position rk
in the 3D gridded space, τk the normalized density derived from prior information for rk , andρ∗k the density at rk . Then, S is computed by
S=−

N
X


ρk ln

k=1

with
ρk =

ρk
τk



ρ∗k
.
N
X
ρ∗k

(2.1)

(2.2)

k=1

S is maximized under constraints C by a method of undetermined Lagrange multipliers,!
NX
Q = S − λC − µρk − 1 ,(2.3)
k=1

where Q is maximized through iterative computations with respect to ρ and two Lagrange
multipliers, λ and µ.
At the optimum solution of MEM, partial derivatives of Eq. (2.3) with respect to any variable
should be 0. Therefore, the entropy and constraint should fulfill the following condition:
∂S
∂C
=λ
+ µ.
∂ρk
∂ρk

(2.4)

Alternatively, they should fulfill the equivalent set of equations in reciprocal space:
∂C
∂S
=λ
.
∂Fi
∂Fi

3

(2.5)

2.2

Optimization Algorithms

The following MEM algorithms are implemented in ERIS:
1. 0th-order single pixel approximation (ZSPA) algorithm [15] .
2. Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm [11].
Electron or nuclear densities determined with these algorithms are similar to each other in mostcases. However, solutions obtained with the ZSPA algorithm do not satisfy true maximumentropy (MaxEnt) conditions [16], whereas the L-BFGS algorithm always gives solutions close
to the true MaxEnt ones. Another algorithm, the Cambridge algorithm also gives solutions
close to the true MaxEnt conditions [16]. Although a variant of the Cambridge algorithm was
implemented in Dysnomia, it was removed in ERIS because the L-BFGS algorithm is much
faster than the Cambridge algorithm while the two algorithm give the same solution.

2.2.1

ZSPA algorithm

In the ZSPA algorithm, ρk in the (i + 1)-th iteration is approximated by
!
(i)
ρk
∂C
(i+1)
exp −λ · (i)
ρk
=
Z
∂ρ

(2.6)

k

with
Z=

Nv
X

exp −λ ·

k=1

∂C
(i)

!
.

(2.7)

∂ρk

Convergence criterion of the ZSPA algorithm is C = 0. Since this algorithm is quite simple
and easy to implement, it is widely utilized in almost all programs for MEM analysis of electron/nuclear density from diffraction data. However, note that this algorithm does not actuallymaximize the information entropy S.

2.2.2

L-BFGS algorithm

The L-BFGS method [11] is a variant of quasi-Newton algorithms, requiring significantly lessmemory than other quasi-Newton methods. Instead of storing a dense n × n approximation (n:the number of variables in the problem) to the inverse Hessian matrix, L-BFGS stores only a
few vectors that represent the approximation implicitly. Let dk = Hk gk be the search direction
at k-th iteration, where Hk is the inverse Hessian matrix of the function, Q(ρ), to be maximized,
gk ≡ ∇Q(ρ) is the gradient of Q, sk = ρk+1 − ρk , yk = gk+1 − gk , and rk = 1/ykT sk . Then, dk iscomputed as follows:4q = gk
For (i = k−1; i >= k−m; i−−){αi = ri sTi q
q = q − α i yi
}
Hk0 =

T s
yk−1
k−1
T y
yk−1
k−1

d = Hk0 q
For (i = k−m; i <= k−1; i++){βi = ri yiT d
d = d + si (αi − βi )}The initial approximate of the inverse Hessian matrix, Hk0 , is represented as a diagonal matrix.
The Lagrange multipliers, λ and µ in Eq. (2.3), cannot be optimized simultaneously with ρ
during the L-BFGS iterations. Then, in the L-BFGS method, λ and µ are gradually changed,
and L-BFGS optimizations are repeated until the two constraints, i.e., Eqs. (2.2) and (3.15),
are fulfilled to achieve convergence. The convergence of the L-BFGS algorithm is judged by
v
u NV


uX  ∂Q 2
∂S
t
ρk
<ε
,
(2.8)
∂ρk
∂ρk
k=1

where ε is a small threshold value, which is set at 1 × 10−3 by default.

2.2.3

Cambridge algorithm

A variant of the Cambridge algorithm [10] was implemented in Dysnomia but removed from ERIS
in favor of the L-BFGS algorithm. Both algorithms converges to the optimal MEM solution but
L-BFGS algorithm is much faster. A brief summary of the Cambridge algorithm is given below
just for the record.
The Cambridge algorithm adopts local quadratic approximation to Eq. (2.3). For simplicity,
Dysnomia utilized only the diagonal elements of the Laplacian, ∇2 Q, and approximate Q(ρ+∆ρ)
by




∂Q
1 2 ∂2Q
Q(ρ + ∆ρ) = Q(ρ) + ∆ρ
+ ∆ρ
.
(2.9)
∂ρ
2
∂ρ2
Then, Q can be optimized by∆ρk = −

∂Q
∂ρk

−1
 2
∂ Q
∂2S
+α· 2
,
∂ρ2k
∂ρk

(2.10)

where α (≥ 0) is the damping factor to ensure that ρ + ∆ρ remains in the trust region of thequadratic approximation [10]. Convergence is judged by comparing the gradient of Eq. (2.3)
with a small threshold value, ε:

NV 
1 X
∂Q 2
< ε.
(2.11)
Nv
∂ρk
k=1

5

2.3

Fourier Transform Algorithms

In MEM analysis, F (hj )’s are calculated by Fourier transform, which is very time-consuming,from electron or nuclear densities, with a result that the CPU time for MEM analysis generally
depends on both N and NF . FFT makes it possible to process huge amounts of data with verylarge values of N and NF at high speed. Therefore, ERIS benefits MEM analyses of crystalline
materials with large unit cells and low symmetry. ERIS is also very suitable for analyzing
neutron diffraction data where nuclear densities tend to be confined in narrower spaces.
In ERIS, FFT is selected if
100N
N a N F nc >,(2.12)
ln N
where Na is the total number of voxels in the asymmetric unit, and nc is a factor to be selected
depending on the presence or absence of an inversion center at the origin (centrosymmetric: 1,
noncentrosymmetric: 2).
Parallel computation
For small-scale problems, a DFT routine making full use of multi-threaded parallel processing
with the OpenMP technology is automatically selected to avoid overhead caused by FFT. Thus,in both FFT and DFT, the processing ability of ERIS increases in a nearly linear fashion withincreasing number of CPU cores on the use of recent multi-core CPUs.

6

Chapter 3

CONSTRAINTS
With ERIS, S in Eq. (2.1) is maximized by imposing a linear combination of the generalized F
and G constraints.

3.1 F Constraint
Let NF be the total number of reflections, Fo (hj ) the observed structure factor, F (hj ) the
calculated structure factor, and σ(hj ) the standard uncertainty of |Fo (hj )|. The F constraintused in MEM analysis from diffraction data is formulated as
CF′ =

NF
X

|∆Fj |2 − NF ,

(3.1)

j=1

where ∆Fj is the normalized residual for the structure factor:∆Fj =Fo (hj ) − F (hj ).
σ(hj )(3.2)

In the actual implementation of the F constraint in ERIS, CF′ is normalized with respect to NF :
CF

=

CF′
NF

=

F
1 X
|∆Fj |2 − 1.
NF

(3.3)
N

(3.4)

j=1

3.2 G Constraint
Information about overlapped reflections in powder diffraction data can be introduced into MEM
analysis through the G constraint [17]:
′
CG

=

NG
X
Goj − Gj 2

σj

j=1

=

NG
X

− NG

|∆Gj |2 − NG

j=1

7

(3.5)

with

L
1/2
j
X

mi |F (hi )|2 Gj =  i=1 L
 .
j


X


mi

(3.6)

i=1

In Eqs. (3.5) and (3.6), NG is the total number of groups comprising overlapped reflections, Goj
is the sum of the integrated intensities of overlapped reflections in group j, Lj is the number ofoverlapped reflections in group j, and mi is the multiplicity of reflection i. Combination of CF′′ affords an integrated constraint, C :and CG
2
C2 =

3.3

′
CF′ + CG.
NF + NG(3.7)

Generalized Constraint

The classical F and G constraints are based on χ2 statistics, whose use assumes that experimental
errors in |Fo (hj )|’s are random with a Gaussian (normal) distribution. Although the use of χ2
constraint is justified by the Gaussian distribution of errors, it is not a suﬀicient condition toensure the Gaussian distribution of residuals between observed and estimated values. In the caseof MEM analysis from diffraction data, some low-Q reflections tend to have very large |∆Fj |’s,which leads to a noisy distribution of electron or nuclear densities estimated by MEM. Such acharacteristic in MEM analysis from diffraction data was theoretically explained by Jauch [18].
To reduce such an undesirable tendency of MEM, Palatinus and van Smaalen [19] proposed touse higher order moments of ∆Fj as a constraint.
A probability distribution of a random variable x is characterized by the values of its central
moments Mn . For the normalized Gaussian distribution, the central moments are defined as
Z ∞Mn (Gauss) =xn (2π)−1/2 exp −x2 /2 dx.
(3.8)
−∞

The moments of odd n are all zero while those of even n arenY
M2n (Gauss) =(2i − 1).

(3.9)

i=1

In the case of N samples of the variable x, the central moments Mn can be computed by
1 X n
xi .
N
N

Mn =

(3.10)

i=1

The generalized F constraint of order n is formulated as
CF′ n =F
X
1|∆Fj |n − NF ,Mn (Gauss)(3.11)

CF n =

CF′ n.
NF(3.12)N

j=1

The classical F constraint corresponds to the generalized constraint of order 2, CF 2 . The classicalG constraint is also generalized as
FX
1|∆Gj |n − NG .
Mn (Gauss)

N

′
CGn

=

j=1

8

(3.13)

The generalized F and G constraints of order n, are combined to give a Cn constraint:
Cn =

3.4

′
CF′ n + CGn.
NF + NG(3.14)

Linear Combination of Generalized Constraints

ERIS adopts a linear combination of the generalized F and G constraints with relative weights,
λn (n = 2, 4, 6, .....):XC=
λn C n(3.15)
n

with


GF
NF
X
n
n X
1

wj |∆Gj |  − Cwn ,Cn =wj |∆Fj | +(NF + NG )Mn (Gauss)
j=1(3.16)

j=1

where wj is the weighting factor (see 4), and Cwn is the criterion for convergence. When wj isunity, the ideal constraint is Cwn = 1 for any even n. Such a rigorous constraint seems to behardly satisfied in actual problems. Accordingly, ERIS always determines Cwn automatically so
as to satisfy Eq. (3.4) or (3.7) on use of the G constraint.
Information about central moments of the normalized residuals, ∆Fj and ∆Gj , is output tofile *.out, for example,
C2 =
C4 =
C6 =
C8 =
C10 =
C12 =
C14 =
C16 =

9.9992285E-01
4.0802412E+00
2.8819103E+01
2.0165493E+02
1.2533774E+03
6.8066503E+03
3.2373854E+04
1.3593284E+05

ln(C2 ) =ln(C4 ) =ln(C6 ) =ln(C8 ) =ln(C10) =ln(C12) =ln(C14) =ln(C16) =-0.000077
1.406156
3.361038
5.306558
7.133597
8.825655
10.385106
11.819916

The first column, C2–C16, lists the central moments of order n for |∆Fj |n and |∆Gj |n normalized
by those for Gaussian distribution. The second column, ln(C2)–ln(C16), gives their natural
logarithms.
If the distributions of ∆Fj and ∆Gj are Gaussian, values in the first column are all 1 while
those in the second column are all 0. Normalized central moments larger than 1 imply that
information in observed data is underestimated and not fully reconstructed by MEM. On thecontrary, normalized central moments smaller than 1 imply the overestimation of data, which
result in overfit to observed data.
In MEM analyses from X-ray diffraction data with traditional second-order constraints, thenormalized central moments of order n > 2 are typically larger than 1, as shown in the above
example. In such a case, increase λn of order 4 or higher to put restraints on them and bindthem to the ideal value. The central moment of a higher order may be smaller than 1 dependingon the type and quality of data, particularly when NF + NG is relatively small. In such a case,set λn such that λ2 = 1 and λn = 0 for n > 2, which corresponds to the traditional χ2 constraint.
On the use of the ZSPA algorithm, substitution of Eq. (3.15) for C in Eq. (2.6) will give rise
to a problem that only the Cn constraint with the highest order practically takes effect at the
beginning of MEM iterations. This is because the highest order Cn constraint can be orders of9

magnitude larger than the C2 constraint at early cycles of MEM iterations. Because the ZSPA
algorithm do not optimize the information entropy, and it is sensitive to calculation route, thefinal results will also be strongly constrained by the highest order Cn . To overcome such a(i)problem specific to the ZSPA algorithm, λn in Eq. (3.15) in the i-th iteration, λn , is adjusted
as follows. If ∆Fj ’s and ∆Gj ’s are kept close to Gaussian distribution during MEM iterations,(i)the order of Cn in the (i)-th iteration is roughly equal to (C2 )n/2 . Then λn is replaced by
λ(i)
n =

λn
(C2 )n/2−1.

(3.17)At any rate, be sure to check whether or not 3D distribution of electron/nuclear densities isphysically and chemically reasonable with VESTA [20].

10Chapter 4

WEIGHTING
4.1

Why weighting of observation is necessary

To obtain reasonable distribution of densities, it is reported that observed diffraction data need
to be weighted with a proper function. Without weighting, residuals of observed and calculated
structure factors, (Fo Fc ), of low-Q (large d-spacing) region tend to be very large.
The necessity of weighting in MEM for crystallography comes from intrinsic nature of crystals. Although the information theory assumes that the probabilities of individual events tooccur are independent of each other, electron/nuclear densities or their Patterson functions in
crystals are not random but smooth, having strong correlation with vicinal positions. Therefore,
the left-hand values of Eq. (2.4) is also smooth in real space. In reciprocal space, such natureof crystals causes systematic decrease of structure factor amplitudes with decreasing d-spacing,
and amplitudes of the left-hand values of Eq. (2.5) also decrease with decreasing d-spacing. On
the other hand, the right-hand values of Eqs. (2.4) and (2.5) are functions of ∆F , which should
not depend on d-spacing but reflect random errors of observations. MEM formalism forces thesetwo different kinds of terms to be equivalent, and this is why a proper weighting is necessary to
suppress systematic dependency of ∆F to d-spacing.

4.2

Weighting based on exponential of d-spacing

4.3

Weighting based on n-th power of d-spacing

Weighting in the F constraint on the basis of the lattice-plane spacing was first proposed byde Vries et al [21]. Its effectiveness was later confirmed by some other researchers including
Hofmann et al. [22].
Let sj be the reciprocal-lattice vector (= ha∗ + kb∗ + lc∗ ) for reflection j, x the real number
for weighting, and dj the lattice-plane spacing (= 1/|sj |). Then, the weighting factors, wj , in
Eq. (3.16) is given by
"N
#
F
X
1
1
wj =
|sj |x|si |x
i=1
"N
#
F
X
x
x
= dj
di .
i=1

If x = 0, no weight is imposed in the same way as with PRIMA [9].

11

(4.1)

This weighting scheme is suitable for X-ray diffraction where parts of low-Q reflections often
have very large ∆Fj values. Values of around 2 are usually recommended in X-ray powder
diffraction; x is increased as the quality of intensity data is improved. In the case of singlecrystal X-ray diffraction, de Vries et al. [21] empirically found x = 4 to be optimum. On theother hand, this approach is not suited for the analysis of neutron diffraction data, where bc
values of constituent elements remain constant regardless Q, particularly those measured at low
temperature. Therefore, x is usually set at 0 in neutron diffraction.
Ad-hoc weighting described above will be effective, particularly, in MEM analysis from Xray diffraction data, where σj (hj )’s tend to be estimated at unreasonably large values for low-Q
reflections.

4.4

Weighting based on multiplicity of reflections

12

Chapter 5

ADJUSTMENT OF STANDARD
UNCERTAINTIES
5.1

Powder Diffraction Data (*.fos)

In powder diffraction, σ(hj )’s are estimated on the basis of the law of propagation of errors incombination with counting statistics:|Fo (hj )|σ(hj ) =
2

(

2 
 )1
σ(s) 2 2
1+,
EIo (hj )s(5.1)

where E is the factor to adjust σ(hj ), Io (hj ) is the observed integrated intensity, s is the scale
factor, and σ(s) is the standard uncertainty of s [6]. Both s and σ(s) results from Rietveld
analysis or whole-pattern fitting with RIETAN-FP. E is selected in such a way that electronor nuclear-density distribution which is physically and chemically reasonable results from MEM
analysis. Convenient bash scripts, MPF_multi.command, for automatic MPF analyses enable
us to change E as specified in *.prf (see 6.2.2).
In the case of angle-dispersive-type powder diffraction where the step width, ∆2θ, is usuallyconstant, E is approximately equal to 1/∆2θ (unit: rad−1 ), which is output as E(SCIO) in *.lst
by RIETAN-FP1 [23]. On the other hand, E depends on the step width, ∆t, of the time-offlight (TOF) in TOF neutron powder diffraction; a utility called Alchemy [24, 25] can be used
to convert files output by GSAS and Fullprof into *.fos or *.mem (see 5.2).

5.2

Single-Crystal Data (*.mem)

The estimation of σ(hj ) is diﬀicult even in single-crystal X-ray or neutron diffraction; σ(hj )’s areoften underestimated or overestimated. With ERIS, σ(hj )’s can be changed with the adjustment
factor, E, according toσ (hj )σ ′ (hj ) = √(5.2)
Ewithout any conversion of σ (hj )’s in *.mem.

1

http://fujioizumi.verse.jp/download/download_Eng.html

13Chapter 6

INSTALLATION AND
EXECUTION
6.1

Installation

ERIS runs on Windows, OS X, and Linux. URLs of their files are as follows:
Windows
http://jp-minerals.org/eris/archives/ERIS.zip
OS X, 64 bit application
http://jp-minerals.org/eris/archives/ERISLinux, 64 bit x86_64
http://jp-minerals.org/eris/archives/ERIS-x86_64.tar.bz2
Linux, 64 bit Arm64
http://jp-minerals.org/eris/archives/ERIS-arm64.tar.bz2
Once the above archive files are downloaded and extracted, the executable file “ERIS” (orERIS.exe) can be directly executed without any additional installation process. To install ERIS
to interoperate with RIETAN-FP, place the extracted files in a specific directory. On Windows, extract the zip files and move the resulting files to C:\Program Files\RIETAN_VENUS\.
On macOS, mount ERIS.dmg by double-clicking its icon and move all the files to folder /Applications/RIETAN_VENUS/. Both versions can be run in combination with a shell script,
MPF_multi.command, written in bash for MPF. MPF_multi.command and its manual are
included in distribution files of the RIETAN-FP–VENUS package.

6.2

Execution

6.2.1

Interactive mode

Run ERIS by double-clicking its icon or entering the name of its executable binary file in acommand line. Then, the program asks questions that must be answered by the user. In what
follows, these questions are represented by boxed texts with gray background.

# The name of the MEM data set file, *.fos or *.mem.

14

Type the absolute path of an input file, for example, C:\users\user_name\sample\sample.fos(Windows).
# The type of the MEM data set file.
# 0. (a) X-ray diffraction data or (b) neutron diffraction data of a compound#containing no element with a negative bc value.
# 1. Neutron diffraction data of a compound containing at least one element#with a negative bc value.
This option is set for the backward compatibility of input files because the file format of *.mem
is different depending on dataset types.
# From which densities will you start?
# 0. Uniform densities.
# 1. Restart from densities recorded in the 3D densities file (*.pgrid).
If you choose the second option, initial densities are input from *.pgrid. This option is usefulwhen you restart MEM analysis after precedent iterations. Note that this option is different
from the choice of prior density τk in Eq. (2.1) in strict sense, although on the use of ZSPA
algorithm, use of non-uniform prior is approximated by restarting calculation from τk . For rigorous analyses using non-uniform prior, use L-BFGS or Cambridge algorithms and specify theprior data on the later question. MEM analysis can only be restarted from densities recorded
in *.pgrid output by ERIS.
# Optimization algorithm.
# 0. 0th order single-pixel approximation (ZSPA).
# 1. The Limited-memory BFGS algorithm (L-BFGS).
# 2. The Cambridge algorithm (Obsolete).
Read 2.3 for details in the above two algorithms.
# Which initial Lagrangian multiplier will you use (LM_type)?
# 0. The value input by user.
# 1. The value calculated by ERIS.
# 2. The value written in the MEM data set file.
This question appears only on use of the ZSPA algorithm, and option 2 appears only when theformat of the MEM dataset file is *.mem.
# The initial Lagrangian multiplier, lambda
This question is asked only when 0 is input in the previous question.
# A parameter x to impose weighting factors on the basis of# lattice-plane spacing.
Parameter x is included in Eq. (4.1). Input x = 0 in classical MEM analysis where no weightis imposed in the F constraint. If x > 0, it is regarded as x in Eq. (4.1). As described in4, typically x = 0 in neutron diffraction, x = 1–2 in X-ray powder diffraction, and x = 4 is

15

optimum in single-crystal X-ray diffraction.

# The coefficient, t, to adjust the Lagrangian multiplier.
This question appears only on use of the ZSPA algorithm. Input a value between 0.05 and 0.1
for fast computation. If λ is too large, MEM iterations diverge and never reach the solution. If
λ is too small, the convergence of the MEM equation is ensured whereas the computation time
may increase considerably, with no convergence achieved on specification of a relatively small
maximum number of cycles. The coeﬀicient t allows us to find the best λ value automaticallythrough multiplying λ by 1 + t in every cycle of MEM iterations until λ becomes too large toconverge the MEM equation, which is different from the manner of changing λ in PRIMA [9].
# The coefficient, E, to adjust estimated standard uncertainty of# structure factors.
For details in E, see 5.1 and 5.2.
# Will you save a feedback data file?
# 0. Yes (output only individual F data).
# 1. Yes (output all the F data including those for grouped reflections#
and estimated for unobserved reflections).
# 2. No.
In MPF, option 1 is usually selected to include F (hj )’s of high-Q reflections whose profiles are
partly lacking.
# Will you save a e(GAUSS) distribution data file?
# 0. Yes (raw data file, *_eps.raw).
# 1. No.
When option 0 is selected, a text file storing histogram of|Fo (hj )| − |F (hj )|σ(hj )
will be output. See also section 8.1.
# Fractions of lambda for generalized constraints.
# (8 parameters; l_2, l_4, l_6, l_8, l_10, l_12, l_14, l_16)
Eight parameters, λn (n = 2 − 16), for the fractions of generalized constraints with orders of n(see 3.4).
# Which prior\index{prior} densities will you use?
# 0. Uniform densities.
# 1. Densities recorded in the 3D densities file (*_prior.pgrid).
This option appears only on the use of the L-BFGS algorithm. When option 1 is selected, τk in
Eq. (2.1) will be read in from 3D densities file. To use non-uniform prior densities in the ZSPA

16

algorithm, restart the calculation using a 3D data file (*.pgrid) storing prior densities.
# Will you save a preferences file?
# 0. No.
# 1. Yes (*.prf is output).
If *.prf is output, all the above parameters are stored in it and it can be reused as a template
file for subsequent MEM analyses.
After answering all the questions, the conditions of MEM analysis are displayed in the screen.
Before starting MEM iterations, ERIS asks the maximum number of iterations.
# Maximum number of cycles.
The iterations will continue until the convergence is achieved (CONSTR ≤ 1). When the convergence is reached, ERIS terminates after outputting CPU times.

6.2.2

Running ERIS using condition files

If the entire conditions of MEM calculations in filename.prf have already been input by
modifying a template file with a text editor, ERIS can be executed in several different ways.
Let the name of *.prf be sample.prf. Then, ERIS can be run from the command line bytypingERIS sample.prf
to read in sample.prf. When no environment variable PATH is set to a directory under which
the program is placed, type the full path of the program, e.g.,"C:\Program Files\RIETAN_VENUS\ERIS.exe" sample.prf
if ERIS.exe is stored in folder “C:\Program Files\RIETAN_VENUS\” on Windows. Of course,
sample.prf has to be placed in the current folder in the above case. A simple batch file (Windows)
or a shell script (OS X or Linux) where such a command is recorded would be more convenient.
The most practical and convenient way of carrying out MPF by ERIS is the use of MPF_multi.command
because E in Eq. (5.1) can be automatically changed during a series of MPF analyses. For
details in this bash script, read the manuals of MPF_multi.command, MPF_multi_Win.pdf(Windows) or MPF_multi_Mac.pdf (OS X), included in a distribution file, documents.zip.1

1

http://fujioizumi.verse.jp/download/download_Eng.html

17Chapter 7

INPUT FILES
7.1

Condition Files• *.eris: An input file storing all the conditions of MEM calculations.
• *.prf: An input file of PRIMA storing conditions of MEM calculations. This file keeps backward compatibility with PRIMA while some new parameters are added for new features
in Dysnomia and ERIS.
• *.alb: An input file of ALBA storing conditions of MEM calculations.

7.2

Intensity Files• *.fcf, *.cif: Standard CIF file format.
• *.fos: Powder diffraction data generated by RIETAN-FP [6].
• *.mem: (a) Single-crystal data or (b) powder data where standard uncertainties of observed
structure factors are given.
• *.baymem: .
• *.ffo: Powder diffraction data generated by RIETAN-FP [6].
• *.hkl: .

7.3

File format of *.eris

7.3.1

title

• Value: string of characters up to 80 characters long
• Default: none
• Description: Title of the calculation.

7.3.2

algorithm• Value: 0/1• Default: 118

• Description: Switch algorithm of MEM calculation.
0. ZSPA algorithm.
1. L-BFGS algorithm.

7.3.3

beam_source• Value: 0/1• Default: 0• Description: From which experiment was the diffraction data obtained? This keyword only
affects on the Wilson plot performed to calculate scale factor in the MEP analysis.
0. X-ray diffraction data.
1. Neutron diffraction data.

7.3.4

cell

• Value: a b c α β γ• Default: none
• Description: Gives unit cell parameters for input data *.hkl, *.ffo, in which unit cell parameters are not recorded.

7.3.5

data

• Value: filename• Default: compulsory keyword – no default• Description: Gives the name of diffraction data file. It can be specified as a relative pathfrom the place of *.eris file.

7.3.6

delta_d• Value: positive real number• Default: 0• Description: Maximum difference in d/Angstrom in grouped reflections. Reflections withthe d-spacing closer than delta_d will be considered as overlapped and treated as groupedreflections.

7.3.7

elements• Value: positive integer, followed by a pair of element name and real number in each line
• Default: 0• Description: Number of chemical species in the unit cell. It is followed by lines of equal
numbers, each line describing a pair of element names and amounts in the unit cell. These
data are used for the Wilson plot performed to calculate scale factor in the MEP analysis.

197.3.8epsilon• Value: positive real number• Default: 0• Description: A small value ε, used as a criterion for convergence of optimization. The
convergence of the L-BFGS algorithm is judged by
v
u NV


uX  ∂Q 2
∂S
t
ρk
<ε
,
(7.1)
∂ρk
∂ρk
k=1

7.3.9

lambda• Value: positive real number• Default: automatically determined• Description: The initial Lagrangian multiplier λ used in the ZSPA algorithm. This keyword
has no effect on the use of the L-BFGS algorithm.

7.3.10lambda_coef• Value: positive real number• Default: 0.05
• Description: A coeﬀicient t, to adjust Lagrangian multiplier λ. This keyword is used onlyin the ZSPA algorithm and has no effect on the use of the L-BFGS algorithm.

7.3.11

max_cycles• Value: positive integer• Default: 10000• Description: Maximum number of MEM cycles.

7.3.12resolution• Value: positive real number• Default: 0.1• Description: Resolution of real space voxels in angstrom.

7.3.13

restart• Value: 0/1• Default: 0• Description: From which densities will you start the calculation?
0. Start from flat (or prior) density.
1. Start from the results of previous run stored in *.pgrid file.

207.3.14

scale_sigma• Value: real value• Default: 1• Description: Scale input σ(F ) as σ(F )′ = scale_sigma × σ(F ).

7.3.15space_group• Value: [space group number] [setting number]• Default: 1 1• Description: Gives the space group number and setting number.

7.3.16scio• Value: real value• Default: 1√• Description: Scale input σ(F ) as σ(F )′ = σ(F )/ scio.

7.3.17

use_prior• Value: 0/1• Default: 0• Description: Which prior densities will you use?
0. Flat density.
1. Prior density recorded in *_prior.pgrid file.

7.3.18weight_cn• Value: eight real numbers• Default: 1 0 0 0 0 0 0 0• Description: Fractions of lambda for generalized constraints.

7.3.19weight_m• Value: real number• Default: 0√• Description: Scale input σ(F ) as σ(F )′ = c × σ(F )/ m, where m is multiplicity of thereflection F and c is a constant value automatically determined so as not to change total
scale of ⟨σ(F )⟩. See section 4.4 for more detail.

21

7.3.20

weight_d• Value: [auto/exp/power] [real number]• Default: auto
• Description: Weighting of observed reflections based on lattice-plane spacing d.

7.4

File format of *.prf

The content of *.prf storing various flags and specifications for MEM analysis is described in
6.2.1, and execution of ERIS and the subsequent input of *.prf in 6.2.2.

7.5

File format of *.alb

7.6

File format of *.mem

A line to give values (POP1, POP2, IPTYP, JPH, JPK, and JPL) related to preferred orientation iscurrently a dummy one. Never analyze powder diffraction data where preferred orientation wasobserved.
Grid numbers along a, b, and c axes should be selected in such a way that symmetry elements
such as mirrors, rotation axes, and inversion centers coincide with intersections of grid lines. In
addition, the grid numbers must be appropriately set by considering lattice parameters, a, b,
and c. On the use of FFT, the best performance is achieved when division number along eachaxis is represented as
2p × 3q × 4r × 5s × 7t × 11u × 13v ,where p, q, r, s, t, u, and v are integers equal to or larger than 0. Any other integers are also
allowed; the calculation time is proportional to N log N even for prime numbers of N .
Beware that the sum, T − , of negative bc ’s in the unit cell should be set at 0.0 even if thesample contains no element with a negative bc value, e.g., H, Li, Ti, or Mn.
In a solid solution where two or more elements occupy the same site, a special procedure isrequired for calculating the sum, T + , of positive bc ’s and/or the sum, T − , of negative bc ’s in theunit cell [26]. Suppose a virtual chemical species with an average coherent-scattering length,b̄c , calculated from the occupancies of the constituent elements. Then, add b̄c multiplied by the
number of the virtual chemical species in the unit cell to T + if b̄c > 0 or T − if b̄c < 0.

7.7

File format of *.fosWith Alchemy [24], text files with the following two formats can be obtained from output files,*.lst, of GSAS:*.fos
Regardless of ID(neg.), both positive and negative values of total scattering amplitudes
are always input. If ID(neg.) = 0, the second amplitude must be 0.0 (dummy).
*.mem
If ID(neg.) = 0, only a positive total scattering amplitude is input, followed by λ inthe same line. If ID(neg.) = 1, both positive and negative values of total scattering
amplitudes are input, followed by λ as well.

22

Chapter 8

OUTPUT FILES
8.1

Output Files*.out
The standard output file of ERIS.
*.pgridThree-dimensional voxel data storing electron or nuclear densities in a binary format. The
content of this file can be visualized in three dimensions using VESTA [20] together with acrystal-structure model if necessary.
*.fba
A list of structure factors calculated by Fourier transform from electron/nuclear densities resulting from MEM analysis. This file is used by RIETAN-FP [27] for whole-pattern fitting in
MPF [3–5].
*.raw
A text file storing histogram of the difference between the absolute Fo (hj ) and F (hj ) values
divided by the standard uncertainty, σ(hj ), of |Fo (hj )| to give a sample of the random variablewith normalized Gaussian distribution, ϵ(Gauss) [19]:ϵ(Gauss) =|Fo (hj )| − |F (hj )|σ(hj )(j = 1−NF ).

This file is useful for evaluating distribution of errors resulting from MEM analysis.

8.2

File format of *.pgrid

Volumetric data are composed of regular grids in 3D space. A volume element, “voxel,” at eachgrid point represents a value on the grid point. Voxels are analogous to pixels, which represent
2D image data.

8.2.1

Periodic grid vs. general grid

In general, two types of formats are used to record volumetric data in files: general and periodic
grids. Figure 8.1 schematically illustrates the general concepts of the two kinds of the formats.

23

The general grid is a uniform one spanned inside a bounding box for molecules and a unit
cell for crystals. For crystal structures, part of data in the general grid are redundant owing to
the periodicity of the data. For example, a numerical value at (1, 1, 1) are equal to that at theorigin, i.e, (0, 0, 0). Grids where these redundant points have been omitted are called periodic
ones.

A

B

: irreducible data points: redundant points (periodic replicas)

Figure 8.1: Two types of the grids for volumetric data on a plane. (A) general gridand (B) periodic grid.

8.2.2

Pseudocode to write *.pgrid

This is a binary file storing a series of physical values in each voxel followed by some header
data. Part of the C++ source code to write the header part is as follows.
char title[80]; // 80 characters of title that ends with '\0'.
int version[4] = {3, 0, 0, 0}; //The current version number is 3 0 0 0
int gType;// 0: general grid (*.ggrid); 1: periodic grid (*.pgrid).
int fType;// 0: raw data; 1: paired record of index and value of voxel.
int nVal;// 1 or 2; Number of data values for each voxel.
int dim;// Only dimension=3 is currently supported.
int nVox[3]; // Number of voxels along a, b, and c axis.
// Total number of voxels to be recorded in the file.
// nAsym = nVox[0]*nVox[1]*nVox[2] in the case of fType = 1.
int nAsym;
float cell[6]; // Unit cell parameters: a, b, c, alpha, beta, gamma// Allocate voxel data
float *rho = new float[nVox[0] * nVox[1] * nVox[2] * nVal];// Set values to the above parameters before writing them.
// Write header
fwrite(version, sizeof(int),
4, fptr);fwrite(title,
sizeof(title), 1, fptr);
fwrite(&gType, sizeof(int),
1, fptr);24

fwrite(&fType,fwrite(&nVal,
fwrite(&dim,
fwrite(nVox,
fwrite(&nAsym,
fwrite(cell,sizeof(int),
1, fptr);sizeof(int),
1, fptr);sizeof(int),
1, fptr);sizeof(int),
3, fptr);sizeof(int),
1, fptr);sizeof(float), 6, fptr);

For the raw data type (when fType = 0), binary data of each voxel is output in the following
order.
if(fType = 0){
for(int k=0; k<nVox[2]; k++){for(int j=0; j<nVox[1]; j++){for(int i=0; i<nVox[0]; i++){m = k*nVox[0]*nVox[1] + j*nVox[0] + i;
for(int n=0; n<nVal; n++){fwrite(&rho[nVal*m + n], sizeof(float), 1, fptr);
}
}
}
}
}
Note that values of nVox are not the same in the case of fType = 0 (*.ggrid) and fType = 1(*.pgrid). When saving periodic data in the former format, i.e., as the general grid format,
each value of nVox is 1 voxel larger than that of the latter format. The difference is due to theredundant data points at x = 1, y = 1, and z = 1, which are recorded in *.ggrid but not in*.pgrid. For non-periodic data, always use *.ggrid instead of *.pgrid.
When the data is periodic and space group symmetry is higher than -1, the file size canbe reduced by only recording data of symmetrically independent voxels. In this case, set theparameter fType = 1 and write a list of symmetry operations before voxel data.
// Additional parameters used when fType = 1.
int nPos; // Number of symmetry operations to be written.
// nCen: A flag if the recorded symmetry operations need to be expanded by// inversion operation.
// 0 for non-centrosymmetric space groups or when all symmetry operations// are recorded.
// 1 for centrosymmetric and only half of symmetry operations are recorded.
// VESTA and ERIS always record full set of symmetry operations and set// nCen=0 to allow arbitrary origin shift.
int nCen;// nSub: Number of lattice points.
// nSub>1 in the case of complex lattices such as I, F, A, B, C etc.
// VESTA and ERIS always record full set of symmetry operations and set// nSub=1 to allow arbitrary origin shift.
int nSub;// A list of lattice points in integer.
// For the P-lattice, or when all symmetry operations are recorded,// subPos[1][3] = {0, 0, 0};25

// subPos[2][3] = {{0, 0, 0},{nVox[0]/2, nVox[1]/2, 0}};C-lattice// subPos[2][3] = {{0, 0, 0},{nVox[0]/2, nVox[1]/2, nVox[2]/2}}; I-lattice// etc.
int subPos[NSUB][3];// A 4x4 matrix class storing symmetry operation in column-major format.
// The first 3x3 part represent rotation.
// The 4th column represent translation.
// An operator (i,j) in the following code is defined to return a value of// i-th raw and j-th column in double.
Matrix *symOP;// A list of indices of voxel positions in asymmetric unit.
int *ijk;// Set values to the above parameters before writing them.
if(fType = 1){
fwrite(&nPos,
fwrite(&nCen,
fwrite(&nSub,

sizeof(int), 1, fptr);
sizeof(int), 1, fptr);
sizeof(int), 1, fptr);// Write symmetry operations// A position X is transformed to X' by X' = symOP[i] * X.
for(i=0; i<nPos; i++){for(j=0; j<3; j++){for(k=0; k<3; k++){int a = (int)(symOP[i](k,j));fwrite(&a, sizeof(int), 1, fptr);}}
for(j=0; j<3; j++){int a = (int)(symOP[i](j,3) * nVox[j]);fwrite(&a, sizeof(int), 1, fptr);}}
for(i=0; i<nSub; i++){fwrite(subPos[i], sizeof(int), 3, fptr);}// Write voxel data in asymmetric unit.
// ijk is 3 indices of position of i_th voxel.
for(i=0; i<nAsym; i++){m = ijk[i*3+2]*nVox[0]*nVox[1] + ijk[i*3+1]*nVox[0] + ijk[i*3];
fwrite(&m,sizeof(int),
1, fptr);for(int n=0; n<nVal; n++){fwrite(&rho[nVal*m + n], sizeof(float), 1, fptr);
}
}
}

26

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Indexcharge flipping, 2
F constraint, 7
G constraint, 7
general grid, 23
Generalized constraint, 8
periodic grid, 23prior, 3, 15, 21
voxel, 2329