Global Parameters in Rietveld Refinement

James A. Kaduk, IIT kaduk@polycrystallography.com

What determines the intensities? (1) Structure Factors

• Atomic positions • Atomic scattering factors • Occupancies • Displacement coefficients • Lattice parameters • Symmetry

What determines the intensities? (2) Global Parameters

• Concentration • Incident intensity • Background • Diffuse scattering • Extinction • Absorption

• Preferred orientation • Multiplicity • Lp factor • Profile function • Diffractometer

parameters • …

Background

• Crucial to get right – affects integrated intensities (and thus the structure) – especially the displacement coefficients

• Interacts with the profile function • Peak tails • Use a few parameters as possible • Crystalline sample – slowly varying • Background parameters may be highly-correlated

Physics of Background

P. Riello, G. Fagherazzi, D. Clemente, and P. Canton, “X-ray Rietveld Analysis with a Physically Based Background”, J. Appl. Cryst., 28(2), 115-120 (1995).

Background Contributions • Air scattering • Incoherent scattering • Diffuse scattering

– Thermal disorder – First-kind lattice disorder

• Amorphous scattering • Thermal diffuse scattering • Short-range order • Small-angle scattering • Scattering off diffractometer parts, specimen holders +

2[1 exp( )]bk inc inc dis coh airi i i i iY K I K ks I Y= + − − +

1/ sin(2 )airiY θ∝

10 15 20 25 30 35 40 45Two-Theta (deg)

0

250

500

750

1000

1250

Inte

nsity

(Cou

nts)

[kadu1406.raw] SRM 660a La B6 (30,10,0.1,2.5,1) JAK[kadu1405.raw] 20307-1-4 w/ Si (30,10,0.1,2.5,1) JAK[kadu1411.raw] Office Snax sugar (30,10,0.1,2.5,1) JAK

Background Functions

Shifted Chebyshev Polynomials of the First Kind

11

( )N

b j jj

I B T x−=

=∑T0(x) = 1, T1(x) = x, T2(x) = 2x2-1, Tn+1(x) = 2xTn(x) - Tn-1(x)

min

max min

2(2 2 ) 12 2

x θ θθ θ

−= −

−

Cosine Fourier Series

12

cos[ ( 1)]N

b jj

I B B x j=

= + −∑

Polynomial

11

0

2 1m

b mm

I BBKPOS

θ=

= −

∑

Diffuse Scattering The Debye Equation

2

4 sinsin

( ) ( ) 2 ( ) ( ) 4 sin

ij

n i jijn i j

r

I f f f r

π θλ

θ θ θ θ π θλ

= +

∑ ∑∑

P. Debye, Annalen der Physik, 351, 809 (1915)

Diffuse Scattering in GSAS (#1)

2sin( ) 1exp2DS

RQI A UQRQ

= −

Q = 2π/d

GSAS Diffuse Scattering Function #1 Terms

2θ, deg

0 20 40 60 80 100 120 140 160

Back

grou

nd In

tens

ity

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

R = 1.6, U = 0.05R = 2.4, U = 0.05R = 1.6, U = 0.2

Differences in Diffuse Scattering Terms

2θ, deg

0 20 40 60 80 100 120 140 160

Back

grou

nd In

tens

ity

0

1e-5

2e-5

3e-5

4e-5

5e-5

1.6/0.05-2.4/0.051.6/0.2-2.4/0.2

χ2 = 1.112

Profile Coefficients

• Crucial to getting the right answers • Structure/intensities/overlap/tails • Valuable information in profile coefficients

Convolution

The Fundamental Parameters approach!

( )( ) ( ) ( ) ( ) ( )f g t f g t d f t g dτ τ τ τ τ τ∞ ∞

−∞ −∞∗ = ⋅ − = − ⋅∫ ∫

http://en.wikipedia.org/wiki/Convolution

Convolution

• Convolution of one function (input) with a second function (impulse response) gives the output of a system

• A weighted moving average • In optics, “blur” is described by convolution

ESRF BM16 (now ID31) Second Monochromator Crystal Rocking Curve

Left; perfect Si(111) Darwin profile. Right: perfect Si(111) reflection convoluted with first crystal strain function. Center: experimental data and fit by convolution of left and right curves. O. Masson, E. Dooryhee, and A. N. Fitch, “Instrument line-profile synthesis in high-resolution synchrotron powder diffraction”, J. Appl. Cryst., 36, 286-294 (2003).

Na2Ca3Al2F14 (921) reflection. From left to right: the incident beam source profile, the transfer function of the monochromator, the pure sample profile, the reflection profile of the analyzer, and the axial divergence asymmetry function. Masson, Dooryhee, and Fitch, in A. Le Bail, “The Profile of a Bragg Reflection for Extracting Intensities”, in R. E. Dinnebier and S. J. L. Billinge, Powder Diffraction: Theory and Practice, RSC Publishing (2008).

Bragg-Brentano Diffractometer

A. Kern, “Profile Analysis”, in A. Clearfield, J. Reibenspies, and N. Bhuvanesh, Principles and Applications of Powder Diffraction, Wiley (2008).

Profile Contributions

Epsilon, degree

-0.10 -0.05 0.00 0.05 0.10

Inte

nsity

(arb

itrar

y un

its)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

source flat surface axial divergence transparency receiving slit

Profile Contributions Effect Equation Range

X-ray Source exp(-k12ε2)

k1 = 1.67(FWHM) -∞ to +∞

Flat Surface |ε|-1/2 -(γ2cotθ)/114.6 to 0 γ = divergence

Axial Divergence |2εcotθ|-1/2 -(δ2cotθ)/(4×57.3) to 0 δ = axial divergence

Transparency exp(k4ε) k4 = (4µR/114.6)sin2θ

-∞ to 0

Receiving Slit -(FWHM)/2 to + (FWHM)/2

H. P. Klug and L. E. Alexander, X-ray Diffraction Procedures, Wiley (1974).

Plus the Cu Kα Profile

J. Hartwig, G. Hölzer, J. Wolf, and E. Förster, “Remeasurement of the Profile of the Characteristic Cu Kα Emission Line with High Precision and Accuracy”, J. Appl. Cryst., 26, 539-548 (1993).

More on Emission Profiles

G. Hölzer, M. Fritsch, M. Deutsch, J. Härtwig, and E. Förster, “Kα1,2 and Kβ1,3 emission lines of 3d transition metals”, Phys. Rev. A, 56, 4554-4568 (1997).

Hugo Rietveld’s low-resolution neutron diffraction peaks were Gaussian (determined mainly by the neutron spectral distribution, the monochromator

response function, and the divergences of the Soller collimators).

He used the “Caglioti” function to describe the widths.

H. M. Rietveld, “A Profile Refinement Method for Nuclear and Magnetic Strutcures”, J. Appl. Cryst., 2, 65-71 (1969).

G. Caglioti, A. Paoletti, and F. P. Ricci, “Choice of Collimators for a Crystal Spectrometer for Neutron Diffraction:,

Nucl. Inst., 3, 223-228 (1958).

FWHM2 = Utan2θ + Vtanθ + W

FWHM2 = Atan2θ + Btanθ + C + Dcot2θ

R. W. Cheary and J. P. Cline, “An Analysis of the Effect of Different Instrumental Conditions on the Shapes of X-ray Powder Line Profiles”, Adv. X-ray Anal., 38, 75-82 (1995).

Specimen Contributions

Size and (micro)Strain

Size Broadening

P. Scardi, “Microstructural Properties: Lattice Defects and Domain Size Effects”, in R. E. Dinnebier and S.J. L. Billinge, Powder Diffraction: Theory and Practice, RSC Publishing (2008)

Integral Breadth

( )

2

2

2 2

2

0

sin ( )( ) 1( )

sin ( )(0) ( )( )s

Nas dsI s ds as Nas

NasI Na DLimas

ππ

βπ

π

∞∞

−∞−∞

→

= = = =∫∫

Convert to 2θ space from reciprocal space:

(2 )cosK

Dβλ

β θθ

=

Scherrer Constants for Various Crystallite Shapes

Shape K (FWHM) K (integral breadth) Sphere 0.89 1.07 Cube 0.83-0.91 1.00-1.16

Tetrahedron 0.73-1.03 0.94-1.39 Octahedron 0.82-0.94 1.04-1.14

J. I. Langford and A. J. C. Wilson, “Scherrer after sixty years: A survey and some new results in the determination of crystallite size”, J. Appl. Cryst., 11, 102-113 (1978)

Shape?

P. Scardi, “Microstructural Properties: Lattice Defects and Domain Size Effects”, in R. E. Dinnebier and S.J. L. Billinge, Powder Diffraction: Theory and Practice, RSC Publishing (2008)

P. Scardi and M. Leoni, “Diffraction line profiles from polydisperse

crystalline systems”, Acta Cryst. Sect. A, 57,

604-613 (2001).

0 2 4 6 8 10 12 140

5

10

15

20

25

30

35

40 TEM WPPM

fre

quen

cy

grain diameter D (nm)

Size Distribution in Ceria Powder

Strain Broadening

Macrostrain

2 sin0 2 sin 2 cos0 2 sin 2 cos

2 2 tan 2 tan

ddd d d

d dd

d

λ θθ θ θθ θ θ

θ θ ε θ

== += ∆ + ∆

∆∆ = − = −

Microstrain

1/ 22(2 ) tanβ θ ε θ∝

Microstrain

P. Scardi, “Microstructural Properties: Lattice Defects and Domain Size Effects”, in R. E. Dinnebier and S.J. L. Billinge, Powder Diffraction: Theory and Practice, RSC Publishing (2008)

Anisotropic Strain P.W. Stephens, “Phenomenological model of anisotropic peak broadening

in powder diffraction”, J. Appl. Cryst., 32, 281-289 (1999)

2 2 22

2 2 21 2 3 4 5 62

2

,

2

2 4 4 4 2 2400 040 004 220

2 2 2 2 3202 022 310 103

1

1

( )

( )

4( ) 3(

) 2(

hkl

hkl

hkl iji j i j

H K Lhkl HKL

HKL

hkl

M Ah Bk Cl Dkl Ehl Fhkd

M h k l kl hl hkd

M MM C

M S h k l

H K LM S h S k S l S h kS h l S k l S h k S

α α α α α α

σα α

σ

σ

= = + + + + +

= = + + + + +

∂ ∂=

∂ ∂

=

+ + =

= + + +

+ + + +

∑

∑

3

3 3 3 3031 130 301 013

2 2 2211 121 112

)

3( )

hlS k l S hk S h l S kl

S h kl S hk l S hkl+ + + +

+ + +

Functional Forms of Size and Strain Broadening

2 theta, degrees0 20 40 60 80 100 120 140

Rel

ativ

e Pe

ak W

idth

0

2

4

6

8

size strain size + strain

Real peaks have both Gaussian and Lorentzian (Cauchy) components

epsilon, deg

-0.2 -0.1 0.0 0.1 0.2

Inte

nsity

0

5

10

15

20

25

30

Gaussian Lorentzian

Same FWHM and area!

Profile Equations

2, 2

2 ln 2 4ln 2expi kk k

IH H

ε −

=

1

2, 2

2 4( 2 1)1i kk k

IH H

επ

− −

= +

Gaussian Lorentzian

S. A. Howard and K. D. Preston, “Profile Fitting of Powder Diffraction Patterns”, in D. L. Bish and J. E. Post, Modern Powder Diffraction (1989), p. 217-275.

SRM 660a LaB6

So use combination of Gaussian and Lorentzian

Voigt (convolution) pseudo-Voigt (sum)

GSAS Profile Function #2 (3-5) pseudo-Voigt

2 22tan tan

cosPU V Wσ θ θθ

= + + +

( )cos cos tancos

X ptec Y stecϕγ ϕ θθ

+= + +

2 cos sin 2tan 2

if asymzero shift trnsθ θ θθ

∆ = + + +

Size Broadening

18000( )iso

inst

KLX X

λπ

=−

18000( )inst

KLX ptec X

λπ

=+ −

18000( )inst

KLX X

λπ⊥ = −

Strain Broadening - Isotropic

( )100%18000 instS Y Yπ

= −

Strain Broadening - Anisotropic

( )100%18000 instS Y stec Yπ

= + −

( )100%18000 instS Y Yπ

⊥ = −

2

( ) 100%18000

H K LS

HKL

dS hkl h k lπ= ∑

Constant Microstrain Surface

Asymmetry (Rietveld)

H. M. Rietveld, “A Profile Refinement Method for Nuclear and Magnetic Structures”, J. Appl. Cryst., 2, 65-71 (1969). C. J. Howard, J. Appl. Cryst., 15, 615-620 (1982).

L. W. Finger, D. E. Cox, and A. P. Jephcoat, “A Correction for Powder Diffraction Peak Asymmetry due to Axial Divergence”,

J. Appl. Cryst., 27, 892-900 (1994).

GSAS Profile Functions #3-5

S/L = sample “half height”/diffractometer radius H/L = slit “half height”/diffractometer radius

6/240 = 0.025

Cubic ZnS, 14.7(1) Å 13 parameters, χ2 = 1.550

18 parameters, χ2 = 1.345

Control of Peak Positions

• Lattice parameters • Specimen displacement • Specimen transparency • (Zero)

Specimen Displacement

36000Rshfts π−

=

2 cosshftθ θ∆ =

Specimen Displacement

Displacement, µm shift a, Å -670 36.79(3) 3.90997(2) -390 21.25(5) 3.91010(2) -100 5.56(4) 3.91016(2) 38 -2.06(3) 3.91025(1) 200 -10.78(4) 3.91007(2) 520 -28.57(4) 3.91025(2) 820 -44.75(5) 3.91025(2)

Average 3.9102(1)

Specimen Transparency

2 sin 2trnsθ θ∆ =

9000eff Rtrns

µπ−

=

70

Specimen Transparency When the specimen is long enough to

intercept the whole beam, and

an additional component of the profile g is generated:

-∞ < ε ≤ 0 (°)

t ≥32.

'sin

µρρ

θ

gR

=

exp

.sin

4114 6

2π ε

θ

71

Transparency

• Significant for thick organic specimens • Additional low-angle asymmetry • Peak shift to low angles

Extra Low-Angle Asymmetry Peak Shift to Low-Angles

5 10 15 20 25 30Two-Theta (deg)

0

50

100

150

SQR

(Cou

nts)

[wong664.raw] DAL0 (30,10,0.2,1.5,1,qzbc,96chan) JAK[wong642.raw] DAL0 (30,10,0.6,2.5,3) JAK

10% RuO2/SiO2 Catalyst

25 30 35 40 45 50 55Two-Theta (deg)

0

2500

5000

7500

Inte

nsity

(Cou

nts)

[kuma001.raw] RuO2:SiO2 (0.1:0.9) 09/05/11) (30,10,0.6,2.5,3) JAK[kuma002.raw] RuO2:SiO2 (0.1:0.9) 09/05/11 (30,10,0.6,2.5,3,qzbc) JAK

Penetration Depth, µm

2θ, ° 28 130 Pure RuO2 22 70

10% RuO2/90% SiO2 100 340

74

Instrument Profiles

“Typical values of Rietveld instrument profile coefficients”, J. A. Kaduk and J. Reid, Powder

Diffraction, 26(1), 88-93 (2011).

Table II. GSAS Function #2 Instrument Profile Parameters for a Variety of LaboratoryDiffractometers.

Diffractometer Date U V W X Y asymX’Pert ProPIXcel/0.04rad Soller

08/2010 0.8048 0 0.5103 2.537 1.946 4.343

D2/Lynxeye 05/2010 1.371 0 2.393 2.183 1.199 2.774D2/Lynxeye 10/2009 2.8329 0 2.695 1.853 2.488 2.194X’Pert ProPIXcel/mono

01/2008 0.7565 0 3.646 2.428 1.902 1.063

X’Pert ProPIXcel/no mon

01/2008 2.6369 0 0 2.778 0 2.486

D8/VANTEC 04/2004 0.2879 0 1.124 2.477 2.103 2.052PAD V 06/2007 1.0270 0 6.640 1.237 2.693 2.109D/MAX-B 06/2002 0.567 0 18.680 2.301 1.960 6.048Miniflex 09/2001 5.568 0 20.47 3.614 0 5.487PW17xx 08/1998 0 0 5.217 0 9.77 7.603

Table III. GSAS Function #3 Instrument Profile Parameters for a Variety of LaboratoryDiffractometers.

Diffractometer U V W X Y S/L H/LX’Pert ProPIXcel/0.04 radSoller

1.423 0 0.5061 2.842 1.509 0.03547 0.00522

D2/Lynxeye 1.376 0 2.640 2.410 0.850 0.02951 0.0005X’Pert ProPIXcel/mono

1.153 -0.928 4.161 2.472 1.814 0.01577 0.0005

X’Pert ProPIXcel/no mon

2.314 0 0 3.040 0 0.02788 0.0005

D8/VÅNTEC 0.3365 0 1.032 2.526 2.051 0.02695 0.0005PAD V 1.103 0 6.412 1.173 2.842 0.03018 0.0005D/MAX-B 3.219 -7.822 24.370 2.460 1.609 0.03858 0.0005

In all of these profile functions, P = 0.

Table IV. GSAS Profile #2 Functions for Several Synchrotron Diffractometers

Instr. Date U V W X Y asym APS

5-BM-C 10/2002 0.1 0 0 0.2505 0.9462 0

APS 5-BM-C 08/2006 17.1 -8.8 1.3 0 0 0

APS 1-ID 02/2002 0.1 0 0 0.2505 0.9462 0.0646

APS 10-ID-B 01/2000 0.3540 0 0.2908 0.3565 0.5177 0.4744

APS 32-ID 12/2004 0.3120 0 0.0104 0.1186 0.4062 0.0419

LNLS D10B 0.8777 -0.1600 0.1063 0.7604 1.1904 0.5157

NSLS X3B1 03/2004 6.427 -1.067 0 0.6102 0.6796 0.6733

Table V. GSAS Profile #3 Instrument Parameters for Several Synchrotron Diffractometers

Inst. Date U V W P X Y S/L H/L

APS 5-BM-C 10/2002 1.212 0 0 0 1.980 0 0.00135 0.00718

APS 1-ID 02/2002 0.1 0 0 0 0.1845 11.190 0.0005 0.00458

APS 10-ID 10/2003 1.212 0 0 0 0.198 0 0.00135 0.00718

APS 11-BMB 02/2009 1.163 -0.126 0.063 0 0.173 0 0.00110 0.00110

APS 32-ID 12/2004 1.212 0 0 0 0.198 0 0.00135 0.00718

AS PD 0.0522 0.5640 0.0621 0 0.293 0.171 0.0000 0.0000

NSLS X7B 0 -125.9 73.3 0 2.03 0 0.0001 0.1000

NSLS X16C 0 0 0 1 3 30 0.014 0.014

GU(GSAS) = 1803.4U(FullProf) (9)GV(GSAS) = 1803.4V(FullProf) (10)GW(GSAS) = 1803.4W(Fullprof) (11)GP(GSAS) = 1803.4IG(FullProf) (12)LX(GSAS) = 100Y(FullProf) (13)LY(GSAS) = 100X(Fullprof) (14)S/L(GSAS) = S_L(Fullprof) (15)H/L(GSAS) = D_L(Fullprof) (16)

fil l i i l i i h d d fil

GSAS/FullProf Conversions

GSAS Instrument Profile Parameters Bruker D2 Phaser, SRM 660a LaB6

Profile 2 3 4 Avg Div/Soller 0.6/2.5 0.2/1.5 0.6/2.5 0.2/1.5 0.6/2.5 0.2/1.5

File 1450 1473 1450 1473 1450 1473 U 2.336 2.976 1.613 2.522 1.718 3.259 2.04(66) V 0 0 0 0 0 0 0 W 3.777 2.718 4.749 3.911 4.751 3.648 3.93(76) P - - 0 0 0 0 0 X 2.718 2.214 2.854 2.609 2.847 2.706 2.66(24) Y 1.868 1.219 0 0.310 - - 0.85(85)

trns 2.408 2.501 1.847 1.104 1.831 2.148 1.97(12) asym 3.759 2.356 - - - - - S/L - - 0.03334 0.01100 0.03325 0.02241 H/L - - 0.0005 0.0005 0.0005 0.0005 eta - - - - 0.900 0.408

Lorentz-Polarization Factor IPOLA Lp Functional Form Description

0

Normal (POLA = 0.5)

1

incident beam monochromator

2

Lp up to the user

3

?

2

2

(1 )cos 22 sin cos

POLA POLAV

θθ θ

+ −

2

2

1 cos 2sin cos

POLAV

θθ θ

+

1V

2

2

(1 )cos 2 sin 22 sin cos

POLA POLAV

θ θθ θ

+ −

83

Texture (Preferred Orientation)

85

Stereographic Projection

http://www.3dsoftware.com/Cartography/USGS/MapProjections/Azimuthal/Stereographic

March-Dollase Function

W. A. Dollase, “Correction of Intensities for Preferred Orientation in Powder Diffractometry:

Application of the March Model”, J. Appl. Cryst., 19(4), 267-272 (1986).

March-Dollase Function 3/ 22

2 2

1

sin1 cospM

jph j

jp

AO Ratio A

M Ratio=

= +

∑

hp = reciprocal lattice vector Mp = multiplicity of hp Aj = angle between specified unique axis and hp Ratio = the refinable parameter “aspect ratio” Cylindrical specimen symmetry assumed

BFDH Morphology - Folic Acid Dihydrate

March-Dollase Function (B-B)

Plates Ratio < 1 Oblate spheroid

Needles Ratio > 1 Prolate spheroid

Check consistency with anisotropic broadening!

Spherical Harmonics Function

R. B. Von Dreele, “Quantitative texture analysis by Rietveld refinement”,

J. Appl. Cryst., 30, 517-525 (1997).

Spherical Harmonics Function

( ) ( ) ( )2

4, 12 1

LN L Lmn m n

p L L LL m L n L

O h y C k h k yLπ

= =− =−

= ++∑ ∑ ∑

Terms depend on crystal and sample symmetry cylindrical 2/m (shear) mmm (rolling) no symmetry

Spherical Harmonics

Texture Index

2

2

112 1

LN L LmnL

L m L n LJ C

L= =− =−

= ++∑ ∑ ∑

J = 1 for random J = ∞ for single crystal

10 20 30 40Two-Theta (deg)

0

1000

2000

3000

4000

5000

6000

Inte

nsity

(Cou

nts)

[ssjr010.raw] Sample #3, 1 (30,10,0.6,2.5,3,qzbc,static) JAK[ssjr018.raw] Sample #3, parallel (30,10,0.6,2.5,3,qzbc,static) JAK[ssjr019.raw] Sample #3, vertical (30,10,0.6,2.5,qzbc,static) JAK

Polymer Dip Tubes

96

Texture in HDPE Pipe

97

A Rietveld Example Nature’s Bounty B-Complex

James A. Kaduk Poly Crystallography Inc.

Naperville IL 60540 kaduk@polycrystallography.com

“Default” Search/Match

0

2500

5000

7500

Inte

nsity

(Cou

nts)

04-013-3344> Brushite - HCa(PO 4)(H2O)2

02-063-2297> C 6H8O6 - L-Ascorbic acid

10 20 30 40 50 60 70Two-Theta (deg)

[kadu1599.raw] Nature's Bounty B-Complex (30,10,0.6,2.5,3,96) JAK

Zoom and Repeat

0

2500

5000

7500

Inte

nsity

(Cou

nts)

04-013-3344> Brushite - HCa(PO 4)(H2O)2

02-063-2297> C 6H8O6 - L-Ascorbic acid

01-075-1520> Monetite - CaHPO 4

10 20 30 40 50 60 70Two-Theta (deg)

[kadu1599.raw] Nature's Bounty B-Complex (30,10,0.6,2.5,3,96) JAK

3-Phase Rietveld Refinement

Pick Peaks Off Difference Plot d, Å I d, Å I

6.0665 280 4.1566 1194 5.9849 491 4.1166 1450 5.8320 365 3.8871 727 4.7189 491 3.7890 1170 4.3332 580 3.4636 617

Search the PDF-4 Organics with SIeve

Phases 4, 5 and 6

10 20 30 40 50 60 70Two-Theta (deg)

0

25

50

75

SQ

R(C

ount

s)

[kadu1599.raw] Nature's Bounty B-Complex (30,10,0.6,2.5,3,96) JAK02-062-0118> C 6H14O6 - D-Mannitol

02-070-4954> C 6H6N2O - Nicotinamide02-072-5617> C 12H17N4OS·NO 3 - Thiamine nitrate

6-Phase Rietveld Refinement

Add Cellulose Iα from 00-056-1719

From the label Phase wt% Vitamin C, ascorbic acid 17.7 Niacin, niacinamide 3.7 D-Ca pantothenate 1.5 Vitamin B-1, thiamin 0.7 Vitamin B-2, riboflavin 0.7 Vitamin B-6, pyroxidine HCl 0.7 Folic acid 0.06 Vitamin B-12, cyanocobalmin 0.04 Biotin 0.04

plus cellulose

dicalcium phosphate stearic acid

and < 2% palm leaf glaze

silica Mg stearate

From the Rietveld Refinement Phase Observed, wt% Expected, wt% Brushite, CaH(PO4)(H2O)2 6.7(1) Monetite, CaHPO4 2.5(2) β-D-mannitol 0.7(1) Ascorbic acid 17.7 17.7 Nicotinamide 0.6(1) 3.7 Thiamine nitrate 0.9(1) 0.7 Cellulose Iα (+ silica?) 17.4(7) Sum 46.5 > 1 unidentified