A Concise Physical Interpretation of 

Several Analytical Grüneisen Formulations 

by Steven B. Segletes 

ARL-TR-3881 August 2006 

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ARL-TR-3881 August 2006 

A Concise Physical Interpretation of 

Several Analytical Grüneisen Formulations 

Steven B. Segletes 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited.
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A Concise Physical Interpretation of Several Analytical Grüneisen Formulations 

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14. ABSTRACT 

A novel method is presented for deriving the Grüneisen formulations of three classical Grüneisen models. Unlike past studies 

which have attempted to differentiate between these Grüneisen models on the basis of Poisson ratio arguments, the current 

approach employs elementary vibration analysis to reach its conclusions. The current approach provides a physical insight into

how the models’ underlying assumptions are responsible for the analytical distinctions between the models. 

15. SUBJECT TERMS

equations of state, lattice dynamics, thermodynamic properties, Slater, Dugdale-MacDonald, free-volume theory 

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Steven B. Segletes 

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Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18 
Contents

1. Introduction 1

2. Classical Gr  ̈uneisen Theories 2

3. Result 4

4. Conclusion 7

5. References 9

Distribution List 11

iii
INTENTIONALLY LEFT BLANK.

iv
1. Introduction

The Gruneisen function,  ̈ Γ, is a key thermodynamic function in the study of materials

subjected to the high pressures of shock loading. Macroscopically, it relates the pressure

changes to changes in internal energy at constant volume, by way of the relation

Γ = V (∂p/∂E)V , (1)

where p is pressure, E is internal energy, and V is the volume of the material’s unit cell.

In general, Γ may be a function of the lattice volume (i.e., spacing of the atomic

oscillators) as well as the temperature (i.e., amplitude of the vibration). However,

Gruneisen theory assumes that the vibrational amplitude is small relative to the lattice  ̈

spacing; thus, the set of oscillators comprising the lattice possess frequencies that are

assumed to vary with volume alone. In terms of this characteristic vibrational frequency

of the lattice, ω, the volume dependence of the Gruneisen function is expressed,  ̈

according to classical theory (1, 2), as

Γ/V = −

dω/dV

ω

. (2)

During the decades in which the classical approach to lattice vibration held sway, a

number of analytical Gruneisen formulations were presented in the literature, with  ̈

several gaining a widely cited reputation (1, 3, 4). Because of the manner in which these

particular models were formulated, Γ was given in terms of both the material’s volume

and cold pressure, pc (i.e., the 0

◦

isothermal pressure). However, since the cold pressure

is, itself, a function of volume alone, these formulations for Γ indeed satisfy the

Gruneisen assumption of temperature independence.  ̈

While these important models retained the key definitional coupling embodied in

equation 2, many later developers of numerical hydrocodes for impact simulation opted

to bypass the formalism of these historical models, in favor of direct empirical

characterization of Γ(V ) according to preferred functional forms. Perhaps the motivation

in this was that empirical shock data provided a material’s shock Hugoniot, and not the

cold-pressure curve demanded of the classical models. Regardless of the motivation,

once analytically decoupled from the mechanics of lattice compression, these wholly

1
empirical formulations for Γ were not always thermodynamically robust. Segletes (5–7)

has shown how these decoupled Gruneisen formulations are susceptible to an unusual  ̈

host of thermodynamic inconsistencies that can lead to nonphysical thermodynamic

instabilities in shock/impact calculations.

There is value, then, in reconsidering the classical Gruneisen methods as an important  ̈

bridge between modern ab initio methods and the decoupled empirical approaches that

still characterize many extensively utilized numerical codes in the shock/impact

community.

2. Classical Gr  ̈uneisen Theories

The classical models by Slater (1), Dugdale and MacDonald (3), as well as the

free-volume theory cited by Vashchenko and Zubarev (4), have been widely studied and

utilized and relate a materials’ Gruneisen behavior to the compression behavior of the  ̈

lattice. While it is somewhat difficult to compare the original derivations of these three

models side by side to ascertain the particular points of departure that bring about their

distinctions, some have attempted to do exactly this.

While the forms originally presented by Slater,

Γ = −

2

3

−

V

2

 

d

2pc/dV 2

dpc/dV !

, (3)

and that of Dugdale and MacDonald,

Γ = −1 −

V

2

 

d

2pc/dV 2 − 10pc/9V2

dpc/dV + 2pc/3V

!

, (4)

bear a notable similarity as presented, it was Vashchenko and Zubarev (4) who were able

to present a unified view of these results, along with the free-volume theory. They

present a single parameterized algebraic expression which accurately captures the forms

of all three of these models, depending upon the value of the model’s parameter. This

expression is given as

Γ = −

4 − 3n

6

−

V

2

d

2

dV 2

(pcV

n

)

d

dV (pcV

n

)

, (5)

2
where n is the model parameter. When n takes on values of 0, 2/3, and 4/3 respectively,

the models of Slater (1), Dugdale and MacDonald (3), and the free-volume theory (4) are

respectively recovered.

To arrive at equation 5, Vashchenko and Zubarev postulated a hypothetical form for the

Poisson ratio, ν, in terms of the cold pressure, volume, and model parameter n:

ν =

1 −

1 − ν0

1 + ν0

 

1 + n

d ln V

d ln pc

!

1 +

1 − ν0

1 + ν0

 

1 + n

d ln V

d ln pc

! . (6)

Coupled with certain relations specifying the manner in which the Poisson ratio relates

to the lattice vibrational frequencies, equation 6 was shown to lead directly to equation 5.

Therefore, if one accepts their postulated form for the Poisson ratio, equation 6, then the

three Gruneisen models of interest (  ̈ 1, 3, 4) follow from equation 5 as specific instances of

the proposed form. Pastine (8) has also utilized an explanation involving the influence of

the Poisson ratio as a means to interpret and differentiate expressions for the Gruneisen  ̈

formulation.

Conceptually, such an approach makes sense, and there is certainly nothing invalid

about it. However, the intuitive understanding that one may obtain from equation 6 is,

at best, quite limited, especially when the Poisson form is, itself, expressed in terms of

volume derivatives of the cold pressure.

What is more, the original derivations of the models of Slater, Dugdale and MacDonald,

and the free-volume theory were not made on the basis of this or any other Poisson ratio

formulation. That researchers could, after the fact, discover the Poisson ratio as a means

to differentiate the models was a useful and convenient resolution to the useful question

of “what scientific explanation can explain the functional difference between these three

models?” But, because the original models were never derived in terms of Poisson ratio,

the Poisson explanation fails to answer the more fundamental question of “what is it

about the formulation of these three models that leads to their functional differences?”

3
3. Result

In this report, a different approach is taken to compare these historical models, based

upon treating the material lattice as a spring-like structure, in which the various models

may be differentiated on the basis of what kind of spring composes the lattice and what

measure is taken to quantify the spring’s characteristic spatial dimension. The key link

in eventually relating the spring’s response back to the Gruneisen function is the  ̈

classical equation 2, which defines the Gruneisen function in terms of the characteristic  ̈

vibrational frequency of the lattice ω.

Consider a lattice with spacing λ such that λ

3 =V . When atomic vibration is not

considered (i.e., at 0

◦

temperature), the potential energy of the lattice network is defined

by the cold energy of the lattice Ec, which will be a function of only a spatial dimension

that is characteristic of the lattice, call it ξ. This characteristic dimension might take on

units of length (as in the lattice spacing λ), though it need not necessarily do so; rather, it

could take the form of volume V or some other spatially related quantity.

For a quantity characterized by an isotropic field potential, such as the lattice energy Ec,

the negative of the gradient of the potential with respect to ξ, in this case −∇Ec(ξ), is an

indicative measure of the field’s “force,” while the Laplacian of Ec, that is ∇2Ec(ξ), is a

measure of the field’s “stiffness.” The terms “force” and “stiffness” are quoted here

because depending on the nature of the characteristic dimension ξ, the physical units of

the characteristic lattice “force” might be other than force and likewise for the “stiffness.”

Vibration analysis indicates that the system’s characteristic stiffness is associated with a

characteristic vibrational frequency ωξ according to the following relation:

∇2Ec(ξ) ∝ ω

2

ξ

, (7)

where the symbol “∝” denotes a direct proportionality. This expression has been used in

the study of lattice mechanics (9, 10). As in the case of the characteristic force and

stiffness, the ξ-based characteristic frequency ωξ may not have the physical units of

1/time that are traditionally associated with frequency, though it will have dimensions of

propagation speed per unit characteristic dimension. However, it may be related to a

characteristic frequency ω that does have traditional units (i.e., 1/time). This connection

4
is made through the definitional relations of vibration theory which connect the

characteristic frequency and propagation velocity through the vibrational wavelength:

ωξξ = C = ωλ . (8)

Thus, combining equations 7 and 8 gives the following proportion:

∇2Ec(ξ) ∝

 

ωλ

ξ

!2

. (9)

Equation 9 is the simple relation from which the results of this report flow. Applying it

to the task of acquiring a relationship for the Gruneisen function is accomplished by  ̈

starting with equation 9 and isolating ω

2

. Take the derivative of this equation with

respect to volume V and divide the resulting equation by the original equation. The

result, in light of equation 2, is given as

Γ = −

V

2

d

dV 

ξ

2

λ

2 ∇2Ec(ξ)

!





ξ

2

λ

2 ∇2Ec(ξ)





. (10)

There are really two parameters that characterize equation 10. The first is the

characteristic lattice dimension ξ, and the second is physical nature of the lattice spring

whose stiffness is characterized by ∇2Ec(ξ). Depending on whether the lattice spring is

considered a simple one-dimensional (1–D) spring or a three dimensional (3–D) radial

spring, the Laplacian operator would be expressed as

∇2Ec(ξ) ∝







d

2Ec/dξ2

(simple 1–D spring)

d

2Ec/dξ2 + (2/ξ)dEc/dξ (3–D radial spring)

. (11)

From equation 10, in light of equation 11, it may be directly shown that Slater’s relation

is recovered when the lattice spring is a simple 1–D spring and the characteristic spatial

dimension ξ of the lattice is taken as its cell volume V . Similarly, the relation of Dugdale

and MacDonald is recovered by taking the lattice spring to be simple 1–D, while taking

the characteristic spatial dimension ξ as the lattice spacing λ. Finally, the free-volume

5
theory may be recovered∗

from equation 10 when the lattice spring is taken as a 3–D

radial spring and the characteristic spatial dimension ξ of the lattice is the lattice spacing

λ. Table 1 categorizes some of the relevant quantities for these three models.

Table 1. Characteristic spring type and spatial dimension for various Gruneisen theories and  ̈

physical units of associated characteristic quantities.

Lattice Spring Characteristic Units of Characteristic

Theory Type Spatial Dimension, ξ Force Stiffness

Slater Simple (1–D) V Pressure Pressure/volume

Dugdale-MacDonald Simple (1–D) λ Force Force/length

Free-volume theory Radial (3–D) λ Force Force/length

These differentiating characteristics are not merely coincidental, which just happen to

produce the given models; rather, they are indicative of the logic and methodologies

employed by the researchers in their original derivations. In the case of Slater’s

derivation, a constant Poisson ratio was assumed. When this is stipulated for a material,

the elastic and bulk moduli remain proportional. This allowed the vibrational frequency

to be cast in terms of volume derivatives of pressure. Thus, it is not surprising that the

characteristic dimension of the Slater model is volume V , leading to a stiffness expressed

in terms of pressure per volume.

In the case of the Dugdale-MacDonald model, the authors’ explicit contention was that

the characteristic lattice velocity is always of a form consistent with a 1–D spring

formulation (which is to say, with a characteristic stiffness having units of force per

length). Plendl (12) has also asserted this relationship. Dugdale and MacDonald even

went so far as to show that Slater’s model failed to give the proper Gruneisen value of  ̈

zero for a harmonic lattice of simple 1–D springs. Thus, wholly consistent with the

model’s underlying principal, a lattice of simply-connected 1–D springs would

necessarily retain λ as the characteristic dimension.

The free-volume theory differs from the Dugdale-MacDonald relation only in that the

lattice spring is taken as a 3–D radial spring rather than as a simple 1–D spring,

reflecting a more complex view of reality in terms of the interatomic potential as a 3–D

field and not as a 1–D spring.

∗Recognition of the compatibility of the free-volume theory with the underlying tenets of equations 2

and 7 was also noted in unpublished work by Shanker et al. (11).

6
It is important to note that when ξ defines an orientation-independent lattice measure

such as λ or V , equation 7 describes a volumetric oscillation, since an ordinary derivative

of Ec, with respect to the characteristic dimension ξ, implies that ξ is changed between

every atom in the lattice network (and in every component direction), in order to

evaluate the resultant change in Ec. Therefore, in such cases, the quantity ωξ may be

justifiably termed a “volumetric frequency,” regardless of whether the springs are

characterized as 1–D or 3–D.

However, it has been shown (13) that when the lattice interactions are restricted to

nearest neighbors only (or alternately when the lattice component stiffness is purely

harmonic in nature), the stiffness associated with a volumetric change is in proportion to

the stiffness associated with a 1–D vibrational wave pulse. Therefore, under the

assumption of nearest-neighbor interactions, this volumetric frequency ωξ is equally

indicative of a traveling-wave vibrational frequency from which the Γ function actually

arises.

In all the models examined in this report (1, 3, 4), the original authors explicitly invoked

nearest-neighbor arguments in their derivations. This widely used nearest-neighbor

assumption (1, 3, 4, 14, 15) serves to justify the use of ωξ in equation 8 as a means to link

the vibrational frequencies of volumetric change to those of traveling waves. It must be

remembered, however, that in the general case when distant interactions of non-nearest

neighbors are considered, the volumetric frequency and vibrational frequency are not

identical (13). As such, equation 7 remains merely an approximation when applied to

the global (as opposed to the pairwise) potential.

4. Conclusion

The simple direct formulation of the Gruneisen function that herein arises from  ̈

elementary vibration analysis serves to provide a clear foundational understanding of

three popular Gruneisen models (  ̈ 1, 3, 4) and their differentiating characteristics. Unlike

the historical explanations (4, 8) that, after the fact, invoked the functional behavior of

the Poisson ratio as the means to functionally differentiate the historical models, the

current approach offers an explanation in terms of two fundamental modeling

assumptions—the nature of the interatomic “spring” and the characteristic spatial

measure of the lattice. Such a differentiation, by its nature, provides a satisfactory

7
answer to the question of “what is it about the formulation of these three models that

leads to their functional differences?” Such understanding is absent with the historical

explanations, where the differentiating feature is cast in terms of the Poisson ratio.

8
5. References

1. Slater, J. C. Introduction to Chemical Physics; 1

st ed.; International Series in Physics,

McGraw Hill: New York, 1939.

2. Rice, M. H.; McQueen, R. G.; Walsh, J. M. Compression of Solids by Strong Shock

Waves. In Solid State Physics, Vol. 6; Seitz, F., Turnbull, D., Eds.; Academic Press:

New York, 1958; pp 1–63.

3. Dugdale, J. S.; MacDonald, D. K. C. The Thermal Expansion of Solids. Phys. Rev.

1953, 89(4), 832–834.

4. Vaschenko, V. Y.; Zubarev, V. N. Concerning the Gruneisen Constant. Sov.

Phys.—Sol. State 1963, 5(3), 653–655.

5. Segletes, S. B. Thermodynamic Stability of the Mie-Grueneisen Equation of State,  ̈

and its Relevance to Hydrocode Computations. J. Appl. Phys. 1991, 70(5), 2489–2499;

Erratum 1992, 71, 1074.

6. Segletes, S. B. Further Examinations on the Thermodynamic Stability of the

Mie-Gruneisen Equation of State.  ̈ J. Appl. Phys. 1994, 76(8), 4560–4566.

7. Segletes, S. B. The Effect of Thermodynamic Constraints Upon the Mie-Gruneisen  ̈

Equation of State. In Constitutive Laws: Experiments and Numerical Implementation;

Rajendran, A. M., Batra, R. C., Eds.; Theory and Engineering Applications of

Computational Methods, CIMNE: Barcelona, 1995; pp 46–51.

8. Pastine, D. J. Formulation of the Gruneisen Parameter for Monatomic Cubic  ̈

Crystals. Phys. Rev. 1965, 138(3A), A767–A770.

9. Maradudin, A. A.; Montroll, E. W.; Weiss, G. H. Theory of Lattice Dynamics in the

Harmonic Approximation. In Solid State Physics; Seitz, F., Turnbull, D., Eds.;

supplement 3 of Advances in Research and Applications, Academic Press: New York,

1963.

10. Hama, J.; Suito, K.; Kawakami, N. First-Principles Calculation of the Shock-Wave

Equation of State of Isotropic Lithium Hydride. Phys. Rev. B 1989, 39(5), 3351–3360.

11. Shanker, J.; Singh, B. P.; Baghel, H. K. Unpublished manuscript, Sept. 2005.

9
12. Plendl, J. N. Some New Interrelations in the Properties of Solids Based on

Anharmonic Cohesive Forces. Phys. Rev. 1961, 123(4), 1172–1180.

13. Segletes, S. B. Classical Methods for Frequency-Based Equations of State. In Recent

Research Developments in Physics and Chemistry of Solids, Vol. 1; Pandalai, S. G., Ed.;

Transworld Research Network: Trivandrum, 2002; pp 1–26.

14. Brillouin, L. Wave Propagation in Periodic Structures; 2

nd ed.; Dover Publications: New

York, 1953 (originally McGraw Hill: New York, 1946).

15. Guinea, F.; Rose, J. H.; Smith, J. R.; Ferrante, J. Scaling Relations in the Equation of

State, Thermal Expansion, and Melting of Metals. Appl. Phys. Let. 1984, 44(1), 53–55.

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 1 UC DAVIS 

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 142432 CHERNOGOLOVKA 

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 ST PETERSBURG 199178 

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 195251 ST-PETERSBURG 

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 5 DEPARTMENTO DE QUIMICA 

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 28040 MADRID 

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 1 UNIVERSIDAD DE CANTABRIA 

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