2.1. Technical construction

A schematic drawing of the heating stage is presented in Fig. 1. The central part of the heating stage is the heating plate (1), which has the shape of a disc. The heating plate is made from a nickel-chromium alloy (Inconel), which shows high stability against oxidation at high temperatures. Defined temperatures of the heating plate are obtained by electrical resistive heating using a Thermocoax heating wire (2) mounted as a spiral at the inner side of the heating plate. The outer side of the heating plate is manufactured as a plain highly polished surface. Thin plate-like samples (3) can be fixed on this side of the heating plates by using clamps (4). The maximum size of the sample is limited by the diameter of the heating plate, which is 28  mm. The temperature is measured at the centre of the heating plate just below the polished surface; an Ni-Cr sheath thermocouple (5) is used. The heating wire and the thermocouple are connected to a temperature controller, which provides the electrical power to heat the heating plate. Temperatures up to 1173  K can be reached by the heating plate. The heating plate is fixed to the chassis (6) by a support (7). The cylindrical support is suspended by elongated holes in order to minimize the heat flow from the heating plate to the chassis. The support is welded to the heating plate as well as the chassis in order to obtain a rigid connection. The thermal isolation between the heating plate and the chassis is achieved by the use of foamed Al₂O₃ (8).

Figure 1 Schematic drawing of the heating stage. The denoted parts are: heating plate (1), heating wire (2), sample (3), clamps (4), thermocouple (5), chassis (6), support of the heating plate (7), foamed Al2O3 (8), dome (9), emission rod (10), rotatable mounting of the cooling system (11), supply hose (12), air-pressure hose (13).

The sample as well as the heating plate is surrounded by a protecting plastic dome (9). The dome has the shape of a half sphere with a diameter of 56  mm and a wall thickness of 0.25  mm. The centre of the half sphere is located at the centre of the heating plate. The half-sphere geometry of the dome allows the primary beam to reach the sample through all possible directions of one hemisphere by passing the same thickness of the wall. The same geometric condition is valid also for the diffracted beam arising from the sample. The dome is made of a high-performance polymer, poly(ether-ether-ketone) (PEEK), in order to provide enough mechanical stability at high temperatures. The dome is fixed to the chassis by screws; it has to be removed from the heating stage when changing samples.

The sample chamber is terminated at one side by the dome and at the other side by the chassis; an O-ring between the dome and the chassis is used to make their connection vacuum tight. A flexible vacuum pipe with a diameter of 6  mm is used to evacuate the sample chamber, a vacuum of 0.5 × 10²  Pa can be obtained within the sample chamber. The sample chamber can be filled with inert gas (e.g. N₂) to avoid any chemical reactions of the sample at elevated temperatures.

The heating stage can be mounted directly to a goniometer or any other sample stage of an X-ray diffraction system. The chassis is fixed by screws directly to the head of the goniometer. The total shape of the chassis is designed as a heat sink with cooling fins for heat dissipation. It is made of aluminium with an outer diameter of 125  mm and a distance from the bottom of the chassis to the top of the heating plate of 20  mm. The chassis carries a cooling system which operates with compressed air. Fine multidirectional streams of air are introduced via the emission rod (10) which is situated above the top of the dome. These air streams build a turbulent flow of air along the dome surface towards the cooling fins of the chassis. This air flow cools the dome as well as the chassis by streaming along the dome wall and passing the cooling fins. The cooling system is rotatable on the chassis, so that the cooling system does not have to follow the movement of the chassis. This rotatable mounting of the cooling system (11) avoids any shadowing of the primary beam and of the diffracted beam by the cooling system. It is possible to remove the cooling system (including the emission rod) from the heating stage. Furthermore, the use of the dome is not mandatory to perform diffraction experiments.

Two different pipes have to be connected to the heating stage; both hoses are necessary for operation of the equipment. The supply hose (12) carries the vacuum line for evacuating the sample chamber, the power supply for the heating wire and the thermocouple for the temperature measurement. The supply hose is fixed to the chassis and has to follow all rotations of the goniometer along the angles , and . Rewinding of the supply hose (at the angle) prevents any intersection of the supply hose with other goniometer parts. The end of the supply hose provides connectors for the thermocouple and for the electrical supply, as well as an adapter for a gas/vacuum system (DN16KF). The air-pressure hose (13) provides the cooling system with compressed air; it is fixed directly to the cooling system. The air pressure hose has to follow only the and movements of the goniometer. The total weight of the heating stage including both hoses is 340  g.

2.2. Beam directions

Fig. 2 shows the notation of the angles , , and of the scattering angle 2 as used in four-circle texture (and stress) goniometers. In addition, some parts of the heating stage are depicted: the heating plate (1), the dome (9) and the emission rod (10) of the cooling system. The angles , and determine the angle of incidence of the primary beam relative to the heating plate (or sample). The rotation axes of the two angles and are drawn as point-dashed lines. At the crossing of these two axes, the direction of the rotation axis of the angle is indicated. The direction of the diffracted beam relative to the primary beam is given by the scattering angle 2. The heating stage allows nearly all possible directions of the primary beam and of the diffracted beam relative to the heating plate. The angular range for is is 0 to 360°. However, no continuous rotation is possible because the supply hose permits only forward and backward rotation of around the initial position of the heating stage. The angular range for is -90 to 90°. However, small limitations are present for the angle and the scattering angle 2 due to the emission rod (10) of the cooling system. A primary beam with an incidence angle of = 82-98° is not possible. The diffracted beam can be shadowed by the emission rod; the inaccessible range of the 2 angle depends on the incidence angle of the primary beam. The inaccessible range of 2 is 82° + > 2 > 98° + . For symmetrical incidence/exit conditions of the primary and the diffracted beam, the angle 2 is limited to values above 164°.

Figure 2 The angles of a four-circle texture goniometer relative to the heating stage. The primary beam covers the angles of incidence , and relative to the heating plate and the diffracted beam takes the scattering angle 2 relative to the primary beam.

3.1. Heating properties

The heat flow away from the heating plate towards colder parts of the heating stage (the chassis and the dome) generates inhomogeneities in the temperature distribution at the heating plate. The temperature distribution at the surface of the heating plate is given in Fig. 3. The temperature of the heating plate was set to 973  K using the thermocouple as temperature sensor. The spatial distribution of the evolving temperature was determined by a CCD infrared camera (Inframetrics Type 760); the dome was removed when taking this picture. At the centre of the heating plate, a temperature of 958  K was observed. Around the centre of the heating plate, the temperature distribution is quite homogenous: approximately 2/3 of the total area shows equal temperature within a tolerance of ±10  K. However, at the outer zone (close to the edge) of the heating plate, the temperature is reduced. A second reason for a temperature inhomogeneity is caused by the heat flow away from the heating plate towards the colder wall of the dome. The main difference between the temperature at the surface of the heating plate and the temperature on top of a sample arises from the temperature gradient through the sample. The temperature gradient along the thickness of the sample depends on the thermal resistance between the bottom of the sample and the surface of the heating plate, and also on the thermal conductivity through the sample. To minimize this temperature error, the thermal contact between the heating plate and the bottom of the sample should be as good as possible: the thermal conductivity of the sample should be as high as possible, thin samples should be used and a highly flat underside of the sample is favourable in order to obtain a large contact surface between sample and the heating plate. In the case of samples with low thermal conductivity, it is recommend to determine the actual temperature at the top surface of the sample prior to the X-ray diffraction experiment.

Figure 3 The temperature distribution at the surface of the heating plate measured with an infrared CCD camera. The temperature of the heating plate was set to 973  K.

The hot stage is designed for 1173  K as the maximum temperature of the heating plate. This temperature can be achieved by evacuating the sample chamber as well as by using inert gas. A power consumption of 110  W (under vacuum) and 130  W (using nitrogen as the inert gas) is necessary to obtain 1173  K. Only 2  min are necessary to reach this temperature from room temperature; it takes a few minutes more for temperature stabilization within an accuracy of ±0.5  K. A heating-plate temperature of 1173  K causes a large heat flux towards the dome and the chassis, which would cause a considerable temperature rise of these components. Therefore, the use of the cooling system is required at such high temperatures. Long-term tests of the heating stage were performed in order to determine the resulting temperatures of the dome and of the chassis. For these purposes, the heating stage was mounted on a Seifert PTS3000 diffractometer. The sample chamber of the heating stage was filled with inert gas. At a heating-plate temperature of 1173  K, stabilized temperature conditions were achieved after 30  min: the outer dome surface reaches a maximum temperature of 320  K; for the chassis 310  K is observed. Results at different temperatures of the heating plate obtained with and without the cooling system are given in Table 1. In all cases the temperature of the goniometer head was the same as the temperature of the chassis. More information about heating characteristics and long-term tests of the heating stage is given by Tamas (2002).

3.2. Thermally induced sample movements

The thermal expansion of the heating plate (1) and of the heating-plate support (7) cause a movement of the surface of the heating plate from the centre of the goniometer. The thermal movement of the heating plate was investigated in the vertical direction by using a micrometer screw with an accuracy of ±10  µm. It was measured at five different positions at the surface of the heating plate; one of these positions was located at the centre, the other four positions were chosen close to the edge of the heating plate. The measurements were performed by setting the micrometer screw (only for a fraction of a second) on the surface of the heating plate. The vertical heights at the five individual positions are given as a function of temperature in Fig. 4. The inset of Fig. 4 gives the exact location of the five investigated positions on the heating plate. The movement of the individual positions is approximately linear with a maximum dislocation of 130  µm at 1173  K. The specific expansion of the heating plate in the vertical direction is determined as 15  µm 100  K^-1.

Figure 4 Vertical movement of the heating plate as a function of temperature measured at five different positions. The five positions are given in the inset.

In order to check the influence of this movement on the diffraction experiment, the 400 reflection of a single-crystal Si(100) wafer was investigated as a function of temperature. The experiments were performed with a Seifert PTS3000 diffractometer in Bragg-Brentano geometry with Cu K radiation. A circular collimator with a diameter of 1  mm was used. /2 scans were performed. From the peak positions of the 400 reflection, the lattice constant (a) of silicon was calculated at different temperatures. These experimental results (filled circles in Fig. 5) were compared with literature data of Dutta (1962) (line in Fig. 5). Some difference in the temperature-dependent behaviour of the lattice constant is observed. However, a vertical dislocation of the sample from the goniometer centre of 15  µm 100  K^-1 is taken into account in order to obtain corrected 2 angles (Klug & Alexander, 1974). Using this correction, a good agreement between the corrected experimental values (open squares in Fig. 5) with the literature data is found for temperatures below 873  K. At higher temperatures, slightly smaller lattice constants are observed experimentally.

Figure 5 The lattice constant of silicon as a function of temperature (filled circles) experimentally determined from the 400 reflection in Bragg-Brentano geometry: corrected data (open squares) and literature data (solid line) from Dutta (1962).

The elevated temperature of the heating stage also induces some torsional movement of the heating plate. As given in Fig. 4, the vertical movement of two positions (open triangles and filled squares) do not have the same tendency compared with the other three positions. Especially at temperatures above 873  K, a difference is clearly visible. A height difference of approximately 0.03  mm is observed between the centre of the heating plate (filled circles) and the position of the `open triangle'. The horizontal distance between the centre and the `open triangle' position was 12  mm. From these two values the torsional movement of the heating plate can be estimated: a value of 0.14° is obtained. In order to confirm this result, diffraction studies of the 400 reflection of a single-crystal Si(100) wafer were performed using the same experimental setup as mentioned above. scans and scans were performed in order to find the shift of the 400 pole as a function of temperature. The detailed experimental results are not shown here, but the maximum shift of the 400 pole was observed at 1173  K with a total torsional angle of 0.18°. More details about the influence of the sample movement on the diffraction experiments are given by Tamas (2002).

3.3. X-ray absorption of the dome

The primary as well as the diffracted beam has to pass the walls of the dome; therefore, an influence of the dome on the X-ray diffraction experiment is expected. The main effect is the reduction of the X-ray beam intensity. The absorption of the beam passing through the dome walls was calculated by using X-ray absorption coefficients and taking into account a wall thickness of 0.25  mm. The absorption coefficients are determined from the mass density and chemical composition of the dome material; the X-ray mass absorption coefficients are taken from Wilson & Prince (1999). Assuming that the X-ray beam passes the dome walls twice, a total absorption of 60% for Cr K, 35% for Cu K and only 5% for Mo K was obtained. The calculated absorption of Cu K was also checked experimentally. For this purpose a textured copper sheet was investigated at ambient temperature using the experimental setup described above. scans of the Cu 111 reflection were performed at room temperature with and without using the dome. The result is depicted in Fig. 6. The transmission of the X-ray beams through the dome wall reduces the intensity to 68%, which means an absorption of 32% (absolute error 3%). The experimentally observed absorption agrees quite well with the calculated value of 35%. The depicted result of Fig. 6 reveals also that the transmission of the beams through the dome reduces only the intensity, the characteristic features of the diffraction pattern remaining unchanged.

Figure 6 scans of the 111 reflection of a textured copper sheet performed with (solid line) and without (filled squares) using the dome.

Another important specification is the homogeneity of the X-ray transmission through the dome walls. The transmission homogeneity is analysed as a function of two parameters: the angles and . The variation of the transmission as a function of the angle can be analysed from the data given in Fig. 6. In the range 0-75°, an absolute variation of ±3% is observed. At high angles, the beam geometry together with the background intensity may have some influence; therefore, the transmission cannot be evaluated from this type of experiment. The transmission homogeneity at different angles was studied with a BaF₂ powder sample at ambient temperature. The heating stage was used without the dome and with the emission rod removed. The 440 reflection of BaF₂, located at a 2 angle of 89.3°, was investigated by using a primary beam in grazing incidence. Using this specific beam geometry and setting the angle to 0°, the diffracted beam passes the dome exactly at the top of the dome. Performing scans, the primary beam will maintain its position for transmitting the dome and the diffracted beam will run along a meridian ( = constant). The intensities along three different meridians, at = 0, 120 and 240°, were measured. The observed intensities of these three scans are depicted in Fig. 7. No differences were observed for the dependences of the three meridians. Owing to the low counting statistics of these experiments, the homogeneity of the transmission through the dome in different directions cannot be given to better than 20%.

Figure 7 scans of the 440 reflection of a BaF2 powder sample at different meridians: = 0° (open circles), 120° (open squares) and 240° (open triangles).