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METHODS AND RESULTS OF STUDYING THE BULK ELASTICITY OF MATTER
M. Kornfeld
Bulk elasticity is the capacity of a substance to change its volume reversibly when the external pressure changes. For small changes in pressure, the change in volume is proportional to the change in pressure. In this case the bulk elasticity may be characterized by the ratio of the change in pressure to the relative change in volume, i.e., by the quantity called the bulk modulus:
\[ K=-\Delta p \frac{V}{\Delta V}. \tag{1} \]
The bulk modulus depends on the nature and state of the substance. Of greatest interest for understanding the phenomenon of bulk elasticity is its dependence on the state of the substance, or, in other words, on pressure and temperature. Data on the dependence on pressure can be drawn from Bridgman’s recent works, published after the war. These works, together with the works of other authors concerning the dependence on temperature, now make it possible to give a sufficiently complete account of the methods and results of studying the bulk elasticity of matter.
I. METHODS OF STUDYING BULK ELASTICITY
In what follows, when speaking of the volume of a substance, we shall mean the quantity
\[ V = v_p / v_0, \tag{2} \]
where \(v_0\) is the volume of the body under test at the initial pressure, and \(v_p\) is that at the pressure under consideration. Usually \(v_0\) corresponds to the volume of the body at atmospheric pressure.
All the methods considered in this article reduce to measuring the volume of a substance under a very slow (“isothermal”) change of pressure. The experimental devices used for this purpose are called piezometers.
The designs of piezometers are substantially different for gaseous, liquid, and solid bodies. These differences are largely erased at high pressures, when the properties of gases approach those of liquids, and the properties of liquids approach those of solid bodies; beginning at thousands of atmospheres, piezometers for gases and liquids become close in design, and at pressures on the order of tens of thousands of atmospheres—those for liquids and solid bodies.
Below is a description of several piezometers most characteristic of modern measurement techniques.
Weight piezometer.
For the study of gases a weight piezometer may be used[^3].
The piezometer consists of a nickel cylindrical vessel (outer diameter 16 mm, inner diameter 5.6 mm, height 50 mm), a threaded nickel plug, and a copper gasket (Fig. 1). After assembly the piezometer is heated to 1200° C; the gasket then melts and solders the plug. For connecting the piezometer to the rest of the apparatus, a hard-drawn nickel capillary, with an outer diameter of 1.6 mm and an inner diameter of 0.25 mm, soldered into the plug with silver solder, is used.
Before soldering, the capillary is subjected from within to a pressure of 2200 kg/cm²; the same treatment is then applied to the piezometer itself. This preliminary “training” makes it possible to carry out measurements at pressures up to 2000 kg/cm² without fear of residual deformations of the piezometer and capillary.
The next preparatory operation is the determination of the volume of the piezometer. The piezometer together with the capillary is weighed, filled with distilled water carefully freed from air, and weighed again. The difference of the weights gives the volume of the piezometer together with the capillary. The volume of the capillary is taken into account on the basis of a calibration performed earlier, up to its soldering into the plug.
Fig. 1. Weight piezometer.
The piezometer is placed in a thermostat, and the free end of the capillary is connected through a precision manometer to a compressor. With the aid of the thermostat and compressor, the required temperature and pressure of the gas under study are established in the piezometer. After this the capillary at the outlet from the thermostat is sealed by soldering in a special device (the section of the capillary intended for sealing is annealed in the flame of a burner before the experiment). Then the capillary is disconnected from the apparatus and immediately ...
is absorbed into the transition sleeve of a nickel cylinder of volume about \(10\ \text{cm}^3\).
The piezometer, together with the capillary and the cylinder, is weighed. The gas is released, for which purpose the capillary is cut with nippers (without loss of metal!). After this a second weighing is carried out. The difference between the weighing results is equal to the weight of the gas contained in the piezometer and in the adjoining section of the capillary (up to the point of sealing).
Denoting this weight by \(Q\) and knowing the density \(\rho\) of the gas at atmospheric pressure, one can find the volume of the substance:
\[ V=\frac{\rho (v+\Delta v)}{Q}, \tag{3} \]
where \(v\) is the total volume of the piezometer and capillary according to the calibration data, i.e., at room temperature and atmospheric pressure, and \(\Delta v\) is the correction for the change in the volume of the piezometer under the pressure and temperature that occurred during the experiment (the change in the volume of the capillary may be neglected).
The correction mentioned may be represented in the form of the sum \(\Delta v=(\Delta v)_p+(\Delta v)_t\). The first term of the sum, which corresponds to the change in the volume of the piezometer under the action of pressure, is estimated by means of the formula\(^4\)
\[ [(\Delta v)_p] \simeq v\frac{p}{E}\, \frac{2d_1^2(1+\sigma)+3d_2^2(1-2\sigma)}{d_1^2-d_2^2}, \tag{4a} \]
where \(d_1, d_2\) are the outer and inner diameters of the piezometer cylinder, \(E, \sigma\) are Young’s modulus and Poisson’s ratio for the piezometer material, and \(p\) is the pressure.
The second term corresponds to the change in the volume of the piezometer under the action of temperature:
\[ (\Delta v)_t = v\alpha (t-t_0), \tag{4b} \]
where \(\alpha\) is the volumetric coefficient of thermal expansion of the piezometer material, \(t\) is the temperature of the experiment, and \(t_0\) is the calibration temperature.
The weighing-piezometer method is applicable in the pressure interval from \(100\) to \(2000\ \text{kg}/\text{cm}^2\). At higher pressures it is, in terms of convenience, inferior to other methods, and at lower pressures it proves insufficiently accurate.
Capillary piezometer
The simplest piezometer for liquids is the capillary piezometer\(^1\), which consists of a glass sphere with a capillary fused to it (Fig. 2). The piezometer is filled with the liquid being studied and is placed in a pressure chamber provided with transparent windows. By increasing the pressure in the chamber (for example,
with the aid of an air compressor) and, observing through windows the displacement of the meniscus in the capillary, it is possible to study the dependence of the volume of the liquid on the pressure. The position of the meniscus is read from a scale marked on the capillary itself. Into the results of the experiment there must be introduced a correction for the change in the dimensions of the piezometer under the action of pressure.
Fig. 2. Capillary piezometer.
Let us denote the volume of the bulb of the piezometer, the cross-section of the capillary, and the position of the meniscus at atmospheric pressure respectively by \(q_0\), \(s_0\), and \(l_0\), and at pressure \(p\) by \(q_p\), \(s_p\), \(l_p\). Then the initial volume of the liquid is equal to
\[ v_0 = q_0 + s_0 l_0 . \tag{5} \]
Denoting the coefficient of volumetric elasticity of the glass by \(k\), we find the volume of the liquid at pressure \(p\). In this case the volume of the piezometer bulb is equal to
\[ q_p = q_0(1 - kp), \]
the cross-section of the capillary
\[ s_p = s_0\left(1 - \frac{2}{3}kp\right), \]
and the height of the liquid column (allowing for the change in the value of a scale division)
\[ l'_p = l_p\left(1 - \frac{1}{3}kp\right). \]
Thus, the volume of the liquid at pressure \(p\) is equal to
\[ v_p = q_0(1 - kp) + s_0 l_p\left(1 - \frac{2}{3}kp\right)\left(1 - \frac{1}{3}kp\right). \]
Using (5), putting \(l_p = l_0 - \Delta l\), and neglecting \(k^2p^2\) in comparison with \(kp\), the preceding expression may be brought to the form
\[ v_p = (v_0 - s_0\Delta l)(1 - kp). \tag{6} \]
Consequently, the volume of the substance is equal to
\[ V = \left(1 - \frac{s_0\Delta l}{v_0}\right)(1 - kp). \tag{7} \]
It follows from this formula that the necessity of introducing a correction for the compression of the glass can be avoided by measuring, at one and the same pressure, the liquid under investigation and some other liquid whose volume at the given pressure is known. Denoting the volumes of these two liquids respectively by \(V_1\) and \(V_2\), and the displacements of the menisci (when using one and the same
and the same piezometer) through \((\Delta l)_1\) and \((\Delta l)_2\), we obtain:
\[ V_1=V_2 \frac{1-\frac{s_0}{v_0}(\Delta l)_1}{1-\frac{s_0}{v_0}(\Delta l)_2}. \tag{8} \]
A capillary piezometer is unsuitable at pressures exceeding \(2000\ \mathrm{kg/cm^2}\), owing to the insufficient strength of the windows and to hysteresis phenomena in the glass of the piezometer.
Linear piezometer
For the study of solids, the linear piezometer[^5] shown schematically in Fig. 3 is convenient.
A cylindrical specimen of the substance under investigation, ranging in length from several centimeters to several tens of centimeters, is placed in a pressure chamber provided with transparent windows. The upper end of the specimen is rigidly connected with the chamber. To its lower end, at the level of the windows, a transparent scale is attached. The chamber is connected to a compression device, and by means of some transmitting medium (air, nitrogen, water, etc.) the required pressure is established in it.
The change in the length of the specimen under the action of pressure is measured through the windows by means of a microscope and the aforementioned scale. In this way the linear compression \(\frac{\Delta L}{L_0}\) along the axis of the specimen is determined, where \(\Delta L\) is the change in the length of the specimen, and \(L_0\) is its initial length at atmospheric pressure.
For isotropic bodies the volume of the substance is calculated from the linear compression by the formula
\[ V=\left(1-\frac{\Delta L}{L_0}\right)^3. \tag{9} \]
Fig. 3. Linear piezometer with visual reading.
For anisotropic bodies, in particular crystals, the situation is somewhat more complicated. The theory of elasticity of anisotropic media shows that a sphere cut from a crystal is transformed, under the action of external pressure, in the general case into an ellipsoid.
For crystals of the cubic system all three axes of the ellipsoid are equal, i.e. the sphere remains a sphere. In crystals belonging to the hexagonal, tetragonal, and rhombohedral systems, the sphere takes the form of an ellipsoid of revolution, whose axis coincides respectively with the hexagonal, tetragonal, or trigonal axis of the crystal. In crystals belonging
to the rhombic system and to all systems following it in decreasing order of symmetry, the sphere is transformed into a triaxial ellipsoid.
Thus, in the first case it is sufficient to investigate a single arbitrarily oriented specimen. The relative volume is then computed, as also for isotropic bodies, by means of formula (9).
In the second case one may restrict oneself to investigating two specimens oriented parallel and perpendicular to the corresponding axis of symmetry. For computing the volume of the substance, the formula used is
\[ V=\left[1-\left(\frac{\Delta L}{L_0}\right)_{\parallel}\right] \left[1-\left(\frac{\Delta L}{L_0}\right)_{\perp}\right]^2, \tag{10} \]
where the index \(\parallel\) denotes the direction parallel to the axis of symmetry, and the index \(\perp\) the direction perpendicular to it.
Fig. 4. Electric micrometer.
In the third case it is necessary to investigate three specimens oriented mutually perpendicular to one another. The volume of the substance is calculated from the formula
\[ V=\left[1-\left(\frac{\Delta L}{L_0}\right)_a\right] \left[1-\left(\frac{\Delta L}{L_0}\right)_b\right]\times \]
\[ \times\left[1-\left(\frac{\Delta L}{L_0}\right)_c\right], \tag{11} \]
where the indices \(a, b, c\) denote mutually perpendicular directions.
A linear piezometer with visual readout is inconvenient at pressures above \(2000\ \text{kg}/\text{cm}^2\), owing to frequent breakage of the glass windows. In this case one must use other devices for measuring the compression of the specimen. The most commonly used of these is Bridgman’s “electric micrometer”\(^{1,6,19}\).
Electric Micrometer
The micrometer consists of a small rod made of an alloy of high electrical resistance, capable of undergoing longitudinal displacements (Fig. 4). The rod is pressed against a contact by means of a light spring. Flexible leads, serving to supply current, are soldered to the ends of the rod. The contact and a third flexible lead are connected to a potentiometer, by means of which the electrical resistance \(r\) of the rod between these two points is measured. Hence, knowing the resistance of the rod per unit length \(R\), it is easy to find the distance between the points under consideration:
\[ l=\frac{r}{R}. \tag{12} \]
The sensitivity of this, at first glance crude, method is very high—the errors in determining the displacement of the rod contact do not exceed one-thousandth of a millimeter.
To achieve such accuracy, it is necessary that the rod, which is a piece of wire about \(0.3\) mm in diameter, be as homogeneous as possible in its electrical resistance and sufficiently smooth. By measuring the electrical resistance of the original wire over small sections and examining its surface with a binocular microscope, it is possible to find pieces satisfying both requirements.
At high pressures, corrections must be introduced into the micrometer readings for the linear compression of the rod and for the change in its electrical resistance per unit length. If the electrical resistance between two points of the rod, located at atmospheric pressure at a distance of \(1\) cm, is equal to \(R_0\), then at pressure \(p\) it becomes equal to
\[ R = R_0 \frac{1 - bp}{1 - ap}, \tag{13} \]
where \(a\) and \(b\) are coefficients characterizing the effect of pressure, respectively, on the length and on the electrical resistance per unit length.
The following table gives numerical values of the coefficients \(a\) and \(b\) (in \(\text{cm}^2/\text{kg}\)) for several high-resistance alloys.
| Alloy | \(a \cdot 10^7\) | \(b \cdot 10^7\) |
|---|---|---|
| Manganin | 2.5 | \(-22.4\) |
| Nichrome | 1.9 | 6 |
| Chromel | 2 | 1.6 |
Sample
Fig. 5. Linear piezometer with an electric micrometer.
Piezometers with an electric micrometer
A piezometer with an electric micrometer for solids \(^{1,6,7,19}\) is shown in Fig. 5. The sample is placed in a vertical position and, with the aid of springs (not shown in the figure), is pressed against the base of the holder. The micrometer rod is rigidly connected with the upper end of the sample, while the micrometer contact is attached to the body of the holder. The entire apparatus, together with the sample, is placed in a pressure chamber. The connection of the micrometer with the current source and the potentiometer is made through special electrodes in the walls of the chamber.
The distance between the micrometer rod points connected to the potentiometer at atmospheric pressure will be denoted by \(l_0\). Then the same distance at pressure \(p\) will be equal to
\[ l = l_0(1-ap) + L_0\left(\frac{\Delta L}{L_0}\right)_{\mathrm{ob}} - L_0\left(\frac{\Delta L}{L_0}\right)_{\mathrm{d}}, \tag{14} \]
where \(L_0\) is the initial length of the specimen, and \(\left(\dfrac{\Delta L}{L_0}\right)_{\mathrm{ob}}\) and \(\left(\dfrac{\Delta L}{L_0}\right)_{\mathrm{d}}\) are the linear compressions of the specimen and of the holder.
According to (12) and (13),
\[ l_0=\frac{r_0}{R_0}, \qquad l=\frac{r}{R_0}\cdot \frac{1-ap}{1-bp}. \]
Substituting these values into (14) and putting \(\Delta r=r-r_0\), we finally obtain:
\[ \left(\frac{\Delta L}{L_0}\right)_{\mathrm{ob}} = \left(\frac{\Delta L}{L_0}\right)_{\mathrm{d}} + \frac{1}{L_0 R_0}\, \frac{1-ap}{1-bp}\, (\Delta r+bpr_0). \tag{15} \]
As a rule, the holder and all the other parts of the instrument are made of ordinary steel. In this case the linear compression of the holder can be calculated by means of Bridgman’s interpolation formula,^2 based on careful measurements of the volume elasticity of iron:
\[ V_{\mathrm{d}} = 1 - 5.826\cdot 10^{-7}p + 0.80\cdot 10^{-12}p^2, \tag{16} \]
where the pressure is expressed in \(\mathrm{kg}/\mathrm{cm}^2\). This formula gives the dependence of volume on pressure at room temperature and pressures up to \(30\,000\ \mathrm{kg}/\mathrm{cm}^2\). The transition from \(V_{\mathrm{d}}\) to \(\left(\dfrac{\Delta L}{L_0}\right)_{\mathrm{d}}\) is carried out by means of formula (9).
When studying slightly compressible substances, one must use a somewhat more complicated construction of the piezometer, namely, place a lever device between the upper end of the specimen and the micrometer rod, as shown in Fig. 6.
Fig. 6. Piezometer with a lever device.
Such a device makes it possible to increase the accuracy of the measurements by approximately a factor of 10.
Piezometers with an electric micrometer for liquids are shown schematically in Figs. 7 and 8.
The first of them—the so-called piston piezometer[^1]—consists of a cylindrical steel vessel with an internal diameter of about 10 mm and a freely moving piston, fitted to the channel with an accuracy of up to several ten-thousandths of a millimeter.
Fig. 7. Piston piezometer.
Fig. 8. Sylphon piezometer.
The displacements of the piston are measured by an electric micrometer, the rod of which is connected to the piston, and the contact—with the cylinder. The vessel is partially filled with the liquid under investigation; the piston is inserted above it, and the entire device is placed in a pressure chamber.
The displacement of the piston under the action of pressure is determined by means of formula (15), where \(L_0\) is taken to be the height of the liquid level above the bottom of the vessel. The expression for the volume of the substance has the following form:
\[ V = \left(1 - \frac{\Delta L}{L_0}\right) V_{\mathrm{d}}^{\prime\,1/3}, \tag{17} \]
where the factor \(V_{\mathrm{d}}^{\prime\,1/3}\) is a correction for the change in the cross-sectional area of the channel under the action of pressure.
The second piezometer, called a sylphon piezometer[^1],[^8], is a sylphon about 17 mm in diameter and 25 mm high, made of sheet brass 0.04 mm thick. The upper end of the sylphon is soldered to a brass disk, the lower end—to a massive brass base. A nickel-silver tube is soldered into the base, through which the sylphon is filled with liquid. Inside the sylphon, along its axis, there is a guiding device consisting of a piston and a cylinder. To reduce—
friction reduction the piston is made of brass, and the cylinder of copper (not the other way round, since brass is compressed under the action of pressure more than copper!).
After assembly the piezometer is calibrated. A graduated glass capillary is attached to the Neusilber tube, and the entire apparatus is carefully filled (under vacuum) with kerosene. By compressing the bellows by means of a micrometer screw and simultaneously measuring the position of the meniscus in the capillary, one can verify that the change in the volume of the bellows is strictly proportional to its axial compression. The proportionality factor is equal to the effective cross section of the bellows.
Before an experiment the piezometer is filled with the liquid under investigation, and the Neusilber tube through which filling was carried out is sealed. The low thermal conductivity of Neusilber protects the liquid from boiling during soldering.
The sealed piezometer is placed in a pressure chamber. The axial compression of the bellows under the action of pressure is measured by an electric micrometer, the rod of which is connected to the upper disk, and the contact to the base. The calculations are carried out using formulas (15) and (17), just as for the piston piezometer, with the only difference that instead of \(V_d\) the corresponding value for brass must enter.
It should be borne in mind that using the piezometer at pressures exceeding the solidification pressure of the liquid under investigation leads to damage of the bellows.
Piezometers with an electric micrometer are suitable up to \(30\,000\ \text{kg}/\text{cm}^2\). At higher pressures any transmitting medium becomes solid or, in any case, very viscous, which disrupts the normal operation of the micrometer*).
Plunger piezometers
The next group of piezometers, covering the pressure range up to \(100\,000\ \text{kg}/\text{cm}^2\), is formed by plunger piezometers.
A plunger piezometer for compressed gases\(^3\) is shown in Fig. 9. Such a piezometer can also be used for liquids; however, special designs of this kind also exist for liquids\(^ {13}\).
The principal element of the piezometer is a compressor consisting of a cylinder and a plunger. The compressor is connected by means of a capillary to the test vessel. All parts of the piezometer, except for the rubber and lead gaskets (blackened in the drawing), are made of steel. The outer diameter of the cylinder is \(5\ \text{cm}\), the inner diameter \(0.65\ \text{cm}\), the height \(8.5\ \text{cm}\). The working stroke of the plung—
*) On the construction of chambers and compressors for such pressures see\(^ {1,2}\). As a transmitting medium at the highest pressures a mixture of isopentane with technical pentane is used.
... 4.7 cm. The internal diameter of the capillary is 0.8 mm. The volume of the test vessel is 1.2 cm\(^3\).
The compressor is placed between the plates of a hydraulic press, and the test vessel is put into a thermostat. To determine the relative volume of a gas at a specified pressure, it is necessary to carry out two experiments: the first with an empty piezometer, the second with a piezometer into which an iron blank has been inserted, reducing its free volume. In all other respects the procedure for carrying out both experiments is the same.
The plunger is moved to the extreme (zero) position, and the piezometer is filled through a branch tube in the upper part of the cylinder with the gas under investigation at a certain definite initial pressure, of the order of 1000 kg/cm\(^2\). Then the plunger, by means of the hydraulic press, is pushed into the cylinder until the pressure, measured by a manganin manometer mounted in the lower part of the cylinder, reaches the value \(p_1\), for which the volume of the substance \(V_1\) is known from some other experiments. After this, by further displacement of the plunger, the pressure \(p_2\) of interest to us is established.
Fig. 9. Plunger piezometer for compressed gases.
Let us denote the volume of the gas and the position of the plunger at pressures \(p_1, p_2\) in the first experiment by \(u_1, u_2\) and \(x_1, x_2\), and in the second experiment by \(u'_1, u'_2\) and \(x'_1, x'_2\). In the same way, let us denote the volume of the test vessel (together with the capillary), the cross-sectional area of the cylinder bore, and the volume of the blank at the same pressures respectively by \(q_1, q_2; s_1, s_2; \varphi_1, \varphi_2\). Then the following equalities hold:
\[ u_1 - u_2 = s_1 (x_1 - x_2) + (s_2 - s_1)(x_1 - x_2) + (q_2 - q_1), \]
\[ u'_1 - u'_2 = s_1 (x'_1 - x'_2) + (s_2 - s_1)(x'_1 - x'_2) + (q_2 - q_1) + (\varphi_1 - \varphi_2), \]
\[ u_1 - u'_1 = s_1 (x_1 - x'_1) - \varphi_1, \]
\[ \frac{u_2}{u_1} = \frac{u'_2}{u'_1} = \frac{V_2}{V_1}. \]
From these equalities it follows that:
\[ V_2=V_1\frac{\varphi_2-s_2(x_2-x_2')}{\varphi_1-s_1(x_1-x_1')}. \tag{18} \]
The quantity \((\varphi_1-\varphi_2)\) is found by means of formula (16), \(V_1\)—on the basis of data obtained, for example, by a weight piezometer, and \(x_1, x_2, x_1', x_2'\)—directly from experiment. The quantity \((s_2-s_1)\) is estimated by means of a formula similar to formula (4a), see \(^{4}\).
Analogously to the preceding, an expression can be obtained that makes it possible to find the volume of the substance \(V_2\) at temperature \(t_2\), if
Fig. 10. Plunger piezometer for pressures up to 50,000 kg/cm².
its volume \(V_1\) at temperature \(t_1\) is known. In this case (assuming the pressure constant)
\[ V_2=V_1\frac{\varphi_2}{\varphi_1-s_1\left[(x_2-x_2')-(x_1-x_1')\right]}. \tag{19} \]
Thus, in both cases it is possible to eliminate from the calculation the volume of the piezometer and its change under the action of pressure and temperature.
INVESTIGATIONS OF THE VOLUMETRIC ELASTICITY OF MATTER
A plunger piezometer of the type described gives satisfactory results up to \(6000\ \mathrm{kg/cm^2}\); at higher pressures, leakage of the substance through the sealing gaskets sharply reduces the accuracy of the measurements.
For work at higher pressures, the piezometer shown in Fig. 10\(^{14,16,18,9}\) is used. A cylinder made of carboloy*) is press-fitted into a steel block having the shape of a truncated cone. The height of the cylinder is \(12\ \mathrm{mm}\), the outside diameter \(16\ \mathrm{mm}\), and the diameter of the channel \(6.35\ \mathrm{mm}\). The cylinder abuts against a hollow screw provided at its end with an insert made of carboloy. The substance under investigation is placed in the channel of the cylinder between two carboloy plungers.
The piston of a hydraulic press forces the block into the conical opening in a massive steel blank. Between the surface of the block and the opening there is placed a thin lead foil, coated on both sides with a layer of paste made of graphite and a mixture of water with glycerin. The use of such a lubricant reduces the coefficient of friction to 0.04. The piston of another hydraulic press simultaneously drives the plunger into the channel of the cylinder. Both presses are connected to a common oil system, the pressure in which is produced by a hand pump.
Fig. 11. Pressures in a block forced into a conical opening.
As follows from Fig. 11, in the block forced into the conical opening there is created a pressure which, for small cone angles and slight friction, is equal to
\[ p \cong \frac{F}{\frac{\pi}{4}\left(d_2^2-d_1^2\right)}. \]
The pressure is transmitted from the outside to the carboloy cylinder, increasing the strength of the carboloy and preventing stretching of the cylinder under the action of the internal pressure in the channel.
In order to maintain a definite clearance between the plunger and the walls of the channel, it is necessary to keep constant the ratio of the external pressure on the cylinder to the internal pressure in the channel. This condition is observed automatically throughout the entire experiment owing to the fact that the pistons pressing on the block and on the plunger are controlled by a common oil system. The best results are obtained when the external pressure is about one quarter of the internal pressure, which is achieved by the corresponding selection of the piston diameters and of the cone angle of the block.
*) A hard alloy corresponding to our pobedit.
The sealing of the plungers, which practically prevents leakage of the substance, is achieved by means of steel rings of triangular cross-section (the weight of such a ring is only 0.015 g). The diameter of the rings is made 0.025 mm greater than the diameter of the channel, so that they are pressed into the channel with some force.
When the substance under investigation is sufficiently plastic, it is placed directly into the cylinder channel in the form of a specimen 6.35 mm in diameter and 3.2 mm thick. Solid substances must be enclosed in a sheath of plastic material. Metallic indium is the most suitable material for this purpose; being highly plastic, it retains this property up to the highest pressures. To make the sheath, a piece of indium weighing about 0.14 g is flattened inside the channel into the shape of a little cup, into which the substance under investigation is then placed. Alkaline-earth metals, which at high pressures readily form chemical compounds with carboloy, must be placed in a cylindrical ampoule made of aluminum. For work with liquid substances, lead ampoules of cylindrical shape are used. The ampoule is filled with the liquid under investigation, after which a ground lead plug, coated with a mercury amalgam of lead, is inserted into its “neck.” After several hours the plug proves to be soldered into the ampoule.
Fig. 12. Piezometer for pressures up to 100,000 kg/cm².
In all cases, in the cylinder channel, between the end faces of the plungers, there is a “tablet” 6.35 mm in diameter and about 3.2 mm high. At first the tablet is compressed by a pressure somewhat exceeding the pressure used in the subsequent measurements. After this the pressure is released and then slowly increased while simultaneously measuring the displacement of the plungers relative to one another (by means of probes connected to a precision dial micrometer).
The volume of the substance is given by the formula \(V=(1-\Delta L/L_0)\), where \(L_0\) is the initial distance between the end surfaces of the plungers, and \(\Delta L\) is its change under the action of pressure.
In principle, corrections should be introduced into the volume of the substance calculated in this way for the increase in the cross-sectional area of the cylinder channel, for the compression (shortening) of the plungers, and for the curvature of their end surfaces. These corrections, however, are small and partially compensate one another, so they may be neglected.
In those cases where the substance under investigation is enclosed in a sheath of indium, in a lead or aluminum ampoule, it is necessary to take into account the volume elasticity of this “ballast.” Let us denote
the volumes of the test body and the ballast body at atmospheric pressure by \(v_1\) and \(v_2\), and at the pressure of interest to us by \(v'_1\) and \(v'_2\). Then the measured, effective volume of the substance will be equal to
\[ V_{\mathrm{eff}}=\frac{v'_1+v'_2}{v_1+v_2}. \]
Hence, taking into account that \(V_{\mathrm{ball}}=\dfrac{v'_2}{v_2}\), it is easy to find the volume of the substance under investigation:
\[ V=V_{\mathrm{eff}}+\frac{v_2}{v_1}\left(V_{\mathrm{eff}}-V_{\mathrm{ball}}\right). \tag{20} \]
A piezometer of the type considered is unsuitable at pressures exceeding \(50\,000\ \mathrm{kg}/\mathrm{cm}^2\), because of frequent failures of the carboloy cylinder and plungers. Better results are achieved if the external pressure on the carboloy cylinder is produced not by the method of pressing in a cone, but directly by placing the entire piezometer in a pressure chamber filled with liquid. When the pressure in the chamber is about \(30\,000\ \mathrm{kg}/\mathrm{cm}^2\), pressures up to \(100\,000\ \mathrm{kg}/\mathrm{cm}^2\) can be obtained in the channel of the cylinder\(^{10,11,13,15,20}\).
The piezometer used in this case is shown schematically in Fig. 12. Into a steel block 11 mm in diameter and 8 mm high a slightly conical carboloy cylinder is pressed. The height of the cylinder is 8 mm, the outside diameter is 8 mm at the top and 7 mm at the bottom, and the channel diameter is 1.57 mm. Both entrances to the channel are widened to 3.14 mm; as a result, the working length of the channel is only 4.5 mm. From both sides carboloy plungers are inserted into the channel of the cylinder. The plungers are pressed into steel guide “plugs,” which enter the widened portions of the channel. In the channel between the plungers is placed the substance under investigation (directly or in a corresponding capsule).
Fig. 13. Diagram of the operation of a piezometer with hydrostatic support.
The piezometer is inserted into a special holder (Fig. 13). The holder is slid onto part \(A\). In this case part \(A\) comes into contact with the movable part \(B\), which carries the rod of an electric micrometer; the micrometer contact is connected with the body of the holder. The entire apparatus as a whole is placed in a pressure chamber filled with pentane. The base of part \(A\) rests against the bottom of the chamber, and the top of the holder rests against the piston sealing the chamber.
The piston is pushed into the chamber by means of a hydraulic press and compresses the liquid contained in it. At the same time the mandrel, overcoming the resistance of the spiral spring supporting it, moves onto part \(A\), as a result of which part \(B\) approaches part \(C\). At the moment when part \(B\) comes into contact with part \(C\), the plungers begin to compress the substance located between them, and the electric micrometer begins to measure the displacement of the plungers relative to one another.
To measure the pressure in the channel of the piezometer, a so-called “grid” \({}^{1)}\) is used. The latter is a thin hardened steel plate provided with counter-slits. Four conductors are soldered to the plate, two of which (at the ends) serve to supply the current, and the other two (in the middle) for connection to the potentiometer. The plate is insulated by two mica sheets and is placed between the base of part \(A\) and the bottom of the chamber. The change in the electrical resistance of the “grid,” measured by the potentiometer, makes it possible to judge the magnitude of the force acting on the plungers and hence the pressure between them.
The principal difficulties of the method, which represents the pinnacle of modern high-pressure technique, are the need for a large number of electrical leads (three for the electric micrometer, three for the “grid,” and one for the manganin manometer measuring the pressure in the chamber) and the procedure of calibrating the “grid.”
II. RESULTS OF THE INVESTIGATION OF VOLUME ELASTICITY
Dependence of the bulk modulus of elasticity on pressure and temperature
Using the experimental methods described in the preceding section, one can measure the dependence of the volume of a substance on pressure.
If \(V_1\) and \(V_2\) denote the volumes corresponding to two close pressure values \(p_1\) and \(p_2\), then the bulk modulus of elasticity \(K\) at the intermediate pressure
\[ p=\frac{p_1+p_2}{2} \]
will be equal to
\[ K=\frac{V_1+V_2}{2}\frac{p_2-p_1}{V_1-V_2}. \tag{21} \]
Applying such an operation to each pair of adjacent pressure values, it is easy to find the dependence of the bulk modulus of elasticity on pressure.
Processing, by means of formula (21), all experimental data known at the present time shows that over a very wide pressure interval, extending at least
tens of thousands of atmospheres, the bulk modulus of elasticity increases linearly with pressure.
This rule is valid for substances of any chemical composition and in any state: gaseous, liquid, crystalline, glassy, polymeric, etc.
As an example, Fig. 14 gives some results pertaining to room temperature and to substances that do not undergo phase transitions in the indicated pressure interval.
Fig. 14. Dependence of the bulk modulus of elasticity on pressure for nitrogen (I), amyl alcohol (II), zinc (III), sulfur (IV), boron glass (V), plexiglass (VI).
Substances in which phase transitions occur, as well as two anomalous substances, will be considered later.
As an analytical expression for the dependence under discussion we shall use the formula
\[ K=\varkappa(p_*+p), \tag{22} \]
where \(p_*\) and \(\varkappa\) are constants characterizing the substance. Table 1 gives numerical values of \(p_*\) and \(\varkappa\) for certain substances at room temperature. As is seen from the table, \(\varkappa\) lies within comparatively narrow limits, from \(\sim 2\) to \(\sim 14\), whereas \(p_*\) varies from zero (approximately) to hundreds of thousands of \(\mathrm{kg/cm^2}\).
Table 1
Numerical values of the parameter \(p_*\) and the coefficient \(\varkappa\) for certain substances
| Substance | Pressure interval \((\mathrm{kg/cm^2})\) | \(p_* \cdot 10^{-3}\) \((\mathrm{kg/cm^2})\) | \(\varkappa\) |
|---|---|---|---|
| Nitrogen \(^{3}\) | 500—5800 | −0.6 | 4.5 |
| Ethyl acetate \(^{18}\) | 1—40 000 | 1.0 | 7.5 |
| \(\mathrm{C_{10}H_{16}}\) \(^{18}\) | 1—40 000 | 1.2 | 8.7 |
| Rubber (butyl) \(^{14}\) | 1—12 000 | 1.5 | 14 |
| \(\mathrm{C_8H_{16}}\) | 1—40 000 | 2.1 | 7.0 |
| Amyl alcohol \(^{12}\) | 1—50 000 | 4.1 | 6.0 |
| Potassium \(^{15,16}\) | 1—100 000 | 5.2 | 3.8 |
| Plexiglas \(^{16}\) | 1—40 000 | 5.9 | 8.5 |
| Ebonite \(^{16}\) | 1—40 000 | 7.3 | 7.5 |
| \(p\)-\(\mathrm{C_6H_4O_2\cdot NCl}\) \(^{16}\) | 1—40 000 | 8.6 | 7.5 |
| \(\mathrm{HCO_3}\) \(^{16}\) | 1—40 000 | 10 | 6.7 |
| \(\mathrm{COOH(CH_2)_2COOH}\) \(^{14}\) | 1—25 000 | 13 | 7.6 |
| Sulfur \(^{13}\) | 1—100 000 | 15 | 6.8 |
| Borosilicate glass \(^{16}\) | 1—40 000 | 28 | 5.0 |
| Potassium alum \(^{16}\) | 1—40 000 | 28 | 6.0 |
| Dextrose \(^{14}\) | 1—25 000 | 31 | 6.4 |
| Quartz (cryst.) \(^{16}\) | 1—40 000 | 42 | 9.4 |
| Sodium chloride \(^{13,14}\) | 1—60 000 | 53 | 4.7 |
| Sodium \(^{15,16}\) | 1—100 000 | 55 | 2.0 |
| Sodium nitrate \(^{13}\) | 1—55 000 | 67 | 3.6 |
| Zinc \(^{16}\) | 1—40 000 | 110 | 5.7 |
| Orthoclase \(^{16}\) | 1—40 000 | 270 | 2.1 |
| Iron \(^{19}\) | 1—30 000 | 450 | 3.8 |
Although the experimental data illustrating the validity of the formula under consideration pertain only to the region of positive pressures, there is every reason to suppose that the formula is also satisfied in the region of negative pressures, corresponding to an all-round stretching of the substance. However, both in that region and in the other, as \((p_*+p)\) approaches zero, deviations from the linear law must be observed, associated with the fact that the bulk modulus cannot take negative values.
Integration of formula (22) gives an equation relating the volume of a substance to pressure:
\[ V=\left(\frac{p_*+p_0}{p_*+p}\right)^{\frac{1}{\varkappa}}, \tag{23} \]
where \(p_0\) is the initial pressure. This equation, being applicable to any substances over an extremely wide range of pressures, may serve as the starting point for a number of technical calculations concerning the behavior of a substance at high pressures.
An experimental investigation of the dependence of the bulk modulus of elasticity on temperature is associated with great difficulties, since modern high-pressure apparatus is unsuitable for work at temperatures substantially different from room temperature. Therefore, in the overwhelming majority of cases, experimental data cover either a very narrow temperature interval over a wide pressure interval, or else a very narrow pressure interval over a wide temperature interval.
Fig. 15. Dependence of the bulk modulus of elasticity on pressure at different temperatures for ethyl alcohol.
Fig. 16. Dependence of the parameter \(p_*\) on temperature for ethyl alcohol.
Only for a few substances are there data relating to comparatively wide intervals of pressure and temperature simultaneously. These substances include:
nitrogen — up to \(5800\ \text{kg}/\text{cm}^2\), from \(-150\) to \(+200^\circ\text{C}\) \(^{3}\);
ethyl alcohol — up to \(1000\ \text{kg}/\text{cm}^2\), from \(-111\) to \(+198^\circ\text{C}\) \(^{21}\);
rubber — up to \(1000\ \text{kg}/\text{cm}^2\), from \(+10\) to \(+85^\circ\text{C}\) \(^{5}\);
lead — up to \(10\,000\ \text{kg}/\text{cm}^2\), from \(+20\) to \(+220^\circ\text{C}\) \(^{7}\),
and some others.
As an example, Figs. 15 and 16 present the corresponding data for ethyl alcohol. As can be seen from Fig. 15, formula (22) remains valid over a wide interva-
Table II
First-order phase transitions in some substances
| Substance | Transition pressure (kg/cm²) | Volume jump |
|---|---|---|
| A. Crystallization¹⁸ | ||
| p-xylene | 343 | 0,9751—0,8111 |
| Cyclohexane | 355 | 0,9674—0,9258 |
| Ethyl bromide | 650 | 0,9667—0,8700 |
| Benzene | 680 | 0,9366—0,8404 |
| o-xylene | 2 300 | 0,9017—0,8310 |
| Styrene | 3 120 | 0,8838—0,8353 |
| Chloroform | 5 500 | 0,8150—0,7476 |
| Chlorobenzene | 7 500 | 0,8172—0,7879 |
| n-amyl ether | 11 140 | 0,7759—0,7549 |
| Methyl chloride | 12 440 | 0,7414—0,7013 |
| n-heptane | 11 450 | 0,7218—0,6798 |
| n-octane | 5 510 | 0,8137—0,7525 |
| n-decane | 3 050 | 0,8630—0,7893 |
| n-dodecane | 1 700 | 0,9161—0,8131 |
| n-hexadecane | 420 | 0,9700—0,8608 |
| B. Allotropic transformations | ||
| Cerium¹⁶ | 12 430 | 0,9264—0,8496 |
| Lanthanum¹⁶ | 23 370 | 0,9245—0,9219 |
| Cesium¹⁶ | 23 300 | 0,6284—0,6224 |
| Cesium¹⁶ | 45 000 | \(V_1 - V_2 = 0,056\) |
| PbSe¹⁵ | 45 000 | 0,917—0,893 |
| KCl¹³ | 20 060 | 0,915—0,803 |
| RbCl¹³ | 5 000 | 0,970—0,830 |
| AgCl¹³ | 90 000 | 0,860—0,844 |
| KNO₃¹³,¹⁴ | 3 650 | 0,9771—0,8871 |
| NaNO₃¹³ | 55 000 | 0,864—0,853 |
| AgNO₃¹³,¹⁴ | 9 500 | 0,9700—0,9573 |
| KCN¹⁶ | 20 340 | 0,8989—0,8115 |
| Cu₂I₂¹⁶ | 14 400 | 0,9661—0,9298 |
| NH₄CHO₂¹⁶ | 11 420 | 0,9253—0,8147 |
| NH₂CONH₂¹⁶ | 5 490 | 0,9589—0,8947 |
| o-C₆H₄O₂NCl¹⁶ | 4 030 | 0,9516—0,9450 |
temperatures. In this case, \(x\) is practically independent, while \(\rho_*\), on the contrary, depends strongly on temperature, decreasing as the latter increases (Fig. 16). These regularities are observed in all the substances studied, with the exception of the anomalous ones already mentioned.
Change of Volume Elasticity at Phase Transitions
In the course of investigating volume elasticity in a number of substances, phase transformations were discovered. The most numerous are phase transitions of the first order, i.e., transitions accompanied by a discontinuous change in the volume of the substance.
Table II gives figures characterizing first-order transitions in some substances at room temperature.
Transitions of the second order, not accompanied by a jump in volume, i.e., detected from a discontinuous change in the modulus of volume elasticity, are considerably less numerous (possibly because of the difficulty of detecting them). Bridgman notes only nine cases of transitions of this kind, and examination of the rest of the experimental data obtained by him permits adding two more substances to this list: selenium and \(\mathrm{NaClO_3}\). All eleven cases, pertaining to room temperature, are given in Table III.
Table III
Second-order phase transitions
| Substance | Transition pressure (kg/cm²) | Substance | Transition pressure (kg/cm²) |
|---|---|---|---|
| Nickel \(^{19}\) | 10 500 | Cobaltite \(^{19}\) | 18 500 |
| Palladium \(^{19}\) | 16 500 | Topaz \(^{19}\) | 26 000 |
| » | 25 500 | Mica \(^{19}\) | 19 500 |
| Magnetite \(^{19}\) | 22 500 | \(\mathrm{NiSO_4\cdot 6H_2O}^{17}\) | 20 000 |
| Andradite \(^{19}\) | 22 500 | Selenium \(^{10}\) | 30 000 |
| Cobaltite \(^{19}\) | 9 500 | \(\mathrm{NaClO_3}^{16}\) | 24 000 |
Among the substances listed in Table III, two have been studied most thoroughly: nickel and \(\mathrm{NiSO_4\cdot 6H_2O}\).
In Fig. 17 the dependence of linear compression on pressure is shown for the case of nickel. The curve leaves no doubt that, near a pressure of \(10\,500\ \mathrm{kg/cm^2}\), a discontinuous change occurs—
of the bulk modulus in the absence of any jump in volume.
It should be noted first of all that formula (22) is valid for all modifications of a substance formed as a result of transitions of both the first and the second kind. Thus, at the point
Fig. 17. Dependence of linear compression on pressure for nickel.
Fig. 18. Change of bulk elasticity at the point of a phase transition of the first kind in chloroform (I), lanthanum (II), cerium (III), Cu\(_2\)J\(_2\) (IV) and NH\(_4\)ClO\(_2\) (V).
of transition there must occur a discontinuous change either of both constants \(p_*\) and \(\chi\), or at least of one of them.
Thus, formally the point of a phase transition may be defined as the point at which both constants \(p_*\) and \(\chi\) (or at least one of them) undergo a discontinuous change. As an illustration, Fig. 18 gives some characteristic cases of change in bulk elasticity at the point of a transition of the first kind, and Fig. 19—at the point of a transition of the second kind.
The definition of the point of a phase transition as a point of discontinuous change of \(p_*\) and \(\chi\) does not exclude the possibility of transitions in which neither the volume nor the bulk modulus experiences a jump. As is known, the existence of phase transitions of this kind is denied by the theory of L. D. Landau \(^{22}\). Nevertheless, a careful examination of Bridgman’s experimental data makes it possible to suppose the presence of such transitions (see, for example, Fig. 20). Of course, since these transitions were not discovered by Bridgman himself and, consequently, were not subjected to special study, one cannot be absolutely certain that they are not accompanied by small jumps in volume or in the bulk modulus.
elasticity, i.e. they are not phase transitions of the first or second kind. In fact, if, for example, in the study of lanthanum (see Table II and Fig. 18, curve II) no large jump in volume at \(23400\ \mathrm{kg/cm^2}\) had been observed, then we would have come to the conclusion that lanthanum undergoes a phase transformation “of the third kind.” Be that as it may, the presence in a number of substances of transitions analogous to those shown in Fig. 20 deserves further study.
Fig. 19. Change in bulk elasticity at the point of a second-order phase transition in \(\mathrm{NiSO_4\cdot 6H_2O}\).
Fig. 20. Transitions not accompanied by jump-like changes in volume and the modulus of bulk elasticity in methylene chloride \((I)\) and AgCN \((II)\).
The list of these substances is given in Table IV.
Table IV
Transitions not accompanied by jumps in volume and in the modulus of bulk elasticity
| Substance | Transition pressure \((\mathrm{kg/cm^2})\) | Jump in coefficient \(\chi\) |
|---|---|---|
| Menthol \(^{14}\) | 10000 | 8.0—2.7 |
| Rubber \(^{14}\) | 12500 | 11—3.4 |
| AgCN \(^{16}\) | 13000 | 8.6—12 |
| \(\mathrm{SiO_2(CH_3)_3}\) \(^{18}\) | 27000 | 8.3—4.1 |
| \(\mathrm{C_6H_5CH(CH_3)_2}\) \(^{18}\) | 27000 | 7.5—10 |
| Methylene chloride \(^{15}\) | 27000 | 7.9—12 |
| Methyl alcohol \(^{12}\) | 31000 | 6.7—13 |
| Ethylene bromide \(^{12}\) | 32000 | 6.3—10 |
| Lead \(^{13,14}\) | 52000 | 4.6—6.0 |
| Sodium chloride \(^{13,14}\) | 60000 | 4.7—2.7 |
M. KORNFELD
Volume Elasticity of Single Crystals
For the description of the volume elasticity of a single crystal, the scalar quantity of the modulus of volume elasticity is insufficient, since, as has already been indicated, linear compressions along different crystallographic axes are different. In the general case the volume elasticity of a crystal is characterized by three linear compressions; in the special case (for crystals of the hexagonal, tetragonal, and rhombohedral systems)—by two linear compressions; and only in crystals
Fig. 21. Dependence of linear compression on pressure in single crystals of zinc, tin, tellurium, and magnesium.
of the cubic system can it be characterized by one linear compression, or, what is the same thing, by the modulus of volume elasticity.
As an example, Fig. 21 gives the dependences of linear compression on pressure for single crystals of zinc, tin, and tellurium, belonging to the hexagonal, tetragonal, and rhombohedral systems[^19]. In these figures the symbol \( \parallel \) denotes the curves corresponding to compressions along the axis of symmetry, and the symbol \( \perp \)—perpendicular to it. It is noteworthy that tellurium is not compressed along the direction of the trigonal axis, but, on the contrary, is “stretched” by all-around pressure.
The most important result of the investigation of the volume elasticity of single crystals is the establishment of the fact that the degree of anisotro-
linear compression turns out to depend most strongly on the nature of the atoms forming the crystal lattice. One can be convinced of this, for example, by comparing the curves in Fig. 21 relating to zinc and to magnesium[^17]. Magnesium, like zinc, belongs to the hexagonal system. At the same time, a magnesium single crystal behaves under hydrostatic pressure as an isotropic body, whereas a zinc single crystal is compressed in one direction approximately 8 times more than in another. This fact shows that in a number of substances, in particular in zinc, the forces of interaction between atoms are not central.
Fig. 22. Dependence of linear compression on pressure for a single crystal of
$\mathrm{NiSO_4 \cdot 6H_2O}$.
Another very interesting feature of single crystals is the anisotropy of phase transitions.
This feature appears, for example, in a second-order phase transition in $\mathrm{NiSO_4 \cdot 6H_2O}$ (see Table III and Fig. 19).
Figure 22 gives the dependence of linear compression on pressure for a single crystal of $\mathrm{NiSO_4 \cdot 6H_2O}$ along the tetragonal axis and perpendicular to it. Along the direction of the crystal axis it behaves normally, whereas in the perpendicular direction, near $20\,000\ \mathrm{kg/cm^2}$, it undergoes a second-order transition.
Anomalies of volume elasticity
Regularities qualitatively different from those set forth above are observed in fused quartz and other glasses with a high content of $\mathrm{SiO_2}$[^16,^15,^23], in cerium[^16], and also, apparently, in chromium[^2].
Figure 23 presents the dependence of the modulus of volume elasticity on pressure for quartz glass at room tempera-
Fig. 23. Dependence of the bulk modulus of elasticity on pressure for quartz glass (I) and cerium (II).
Axis labels:
\(K \times 10^{-3}\ \mathrm{kg/cm^2}\)
\(p \times 10^{-3}\ \mathrm{kg/cm^2}\)
Fig. 24. Dependence of the bulk modulus of elasticity of quartz glass on pressure at different temperatures.
Axis labels:
\(K \times 10^{-3}\ \mathrm{kg/cm^2}\)
\(p \times 10^{-3}\ \mathrm{kg/cm^2}\)
Curve labels: \(390^\circ\mathrm{C}\), \(247^\circ\mathrm{C}\), \(100^\circ\mathrm{C}\), \(11^\circ\mathrm{C}\).
type. At first, over a range of approximately \(30\,000\ \mathrm{kg/cm^2}\), the bulk modulus decreases linearly with pressure; then the glass undergoes a phase transition of the second order, i.e., a transition not accompanied by a jump in volume, after which its behavior becomes normal. An analogous picture is observed for cerium, with the only difference that here the boundary of the anomaly is a phase transition of the first order (see Table II).
Quartz glass also possesses an anomalous dependence of bulk elasticity on temperature: its bulk modulus increases with increasing temperature (Fig. 24)\(^7\).
In the case of glass, the anomalies under consideration could be ascribed to the thermodynamic nonequilibrium of the substance. Such an explanation is probably inapplicable to cerium. In this connection it is interesting to note one feature of the phase transformation that converts cerium from the anomalous into the normal state. At atmospheric pressure cerium has a face-centered cubic crystal lattice.
As an X-ray study shows,\(^{24}\) after the transformation, accompanied by a 10% decrease in volume, the cerium lattice remains, as before, face-centered cubic.
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