Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6
https://doi.org/10.1186/s40645-021-00463-2

RESEARCH ARTICLE

Progress in Earth andPlanetary Science

Open Access

Formation, preservation and extinctionof high‑pressure minerals in meteorites:
temperature effects in shock metamorphismand shock classification
Jinping Hu1,2*

and Thomas G. Sharp2

Abstract
The goal of classifying shock metamorphic features in meteorites is to estimate the corresponding shock pressure
conditions. However, the temperature variability of shock metamorphism is equally important and can result in adiverse and heterogeneous set of shock features in samples with a common overall shock pressure. In particular, highpressure (HP) minerals, which were previously used as a solid indicator of high shock pressure in meteorites, requirecomplex pressure–temperature–time (P–T–t) histories to form and survive. First, parts of the sample must be heated
to melting temperatures, at high pressure, to enable rapid formation of HP minerals before pressure release. Second,the HP minerals must be rapidly cooled to below a critical temperature, before the pressure returns to ambient conditions, to avoid retrograde transformation to their low-pressure polymorphs. These two constraints require the sample
to contain large temperature heterogeneities, e.g. melt veins in a cooler groundmass, during shock. In this study, we
calculated shock temperatures and possible P–T paths of chondritic and differentiated mafic–ultramafic rocks forvarious shock pressures. These P–T conditions and paths, combined with observations from shocked meteorites, areused to constrain shock conditions and P–T–t histories of HP-mineral bearing samples. The need for rapid thermal
quench of HP phases requires a relatively low bulk-shock temperature and therefore moderate shock pressures below~ 30 GPa, which matches the stabilities of these HP minerals. The low-temperature moderate-pressure host rock
generally shows moderate shock-deformation features consistent with S4 and, less commonly, S5 shock stages. Shockpressures in excess of 50 GPa in meteorites result in melt breccias with high overall post-shock temperatures thatanneal out HP-mineral signatures. The presence of ringwoodite, which is commonly considered an indicator of the S6
shock stage, is inconsistent with pressures in excess of 30 GPa and does not represent shock conditions different fromS4 shock conditions. Indeed, ringwoodite and coexisting HP minerals should be considered as robust evidence formoderate shock pressures (S4) rather than extreme shock (S6) near whole-rock melting.
Keywords: Chondrite, Martian meteorite, Shock metamorphism, High-pressure minerals*Correspondence: jinping@caltech.edu
1Division of Geological and Planetary Sciences, California Instituteof Technology, Pasadena, CA 91125, USA
Full list of author information is available at the end of the article

1 Introduction
The study of shock metamorphism in natural samples
is motivated by its usefulness for constraining planetary and terrestrial impact conditions, including thevelocity, size and material of the impacting objects.
High-pressure (HP) minerals have been recognized
as unambiguous evidence of shock metamorphism inmeteorites and terrestrial impact structures since the© The Author(s) 2021. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
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Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

1960s (e.g. Chao et al. 1960, 1962; Binns 1967; Binns
et al. 1969). In ordinary chondrites and shergottites,
for instance, the presence of HP minerals, combinedwith mineral recrystallization and the results of shock
recovery experiments, was proposed as evidence for avery high shock stage (S6) near the whole-rock melting regime (e.g. Stöffler et al. 1986, 1991), which has
been widely used by meteoriticists thereafter. In thispaper, we will be discussing the controversy caused bythis classification. Ringwoodite, the polymorph of olivine with spinel structure, is quite commonly found in
the melt veins of highly shocked ordinary chondrites
and Martian meteorites and occurs as deep blue grainsthat provide easily recognized evidence of high shock(Stöffler et al. 1991). A variety of HP minerals, including wadsleyite, majorite, akimotoite, bridgmanite, lingunite, etc. have provided natural examples of minerals
thought to be present deep in Earth’s mantle (see Gillet
and El Goresy 2013; Tomioka and Miyahara 2017 andreferences therein). Ahmed El Goresy has made a large
contribution to the field of shock metamorphism andHP minerals through detailed petrographic and microanalytical studies of shocked samples and the discovery of new HP minerals (e.g. El Goresy and Chao 1976;
Chen et al. 1996; El Goresy et al. 2004; Miyahara et al.
2019). Throughout his career, Professor El Goresy has
been a strong proponent of using HP minerals to estimate shock pressures.
Although HP minerals provide clear evidence of highshock pressures in meteorites, their usefulness for constraining shock pressures is still debated because thepressures inferred from HP-mineral stabilities are far lessthan whole-rock melting and those inferred from deformational shock features and shock-recovery experiments(e.g., Stöffler et al. 1991; Sharp and De Carli 2006; Gillet
and El Goresy 2013; Fritz et al. 2017).
HP minerals in shocked meteorites form by the crystallization of shock melt and by solid-state transformation of mineral fragments entrained in or in contact
with shock melt (e.g. Binns et al. 1969; Chen et al 1996).
Chen et al. (1996) were the first to recognize that themelt-vein crystallization assemblage, in the L6 chondrite Sixiangkou, matched the liquidus assemblage inthe high-pressure melting experiments on Allende (Ageeet al., 1995) and peridotite (Zhang and Herzberg, 1994).
They concluded that the melt veins in Sixiangkou crystallized at 20–24 GPa and that this represented the equilibrium shock pressure for the sample. Since then, many
have used shock-melt crystallization assemblages to constrain shock pressure or pressure during decompression(e.g. Sharp et al. 1997; Langenhorst and Poirier 2000;
Beck et al. 2004; Ohtani et al. 2004; Xie and Sharp 2004;
Xie et al 2006a, 2006b; Chen and Xie 2008; Fritz andPage 2 of 22

Greshake 2009; Miyahara et al. 2011a; Walton et al. 2014;
Hu and Sharp 2017; Pang et al. 2016; Li and Hsu 2018
and many more), but this approach has not been universally adopted because of questions regarding the use of
equilibrium phase diagrams to interpret shock pressure.
Although supercooling and metastable crystallization
can occur within the shock melt (Hu and Sharp 2017), thecrystallization assemblages in most chondritic samples
are consistent with the liquidus assemblages produced
in static experiments (Zhang and Herzberg 1994; Ageeet al. 1995; Herzberg and Zhang 1996; Trønnes and Frost
2002). If the shock melt remains in a liquid state afterpressure release, it crystallizes a low-pressure assemblage(Xie et al. 2006b), such as olivine, orthopyroxene plus plagioclase, mostly observed in for example martian meteorites, which generally show evidence of shock pulses(~ 10 ms) that are 1–2 orders of magnitude shorter thanthose inferred for shocked L chondrites (Langenhorst
and Poirier 2000; Ohtani et al. 2004; Beck et al. 2005;
Walton et al. 2014; Sharp et al. 2015;). Solid-state transformations of olivine and pyroxene only occur in association with shock melt and are strongly dependent on large
temperature heterogeneities. Because of the short duration of shock-induced high-pressure pulses (milliseconds
to seconds) in hypervelocity impacts, particularly for
Martian meteorites (e.g. Beck et al. 2005; Walton et al.
2014; Sharp et al. 2015, 2019), the reconstructive phasetransformations of the mineral constituents to their HP
polymorphs must occur very quickly. Transformation
temperatures close to melting are required to achieve thenecessary reaction rates during the shock pulse (Xie andSharp 2007). This explains the nearly exclusive association of HP minerals with shock-melt veins and pockets inmeteorites.
Many other features are recognized as indicators ofshock in meteorites, including planar fractures, planar
deformation features, mosaicism, staining and recrystallization of olivine (e.g. Bauer 1979; Dodd and Jarosewich
1979; Stöffler et al. 1991; Takenouchi et al. 2019) plus
blackening of silicates by disseminated metal or sulfide(e.g. Heymann 1967; Rubin, 1992; Moreau et al. 2017),
maskelynite and feldspathic normal glass (e.g. Milton
and DeCarli 1963; Stöffler 1984; Ferrière and Brandstätter 2015) and pervasive shock melt (e.g. Fredriksson et al.
1963; Dodd and Jarosewich 1979). The shock classification systems of Stöffler et al. (1991, 2018) use progressive shock effects to classify shocked meteorites as shockstages S1 (unshocked, < 5 GPa) through S6 (very highly
shocked, < 75 GPa; the pressure thresholds vary withrock type) and whole-rock melting. The shock pressuresneeded to produce these shock effects are calibrated
against shock effects observed from shock-recovery
experiments (e.g. Milton and DeCarli 1963; Stöffler 1972;

Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

Kieffer et al. 1976; Bauer 1979; Jeanloz 1980; Ostertag
1983; Stöffler and Langenhorst 1994).
A variety of shock features, including shock melt, commonly co-exist in the so-called impact melt breccias andstrongly shocked martian meteorites, which are rich inquenched impact melt and contain lithic and mineralclasts of the impacted rock. However, HP minerals are
consistently absent in these melt-rich breccia and achondrite samples. This suggests that either the pressures inthese samples were not sufficiently high to generate HP
minerals, or the post-shock temperature was too highfor the preservation of metastable high-pressure minerals (Stöffler 1974). The former is unlikely because theimpact breccias (Hu 2016) and strongly shocked shergottites (Walton and Herd 2007) generally contain morepervasive mosaicism, extensive feldspar glass, silicate
blackening and more melting than HP-mineral bearing
S6 samples. It is more likely that the shock features andhigh-pressure minerals are annealed by high post-shock
temperatures during slow cooling.
Unlike olivine recrystallization that can be unambiguously recovered from experimental shock at high pressure (e.g. Bauer 1979), HP minerals were first proposed
and used as indicator of S6 extreme shock (Stöffler et al.
1991) because of the excessive difficulty in forming themby shock-synthesis (Stöffler 1974; Syono et al. 1981;
Tschauner et al. 2009). More recently, HP minerals were
excluded from a revised shock-stage scheme (Stöffler
et al. 2018) because their complex formation conditions
require specific P–T–t paths instead of just certain shockpressure (Fritz et al. 2017). This leaves out an importantsource of information about shock conditions. If shockpressure alone cannot describe the formation of HP minerals, it is worth reviewing exactly what shock conditions they can constrain in a multi-parameter space. Highpressures and temperatures in the shock state are necessary, but not fully sufficient, to produce observable HP
minerals in shocked meteorites. Cooling of the meteorite before or during pressure release is equally important
for the preservation of HP phases (e.g. Walton 2013; Hu
and Sharp 2017; Sharp et al. 2019; Hu et al. 2020). The HP
phases formed during the initial shock pulse are metastable at low pressure and they will be transformed backto low-density phases if the temperature is sufficiently
high for retrograde back transformation after pressure
release. HP minerals are only preserved if the sample
cools through a critical temperature before the pressure
release makes them metastable. That critical temperature is dependent on the thermodynamic stability of themineral and on the reaction rates for back transformation(Reynard et al. 1996; Kimura et al. 2003; Hu and Sharp2017), which sets an upper bound for the post-shock
temperature of the sample. In summary, the HP phases

Page 3 of 22

observed in meteorites require a pressure–temperature–time (P–T–t) path that involves very high shock
temperatures to drive high-pressure reactions, combinedwith rapid thermal quench during decompression to prevent back-transformation reactions at low-pressure. An
understanding of P–T–t histories of HP-mineral bearing samples provide an opportunity to further constrain
shock conditions in highly shocked meteorites if shock
temperatures can be accurately estimated.
Overall, natural shock features, particularly HP minerals, in meteorites result from shock-induced high pressures plus complex and heterogeneous thermal histories.
In this study, we calculated the bulk shock temperaturesand release paths of common rocks and meteorites anduse these to re-evaluate the shock conditions and P–T–t
paths in highly shocked chondrites and achondrites in
the context of progressive shock-stage classification. We
are mostly focused on the temperatures of shock metamorphism, the formation condition of HP minerals andthe P–T–t paths necessary for their preservation. For
reviews on shock physics of geological materials, impact
processes, HP mineral physics, shock stage classifications
and summary of HP minerals in meteorites, the readers
are referred to Melosh (1989), Stöffler et al. (1991), Sharp
and DeCarli (2006), Gillet and El Goresy (2013), Asimow(2015), Fritz et al. (2017), Tomioka and Miyahara (2017)
and Stöffler et al. (2018).

2 Calculation of shock temperatures
Since the 1950s, shock experiments have been used toobtain high-pressure equation of state (EOS) data forvarious materials. Commonly, those experiments measure the velocity histories of the sample material in theshock state (e.g. Goranson et al. 1955; Walsh and Christian 1955; Jones et al. 1966). The corresponding pressure and density of the shocked material can be precisely
calculated with the Rankine–Hugoniot relations for a
simple one-wave shock. However, the measurement ofshock temperatures is more complicated and the available techniques only work for limited types of samples(e.g. Asimow 2018 and references therein). Moreover, thecalculation of shock temperatures requires high orderequations of state with considerable assumptions andapproximations. In order to discuss the temperatureeffects in naturally shocked meteorites, we introduce
two methods for calculating shock temperatures. The
first method employs the Birch-Murnaghan and MieGrüneisen equation of state. The second method uses anintegral approximation along the Hugoniot of the sample.
2.1 Shock temperature in EOS

Assuming the pre-shock velocity of the sample is zero,from conservation of mass, momentum and energy inHu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

Page 4 of 22

shock state, the Rankine–Hugoniot equations can be
written as:ρo Us = ρ Us − up
(1)

(P − Po ) = ρo Us up(2)

E − Eo = (P + Po )(V − Vo )/2(3)

where ρ is the density, Us is the shock-wave velocity, up
is the particle velocity, P is the pressure, V is the specificvolume (inverse density ρ) and E is the internal energy
per specific volume. The subscript 0 denotes the preshock state of the sample. The Rankine–Hugoniot equations work for time-invariant systems. In the case thatthe sample has an initial velocity of u, up in Eqs. (1)–(3)needs to be replaced by up − u. Correlations between thestate functions P, V, E, up and Us in the shock state arecalled the Hugoniot curve of the material, or simply the
Hugoniot.
For a plastic wave propagating in a single-phasesolid material, Us and up yield an empirical linear
relationship:

Us = Co + s up

(4)where Co is the zero-pressure sound speed of the material and s is a dimensionless factor. This form of Hugoniot
is convenient because Co and s are obtained by fitting thevelocity data measured in shock experiments (Table 1).
For demonstrating the shock behavior of material, it ismore straightforward to show the pressure–volume correlation (Fig. 1) calculated by Eq. (1), (2) and (4).
If the sample undergoes a phase transformation in
the shock state, separate Hugoniot curves for the lowdensity and high-density phases are needed to describe
the shock behavior. Porosity also complicates the shock
behavior and Hugoniot of a material. In the low-pressure regime, pores are collapsed by an infinitely weak
shock, leading to a gentle slope on the P–V Hugoniot(Fig. 1). In practice, it is not uncommon for polymineralic porous rocks to have nearly linear Us-up Hugoniot (Table 1) in a wide range of pressures above the
Hugoniot elastic limit (e.g. Ahrens and Gregson 1964;
Anderson and Kanamori 1968; Marsh 1980; Anderson
and Ahrens 1998; Sekine et al. 2008).
Although shock wave propagation can be considered as adiabatic at large spatial scales, the discontinuity between pre-shock and shock states increases
the entropy of the system. Derivation of the Rankine–
Hugoniot equations show that the entropy increase
along the Hugoniot is of third order (McQueen 1989).
In other words, the first- and second-order pressure/volume derivative on the isentrope and Hugoniot are

the same at zero pressure. Therefore, at low pressure,the Hugoniot of a non-porous material and its isentrope are close to each other (Jeanloz 1989; McQueen
1989). To obtain the temperature on the Hugoniot, it
is useful to first calculate the temperature on the isentrope and make a correction towards the Hugoniot.
To demonstrate the pressure–volume correlation ofisentropic compression, the third order Birch-Murnaghan
finite-strain equation of state is widely used (Jeanloz 1989)
in its isentropic form:

 

5
3 ′
2
K −4 f1+
PS = 3KoS f 1 + 2f
(5)
2 oS

PS is the pressure on the principle isentrope, KoS is thezero-pressure isentropic bulk modulus, KoS′ is the pressure derivative of KoS and  2
Vo 3
1
−1
f =
(6)
2
V

For porous samples, the Vo and KoS need to be the zeropressure specific volume and bulk modulus of the material in the condensed state. With the limit of zero strain(Ruoff 1967), KoS and K’oS are obtain by

KoS = Co2 ρo′KoS = 4s−1

(7)
(8)

although with high finite strain, the effect of KoS″
becomes significant and causes the KoS′ to deviate from4s − 1 by 20% (Jeanloz 1989).
Since our goal is to determine the temperature, thetotal differential of entropy is written as a function of V
and T and combined with the Maxwell’s relation:




∂P
∂S
dT +
dV
dS(V , T ) =
(9)
∂T V
∂T V
On the isentrope, dS = 0 leads to ∂S
dT = −dV
∂P V

(10)

Temperature and volume are two independent variables in Eq. (10). However, the solution for the differential
dT and dV is difficult because pressure and entropy are
two parameter (V, T) variables.
To solve the differential equation, we use the wellknown Grüneisen parameter (γ). It is a parameter todescribe the thermal pressure at constant volume, by its
definition:

Hu and Sharp ﻿Progress in Earth and Planetary Science

γ =V



∂P
∂E



(2022) 9:6

Page 5 of 22

(11)

V

The Grüneisen parameter has the advantage of being
well-approximated as a function of volume only and canbe used to relate other thermodynamic parameters, e.g.
S and P in Eq. (10). The γ − V correlation is commonly
expressed as: qVγ (V ) = γo(12)Vo
where γo is the initial Grüneisen parameter at volume Vo
and q is an exponential factor. Applying the first law ofthermodynamics, dE = TdS − PdV, to Eq. (11), we get
 
∂P
γT
=
(13)
∂S V
V
The combination of Eqs. (10), (12) and (13) demonstrates that the temperature (TS) on the isentrope at agiven V is solved as

TS = To e

γo
q



1−



V
Vo

q 

(14)

and To is the starting temperature.
Knowing the isentropic pressure (PS) and temperature (TS) at a certain volume V, correction to the
Hugoniot pressure (PH) and temperature (TH) at thisgiven volume V can be done by solving the Grüneisen
Eq. (11). We get:
�EV (V ) = EH (V ) − ES (V ) =V(PH (V ) − PS (V ))γ (V )(15)
where ΔEv is the energy difference (Fig. 1) between thesystem energy on the Hugoniot (EH) and isentrope (ES).

Fig. 1 Pressure–volume Hugoniots of L chondrite from mineralmixing models (Schmitt et al. 1994; Xie et al. 2006b) and shock
experiment (Anderson and Ahrens 1998) along with a calculateddense isentrope. ΔEV is the internal energy difference between theisentropic and Hugoniot state of the same specific volume

This is referred to as the Mie-Grüneisen equation ofstate. Thus, the temperature on the Hugoniot is given by

TH = TS + �TV = TS +

�EV (V )CV(16)where ΔTv is the conceptional temperature difference
between the Hugoniot and isentrope at a given volume
and CV is the isochoric heat capacity defined by the temperature partial derivative of internal energy at constantvolume.

Table 1 Hugoniot data for basalt, gabbro and chondrites
Kinosaki ­basalt1

Murchison2 CM2Bruderheim2 L6

San
Macros­Gabbro3

ρo, kg/m3

2700

2244

3337

2941

porosity, vol%

4–7%

23%

6–8%

–

KoS, GPa

50

24

32

32

KoS’

4.2

4.9

5.5

4.6

γo

1.39

1

2.00

1.37

q

–

0

–

0.94

Co, km/s

3.5

1.87

3.11

3.3

s

1.3

1.48

1.62

1.41

ρo: initial bulk density, KoS: zero-pressure isentropic bulk modulus, KoS’: pressure derivative of the bulk modulus, γo and q: Grüneisen parameter at initial crystal density
and the exponent factor, Co and s: bulk sound speed and the dimensionless parameter in the US-uP Hugoniot
1

Sekine et al. (2008); 2Anderson and Ahrens (1998); 3Boslough (1984)

Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

Page 6 of 22

As mentioned, TS and TH are close for condensed
material at below 10 GPa (Fig. 2). The effect of porosity
in the sample will be discussed in Sect. 2.3.
2.2 Shock temperature by integral approximation
along the Hugoniot

The application of the Birch-Murnaghan EOS requires
the ability to fit the shock velocity data to the condensed isentropic bulk modulus. For a polymineralicrock of heterogeneous textures, this fitting can haveconsiderable uncertainty if the experimental data is too
limited (Fig. 1). An alternative method for estimating
the shock temperature is to integrate the infinitesimaltemperature change along the Hugoniot to the pressure of interest (McQueen 1989). The advantage of thismethod is that it makes use of the measured Hugoniot
curve and does not required fitting of the data to linear
US-up correlations.
From the Hugoniot energy Eq. (3) and the first lawof thermodynamics, the total differential of internalenergy on the Hugoniot isdE =1[(Vo − V )dP − PdV ] = T dS − PdV
2(17)

Using the isochoric heat capacity and Grüneisenparameter to replace the partial derivatives in Eq. (9),the differential of entropy is written asdS(V , T ) =

CV
γ CV
dT +
dV
T
V

(18)

Substituting the dS in Eq. (17), we get

dT = −Tγ
1dV +[(Vo − V )dP + PdV ]
V
2CV

(19)

Fig. 2 The shock temperature versus pressure for porous and dense
L chondrite, calculated by the equation of state (EOS) and integral
approximation, using the Hugoniot data in Table 1

two methods generally agree well (Fig. 2) and both givereasonable estimates of shock temperature.
2.3 Shock temperature and porosity of chondritic
and differentiated mafic rocks

The Hugoniot of common rocks and meteorites used
for temperature calculation is summarized in Table 1.
The Us-up data of the Bruderheim L6 chondrite does
not perfectly fit a single line (Anderson and Ahrens
1998). At 25–65 GPa, the Hugoniot seems to be a mixture of low-density and high-density phases. Also,at ~ 13 GPa the shock wave shows a two-wave structure, commonly interpreted as a low velocity shock
wave overlapping with an elastic precursor. However,the velocity is extraordinarily high for an elastic precursor of silicate rock and, alternatively, may result
from a sluggish phase transformation (Fig. 1; Anderson and Ahrens 1998). The data were not enough to fit
separate Hugoniots for the phase transitions. Nevertheless, the low- and high-density Hugoniots are notdrastically different, probably because either one is acombination of low- and high-density polymorphs of

A general solution for T in this differential equation
is mathematically difficult. However, in practice, we canreplace the infinitesimal dV and dP by a small ΔV andΔP on the Hugoniot (McQueen 1989). Assuming i andi − 1 are two closely adjacent points on the Hugoniot,
Eq. (19) is approximated by





γ V −VTi−1 1 − ( i2V i−1 ) + 2C1 Vo − V (Pi − Pi−1 ) + P(Vi − Vi−1 )VTi =γ V −V
1 + ( i2V i−1 )where the overbar denotes the mathematical average ofV, P, γ and CV between points i and i − 1.
Unlike the B–M equation of state, the integral
approximation is not a function of the final state on the
Hugoniot. Every shock temperature on the Hugoniot is
important for calculating the temperature of the next
points. For condensed material, the results from the(20)

rock-forming minerals. The simplified fitted Hugoniot
generally matches the data points except at low-pressure < 13 GPa (Fig. 1). Given that, and the very limited
Hugoniot data for L chondrites, we still use the linear
Hugoniot that reasonably fits the HP data points for our
calculation. Hugoniot of terrestrial mafic rocks, such as
dunite, peridotite and gabbro do not seem to resemble

Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

the chondrite data in phase transition (Marsh 1980).
Mineral-mixing models of chondrite can also be used
to fit and interpret the Hugoniot data. A mineral-mixing model by Schmitt et al. (1994) shows a Hugoniot
above the measurements. In contrast, the mixing model
for Tenham L6 chondrite used by Xie et al. (2006b) and
Sharp and DeCarli (2006) agrees with the fitted Hugoniot (Fig. 1). This suggests that the fitted Hugoniot andthe model by Xie et al. (2006b) are likely suitable fortemperature calculations.
The Grüneisen parameter of rocks is an essential partin temperature calculations. The zero-pressure γo canbe calculated by

γ =

V αKT
CV

(21)

where α is thermal expansion and KT is the isothermalbulk modulus (Asimow 2015). The high-pressure γ andexponential factor q can be determined by shocking
material of different initial conditions to the same volume(e.g. Luo et al. 2002). If experimental data is not available,
it is reasonable to assume q = 1, thus ργ = ρoγo, for solids(Asimow 2012).
The isochoric heat capacity CV is also needed for the
calculations (Eq. 16 and 20). In experiments, the isobaric
heat capacity CP is commonly measured. They have thecorrelation ofCP − CV = TV α 2 KT

(22)

For typical solids, the difference between CV and CP areon the order of 10 J/kg K (Navrotsky 1994). The pressure
effect on heat capacity has not been fully investigated.
Data on olivine suggest that at the pressure of interest theheat capacity decreases on the order of 10 J/kg K compared to that of zero pressure. The measured CP values at
low pressure are in the range of 800–1400 J/kg K at 300–
3000 K for common rocks and silicate minerals (Watanabe 1982; Richet and Fiquet 1991; Waples and Waples
2004). We use these values for our calculations although
we understand that they are somewhat different from theneeded high-pressure CV.
Natural samples, other than deep intrusive rocks, haveconsiderable initial porosity (Table 1). A porous Hugoniot can be calculated from the Hugoniot of a condensed
material using the following equation (e.gAhrens andO’Keefe 1972; Tyburczy et al. 2001):
1 − γ2 VVod − 1
Pp = Pd(23)
V
1 + γ2 Vop − 1

Page 7 of 22

where Pp and Pd are the shock pressure on porous anddense Hugoniot at volume V. Vod and Vop are the initial
volume of dense and porous sample. We also use thisequation and the Hugoniot of naturally porous samples
to estimate the zero-pressure bulk modulus of the samematerial in the condensed state (Table 1).
Figure 1 shows the Hugoniot of a naturally porous L
chondrite plus the calculated (dense) reference isentrope.
The dense isentrope matches with the pore collapse volume predicted by the mineral-mixing model from Xie
et al. (2006b). The corresponding shock temperatures areincluded in Fig. 2, calculated by the two methods introduced in Sects. 2.1 and 2.2. The temperature estimates ofdense material from the two methods are close to eachother. The small difference likely suggests the uncertainty
of bulk modulus calculated from the simplified Hugoniot.
In contrast, the EOS calculation for a porous Hugoniot
shows a temperature jump in the pore collapse regime.
This is because the pore collapse by the infinite weak
shock deposits large amount of energy to the system andincreases the shock temperature. In the Mie-Grüneisen
EOS, this process is quantitatively demonstrated by thefact that the Hugoniot pressure and isentropic pressure(infinite weak) at a given volume are significantly different for porous material (Fig. 1). Thus, the ΔEV and TV
in Eqs. (15) and (16) are greater than that of condensed
material. The integral approximation does not show sucha temperature jump because this method relies on the
shape of the Hugoniot curve. The simplified Hugoniotof L chondrite (Table 1) does not include a pore-collapse
transition, which is thus neglected in the temperature
calculation. In the integral method, each temperature
is dependent on the temperature of the adjacent point.
Therefore, the underestimation of shock temperaturein the pore collapse regime is propagated into the highpressure temperatures. This explains why the integral
temperatures are below the EOS temperatures for the
porous material. Because there is not enough data toconstrain the shock behavior of a chondrite through thepore-collapse and the elastic regimes, neither of the two
methods is perfectly accurate. Nevertheless, both methods provide reasonably consistent shock temperatureapproximations of porous materials.
Shock temperatures for common rocks calculatedby the integral method, and all starting with 298 K, areshown in Fig. 3. The Murchison CM2 chondrite has adistinctly higher porosity (Table 1) and therefore higher
shock temperature. The P–T Hugoniot crosses the chondrite liquidus at 40 GPa. For other rock types, whole-rockmelting does not occur below 50 GPa, but melting may
still occur locally, forming melt veins and pockets. The
shock temperatures roughly increase in the sequence of L
chondrite, dense gabbro, basalt and porous gabbro. ThisHu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

sequence agrees with increasing porosity and compressibility of these rocks. These results are consistent withresults using similar calculation methods by Gillet and El
Goresy (2013).

3 Temperatures on release adiabat
Post-shock temperature can have a significant effect onannealing high-pressure signatures in shocked meteorites (Rubin 2002, 2004; Kimura et al. 2003; Walton 2013;
Hu and Sharp 2017; Fukimoto et al. 2020). Annealing
used in the context of shock metamorphism refers to
the retrograde metamorphism of shock features resulting from post-shock heating (Bauer 1979; Jeanloz 1980).
The post-shock temperature for a given shock pressure
can be calculated from the amount of waste heat added
to the system, defined by the work from shock loading along Rayleigh-line minus the work from decompression (Schmitt et al. 1994; Sharp and DeCarli 2006).
However, the final post-shock temperature is not the
only important factor in the retrograde metamorphismof shock features. On the release path, a high-T moderate-P condition may also modify the signature of peak
shock pressure (Hu and Sharp 2017). In that case, it isuseful to calculate the Prelease-Trelease path of continuous
decompression.
As the release wave propagates, decompression occurs
as a continuous process in a time-variant system (Asimow 2015). Several techniques are used to measure therelease states in shock experiments, e.g. using a series of
release buffers that have lower shock impedance than thesample to get a series of up-P points on the release path
(Anderson and Ahrens 1998; Luo et al. 2003). The volume
on the release path can be calculated with the Riemannintegral
upr

dup

dup

Vr = VH + ∫upH dP

Page 8 of 22

equation of release (Eq. 24). The interpretation can bethat melting occurs more extensively in the sample than
expected even at below 50 GPa (Fig. 3). In this scenario,
for the unmelted portion, the final release volume should
only be slightly greater than the initial condensed volume. With the limited data, it is difficult to build a complete release path. Instead, we use Eq. (25) to construct a
V–T path of release for solid and melt. At the temperature of 500–800 K for the solid material, the expansionof major chondritic minerals is only on the order of 1­ 0–3(Suzuki 1975) and we can use the initial volume as reference for release volume without making a noticeable
error in the calculation of post-shock temperatures. The
P–T release path is constructed by comparing the release
path to the Hugoniot using the Mie-Grüneisen equation:

Pr (V ) = PH (V ) +(Tr − TH )CV γ
V(26)

where H and r denotes the points on the Hugoniot andrelease isentrope at volume V. The P–T release path ofan L chondrite and a basalt from 25, 40 and 50 GPa areshown in Fig. 5 and 6. The slope of release is gentle at the
beginning and becomes steeper with decreasing pressure.
The complete temperature drop upon release is 300 (for25 GPa)–900 (for 50 GPa) Kelvin in the bulk rock, making the post-shock temperature 500–900 K.
In meteorites, many shock features, particularly thehigh-pressure minerals, are associated with local shock(24)

where H and r denotes the points on the Hugoniot andrelease path. The release adiabat is approximately isentropic so that the temperature upon decompression canbe calculated with a modification of Eq. (14):Tr = TH eγ (VH )
q



1−



V
VH

q 

(25)

This equation is the same as (14) except the starting
point is the shock state.
Anderson and Ahrens (1998) obtained three points on
the release state of the Bruderheim L chondrite (Fig. 4).
The data show that the sample has a higher particle velocity on the release path than that on the reflected Hugoniot. This indicates that the release volume is greaterthan the volume on the Hugoniot, based on the RiemannFig. 3 The shock temperatures of common rocks and meteorites,
calculated by the Hugoniot integral approximation, using theparameters in Table 1. SMG is San Marcos gabbro. The representative
solidus and liquidus of ultramafic/mafic rocks are for chondrites (OC)from Agee et al. (1995) and basalt (B) from Hirose et al. (1999) andAndrault et al. (2011)

Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

melt (e.g. Sharp and DeCarli 2006; Tomioka and Miyahara 2017 and references therein). The P–T release pathof shock melt is essential for understanding the formation of shock features. However, local melting represents a deviation from the bulk Hugoniot that can bevery large at modest shock pressures. In this case, it is
not possible to constrain the temperature and density ofshock melt vein at < 50 GPa using the bulk Hugoniot of
the rock. Qualitatively, we assume that the shock meltreaches pressure equilibrium with the bulk rock and has
a shock temperature slightly higher than the liquidus forthe sample (Figs. 5 and 6). In practice, the temperature ofshock melt can be either significantly above the liquidus,
resulting in further melting of adjacent host rock (Sharp
et al. 2015), or it can be even slightly below the liquidus,
because of metastable melting of low-pressure phases athigh pressure.
To calculate the P–T release path of shock melt,
we use the Grüneisen parameter and density of melt
(Table 2) for Eq. (25). We refer to the density model ofperidotite melt (Asimow 2018) to represent the chondritic melt. The data for basaltic melt are taken fromab initio simulations (Bajgain et al. 2015) as it cannot beeasily measured by shock experiments. The Grüneisenparameter of a liquid is very different from that of asolid and commonly has a negative exponential factor
q (Eq. 12; Asimow 2012). Parameters used for the calculations are listed in Table 2. The release paths from25, 40, and 50 GPa (Fig. 5 and 6) show a moderate temperature drop and are mostly more gentle than the lowpressure liquidus of chondrite and basalt, indicating nocrystallization upon adiabatic decompression in mostcases. Although the chondrite path of 25 GPa crosses
the liquidus between 4 and 13 GPa, crystallization is
likely to be limited during adiabatic decompression
because the release of latent heat of crystallization. For
the basalt, the release path sits well above the liquidus
through the pressure range (Fig. 6).

4 Temperature effects in shocked meteorites
Our calculations of shock and release temperatures allow
for the interpretation of shock effects in meteorites interms of pressure–temperature–time (P–T–t) histories. The shocked meteorites that we will discuss include
HP-mineral bearing (previously classified as S6 based on
Stöffler et al. 1991) samples, partially annealed “S6” samples and melt breccias (some also represent S6) that haveabundant shock melt but lack HP minerals. The P–T conditions of HP-mineral formation as well as P–T paths ofquench and annealing are illustrated in Fig. 7. This discussion will walk through the representative samples,

Page 9 of 22

Fig. 4 Measured Hugoniot and release state of the Bruderheim L6
chondrite in particle velocity–pressure space, after Anderson andAhrens 1998

their corresponding paths in Fig. 7 and how their pressures deviate from the actual shock stage S6.
4.1 HP‑mineral formation constraints on P–T

The majority of high-pressure minerals in shocked meteorites are formed by melt-crystallization and reconstructive phase transformations exclusively in melt regions
and are mostly used to identify the S6 shock stage, basedon Stöffler et al. 1991. The HP-mineral assemblage inmost of these samples consists of polymorphs of common silicate, oxide and phosphate minerals (Sharp and
DeCarli, 2006; Gillet and El Goresy 2013; Tomioka andMiyahara 2017 and references therein). Olivine clasts,
entrained in the shock melt, are commonly transformed
to ringwoodite or in some cases wadsleyite (Fig. 8a-b).
The transformation of enstatite to its HP polymorphs
majorite, akimotoite and bridgmanite are relatively lesscommonly reported. Shock melt in S6 L chondrites commonly crystallizes an assemblage that contains majoritic garnet solid solutions along with ringwoodite and/or magnesiowüstite (Fig. 8c) and in some cases bridgmanite or akimotoite. Such HP-mineral assemblages
provide three major constraints on the pressure–temperature conditions of shock, as outlined in the following
paragraphs.
First, the crystallization assemblages in these nominallyhighly shocked (S6) samples are consistent with crystallization at approximately 20–25 GPa, as indicated by static
high-pressure experiments on chondritic compositions
and the resulting liquidus phase-relations (Agee et al.
1995; Asahara et al. 2004). If a high-pressure crystallization assemblage in the sample is constant across themelt veins and melt pockets, it can be inferred that thepressure did not drop significantly during crystallization

Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

Page 10 of 22

Fig. 5 The adiabatic release paths of L chondrite bulk rock (cyan) and shock melt (orange) at 25, 40 and 50 GPa superimposed on the phase
diagram of Agee et al. (1995). The P–T Hugoniot of L chondrite is the same as Figs. 2 and 3. The calculation parameters are in Tables 1 and 2

and that the crystallization pressure is the equilibration
shock pressure (blue curve in Fig. 7; Sharp and De Carli
2006; Miyahara et al. 2011a; Hu and Sharp 2017). A constant or nearly constant crystallization pressure implies
that the sample was in the isobaric zone of the impact
where the separation between the shock wave and releasewave results in a period of nearly constant equilibrium
shock pressure. This isobaric zone is in contrast to areas
shocked by an attenuated wave, which experiences animmediate decay in shock pressure instead of an initially constant pressure (Asimow 2015; Fritz et al. 2017).
If the melt matrix includes mostly wadsleyite instead of
ringwoodite, such as in LL chondrite NWA 757 (Hu andSharp 2017), the crystallization pressure is inferred to
be lower than 20 GPa. Even though metastable melting
and crystallization may occur, the crystallization pressure

can still be constrained from the liquidus and sub-solidus
phase-assemblages in the melt matrix (Hu and Sharp2017). In contrast to the long pulse of up to seconds in
L chondrites, the relatively short pulse recorded in martian meteorites results in rapid pressure release before
complete crystallization of the shock melt, leading to
low-pressure crystallization assemblages and even vesiculated melt pockets (e.g. Walton et al. 2014; red melt
release path in Fig. 7). It is worth noting that the shortpulse does not necessarily imply small source craters on
Mars. Shock dwell-time models by Bowling et al. (2020)
indicate that dwell times of 10–100 ms are consistent
with craters of 14–104 km diameter for a typical impact
velocity of 13.1 km/s. Therefore, the 3–60 km potential
meteorite-launching young (< 10 Ma) craters on Mars
(Lagain et al. 2021) are consistent with relatively shortHu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

Page 11 of 22

Fig. 6 The adiabatic release path of bulk basalt (light purple) and basaltic melt (orange) superimposed on a phase diagram based on MORB from
Fei and Bertka (1999) and Hirose et al. (1999). The P–T Hugoniot is the same as Fig. 3. Parameters used for the calculations are listed in Table 1 and 2

Table 2 Parameters and temperatures of shock melt
P, GPaChondrite

Basalt

25–50

25–40

γo

0.81

0.356

q

− 1.41− 1.63

Source11

ρ, g/cm3

3.7–4.1

3.9–4.22, 3

ρo, g/cm3

2.8

2.5

2, 3

Tshock, K

2400–3300

2700–3200

Tps, K

1815–2143

2140–2394

γo and q: zero-pressure Grüneisen parameter and the exponent factor. ρ andρo: density at high and zero pressure. The shock temperature Tshock of melt
is assumed to be slightly above the liquidus. Post-shock temperature Tps iscalculated using the listed parameters
1

Asimow (2012); 2 Asimow (2015); 3 Bajgain et al. (2015)

pulse durations of tens of milliseconds. Nevertheless, themartian basalt NWA 8159 has a constant crystallizationassemblage in a relatively thick melt vein that suggests
a ~ 100 ms shock pulse (Sharp et al. 2019) and a somewhat larger source crater.
The large temperature difference between the superliquidus shock melt and relatively cool host rock drives
the rapid high-pressure quench of the shock melt by heat
conduction. The heat flux is proportional to temperature
gradient, based on Fourier’s law. The heat transfer is alsoenhanced by turbulence of the shock melt that can transport hotter material to the interface with the colder bulk
host (Adcock et al. 2017). The shock melt starts with asuper liquidus temperature at ~ 2400 K (red dot in Fig. 7).
The bulk shock-temperature of the host rock is only
500–700 K (cyan path) at 20–25 GPa, corresponding toHu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

Page 12 of 22

Fig. 7 The cooling paths of typical moderately to highly shocked meteorites. The adiabatic release paths (cyan, red and purple lines) are from
Fig. 5. HP and LP melt refer to shock melt that crystallizes before and after pressure release, respectively. The vertical line represents pure heattransfer from shock melt veins (SMV) to solid groundmass without synchronous pressure drop by decompression. The dark yellow line shows
the approximated temperature of triggering rapid HP transformation. The grey area represents the conditions for destruction of akimotoite (ilm),
ringwoodite (rwd) and garnet (gt) by amorphization or back transformation. The blue and green lines represent combinations of decompression
and synchronous heat-transfer, where blue corresponds to rapid thermal quench with crystallization at high pressure and green corresponds to agreater contribution for decompression, resulting in crystallization at slightly lower pressure and partial annealing in the grey area. The shown P–T
paths are technically for chondritic/ultramafic rocks but the concepts also work for mafic rocks. The top x-axis shows shock stage classification with
deformation features of chondrite (C) and mafic meteorite (M) such as basaltic shergottite (Stöffler et al. 1991, 2018)

Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

a temperature difference of up to 1900 K. This large temperature gradient between the shock melt and the surrounding host rock results in the rapid transfer of heat
into the surrounding host rock (blue P–T path in Fig. 7)
and rapid thermal quench of the shock melt (Langenhorst
and Poirier 2000; Xie et al. 2006b). If the shock pulse islonger than the time required to quench the melt, themelt will crystallize a high-pressure assemblage (Sharp
and De Carli 2006). If such a long shock pulse occurs forsamples within the isobaric shock zone, there would be
no significant decompression during thermal quench andthe resulting isobaric crystallization (blue path) would
result in a constant crystallization assemblage throughout the quenched melt. This is apparently the scenario
for many highly shocked L chondrites that have constantor nearly constant crystallization assemblages, implyingthat the P–T–t path and crystallization was dominated bythermal quench at high pressure and nearly isobaric (e.g.
Chen et al. 1996; Ohtani et al. 2004; Sharp et al. 2015).
Second, the chondrite fragments entrained in theshock melt are heated rapidly to enable reconstructive transformation to HP polymorphs during theshock pulse (Fig. 8a). Although the duration of the

Page 13 of 22

shock pulse varies depending on the size of the impacting body and the location of the sample relative to theimpact, the pulse duration is thought to be on the orderof 1 s for highly shocked L chondrites (Chen et al. 2004;
Ohtani et al. 2004; Beck et al. 2005; Sharp et al. 2015).
Transformation of olivine on such a time scale requires
a temperature in excess of 1600 K (Xie and Sharp 2007),
illustrated by the dark yellow line and upward arrow in
Fig. 7. The transformation of enstatite clasts to majorite, akimotoite or bridgmanite requires even higher
temperatures because of the slow kinetics of enstatitetransformations (Hogrefe et al. 1994; Lockridge 2015).
The upward dark yellow arrow in Fig. 7 illustrates the
rapid heating of an entrained mineral fragment to nearsolidus temperatures. Only material within the shock
melt or immediately adjacent to the melt reach sufficiently high temperatures for reconstructive transformation during the shock pulse (Fig. 8a–c). The muchrarer displacive transformation, such as zircon to reidite, can occur without extensive melting althoughhigher local temperature would still greatly enhance thereaction (Leroux et al. 1999; Xing et al. 2020).

Fig. 8 Variable shock features in L chondrites previously classified as shock stage S5-6. a RC 106 (L6) thin section with a thick shock melt vein.
The insert is a plane-polarized light (PPL) micrograph of the shock melt vein containing blue ringwoodite (rwd). b PPL micrograph of a shock meltvein in Mbale (L5-6). The clasts in the thin part of vein are wadsleyite (wads) aggregates and the thicker vein contains polycrystalline olivine. cbackscattered electron (BSE) image of a polycrystalline olivine (ol) aggregate with trace wadsleyite in contact with majorite garnet (gt) in quenched
Mbale shock melt. d Shock-darkened impact melt breccia Chico (L6). e BSE image of a barred olivine chondrule in melt breccia NWA 091 (L6). The
troilite (tr) veinlets are disseminated in the olivine. f BSE image of recrystallized sodic pigeonite (pg)-plagioclase (pl)-olivine (ol) aggregate in Chico.
The insert is another recrystallized olivine-plagioclase aggregate. Figures modified after Sharp et al. (2015), Hu (2016) and Hu and Sharp (2017)

Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

Thermal history is the key difference between HP
mineral regions (previously S6) and the bulk host rock.
Observable transformation of olivine or pyroxene cannot
occur in the host-rock away from the shock melt because
the bulk-shock temperature is only 600–800 K at 25 GPa(cyan path in Fig. 7). The relatively low temperature of
the host rock limits the shock metamorphic effects tomosaicism (S4–S5), planar deformation features (S4–S5) and maskelynite (S5) outside the shock melt (Stöffler et al. 1991). In the recently revised scheme (Stöffler
et al. 2018), these shock features are re-classified as S3-S5(Fig. 7). This is consistent with shock classification of
2280 samples by Bischoff et al. (2019), who found that
70% of their 53 chondrite samples that contain ringwoodite are classified as shock stage up to S4 in regions distantfrom the shock melt. Stöffler et al. (1991) proposed thatS6 shock features may result from local pressure–temperature excursions and therefore higher shock pressure
than the bulk rock. Although significant pressure heterogeneities are expected during compression, pressure variations “ring down” to the equilibrium shock pressure in avery short time relative to the duration of the shock pulse
and the formation of HP minerals (Sharp and De Carli
2006). Hydrocode simulations of mesoscale shock effects
in chondrites confirm that local pressure excursions
occur around high-impedance components (Moreau
et al. 2017, 2018) and that shock pressures equilibrate
in 10–20 ns. Although local temperature excursions are
required to generate S6 features, the HP minerals form
after the pressure heterogeneities have equilibrated.
Third, the HP minerals must cool below a critical temperature before pressure release in order to survive. At
ambient pressure, ringwoodite and wadsleyite break downat temperatures above 900 K and back-transform to olivine
within seconds at T > 1200 K (Hu and Sharp 2017). The
critical temperature for retrograde back transformation
depends on the stability of the mineral and the kinetics of
its back-transformation reaction. The critical temperaturefor garnet is ~ 1300 K, which makes garnet more resistant to back transformation than the olivine polymorphs(Thiéblot et al. 1998), and therefore a robust indicator ofhigh pressure in partially annealed samples. Pyroxene’s
other two polymorphs, akimotoite and bridgmanite, break
down at significantly lower temperatures of 700 and 400 K,
respectively (Ashida et al. 1988; Durben and Wolf, 1992)
and bridgmanite is generally transformed to glass during
or after decompression (Sharp et al, 1997). The grey area in
Fig. 7 represents a critical pressure–temperature region of
annealing for akimotoite through garnet. In highly shocked
samples that retain HP minerals, heat transfer from the meltzones to the relatively cool host rock is the major mechanism for quenching shock melt and for cooling HP minerals
sufficiently to survive under reduced pressure conditions.

Page 14 of 22

Therefore, the P–T–t paths for quench of HP minerals must
be steep (high dT/dP) in order to cool below the P–T conditions of back transformation (blue path in Fig. 7).
4.2 Partially annealed samples

The back-transformation of HP minerals in some classic
S6 samples implies shallow P–T paths (lower dT/dP) andinsufficient cooling during pressure release. For example, EETA 79001 (shergottite) and Mbale (L chondrite)
are highly shocked samples that contain fine-grained olivine aggregates, with remnants of ringwoodite or wadsleyite that are surrounded by radiating fractures frompost-shock volume expansion (Walton 2013; Hu andSharp 2017). These textures provide evidence of backtransformation of ringwoodite and wadsleyite and provide a mechanism for the origin of polycrystalline olivine
aggregates. Remnants of high-pressure phases in theshock melt indicate that melt started at high pressure, butfollowed shallower P–T paths (green path in Fig. 7) thatcrossed through conditions of ringwoodite back transformation. The clasts in the shock melt were heated to temperatures sufficiently high to enable rapid transformationof olivine to its HP polymorphs (Fig. 8b, c) but did notcool below the critical temperature for back transformation before low-pressure conditions were reached. In
these cases, rapid decompression occurred while heat
was being transferred from the shock melt to the surrounding host rock. In Mbale (L5-6), thinner shock veins
contain wadsleyite while thick veins contain back transformed olivine (Fig. 8 b and c), indicating that the slowercooling thick vein had a shallower P–T path than thethinner vein (Hu and Sharp 2017).
The P–T–t path followed by any particular part of thesample will depend on the abundance of shock melt andthe duration of the shock pulse. Graphically, the corresponding P–T paths (blue and green paths in Fig. 7) will
have variable dT/dP slopes, depending on the amountof shock melt that is cooling during the shock pulse andrelease. For example a small melt vein or a melt vein edge
would cool quickly by conduction and turbulence to thecooler host rock and would follow a relatively steep P–T
path (high pressure portion of blue and green lines inFig. 7). Similarly, a larger volume of melt contains higher
internal energy that takes longer to dissipate and would
therefore follow a relatively shallow P–T path (green
line in Fig. 7) that may traverse the back-transformation
regime of low-P and high-T (grey region). If the sample
quenched in the isobaric shock zone, the paths would
start with steep isobaric cooling, followed by shallowing of the P–T–t path. The extent of the isobaric cooling trend is inversely related to the amount of melt being
cooled by conduction, such that the smallest volume of
melt would produce a most stable isobaric segment forHu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

quenching before decompression. This is the case formany shocked L chondrites. In samples experiencing ashort isobaric shock pulse, the P–T–t would start witha steep slope and quickly turn to a shallower slope (blueand green lines in Fig. 7). Outside of the isobaric zone,
all P–T–t paths would be non-vertical decompression
paths with the slope determined by the competing effects
of decompression and thermal conduction to the cooler
host rock. In shergottites, such as EETA 79,001, the shock
pulse is inferred to be short and anisobaric, resulting
in a P–T path through the back-transformation regime(Walton 2013). The back-transformed Mbale L chondrite, which crystallized a relatively small volume of melt
at 14 – 18 GPa, is unlikely to have a lengthened coolingtime caused by a high bulk-shock temperature (Hu andSharp 2017). We therefore infer that Mbale experienced
a shorter shock pulse than most HP-mineral bearing L
chondrites (Hu and Sharp 2017). Furthermore, morepervasive melting and higher shock and post-shock temperature, as in impact melt breccias, would lead to more
extensive back transformation and post-shock annealing.
Larger shock-melt volumes produce shallow P–T–t
paths that are dominated by decompression. The red line
in Fig. 7 illustrates the adiabatic cooling path for a shock
melt at 25 GPa that decompresses without dissipation ofheat to the host rock. This is not the case for HP-mineralbearing “S6” L chondrites, which have small amounts ofshock melt and steep P–T–t paths dominated by heattransfer. However, for small high-velocity impacts onplanets, the shock pulse could be much shorter, resulting in decompression dominating the P–T–t quench path
(Beck et al. 2005). Several shergottites have melt crystallization assemblages with olivine and pyroxene or glass
coexisting with HP products of olivine transformation
at ~ 25 GPa (Miyahara et al. 2011b, 2016; Walton 2013;
Walton et al. 2014). In Tissint (olivine-phyric shergottite) for instance, only the micron thin veins crystallize
and preserve stishovite and ringwoodite whereas the mm
thick veins consist of olivine and clinopyroxene (Walton et al. 2014). The low-pressure assemblage of their
quenched shock melts indicates that the corresponding
P–T–t paths of the melt are closer to the adiabatic release
path of melt (red line in Fig. 7).
4.3 Impact melt breccias

Impact melt breccias have abundant shock features andevidence of high temperature, but they lack direct evidence of high pressure phase transitions. Sample suchas NWA 091 and Chico (Bogard et al. 1995; Yolcubal
et al. 1997; Hu 2016), have shock features that include
recrystallization of olivine, blackening of silicates by disseminated metal-sulfide, feldspathic normal glass andpervasive impact melt (Fig. 8d–f ). These samples were

Page 15 of 22classified as highly shocked S6 samples by Stöffler et al.
(1991), although they lack typical S6 shock veins andHP minerals. Bischoff et al (2019) classified Chico as
an L6 breccia that contains fragments of melt breccia
and we have studied a portion of melt breccia. Unlike ahand sample with pressure equilibrium in nanoseconds,
the 105 kg Chico can have portions experiencing different levels of shock. The direct evidence for strong shockin these samples includes common recrystallization ofrock-forming minerals and disseminated troilite thatindicates mobile sulfide melt throughout the sample (Hu2016). The troilite in these samples occurs as strings oftiny inclusions in olivine and pyroxene rather than veins(Fig. 8e). We interpret this texture to indicate that thesilicate minerals were fractured during shock and thefractures were filled by sulfide melt. During subsequentannealing, the fractures healed, leaving strings of sulfide
inclusions where fractures had been (Rubin 1992).
The severity of shock recorded in impact melt breccias
is not well defined, but pervasive melting implies significantly higher shock pressures than those of S6-bearing
chondrites. Moreau et al. (2017, 2018) used mesoscalehydrocode simulations to investigate the shock behaviorof olivine-troilite-iron mixtures that resembles ordinarychondrites. The simulations at bulk shock pressures of
40–50 GPa result in considerable post-shock melting oftroilite and the onset of local olivine melting. Moreau
et al. (2017) inferred that > 35 GPa is the shock pressure
range for pervasive silicate blackening. This is consistent
with our interpretation that extensive darkening requires
higher pressures than those of HP-mineral bearing “S6”samples (< 30 GPa). In blackened chondrite samples
such as Chico, the sulfide particles and veinlets are muchfiner-grained than the common crack-filling sulfide inother chondrites and the sulfide particles occur in strings
without co-existing fractures. We suggest that somechondrite melt breccias, including Chico and NWA 091,
underwent high degrees of recrystallization and annealing at high post-shock temperatures (Hu 2016).
Shock pressures of 40–50 GPa would result in abundant shock melt, shallow P–T–t paths and post-shock
temperatures too high for preservation of HP minerals(Fig. 7). Chico contains polycrystalline olivine-pyroxeneplagioclase aggregates (Fig. 8e, f ) that appear to result
from a combination of back-transformation, recrystallization and low-pressure melt crystallization (solid anddashed purple lines in Fig. 7). During a shock pulse to
40–50 GPa, the olivine single crystals in the shock melt
would transform to polycrystalline aggregates of wadsleyite, ringwoodite or bridgmanite plus oxide (Ito and Takahashi 1989). During pressure release from 40 to 50 GPa,these HP minerals would follow a shallow P–T path
between that of melt and the bulk rock (purple lines inHu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

Fig. 7). Based on observations in partially annealed samples, this would result in post-shock temperatures highenough to back-transform ringwoodite to olivine andcause recrystallization of back-transformed aggregates(Fig. 8f ). Shock pressure greater than ~ 50 GPa would
result in large melt fractions, generally shallower P–T
decompression paths (Hu and Sharp 2017), and postshock annealing out of high-shock indicators (grey areain Fig. 7). It is worth noting that, although the melt breccia may still contain thin shock veins that experienced
relatively fast cooling with steeper P–T curves, the bulk
post-shock temperature of the sample sits in the backtransformation regime and destroys the HP minerals.
Shock to 50 GPa results in a bulk release P–T–t pathand a post shock temperature that is too hot for the preservation of most HP minerals. The bulk release path of
50 GPa (solid purple line in Fig. 7), which ends in the grey
back-transformation area in Fig. 7, represents the lowest
thermal history possible during and right after shock insuch a sample. This illustrates that even the coldest partsof such a sample would experience back transformationof HP minerals and recrystallization of low-pressuremineral aggregates (Fig. 8e and f ). The release paths forpure melt (dashed purple lines in Fig. 7) are above the liquidus and would result in crystallization at low pressure
after pressure release. In natural melt-breccia samples,
the release paths would be between these two end-members, with shock melt following a very shallow P–T pathclose to the liquidus. In this scenario, shock-melt crystallization may occur at elevated pressures, but at pressuresgenerally too low to produce typical HP minerals and any
HP minerals formed by transformation would not survive
the high post-shock temperatures.

5 Shock pressure estimation by P–T–t paths
Realistic P–T–t paths of shock metamorphism in chondrites are important for the interpretation of heterogeneous shock effects and conditions that are well off the bulk
Hugoniot, particularly for the temperatures. The formation of shock melt veins results in local regions that are
far hotter than that predicted by the bulk-rock Hugoniot
for a given shock pressure. Formation of these local meltzones likely involves transient local pressure excursions,as described by Stöffler et al. (1991) but these transientpressure variations ring down to the equilibrated shockpressure at least 1­06 times faster than the crystallization of even micron-sized shock melt (Langenhorst andPoirier 2000). More importantly, these local excursions,caused by pore collapse and shear during compression, result in temperature heterogeneities that are far
above the temperature of the bulk-rock Hugoniot (Sharp
and DeCarli 2006; Moreau et al. 2018). These temperature heterogeneities not only produce local melting andPage 16 of 22

HP minerals; they also enhance thermally activateddeformational processes. For example, the pressure forsolid-state amorphization of plagioclase is strongly temperature dependent (Daniel et al. 1997; Tomioka et al.
2010; Kubo et al. 2010; Sharp et al. 2019) and the pressure at which PDFs and mosaicism occur decreases withincreasing temperature (Huffman et al. 1993; Bauer 1979;
Bowden 2002). Moreover, ignoring temperature heterogeneities would lead to difficulties explaining the significant discrepancy between co-existing shock features inmany samples, such as sharp extinction of plagioclase in
S4-S6 chondrites (Hu and Sharp 2017; Fritz et al. 2017).
The thermal dependence of shock metamorphic effects
explains why they are strongest (S6) in and adjacentto shock veins, but decrease to S5 and S4 with increasing distance from shock veins in some samples (Bischoff
et al. 2019).
Given that some parts of a shocked sample deviate
from the bulk Hugoniot conditions and are therefore“non-equilibrium” shock features (Stöffler et al.1991),
they are still useful for estimating shock pressure because
they are hot enough to drive thermally activated mineral reactions. For a typical HP-mineral bearing sample, the shock pressure needs to be roughly in the range
as the stability field of the observed HP minerals, i.e.
15–30 GPa. In principle, HP minerals can form metastably at lower pressure than equilibrium under high strain(Melosh 1989) or by a metastable reaction (Walton et al.
2014; Kubo et al. 2015). The shock melt must experience
sufficiently high pressure to crystallize HP minerals andcontain enough heat to transform entrained clasts to HP
phases. In the 15–30 GPa pressure range, the bulk shock
temperature (Hugoniot temperature) is low enough forthe bulk sample to act as a very large heat sink that drives
rapid quench of the shock melt before and during decompression. These local “non-equilibrium” S6 shock features
are essential for creating and preserving mineralogicalevidence of high pressure.
If shock pressure were significantly higher than the
15–30 GPa indicated by HP mineral stability, the mineralogical evidence would not be well preserved (Fig. 7).
For a shock pressure between 25 and 35 GPa, we would
expect bridgmanite and magnesiowüstite to be the dominant HP minerals to crystallize from the shock melt andto form from the transformation of entrained mineralfragments. However, bridgmanite has the lowest preservation temperature of all of the HP minerals and readily
decomposes at ambient pressure and temperatures inexcess of ~ 400 K (Durben and Wolf 1992). This preservation temperature is lower than the post-shock releasetemperatures from 25 GPa (Fig. 6). Although bridgmanite can form in shocked meteorites, the shock conditions
where bridgmanite should dominate the mineralogyHu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

cannot preserve bridgmanite after pressure release. This
explains why crystalline bridgmanite is so rare in shockedchondrites, having only been found as a trace component
in Tenham (L6; Tomioka and Fujino 1997; Tschauneret al. 2014). Instead, glassy grains with bridgmanite stoichiometry have been reported to coexist with akimotoite
and ringwoodite in the quenched shock melt of Acfer 040
and Tenham (L6; Sharp et al. 1997; Xie et al. 2006b). The
texture, composition and amorphous state provide strong
evidence that these glassy grains were bridgmanite thatcrystallized from the shock melt and decomposed uponpressure release. Also, evidence for the transformationof olivine to bridgmanite (now pyroxene glass) plus magnesiowüstite has been reported from Martian meteorites, including Tissint (Walton et al, 2014; Miyahara et al.
2016; Ma et al. 2016) and Dar al Gani 735 (Miyahara et al.
2011b). If highly shocked meteorites were shocked topressures above 25 GPa, we would expect to find abundant evidence for amorphous or transformed bridgmanite in S6 samples. The rarity of this evidence suggeststhat most shocked meteorites with S6 shock effects did
not experience equilibrated shock pressures in excess ofabout 25 GPa. For a shock pressure of 40–50 GPa, eventhe bulk release temperature would be high enough todestroy all HP minerals (solid purple line in Fig. 7). In
samples shocked to these pressures, the primary evidence for shock metamorphism would be extensive melting, darkening and recrystallization, as in the impact melt
breccias.
5.1 S6 shock‑pressure inconsistencies and the use of HP
minerals for evaluating shock stage classification

Shock deformation features in the unmelted portion of
the meteorites have been the gold standard for shockclassification and pressure estimation since Stöffler
et al (1991) published their shock classification. Shockrecovery experiments are an important technique forcalibrating the corresponding shock pressures of thesedeformational features (e.g. Milton and DeCarli 1963;
Stöffler 1972; Kieffer et al. 1976; Bauer 1979; Jeanloz
1980; Ostertag 1983; Stöffler and Langenhorst 1994) andthey provide the basis for most of the Stöffler et al (1991)
pressure calibration. However, S6 features, particularly
HP minerals, are not well calibrated by shock recovery
experiments. S6 shock effects, which are closely associated with shock melting, are interpreted by Stöffler et al(1991) to represent shock pressures from 50 to 90 GPa,with ringwoodite requiring 80–90 GPa to form. However,pinning of the pressure scale to 80–90 GPa for ringwoodite is not based on the formation of ringwoodite in shock
recovery experiments, but rather on the lack of suchshock synthesis. Without well calibrated pressures for S6

Page 17 of 22

shock effects, this interpretation of S6 shock conditions,based on high-pressure minerals, is not well supported.
Chen et al. (1996) used TEM to characterize garnetbearing melt-vein assemblages in Sixiangkou (L6) andthe entrained ringwoodite and majorite that formed fromolivine and enstatite. In that paper, the majorite-pyrope
garnet plus magnesiowüstite assemblage was interpreted
to represent melt-vein crystallization at 20–24 GPa andbetween 2050 and 2300 °C. This shock-pressure estimate
was far lower than the 80–90 GPa estimate of ringwoodite formation of Stöffler et al. (1991) and started a heateddiscussion over the meaning of S6 shock effects and theuse of HP mineral stability to constrain shock pressure.
Here we use our modeling of shock temperatures anddiscussions of P–T–t cooling paths to provide insight
into the importance of HP minerals and the S6 classification of shocked meteorites.
The central issue in the discussion of S6 features is theinterpretation of shock-melt crystallization pressure versus the equilibrium shock pressure. Stöffler et al. (1991)
state that S6 features represent local pressure–temperature excursions, within and near shock melt, but they
are less clear about how to interpret those excursions in
the context of equilibrium shock pressure. S6 features,
especially blue ringwoodite aggregates in shock veins,
have become an easily observed indicator of S6 shockstage. More recently, Stöffler et al. (2018) stated that HP
minerals in shocked meteorites are duration-dependentand represent an extended pressure release regime thatshould be ignored in the interpretation of equilibrium
shock pressures and they removed the use of HP minerals from S6 criteria and from most of the revised classification schemes. In particular, HP minerals are excluded
from ultramafic and mafic samples (e.g. shergottites,
eucrites etc.) and only mentioned as a minor possibility for S5-6 in chondrites (> 35 GPa), in contrast to an
exclusively S6-defining feature in Stöffler et al. (1991).
Although HP minerals do not support the pressure calibration of S6 in Stöffler et al (1991) or S5-6 in Stöffler
et al. (2018), they do provide valuable constraints on
shock pressure. The durations of non-equilibrium pressure excursions are on the order of 10–20 ns, which is too
short for significant HP-mineral crystallization or transformation at extreme pressure. Instead, the high temperatures generated are much more important in creating
HP-mineral signatures because they persist long afterpressure equilibration. These high temperature zones,
which are far off the bulk Hugoniot, are nearly the only
parts of the samples hot-enough to record a mineralogical signature of shock pressure.
Because shock deformation features, such as fracturingand mosaicism, can occur in a wide range of shock pressures (e.g. Bauer 1979), HP mineral assemblages provideHu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

complementary constraints on shock pressure, temperature and duration that should not be ignored. The timingof shock-melt crystallization relative to pressure releaseis critical for interpreting pressure constraints from HP
minerals. Many highly shocked L6 chondrites have nearly
constant shock-vein crystallization assemblages thatsuggest a narrow range of crystallization pressure (e.g.
< 15–25 GPa; Chen et al. 1996; Sharp and DeCarli 2006;
Chen and Xie 2008; Miyahara et al. 2011a; Hu and Sharp2017). This implies that the shock-veins in most highly
shocked L chondrites represent crystallization at equilibrium shock pressures between 15 and 25 GPa. Crystallization during decompression cannot reconcile thedifference between very high pressures for S6 featuresof Stöffler et al. (1991, 2018) and the 15–25 GPa crystallization pressures recorded in L chondrites. The P–T–t
paths for S6 shock veins are driven by a combination of
decompression and heat transfer to the surrounding bulk
rock (blue and green lines in Fig. 7). It is highly unlikelythat shock-melts formed at pressures in excess of 50 GPa
and temperatures over 3000 K would all follow P–T–t
paths that consistently cross the liquidus between 15 and
25 GPa regardless of the size and cooling rate of the meltzone. Such a large drop in crystallization pressures would
result in a large range of crystallization assemblages in agiven sample and between samples. Moreover, a quench
path from 50 GPa that crosses the liquidus at 15–25 GPawould result in P–T–t conditions and post-shock temperatures that would destroy all HP minerals (solid purple lines in Fig. 7). The narrow pressure ranges recorded
in many S6 L chondrites are consistent with shock in theisobaric zone (Fritz et al. 2017) of the large impact thatbroke up the L-chondrite parent body.
A similar scenario of HP minerals associated withshock melt occurs in the ultramafic–mafic achondrites,such as shergottites and eucrites, but these meteorites do
not all record a narrow range of crystallization pressures.
Martian meteorites commonly contain both high- andlow-pressure features that represent crystallization andquenching through a range of release paths and shockoutside of the isobaric zone (Walton et al. 2014; Fritz
et al. 2017). This complication does not jeopardize theshock-pressure estimation from HP minerals but rather
provides more information for constructing possible P–
T–t paths for various parts of a given sample (green andred paths in Fig. 7). A HP mineral assemblage in the shergottite Zagami was first reported by Langenhorst andPoirier (2000) and majorite plus ringwoodite/wadsleyite
assemblages, similar to those in shocked chondrites, werefirst reported in SNCs by Malavergne et al. (2001). Since
then, many HP minerals have been recognized in mafic/
ultramafic achondrites, indicating shock pressures of
15–30 GPa (Tomioka and Miyahara 2017 and references

Page 18 of 22

therein). Even the post-stishovite silica polymorph seifertite (Sharp et al. 1997; El Goresy et al. 2008; Miyahara
et al. 2013), which has a stability field of > 100 GPa (Grocholski et al. 2013), can form metastably from cristobalite
within the 15–30 GPa pressure range (Kubo et al. 2015).
This pressure range is in good agreement with the moderate deformational features, such as partial to full diaplectic glasses and weak mosaicism in mafic minerals, foundin Martian meteorites and eucrites (e.g. Langenhorst
and Poirier 2000; Pang et al. 2016; Sharp et al. 2019). As
mentioned above, shocked achondrites generally record
shorter HP pulses, which makes the back-transformationand decompression sequence clearer in these samples(e.g. Walton 2013; Sharp et al. 2019; Hu et al. 2020). Theseoccurrences are inconsistent with HP minerals originating from local/transient pressure excursions (Fritz andGreshake 2009). The samples truly shocked to ~ 40 GPa
and above, would have high bulk temperature, pervasive
melting and no HP minerals (e.g. ALHA 77,005; Waltonand Herd 2007), similar to chondritic melt breccias. The
claim that all the meteoritical HP minerals formed during
pressure release from extreme pressures is not supported
by observed assemblages. Therefore HP minerals can andshould be used as shock stage indicator of moderate pressure in ultramafic–mafic achondrites.
The reconciliation of S6 conditions in shocked meteorites comes from the heterogeneous nature of shock
effects and shock stage in these samples. Bischoff et al.
(2019) reported shock classification results from 2280
shocked ordinary chondrites and specifically addressed
the issue of shock-stage heterogeneity and S6 classification. They found that among 52 L chondrites and one
LL chondrite with ringwoodite, 23 had more crystalline
than amorphous plagioclase and an additional 16 had
more than 25% crystalline plagioclase. This indicatesthat more than 70% of the ringwoodite-bearing chondrites belong to the shock stage S4 (Bischoff et al, 2019).
They also stress the importance of sampling when classifying heterogeneous samples and the need to look at
more than one small thin section in classifying shockstage. For example, Tenham, the iconic S6 chondrite
which retains crystalline bridgmanite (Tschauner et al.
2014), has melt-vein free material that is classified as S4(Bischoff et al. 2019). These results indicate that the S6classification based on HP minerals is not a valid partof a progressive series of shock effects, but rather the
result of local hot zones in samples shocked to predominantly S4 conditions. The 15–30 GPa (majorite predominant) shock pressures inferred from the crystallizationof shock melt in previous S6 samples is well within therange of S4 conditions originally proposed by Stöffler
et al. (1991). The presence of HP minerals, including theeasily recognized presence of ringwoodite transformed

Hu and Sharp ﻿Progress in Earth and Planetary Science(2022) 9:6

from olivine, are not an indication of very high shockpressures, but rather an indication of moderate S4 shockpressures. Therefore, HP minerals in shock veins should
not be ignored in shock classification because they
provide evidence of moderate shock pressure and constraints on the P–T–t history of the shock melt throughquench and decompression.

6 Conclusions
High-pressure minerals are important signatures ofshock in meteorites, but they are not necessarily anindication of very high shock pressures. Their production and preservation requires complex but wellconstrained P–T–t paths from very high temperaturesduring shock to low temperature after pressure release.
Calculations of shock and release temperatures provide
the context for constraining these complex P–T–t paths.
Observed high-pressure minerals in shocked meteorites
correspond to shock pressures of 15–30 GPa. The corresponding HP liquidus for chondrites is approximately
2300 K, which is hot enough to enable the transformation of low-pressure minerals, in association with shock
melt, to their high-pressure polymorphs. At these pressures, the bulk shock temperature is low enough for thehost sample to act as a heat sink to rapidly cool shockmelt veins and pockets, resulting in rapid quench (highdT/dP) of these hot zones before and during decompression. The bulk meteorite is also sufficiently cool sothat only deformational shock features occur outside
the melt veins. In contrast, if a meteorite sample experiences even slightly hotter background temperature and/or shorter shock pulse, the quench of shock melt duringdecompression would have a gentle dT/dP slope which
can leave the newly-formed HP assemblage too hot tosurvive at reduced pressure. The shock veins in samples with abundant melt, such as melt breccias, generally crystallize low-pressure assemblages and anneal out
many shock deformation effects. At 40–50 GPa, the bulk
shock temperature is too high for HP minerals to cool
sufficiently to survive at low pressure. Samples shocked
to such high pressures will only retain high-temperature features, such as extensive melting, darkening andrecrystallization. The presence of ringwoodite in manychondritic samples, which was commonly considered as
evidence of shock stage S6, is not an indication of veryhigh shock pressure, but rather an indication of moderate shock pressures consistent with shock stage S4.
Abbreviations
HP: High pressure; SMV: Shock melt vein; PDF: Planar deformation features;
MORB: Mid-ocean ridge basalt; SNC: Shergottites, nakhlites, chassignites.

Page 19 of 22

Acknowledgements
We thank Dr. Naotaka Tomioka and an anonymous reviewer for their careful
review and constructive comments. We also thank Dr. Erin Walton and an
anonymous reviewer for reviewing an earlier version of this manuscript. We
acknowledge the late Prof. Ahmed El Goresy for his many contributions to thestudy of HP minerals in shocked meteorites. We miss and gratefully thank thelate Paul DeCarli who shared his wealth of knowledge on shock and mineral
physics with us since early 2000s. We thank Dr. Paul Asimow for the guidanceon thermodynamic calculations.
Authors’ contributions
TGS conceived the initial concepts of this study. JH and TGS designed thisstudy. JH performed the thermodynamic calculations. JH and TGS wrote themanuscript. All authors read and approved the final manuscript.
Funding
The study was initially funded by NASA Cosmochemistry Grant 12-COS120002. JH is supported by NASA Solar System Workings Grant 80NSSC18K0532
and NSF Award 1725349. TGS is supported by NASA Emerging Worlds Grant
17-EW17_2-0090.
Availability of data and materials
All data generated or analyzed during this study are included in this publishedarticle.

Declarations
Competing interests
The authors declare that they have no competing interests.
Author details
1
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA. 2 School of Earth and Space Exploration,
Arizona State University, Tempe, AZ 85287, USA.
Received: 28 August 2021 Accepted: 18 December 2021

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