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Arithmetical determination of the resolution function for surface structure analysis
113
Surface Science 211/212 (1989) 113-119 North-Holland. Amsterdam
ARITHMETICAL DETERMINATION OF THE RESOLUTION FUNCTION FOR SURFACE STRUCTURE ANALYSIS C. SCHAMPER,
H. SCHULZ
Institut fiir Krisfailographie, Fed. Rep. of Germany Received
and D. WOLF
Uniuersttiit Miinchen,
16 June 1988; accepted
for publication
Therestenstrasse
26 October
41, D-8000 Miinchen 2,
1988
All experimental diffraction data have to be corrected for the instrumental resolution. For the case of X-ray diffraction from crystal surfaces we propose a numerical method to calculate the instrumental resolution function at any diffraction conditions. The resolution is calculated as a function of the wavelenght and the divergence of the primary beam. The instrument profile of the diffracted beam is calculated as well. Since no analytical approximations are made, the algorithm is valid for all diffraction geometries and arbitrary angle profile and wavelength distribution of the primary beam. The kinematic scattering theory is used and effects by total external reflection have been neglected. A simulation experiment on the rod-like diffuse scattering of polymorphic Sic verified the results of a model calculation.
1. Introduction Unlike the diffraction from three-dimensional crystals, the diffraction from crystal surfaces results in a set of diffracted beams at all diffraction conditions. The usual rotation of the crystal for the measurement of the integral intensity is not necessary in the 2D case because the lattice rod always intersects the Ewald sphere. The divergence and finite energy resolution of the primary beam causes the excitation of an extended zone AL, on the rod in the reciprocal lattice so that the resolution in real space in the direction normal to the surface is limited (fig. 1). This effect becomes essentially important at diffraction conditions where the diffracted beam leaves the crystal at small angles to the surface. An additional beam broadening arises through the mosaic spread and the extension of the illuminated surface area. These effects are not considered here. Here only the profile of the diffracted beam as a function of the scattering geometry and the characteristics of the primary beam assuming a perfect crystal is discussed. In that case the integral intensity of the diffracted beam is given by [l]
1, = ‘,p
1
io(Ko)I FHKLI *
sin*rN, h sin*rN, k sm*rh
dK,
dK,
sin*rk
0039-6028/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
dt,
114
C. Srhamper
et ~1. / Determinatron
of the resolution
Jiinctim
Fig. 1. Schematic representation in reciprocal space of the diffraction by a surface (Ewald construction) for a polychromatic parallel primary beam (a) and a monochromatic divergent primary beam (b). In both cases the excited zone AL,, has a finite length and the divergence k changed by the diffraction. In (b) the definition of a visible zone AL,. determined by the detector aperture is shown.
where r, = e’/mc”, P the polarisation factor and t the measuring integration has to be taken over the primary and the diffracted integration over K, with a fixed wave vector of the primary beam standard formula for the integrated intensity. L is a continuous two-dimensional diffraction and the excited zone for a general geometry depends in a complicated manner on the divergence of
time. The beam. The leads to the variable for diffraction the primary
C. Schamper
et al. / Determrnation
of the resolutron function
115
beam. In the following an algorithm is described to calculate the resolution function numerically. The continuous change of the diffraction geometry during a rod scan shifts the excited zone along the rod, this corresponds to a convolution of the structure factor with the resolution function. The resolution function itself changes rapidly with small angles between the surface and the outgoing beam. A finite detector aperture usually influences the profile of the diffracted beam (fig. lb). In this case a reduced intensity 1,’ is measured. The relevant rod zone (visible zone) is a defined subrange (with length AL,) of the excited zone. With a small detector aperture the resolution is improved. However, in order to get the true integral intensity of the diffracted beam, as required in the structure analysis, the measured intensity has to be corrected. Using X-ray surface diffractometers [2,3] not limited to detector settings in a horizontal plane, this method offers a universal three-dimensional treatment for inclined scattering planes combined with arbitrary sample orientations. In the numerical calculation of the resolution function the illuminated area of the surface is neglected and the primary beam is assumed to be focused on the surface. The resolution is then determined only by the wavelength range and the divergence of the primary beam and the orientation of the crystal, provided the area of the detector is large enough to receive the total diffracted intensity. Primary beams containing radiation of several discrete energies, e.g. Ka, and Ka,, are simply treated by superposition. Calculations for arbitrary angle distributions of the primary beam are possible as well.
2. Calculation
of the resolution
and the instrument
profile
The introduction of three specific coordinate systems in reciprocal space allows a convenient treatment of the three-dimensional problem. Their relationship is shown in fig. 2 and it is useful to choose an absolute A-’ scale. The reciprocal surface system (s) is defined as a Cartesian coordinate system with the z,-axis parallel to the rods. The second system is the labsystem (1). Its origin is identical to that of the s-system and the directions of its axes are defined by the primary beam: z, is antiparallel to the central primary beam and x, and y, are adapted to the rectangular shape of the window of the angular width of the primary beam. The transformation operation for the transition from the l- to the s-system can be described as the product of three rotations about the axes:
M,, = R,(Y) * R,Jb) - R,(a). The rotation angles (Y, p and y describe respect to the central primary beam.
(2) the orientation
of a crystal
with
116
/) I)’ \( ,
A,’ zc
, ,”
_. Yr ,:_’
1 21 * zs= c*
/
‘\
\
,,” ‘\
XS
x:
Fig. 2. Ewald-construction
and relationship
hetween
the s-. l- and c-system.
To take account of the divergence of the primary beam the deviation of a single photon from the z, direction, 6, and a,, is assumed and the calculation of the scattering event of this photon has to be done with the orientation angles (Y+ 6,. ,LI+ 6,. and y. A lattice rod usually intersects the Ewald-sphere twice. Here only one point corresponding to a reflected beam is of interest. In the third system, the c-system, (fig. 2) this point is found by a simpe triangle calculation. The origin of the c-system is identical to the centre of the Ewald-sphere and zC is parallel to the rods. The x,-z,-plane is defined by the rod and the origin. Therefore a vector V, in the s-system is given in the c-system by ~=R,(u)*V,-tK,,=M,,*V,.
(3)
K,, must be used in coordinates of the s-system. The rotation angle u is given by (T= 71/2 + arctan( H/K). H, K, and L are the components of the momentum transfer in the s-system (scattering vector). The transformation of the vector (H,K,O) into the c-system allows the calculation of the intersection point in coordinates of the c-system. Retransforming this intersection point into the s-system and the l-system yields the scattering vector Q = (H, K, L) and the direction of the diffracted photon K, in the l-system: (in eq. (5) K,, has to be used in coordinates of the l-system) I Q = MI,’
* \
K, =Ko+
M1,l.Q.
(5)
C. Schamper et al. / Determtnation
of the resolution function
117
where V, is given by
The excited zone and the two-dimensional profile of the diffracted beam is obtained by choosing a two-dimensional grid of primary beams K, (,,) and calculating the corresponding scattering vectors Q(,)) as well as the resulting momenta K, (,,). If only the resolution in reciprocal space is required the maximum and minimum L-values are found much faster by calculating the L-value for an arbitrary K, (,,) and tracing fgrad L( K,) until the extreme values are reached. of the primary Weigthening the K, (, ,) according to an intensity distribution beam is possible. For any photon a check is made whether it can pass the detector aperture (fig. 1). The measured intensity (I,‘),,, multiplied by the calculated ratio of the total received intensity (Z;)ca,c and the total diffracted intensity (Z,)ca,c yields the true integrated intensity of the diffracted beam (Z1)exp: (6)
( Z, Lp = ( ZI/ZI’ Llc (ZI’ Lxp. Since the calculation is based on the elastic primary beam the total energy is conserved
scattering
of every photon
( 1,Lk = ( 10Lk. Estimates of the experimental characteristic of the primary relevant parameters.
of the
(7) error resulting beam are easily
from done
an unprecisely by a variation
known of the
3. A simulation experiment with polymorphic SIC An important feature of polymophic crystals like SIC is a rod lattice in reciprocal space. Since the c-axis is parallel to the rods the situation is analogous to a surface and experimental studies of the instrumental resolution can easily be performed. The (O,l)-rod was scanned in the y,-z,-plane by rotating the crystal around the x,-axis. The primary beam (Cu KLY, radiation) has been focused on the sample and the total diffracted intensity has been measured. In this case the resolution is very sensitive to the sample orientation. Fig. 3a shows the calculated length of the excited zone as a function of the angle of incidence a.
118
superstructure period 0660
60
Fig. 3. Resolution relevant calculated
0726
0853
65
092L
70
n
t "1
limit: Measured intensity profile for a scan of the SIC (O,l)-rod (b) and the length of the excited zone (a). The scattering geometry is shown in the insert of (a).
An approximate limit of the resolution of the periodic modulation is reached, when the period becomes equal to the length of the excited zone. Since the period here is 3.3 X 10e3 k' an insufficient resolution can be expected above L = 0.726 A-‘. In the range from 0.858 to 0.924 k’ the resolution is better and the superstructure should be resolved. (By theory there is no intensity in the range between 0.726 and 0.858 k’.) The experimental results (fig. 3b) verify the calculations. The deviation from the predicted resolution limit is mainly caused by experimental reasons (e.g. unprecise determination of the divergence).
4. Outlook The formalism discussed here can be used as a part of a Monte Carlo simulation to determine the resolution of a complete experimental setup for
C. Schamper
et al. / Derermnu~ion
of the resolution
funcrion
119
X-ray surface diffraction. The advantage of this method is its flexibility and the possibility to perform the calculation with many parameters such as the illuminated surface area and the crystal mosaic spread. Such a program is in development now.
Acknowledgement We gratefully acknowledge the support Forschungsgemeinschaft” (SFB 128).
of this work
by the “Deutsche
References [l] B.E. Warren, X-ray Diffraction (Addison-Wesley, Reading, 1969). [2] F. Kretschmar, D. Wolf, H. Schulz, H. Huber and H. PlGckl, Z. Krist. 175 (1987) 130. [3] E. Vlieg, A. van ‘t Ent, A.P. de Jongh, H. Neerings and J.F. van der Veen, Nucl. Methods A 262 (1987) 522.
Instr.
Surface Science 211/212 (1989) 113-119 North-Holland. Amsterdam
ARITHMETICAL DETERMINATION OF THE RESOLUTION FUNCTION FOR SURFACE STRUCTURE ANALYSIS C. SCHAMPER,
H. SCHULZ
Institut fiir Krisfailographie, Fed. Rep. of Germany Received
and D. WOLF
Uniuersttiit Miinchen,
16 June 1988; accepted
for publication
Therestenstrasse
26 October
41, D-8000 Miinchen 2,
1988
All experimental diffraction data have to be corrected for the instrumental resolution. For the case of X-ray diffraction from crystal surfaces we propose a numerical method to calculate the instrumental resolution function at any diffraction conditions. The resolution is calculated as a function of the wavelenght and the divergence of the primary beam. The instrument profile of the diffracted beam is calculated as well. Since no analytical approximations are made, the algorithm is valid for all diffraction geometries and arbitrary angle profile and wavelength distribution of the primary beam. The kinematic scattering theory is used and effects by total external reflection have been neglected. A simulation experiment on the rod-like diffuse scattering of polymorphic Sic verified the results of a model calculation.
1. Introduction Unlike the diffraction from three-dimensional crystals, the diffraction from crystal surfaces results in a set of diffracted beams at all diffraction conditions. The usual rotation of the crystal for the measurement of the integral intensity is not necessary in the 2D case because the lattice rod always intersects the Ewald sphere. The divergence and finite energy resolution of the primary beam causes the excitation of an extended zone AL, on the rod in the reciprocal lattice so that the resolution in real space in the direction normal to the surface is limited (fig. 1). This effect becomes essentially important at diffraction conditions where the diffracted beam leaves the crystal at small angles to the surface. An additional beam broadening arises through the mosaic spread and the extension of the illuminated surface area. These effects are not considered here. Here only the profile of the diffracted beam as a function of the scattering geometry and the characteristics of the primary beam assuming a perfect crystal is discussed. In that case the integral intensity of the diffracted beam is given by [l]
1, = ‘,p
1
io(Ko)I FHKLI *
sin*rN, h sin*rN, k sm*rh
dK,
dK,
sin*rk
0039-6028/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
dt,
114
C. Srhamper
et ~1. / Determinatron
of the resolution
Jiinctim
Fig. 1. Schematic representation in reciprocal space of the diffraction by a surface (Ewald construction) for a polychromatic parallel primary beam (a) and a monochromatic divergent primary beam (b). In both cases the excited zone AL,, has a finite length and the divergence k changed by the diffraction. In (b) the definition of a visible zone AL,. determined by the detector aperture is shown.
where r, = e’/mc”, P the polarisation factor and t the measuring integration has to be taken over the primary and the diffracted integration over K, with a fixed wave vector of the primary beam standard formula for the integrated intensity. L is a continuous two-dimensional diffraction and the excited zone for a general geometry depends in a complicated manner on the divergence of
time. The beam. The leads to the variable for diffraction the primary
C. Schamper
et al. / Determrnation
of the resolutron function
115
beam. In the following an algorithm is described to calculate the resolution function numerically. The continuous change of the diffraction geometry during a rod scan shifts the excited zone along the rod, this corresponds to a convolution of the structure factor with the resolution function. The resolution function itself changes rapidly with small angles between the surface and the outgoing beam. A finite detector aperture usually influences the profile of the diffracted beam (fig. lb). In this case a reduced intensity 1,’ is measured. The relevant rod zone (visible zone) is a defined subrange (with length AL,) of the excited zone. With a small detector aperture the resolution is improved. However, in order to get the true integral intensity of the diffracted beam, as required in the structure analysis, the measured intensity has to be corrected. Using X-ray surface diffractometers [2,3] not limited to detector settings in a horizontal plane, this method offers a universal three-dimensional treatment for inclined scattering planes combined with arbitrary sample orientations. In the numerical calculation of the resolution function the illuminated area of the surface is neglected and the primary beam is assumed to be focused on the surface. The resolution is then determined only by the wavelength range and the divergence of the primary beam and the orientation of the crystal, provided the area of the detector is large enough to receive the total diffracted intensity. Primary beams containing radiation of several discrete energies, e.g. Ka, and Ka,, are simply treated by superposition. Calculations for arbitrary angle distributions of the primary beam are possible as well.
2. Calculation
of the resolution
and the instrument
profile
The introduction of three specific coordinate systems in reciprocal space allows a convenient treatment of the three-dimensional problem. Their relationship is shown in fig. 2 and it is useful to choose an absolute A-’ scale. The reciprocal surface system (s) is defined as a Cartesian coordinate system with the z,-axis parallel to the rods. The second system is the labsystem (1). Its origin is identical to that of the s-system and the directions of its axes are defined by the primary beam: z, is antiparallel to the central primary beam and x, and y, are adapted to the rectangular shape of the window of the angular width of the primary beam. The transformation operation for the transition from the l- to the s-system can be described as the product of three rotations about the axes:
M,, = R,(Y) * R,Jb) - R,(a). The rotation angles (Y, p and y describe respect to the central primary beam.
(2) the orientation
of a crystal
with
116
/) I)’ \( ,
A,’ zc
, ,”
_. Yr ,:_’
1 21 * zs= c*
/
‘\
\
,,” ‘\
XS
x:
Fig. 2. Ewald-construction
and relationship
hetween
the s-. l- and c-system.
To take account of the divergence of the primary beam the deviation of a single photon from the z, direction, 6, and a,, is assumed and the calculation of the scattering event of this photon has to be done with the orientation angles (Y+ 6,. ,LI+ 6,. and y. A lattice rod usually intersects the Ewald-sphere twice. Here only one point corresponding to a reflected beam is of interest. In the third system, the c-system, (fig. 2) this point is found by a simpe triangle calculation. The origin of the c-system is identical to the centre of the Ewald-sphere and zC is parallel to the rods. The x,-z,-plane is defined by the rod and the origin. Therefore a vector V, in the s-system is given in the c-system by ~=R,(u)*V,-tK,,=M,,*V,.
(3)
K,, must be used in coordinates of the s-system. The rotation angle u is given by (T= 71/2 + arctan( H/K). H, K, and L are the components of the momentum transfer in the s-system (scattering vector). The transformation of the vector (H,K,O) into the c-system allows the calculation of the intersection point in coordinates of the c-system. Retransforming this intersection point into the s-system and the l-system yields the scattering vector Q = (H, K, L) and the direction of the diffracted photon K, in the l-system: (in eq. (5) K,, has to be used in coordinates of the l-system) I Q = MI,’
* \
K, =Ko+
M1,l.Q.
(5)
C. Schamper et al. / Determtnation
of the resolution function
117
where V, is given by
The excited zone and the two-dimensional profile of the diffracted beam is obtained by choosing a two-dimensional grid of primary beams K, (,,) and calculating the corresponding scattering vectors Q(,)) as well as the resulting momenta K, (,,). If only the resolution in reciprocal space is required the maximum and minimum L-values are found much faster by calculating the L-value for an arbitrary K, (,,) and tracing fgrad L( K,) until the extreme values are reached. of the primary Weigthening the K, (, ,) according to an intensity distribution beam is possible. For any photon a check is made whether it can pass the detector aperture (fig. 1). The measured intensity (I,‘),,, multiplied by the calculated ratio of the total received intensity (Z;)ca,c and the total diffracted intensity (Z,)ca,c yields the true integrated intensity of the diffracted beam (Z1)exp: (6)
( Z, Lp = ( ZI/ZI’ Llc (ZI’ Lxp. Since the calculation is based on the elastic primary beam the total energy is conserved
scattering
of every photon
( 1,Lk = ( 10Lk. Estimates of the experimental characteristic of the primary relevant parameters.
of the
(7) error resulting beam are easily
from done
an unprecisely by a variation
known of the
3. A simulation experiment with polymorphic SIC An important feature of polymophic crystals like SIC is a rod lattice in reciprocal space. Since the c-axis is parallel to the rods the situation is analogous to a surface and experimental studies of the instrumental resolution can easily be performed. The (O,l)-rod was scanned in the y,-z,-plane by rotating the crystal around the x,-axis. The primary beam (Cu KLY, radiation) has been focused on the sample and the total diffracted intensity has been measured. In this case the resolution is very sensitive to the sample orientation. Fig. 3a shows the calculated length of the excited zone as a function of the angle of incidence a.
118
superstructure period 0660
60
Fig. 3. Resolution relevant calculated
0726
0853
65
092L
70
n
t "1
limit: Measured intensity profile for a scan of the SIC (O,l)-rod (b) and the length of the excited zone (a). The scattering geometry is shown in the insert of (a).
An approximate limit of the resolution of the periodic modulation is reached, when the period becomes equal to the length of the excited zone. Since the period here is 3.3 X 10e3 k' an insufficient resolution can be expected above L = 0.726 A-‘. In the range from 0.858 to 0.924 k’ the resolution is better and the superstructure should be resolved. (By theory there is no intensity in the range between 0.726 and 0.858 k’.) The experimental results (fig. 3b) verify the calculations. The deviation from the predicted resolution limit is mainly caused by experimental reasons (e.g. unprecise determination of the divergence).
4. Outlook The formalism discussed here can be used as a part of a Monte Carlo simulation to determine the resolution of a complete experimental setup for
C. Schamper
et al. / Derermnu~ion
of the resolution
funcrion
119
X-ray surface diffraction. The advantage of this method is its flexibility and the possibility to perform the calculation with many parameters such as the illuminated surface area and the crystal mosaic spread. Such a program is in development now.
Acknowledgement We gratefully acknowledge the support Forschungsgemeinschaft” (SFB 128).
of this work
by the “Deutsche
References [l] B.E. Warren, X-ray Diffraction (Addison-Wesley, Reading, 1969). [2] F. Kretschmar, D. Wolf, H. Schulz, H. Huber and H. PlGckl, Z. Krist. 175 (1987) 130. [3] E. Vlieg, A. van ‘t Ent, A.P. de Jongh, H. Neerings and J.F. van der Veen, Nucl. Methods A 262 (1987) 522.
Instr.