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Thermomechanical response of thickly tamped targets and diamond anvil cells under pulsed hard x-ray irradiation

In the laboratory study of extreme conditions of pressure, temperature, and timescale, the irradiation of matter by high intensity sources has been of central importance. Here we investigate the performance of layered targets in experiments involving very high intensity hard x-ray irradiation, motivated by the advent of extremely high brightness x-ray sources such as free electron lasers. These sources enable massive ($\mu$m to mm) scale targets with confinement of the directly irradiated regions by thick, x-ray transparent tampers. The thermal and mechanical response of these layers, from femtosecond-picosecond energy delivery, to mechanical relaxation on nanosecond timescales and heat flow and dissipation on microseconds, is controlled by the differential energy deposition, shock and release wave production, and thermal transport properties. Sample survival over one or more exposures can depend on damage arising from heating or mechanical stresses induced by heating, depending on the materials used and target geometry. This study also doubles as an investigation into the properties of diamond anvil high pressure cells, long used in conjunction with x-ray synchrotron sources, at new high intensity x-ray beamlines at free electron lasers and upgraded synchrotrons. Configuring a tamped target as a high pressure cell confers certain advantages, such as resistance to expansion and thermal stresses, and sample confinement, in conjunction with efficient control of heat. This study suggests new routes to studying thermodynamic equilibrium states of high energy density matter via the capacity to confine extreme states for particularly long durations.


I. INTRODUCTION
The characterization of matter at high densities (similar to the solid state, and higher), intermediate temperatures (a thousand to a million degrees K), and fast (microsecondnanosecond) to ultrafast (picosecond to femtosecond) timescales, is central to our understanding of planetary and stellar interiors, fusion energy technologies, and the fundamental interactions in matter at extreme conditions. These often-called 'warm dense matter' states appear at conditions that fall between those described by traditional condensed matter and plasma physics, where theoretical simplifications are often not available. The onus of developing a physical understanding of this regime thus falls particularly on laboratory experiments. The creation and probing of these warm and dense states of matter in the laboratory often relies on central facilities capable of delivering high-brilliance irradiation, which can rapidly generate extreme temperatures in dense (i.e. solid) targets by ultrafast (fs-ps) 'isochoric' heating, or by production of dynamic compression waves within the target facilitated by the expansion of heated matter on longer timescales (ps-ns) 1 . Ultrafast techniques have been widely employed to study the case of isochoric heating, at timescales from femtosecond energy delivery to electrons, to picosecond heating of the lattice ions and hydrodynamic expansion into a vapor [2][3][4] .
A common strategy uses electromagnetic radiation, typically in the optical range, to deliver the intense energy burst. In such photonic experiments energy is delivered directly to electrons, which then transfer energy to the ions (lattice) as the system relaxes toward a state of local thermal (and thermodynamic) equilibrium (LTE). The timescale of equilibration between the ions and electrons is typically in the range of ps to ns [4][5][6][7] such that equilibration can occur roughly coincident with expansion, melting, and vaporization, leading to loss of high-density conditions and sample confinement before LTE is achieved and ionic structures adjust to the equilibrium state. This makes it practically challenging to use isochoric heating to study equilibrium warm dense matter, as needed to accurately simulate the true conditions of many high energy density systems in nature and technology. Other often-used methods of irradiative volume or bulk energy deposition providing access to similar states of matter include intense proton 2,8 , heavy ion 9 , and electron 5 beams or other modes of fast electron deposition 3,10,11 . Dynamic compression, the driving of compression waves travelling at near sound velocities (∼1-10 µm/ns), is a somewhat slower form of volumetric energy delivery, while diffusive (as opposed to ballistic) heat conduction is even slower.
One strategy to extend the lifetime of an irradiation-driven warm dense state is to provide additional material around samples through which energy may be deposited and which delay, prevent, or otherwise control expansion [7][8][9][12][13][14] , such as by extending the time it takes pressure release waves and cracks to propagate through the heated target. This 'tamping' approach can even confine the heated region entirely, allowing recovery of samples quenched from conditions that would normally lead to vaporization 12 and further experiments on the same target. For optical radiation, tamping can be achieved by placing an absorptive (i.e. metal) layer between transparent (i.e. dielectric) tamper materials 7,8,13 , by tightly-focussing the beam within the tamper itself 12 , or other configurations. However, tamping using highpower optical laser irradiation is limited by the need to deliver sufficient energy through the tamper to the sample, and is thus limited by the optical transmission of the material under high brightness radiation, often requiring thinner tampers (of micron thickness) and frequently producing similarly short-lived states. Similarly the thickness of the excited region is typically limited to ∼tens of nm using these strategies, with tampers up to ∼µm thickness being possible, while spot sizes are diffraction-limited to several µm diameter.
Intense x-rays can also rapidly heat samples 6,8,[15][16][17][18][19] . This energy deposition may be introduced deliberately (e.g. to heat or otherwise excite electrons in a sample, or to filter beam energy) or may be a side effect of probing samples with a high intensity x-ray beam.
Such heating does not depend on nonlinear optical damage thresholds that affect optical radiation transmission, but instead depends more linearly on the x-ray absorption properties of the target materials, with x-ray absorptivity dependent on atomic number Z (i.e. the number of electrons in the atoms). For deliberate heating strategies, the potentially larger absorption lengths enable more homogenous heating compared to optical lasers 16 and scaling up of targets. X-ray heating performed with intense laser-generated x-rays on large laser facilities 8 , Z-pinch 14 , and soft x-ray free electron lasers 6,8,15,16,18,19 have been demonstrated, though typically at lower x-ray energies (hundreds of eV to several keV) which can limit the thickness and materials of target components including tampers, and often comprise divergent sources which limit experimental geometries and brightness.
With x-ray absorption lengths reaching the ∼mm level at ∼10 keV in common lower-Z materials, the possibility of massive, thick tampers and samples may be realised with hard x-ray sources. The increasing availability of free electron lasers (FELs) operating in the hard x-ray regime above 10 keV as well as other high-brightness hard x-ray sources including upgraded synchrotron light sources (Table I) can thus enable new kinds of heating strategies. For hard x-ray FELs, the high potential pulse energy (∼ 1 mJ, or 10 12 photons), fast timescale (10-100 fs), and high brightness ( 10 18 W/cm 2 ) is comparable to typical optical laser systems. Similar advancements at synchrotron radiation facilities achieving comparable pulse energies in the hard x-ray regime, but somewhat longer pulse timescales ( 100 ps), are also forthcoming (Table I).
High Power X-ray Sources  Brilliant x-ray irradiation from current and emerging central facilities can thus lead to significant energy deposition in targets in the fs-ns timescale, with one result being thermal energy deposited differentially in the target depending on local x-ray absorption behaviour 8 .
Due to differences in x-ray absorption cross sections and the high transparency of lower-Z matter to hard x-rays, substantial energy deposition differences in light element (low-Z) and heavy element (high-Z) materials suggests particular target designs using tamped configurations that can be used to control the effects of irradiation. Specifically, delivering the energy through light element (low-Z) tampers to samples containing heavy elements (high-Z) is possible for such sources.
Energy deposition can be effectively adiabatic during and after energy delivery on femtosecond to picosecond timescales for target dimensions in the range of microns and larger.
The hot region can be inertially confined up until the timescale of stress wave propagation through the target, which is of order nanoseconds. Heat can be confined by the relatively slower rate of heat transfer, on a timescale of order microseconds, setting an upper limit on adiabatic conditions in the sample. Assuming hot regions can be confined by massive tampers which prevent significant expansion, the primary cooling process in the sample will be via heat transfer, controlled mainly by phonon and electron heat diffusion rates, given the inefficiency of radiative transfer on short timescales 22 . This can allow a sudden creation of high-temperature conditions inside of a bulk target where the extreme state produced is both inertially and thermally confined, and the sample is unable to undergo significant expansion from its initial density (i.e. from solid density) due to the large amount of material surrounding it; and it is unable to cool significantly either through the adiabatic expansion or by heat transport out of the hot region.
Many interesting and poorly-understood phenomena at 'warm dense matter' conditions accessible by such rapid heating are found only at elevated densities, such as fluid-fluid phase transitions , phase separation of noble gases , and metallization in molecular materials 23 .
Knowledge of how increasing density influences fundamental interactions including rates of bond dissociation and electron-ion thermal equilibration are also needed 5,6,16 . The ability to employ confining materials of near arbitrary thickness and sufficiently low-Z brings to mind static high pressure devices which employ thick anvils to pre-compress samples prior to probing, and which are compatible with hard x-ray radiation.
The purpose of this study is several-fold, and motivated by the increasing brightness of hard x-rays from free electron lasers, upgraded synchrotrons, and laser driven 'backlighter' x-rays from large laser facilities. These can provide fast pulsed (nanosecond to femtosecond) hard x-rays (to tens of keV) at high brightness (10 11 -10 12 photons per pulse). Our main objective is to explore the longer duration thermal evolution of pulse-irradiated tamped targets involving particularly thick tampers. This is a configuration suggested by the ability of hard x-rays to pass unimpeded through low-Z tampers to a high-Z target layer confined within, to which energy is delivered. We also aim to characterize the heat dissipation in solid layered targets which may be of practical use as, e.g., beamline optics 28 and detectors 29 at x-ray facilities. The survival of these components often depends on their heat and stress dissipation capabilities. For example, current schemes utilise robust, high thermal conductivity materials such as diamond 28,29 , a material examined here.

II. METHODOLOGY
The x-ray thermomechanical response of these targets evolves on several different timescales. Initially, there is the delivery of energy on timescales of ∼ 100 fs, and associated approach to local thermal equilibrium between electrons, which receive the energy, and ions, which equilibrate with the electrons, occurring on somewhat longer timescales in the range of ∼ 1-10 ps. For the purposes of this study, we are primarily interested in the subsequent effects in the microns-millimeters scale target. First, pressure waves, taking the form of shock and release waves, adiabatically mediate the evolution of stress, strain, and temperature in the differentially heated target, on roughly ns timescales. Second, heat conduction as the adiabatic approximation breaks down further modifies the temperatures and ultimately cools the heated areas toward their initial conditions. To study heating, we introduce an accurate two-dimensional finite element model which handles both conduction along and lateral to the heating beam, both effects being crucial to include on these longer timescales for tightly focussed radiation (Sec. II A). As finite element models are not well suited to study stress waves of larger magnitude, which can take the form of shock discontinuities, we separately employ 1D hydrodynamic models (using hyades) to study the stress evolution of the system for the first few ns (Sec. II B), where the 1D approximation is suitable as, for an irradiation spot size greater than the thickness of the relevant layers, a 1D model can capture the initial evolution of the sample accurately.
A. Finite element models

General Approach
In this study we the examine the thermal response of tamped samples to high-brightness monochromatic hard x-rays. Our samples generally consist of a sample layer or layers (µm thickness) between thick (mm thickness) tampers. The advantages of this configuration are: (1) exceptionally long confinement of samples at extreme conditions, enabling the approach  Table II. to and properties of thermodynamic equilibrium states of high density and temperature to be studied (2) efficient control of sample temperature by using high thermal conductivity tampers, enhancing sample stability and promoting sample survival after irradiation. (3) the ability to pre-compress samples with strong tampers, and resist thermomechanical stresses.
In order to describe a single-pulse x-ray heating and cooling of a tamped sample configuration, we used a simulation software (comsol Multiphysics) based on Finite Element Analysis (FEA) to implement a two-dimensional, time-dependent heat transfer model 30 , with semitransparent materials exhibiting a bulk absorption of the x-ray radiation. We simulate the case of a single intense x-ray pulse of ∼100 fs duration striking a sample initially at room temperature (300 K). These simulations can be extended to describe the effects of pulse trains, and pulses of different lengthscales and diameters.
Assuming a multilayer target of layers perpendicular to the incident x-ray beam (  Table II. This configuration is also symmetric about a plane through the middle of the sample layer. The pulsed x-ray beam propagates in the +z direction of the geometry and centred at r = 0. One tamped sample configuration of particular interest comprises two thick diamond tampers with a thin sample contained within. This is the configuration of a diamond anvil cell, a device able to produce very high initial pressure and density states in samples prior to x-ray probing and/or thermal excitation. Long used at synchrotron facilities, and compatible with hard x-ray illumination as either a non-intrusive probe or as a pump, the diamond anvil cell offers the possibility to creating and probing high energy density states at very high pressure, temperature, and material density when coupled to fast x-ray sources. This particular configuration is discussed in Sec. IV F. In order to describe the dynamical temperature evolution inside the DAC under various heating conditions we used a finite-element solution of the time-dependent energy transfer equation. The volumetric heat source Q(r, z; t) (the net energy generated per unit volume and time) representing the radiative energy absorbed within the target is given as where T is the temperature, t is the time, k is the thermal conductivity, ρ is the density, and C P is the heat capacity at constant pressure. For constant physical properties and assuming heat deposition occurs instantaneously compared to heat transport, this reduces to where κ is the thermal diffusivity, Radiative (photon) heat transfer is generally negligible compared to diffusive (phonon and electron) heat conduction at the presently examined temperatures and timescales 22 , and is not included.
The source term Q(r, t) (typical units of W/m 3 ) (Eq. 1) is the volumetric heat generation when the incident x-ray beam passes through and is absorbed within the semi-transparent materials, and due to this absorption the beam intensity decays with depth following an exponential behavior (Lambert-Beer law). At the considered x-ray energies, the contribution of scattering to total attenuation is small and is neglected in our calculations, as is coherent scattering (Bragg diffraction). It should be noted that Bragg contributions to the attenuation relevant to thick tampers could affect beam intensity and even cause significant redirection of beam energy within targets, e.g. where single crystals are used as tampers, but can be avoided in practice 31 . At interfaces, this beam can also be partially reflected backwards, though for x-ray radiation reflectivities are exceedingly small, of order R ∼ 10 −9 − 10 −13 , and may also be neglected. Thus for these models the attenuation of x-rays as well as the energy deposition is entirely due to absorption.
Thus, the energy deposition in a given homogenous layer in a target can be written as where α is the absorption coefficient, constant in the layer, z s is the z position of the layer surface the radiation is incident on, R s is the reflectivity of the leading surface or interface, The absorption in the target is given by computing the sequential absorption in several such layers. At the opposite face to the incident beam direction, boundary condition establish that any light reaching that boundary will leave the domain and pass to the next layer and this is repeated until the beam reaches the opposite target face and leaves the geometry.
For example, in the center of the sample (and target), we have where S, M , and T refer to the sample, medium, and tamper values, respectively, I(r, t) is the incident intensity on the target assembly, d refers to the thickness of particular layers, and z c refers to the center of the sample (and target assembly).
The model considers induced heating for a ∼100 fs duration x-ray pulse as incident heat source, and heat transfer following the rapidly, imposed temperature distribution. The heating pulse intensity is assumed to follow a Gaussian distribution in time and space, with incident intensity I(r, t) (Eq. 6) reaching a maximum, I max , at t = µ and r = 0 as where σ r is a Gaussian radius parameter, where the FWHM diameter of the pulse is 2 √ 2 ln 2σ r , and σ t defines the temporal width of the pulse as 2 √ 2 ln 2σ t (FWHM). For the parameters of this simulation (Table III) the spot size is then ∼12 µm, and the pulselength ∼240 fs. The incident peak intensity I max can be related to the net energy of the single pulse E pulse (in J), the peak incident power P max (in J/s, and occuring at t = µ), and the peak areal energy density Λ max (in J/m 2 , and occurring at r = 0) as The number of photons per pulse N , is equivalent to ∼10 12 for the peak energy per pulse and x-ray energy simulated here (3.5 mJ). In our models we specify E pulse (Eq. 7), which when integrated over the pulse duration (Eqs 5 and 6) leads to Q(r, z; t >> µ) independent of the pulse length, such that T (r, z) immediately after the pulse depends only on total pulse energy and its spatial distribution.
The initial temperature of the entire system is assumed to be at ambient (300K). As a boundary condition, the external surface of the simulation cell shown in Fig. 1 was given by natural heat exchange with a surrounding atmosphere (air), with the external temperature fixed at 300 K, and heat loss from the surface determined as where q 0 is the convective heat flux and h is the convective heat transfer coefficient (h = 5 W/m 2 /K, for natural convection in air).
We use a free triangular mesh which is extremely fine at interfaces due to the large temperature gradients at interfacial regions at heating times 10 −13 to 10 −12 s; the heat transfer starts at approximately on 10 −9 s time scales. A coarser mesh is used away from the interfaces.
As the simulations seek to establish general trends for the effects of target composition, geometry, and beam parameters, a number of physical assumptions are made in our calculations.
We assume a direct relationship between the amount of x-ray energy deposited in the target at a given location and the amount of heating at this location, and further that thermal equilibrium in the sample (i.e. between electrons, which initially absorb energy, and ions, which heat more gradually on the timescale of electron-ion equilibration, or roughly picoseconds) occurs instantaneously. That is, our simulations should be approximately accurate in the long duration limit, at timescales of roughly 10 −12 and longer, i.e.sufficiently greater than electron ion equilibration times. Implicitly, we also assume localisation of hot electrons during the equilibration period, i.e. that any hot electrons produced ultimately equilibrate with nearby ions; ballistic electron transport in the samples could be relevant in causing a redistribution of thermal energy, e.g. before lattice equilibration occurs, which may need to be accounted for in more detailed simulations. However, the typical mean free path of ballistic hot electrons in condensed matter tends to be tens of nm 8,16,32 , which is quite smaller than the sample dimensions simulated in this study (1-10 3 µm), so no major errors are anticipated for the present simulations; that is, a diffusive heat transfer model is considered sufficiently accurate on these time and lengthscales. However, recent modelling studies on multi-layer targets at high density and temperature 11 suggest hot electron transport and nonlocal energy deposition may play a role on longer timescales and over larger distances, suggesting this issue should be examined experimentally. In summary, local thermal equilibrium is assumed between ions and electrons, but thermal equilibrium is not assumed in the target itself: we model the macroscopic temperature evolution out of equilibrium.
With a propagation time across the entire target of ∼10 −11 s, it suffices for our purposes to assume the x-ray beam is propagated instantaneously across the target, since conduction processes occur on longer timescales.
Nonlinearities that may occur directly as a result of the high x-ray fluence or rapid timescale, e.g. resulting from ejection of core electrons 17  In these simulations interface temperatures between differentially heated surfaces are effectively constant on shorter (adiabatic) timescales. This is because, instantly upon illumination, the interface temperature reaches an intermediate temperature between the adjacent materials, defined in part by the relative temperatures and in part by their thermal conductivities. This follows from the expectation of constant interfacial temperatures following the rapid emplacement of an interfacial temperature discontinuity 34,35 . For assumed constant layer thermal conductivities (Sec. II A 2), interface temperature T i is given as where number subscripts indicate properties for the contacting layers. This correctly predicts the simulated constant interface temperatures before heat conduction activates on timescales of ∼10 −8 s. Accurate modelling of the interfacial temperature on shorter timescales required especially fine zoning of the simulation near the interface; if this was not performed, significant anomalies in temperatures (short timescale rapid variations) appeared.

Materials Parameters
A suite of materials with varying properties are included in the models to examine the possible range of heating and cooling behavior under x-ray irradiation. As the degree of x-ray absorption in a substance is roughly given as where ρ is the density, x-ray energy is E, atomic number and mass are Z and A respectively, we sought to explore samples over a wide range of Z, and lesser variances in the possible surrounding low-Z materials. Materials properties are assumed to be constant with temperature to provide a representative and simplified picture of material response for a range of material properties, to enable material selection for target design, and, as required, for more detailed modelling including temperature (and pressure) sensitivity of parameters, effects of phase transformations, effects of electronic excitations (e.g. electronic heat capacity), as well as short time-scale effects otherwise left out of our simulations, as discussed above. These simulations thus provide a representative picture of how strongly tamped targets respond to rapid irradiation. All material properties are taken to be isotropic.  one by one the x-ray energy (Tables IV and V ), beam power (Table IV), the materials comprising the sample, medium, and tamper (Tables IV and VI), and the medium thickness (Tables IV). Representative thermo-physical and optical bulk material parameters (Tables   V and VI) were taken from values measured at ambient pressure and temperature, unless otherwise noted. The additional material at the outside edge of the sample area, referred to as a gasket, is composed of rhenium. X-ray photon energies were taken from the hard x-ray regime, and are used in modern x-ray diffraction and absorption measurements. Pulse intensities were taken to be near the maximum presently available at facilities.
Sample materials was chosen to represent a range of possible x-ray absorption levels, including a range of metals across a range of Z (Fe, Mo, Pb), a representative low-Z material (H 2 O) which is also an insulator, and a representative high-Z insulator (gadolinium gallium garnet, GGG).
Diamond was selected as an ideal tamper due to its high x-ray transparency, high thermal conductivity, and high strength to withstand mechanical stresses generated by heating and to act as an anvil for applying initial stress and pre-compressing samples, as in a diamond anvil cell 33 . Diamond has an extremely high mechanical damage threshold beyond that of all known substances 37 with ability to withstand localised stresses exceeding a TPa 38 . It has the highest thermal conductivity of all known bulk matter, allowing it to act as an excellent heat sink which, when properly configured, allows the tamper to remain at very low temperature even when adjacent to very high temperature matter 26,33  The interfacial layer (medium) between the tamper and sample could act as a pressure-

Experimental Parameters
Material Thermodynamic Photo absorption  or thermal 26 graphitization of the tamper surface.

B. Hydrodynamic models
As temperature is increased in targets stresses will develop which are roughly proportional to the amplitude of the temperature change. On short timescales (fs-ps), heating is isochoric, or nearly so, producing large thermal stress (thermal pressure assuming hydrostatic conditions) within the directly heated regions. On the longer term (ps-ns), target expansion and the concomitant production of stress waves will occur. In the limiting case of isochoric heating and assuming hydrostatic stress and thermodynamic equilibrium where α and K T are the thermal expansivity and isothermal bulk modulus, respectively. In terms of a isochoric thermal pressure change ∆P V for a given imposed temperature change Thus thermal pressures on the order of the bulk modulus are achieved at ∆T 1/α. Taking a typical value of α 10 −5 K −1 for condensed matter, this characteristic temperature would be roughly 10 5 K, at the upper limit of the conditions studied here. With K T of order 1 -10 3 GPa, stresses produced in such experiments would be compatible with the creation of high pressure shock waves.
The corresponding deviation from isochoric conditions will reduce the amplitude of stress, to zero for free expansion (ignoring tensile or negative stress states that might occur in high tensile strength materials). However, for a tamped target free expansion is prevented leaving a degree of thermal stress in place until the sample has time to release. In the system considered here, a crude estimate of this timescale is about twice the time it takes for a sound wave to transit the tamper, or ∼ 1 µs, ignoring the possibility of cracks or, as we will see below, voids formed due to strong tensile stresses. The stress evolution of such a system could be complex, coupling to thermal conduction on similar timescales. However, we can outline the nature of these stresses and the manner in which they might affect experiments.
We employ the hyades hydrocode 40 to study the 1D evolution of the stress, strain, and temperature in the adiabatic initial part of the heating cycle. Local thermodynamic equilibrium is assumed (ion and electron temperatures identical). Experiments are initialized at T =300 K and ambient pressure and density. X-ray heating is assumed to be take the

Baseline simulation
The baseline simulation, from which other similations represent a modification, uses the standard target materials arrangement, radiation of 25 keV and a pulse energy of 0.35 mJ (Fig. 2). A close-up of the sample region shows the development of temperature gradients from an initial state of nearly homogenous axial temperature within layers and sharp gradients between layers (Fig. 3). The diamond tamper in this case, by virtue of its high thermal conductivity, provides rapid quenching of the tamper itself by radial heat flow, while the sample region remains hot on longer timescales (Fig. 3). Initial radial gradients (imposed by the assumed Guassian beam profile) are roughly preserved and broadened with time (Fig.   2C).

Radiation Variance: X-ray Intensity
Varying the beam intensity proportionally shifts the thermal response of the target components, as can be expected for a linear absorption process and ignoring the temperature sensitivity of the components. Thus, as an approximate rule of thumb the temperature in any simulation T sim shown here can be scaled by the ratio of the power, i.e.
to determine the level of heating for different power levels I max .

Radiation Variance: X-ray Photon Energy
Varying the x-ray wavelength (and equivalently, energy) through the typical hard x-ray ray range will vary the differential absorption in samples, and the temperature gradients established. Crudely, for lower energies (∼5 keV) the x-ray will be absorbed entirely by the initial material it passes through (the tamper) whereas harder x-rays (∼25 keV) will largely pass through the sample assembly without generating much heating. For the configuration       discussed here, the 15 keV energy provides maximum sample heating, nearly homogeneous temperature in the sample and moderate but potentially survivable heating in the tamper.
The homogeneity of heating depends on the x-ray energy, with harder x-ray producing superior homogeneity in the adiabatic interval ( Fig 5A).

Geometry Variance: Medium Thickness
An interfacial layer between the sample and tamper can play several roles in experiments.
It can act: as a protective layer, preventing direct heating of the tamper and absorbing thermal stress when resisting hydrodynamic expansion; as an insulating layer to extend the experimental duration by limiting cooling of the sample; and as a hydrostatic pressure medium, in cases where the target is configured as a high pressure cell.
Without this interfacial layer the temperature of the tamper is maximized, immediately after heating, and the sample cools rapidly while the tamper interfacial region remains relatively hot (Fig. 6). Addition of even a thin medium layer preserves the sample heat maximally, heat retention being insensitive to tamper layer thickness; the tamper interfacial   region temperature (initially identical) reduces more rapidly for a thicker tamper, though a transient rise in temperature due to arrival of the heat wave from the sample (Fig. 6B) can briefly drive it higher, possibly to above the initial temperature. Inasmuch as damage to the tamper may be thermal in nature, it suffices that addition of even a thin tamper layer is beneficial.

Material Variance: Sample
The samples generally show the strongest heating in the targets, as these are typically higher Z materials, which scale strongly with absorption of each. The insulators modelled have reduced thermal conductivities, compared to the metals, which slow their thermal evolution during the experiments, effectively maintaining the sample temperature even after 200 ns. GGG has an initial thermal response similar to Fe. Water has an initially very low absorption of heat, and rather than lose heat to the surroundings heat conducts into the sample from the hotter medium layers. At this x-ray energy (25 keV) the sample does not strongly perturb the beam intensity and only weakly influences the downstream temperature   distribution in the sample; more pronounced asymmetries in temperature in the sample area occur for higher Z materials (Fig. 7), as well as lower energies (see Fig. 5).

Material Variance: Tamper
Tampers show generally comparable x-ray transparency, with the exception of alumina which has somewhat reduced transmission and hence lower sample temperature. There is significant variance in the temperature evolution of the tamper, but sample conditions do

Material Variance: Medium
The choice of medium can influence the temperature by controlling the rate of sample cooling, which is most notable on longer (µs) timescales. The dominant effect is from the thermal conduction properties of this layer. Notably the cooling is most sluggish for the lowest thermal conductivity medium (LiF) even though the initial temperature of this layer is also the lowest.

General features of thermal evolution / general observations
We find that simulations for targets of the length scales described here are effectively adiabatic on timescales up to 1-10 ns. As samples are effectively adiabatic (excluding the energy deposited by the x-ray beams) on the timescales of FEL (10-100 fs) or synchrotron bunch (10-100 ps) radiation, there should be little difference between the thermal evolution of samples subjected to either form of illumination, once local thermal equilibrium is achieved; the differences will appear only in the initial heating rate (and related shorttimescale phenomena not included in this analysis). That is, pulsed x-ray heating in the fs-ps range (Table I) will have essentially similar effects on heating, as these timescales do not allow significant cooling during the energy deposition phase. Thus for ultrafast sources, the relevant parameter for assessing the temperature following x-ray illumination and LTE is the total pulse energy and its spatial distribution. Therefore the thermal evolution calculations made here are relevant for pulses of any length, up to the adiabatic limit of ∼ 10 ns. This clearly changes for much thinner targets 8 , thus adiabaticity depends crucially on length and time scales.
For samples involving an additional low-Z (medium) layer between the sample and the tamper surface, a late pulse in temperature occurs as the heat wave from the high-Z sample reaches the tamper surface. This leads to a sudden increase in temperature. It is often relatively minor, even where extreme sample temperatures are reached (e.g. of 55000 K in an Pb sample, Fig. 7, where this pulse is only from ∼400 to ∼650 K at the alumina-diamond interface). This heat pulse is equivalent to that described for a two-layer differentially heated target in an XFEL 8 . The timing and amplitude of the heat pulse is correlated with many properties of the system, showing, for example, a direct correlation with the thermal conduction properties of the materials. It can be observed that the arrival time of this pulse increases systematically with thermal diffusivity of the medium ( Fig. 9 and Tables V and VI), i.e. it is fastest for a layer of dense argon (κ = 1.9 × 10 −5 m 2 /s), slowest for LiF (κ = 2.7 × 10 −6 m 2 /s), and intermediate for alumina (κ = 1.5 × 10 −5 m 2 /s). The pulse amplitude is lowest for higher thermal conductivity tampers and highest for the insulating tamper (Fig. 8).
Comparison of the temperature of the sample center and at the interface between the sample and its surroundings provides some indication of the temperature gradient occurring in the sample. On shorter timescales the temperature distribution in the sample is defined exclusively by the absorption profile (Fig. 5) with a large gradient in initial temperature possible in low keV experiments (Fig. 5) or high-Z samples (Fig. 7). In the long duration limit, the sample temperature along the beam axis is at its lowest at the interface, whereas temperature in the sample centre remains close to the maximum.
For harder x-rays (15 keV and above), peak temperatures in the low-Z tamper are generally produced adjacent to the sample layers, either immediately upon heating (due to interfacing with a higher-Z medium (Fig. 9) or sample (Fig. 6), or after the heat wave from the cooling sample reaches the tamper interface (Figs. 6B, 7B, 8B). At lower keV, the hottest portion of the tamper is the leading free surface due to efficient absorption of the beam, however only at the lowest value simulated (5 keV) is the tamper hotter than the sample (indeed, there is negligible heating in the sample in this instance).
More accurate calculations would include also the effects of adiabatic heating and cooling from shock and release processes, in the first picoseconds to nanoseconds of the simulation, which would lead to some degree of additional temperature change before large-scale heat conduction (Sec. III B).

B. Hydrodynamic Model Results
A representative hydrodynamic model (Fig. 10) is shown for a strongly heated (Mo) foil contained by an alumina medium and diamond tamper (see also Fig. 7). The central high-Z foil is heated to peak temperature near 2 × 10 4 K, producing an initial increase in pressure to ∼ 55-70 GPa, whereas minor heating in the surrounding layers produces weaker initial pressurization. Release reduces the pressure on the foil (though not to zero due to the presence of the medium). Due to the differential heating and resulting differential pressures, the central foil expands -from its surfaces inward via a release wave, while driving shock waves outward through the medium, toward the tamper. The inward moving release waves interact in the target centre, producing beyond ∼0.5 ns a stress minimum in the sample which nearly restores the initial (zero) pressure condition. Depending on how the sample is configured, the state produced by these interacting release waves can be one of tensile stress in the target 4 , or spall and zero pressure (see Fig. 11).
The outward moving shocks reflect off the tamper and back toward the sample (at sim0.6 ns), producing a stress maximum on the tamper comparable in magnitude to the initial thermal stress induced in the sample (Fig. 10D). It is this stress maximum which would be a likely point of mechanical failure of the tamper, though the amplitude of this stress (∼30 GPa) is within the elastic yield strength of diamond 37 , at least. Consideration of 2D effects could reduce the amplitude of the pressure wave at the tamper further.
Heating, compression and release is nearly symmetric about the sample center in this case due to near-homogenous heating at 25 keV, but strong asymmetries occur for inhomogeneous heating including when testing lower x-ray energies.
Conditions are adiabatic during these experiments, with adiabatic (shock) compression and release (isentrope) processes modifying the initial temperature in the target (Fig 10A-B). The most pronounced effect are temperature reductions during foil expansion. The temperature at the tamper interface is somewhat increased during shock, however, it remains below thermal damage thresholds.

A. Pulse train Response
Some emerging x-ray sources involve high repetition rate pulse trains, up to the MHz level (Table I). For repeated pulses, the degree of cooling between pulses (i.e. initial temperature at the time of a given pulse) is important. For sources operating in high repetition rates faster than the thermal relaxation time of samples (of order 10 µs based on these simple models), accumulation of thermal energy may occur very rapidly. Given the thermal inertia of the samples examined here, the repetition rates of many high-power x-ray pulses suggest that significant accumulation of heat during a pulse series is possible. For example, even for the lowest level of heating studied on our standard sample arrangement, for a 0.0035 mJ pulse (Fig. 4, ∆T =50 K initially), the cooling after ∼220 ns (the timing of a subsequent 4.5 MHz bunch pulse on European XFEL) reduces the sample peak temperature to ∼∆T =30 K, which is the starting point for the next pulse. In this scenario it would take roughly 50 pulses for the sample to be driven to its melting point (1811 K) from room temperature, in ∼11 µs. For comparison, European XFEL pulse trains will enable up to 2700 pulses, with peak temperature achieved being ∼7 eV over 600 µs, assuming the sample could survive P 0 11. Impedance match construction for the mechanical evolution of the x-ray heated sample (pressure P vs. mass velocity U P ). Shocks and releases are approximated as linear elastic (i.e. ∆P = ρ 0 c S ∆U P , where ρ 0 is the initial density and c S is a constant pressure wave velocity).

Uniform heating in each target layer (a sample surrounded by two infinite tampers) is assumed. A,
Compression and release response of the high-Z sample (S) and a low-Z tamper (T), where tamper is assumed to also have lower impedance. Lines indicate achievable states on compression from initial state P = 0, U P = 0, with release states also on this curve; dot represents a particular compressed state. B, Case of a freestanding sample layer in vacuum under x-ray heating. The sample foil is immediately driven to a high thermal pressure at zero velocity, and releases from both sides (Fig.   10), driving each side of the target to plus or minus a particle speed and zero pressure. These release waves converge at target center, causing a further stress reduction equivalent to the initial thermal pressure. The interacting release waves produce tension, and, if this tension exceeds the tensile strength of the material, spallation. C, Case of a tamped sample, with only a partial initial pressure reduction due to confinement by surrounding material, and a reduced but not eliminated tension state, which could still result in spallation (tension is prevented if the sample and tamper have closer impedances). D, The preceding scenarios apply for a typical laboratory condition with an initial pressure P 0 much less than the dynamic pressure (e.g. vacuum initial conditions). Tension can be prevented by applying an initial hydrostatic pressure to this system (P 0 > 0) comparable in magnitude to the initial dynamic pressure.
e.g. if configured as an anvil cell with eV capability on microsecond timescales 26 . Higher pulse power could reach such levels much faster. It is thus crucial to consider this energy deposition for serial crystallography applications even at lower (non-invasive) power levels.

C. Thermal effects of the beam
Many relevant effects within the targets can be considered entirely due to thermal effects, e.g. melting of simple materials, thermal transformation of diamond to graphite due to its metastable nature, and sublimation of graphite. Primary thermal effects could include reaction, irreversible and reversible phase transformation, melting, strength loss, and for free surfaces, the possibility of vaporization.
In reporting the results of our models, we have focussed on conditions at surfaces of the tamper, as these will characterize the peak temperatures to which the tamper is subject, and thus provide an estimate of the ability of the tamper to survive the thermo-mechanical cycle and successfully confine the sample throughout. For example, diamond will undergo graphitisation (or oxidation) near ambient pressure and melting under pressure 39 at conditions on the order of ∼1000 K or more, so ensuring the maximum temperature encountered in diamond does not exceed this limit is crucial for assessing its survival through experiments.
This includes the tamper surfaces facing the sample, heated by direct contact with the sample-medium layer, and the free surface facing the beam, which is heated most strongly by the x-rays directly.
Many of the temperature conditions modelled in these simulations are in principal such that targets (in particular their tampers) can survive irradiation. Except for softer x-rays ( Fig. 5), low-thermal conductivity tampers (Fig. 8) or no interfacial layer (Fig. 6) where D is an effective spot size. Such ablation is possible at free surfaces where unconfined atoms may easily escape the target, so is most directly relevant to the tamper. However, for the low-Z tampers considered here such as Be and C varieties, this limit is not easily For diamond, however, intensity-induced breakdown of diamond to graphite is anticipated to occur at lower energy, ∼ 0.7 eV per atom 29,32 , and we turn to this as one likely damage threshold for a diamond tamper specifically. Use of such a criterion (Eq. 17) can lead to rough estimates for acceptable beam parameters, such as adding further constraints on beam operating conditions at various radiation energies (Fig. 12), such as placing a lower limit on beam diameter (Fig. 13). In detail, consider the case of a diamond tamper at variable x-ray energy and pulse power (Fig. 12). Fluences remain below the predicted nonthermal radiation damage threshold for diamond 32 except possibly at the lowest energy (5 keV); we note that at such energies the overall damage threshold is likely lower as damage could also be thermal in nature due to the considerable absorption of the x-ray beam (Fig. 5). Based on the predicted nonthermal limit at 25 keV, a diamond tamper could enable iron sample temperatures up to ∼5 × 10 5 K (∼ 40 eV) in a single pulse without inducing radiation damage to the diamond (Fig. 12B); meanwhile direct heating of the diamond is roughly two orders of magnitude less than the iron ( Fig. 4 and Fig. 5), and so roughly a few thousand K (possibly survivable for a short interval), while the heat wave from the hot sample doesn't add more heat if the target is configured with a suitable interfacial layer (Figs 4,6,9). Using a heavier element sample could extend this peak temperature limit considerably, by possibly another order of magnitude (Fig. 7), accessing temperatures in the 100 eV range. These considerations are understandably speculative, and importantly, do not account for thermal pressure, which at the above temperatures would be significant.

E. Mechanical Stresses
The large mechanical stresses associated with target heating can introduce sudden or cumulative damage to targets over time, such as interfacial delamination, fracturing, and gap formation. Melting, flow, and vaporisation into gaps are possible. Thus, target survival after a single pulse or series of pulses will depend on the integrity of the target under mechanical stresses as temperature is raised.
The generated stress within targets can be large, but as this stress relaxes, relatively lower stress is applied to the surrounding materials and tampers. For example, for a 70 GPa initial stress in the Mo sample corresponding to a temperature of 17,000 K (Fig. 10), shock waves forming in conjunction with this pressure and striking the tamper are only a few tens of GPa in amplitude, and while producing a higher-stress reshock state upon hitting the tamper in this case, this still is well below the yield point of the tamper (diamond in this instance). Perhaps counterintuitively, the greatest mechanical risk to the integrity of the targets may be the tensile stress applied within strongly heated layers as they expand outwards, potentially tearing them apart with a stress reduction equivalent to the twice the initial increase (Fig. 11). This can create gaps even in an otherwise well confined target.
Notably, this can also cause extremely rapid removal of the most severe thermal pressures in the experiment.

F. Anvil Cell Configuration
The need to limit such damage suggests that a degree of target confinement could enhance survival, and consideration of an 'anvil cell' type design, where the sample is configured to withstand high stresses in the sample area via confinement by thick, hard materials, is a promising route to achieve this that is basically identical to the tamped experimental system explored in this study. High potential and applied confining stress can inhibit cracking and delamination (or any gaps in targets generally), miminize expansion i.e. by limiting thermal expansion of the condensed state 33 26,27,33 , and this is instructive for extending the lifetime of the extreme state in the present x-ray heating approach.
In one possible experiment, a tamped sample of an anvil-cell design could be placed under some small initial stress (to ensure good initial confinement) and thermal stresses introduced by x-ray heating could be controlled by the anvil's high strength and peak potential confining stress. So long as the confinement can withstand the additional thermal stresses (on the order of GPa or higher for conditions considered here, which is readily achieved) (Figs 10 and 11) then the target could be stabilized.
The ability to pre-compress samples to elevated densities can also offer, in conjunction with x-ray heating, a new route to studying warm dense matter at conditions in excess of solid density, and on a range of timescales, accessing phenomena from electron-ion thermalisation to structural transitions. Static pre-compression of matter to hundreds of GPa confining pressure, or larger using modern double-stage anvils 38 , is a common methodology in producing extreme states of pressure, density and temperature by manipulation of density by static pre-compression prior to excitation and probing. This offers a potential complementary approach to dynamic compression for exploring relevant ultra-dense states of matter. Addition of pressure could, at least for the sample interfacing region, elevate the damage thresholds for the diamond tamper, both in terms of its thermal resistance and mechanical resistance. Thermal graphitization is prevented above ∼13 GPa where diamond becomes the stable structure of carbon, whereas the melting temperature of diamond at these conditions exceeds 4000 K 39 . Confining pressure also increases the the strength of diamond 37 , a fact employed in modern anvil cell designs to enhance the potential stress resistance 38 . Additionally, as tensile stress is generated for foil irradiation and release under normal conditions (Fig. 10), pre-compression in a diamond cell offers a strategy to maintain the perature behaviour at these early times. A comparison of the temperature history at the sample center shows notable differences in simulated temperature only during heating (a shorter pulse was assumed for the DAC simulation), and very late in the cooling phase. The latter difference is due to the larger heat sink provided by the full-size target assembly, resulting in lower limiting temperature. The cylindrical system modelled is found to describe the principal thermal phenomena of the more complex target assembly accurately, and at lower computational cost. Slight differences in beam diameter between the simulations are also noted. layer integrity, allowing stress reduction but not tension by beginning at a static pressure exceeding the peak pressure, inasmuch as, to first approximation, release can produce a pressure reduction of the same magnitude as the initial increase (Fig. 11). This would eliminate any tension, spall, and vaporization at the internal interface, maintaining target confinement.
The ∆P V assessed in the earlier manner (Sec. II B) gives an upper limit on the mechanical stress that might be encountered in such experiments. Reduction from this value occurs due to the hydrodynamic expansion (to zero stress for free expansion) and decreasing temperatures away from the hotspot need also be considered. Typical 'equilibrium' thermal pressures in a laser-heated diamond anvil cell having similar temperature distributions to those obtained here are ∼ 30% of ∆P V 33 , which accounts for the partial confinement of the sample in the long duration limit after hydrodynamic processes have occurred. Indeed, the anvil cell provides a built in way to safely relieve thermal stresses in samples to this mechanical equilibrium confinement state without hydrodynamic expansion, solving a principal issue in tamped laser-driven targets that may only be partially mitigated by tamping alone.
Finally, heating via x-rays in a diamond anvil cell, as an alternative to optical laser heating, could allow heating through anvils that are not optically transparent and provide better initial spatial homogeneity in heating (e.g. if the beam profile was engineered as a flat top rather than as a Gaussian (Fig. 2C), though temperature gradients would rapidly develop after the x-ray pulse regardless of initial heating uniformity (Fig. 3).

V. CONCLUSIONS
This study outlines the long-duration response of thickly-tamped targets to thermomechanical excitation by an intense x-ray pulse, similar to those produced by current generation free electron lasers. Going beyond the early dynamic evolution of electrons and lattices in typical isochoric heating, our study examines the principal thermal and mechanical evolution that follows these relatively faster processes.
Conditions achieved at the most extreme temperatures fall within the regime of warm dense matter, i.e. conditions near solid density and temperatures exceeding several eV where ratios of Coulomb interaction energy to thermal kinetic energy Γ (the coupling parameter) and of Fermi energy to thermal energy Θ (the degeneracy parameter) approach unity. That these conditions could be sustained for up to microsecond timescales offers a potential way to study properties of warm dense matter under total thermodynamic equilibrium conditions, on timescales exceeding those of modern laser-driven shock experiments as well as traditional isochoric heating experiments. Pre-compression and tamping by thick anvils can enable further exploration of novel regimes of density, temperature, and timescale in warm dense matter.
Stresses introduced by thermal energy deposition, on the order of tens of GPa, are sufficient to produce high amplitude stress waves in samples, including shock waves, which in addition to affecting the thermal state can play a significant role in stress evolution and damage. We conclude that stresses induced by the heating and sudden thermal expansion, rather than thermal effects directly, will be a principal mode of failure for tamped targets. Hence, the survival limit is strongly defined by the targets potential resistance to thermal-mechanical stress, shock, and tension, potentially more than by any direct thermal or radiation damage in many instances considered. The initial stress waves emerging from a directly heated sample would likely be a likely cause of first failure, so the mechanical strength of tamper remains a critical parameter. Using a low strength and compressible material surrounding (or comprising) the heated part of the target (to dissipate and distribute thermal stresses is one way to alleviate these stresses.
We find that consideration of heating effects is necessary even for nominally noninvasive x-ray probing experiments using modern, high repetition rate, high brightness x-ray sources.
Even when reduced to power levels orders of magnitude below facility maximums, accumulation of heat over pulse trains can be substantial. Several strategies have been explored which can minimize heating where non-invasive x-ray probing is desired. Clearly, maintaining total pulse energy at low levels, use of broader focal spots, and avoiding higher-Z materials is the simplest approach. However, use of high thermal conductivity tamper materials surrounding samples has little impact on minimizing achieved temperatures, as the timescale of heat deposition is so fast that there is no time to remove heat from the system while peak temperature is reached.
The general thermal behavior of the targets determined in our simulations (such as peak temperature and cooling behavior) would be essentially unchanged for x-ray pulses of other durations than the femtosecond pulses used in our study. Sample layers are effectively adiabatic following illumination up to timescales of nanoseconds, so pulses of durations less than a few nanoseconds with the same energy and spatial distribution will produce the same peak temperature. This principle could be used to control thermal stress conditions in samples for a given required total pulse energy e.g. by providing more gradual stress changes and minimizing shock formation.
The multilayer target configuration discussed here is informed by, and mimics, the configuration of a high pressure anvil system, of which the diamond anvil cell uniaxial press is the most well recognized. There are several reasons for this. First, anvil cells have the ability of generating initial states of elevated density and pressure in samples, prior to excitation to more extreme states; this method of 'pre-compression' has been widely advantageous for preparing samples for adiabatic shock-wave compression 42,43 and near-isochoric optical laser heating 26,27,33 , providing access to otherwise unreachable states of matter as well as access to these states by a range of radiation probing via the high transparency of the anvils. In Complete treatment of the present ultrafast heating would include not only longer duration phenomena at thermodynamic and statistical equilibrium (the conditions under which the target evolves to a steady state on timescales of nanoseconds and longer), but also nonequilibrium initial states produced by extreme irradiation, including, transfer of energy from hot electrons to the lattice, electronic excitations of core electrons or from valence to conduction bands in dielectrics, ballistic electron heat transfer on short timescales, ultrafast phase transformations, and many other potentially relevant phenomena affecting thermal and inertial transport. For our purposes, we assume that these processes occur relatively rapidly and that the ultimate thermal state is still adiabatic but has reached an equilibrium condition indistinguishable from that had the system been heated by more gradual means at along a thermodynamic path (i.e. at thermodynamic equilibrium). This assumption gives a reference condition for the plausible thermal evolution of samples after ultrafast-timescale effects have dissipated, from which more complex models could be developed. Ultimately experiments must be performed to assess the accuracy of the models developed here, as will soon be feasible on modern x-ray facility sources.