Chapter 4

Compressibility

One of the simplest physical phenomena is compression, the response of the time region equilibrium to external forces impressed upon it. With the benefit of the information developed earlier, we are now in a position to begin an examination of the compression of solids, disregarding for the present the question of the origin of the external forces. For this purpose we introduce the concept of pressure, which is defined as force per unit area.

In many cases it will be convenient to deal with pressure on a volume basis rather than on an area basis. We therefore multiply both force and area by distance, s, which gives us the alternative equation:

In the region outside unit distance, where the atoms or molecules of matter are independent, the total energy of an aggregate can thus be expressed in terms of pressure and volume as

As we will find in the next chapter when we begin consideration of thermal motion, a condition of constant temperature is a condition of constant energy, other things being equal. Equation 4-3 thus tells us that for an aggregate in which the cohesive forces between the atoms or molecules are negligible, an ideal gas, the volume at constant temperature is inversely proportional to the pressure. This is Boyle's law, one of the well-established relations of physics.

For application to the time region in which the solid equilibrium is located, the second power of the volume must be substituted for the first power, in accordance with the general inter-regional relation previously established. The time region equivalent of Boyle’s Law is therefore

In terms of volume this becomes

This equation tells us that at constant temperature the volume of a solid is inversely proportional to the square root of the pressure. The pressure represented by the symbol P in this equation is, of course, the total effective pressure; that is, the pressure equivalent of all of the forces acting in opposition to the rotational forces of the atom. The force due to the progression of the natural reference system opposes the rotational forces, and acts in parallel with the external compressive forces, but it has the same magnitude regardless of whether or not any such external forces are present. It therefore exerts what we may call an internal pressure, an already existing level of pressure to which an external pressure becomes an addition. In order to conform to established usage and to avoid confusion, the symbol P will hereafter refer to the external pressure only, the total pressure being expressed as P₀ + P. On this basis, quation 4-5 may be restated as

Compression is normally expressed in terms of relative rather than absolute volumes, the reference volume being the volume at zero external pressure, where equation 4-6 has the form

Dividing the equation 4-6 by equation 4-7, and rearranging, we obtain

As this equation brings out, the internal pressure, P₀, is the key factor in the compression of solids. Inasmuch as this pressure is a result of the progression of the natural reference system which, in the time region, is carrying the atoms inward in opposition to their rotational forces (gravitation), the inward force acts only on two dimensions (an area), and the magnitude of the pressure therefore depends on the orientation of the atom with respect to the line of the progression. As indicated in connection with the derivation of the inter-regional ratio, there are 156.44 possible positions of a displacement unit in the time region, of which a fraction az represents the area subjected to pressure, a and z being the effective displacements in the active dimensions. The letter symbols a, b, and c, are used as indicated in Chapter 10, Volume I. The displacement z is either the electric displacement c or the second magnetic displacement b, depending on the orientation of the atom.

From the principle of equivalence of natural units it follows that each natural unit of pressure exerts one natural unit of force per unit of cross-sectional area per effective rotational unit in the third dimension of the equivalent space. However, the pressure is measured in the units applicable to the effect of external pressure. The forces involved in this pressure are distributed to the three spatial dimensions and to the two directions in each dimension. Only those in one direction of one dimension–one sixth of the total–are effective against the time region structure. Applying this 1/6 factor to the ratio az/156.444, we have for the internal pressure per rotational unit at unit volume,

This expression may now be generalized to apply to y rotational units and V units of volume, as follows:

The force exerted by the progression of the natural reference system is independent of the geometrical arrangement of the atoms, and the volume term in equation 4-10 refers to what we may call the three-dimensional atomic space, the cube of the inter-atomic distance, rather than to the geometric volume. We will therefore replace V by S₀³. This gives us the internal pressure equation in final form:

The value derived from this equation is the magnitude of the internal pressure in terms of natural units. To obtain the pressure in terms of any conventional system of units it is only necessary to apply a numerical coefficient equal to the value of the natural unit of pressure in that system. This natural unit was evaluated in Volume I as 5.282 x 10¹² dynes/cm². The corresponding values in the systems of units used in the reports of the experiments with which comparisons will be made in this chapter are:

1.554 x 10⁷ atm
1.606 x 10⁷ kg/cm²
1.575 x 10⁷ megabars

In terms of the units used by P.W. Bridgman, the pioneer investigator in the field, in most of his work, equation 4-11 takes the form

The internal pressure thus calculated for any specific substance is not usually constant through the entire external pressure range. At low total pressures, the orientation of the atom with respect to the line of progression of the natural refe- rence system is determined by the thermal forces which, as we will see later, favor the minimum values of the effective cross-sectional area. In the low range of total pressures, therefore, the cross-section is as small as the rotational displacements of the atom will permit. In accordance with Le Chatelier’s Principle, a higher pressure, either internal or external, applied against the equilibrium system causes the orientation to shift, in one or more steps, toward higher displacement values. At extreme pressures the compressive force is exerted against the maximum cross-section: 4 magnetic units in one dimension and 8 electric units in another. Similarly, only one of the magnetic rotational units in the atom participates in the radial component y of the resistance to compression at the low pressures, but further application of pressure extends the participation to additional rotational units, and at extreme pressures all of the rotational units in the atom are involved. The limiting value of y is therefore the total number of such units. The exact sequence in which these two kinds of factors increase in the intermediate pressure range has not yet been determined, but for present purposes a resolution of this issue is not necessary, as the effect of any specific amount of increase is the same in both cases.

Helium and neon, the first two of the inert gases, the elements that have no effective rotation in the electric dimension, take the absolute minimum compression factors: one rotating unit with one effective unit of displacement in each of the two effective dimensions. The azy factors for these elements can be expressed as 1-1-1. In this notation, which we will use for convenience in the subsequent discussion, the numerical values of the compression factors are given in the same order as in the equations. It should be noted that the absolute minimum compression, that applicable to the elements of least displacement, is explicitly defined by the factors 1-1-1. The value of the factor a increases in the higher members of the inert gas series because of their greater magnetic displacement.

Because of their negative displacement in the electric dimension, which, in this context, is equivalent to the zero displacement of the inert gases, the electronegative elements follow the inert gas pattern, taking the minimum 1-1-1 factors in the lowest members of the lowest rotational groups, and values that are higher, but still generally well below those of the corresponding electro-positive elements, as the displacement increases in either or both of the atomic rotations. None of the elements of the electronegative divisions below electric displacement 7 has the 4-8 az factors initially, although they are capable of these high levels, and can eventually reach them under appropriate conditions.

All of the electropositive elements studied by Bridgman have the full 4 units in one dimension; that is, a = 4. The value of z for the alkali metals is equal to the electric displacement, one unit, and since y takes the minimum value under low pressure conditions, the compression factors for these elements are 4-1-1. The displacement 2 elements (calcium, etc.) take the intermediate values 4-2-1 or 4-3-1. The greater displacements of the elements that follow have a double effect. They increase the internal pressure directly by enlarging the effective cross-section, and this higher internal pressure then has the same effect as a greater external pressure in causing a further increase in the compression factors. Most of these elements therefore utilize the full displacements of the active cross-section dimensions from the start of compression; that is, 4-4-1 (az = ab, two magnetic dimensions) in some of the lower group elements and the transition elements of Group 4A, and 4-8-1, or 4-8-n (az = ac, one magnetic and one electric dimension) in the others.

The factors that determine the internal pressures of the compounds that have been investigated thus far fall mainly in the intermediate range, between 4-1-1 and 4-4-1. NaCl, for instance, has 4-2-1 initially, and shifts to 4-3-1 in the pressure range between 30 and 50 M kg/cm². AgCl has 4-3-1 initially, and carries these factors up to a transition point near Bridgman's pressure limit of 100 M kg/cm². CaF₂ has the factors 4-4-1 from the start of compression. The initial values of the internal pressure of most of the inorganic compounds examined in this investigation are based on one or another of these three patterns. Those of the organic compounds are mainly 4-1-1, 4-2-1, or an intermediate value 4-1½-1

Compression is ordinarily measured in terms of relative volume, and most of the discussion in this chapter will deal with the subject on this basis, but for some purposes we will be interested in the compressibility, the rate of change of volume under pressure. This rate is obtained by differentiating equation 4-8.

The compressibility at P₀, the initial compressibility, is of particular interest. For all practical purposes it is the same as the compressibility at one atmosphere, this pressure being only a small fraction of the internal pressure P₀. The initial compressibility may be obtained from equation 4-13 by letting P equal zero. The result is

Since the initial compressibility is a quantity that can be measured, its simple and direct relation to the internal pressure provides a significant confirmation of the physical reality of that theoretical property of matter. Initial compressibility factors derived theoretically for those elements on which consistent compressibility data are available for comparison, the internal pressures calculated from these factors, and the initial compressibilities corresponding to the calculated internal pressures are listed in Table14, together with measured values of the initial compressibility at room temperature. Two sets of experimental values are given, one from Bridgman and one from a more recent compilation. Values of S₀³, except those marked with asterisks, are computed from the inter-atomic distances (S₀) in the tables of Chapter 2. Where the structure is anisotropic the S₀³ value shown is the product of one of the distances given in the earlier tabulations by the square of the other. The reason for the occurrence of the indicated deviations from the Chapter 2 values will be explained later.

S03			Comp. Factors			P0		Initial Compressibility x 106
			a	z	y	(M kg/cm2)		Calc.		Obs.3	Obs.4
	Li	1.151	4	1	1	59	.5		8.42	8.41	8.46
	Be	0.482	4	4	1	568			0.88	0.87	0.98
	C(dia.)	0.147	4	6	1	2793			0.18	0.18	0.18
	Na	2.048	4	1	1	33	.4		14.97	15.1	14.42
	Mg	1.291	4	4	1	212			2.36	2.86	2.77
	Al	0.915	4	5	1	374			1.34	1.30	1.36
	Si	0.497	4	4	1	551			0.91	0.31	0.99
	K	3.659	4	1	1	18	.7		26.74	31.0	30.4
	Ca	2.588	4	3	1	79	.3		6.31	5.51	6.45
	Ti	1.033	4	8	1	530			0.94	0.77	0.93
	V	0.729	4	8	1	751			0.67	0.59	0.61
	Cr	0.603	4	8	1	908			0.55	0.50	0.52
	Mn	0.705	4	8	1	777			0.64	0.76	1.65
	Fe	0.603	4	8	1	908			0.55	0.57	0.58
	Co	0.603*	4	8	1	908			0.55	0.52	0.51
	Ni	0.603*	4	8	1	908			0.55	0.50	0.53
	Cu	0.652	4	6	1	630			0.79	0.70	0.72
	Zn	0.903	4	4	1	303			1.65	1.64	1.64
	Ge	0.603	4	4	1	454			1.10	1.33	1.27
	Rb	4.616	4	1	1	14	.8		33.78	38.7	31.4
	Sr	3.268	4	3	1	62	.8		7.96	7.9	8.46
	Zr	1.306	4	6	1½	472			1.06	1.06	1.18
	Nb	0.921	4	8	1½	892			0.56	0.55	0.58
	Mo	0.764*	4	8	2	1433			0.35	0.34	0.36
	Ru	0.764*	4	8	2	1433			0.35	0.34	0.31
	Rh	0.764	4	8	2	1433			0.35	0.36	0.36
	Pd	0.823	4	8	1½	998			0.50	0.51	0.54
	Ag	0.956	4	8	1	573			0.87	0.96	0.97
	Cd	1.118	4	4	1	245			2.04	1.89	2.10
	In	1.165*	4	4	1	235			2.13		2.38
	Sn	0.913*	4	4	1	300			1.67	1.64	0.80
	Sb	1.325*	4	4	1	207			2.42	2.32	2.56
	Cs	5.774	4	1	1	11	.9		42.0	59.0	49.1
	Ba	2.686*	4	2	1	51	.0		9.80		9.78
	La	2.044	4	4	1	134			3.73	3.39	4.04
	Ce	1.893	4	4	1	145			3.45	3.45	4.10
	Pr	1.758*	4	4	1	156			3.21		3.21
	Nd	1.758*	4	4	1	156			3.21		3.00
	Sm	1.758*	4	4	1	156			3.21		3.34
	Gd	1.346*	4	4	1	203			2.46		2.56
	Dy	1.346	4	4	1	203			2.46		2.55
	Ho	1.346*	4	4	1	203			2.46		2.47
	Er	1.346*	4	4	1	203			2.46		2.38
	Tm	1.346*	4	4	1	203			2.46		2.47
	Yb	2.167*	4	2	1	63	.2		7.92		7.38
	Lu	1.346*	4	4	1	203			2.46		2.38
	Ta	1.027*	4	8	2	1066			0.47	0.47	0.49
	W	0.953*	4	8	3	1723			0.29	0.28	0.30
	Ir	0.823	4	8	3	1996			0.25		0.28
	Pt	0.823	4	8	2	1330			0.38	0.35	0.35
	Au	0.953	4	8	1½	862			0.58	0.56	0.57
	Ti	1.631	4	4	1	168			2.98	3.31	2.74
	Pb	1.249*	4	4	1	219			2.25	2.29	2.29
	Bi	1.249	4	3	1	164			3.05	2.71	3.11
	Th	1.758	4	8	1	311			1.61		1.81
	U	0.984	4	8	1	556			0.90	0.94	0.99

In most cases the difference between the calculated and measured compressibilities is within the probable experimental error. Substantial deviations from the calculated values are to be expected in the case of low melting point elements such as the alkali metals, unless corrections have been applied to the empirical data, as there is an additional component in the initial volume of such substances. Elsewhere, the differences between the calculated compressibilities and either of the two sets of experimental values are, on the average, no greater than the differences between the experimental results. This process is repeated at successively higher pressure levels until the maximum compression factors for the element are reached.

Because of the nature of this compression pattern, a convenient method of analyzing the experimental values of the volume of various substances under compression can be made available by expressing equation 4-8 in the form

According to this equation, if we plot the reciprocals of the squares of the relative volumes against the corresponding total pressure ratios we should obtain a straight line intersecting the zero pressure ordinate at the reference volume 1.00. The slope of the line is determined by the magnitude of the internal pressure, P₀ Fig.1(a) is a curve of this kind for the element tin, based on Bridgman’s experimental values.

Figure1: Compression Patterns

Where there is a transition to a higher set of compression factors within the experimental range, and the magnitude of P₀ changes, the volumes diverge from the original line and follow a second straight line, the slope of which is determined by the new compression factors. On preparing curves of this kind for the other elements investigated by Bridgman, we find that about two-thirds of them actually do conform to a single straight line up to the 30,000 kg/cm² pressure limit of his earlier work. His studies of the less compressible substances, such as the higher elements of the electropositive divisions, were not carried beyond this level, but he measured the compression up to
100,000 kg/cm² on many other elements, and most of them were found to undergo a transition in which the effective internal pressure increases without any volume discontinuity. The compression curve for such a substance consists of two straight line segments connected by a smooth transition curve, as in Fig.1(b), which represents Bridgman's values for silicon.

In addition to the changes of this type, commonly called second order transitions, some solid substances undergo first order transitions in which there is a modification of the crystal structure and a volume discontinuity at the transition point. The effective internal pressure generally changes during a transition of this kind, and the resulting volumetric pattern is similar to that of KCl, Fig.1(c). With the exception of some values which are rather erratic and of questionable validity, all of Bridgman's results follow one of these three patterns or some combination of them. The antimony curve, Fig.1(d), illustrates one of the combination patterns. Here a second order transition between 30,000 and 40,000 kg/cm² is followed by a first order transition at a higher pressure. The numerical values corresponding to these curves are given in the tables that follow.

The experimental second order curves are smooth and regular, indicating that the transition process takes place freely when the appropriate pressure is reached. The first order transitions, on the other hand, show considerable irregularity, and the experimental results suggest that in many substances the structural changes at the transition points are subject to a variable amount of delay due to internal conditions in the solid aggregate. In these substances the transition does not take place at a well-defined pressure, but somewhere within a relatively broad transition zone, and the exact course of the transition may vary considerably between one series of measurements and another. Furthermore, there are many substances which appear to experience similar delays in achieving volumetric equilibrium even where no transitions take place. The compression curves suggest that a number of the reported transitions are actually volume adjustments which merely reflect delayed response to the pressure applied earlier. For example, in the barium curve based on Bridgman's results there are presumably two transitions, one between 20,000 and 25,000 kg/cm², and the other between 60,000 and 70,000 kg/cm². Yet the experimental volumes at 60,000 and 100,000 kg/cm² are both very close to the values calculated on the basis of a single straight line relation. It is quite probable, therefore, that this element actually follows one linear relation at least up to the vicinity of 100,000 kg/cm².

The deviations from the theoretical curves that are found in the experimental volumes of substances with relatively high melting points are generally within the experimental error range, and those larger deviations that do make their appearance can, in most cases, be explained on the foregoing basis. The compression curves for substances with low melting points show systematic deviations from linearity at the lower pressures, but this is a normal pattern of behavior resulting from the proximity of the change of state. As will be brought out in detail in our examination of the liquid state, the physical state of matter is basically a property of the individual atom or molecule. The state of the aggregate merely reflects the state of the majority of its individual constituents. Consequently, a solid aggregate at any temperature near the melting point contains a specific proportion of liquid molecules. Since the volume of a liquid molecule differs from that of a solid molecule, the volume of the aggregate is modified accordingly. The amount of the volume deviation in each case can be calculated by methods that will be described in the subsequent discussion of the liquid volume relations.

Table 15 compares the results of the application of equation 4-8 with Bridgman’s measurements on some of the elements that maintain the same internal pressure all the way up to his pressure limit of 100,000 kg/cm². In many cases he made several series of measurements on the same element. Most of these results agree within about 0.003, and it does not appear that listing all of the individual values in the tabulations would serve any useful purpose. The values given in Table15, and in the similar tables that follow, are those obtained in experiments that were carried to the full 100,000 kg/cm² pressure level. Where the high pressure measurements were started at some elevated pressure, or where the measurement interval was greater than usual, the gaps have been filled from the results of other Bridgman experiments.

Table 15: Relative Volumes Under Compression

Table16 extends the volume comparisons to representative elements of the classes that are subject to transitions within the experimental range of pressures. Transitions reported by the investigator or indicated by the theoretical calculations are shown by horizontal lines in the appropriate columns. In these tabulations the position of the upper branch of each curve has been fixed by using the experimental volume at a selected pressure in the straight line segment above the transition (identified by the symbol R) as a reference point. Thus the slope of this upper branch of the curve is determined theoretically, but its position relative to the 1/V² scale is empirical. Some work has been done toward extension of the theoretical development to a determination of the exact position of the upper section of each curve, but this project is not far enough advanced to justify any discussion of it at this time.

Table 16: Relative Volumes Under Compression