Particle size analysis -- Small angle X-ray scattering method

Analyse granulométrique -- Méthode de dispersion par rayons X sous angle faible

Sejalna analiza - Metoda disperzije rentgenskih žarkov pod ostrim kotom

General Information

Status
Withdrawn
Publication Date
14-Mar-2001
Withdrawal Date
14-Mar-2001
Current Stage
6060 - International Standard published
Start Date
13-Feb-2001
Completion Date
15-Mar-2001

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TECHNICAL ISO/TS
SPECIFICATION 13762
First edition
2001-03-15
Particle size analysis — Small angle X-ray
scattering method
Analyse granulométrique — Méthode de dispersion par rayons X sous
angle faible
Reference number
ISO/TS 13762:2001(E)
ISO 2001
---------------------- Page: 1 ----------------------
ISO/TS 13762:2001(E)
PDF disclaimer

This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but shall not

be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In downloading this

file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat accepts no liability in this

area.
Adobe is a trademark of Adobe Systems Incorporated.

Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation parameters

were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In the unlikely event

that a problem relating to it is found, please inform the Central Secretariat at the address given below.

© ISO 2001

All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic

or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISO's member body

in the country of the requester.
ISO copyright office
Case postale 56 � CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.ch
Web www.iso.ch
Printed in Switzerland
ii © ISO 2001 – All rights reserved
---------------------- Page: 2 ----------------------
ISO/TS 13762:2001(E)
Contents Page

Foreword.....................................................................................................................................................................iv

Introduction.................................................................................................................................................................v

1 Scope ..............................................................................................................................................................1

2 Normative references ....................................................................................................................................1

3 Symbols and abbreviations..........................................................................................................................2

4 Principle..........................................................................................................................................................2

5 Sample preparation and requirement..........................................................................................................4

6 Apparatus .......................................................................................................................................................4

7 Procedure .......................................................................................................................................................5

8 Calculation and expression of results.........................................................................................................5

9 Verification .....................................................................................................................................................7

10 Reporting of results.......................................................................................................................................7

Annex A (normative) Determination of slit-height weighting function F(t) ...........................................................8

Annex B (normative) Calculation of coefficient, a .................................................................................................9

Annex C (informative) Equipment for SAXS measurement ..................................................................................10

Annex D (informative) Reproducibility and accuracy............................................................................................12

Annex E (informative) Verification and examples for determination of particle size distribution of

ultra-fine powders by SAXS method..........................................................................................................15

Bibliography..............................................................................................................................................................22

© ISO 2001 – All rights reserved iii
---------------------- Page: 3 ----------------------
ISO/TS 13762:2001(E)
Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO

member bodies). The work of preparing International Standards is normally carried out through ISO technical

committees. Each member body interested in a subject for which a technical committee has been established has

the right to be represented on that committee. International organizations, governmental and non-governmental, in

liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical

Commission (IEC) on all matters of electrotechnical standardization.

International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.

The main task of technical committees is to prepare International Standards. Draft International Standards adopted

by the technical committees are circulated to the member bodies for voting. Publication as an International

Standard requires approval by at least 75 % of the member bodies casting a vote.

In other circumstances, particularly when there is an urgent market requirement for such documents, a technical

committee may decide to publish other types of normative document:

� an ISO Publicly Available Specification (ISO/PAS) represents an agreement between technical experts in an

ISO working group and is accepted for publication if it is approved by more than 50 % of the members of the

parent committee casting a vote;

� an ISO Technical Specification (ISO/TS) represents an agreement between the members of a technical

committee and is accepted for publication if it is approved by 2/3 of the members of the committee casting a

vote.

An ISO/PAS or ISO/TS is reviewed every three years with a view to deciding whether it can be transformed into an

International Standard.

Attention is drawn to the possibility that some of the elements of this Technical Specification ISO/TS 13762 may be

the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights.

ISO/TS 13762 was prepared by Technical Committee ISO/TC 24, Sieves, sieving and other sizing methods,

Subcommittee SC 4, Sizing by methods other than sieving.

Annexes A and B form a normative part of this Technical Specification. Annexes C to E are for information only.

iv © ISO 2001 – All rights reserved
---------------------- Page: 4 ----------------------
ISO/TS 13762:2001(E)
Introduction

The size range for which the small angle X-ray scattering (SAXS) method is applicable is approximately in the

range of 1 nm to 300 nm. The success of the technique is mainly based on the fact that SAXS effect results from

the difference of electron density between particles and their surroundings so that size X always indicates the

SAXS

size of a primary particle rather than the internal crystallite or external agglomerate size; in other words, the

requirement of particle dispersion of a sample for SAXS analysis is not as strict as that for other methods.

However, the SAXS method has its limitations: firstly, it cannot distinguish pores from particles; secondly, the

interference effect between particles will arise as the sample is available only in concentrated form.

SAXS measurements and the interpretation of the data are currently not uniform. The purpose of this Technical

Specification is to facilitate comparisons of size analysis made in different laboratories.

It is well known that X-rays can kill human tissue. However, this Technical Specification does not purport to address

all the safety problems associated with the use of the SAXS method; it is the responsibility of the user to establish

appropriate safety and health practices prior to its use.
© ISO 2001 – All rights reserved v
---------------------- Page: 5 ----------------------
TECHNICAL SPECIFICATION ISO/TS 13762:2001(E)
Particle size analysis — Small angle X-ray scattering method
1 Scope

This Technical Specification specifies the method for determining particle size distribution of ultra-fine powders by

the small angle X-ray scattering technique. It is applicable to particle sizes ranging from 1 nm to 300 nm. In the

data analysis, it is assumed that particles are isotropic and spherically shaped.

The method described in this Technical Specification is also applicable to particle suspensions.

This Technical Specification does not apply to:

a) powders containing particles whose morphology is far from spherical, except by special agreement;

b) powders consisting of porous particles;
c) mixtures of powders.
2 Normative references

The following normative documents contain provisions which, through reference in this text, constitute provisions of

this Technical Specification. For dated references, subsequent amendments to, or revisions of, any of these

publications do not apply. However, parties to agreements based on this Technical Specification are encouraged to

investigate the possibility of applying the most recent editions of the normative documents indicated below. For

undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC

maintain registers of currently valid International Standards.

ISO 9276-1, Representation of results of particle size analysis — Part 1: Graphical representation.

ISO 9276-2:— , Representation of results of particle size analysis — Part 2: Calculation of average particle

sizes/diameters and moments from particle size distributions.
1) To be published.
© ISO 2001 – All rights reserved 1
---------------------- Page: 6 ----------------------
ISO/TS 13762:2001(E)
3 Symbols and abbreviations
SAXS small angle X-ray scattering
x particle size, diameter of a sphere
�x length of particle size interval, �xxx��
i iii�1
n total number of size classes
q average density distribution of the class �x by volume or mass
3,i
average density distribution in a representation with logarithmic abscissa
3,i�
Q cumulative volume fraction of particle sizesxxu
3,i i
scattering angle
�Q volume fraction within the class �x ,��Qq
3,i 3,ii3, ��x
scattering function of a spherical particle:
��()
3221/2
��()��3(sin� �cos�)/� ,�� (�� t )
wavelength of incident X-ray beam
t angle variable along the slit-height direction
F(t) slit-weighting function along the slit-height direction
4Principle

As a narrow beam of X-ray passes through a powder layer containing ultra-fine particles, it will be dispersed around

the incident beam at a small angle range resulting from the electron scattering in the particles. The distribution of

scattered intensity is closely related to the particle size distribution.

The scattered intensity I(�) from a dilute spherical particle system can be expressed by the integral equation:

Ic()��� F(t)dt (x)x� (�)dx (1)
�� x
where
x is the size below which there are no particles;
x is the size above which there are no particles;
�(x)is the size distribution function by volume (without normalization);
c is a synthetic constant.
2 © ISO 2001 – All rights reserved
---------------------- Page: 7 ----------------------
ISO/TS 13762:2001(E)

The size range of x to x can be divided into n intervals and the lengths of the intervals, �x, increase with x

increasing. Let � represent the average distribution function of the class �x , and then measure scattering

j j

intensities at a series of chosen angles to obtain n values of I(�). The angles are specified by the equation:

����25 / (xx� ) (i = 1, 2, 3, ....., n)(2)
ii�1i

Approximately, the integral equation (1) is transformed into a set of a linear equations:

la()��� (i = 1, 2, 3, ....., n)(3)
ii�jj
j�1
where
�� j
aF� ()t dt x��( )dx (4)
�� x
j�1

All the coefficients, a , in the set of linear equations are a group of constants for given �, collimation geometry,

classes and specific angles. We may calculate them one by one by using a numerical integration method as long

as the slit-height weighting function, F(t), has been measured (annex B). Thus, the values of distribution function

corresponding to each class can be obtained from solving equation (3). Based on the solutions, the average

density distribution, volume fractions, cumulative volume fractions and the mean size x of the particles can be

calculated as follows:
qx����/ ( j = 1, 2, ..., n)(5)
3,jj �k k
k�1
��Qq�x ( j = 1, 2, ..., n)(6)
3,jj3, j � 100 %
QQ�� ( j = 1, 2, ..., n) (7)
3,jk� 3,
k�0
xQ��()x �x/2 ( j = 1, 2, ..., n)(8)
3,jj�1 j

The solution of equation (3) involves the inversions of an ill-conditioned problem but it can be improved through

optimizing the coefficient matrix and introducing a diagonal matrix B and a stabilizer �. In matrix form, equation (3)

can be modified as:
[]A����BI (9)
where
Aa�() ;
ij n � n
Ba� diag( ,a ,...,a );
11 22 nn
0<� <0,3
���(,�,...,� )
12 n
II�(,I ,...,I )
12 n
© ISO 2001 – All rights reserved 3
---------------------- Page: 8 ----------------------
ISO/TS 13762:2001(E)
5 Sample preparation and requirement
5.1 Preparation of collodion solution

The analytically pure acetone and collodion without small-angle scattering should be used to prepare the solution.

In general, the concentration of the solution may be approximately 5 % to 10 %.
5.2 Requirements and preparation of the sample plate
5.2.1 Requirements of the sample plate

The sample plate used for scattering measurement should meet the following requirements:

a) the volume concentration of the test powder in the plate should be less than 1 %;

b) to achieve maximum scattering intensity, the thickness of the sample plate shall be chosen to an optimum

value where the monochromatic X-ray beam will be attenuated to 37 %;

c) the sample plate should be uniform, flat, crack free and should ensure that the powder particles are separated

as well as possible;

d) the sample plate should have suitable dimensions, for example a length of 20 mm and a width of 10 mm.

5.2.2 Preparation of the sample plate

Weigh out a given mass of test powder and pour onto it an appropriate volume of collodion solution. The quantities

requested for the powder and solution may be estimated from the density of each and from their X-ray absorption

coefficient, in order to meet the requirements in 5.2.1.

Stir the suspension in a small cup. A drop of wetting agent shall be added to the suspension if the solution does not

readily wet the powder.

The dispersion of the powder in the suspension shall be made by the addition of dispersing agent and/or an

ultrasonic treatment.

Place the cup containing the suspension into an oven at the temperature of 20 °Cto 50 °C and with a relative

humidity of below 50 %, until a dry sample plate has formed.
5.3 Preparation of dry powder sample

Place the dry powder in a special sample holder with a slot. If the requirements in 5.2.1 are met in this way, it shall

be directly used for SAXS measurement.
5.4 Preparation of colloidal solution

The concentration of the colloidal solution should be less than 1 %. An appropriate surfactant may be added to

prevent the colloids from agglomerating.

When preparing the sample for measurement, the method chosen should not affect the particle morphology.

6 Apparatus
The apparatus consists of:

� an X-ray generator combined with electronic circuit panel, detector and recording;

� a system, with synthetic stability superior to 1 %;
4 © ISO 2001 – All rights reserved
---------------------- Page: 9 ----------------------
ISO/TS 13762:2001(E)
� a small angle scattering goniometer;
� a computation unit;

� an ordinary laboratory apparatus, such as an analytical balance, ultrasonator and oven.

Seealsoannex C.
7 Procedure
7.1 Switch the X-ray diffractometer on and allow it to warm up.

7.2 Adjust or check the small angle scattering goniometer according to the manual.

7.3 Once the conditions of the instrument are stable, adjust the goniometer according to the instruction manual.

Insert an appropriate number of absorber plates in front of the detector then set the goniometer at 0 scale of the

angle. Register the intensity of the primary X-ray beam as an intensity of the goniometer’s 0 degree.

7.4 Fix the sample plate onto the sample holder, or inject the colloidal suspension sample into a special cell and

then insert the holder with the sample into the shaft hole of the goniometer. Pull out the absorber plates.

7.5 Measure the scattering intensities, I (� ), at the angles specified by equation (2). The total pulse counts

a i

measured should reach at least 5 000 at each angle. The measured intensity, I (� ), is composed of the small-angle

a i
scattering from the sample as well as the scattering background.

7.6 Take out the sample. Reinsert the absorber plates, reset the goniometer to 0� and then measure the intensity

of the primary beam. The relative deviation between this value and the intensity registered in 7.3 shall be less then

15 %; if this is not the case, repeat the procedures starting from 7.2.

7.7 Place the sample between the X-ray window and the first slits of the goniometer. Measure the scattering

background, I (� ), under the same instrumental conditions and at the same angles as in 7.5. However, the pulse

b i
counts for I (� ) need not be specified.
b i

7.8 Subtract the intensity measured in 7.7 from that measured in 7.5, i.e. the small-angle scattering intensity,

I(�)= I (� )� I (� ).
i a i b i
8 Calculation and expression of results
8.1 Calculation
8.1.1 Evaluation of equation (3)

Substitute the values obtained in 7.8 and 7.9 in equation (3); each � can be calculated by means of a

microprocessor. If the solution is not stable or reasonable enough, the equation (9) shall be used.

8.1.2 Calculation of average density distribution and volume fraction

The average densities distribution, q , and the volume fractions, �Q , for each class shall be calculated from the

3j 3,j
values of� according to equation (5) and equati
...

SLOVENSKI STANDARD
SIST ISO/TS 13762:2002
01-junij-2002
Sejalna analiza - Metoda disperzije rentgenskih žarkov pod ostrim kotom
Particle size analysis -- Small angle X-ray scattering method
Analyse granulométrique -- Méthode de dispersion par rayons X sous angle faible
Ta slovenski standard je istoveten z: ISO/TS 13762:2001
ICS:
19.120 Analiza velikosti delcev. Particle size analysis. Sieving
Sejanje
SIST ISO/TS 13762:2002 en

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------
SIST ISO/TS 13762:2002
---------------------- Page: 2 ----------------------
SIST ISO/TS 13762:2002
TECHNICAL ISO/TS
SPECIFICATION 13762
First edition
2001-03-15
Particle size analysis — Small angle X-ray
scattering method
Analyse granulométrique — Méthode de dispersion par rayons X sous
angle faible
Reference number
ISO/TS 13762:2001(E)
ISO 2001
---------------------- Page: 3 ----------------------
SIST ISO/TS 13762:2002
ISO/TS 13762:2001(E)
PDF disclaimer

This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but shall not

be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In downloading this

file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat accepts no liability in this

area.
Adobe is a trademark of Adobe Systems Incorporated.

Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation parameters

were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In the unlikely event

that a problem relating to it is found, please inform the Central Secretariat at the address given below.

© ISO 2001

All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic

or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISO's member body

in the country of the requester.
ISO copyright office
Case postale 56 � CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.ch
Web www.iso.ch
Printed in Switzerland
ii © ISO 2001 – All rights reserved
---------------------- Page: 4 ----------------------
SIST ISO/TS 13762:2002
ISO/TS 13762:2001(E)
Contents Page

Foreword.....................................................................................................................................................................iv

Introduction.................................................................................................................................................................v

1 Scope ..............................................................................................................................................................1

2 Normative references ....................................................................................................................................1

3 Symbols and abbreviations..........................................................................................................................2

4 Principle..........................................................................................................................................................2

5 Sample preparation and requirement..........................................................................................................4

6 Apparatus .......................................................................................................................................................4

7 Procedure .......................................................................................................................................................5

8 Calculation and expression of results.........................................................................................................5

9 Verification .....................................................................................................................................................7

10 Reporting of results.......................................................................................................................................7

Annex A (normative) Determination of slit-height weighting function F(t) ...........................................................8

Annex B (normative) Calculation of coefficient, a .................................................................................................9

Annex C (informative) Equipment for SAXS measurement ..................................................................................10

Annex D (informative) Reproducibility and accuracy............................................................................................12

Annex E (informative) Verification and examples for determination of particle size distribution of

ultra-fine powders by SAXS method..........................................................................................................15

Bibliography..............................................................................................................................................................22

© ISO 2001 – All rights reserved iii
---------------------- Page: 5 ----------------------
SIST ISO/TS 13762:2002
ISO/TS 13762:2001(E)
Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO

member bodies). The work of preparing International Standards is normally carried out through ISO technical

committees. Each member body interested in a subject for which a technical committee has been established has

the right to be represented on that committee. International organizations, governmental and non-governmental, in

liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical

Commission (IEC) on all matters of electrotechnical standardization.

International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.

The main task of technical committees is to prepare International Standards. Draft International Standards adopted

by the technical committees are circulated to the member bodies for voting. Publication as an International

Standard requires approval by at least 75 % of the member bodies casting a vote.

In other circumstances, particularly when there is an urgent market requirement for such documents, a technical

committee may decide to publish other types of normative document:

� an ISO Publicly Available Specification (ISO/PAS) represents an agreement between technical experts in an

ISO working group and is accepted for publication if it is approved by more than 50 % of the members of the

parent committee casting a vote;

� an ISO Technical Specification (ISO/TS) represents an agreement between the members of a technical

committee and is accepted for publication if it is approved by 2/3 of the members of the committee casting a

vote.

An ISO/PAS or ISO/TS is reviewed every three years with a view to deciding whether it can be transformed into an

International Standard.

Attention is drawn to the possibility that some of the elements of this Technical Specification ISO/TS 13762 may be

the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights.

ISO/TS 13762 was prepared by Technical Committee ISO/TC 24, Sieves, sieving and other sizing methods,

Subcommittee SC 4, Sizing by methods other than sieving.

Annexes A and B form a normative part of this Technical Specification. Annexes C to E are for information only.

iv © ISO 2001 – All rights reserved
---------------------- Page: 6 ----------------------
SIST ISO/TS 13762:2002
ISO/TS 13762:2001(E)
Introduction

The size range for which the small angle X-ray scattering (SAXS) method is applicable is approximately in the

range of 1 nm to 300 nm. The success of the technique is mainly based on the fact that SAXS effect results from

the difference of electron density between particles and their surroundings so that size X always indicates the

SAXS

size of a primary particle rather than the internal crystallite or external agglomerate size; in other words, the

requirement of particle dispersion of a sample for SAXS analysis is not as strict as that for other methods.

However, the SAXS method has its limitations: firstly, it cannot distinguish pores from particles; secondly, the

interference effect between particles will arise as the sample is available only in concentrated form.

SAXS measurements and the interpretation of the data are currently not uniform. The purpose of this Technical

Specification is to facilitate comparisons of size analysis made in different laboratories.

It is well known that X-rays can kill human tissue. However, this Technical Specification does not purport to address

all the safety problems associated with the use of the SAXS method; it is the responsibility of the user to establish

appropriate safety and health practices prior to its use.
© ISO 2001 – All rights reserved v
---------------------- Page: 7 ----------------------
SIST ISO/TS 13762:2002
---------------------- Page: 8 ----------------------
SIST ISO/TS 13762:2002
TECHNICAL SPECIFICATION ISO/TS 13762:2001(E)
Particle size analysis — Small angle X-ray scattering method
1 Scope

This Technical Specification specifies the method for determining particle size distribution of ultra-fine powders by

the small angle X-ray scattering technique. It is applicable to particle sizes ranging from 1 nm to 300 nm. In the

data analysis, it is assumed that particles are isotropic and spherically shaped.

The method described in this Technical Specification is also applicable to particle suspensions.

This Technical Specification does not apply to:

a) powders containing particles whose morphology is far from spherical, except by special agreement;

b) powders consisting of porous particles;
c) mixtures of powders.
2 Normative references

The following normative documents contain provisions which, through reference in this text, constitute provisions of

this Technical Specification. For dated references, subsequent amendments to, or revisions of, any of these

publications do not apply. However, parties to agreements based on this Technical Specification are encouraged to

investigate the possibility of applying the most recent editions of the normative documents indicated below. For

undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC

maintain registers of currently valid International Standards.

ISO 9276-1, Representation of results of particle size analysis — Part 1: Graphical representation.

ISO 9276-2:— , Representation of results of particle size analysis — Part 2: Calculation of average particle

sizes/diameters and moments from particle size distributions.
1) To be published.
© ISO 2001 – All rights reserved 1
---------------------- Page: 9 ----------------------
SIST ISO/TS 13762:2002
ISO/TS 13762:2001(E)
3 Symbols and abbreviations
SAXS small angle X-ray scattering
x particle size, diameter of a sphere
�x length of particle size interval, �xxx��
i iii�1
n total number of size classes
q average density distribution of the class �x by volume or mass
3,i
average density distribution in a representation with logarithmic abscissa
3,i�
Q cumulative volume fraction of particle sizesxxu
3,i i
scattering angle
�Q volume fraction within the class �x ,��Qq
3,i 3,ii3, ��x
scattering function of a spherical particle:
��()
3221/2
��()��3(sin� �cos�)/� ,�� (�� t )
wavelength of incident X-ray beam
t angle variable along the slit-height direction
F(t) slit-weighting function along the slit-height direction
4Principle

As a narrow beam of X-ray passes through a powder layer containing ultra-fine particles, it will be dispersed around

the incident beam at a small angle range resulting from the electron scattering in the particles. The distribution of

scattered intensity is closely related to the particle size distribution.

The scattered intensity I(�) from a dilute spherical particle system can be expressed by the integral equation:

Ic()��� F(t)dt (x)x� (�)dx (1)
�� x
where
x is the size below which there are no particles;
x is the size above which there are no particles;
�(x)is the size distribution function by volume (without normalization);
c is a synthetic constant.
2 © ISO 2001 – All rights reserved
---------------------- Page: 10 ----------------------
SIST ISO/TS 13762:2002
ISO/TS 13762:2001(E)

The size range of x to x can be divided into n intervals and the lengths of the intervals, �x, increase with x

increasing. Let � represent the average distribution function of the class �x , and then measure scattering

j j

intensities at a series of chosen angles to obtain n values of I(�). The angles are specified by the equation:

����25 / (xx� ) (i = 1, 2, 3, ....., n)(2)
ii�1i

Approximately, the integral equation (1) is transformed into a set of a linear equations:

la()��� (i = 1, 2, 3, ....., n)(3)
ii�jj
j�1
where
�� j
aF� ()t dt x��( )dx (4)
�� x
j�1

All the coefficients, a , in the set of linear equations are a group of constants for given �, collimation geometry,

classes and specific angles. We may calculate them one by one by using a numerical integration method as long

as the slit-height weighting function, F(t), has been measured (annex B). Thus, the values of distribution function

corresponding to each class can be obtained from solving equation (3). Based on the solutions, the average

density distribution, volume fractions, cumulative volume fractions and the mean size x of the particles can be

calculated as follows:
qx����/ ( j = 1, 2, ..., n)(5)
3,jj �k k
k�1
��Qq�x ( j = 1, 2, ..., n)(6)
3,jj3, j � 100 %
QQ�� ( j = 1, 2, ..., n) (7)
3,jk� 3,
k�0
xQ��()x �x/2 ( j = 1, 2, ..., n)(8)
3,jj�1 j

The solution of equation (3) involves the inversions of an ill-conditioned problem but it can be improved through

optimizing the coefficient matrix and introducing a diagonal matrix B and a stabilizer �. In matrix form, equation (3)

can be modified as:
[]A����BI (9)
where
Aa�() ;
ij n � n
Ba� diag( ,a ,...,a );
11 22 nn
0<� <0,3
���(,�,...,� )
12 n
II�(,I ,...,I )
12 n
© ISO 2001 – All rights reserved 3
---------------------- Page: 11 ----------------------
SIST ISO/TS 13762:2002
ISO/TS 13762:2001(E)
5 Sample preparation and requirement
5.1 Preparation of collodion solution

The analytically pure acetone and collodion without small-angle scattering should be used to prepare the solution.

In general, the concentration of the solution may be approximately 5 % to 10 %.
5.2 Requirements and preparation of the sample plate
5.2.1 Requirements of the sample plate

The sample plate used for scattering measurement should meet the following requirements:

a) the volume concentration of the test powder in the plate should be less than 1 %;

b) to achieve maximum scattering intensity, the thickness of the sample plate shall be chosen to an optimum

value where the monochromatic X-ray beam will be attenuated to 37 %;

c) the sample plate should be uniform, flat, crack free and should ensure that the powder particles are separated

as well as possible;

d) the sample plate should have suitable dimensions, for example a length of 20 mm and a width of 10 mm.

5.2.2 Preparation of the sample plate

Weigh out a given mass of test powder and pour onto it an appropriate volume of collodion solution. The quantities

requested for the powder and solution may be estimated from the density of each and from their X-ray absorption

coefficient, in order to meet the requirements in 5.2.1.

Stir the suspension in a small cup. A drop of wetting agent shall be added to the suspension if the solution does not

readily wet the powder.

The dispersion of the powder in the suspension shall be made by the addition of dispersing agent and/or an

ultrasonic treatment.

Place the cup containing the suspension into an oven at the temperature of 20 °Cto 50 °C and with a relative

humidity of below 50 %, until a dry sample plate has formed.
5.3 Preparation of dry powder sample

Place the dry powder in a special sample holder with a slot. If the requirements in 5.2.1 are met in this way, it shall

be directly used for SAXS measurement.
5.4 Preparation of colloidal solution

The concentration of the colloidal solution should be less than 1 %. An appropriate surfactant may be added to

prevent the colloids from agglomerating.

When preparing the sample for measurement, the method chosen should not affect the particle morphology.

6 Apparatus
The apparatus consists of:

� an X-ray generator combined with electronic circuit panel, detector and recording;

� a system, with synthetic stability superior to 1 %;
4 © ISO 2001 – All rights reserved
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SIST ISO/TS 13762:2002
ISO/TS 13762:2001(E)
� a small angle scattering goniometer;
� a computation unit;

� an ordinary laboratory apparatus, such as an analytical balance, ultrasonator and oven.

Seealsoannex C.
7 Procedure
7.1 Switch the X-ray diffractometer on and allow it to warm up.

7.2 Adjust or check the small angle scattering goniometer according to the manual.

7.3 Once the conditions of the instrument are stable, adjust the goniometer according to the instruction manual.

Insert an appropriate number of absorber plates in front of the detector then set the goniometer at 0 scale of the

angle. Register the intensity of the primary X-ray beam as an intensity of the goniometer’s 0 degree.

7.4 Fix the sample plate onto the sample holder, or inject the colloidal suspension sample into a special cell and

then insert the holder with the sample into the shaft hole of the goniometer. Pull out the absorber plates.

7.5 Measure the scattering intensities, I (� ), at the angles specified by equation (2). The total pulse counts

a i

measured should reach at least 5 000 at each angle. The measured intensity, I (� ), is composed of the small-angle

a i
scattering from the sample as well as the scattering background.

7.6 Take out the sample. Reinsert the absorber plates, reset the goniometer to 0� and then measure the intensity

of the primary beam. The relative deviation between this value and the intensity registered in 7.3 shall be less then

15 %; if this is not the case, repeat the procedures starting from 7.2.

7.7 Place the sample between the X-ray window and the first slits of the goniometer. Measure the scattering

background, I (� ), under the same instrumental conditions and at the same angles as in 7.5. However, the pulse

b i
counts for I (� ) need not be specified.
b i

7.8 Subtract the intensity measured in 7.7 from that measured in 7.5, i.e. the small-angle scattering intensity,

I(�)= I (� )� I (� ).
i a i b i
8 Calculation and expression of results
8.1 Calculation
8.1.1 Evaluation of equation (3)

Substitute the values obtained in 7.8 and 7.9 in equation (3); each � can be calculated by means of a

microprocessor. If the solution is not stable or reasonable enough, the equation (9) shall be used.

...

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