苏 玉 长
衍射花样指数化
苏 玉 长
The first step in analysing unknown powder
pattern is often an attempt to find a unit cell
that explains all observed lines in the spectrum.
You do not need additional crystallographic
data,although if it exists it makes for faster and
more reliable results,The material to be
analysed must be single phased and the
experimental material must be very accurate.
苏 玉 长
Indexing programs use only the positional
information of the pattern and try to find a set
of lattice constants (a,b,c,a,b,g) and
individual Miller indices (hkl) for each line,
The form of equations to solve is complicated
for the general case (triclinic) in direct space
but is straightforward in reciprocal space,In
the latter the set of equations is：
苏 玉 长
Q = h2A + k2B + l2C+ hkD + hlE + klF
where the Q-values are easily derived from the
diffraction angle Q,This set has to be solved for
the unknowns,A,B,C,D,E,F,which are in a
simple way related to the lattice constants,Finding
the proper values for the lattice parameters so that
every observed d-spacing satifies a particular
combination of Miller indices is the goal of
indexing,It is not easy even for the cubic system,
but it is very difficult for the triclinic system.
苏 玉 长
? There are two general approaches to indexing,the
exhaustive and the analytical approach,Both of
these approaches require very accurate d-spacing
data,The smaller the errors,the easier it is to test
solutions because there are often missing data
points due to intensity extinctions related to the
symmetry or the structural arrangement or due to
lack of resolution of the d-spacing themselves,
The earliest approaches were of the exhaustive
type and were done by graphical fitting or
numerical table fitting.
苏 玉 长
? Indexing Programs
? The methods currently implemented are
shown bold,They are selected through the
item Indexing in the main menu.
? Program Author Type
? ITO Visser analytical
? TREOR Werner exhaustive
? POWDER
? DICVOL
? CUBIC
苏 玉 长
? The programs use a set of common
parameters,e.g,the wavelength and a
method specific set,After you have clicked
on a program with the left mouse button
indexing is immediately started with the
active parameter set,Depending on the
problem and computer type the program run
can take from seconds to many minutes,All
solutions are computed and stored internally,
The "best" one is displayed at the top of the
screen,the next to best at the bottom,For an
overview of all solutions select Solutions in
the main menu.
苏 玉 长
? Miller Indices
Triplet of integer numbers uniquely assigned
to a Bragg reflection,The notation is
usually in the form of (hkl),Formally
spoken,Miller Indices represent coordinates
of lattice points in reciprocal space,
苏 玉 长
? Reciprocal Space
Direct space is composed of unit cells and
its contents,whereas reciprocal space is a
lattice whose lattice points are Bragg
reflections,Direct and reciprocal space are
tightly coupled.
苏 玉 长
Lattice constants
A set of maximally six floating numbers
representing the unit cell,As crystal
symmetry grows the number of lattice
constants needed to describe the metrics of
a unit cell reduce,In the cubic system there
is only one constant.
苏 玉 长
Q-value
The Q-value is one of several possible forms to
represent the positional information of peaks in
powder pattern,They are defined as:
Q = 1/d2= ( 4 sin2Q)/l2
Q can be represented as a quadratic form of the
(hkl)'s.
Q = h2A + k2B + l2C + hk D + hl E + kl F,
where A,B,C,D,E,F are related to the lattice
constants.
In ITO each Q-value is multiplied by 10000.
苏 玉 长
d-Spacing
The d-spacing is one of several possible forms
to represent the positional information of
peaks in powder pattern,Other forms are the
angle Q,or 2Q,or Q-values,Win_DIFFRAC
assumes input in the form of d-spacings,They
are internally converted,e.g,ITO works only
with Q's.
苏 玉 长
TREOR
This program uses the exhaustive approach,It is
widely used for indexing of powder pattern
throughout the world and is believed to be one of
the most powerful engines for solving high and
medium symmetries,It is useful as well for low
symmetry at the cost of rather high time
consumption.
苏 玉 长
We recommend that you start with a diffraction
pattern whose result is well known to you,If
you are already familiar with the TREOR
program you may decide to straightforward
indexing,using the default parameters,In case
you are a novice user you should swiftly go
through the parameters and change settings
only when they are obviously wrong,With the
results obtained you may then turn to the more
sophisticated parameters,e.g,Select Baseline
Set.
The parameters are described in detail under
general and TREOR-specific parameters.
苏 玉 长
In contrast to ITO a multitude of non-systematic
extinctions among the first lines may not
appreciably affect the power of trial-and-error
methods,However,'powder indexing is not like
structure analysis,which works well on good
data,and will usually get by on poor data given
a little more time and attention,Powder indexing
works beautifully on good data,but with poor
data it will usually not work at all.' (op.cit
Ref.[4]),
苏 玉 长
The standard procedure using TREOR is to start
with the high symmetries,cubic,tetragonal,
hexagonal and orthorhombic in one run,Next
the monoclinic symmetry should be tried,More
than one run may be necessary,successively
increasing the number of baseline sets,the cell
volume and cell edge,If the formula weight and
density are known,they should be used,The
CPU-time needed will then usually be strongly
reduced,Unfortunately they are mostly not well
known and therefore not often usable,If all
previous tests have failed the triclinic test
remains for which in general we recommend
ITO.
苏 玉 长
Analytical Approach
The crystal is assumed to be triclinic,Solutions
are approached in three steps,In the first,one
tries to find reasonable zones from the diffraction
data,Each pair of lines,together with the origin
forms a crystallographic zone,A zone is relevant
to the solution finding algorithm if some other
diffraction lines lie on it,if not it is discarded,In
the second step the best zones are then tried in
combinations to find complete lattices,This
method is known as the ITO method,Once a
lattice is found that satisfies the experimental d-
spacing data,the geometry of the lattice is
examined for possible higher metric symmetry,
This is the final step.
苏 玉 长
衍 射分 析应 用
物质对射线的衍射产生了衍
射花样或衍射谱, 对于给定的单
晶试样, 其衍射花样与入射线的
相对取向及晶体结构有关；对于
给定的多晶体也有特定的衍射花
样 。 衍射花样具有 三要素,衍射
线 （ 或衍射斑 ） 的位置, 强度和
线型 。 测定衍射花样三要素在不
同状态下的变化, 是衍射分析应
用的基础 。
苏 玉 长
衍 射分 析应 用
? 基于衍射位置的应用
⑴ 点阵参数的精确测定，膨胀系数的测定；
⑵ 第一类 （ 即宏观残余 ） 应力的测定；
⑶ 由点阵参数测定相平衡图中的相界 ；
⑷ 晶体取向的测定；
⑸ 固溶体类型的测定, 固溶体组分 的测定；
⑹ 多晶材料中层错几率的测定；
⑺ 点缺陷引起的 Bragg峰的漂移 。
苏 玉 长
衍 射分 析应 用
? 基于衍射强度测量的应用
（ 1）物相的定量分析，结晶度的测定
(2) 平衡相图的相界的测定；
(3) 第三类应力的测定；
(4) 有序固溶体长程有序度的测定；
(5) 多晶体材料中晶粒择优取向的极图、
反极图和三维取向分布的测定；
(6) 薄膜厚度的测定。
苏 玉 长
衍 射分 析应 用
? 基于衍射线型分析的应用
(1) 多晶材料中位错密度的测定，层错能
的测定，晶体缺陷的研究；
(2) 第二类（微观残余）应力的测定；
(3) 晶粒大小和微应变的测定；
苏 玉 长
衍 射分 析应 用
? 基于衍射位置和强度的测定
(1) 物相的定性分析
(2) 相消失法测定相平衡图中的相界 ;
(3) 晶体 (相 )结构,磁结构,表面结构,界面结
构的研究
苏 玉 长
衍 射分 析应 用
? 同时基于衍射位置、强度和线型的
Rietveld多晶结构测定
需输入原子参数（晶胞中各原子的坐标、
占位几率和湿度因子）、点阵参数、波长、偏
正因子、吸收系数、择优取向参数等。
苏 玉 长
衍 射分 析应 用 ? Rietveld profile refinement
?What is the Rietveld method?
?A comprehensive,computer based method
fro the analysis of X-ray and neutron powder
diffraction patterns by least squares fitting of
a calculated diffraction pattern to the
observed data.
?What is special for Rietveld method?
?This method does not use integrated powder
diffraction intensities,but employs directly
the profile intensities obtained from step-
scanning measurements of the powder
diagram.
苏 玉 长
衍 射分 析应 用
? Rietveld profile refinement
?How useful of Rietveld method?
?The Rietveld method has been developed into a
valuable method for structure analyses of nearly
all class of crystalline materials (Young 1995) by
a number endeavours.It has been extensively used
to investigate the structure and phase problems,A
lot of useful information can be retrieved by
Rietveld method.
苏 玉 长
苏 玉 长
苏 玉 长
对于大多数固溶体，其点阵参数随溶质原子的浓度呈近似线
性关系，即服从费伽 (Vegard)定律：
式中,aA和 aB分别表示固溶体组元 A和 B的点陈参数。因此，
测得含量为 x的 B原子的因溶体的点阵参数工 ax，用上式即求
得固溶体的组分。
实验表明，固溶体中点阵参数随溶质原子的浓度变化有不
少呈非线性关系，在此情况下应先测得点阵参数与溶质原子
浓度的关系曲线。 实际应用中，将精确测得的点阵参数与
已知数据比较即可求得固溶体的组分。
%100????
AB
Ax
aa
aax
衍射花样指数化
苏 玉 长
The first step in analysing unknown powder
pattern is often an attempt to find a unit cell
that explains all observed lines in the spectrum.
You do not need additional crystallographic
data,although if it exists it makes for faster and
more reliable results,The material to be
analysed must be single phased and the
experimental material must be very accurate.
苏 玉 长
Indexing programs use only the positional
information of the pattern and try to find a set
of lattice constants (a,b,c,a,b,g) and
individual Miller indices (hkl) for each line,
The form of equations to solve is complicated
for the general case (triclinic) in direct space
but is straightforward in reciprocal space,In
the latter the set of equations is：
苏 玉 长
Q = h2A + k2B + l2C+ hkD + hlE + klF
where the Q-values are easily derived from the
diffraction angle Q,This set has to be solved for
the unknowns,A,B,C,D,E,F,which are in a
simple way related to the lattice constants,Finding
the proper values for the lattice parameters so that
every observed d-spacing satifies a particular
combination of Miller indices is the goal of
indexing,It is not easy even for the cubic system,
but it is very difficult for the triclinic system.
苏 玉 长
? There are two general approaches to indexing,the
exhaustive and the analytical approach,Both of
these approaches require very accurate d-spacing
data,The smaller the errors,the easier it is to test
solutions because there are often missing data
points due to intensity extinctions related to the
symmetry or the structural arrangement or due to
lack of resolution of the d-spacing themselves,
The earliest approaches were of the exhaustive
type and were done by graphical fitting or
numerical table fitting.
苏 玉 长
? Indexing Programs
? The methods currently implemented are
shown bold,They are selected through the
item Indexing in the main menu.
? Program Author Type
? ITO Visser analytical
? TREOR Werner exhaustive
? POWDER
? DICVOL
? CUBIC
苏 玉 长
? The programs use a set of common
parameters,e.g,the wavelength and a
method specific set,After you have clicked
on a program with the left mouse button
indexing is immediately started with the
active parameter set,Depending on the
problem and computer type the program run
can take from seconds to many minutes,All
solutions are computed and stored internally,
The "best" one is displayed at the top of the
screen,the next to best at the bottom,For an
overview of all solutions select Solutions in
the main menu.
苏 玉 长
? Miller Indices
Triplet of integer numbers uniquely assigned
to a Bragg reflection,The notation is
usually in the form of (hkl),Formally
spoken,Miller Indices represent coordinates
of lattice points in reciprocal space,
苏 玉 长
? Reciprocal Space
Direct space is composed of unit cells and
its contents,whereas reciprocal space is a
lattice whose lattice points are Bragg
reflections,Direct and reciprocal space are
tightly coupled.
苏 玉 长
Lattice constants
A set of maximally six floating numbers
representing the unit cell,As crystal
symmetry grows the number of lattice
constants needed to describe the metrics of
a unit cell reduce,In the cubic system there
is only one constant.
苏 玉 长
Q-value
The Q-value is one of several possible forms to
represent the positional information of peaks in
powder pattern,They are defined as:
Q = 1/d2= ( 4 sin2Q)/l2
Q can be represented as a quadratic form of the
(hkl)'s.
Q = h2A + k2B + l2C + hk D + hl E + kl F,
where A,B,C,D,E,F are related to the lattice
constants.
In ITO each Q-value is multiplied by 10000.
苏 玉 长
d-Spacing
The d-spacing is one of several possible forms
to represent the positional information of
peaks in powder pattern,Other forms are the
angle Q,or 2Q,or Q-values,Win_DIFFRAC
assumes input in the form of d-spacings,They
are internally converted,e.g,ITO works only
with Q's.
苏 玉 长
TREOR
This program uses the exhaustive approach,It is
widely used for indexing of powder pattern
throughout the world and is believed to be one of
the most powerful engines for solving high and
medium symmetries,It is useful as well for low
symmetry at the cost of rather high time
consumption.
苏 玉 长
We recommend that you start with a diffraction
pattern whose result is well known to you,If
you are already familiar with the TREOR
program you may decide to straightforward
indexing,using the default parameters,In case
you are a novice user you should swiftly go
through the parameters and change settings
only when they are obviously wrong,With the
results obtained you may then turn to the more
sophisticated parameters,e.g,Select Baseline
Set.
The parameters are described in detail under
general and TREOR-specific parameters.
苏 玉 长
In contrast to ITO a multitude of non-systematic
extinctions among the first lines may not
appreciably affect the power of trial-and-error
methods,However,'powder indexing is not like
structure analysis,which works well on good
data,and will usually get by on poor data given
a little more time and attention,Powder indexing
works beautifully on good data,but with poor
data it will usually not work at all.' (op.cit
Ref.[4]),
苏 玉 长
The standard procedure using TREOR is to start
with the high symmetries,cubic,tetragonal,
hexagonal and orthorhombic in one run,Next
the monoclinic symmetry should be tried,More
than one run may be necessary,successively
increasing the number of baseline sets,the cell
volume and cell edge,If the formula weight and
density are known,they should be used,The
CPU-time needed will then usually be strongly
reduced,Unfortunately they are mostly not well
known and therefore not often usable,If all
previous tests have failed the triclinic test
remains for which in general we recommend
ITO.
苏 玉 长
Analytical Approach
The crystal is assumed to be triclinic,Solutions
are approached in three steps,In the first,one
tries to find reasonable zones from the diffraction
data,Each pair of lines,together with the origin
forms a crystallographic zone,A zone is relevant
to the solution finding algorithm if some other
diffraction lines lie on it,if not it is discarded,In
the second step the best zones are then tried in
combinations to find complete lattices,This
method is known as the ITO method,Once a
lattice is found that satisfies the experimental d-
spacing data,the geometry of the lattice is
examined for possible higher metric symmetry,
This is the final step.
苏 玉 长
衍 射分 析应 用
物质对射线的衍射产生了衍
射花样或衍射谱, 对于给定的单
晶试样, 其衍射花样与入射线的
相对取向及晶体结构有关；对于
给定的多晶体也有特定的衍射花
样 。 衍射花样具有 三要素,衍射
线 （ 或衍射斑 ） 的位置, 强度和
线型 。 测定衍射花样三要素在不
同状态下的变化, 是衍射分析应
用的基础 。
苏 玉 长
衍 射分 析应 用
? 基于衍射位置的应用
⑴ 点阵参数的精确测定，膨胀系数的测定；
⑵ 第一类 （ 即宏观残余 ） 应力的测定；
⑶ 由点阵参数测定相平衡图中的相界 ；
⑷ 晶体取向的测定；
⑸ 固溶体类型的测定, 固溶体组分 的测定；
⑹ 多晶材料中层错几率的测定；
⑺ 点缺陷引起的 Bragg峰的漂移 。
苏 玉 长
衍 射分 析应 用
? 基于衍射强度测量的应用
（ 1）物相的定量分析，结晶度的测定
(2) 平衡相图的相界的测定；
(3) 第三类应力的测定；
(4) 有序固溶体长程有序度的测定；
(5) 多晶体材料中晶粒择优取向的极图、
反极图和三维取向分布的测定；
(6) 薄膜厚度的测定。
苏 玉 长
衍 射分 析应 用
? 基于衍射线型分析的应用
(1) 多晶材料中位错密度的测定，层错能
的测定，晶体缺陷的研究；
(2) 第二类（微观残余）应力的测定；
(3) 晶粒大小和微应变的测定；
苏 玉 长
衍 射分 析应 用
? 基于衍射位置和强度的测定
(1) 物相的定性分析
(2) 相消失法测定相平衡图中的相界 ;
(3) 晶体 (相 )结构,磁结构,表面结构,界面结
构的研究
苏 玉 长
衍 射分 析应 用
? 同时基于衍射位置、强度和线型的
Rietveld多晶结构测定
需输入原子参数（晶胞中各原子的坐标、
占位几率和湿度因子）、点阵参数、波长、偏
正因子、吸收系数、择优取向参数等。
苏 玉 长
衍 射分 析应 用 ? Rietveld profile refinement
?What is the Rietveld method?
?A comprehensive,computer based method
fro the analysis of X-ray and neutron powder
diffraction patterns by least squares fitting of
a calculated diffraction pattern to the
observed data.
?What is special for Rietveld method?
?This method does not use integrated powder
diffraction intensities,but employs directly
the profile intensities obtained from step-
scanning measurements of the powder
diagram.
苏 玉 长
衍 射分 析应 用
? Rietveld profile refinement
?How useful of Rietveld method?
?The Rietveld method has been developed into a
valuable method for structure analyses of nearly
all class of crystalline materials (Young 1995) by
a number endeavours.It has been extensively used
to investigate the structure and phase problems,A
lot of useful information can be retrieved by
Rietveld method.
苏 玉 长
苏 玉 长
苏 玉 长
对于大多数固溶体，其点阵参数随溶质原子的浓度呈近似线
性关系，即服从费伽 (Vegard)定律：
式中,aA和 aB分别表示固溶体组元 A和 B的点陈参数。因此，
测得含量为 x的 B原子的因溶体的点阵参数工 ax，用上式即求
得固溶体的组分。
实验表明，固溶体中点阵参数随溶质原子的浓度变化有不
少呈非线性关系，在此情况下应先测得点阵参数与溶质原子
浓度的关系曲线。 实际应用中，将精确测得的点阵参数与
已知数据比较即可求得固溶体的组分。
%100????
AB
Ax
aa
aax
