Characterisation of materials using x-ray diffraction and X-ray powder diffraction. Cristina Mercandetti Nicole Schai

Transcription

1 P1 and P2 Characterisation of materials using x-ray diffraction and X-ray powder diffraction Cristina Mercandetti Nicole Schai Supervised by Taylan Oers and Pawel Kuczera Report ETH Zurich 2012

2 TABLE OF CONTENT TABLE OF CONTENT 1 ABSTRACT INTRODUCTION PRODUCTION AN HANDLING OF X- RAYS THE LAUE METHOD DEBYE SCHERRER METHOD... FEHLER! TEXTMARKE NICHT DEFINIERT. 2.4 POWDER DIFFRACTION... FEHLER! TEXTMARKE NICHT DEFINIERT Qualitative Phase Analysis Identification of the crystallite size MATERIAL AND METHOD P1 CHARACTERISATION OF MATERIALS USING X- RAY DIFFRACTION Laue diffraction pattern of a silicon waver Diffraction pattern of PET, NaCl and aluminium foil P2 POWDER DIFFRACTION Qualitative Phase Analysis Determination of the crystallite size of Ce 0.9 Gd 0.1 O RESULTS P1 - CHARACTERISATION OF MATERIALS USING X- RAY DIFFRACTION Laue diffraction pattern of a silicon waver Diffraction patterns of PET, NaCl and aluminium foil P2 - POWDER DIFFRACTION Qualitative Phase Analysis Determination of the crystallite size of Ce 0.9 Gd 0.1 O DISCUSSION P1 CHARACTERISATION OF MATERIALS USING X- RAY DIFFRACTION P2 POWDER DIFFRACTION Qualitative Phase Analysis Determination of the crystallite size of Ce 0.9 Gd 0.1 O CONCLUSION REFERENCES LITERATURE WEBSITES Seite 2/ 19

3 1. ABSTRACT 1 ABSTRACT There were two experiments made. P1 was all about material characterisations by x-ray diffraction. In order to find some of the inherent symmetries of the crystal structure of a silicon wafer, the Laue method was applied. X-rays from an x-ray tube were released on the wafer. The diffracted beam was detected on a photo plate. As the wafer was orientated in the [100] direction, the 4-fold axis as well as the mirror axis can be seen. As a second part, the crystallinity of aluminium, NaCl and a PET bottle was examined. One could see how the production of a material (i.e. blowing up the PET moulding or rolling the aluminium foil) had an impact on the crystallites size and orientation. The NaCl crystal examined was not quite a single crystal but close to it. The second experiment P2 dealt with powder diffraction. The experiment consisted of two different tasks. In the first task, an unknown substance had to be determined using powder diffraction. To do so, the substance was analysed by comparing the observed powder diffraction pattern to the diagrams of the database. It could be concluded that the unknown substance was a mixture of two polymorphs of calcium carbide namely aragonite and calcite. In the second task, the order of magnitude of the crystallite size of the crystallites of Ce 0.9 Gd 0.1 O 1.95 was determined using powder diffraction. It is not possible to give an exact size of the crystallites as it depends on the direction from where it is looked at. But after the measurements it could be concluded that the crystallites have a the size of about 200 A. Seite 3/ 19

4 2. INTRODUCTION 2 INTRODUCTION Two different experiments with x-rays were made. The P1 experiment deals with the symmetry of single crystals as well as polycrystalline material. P2, on the other hand, was all about identifying a certain substance by powder diffraction. All of the samples were exposed to x-rays, which were created in an x-ray tube. 2.1 Production an handling of X-Rays X-radiation or just x-rays are part of the electromagnetic spectrum and have a wavelength between 0.01 and 1 nm, which is shorter than the one of visible light and UV rays but longer than the one of gamma rays. Due to the short wavelength, x-rays have high energies and can therefore have damaging effects on living tissue, because they can disrupt molecular bonds by ionization. The wavelength corresponds approximately to the distance between two atoms and as a result they can be used to make different experiments on materials. X-rays can be produced in x-ray tubes. An x-ray tube is a vacuum tube that consists of a cathode that emits electrons and an anode that attracts the electrons. The anode is made of a specific material that will affect the wavelength of the produced x-rays. Commonly used materials are molybdenum and copper. The flow of the electrons between the electrodes creates an electrical current and is accelerated by an applied voltage. When the emitted electrons hit the anode material, they interact with other electrons in the anode and this process generates energy. Most of this energy is released as heat, but a small part is emitted in form of radiation. There are two different processes that happen, when the accelerated electrons hit the anode material: 1. An accelerated electron can knock an electron out of the metal and create a vacancy in an inner electron shell. As a result, an electron from a higher shell will fall down into the lower energy state to fill the vacancy. This change of energy level causes an emission of an x-ray photon. The wavelength of this emission depends on the energy difference of the shells and can therefore only be a discrete value depending on the metal. These are also called characteristic lines, which are usually named after the shell, which the electrons fall into (K lines, L lines etc.). These lines can be seen in the spectrum as spikes. Fig. 1: X-ray spectrum; the wavelength is plotted against the intensity of the radiation. The K lines are the spikes and have discrete wavelengths; the Bremsstrahlung is the continuous curve [5] 2. If the electron has not enough energy to knock out another electron, they will just be scattered by the strong electric field around the nucleus of the atom and decelerated. During this process the electrons lose energy in form of a continuous radiation. These kinds of x-rays are called the Bremsstrahlung. Seite 4/ 19

5 2. INTRODUCTION The interaction of the x-rays with a material can be very different. On one side there is for example beryllium that is very permeable for x-radiation because it only has four electrons per atom. On the other side there is for example lead that is very dense and is able to insulate from x-rays. Both materials have different uses in the handling with x-rays. Beryllium is used to build the windows of the x-ray tubes, which leave the produced x-rays out and lead is used to protect from x-rays. 2.2 The Laue Method The Laue Method is a long known method to identify the symmetry elements in a certain direction of a single crystal. However, using the Laue Method, one cannot determine the lattice parameters as the x-ray waves are not monochromed. The crystal is exposed to the x-rays. The important formula for any calculations and pictures taken by a method using x-rays is the Bragg equation. It explains how the initial rays result in constructive and destructive interference. In a single crystal, the waves hit the surface. As the wavelength of x-rays is similar to the distance between two atoms in a crystal (around 1 Angstrom), the waved propagate through the crystal until it hits an atom. Two waves reflected at two planes parallel to each other (a set of planes) interfere. If the additional distance travelled by the second wave is the multiple of the wavelength of a wave diffracted on a plane above, the waves interfere constructively and leave a spot on the photographic plane set up behind the single crystal. Bragg equation: nλ = 2d sin(θ) Fig. 2 Constructive interference. Constructive interference exists as soon as the Bragg equation is fulfilled. 2δ is the additional way travelled by beam number 2. If 2δ is a multiple of the wavelength e.g. 2δ= nλ - n being a whole interger. δ is defined by the angle θ (which is fix for every set of plane in the laue method) and d (= lattice parameter). Using the Laue Method, the wavelength is unknown and therefore, d cannot be determined. Using the Laue method, the single crystal is fixed and not moving during the experiment. Consequently the Bragg angle θ is fixed for every set of planes in the crystal. Each set of planes diffracts the one particular wavelength which fullfills the Bragg equation. Using white x-rays (not monochromised), the whole spectrum of wavelength is applied tot he sample. However, the angle for every set of planes is fixed. This combination makes it possible to picture every multiple set of planes in a single crystal in just one experiment. However, one only sees the symmetry elements of the crystal of the orientation in which the single crystal has been exposed to the x-ray. Seite 5/ 19

6 2. INTRODUCTION 2.3 Single Cristal Diffractometer In contrast to the Laue Camera Method, a single crystal diffractometer only uses monochromed x- ray waves. Consequently, λ is known and the lattice parameter can be calculated. However, as there are no more different wave length in order to coherently interfere at different multiple sets of planes, in one scan one can only see a part of the symmetry elements of a single crystal. In order to obtain a full pattern several frames with different theta, chi and phi values have to be made. As in the Laue Method, the reflection of one multiple set of planes of one small crystal leave one spot on the detector. However, as the crystals are orientated randomly, the reflection angle for one set of planes in different crystallites varies. Consequently, in a polycrystalline material with randomly orientated crystals, the reflects of one set of planes are arrayed and lining up to a circle on the detector. The production and treatment of the material before the exposure is very important to the resulting picture. Stress applied to a material might lead to an orientation of the crystallites. They are no more orientated randomly and the initial complete circles in the picture have now gaps. Heat treatment leads to a recrystallization and healing out of defects of a polycrystalline material. As a result of the larger, but fewer crystallites, one only sees spots. 2.4 X-ray powder diffraction The X-ray powder diffraction is another method to characterize different materials using x-rays. The advantage of this method is that the sample does not have to be a single crystal. The powder diffraction has many different applications, but the characteristic thing about this method is that the sample is always a powder with crystallites of about 0.1 to 10 μm. Like this, every possible crystalline orientation is represented equivalently. The apparatus of this method is similar to the other x-ray diffraction methods. The x-rays are produced in an x-ray tube, monochromatized using a monochromator crystal and slits and emitted on the capillary that is filled with the powder that has to be analysed. The capillary is fixed on a goniometer head. Another important part of the apparatus is the detector. While the sample is fixed, the detector moves around it and measures the intensity of the scattered radiation at different angles. The result of this method is a X-ray powder diffraction pattern that plots the intensity against the angle. The obtained X-ray powder diffraction pattern is a graph with many peaks at specific positions. This diagram is very helpful to characterize the material. By analysing the position, the width and the intensity of the peaks, characteristics like the dimension of the unit cell, the crystal structure, crystallite size, inner tensions and the texture can be defined. The X-ray powder diffraction has many advantages. It is a very rapid method that allows to analyse unknown substances and to characterize materials as for example medicines. This method is used in different fields of science and finds a great importance in pharmaceutical sciences. In this experiment the X-ray powder diffraction is first used to make a qualitative phase analysis and on another sample it is used to define the crystallite size. Seite 6/ 19

7 2. INTRODUCTION Qualitative Phase Analysis The position of the peaks depends on the size of the unit cell, and the intensity of these peaks depends on the position and kind of the atoms in the unit cell. So, these two properties together can be used to characterize the chemical compound or phase. Therefore, each material has its own fingerprint. Using the Bragg s equation, the lattice plane distance (d) can be calculated: d = λ 2 sin (θ) where λ is the wavelength of the x-rays and θ half of the diffraction angle. The d-values are characteristic for the crystal structure and therefore an important material property. As the X-ray powder diffraction has a long history, plenty of different materials have already been analyzed over the time. The different d-values have been calculated from the X-ray powder diffraction patterns and put together in databases, which make it easy to identify the material in the sample Identification of the crystallite size The shape of the peak is an effect of different influences. On one side the influences can occur from the instrument used, on the other side they can occur from the sample (crystallite size, defects, inner tensions). If the influences of the instrument are known, the one from the sample can be calculated using the following formula: β!!" =! β!"#$%&"'! β!"#$%&'("$ The influences of the sample are good information about the condition of the sample. For example it can be used to calculate the crystallite size applying the Scherrer Equation: τ!!! = λ β!!" cos (θ) where τ!!" is the size of the crystallite along the direction [hkl], λ is the wavelength oft he x-rays, β!!" is the halfwidth of the peak in radians and θ is half oft he diffrection angle. Seite 7/ 19

8 3. MATERIAL AND METHOD 3 MATERIAL AND METHOD 3.1 P1 Characterisation of materials using x-ray diffraction Laue diffraction pattern of a silicon waver In order to understand the functioning and concept of x-ray diffraction, a diffraction pattern of a silicon wafer was taken. The waver was orientated along the [100] axis by the assistant. As the crystal structure of is F4 1 /d 3 2/m [2] and the first direction is the [100] direction, we expect to see a 4-fold rotation axis. In Laue diffraction patterns an inversion centre is always added to the group. By just looking at a Laue diffraction pattern, one cannot identify whether the crystal has a 4-fold rotation axis or a 4. Material: - X-ray set up (tube, lead containing glass and blanket) - Silicon waver - watch - gloves - photo plate - black box The x-rays used were produced in an x-ray tube. It was placed in a box of led containing glass. Additionally, the backside was covered with a led-blanket. Before starting the experiment, one has to adjust the led-blanket to insure, that no x-ray can leave the box. As led is poisonous it should only be touched with gloves. The electrical set up for the tube was set to 50 kw and 35 ma. The silicon waver is placed in front of the x-ray tube. Behind it, the box containing the photo plate is set up. One has to be careful to not expose the photo plate to the light. A beam stop in order to stop the beam going directly through the material had to be put in place. A shutter on the x-ray tube was opened manually, releasing the beam from the tube. It was directly projected on the [100] direction of the silicon waver for exactly 4 minutes. After closing the scatter, the black box containing the photo plate was removed and the Laue diffraction patter on the photograph was scanned Diffraction pattern of PET, NaCl and aluminium foil Three different materials all different concerning their treatment before the measurement and grade of cristallinity have been measured with an x-ray diffractometer. Other than in the Laue diffraction, a single crystal diffractometer using monochrome x-rays was used. A detector instead of a conventional photographic film was placed behind the probe to detect the diffracted waves. Like in the Laue method, a beam stop prevented the detector from being destroyed by the beam passing directly through the sample. The diffractometer was used with an electrical set up of 50 kw and 40 ma. Additionally, the detector was cooled down to -40 C in order to minimise background noise. Material: - X-ray diffractometer - Samples: PET neck and flask, NaCl crystal, aluminium foil - lighter Seite 8/ 19

9 3. MATERIAL AND METHOD First, a piece of the neck of a PET bottle was placed in the diffractometer. It was exposed to the x- rays for only 10 seconds. The detected diffraction pattern was visualised on a computer. Second, a piece of the bottle itself was measured. It was placed upright in the diffractometer (same way as the bottle would be standing on a table). In contrast to the amorphous PET material, a NaCl crystal was measured. It was unknown whether the crystal was a single crystal or a polycrystalline material. As a third sample aluminium foil from the super market was placed in the diffractometer. By heating the foil with a lighter and measuring it again, the effect of heat treatment could be shown. As a last sample, the heat treated foil was exposed to mechanical stress by scratching the surface with a spade. The scratched sample was measured again. For all the diffraction patterns taken, the same set up was used. The samples were exposed to the x- ray beam for 10 seconds. 3.2 P2 X-ray powder diffraction Qualitative Phase Analysis In this experiment an unknown sample was analyzed using the x-ray powder diffraction method. Comparing the obtained x-ray powder diffraction pattern to the database, it could be concluded what material corresponded to the unknown sample. Material: - Agate mortar - Capillary (0.5 mm) - One drop of wax - Diffractometer (transmission, Cu radiation: 1.54 A ) First, the unknown substance was pestled to avoid agglomerates that could plug the capillary. Then, the substance was filled into the capillary, which was then broken off at about 3-4 cm. The capillary was stuck on a sample holder using wax and then fixed on the goniometer head of the diffractometer. The goniometer head was adjusted, so that the capillary was positioned perfectly. Before starting the measurement it was very important to mount a beam stop in order to avoid damages on the detector. After closing all the sliding windows the measurement was started from the computer. The angle started from 27, increased with steps of 0.01 and ended at 60. Each position was measured for 7 sec. After the measurement, the data was displayed on the computer. The background was removed and using a program the position and the intensity of the main peaks were determined and then compared with similar substances from the database Determination of the crystallite size of Ce 0.9 Gd 0.1 O 1.95 In this part of the experiment another X-ray powder diffraction was conducted, but this time with a known substance. Using the X-ray powder diffraction pattern and the formulas (see chapter 2.4.2) the crystallite size was determined. Material: - Ce 0.9 Gd 0.1 O Diffractometer Seite 9/ 19

10 3. MATERIAL AND METHOD A X-ray powder diffraction of the substance was made and then the peaks were analyzed using a computer program. The four main peaks were chosen and the half width and the position were determined. Using the Scherrer Equation the crystallite size could be determined. Seite 10/ 19

11 4. RESULTS 4 RESULTS 4.1 P1 - Characterisation of materials using x-ray diffraction Laue diffraction pattern of a silicon waver The Laue diffraction pattern measured with a conventional, old-fashioned x-ray diffractometer was scanned and compared to a pattern taken from the literature. a ) Fig. 3 Laue diffraction pattern of a silicon waver in [100] direction. a) Pattern from the experiment; b) Pattern taken from literature [4] Diffraction patterns of PET, NaCl and aluminium foil The diffraction patterns of amorphous PET have been measured in an single crystal diffractometer. As the detector was not completely cooled down to -40 C, the background noise is higher in the first patterns taken. a) b) Fig. 4 Diffraction pattern of a PET bottle, taken with monochromed x-rays. a) PET neck; b) PET bottle Seite 11/ 19

12 4. RESULTS In contrast to the amorphous PET, the x-ray beam was diffracted on a 150 micrometre NaCl crystal. It was unknown, whether the crystal is a single crystal or a polycrystalline. Fig. 5 Diffraction pattern of a NaCl crystal, taken with an single crystal diffractometer using monochromed x-rays. Finally, aluminium foil after different mechanical stress was measured. The material seems to have differently orientated and sized crystallites, depending on how they have been treated before the measurements. For all the measurements, the same sample has been used. a) b) c) Fig. 6 Diffraction pattern of aluminium foil from the supermarket. a) before heating, straight from the role; b) after heating with a lightener; c) after scratching the surface of the foil with a spade. Seite 12/ 19

13 5. DISCUSSION 4.2 P2 - X-ray powder diffraction Qualitative Phase Analysis Fig. 7: X-ray powder diffraction pattern of the unknown substance after removing the background. The x-axis is corresponds to the values of 2θ and the y-axis corresponds to the intensity. Fig. 8: The black peaks belong to the unknown substance; the red ones were taken from the database and correspond to aragonite Seite 13/ 19

14 5. DISCUSSION Fig. 9: The black peaks belong to the unknown substance; the red ones were taken from the database and correspond to calcite Determination of the crystallite size of Ce 0.9 Gd 0.1 O 1.95 Fig. 10: X-ray powder diffraction pattern of Ce 0.9Gd 0.1O 1.9; four main peaks can be recognized and these are used to calculate the crystallite size To calculate the crystallite size, the formulas from chapter were used. The position and the half width of the four main peaks were taken from the experimental data. The other factors are determined below: Seite 14/ 19

15 5. DISCUSSION β!"#$%&'("$ = 0.08 λ = 1.54 A Table 1: Calculation of the crystallite size using the data of the four main peaks Peak Position 2θ [ ] β!"#$%&"' [ ] β!!" [rad] τ!!" [A ] ~ ~ ~ ~ 190 Fig. 11: Close-up of the first peak with an approximated gauss function to calculate the half width Seite 15/ 19

16 5. DISCUSSION 5 DISCUSSION 5.1 P1 Characterisation of materials using x-ray diffraction The Laue diffraction pattern of the silicon wafer corresponds well to a pattern taken from literature. The 4-fold rotational axis can be seen well. Additionally, there are mirror planes due to the rotational axis in the horizontal and vertical line. Finally there are two mirror planes along the diagonal. The white shade on the patterns can be identified as the shade of the beam stop. Another interesting fact can be seen on the diffraction pattern. There are slightly darker grey shades on the pattern. They might be an indication, that the silicon waver is not a perfect single crystal. Another explanation could be thermal diffuse scattering. The shadow origins from the correlated thermal vibration of the atoms from their ideal lattice points. In contrast to the clear spots of the Laue diffraction pattern of a single crystal, the diffraction pattern (made from monochromed x-rays) of an amorphous polymer like PET shows diffuse circles. The ring pattern is probably the result of the side chains of the PET, which are similarly distributed in every chain. However, the width of the circles is much too large in order assume the polymer to be a polycrystalline material with randomly distributed crystallites. If this were the case, the lines of the circles would be a lot thinner and more precise. This assumption is supported by the second diffraction pattern of the PET bottle. It shows two half moon shaped intensities. This special pattern and difference to the first pattern of the neck can be explained by the production of a PET bottle The PET bottle starts form a PET moulding blank, which is then blown up to a normally sized bottle. During this process, the polymer chains are aligned along the stress application. The stress is the highest along the long axis. The neck on the other hand is not deformed or changed during the blowing up and keeps the relaxed state obtained after the moulding. How production affects the diffraction pattern can also be observed in the diffraction patterns of aluminium. The first pattern shows many, quite sharp, half moon shaped intensities. The sharpness of the intensities indicates, that the aluminium foil is a polycrystalline material. However, they do not form complete circles. One therefore has to assume, that the crystallites are not randomly distributed but slightly arranged in order. The order is obtained during the production. Aluminium is rolled many times, until it is really thin and ready to be used as a foil. The pressure on the aluminium sheet arranges the crystallites. The clear spots obtained in the pattern after heat treatment indicate even higher order of the crystallites. The heat allowed vacancies, displacements and other defects to heal out. Additionally, recrystallization takes place. Bigger grains grow at the expense of smaller grains, which finally disappear. The aluminium is still polycrystalline, however, the grains are bigger but fewer. The spots are sharper but fewer and do not add up to a complete circle anymore. The third pattern of aluminium shows a pattern more similar to the one of untreated aluminium foil. The scratching of the surface destroyed some of the grown crystallites. As the scratching was done in only two directions and not randomly, the spots do not add up to a whole circle but show again the pattern of orientated crystallites (half moons). The NaCl crystal left few intensities on the detector. As already seen in the aluminium foil, the bigger the crystallites are, the fewer spots are obtained in a diffraction pattern. A single crystal only leaves very few spots, as only few set of plane satisfies the Bragg equation (because of the monochromed x- rays). The NaCl crystal in the experiment left too many spots in order to be just one single crystal. Seite 16/ 19

17 5. DISCUSSION Additionally, some spots are too close each other. As a conclusion, the NaCl crystal was not a single crystal, but was only built from few grains. 5.2 P2 X-ray powder diffraction Qualitative Phase Analysis By comparing the peaks of our sample with different peaks from materials of the database, we were able to conclude that the unknown substance was calcium carbonate (CaCO 3 ). Calcium carbonate has three different polymorphs: aragonite, calcite and vaterite. Calcium carbonates are an important component of the earth s crust and are therefore important rock-forming minerals. But calcium carbonates can also be found in plants, animals and water and therefore almost everywhere in nature. [6] Table 2 Crystallographic data of aragonite and calcite Polymorph Crystal Structure a [A ] b [A ] c [A ] Cell Volume [A! ] Calcite Trigonal Aragonite Orthorhombic Some peaks of our x-ray diagram are covert by the ones from aragonite and some are covert by the ones from carbonate. Therefore the calcium carbonate must have been a mixture of its two polymorphs aragonite and calcite. By comparing the peaks one can still see that the black peaks of the substance are slightly shifted to the right. A shift to the right is an increase of the angle. Using the Bragg s equation we can conclude that an increase in the position of the peak leads to a smaller lattice plane distance d. Therefore, we can conclude that the unknown substance was a mixture of two polymorphs of calcium carbonate namely aragonite and calcite, but where the calcite part has a slightly smaller unit cell as in the database. The reason could be the difference of crystallographic data between the two polymorphs. The intensities of the peaks do not always correspond to the database, but this is probably because the database diagrams were taken with ideal samples from the laboratory Determination of the crystallite size of Ce 0.9 Gd 0.1 O 1.95 As the crystallites do not have a perfect spherical shape, it is not possible to determine an exact size (like a radius) of the crystallite. The size depends on the direction from where it is looked at. Therefore it is only possible to determine an approximate magnitude of the crystallite. In the case, the crystallites of Ce 0.9 Gd 0.1 O 1.95 are around 200 A. Seite 17/ 19

18 6. CONCLUSION 6 CONCLUSION In conclusion, one can say x-ray diffraction as well as powder methods are widely used, fast and quite cheap methods to characterise different materials. X-ray diffraction gives an insight to the materials structure, whereas X-ray powder diffraction is often used in order to determine, whether the wanted product is actually in the sample. As these methods have already been known for a long time, a lot of information for various substances is collected in databases. The experiments we made were really interesting. The theory about Laue patterns, x-ray diffraction and powder methods is much easier to understand, once one has actually seen an x-ray tube, the set up as a whole and a real-time measurement. It was impressing, how much knowledge is collected in databases and how many details one can pull out of a diffraction pattern. The set up for the Laue method was perfect to see, how simple the whole measurement actually is. On the other hand, the diffractometer in the second part of P1 showed, how the method has been improved over the past years. Especially we would like to thank the assistants, who were really motivated. It was a real pleasure to listen to them! Seite 18 /19

19 7. REFERENCES 7 REFERENCES 7.1 Literature [1] STUDIENGANG MATERIALWISSENSCHAFT ETH ZÜRICH SKRIPT VERSUCH P1/P2 [2] STEURER W. KRISTALLOGRAPHIE SKRIPT Websites [3] WIKIPEDIA, BRAGG-GLEICHUNG, OFFENE AUTORENSCHAFT [4] INTERNATIONAL UNION OF CRISTALLOGRAPHY; LAUE MICRO-BEAM X-RAY DIFFRACTION; [5] X-RAY SPECTRUM, ( ) [6] INFORMATION ABOUT CALCITE AND ARAGONITE ( ) Seite 19 /19