Growth of, and diffusion in, olivine in ultra-fast ascending basalt magmas from Shiveluch volcano

Complex core-rim zoning of Mg-Fe-Ni-Ca-Cr-Al-P in high-Mg olivine crystals from a tuff ring of Shiveluch volcano, Kamchatka, enables reconstruction of the entire olivine crystallization history from mantle conditions to eruption. Bell-shaped Fo86–92 and Ni profiles in crystal cores were formed by diffusion after mixing with evolved magma. Diffusion proceeded to the centres of crystals and completely equilibrated Fo and Ni in some crystals. Diffusion times extracted from Fo and Ni core profiles range from 100 to 2000 days. During subsequent mixing with mafic mantle-equilibrated melt, the cores were partially dissolved and overgrown by Fo90 olivine. Times extracted from Fo and Ni diffusion profiles across the resorption interface between the core and its overgrowth range within 1–10 days, which corresponds to the time of magma ascent to the surface. The overgrowth shows identical smooth Fo-Ni decreasing zoning patterns for all crystals towards the margin, indicating that all crystals shared the same growth history after last mixing event prior to eruption. At the same time, Ca, and to an even greater extent Cr, Al, and P have oscillatory growth patterns in the crystals overgrowth. Our data show that magma ascent can be extremely short during maar/tuff ring eruption.


SM1.1. Mineral microprobe analyses
All mineral compositions, geochemical profiles and elemental maps measured on olivine were determined using a JEOL JXA 8900RL microprobe equipped with 5 wavelength dispersive spectrometers (WDS) installed at the GZG (Geowissenschaftliches Zentrum Göttingen) of the Georg-August Göttingen University. The program to analyse olivine compositions with high precision for trace elements was modified based on a published approach 1 . Olivine measurements were run at 20 kV accelerating voltage and a focused beam of 300 nA and 1-5 μm diameter. For standardization of major and trace elements, we used a set of synthetic and natural standards. Peak counting times for major elements were 15 s, for Mn -60 s, for Cr, Zn and Ca -120 s, for Co -180 s, for Ni and Al -260 s, and for P -300 s (  (Table SM1.2). For other elements, no systematic offsets within the errors by counting statistics and the uncertainties of matrix correction procedures were observed. Thus, the published reference values were used for MgO, MnO, SiO2 3 ; for Cr2O3 4 ; for CaO 5 ; for Al2O3, P2O5, CoO, ZnO 1 . SC-Goe was analysed during all measurements after 30-50 analyses of unknowns. The quality of analyses was also controlled by mineral stoichiometry for the olivine. The absolute average errors (2σ) by counting statistics for major and trace elements on the SC-Goe olivine are (mass-%): SiO2 ~ 0.24, MgO ~ 0.3, FeO ~ 0.12 and NiO ~ 0.003, MnO ~ 0.004, CaO ~ 0.0016, Cr2O3 ~ 0.0022, Al2O3 ~ 0.0013, ZnO ~ 0.0033, CoO ~ 0.0017, and P2O5 ~ 0.0017. The standard deviations calculated from n=63 total measurements are 1.2-to 3-times larger than the predicted error by counting statistics, except for Al2O3, where the standard deviation of 63 measurements of San Carlos olivine is 6.7 times larger. Relative deviations from recommended values are better than 3% for all oxides except for Co, P (5%), Cr (9%), and Zn (~48%). Due to a non-linear behavior of the background signal near the Co-Kα-line resulting from the nearby Fe-Kβ interference, a linear correction factor was applied to correct for the systematic underestimation of the raw net-signals. Statistics for the olivine analyses are given in Table SM1.1-B. Detection limits are calculated from the error by counting statistics of the background noise and given as 2-sigma values in Table SM1.1-B.
To increase the analytical precision for aluminum in olivine for the application of the sp-ol geothermometer 6 , a specially tuned analytical program for Al was designed where Al2O3 was simultaneously measured on two spectrometers. This technique allowed to an improvement to the 2 sigma error by counting statistics to ±8.4 µg•g -1 and a reduction in the detection limit down to 5.6 µg•g -1 .
The compositions of other rock-forming minerals in basaltic tephra (spinel, pyroxene, and plagioclase) were determined using routine procedures at 15-20 kV and 15-20 nA with a 1-10 μm beam diameter. Peak counting times for major elements were 15-30 s, and for trace elements 60 s. Analytical precision was better than 5% for most major elements and 7% for K and P 7 .

SM1.2. Element distribution maps
Element distribution maps of Fe, Mg, Ni, Ca, Cr, Al, and P in olivine were carried out on whole olivine grains or on some sections or individual zones of olivine crystals with excitation conditions similar to those used for the high-precision quantitative analyses -20 kV and 300 nA. Depending on the size of the analyzed area and on the required spatial resolution, step width and dwell time per pixel were changed from 0.1 to 1.8 μm and from 100 to 210 ms, respectively. The covered areas range between 300 and 800 μm 2 and the total acquisition time range from 5 to 15 hours per map. A focused beam was used for all element maps. The full measurement conditions for 5 elemental maps can be found in Table SM.1-C.

SM1.3. EBSD analysis
Diffusion modelling in olivines requires the exact knowledge of crystal orientation due to the strong anisotropic diffusion of Fo-Ni in olivine [8][9][10] , which can be described as Da=Db=Dc/6. Therefore, the orientation of every crystal must be measured to obtain correct diffusion times. The crystal orientation was determined via electron backscatter diffraction 11 (EBSD) on a Quanta 200 F instrument at the Crystallography Department at the GZG, Georg-August Göttingen University.
For EBSD analysis, the thin sections were polished in addition to the normal procedure for EMS to a final polishing fineness of 0.05 μm Al2O3. Every sample was covered with a very thin carbon layer to minimize electrostatic charge due to the high vesicularity of the samples. Several points were measured along the compositional profile of every grain to ensure that crystal orientation does not change within the crystal due to cracks or deformation.
The software OIM Analysis 12 was used to calculate the angles between the line profile and the crystallographic axes obtained from EBSD. These angles can be used to define a diffusion coefficient along the measured profile DPr using an equation 8,13 , which is: with Da, Db, and Dc as diffusion coefficients along the crystallographic axes a, b, and c, and , , and  as angles between the measured profile and a-, b-, and c-axis.

SM2. Profile measurements
This part of supplementary materials presents results of measuring elemental concentrations and crystal orientations along the profiles in olivine grains. We measured 27 profiles across 19 carefully selected olivine grains in eight thin sections from two samples. Table SM2-A contains the raw microprobe data. Table SM2-B contains raw orientation data.

SM2.1. Profile SHIV-08-05 17 Ol-8-2
We will demonstrate the scheme of analyzing the measured compositional profiles in olivine crystals using as example profile SHIV-08-05 17 Ol-8-2. The distribution of Fo, NiO and Cr2O3 are shown in Fig. SM2.1a, a Fo-Ni diagram is shown in Fig. SM2.1b. Table SM2 (Table SM2-C).  Objects and valuesshows the rows for the Bunge's form of three Euler angles 12 ; the row for starting and ending coordinates of the profile.  Rotation matrixthe components of rotation matrix for calculating of crystal's axes orientations.  Axesshows the three crystal axes rows and measured profile row.  x and ycoordinates of stereographic projection to lower hemisphere of the three crystal axes and measured profile.  cos 2the squared cosines of angles between the measured profile and a-, b-, and c-axis of crystals. The "Profile" raw shows the check sum of three previous squared cosines: should be equal 1.  Geometric factor APr, i.e. the components of the diffusion coefficients along the a-, b-and caxes 8 and the resulting geometric factor APr of the diffusion coefficients along the profile.  are plotted on the right axis of the plots  Plots for Ca, Co, Al, P, Zn -CaO are plotted on the left axis, CoO, Al2O3, P2O5, and ZnO are plotted on the right axis of the plot  SEIimages of grains from microprobe in SEI mode  COMPO and profilesimages of grains from microprobe in COMPO mode. The positions of measured points are represented by red dots  Projectionslower hemisphere of stereographic projection of a-, b-and c-axes of crystal

SM3. Estimating P-T-fO2 conditions of olivine crystallization
Because Mg, Fe, and Ni are highly diffusive elements in olivine, any temperature estimate for olivine studied here would be not correct using thermometers based on these elements. Therefore we used Al-in-olivine thermometry 6 , which is based on Al-Cr distribution between olivine and spinel. Both elements have low diffusivity in both phases 14,15 . Another advantage of this method is that today it is possible to measure low concentrations of Al-in-olivine by electron microprobe at high spatial resolution and with high precision 1,4,5 . The measured data are represented in Table  SM3-A. Oxygen fugacity was estimated using the approach 16 based on the improved Ballhaus-Berry-Green ol-opx-sp oxybarometer. Fugacity estimates were done on the same ol-sp pairs, which were also used for thermometry. In all cases, ol-sp pairs were measured closely at the contact of the olivine to their spinel inclusions in the different parts of the phenocrysts.
Pressure was estimated from clinopyroxene compositions and clinopyroxene-melt equilibria (Eq. 30, 31, 32b and 32c in ref. 17 ). Fe-Mg diffusion in clinopyroxenes is much slower compared to olivines and even compared to orthopyroxenes 18,19 although can be complicated by strong compositional zoning. Thereby the clinopyroxene-melt barometer does not suffer from Mg-Fe diffusion. To calculate the pressure using the barometers 17 , we used H2O estimates 20 in the melt as 4%. Olivine compositions from the overgrowths (Fo88-90.5, SM2) are in equilibrium with the whole rock (a proxy for their host melt). Six cpx grains, however, show variable compositions with Mg# 68-87 (Table SM3-B) and none was in equilibrium with the whole rock composition. However, after subtracting 5% of olivine Fo90 from the whole rock composition, the cpx compositions passed the equilibrium test 17 . The pressure thus determined from clinopyroxene-melt compositions was 6 to 10 kbar, consistent with the results from cpx-only barometry -6 kbar. These data are in good agreement with previously published estimates in 7-9 kbar 21 .
Temperatures estimated for the inner cores of Shiveluch olivine crystals range from 1230 to 1260 °C and are lower in outer cores (1170-1190 °C) and gradually decrease from the transition zone to the rim: 1150-1220 °C in the transition zone, 1160-1200 °C in the overgrowth and 1130-1155 °C at the rim. There are no spinel inclusions in the outermost rims, but the pre-eruption temperature determined for Shiveluch lavas of a similar composition was around 1100 °C, i.e. still slightly lower than and thus in accordance with temperatures estimated towards the rim of olivines.
Table SM3-C also contains the calculation of diffusion coefficients 14,22 for all zones of the crystals for which the temperatures were determined individually. We chose the conditions for lowest and for fastest diffusion coefficients for every zone. The diffusion coefficients of Fo and Ni will be used in subsequent supplementary materials to estimate the timescales and durations of diffusive processes.

SM4. Outer core diffusion times estimation
Olivine crystals from group 1 have experienced diffusion in the outer cores. During this diffusion stage, the maximum values for Fo and NiO in the inner core remained nearly constant and only outer core parts were affected by diffusion. Such a case, outer core diffusion, permits a onedimensional model description of the diffusion process and it is possible to calculate the diffusion time from a simple analytical solution.
The inner cores of olivines from group 1 are in equilibrium with high-Mg melt and have compositions of Focore=91.8-92.1 mol. %, and NiOcore=0.45-0.49 wt. % (Fig. 8a, Fig. 8b). The rims of these crystals suffered diffusion at the contact to a more evolved melt in equilibrium with Fodm=88.7 mol. % and NiOdm=0.23 wt. % (Fig. 5). At these parameters, there is a linear dependency between the left and right sides of the equation Eq. 3. The slope of correlation line in Fig. 5c controls the ratio DNi/DFo=0.86. In this case, the concentrations of forsterite Fo and nickel NiO along the profile will be described by following equations (see Eq. 3.13 for semi-infinite medium in ref. 23 ): The characteristic dimensions of the diffusion zones ΔNi, ΔFo and position of xdm are calculated by the least square method from equations Eq. SM4.1-SM4.3. Note that xdm does not mark the margin of present crystal but the original margin of the now partially resorbed core when it was exposed to a more evolved melt. Thus, the values ΔNi, ΔFo and xdm need to be evaluated from only partially preserved profiles after the resorption event.
Five profiles on three olivine grains were used for estimating the duration of outer core diffusion.
The calculated values for ΔNi, ΔFo and xdm are shown in Table SM4 Because the outer core diffusion occurred after the olivine crystals were moved into a more evolved melt, the P-T values specific of the transition zone (raw 3 in Table SM3-C) were used for DFo and DNi calculations. Correction for crystal anisotropy from Table SM2-B was also applied.
Contrary to our estimate for DNi/DFo=0.86 (Fig. 5c), independent data for the same temperature and compositions 14 (raw 3 in Table SM3-C) give a DNi/DFo value about 1.5. Taking into account the uncertainties in the determination of diffusion coefficients as well as the effect of P-T-fO2, the diffusion time is estimated at 400-1800 days (last columns in Table SM4-A, and blue line in Fig.  7).

SM5. Advanced core diffusion times estimation
Olivine cores that were affected by diffusion from margin to centre are identified by a corresponding decrease in their maximum Fo and NiO in the core. These grains were affected by what we call advanced core diffusion. To describe the diffusion effects through the entire crystal, a diffusion model should take into account the following parameters: the three-dimensional pattern of diffusion, anisotropy of the diffusion, and the complex shape of the original crystal (which is not preserved because of resorption).
A series of calculations with anisotropic crystals of variable shapes shows rather large variations in diffusion times. It is also obvious that we cannot constrain the parameters above with any confidence for the resorbed cores. For this reason our diffusion time estimation can only be an "order-of-magnitude" approximation using the simplest model where the olivine grain is represented as a sphere with radius R. In this model the olivine crystal anisotropy is controlled by the additional geometric factor APr to the diffusion coefficient along the profile.
For example, consider the following scenario: a uniform sphere with radius Rdm (dmdissolved margin, representing the original margin of the crystal that will be later dissolved) containing Focore and Nicore components is moved into a melt that is in equilibrium with Fodm and Nidm. If diffusion coefficients are DFo and DNi for Fo and Ni, respectively, then for the point with coordinate r inside the sphere at time t the solution for Fo and NiO can be described as: Here F is the solution of the diffusion equations in a sphere with constant initial conditions and given compositional values at the boundary (Eq. 6.18 in ref. 23 ). It is assumed that all initial crystals have the same composition Focore=92.16 and NiOcore=0.48 wt. % (the composition of the most magnesian core in crystal SHIV-08-05 17 Ol-7 from group 1), and the melt around is in equilibrium with Fodm=86.4 and NiOdm=0.22 wt. % (in equilibrium with olivine cores from group 4). The diffusion coefficient ratio was estimated from the diffusion trend of group 1 to group 4 core compositions (purple line in Fig. 2b) as DNi/DFo=0.74. In this case, using equations Eq. SM5.1 and Eq. SM5.2 by the least square method for each profile allows to calculate the values of Rdmolivine radius before its melting, and ttime of diffusion.
Because the advanced core diffusion (as in case with outer core diffusion) occurred after the olivines were moved to more evolved melt, the P-T values characteristic of the transition zone (raw 3 in Table SM3-C) were used for DFo and DNi calculations. Correction for crystal anisotropy from Table SM2-B was also applied.
18 profiles on 15 olivine grains were used to estimate the duration of advanced core diffusion. Rdm and t were calculated from the measured profiles by the least square method (Table SM5-A).
Representative profiles together with locations of dissolved margins and the approximation curves are shown in Fig. SM5.1.
The calculated coordinates of dissolved margins show some variation for small crystals (Fig.  SM5.2). Larger crystals show consistent results, suggesting that 0.07-0.09 mm of the original crystals were dissolved. Coordinates of the dissolved margins, calculated by the outer core diffusion model (red dots on the Fig. SM5.2) and by the advanced core diffusion model (red line on the Fig. SM5.2), are close each other. This supports the validity of our assumptions and models. Our estimate of DNi/DFo=0.74 differs from published data 14 for olivines of the same composition and conditions of formation (raw 3 in Table SM3-C) which indicate a DNi/DFo ratio of about 1.5, (SM4). Taking into account the uncertainties in the definition of diffusion coefficients as well as P-T-fO2 conditions, the diffusion time interval is in the range of 100 to 2000 days (last columns in Table SM5-A, and green line in Fig. 7).
Outer core diffusion is observed only in larger crystals (Fig. SM5.2), which had no time to diffuse into the core during the estimated diffusion time. In fact the outer core diffusion is only the special case of diffusion for large crystals whereas smaller crystals have partly or completely equilibrated during the same diffusion time. This explains nicely the close agreement between outer core diffusion times (blue line in Fig. 7) and advanced core diffusion times (green line in Fig. 7), estimated by two methods. Some discrepancies may be related to differences in Fodm, Nidm and ratio DNi/DFo estimates.

SM6. Сore-overgrowth diffusion times estimation
The transition zone with drastic changes of Fo and Ni occurs between the core and the overgrowth of the crystal. Since the transition zone was affected by diffusion, the size of the transition zone for NiO depends on the coefficient DNi and the size of the transition zone for Fo depends on the coefficient DFo, and in both cases the size is determined by the action time of diffusion. Thus, the width of the diffusion zone at the core-overgrowth boundary allows us to determine the time elapsed between the overgrowth formation and the crystal quenching after eruption to the surface.
Here we apply an analytical solution of the diffusion equation for times estimations through Fo and Ni distributions across the transition zones in group 4 olivines.
The transition zone of the olivine crystals from group 4 represent a convenient case for the evaluation of the diffusion time in the system. The cores of these olivines come to equilibrium and have flat Fo86-88, and NiO=0.2-0.3 wt. % distributions (Fig. 3a, Fig. 8b). After mixing with hot magnesian melt the olivines were partly dissolved (Fig. 8c) and then overgrown by high-Mg high-Ni olivine with Fo89.5-90 and NiO=0.4-0.45 wt. % (Fig. 8d). Diffusion of Fo and Ni across this boundary forms the transition zone (Fig. 8e).
To model the diffusion profile through the transition zone all measured points of the core were considered because for group 4 olivines the Fo and NiO values inside the core are nearly constant (filled circles in Fig. SM6 where subscripts 1 and 2 correspond to parameters at both sides of the transition zone, Erferror function, x0initial location of the contact boundary, ΔFo and ΔNithe characteristic halfdimension of the diffusion zone for Fo and Ni, respectively. Unknown parameters in Eq. SM6.1 and Eq. SM6.2 were calculated by the least square method that requires the solution of a system of nonlinear equations. The position of the diffusion interface, i.e. the contact boundary between the resorbed core and the overgrowth, must be the same for Fo and Ni. However, it is possible to determine this parameter for Fo and Ni separately, and after comparing them with each other, to control the quality of the model. Thus, for each Fo and Ni profile 4 parameters were determined: the compositional value to the left, the value to the right, the position of diffusion boundary and the characteristic half-dimension of the diffusion zone. The evaluated half-dimensions of the diffusion zone are related to the diffusion time as: (Eq. SM6. 3) were DFo and DNi are diffusion coefficients. Because diffusion in the transition zone occurred after the overgrowth was arise on olivine core, the P-T-fO2 values characteristic of the overgrowth (raw 4 of Table SM3-C) were used for D calculations. Correction for crystal anisotropy from Table  SM2-B was also applied. The ratio of half-dimension of the diffusion zones is defined by the ratio of DFo and DNi: Nine olivine profiles (all from group 4) were used for diffusion time estimates. All relevant parameters for modeling these profiles shown in Table SM6 Table  SM6-A show that the dimension of the diffusion zone for Fo is always larger than the dimension of the diffusion zone for Ni, (8 definitions of 9). On average, the ratio (width of the diffusion zone for Ni relative to the width of the diffusion zone for Fo) is 0.6. Using this data and the equation Eq. SM6.4, the DNi/DFo ratio is estimated at about 0.4.
We simultaneously measure and model diffusion of Fo and NiO across a diffusion zone inside olivine crystals. This allows, for the first time, to directly assess their relative diffusion coefficients.
In previous studies, such diffusion modeling was done separately either for Fo (e.g. 24 ) or for NiO (e.g. 25 ). Modeling of olivine profiles with simultaneous exchange Fo and NiO previously were described only for the rims of crystals. However, as we show in this study, such rims maybe affected by both diffusion and crystal growth, and comparing results for Ni and Fo may not be advised in this case.
The model curves depend strongly on the relative diffusivities of Ni and Mg-Fe. We found the best fit between observations and model for DNi<DFo. This is in conflict with published 14 the diffusion coefficients that indicate DNi>DFo However, it is known that at high P-T conditions the "Fe-Mg interdiffusion, Ni diffusion and Mn diffusion are almost identical within the uncertainties" 26 . We thus argue that Fo and Ni diffusion coefficients are not well enough constrained for the P-T conditions relevant here and that the apparent contradiction is within the uncertainty.
Despite our estimate of DNi/DFo=0.4 for core-overgrowth diffusion, the approximation 14 for the same conditions (raw 3 in Table SM3-C) give a DNi/DFo ratio about 1.6. Taking into account the uncertainties in the definition of diffusion coefficients as well as P-T-fO2 conditions, the diffusion time interval is 1-10 days (last columns in Table SM6-A, and red line in Fig. 7).