Relationship between the Texture and Composition of Titanomagnetite in Hannuoba Alkaline Basalt: A New Geospeedometer

Abstract

The nucleation and growth of crystals in igneous rocks is usually thought to occur under thermodynamic equilibrium conditions. However, recent studies on igneous textures and mineral compositions have shown that these processes probably occur under thermodynamic disequilibrium conditions. Titanomagnetite with variable crystal sizes can be observed in Hannuoba alkaline basalt, indicating disequilibrium crystallization processes (different cooling rates). The ratio of the maximum particle size to the area abundance of titanomagnetite, as determined by an analysis of previous studies on the texture of minerals, was negatively correlated with the apparent cooling rate. We analyzed the chemical composition and crystal size distribution of titanomagnetite in ten Hannuoba alkaline basalt samples to determine the connection between the apparent cooling rate and titanomagnetite composition. In Hannuoba samples, the cooling rate was found to affect cationic substitution in the titanomagnetite solid solution, and an increase in cooling rate led to a decrease in Ti⁴⁺ and an increase in Fe³⁺. The partition coefficient of Ti between titanomagnetite and the melt (DTi) is negatively correlated with the apparent cooling rate. These findings are consistent with those in experimental petrology and help us propose a better, more general geospeedometer. The cooling rate also impacted Mg²⁺ and Al³⁺, but they were more impacted by the melt composition and crystallinity of the coexisting melt. Therefore, a new geospeedometer was calibrated by considering the titanomagnetite composition, melt composition and the content of the clinopyroxene.The cooling rates of the Hannuoba basalt samples measured using the new geospeedometer calibrated in this study range from 0.7 to 7.0 (±0.5) °C/min. It cannot accurately predict the cooling rate from titanomagnetite in intermediate rock, felsic rock or Fe-rich basaltic melts. The new titanomagnetite geospeedometer can better measure the cooling rate of alkaline basalt and may help identify the effects of kinetically controlled crystallization on isotope fractionation, evaluate mineral thermobarometers and better recognize thermal remanence magnetization and ancient magnetic fields.

1. Introduction

The study of the cooling rate in igneous rocks is an effective supplement to the dynamics of igneous rocks and can further improve and constrain the dynamics of the magmatic process. The chemical compositional characteristics of rock-forming minerals have long been used to estimate temperature and pressure in igneous systems under thermodynamic equilibrium conditions [1].

However, recent studies suggest that major and minor element diversities in many rock-forming minerals, such as olivine [2,3], titanomagnetite [4,5,6,7], plagioclase [8,9,10] and clinopyroxene [11,12,13,14], could be significantly affected by cooling rate variations, which is a challenge for traditional geothermobarometers. In addition, if the cooling rate of magnetite is ignored, extending to geomagnetism can affect the intensity of the thermal remanence magnetization (T.R.M.) acquired by rocks during cooling to ambient temperature, which may lead to an underestimation or overestimation of the intensity of ancient magnetic fields [7,15,16]. Therefore, this encourages us to develop a mineral geospeedometer that records the cooling rate as magma rises [5,6,17,18,19].

In many igneous rocks, titanomagnetite is a common accessory mineral [20,21,22]. It can incorporate many major and trace elements, and cation redistribution in a titanomagnetite solid solution frequently occurs [4,6,23]. Chemical changes in titanomagnetite can reflect the variations in coexisting melt composition, temperature, pressure, oxygen fugacity, sulfur fugacity, undercooling, cooling rate and fluid compositions during the magma solidification process [6,18,24,25,26,27]. Chemical changes in titanomagnetite make it a useful petrogenetic indicator and geological processes chronicler mineral for mineralogy, petrology and geochemical studies [6,7,21,23,28,29,30,31,32,33,34,35]. Studying the distribution of cations in titanomagnetite can help us understand the magmatic environment in which minerals are formed.

Previous studies have found that pillows of mid-ocean ridge basalt (MORB) show a substantial change in textural and titanomagnetite mineralogical features as a function of cooling rate. These variations are of great importance to the characteristics of the MORB’s magnetization [7,15,36,37]. Recent experimental studies on trachybasalts, basalts and basaltic trachyandesites have shown that the cooling rate significantly affects the texture and chemical composition of titanomagnetite [5,6,7,38,39,40,41]. In these studies, when the cooling rate becomes faster, the particle size of titanomagnetite gradually decreases, and some titanomagnetites show dendritic morphology [5,6,7,40]. Compatible elements gradually replace other elements in the crystal lattice if the cooling rate is slow. The results of experimental petrology studies [6,34,42] show that, in the case of increasing cooling rates, titanomagnetite in basalt is rich in Mg²⁺, Al³⁺ and Fe³⁺ but poor in Ti⁴⁺ and Fe²⁺, and its ulvöspinel (Usp) end member is greatly reduced during the kinetically controlled crystallization process.

One titanomagnetite geospeedometer has been developed using cation redistribution from experiments conducted under suitable fO₂ conditions to estimate the cooling rate of a dike outcropping on Mt Etna volcano [6]. The cooling rate from the innermost to the outermost part of the dike is from 0.02 to 1.13 °C/min [5,6]. However, the geospeedometer above does not take into account the composition of the melt (in Hannuoba basalt, Mg²⁺ and Al³⁺ in titanomagnetite are affected by melt composition) or other possible factors [6]. Melt composition may have an influence on titanomagnetite composition, so the geospeedometer should consider its influence on titanomagnetite. We need to find a geospeedometer suitable for experimental petrology and most field samples.

Zhou et al. [7] explored whether the Ti content in titanomagnetite is directly proportional to the titanomagnetite particle size in MORB. The results show that the rapid cooling rate leads to small particle sizes and a decrease in the Ti content in titanomagnetite. Crystal size may be an indicator of apparent cooling rate: when the cooling rate is slow, the nucleation rate is slow (relative to the crystal growth rate), and large crystals are easily formed. In contrast, when the cooling rate is fast, the nucleation rate is fast, and tiny crystals are easily formed. These phenomena are observed in field samples and products of high-temperature experiments, showing that the cooling rate is generally inversely proportional to the size of the crystals [5,6,7,13,43,44]. However, the crystal size is determined by various factors, such as the crystallization time, nucleation rate, cooling rate, magma mixing, and overgrowth [34,45,46,47,48,49]. Previous experimental studies explored the relationship between the cooling process and mineral quantitative textural parameters (e.g., [34,43]), indicating that the crystal size may not accurately reflect the cooling rate. So, more theoretical and case studies on the relationship between cooling rate, titanomagnetite texture and chemical composition are needed.

Hannuoba basalt is representative of Cenozoic basalt in eastern China [50,51,52,53,54,55,56,57,58]. Titanomagnetite with variable crystal sizes and texture can be observed in Hannuoba alkaline basalt, indicating kinetically controlled crystallization processes (different cooling rates).

In this study, we determine better parameters to describe the apparent cooling rate using the crystal size distribution of titanomagnetite in Hannuoba basalt. We found that the ratio of the maximum particle size to the area abundance of titanomagnetite could represent an apparent cooling rate in Hannuoba alkaline basalt. Furthermore, this study attempts to determine how cooling rate affects cation redistribution in titanomagnetite by comparing the apparent cooling rate of natural samples and the chemical composition of titanomagnetite with experimental petrology. The study shows that, although rock content and mineral content can influence cation redistribution, the cooling rate still plays a critical role in the composition of titanomagnetite despite other geological factors. The new geospeedometer obtained in this study is suitable for mafic rocks, especially alkaline basalt.

2. Geological Setting

The Hannuoba basalt is one of many Cenozoic basalts in eastern China (~200 km NW of Beijing) and includes alkaline and subalkaline basalts [50,51,52,53,54,55,56,57,58] (Figure 1a). Overall, Hannuoba basalt is a fissure overflow basalt, a vast lava slab that has unconformity above the Archean, Jurassic, Cretaceous, and Yanshanian igneous rocks and earlier intrusive rocks. Hannuoba basalt is a multicycle product with at least two large magma cycles at approximately 26 Ma in the early–middle Miocene. Its area is up to 1700 km², where alkaline and subalkaline basalts are interlayered, forming as much as 90% of the outcrop [54]. Mantle xenoliths are widely found in Hannuoba alkaline basalt. Previous studies suggested that Hannuoba xenoliths bearing alkaline basalt derived from isotopically homogeneous sources and crustal contamination are negligible [53,54], although the nature of the source rocks remains under debate [53,54,59,60]. High-pressure eclogite fractionation may cause compositional diversity in basalts [54].

3. Methods

3.1. Quantitative Mineral Texture Analysis

In recent years, quantitative rock structure analysis has gradually become an important research focus in igneous rock [34,43,44,61,62,63]. By quantifying a series of structural parameters, such as mineral or porosity content, three-dimensional morphology, degree of self-shape, grain size distribution, spatial distribution, degree of orientation and dihedral angle [42,45,47,63,64,65,66,67,68], we can quantitatively understand the diversity of igneous rock structure characteristics and the dynamic process of magma crystallization [4,42,43,44,45,46,47,66,69,70].

Marsh [45] introduced crystal size distribution (CSD) theory into the study of magmatic systems. In recent years, this theory has been widely used to discuss the crystallization kinetics of igneous rock systems (e.g., [4,43,44,46,61,64,69,70,71,72,73,74,75,76,77,78,79,80,81,82]). In simple terms, CSD theory refers to the representation of a mineral in a rock in the covariant diagram of ln(n) (the natural logarithm of population density) and L (crystal size). Equation (1) clearly shows that if the crystal growth rate is independent of the crystal size and the nucleation rate exponentially increases with time, a closed system produces a semi-logarithmic linear relationship between crystal density and crystal length [45,47,62]:

$$ln\left(n\right)=ln\left(n^0\right)+\left(\frac{-1}{G\tau }\right)L$$

(1)

In Equation (1), G is the crystal growth rate, and τ is the crystallization time. The intercept is $ln(n^0)$ (the natural logarithm of the final nucleation density of the crystal), and the slope is $\left(\frac{-1}{G\tau }\right)$.

In this study, we used a two-dimensional thin section method to obtain quantitative data on the crystal sizes of igneous rocks [45,47,62]. Appropriate microscope magnification was selected to obtain reflection photomicrographs according to the crystal size of titanomagnetite. Ensuring that each sample had at least 200 crystals was conducive to obtaining the three-dimensional shape parameters of the crystal, avoiding interruptions in the crystal size interval and reducing errors [83]. Due to observation accuracy and image resolution limitation, titanomagnetite with a particle size less than 0.01 mm should be omitted. Photoshop (Adobe Photoshop, Adobe Systems Incorporated, San Jose, CA, USA) and CorelDRAW (CorelDRAW Graphics Suite, Corel, Ottawa, ON, Canada) were used to distinguish the mineral crystals and manually depict the mineral boundaries. The images were output at the original size in TIFF format and imported into ImageJ software (Image J, National Institutes of Health, Bethesda, MD, USA); the measuring scale and statistical parameters were set, and the area of titanomagnetite, the aspect ratio of the fitting ellipse and roundness were then analyzed. To obtain 3D texture characteristics of titanomagnetite, a more convenient, fast and accurate conversion method was realized by 2D length (l) and width (w), which is widely used at present [66]. Furthermore, based on the formula above, the cut-section effect and intersection–probability effect were also considered. We then imported the abovementioned data into CSD corrections 1.38 ((Higgins, 2000), Canada) [62]. In this software, the textural characteristics, morphological characteristics, roundness and other parameters of titanomagnetite were set and calculated, and the characteristic size distribution curve of titanomagnetite was obtained.

3.2. Sampling and Mineral Identification

We selected ten representative samples based on extensive petrographic observations, whose matrix minerals have different grain size characteristics. They are all Miocene basalts. The location of each sample is shown in Figure 1b.

BSE images of ten samples were obtained by a TESCAN MIRA3 high-performance field emission scanning electron microscope in Resources Exploration Experiment and Training Center–Research Center of Genetic Mineralogy, China University of Geosciences, Beijing. Representative images are shown in Figure 2. The accelerating voltage was 20 kV. The current was about 260 pA, and the beam size was 85.0 nm.

Different minerals (glass, olivine, clinopyroxene, titanomagnetite, plagioclase) and their respective contents can be distinguished by the gray range of the BSE photographs (Figure 2a,b). Clinopyroxene (Figure 2c), titanomagnetite (Figure 2d) and other minerals can be distinguished by grayscale, and ImageJ Threshold tools can measure their contents. This method can quickly distinguish minerals and estimate the mineral content, but the mineral outline image obtained by this method has a little more noise, and the accuracy of mineral morphology is not very high, so it cannot be used for CSD analysis.

3.3. Rock Major Element Analysis

Analyses of the major elements of whole rocks were carried out in the Laboratory of Geochemical Micro area of the State Key Laboratory of Geological Processes and Mineral Resources, China University of Geoscience, Beijing. The instrument model was a Shimadzu XRF-1800 wavelength dispersive X-ray fluorescence spectrometer. Samples were prepared by a vitreous melting method. In a platinum crucible, analytical samples (0.7 g) with a crystal size of at least 200 mesh, a flux (lithium metaborate and lithium tetraborate mixed flux, 7 g) and a release agent (lithium bromide) were weighed at a ratio of 1:10. After thorough mixing, the crucible was placed in an HMS-II-MXZ oven and melted for 30 min at 950~1100 °C. Then, the fused beads were loaded into the XRF instrument for determination for 30 min. The prepared fuses were placed in an X-ray fluorescence spectrometer for measurement, and the calibration curve method was used to quantitatively analyze the major elements of the rock. The produced measurement and data quality were monitored by repeat analyses of international basalt standards BCR-2 and BHVO-2. The analytical precision (RSD, relative standard deviation) and accuracy (RE, relative error between measured and recommended values) are better than 5% for major elements, with many elements agreeing to within 2% of the reference values.

3.4. Mineral Major Element Analysis

Electron microprobe analysis was performed to determine the major elements in titanomagnetite from the ten samples. In each sample, 15 particles of different sizes were randomly selected to determine the overall chemical characteristics of titanomagnetite. Two independent tests were conducted to obtain two sets of data, with 30 points in total. The aim was to test the reliability of statistical rules. All experiments were carried out in the Electron Probe Laboratory of the Beijing Research Institute of Uranium Geology. A JEOL-JXA-8100 electron probe microanalyzer (EPMA) was used, and the following standards were adopted for the various chemical elements: jadeite (Si), corundum (Al), forsterite (Mg), andradite (Fe), chromite (Cr), calcium vanadate (V), nickel oxide (Ni), rutile (Ti), labradorite (Ca), and spessartine (Mn). The accelerating voltage was 20 kV, beam current was 10-8 A, electron probe beam size was 2 μm, and exit angle was 40°. Peak and background counting times were 10 and 5 s for Mg, Al, Fe and Ti, 20 and 10 s for V, Cr, Si, Ca, Mn and Ni. The standards used were SPI standard minerals and were analyzed as internal standards to monitor data quality. Analyses are accurate up to 1%~2% for major elements (>10%) and 2%~10% relative for minor elements (0.5%~10%).

4. Results

4.1. Petrology

We studied Miocene Hannuoba xenolith bearing alkaline basalt. The basalt sample has a porphyritic texture overall. The phenocrysts are olivine and clinopyroxene. The fine-grained groundmass contains olivine, plagioclase, clinopyroxene, titanomagnetite and fresh glass (Figure 3). In the reflected light (Figure 4a), we identified the opaque mineral as titanomagnetite. SEM allows us to easily segment different minerals and glasses to obtain the crystalline volume content of different minerals. We investigated ten alkaline basalt samples with significant differences in the crystal size of the groundmass titanomagnetite for analysis. The fine- and coarse-grained structures are relative concepts in this article, the coarse-grained titanomagnetite denotes grains between 0.1 mm and 0.2 mm, and the fine-grained titanomagnetite denotes grains between 0.03 mm and 0.1 mm. Titanomagnetite crystals (Figure 3) are primarily granular, and skeleton crystals are visible in the fine-grained samples. There are a small number of euhedral and subhedral titanomagnetite grains. There is no composition zoning in titanomagnetite. We assume that each titanomagnetite grain is just one crystal to ensure the analysis accuracy. The modal abundance of titanomagnetite ranges from approximately 3% to 5%. The crystal size ranges from 10 μm to 100 μm. Titanomagnetite in the samples accords with the description of the characteristics of titanomagnetite in Hannuoba alkaline basalt in previous research [54].

4.2. Quantitative Textural Parameters

In these samples, clinopyroxene and plagioclase are typical mineral phases, and some samples have a small amount of olivine and apatite. Titanomagnetite is about 4.5%, clinopyroxene is from 18.8 to 28.7%, plagioclase is from 28.0 to 49.4%. The phase abundance of other major minerals is presented in

Table 1. Phase abundances of major minerals (%).

Sample	ZHT07	ZHT08	ZHT09	ZHT12	ZHT13	17JSB01	17XCNG01	SQB06	17XHK01	TL11Y1
Cpx	27.1 (0.9)	27.3 (0.9)	21.2 (1.9)	23.3 (1.3)	18.8 (0.4)	28.7 (0.7)	24.0 (0.4)	22.2 (0.5)	22.4 (0.7)	20.8 (0.7)
Pl	28.0 (1.9)	33.2 (0.4)	48.8 (1.4)	46.3 (1.5)	49.4 (1.9)	35.8 (0.7)	35.5 (1.0)	48.4 (0.9)	47.0 (1.0)	38.9 (0.9)
Ol	-	-	0.5 (0.2)	1.0 (0.1)	8.6 (0.2)	6.0 (0.4)	-	10.3 (0.3)	7.5 (0.7)	13.7 (0.7)
Ap	2.1 (0.2)	1.9 (0.2)	1.6 (0.1)	1.1 (0.1)	-	1.5(0.2)	2.1 (0.4)	1.0 (0.1)	1.0 (0.1)	1.0 (0.1)
Timt	5.4 (0.1)	4.5 (0.2)	4.2 (0.2)	4.5 (0.1)	4.8 (0.2)	3.9 (0.1)	4.8 (0.2)	4.2 (0.1)	3.2 (0.1)	5.3 (0.1)
Glass	37.0 (0.9)	35.1 (0.5)	24.5 (0.5)	23.5 (1.4)	19.5 (1.6)	25.7 (0.5)	34.1 (1.1)	13.2 (0.6)	20.1 (0.7)	21.2 (1.2)

Notes: Cpx: Clinopyroxene, Pl: Plagioclase, Ol: Olivine, Ap: Apatite, Timt: Titanomagnetite. “-“ means the phase abundances is lower than 0.5%. The standard deviation of measurement is in parentheses.

.

Outlined images of the titanomagnetite crystals for all the samples are provided in Appendix A Figure A1. Photoshop and CorelDRAW were used to distinguish the mineral crystals and manually depict the mineral boundaries. One example is shown in Figure 4. The textural parameters of all the titanomagnetite samples by the images are presented in

Table 2. Titanomagnetite textural parameters of the Hannuoba alkaline basalt samples.

Sample	ZHT07	ZHT08	ZHT09	ZHT12	ZHT13	17JSB01	17XCNG01	SQB06	17XHK01	TL11Y1
Total particles	1618	1412	2689	2085	1239	1888	2104	399	1185	1026
Area/mm2	4.5205	4.4574	4.3363	2.1913	3.924	4.2866	2.0144	4.3718	26.4207	4.5352
Roundness	0.5761	0.6054	0.6336	0.7225	0.6628	0.6986	0.7144	0.6299	0.6043	0.6422
Simulated 3D morphology	1:1.25:1.8	1:1.25:1.8	1:1.25:1.8	1:1.15:1.5	1:1.3:2.0	1:1.15:1.6	1:1.15:1.6	1:1.2:1.8	1:1.4:2.3	1:1.25:1.9
Area abundance/%	4.1655	4.2029	3.5122	3.2812	3.8075	3.4013	3.1572	3.4562	2.6990	4.0373
CSD volume/%	6.73	6.73	5.46	5.18	6.17	5.15	4.93	5.36	4.19	6.79
1 σ/%	0.78	0.62	0.44	0.43	1.67	0.50	0.43	0.97	0.52	1.01
Intercept/mm−4	15.29	15.14	17.12	17.83	15.03	16.21	18.11	12.56	11.63	14.57
1 σ/mm−4	0.06	0.07	0.06	0.06	0.04	0.18	0.06	0.15	0.05	0.06
Slope/mm−1	−102.0	−96.8	−165.0	−211.0	−74.6	−138.0	−225.0	−53.8	−42.2	−83.1
1 σ/mm−1	2.0	2.5	3.0	5.0	1.5	6.0	5.0	2.8	0.8	1.7
Maximum particle size/mm	0.0623	0.0532	0.0384	0.0310	0.1358	0.0542	0.0291	0.0929	0.1322	0.0765
1 σ/mm	0.0043	0.0037	0.0064	0.0027	0.0065	0.0129	0.0014	0.0187	0.0039	0.0064
R	0.842	0.816	0.858	0.821	0.790	0.950	0.919	0.865	0.830	0.905
A.F.	0.03	0.02	0.01	0.11	0.16	0.12	0.13	0.08	0.10	0.05

Notes: Area: the area of titanomagnetite particles under a two-dimensional thin sheet; Roundness: average roundness of the grains analyzed; Simulated 3D morphology: the three-dimensional shape of titanomagnetite estimated by CSDslice 5; Area abundance: percentage of titanomagnetite 2D area by manually delineating the minerals; CSD volume: volume content calculated by CSD corrections 1.38; Intercept and Slope: the intercept and slope of the CSD regression line; Maximum particle size: the average length of the four largest long axes in a two-dimensional thin sheet (e.g., [44,64,80]); R is the aggregation degree of titanomagnetite: an R value less than 1 represents the aggregation distribution; an R value greater than 1 represents an ordered distribution; R=1 indicates a random distribution; A.F. (alignment factor): the degree of orientation of titanomagnetite. The closer this value is to 1, the higher the degree of orientation is.

. The CSD curves of the ten samples by the parameters are plotted in Figure 5 with negative slopes, as shown on a semilogarithmic CSD diagram. The CSD curves of all the samples have negative slopes, and three types of CSD curves (Higgins, 2006a) are shown in Figure 5: S-CSD (semilogarithmic CSDs) lines (17JSB01, ZHT07), L-CSD (lognormal CSDs) lines (17XCNG01, SQB06, ZHT08, ZHT09, ZHT12) and F-CSD (fractal CSDs) lines (17XHK01, TL11Y1, ZHT13).

4.3. Rock Major Element

The whole-rock data for the major elements in each sample are presented in

Table 3. Whole-rock major element data for the Hannuoba alkaline basalt (wt%).

Sample	ZHT07	ZHT08	ZHT09	ZHT12	ZHT13	17JSB01	17XCNG01	SQB06	17XHK01	TL11Y1
SiO2	46.16	45.56	49.49	49.88	50.50	46.82	48.62	46.63	48.77	46.52
TiO2	2.79	2.64	2.35	2.41	2.47	2.66	2.21	2.65	2.20	2.78
Al2O3	13.94	15.88	16.38	15.82	15.44	14.07	13.97	13.67	13.88	13.40
FeO	13.20	12.37	11.32	11.44	11.27	12.70	11.92	12.31	11.40	12.73
MnO	0.18	0.16	0.15	0.14	0.14	0.18	0.19	0.18	0.17	0.19
MgO	6.90	7.17	5.61	5.62	5.33	9.04	6.60	9.38	8.31	8.63
CaO	9.42	9.24	7.52	7.20	7.66	9.69	8.43	9.37	8.78	9.78
Na2O	4.69	4.43	4.03	4.38	4.05	2.92	4.24	3.71	4.13	3.40
K2O	1.39	1.30	2.35	2.28	2.31	1.20	2.60	1.36	1.83	1.71
P2O5	1.34	1.26	0.82	0.83	0.83	0.72	1.23	0.74	0.55	0.87
LOI	3.49	3.82	3.11	3.24	3.13	0.43	1.87	1.40	1.97	1.79

. For whole-rock major elements, the ranges of SiO₂, Al₂O₃, FeO, CaO, and MgO are 46.16~50.50 wt%, 13.40~16.38 wt%, 11.27~13.20 wt%, 7.20~9.78 wt% and 5.33~9.38 wt%, respectively. The samples are all alkaline basalt, and most of the samples are trachybasalt or very close to trachybasalt in this study (Figure 6).

4.4. Mineral Major Element

The major chemical titanomagnetite composition of all particles is reported in Supplementary Materials Table S1. Two independent tests were conducted to obtain two sets of data (15 particles for each test), marked as ‘sample-a’ and ‘sample-b’, respectively. The average major chemical data of each test in each sample are shown in

Table 4. Major elements in titanomagnetite from the Hannuoba alkaline basalt (wt%) and the number of cations based on four oxygen atoms.

Sample	17JSB01a	17JSB01b	17XCNG01a	17XCNG01b	SQB06a	SQB06b	17XHK01a	17XHK01b	TL11Y1a	TL11Y1b
MgO	3.61 (0.50)	3.17 (0.29)	1.03 (0.53)	0.71 (0.34)	3.54 (0.39)	3.17 (0.29)	2.81 (0.43)	3.00 (0.22)	2.38 (0.65)	2.30 (0.33)
Al2O3	2.77 (0.85)	2.02 (0.32)	0.55 (0.45)	0.28 (0.16)	3.67 (0.20)	2.02 (0.11)	1.64 (0.28)	1.74 (0.31)	1.97 (1.00)	1.80 (0.54)
FeOT	62.39 (1.57)	63.79 (0.41)	68.17 (1.14)	68.28 (1.15)	61.50 (0.94)	63.79 (0.26)	64.63 (1.01)	66.44 (0.87)	64.82 (1.60)	66.87 (1.02)
CaO	0.06 (0.14)	0.11 (0.08)	0.12 (0.15)	0.30 (0.28)	0.02 (0.04)	0.11 (0.08)	0.00 (0.00)	0.07 (0.05)	0.03 (0.05)	0.11 (0.06)
SiO2	0.00 (0.00)	0.00 (0.00)	0.01 (0.02)	0.04 (0.12)	0.00 (0.00)	0.00 (0.00)	0.00 (0.00)	0.00 (0.00)	0.00 (0.00)	0.04 (0.15)
NiO	0.06 (0.08)	0.08 (0.04)	0.04 (0.06)	0.07 (0.04)	0.08 (0.10)	0.08 (0.05)	0.07 (0.09)	0.13 (0.06)	0.03 (0.06)	0.09 (0.05)
MnO	0.65 (0.07)	0.06 (0.03)	0.73 (0.07)	0.76 (0.03)	0.64 (0.06)	0.60 (0.02)	0.69 (0.07)	0.65 (0.03)	0.67 (0.05)	0.67 (0.03)
TiO2	25.75 (1.03)	25.13 (0.47)	22.28 (1.06)	22.50 (0.77)	26.60 (0.62)	25.13 (0.29)	27.26 (0.53)	26.27 (0.65)	25.84 (1.09)	25.96 (0.46)
Cr2O3	0.56 (0.87)	0.20 (0.04)	0.39 (0.37)	0.13 (0.13)	0.18 (0.12)	0.20 (0.04)	0.10 (0.06)	0.07 (0.03)	0.29 (0.36)	0.07 (0.03)
V2O3	0.48 (0.09)	0.51 (0.03)	0.43 (0.09)	0.39 (0.05)	0.51 (0.08)	0.51 (0.04)	0.47 (0.07)	0.54 (0.04)	0.46 (0.09)	0.46 (0.03)
Fe2O3	15.46 (1.70)	20.99 (0.70)	22.28 (1.70)	25.37 (1.51)	13.18 (1.07)	20.99 (0.25)	14.77 (1.08)	21.59 (1.21)	16.00 (1.52)	21.29 (1.02)
FeO	48.48 (1.15)	44.91 (0.71)	48.12 (1.51)	45.45 (0.84)	49.63 (0.88)	44.91 (0.27)	51.33 (0.73)	47.01 (0.60)	50.42 (1.55)	47.71 (0.38)
Tot	96.33 (0.92)	95.62 (0.64)	93.75 (1.49)	93.46 (0.64)	96.73 (0.61)	95.62 (0.47)	97.66 (0.55)	98.91 (0.34)	96.48 (0.10)	98.37 (0.37)
Tot*	97.88 (0.10)	97.72 (0.63)	95.98 (1.49)	96.00 (0.66)	98.05 (0.59)	97.72 (0.46)	99.14 (0.57)	101.07 (0.31)	98.08 (0.96)	100.51 (0.40)
Mg2+	0.2027	0.1804	0.0604	0.0421	0.1978	0.1760	0.1579	0.1659	0.1340	0.1284
	(0.0262)	(0.0153)	(0.0306)	(0.0199)	(0.0214)	(0.0155)	(0.0233)	(0.0118)	(0.0355)	(0.0179)
Al3+	0.1227	0.0910	0.0257	0.0131	0.1623	0.1530	0.0730	0.0759	0.0857	0.0795
	(0.0355)	(0.0138)	(0.0208)	(0.0073)	(0.0085)	(0.0039)	(0.0122)	(0.0131)	(0.0439)	(0.0234)
Ca2+	0.0025	0.0046	0.0051	0.0128	0.0007	0.0041	0.0000	0.0029	0.0013	0.0045
	(0.0059)	(0.0033)	(0.0066)	(0.0123)	(0.0017)	(0.0032)	(0.0000)	(0.0021)	(0.0022)	(0.0025)
Si4+	0.0000	0.0000	0.0003	0.0018	0.0000	0.0000	0.0000	0.0000	0.0000	0.0015
	(0.0000)	(0.0000)	(0.0007)	(0.0047)	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0057)
Ni2+	0.0018	0.0024	0.0013	0.0022	0.0023	0.0017	0.0020	0.0037	0.0012	0.0026
	(0.0024)	(0.0013)	(0.0019)	(0.0014)	(0.0029)	(0.0014)	(0.0026)	(0.0018)	(0.0018)	(0.0014)
Mn2+	0.0207	0.0195	0.0247	0.0258	0.0202	0.0200	0.0220	0.0205	0.0217	0.0212
	(0.0024)	(0.0008)	(0.0025)	(0.0012)	(0.0019)	(0.0006)	(0.0022)	(0.0011)	(0.0017)	(0.0009)
Ti4+	0.6497	0.6419	0.5918	0.6016	0.6669	0.6448	0.6879	0.6517	0.6606	0.6505
	(0.0308)	(0.0128)	(0.0284)	(0.0209)	(0.0140)	(0.0047)	(0.0132)	(0.0156)	(0.0306)	(0.0128)
Cr3+	0.0165	0.0060	0.0121	0.0040	0.0054	0.0025	0.0029	0.0021	0.0082	0.0022
	(0.0250)	(0.0011)	(0.0113)	(0.0040)	(0.0035)	(0.0013)	(0.0016)	(0.0009)	(0.0107)	(0.0010)
V3+	0.0146	0.0157	0.0137	0.0126	0.0154	0.0145	0.0144	0.0160	0.0142	0.0140
	(0.0025)	(0.0009)	(0.0027)	(0.0016)	(0.0023)	(0.0012)	(0.0021)	(0.0012)	(0.0027)	(0.0008)
Fe3+	0.5467	0.6034	0.7643	0.7635	0.4830	0.5404	0.5339	0.6027	0.5707	0.6006
	(0.0456)	(0.0202)	(0.0466)	(0.0447)	(0.0289)	(0.0088)	(0.0287)	(0.0347)	(0.0422)	(0.0302)
Fe2+	1.4220	1.4350	1.5006	1.5204	1.4460	1.4429	1.5059	1.4586	1.5024	1.4953
	(0.0440)	(0.0264)	(0.0513)	(0.0299)	(0.0256)	(0.0136)	(0.0263)	(0.0200)	(0.0574)	(0.0193)
TOTAL	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000
Sample	ZHT07a	ZHT07b	ZHT08a	ZHT08b	ZHT09a	ZHT09b	ZHT12a	ZHT12b	ZHT13a	ZHT13b
MgO	1.78 (0.42)	1.77 (0.20)	1.87 (0.25)	1.76 (0.39)	1.89 (0.26)	1.93 (0.21)	1.29 (0.74)	0.92 (0.31)	2.79 (0.46)	2.7 (0.35)
Al2O3	1.18 (1.05)	1.51 (0.94)	2.12 (1.22)	1.60 (0.99)	0.66 (0.24)	0.62 (0.04)	0.91 (0.77)	0.31 (0.36)	2.61 (0.36)	2.62 (0.35)
FeOT	71.05 (1.73)	70.4 (0.72)	64.48 (1.38)	69.23 (1.07)	69.03 (2.48)	69.74 (0.90)	64.83 (1.55)	65.74 (1.5)	64.48 (1.38)	66.84 (1.27)
CaO	0.20 (0.21)	0.16 (0.07)	0.06 (0.11)	0.16 (0.08)	0.11 (0.14)	0.16 (0.07)	0.11 (0.12)	0.21 (0.09)	0.01 (0.02)	0.05 (0.02)
SiO2	0.00 (0.00)	0.00 (0.00)	0.06 (0.13)	0.03 (0.12)	0.01 (0.04)	0.03 (0.12)	0.08 (0.12)	0.17 (0.23)	0.00 (0.01)	0.02 (0.09)
NiO	0.03 (0.05)	0.07 (0.04)	0.04 (0.07)	0.06 (0.05)	0.02 (0.05)	0.08 (0.03)	0.05 (0.06)	0.07 (0.04)	0.08 (0.10)	0.12 (0.06)
MnO	0.71 (0.08)	0.66 (0.02)	0.68 (0.06)	0.65 (0.03)	0.63 (0.07)	0.61 (0.03)	0.76 (0.27)	0.72 (0.21)	0.59 (0.04)	0.57 (0.03)
TiO2	22.31 (1.11)	22.00 (0.90)	21.81 (0.96)	21.03 (1.11)	23.35 (1.03)	23.25 (0.81)	24.84 (1.18)	24.86 (0.78)	25.17 (1.37)	24.94 (1.22)
Cr2O3	0.20 (0.20)	0.14 (0.09)	0.35 (0.32)	0.13 (0.12)	0.19 (0.15)	0.21 (0.15)	0.33 (0.31)	0.07 (0.07)	0.05 (0.04)	0.04 (0.05)
V2O3	0.38 (0.08)	0.38 (0.09)	0.39 (0.10)	0.36 (0.05)	0.38 (0.10)	0.40 (0.05)	0.48 (0.09)	0.4 (0.05)	0.65 (0.10)	0.71 (0.09)
Fe2O3	25.19 (2.33)	28.16 (1.11)	24.12 (1.11)	28.32 (1.75)	22.57 (3.08)	26.75 (1.59)	16.44 (2.10)	20.64 (1.85)	16.82 (2.55)	22.42 (1.98)
FeO	48.39 (0.98)	45.06 (0.71)	47.93 (1.02)	43.74 (1.17)	48.73 (0.93)	45.67 (0.64)	50.04 (1.59)	47.17 (0.68)	49.35 (1.13)	46.66 (0.78)
Tot	97.85 (0.59)	97.08 (0.45)	96.44 (0.71)	95.01 (0.61)	96.28 (2.04)	97.03 (0.73)	93.69 (2.04)	93.47 (0.93)	96.44 (0.71)	98.61 (0.45)
Tot*	100.37 (0.65)	99.9 (0.44)	99.44 (0.96)	97.84 (0.61)	98.54 (2.28)	99.71 (0.77)	95.33 (2.13)	95.54 (1.04)	98.12 (0.69)	100.86 (0.37)
Mg2+	0.0998	0.0999	0.1050	0.1013	0.1082	0.1096	0.0756	0.0553	0.1575	0.1492
	(0.0233)	(0.0105)	(0.0128)	(0.0211)	(0.0140)	(0.0114)	(0.0426)	(0.0185)	(0.0247)	(0.0189)
Al3+	0.0521	0.0670	0.0937	0.0724	0.0301	0.0277	0.0420	0.0147	0.1165	0.1142
	(0.0461)	(0.0414)	(0.0534)	(0.0441)	(0.0107)	(0.0020)	(0.0349)	(0.0170)	(0.0152)	(0.0151)
Ca2+	0.0080	0.0065	0.0026	0.0068	0.0045	0.0064	0.0047	0.0089	0.0003	0.0018
	(0.0084)	(0.0028)	(0.0046)	(0.0033)	(0.0058)	(0.0031)	(0.0054)	(0.0039)	(0.0009)	(0.0009)
Si4+	0.0000	0.0000	0.0024	0.0012	0.0005	0.0012	0.0033	0.0067	0.0000	0.0008
	(0.0000)	(0.0000)	(0.0053)	(0.0046)	(0.0016)	(0.0046)	(0.0049)	(0.0093)	(0.0002)	(0.0033)
Ni2+	0.0010	0.0021	0.0013	0.0018	0.0007	0.0025	0.0017	0.0023	0.0025	0.0036
	(0.0014)	(0.0013)	(0.0020)	(0.0017)	(0.0017)	(0.0010)	(0.0018)	(0.0013)	(0.0029)	(0.0018)
Mn2+	0.0225	0.0211	0.0217	0.0211	0.0206	0.0197	0.0259	0.0244	0.0191	0.0179
	(0.0025)	(0.0008)	(0.0020)	(0.0011)	(0.0022)	(0.0010)	(0.0095)	(0.0070)	(0.0014)	(0.0011)
Ti4+	0.5617	0.5573	0.5512	0.5434	0.6004	0.5923	0.6620	0.6669	0.6380	0.6181
	(0.0295)	(0.0261)	(0.0284)	(0.0317)	(0.0355)	(0.0211)	(0.0401)	(0.0234)	(0.0341)	(0.0295)
Cr3+	0.0060	0.0043	0.0104	0.0039	0.0056	0.0064	0.0102	0.0023	0.0016	0.0011
	(0.0059)	(0.0027)	(0.0094)	(0.0035)	(0.0044)	(0.0046)	(0.0096)	(0.0021)	(0.0013)	(0.0015)
V3+	0.0115	0.0114	0.0113	0.0111	0.0116	0.0121	0.0153	0.0128	0.0199	0.0212
	(0.0023)	(0.0027)	(0.0041)	(0.0016)	(0.0029)	(0.0014)	(0.0029)	(0.0017)	(0.0030)	(0.0025)
Fe3+	0.8069	0.8027	0.7775	0.8233	0.7511	0.7669	0.6021	0.6228	0.5860	0.6256
	(0.0633)	(0.0315)	(0.0315)	(0.0500)	(0.0721)	(0.0441)	(0.0509)	(0.0522)	(0.0681)	(0.0565)
Fe2+	1.4304	1.4276	1.4229	1.4136	1.4668	1.4552	1.5574	1.5828	1.4587	1.4465
	(0.0332)	(0.0314)	(0.0381)	(0.0484)	(0.0404)	(0.0239)	(0.0664)	(0.0235)	(0.0374)	(0.0257)
TOTAL	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000

Notes: FeO^T is total iron as FeO. Tot* is the total content of major elements, corrected by dividing FeO^T into Fe₂O₃ and FeO. We calculated the molar proportions of the cations in all samples, and the cations were normalized in a formula unit of 3 sites. The result of Fe ion in electron microprobe analysis is presented as FeO, so using the charge deficiency balance, we can convert Fe ion to Fe²⁺ and Fe³⁺. Each titanomagnetite composition in this table is the average of 15 data points in each sample. The standard deviation is in parentheses.

. In these samples, there is no statistical difference in the average composition of major elements between the two sets of data. The two sets of data represent the average composition characteristics of major elements of titanomagnetite, and these average data are consistent with the statistical rules. For titanomagnetite mineral major elements, Al₂O₃ are 0.28~2.77 wt%, FeO are 43.74~51.33 wt%, Fe₂O₃ are 13.18~28.32 wt%, MgO are 0.71~3.61 wt%, and TiO₂ are 21.03~27.26 wt%. In the FeO-Fe₂O₃-TiO₂ ternary diagram, the main oxide was near the titanomagnetite series, as Mollo et al. [6] reported (Figure 7). Compared with data in Mollo et al. [6], titanomagnetite in Hannuoba samples is more closed to Usp unit.

5. Discussion

5.1. Magma Cooling Process

It is vital to understand the magmatic ascent and flow process in Hannuoba, especially because the cooling rate may change during this process. CSD data are appropriate for exploring the fundamental aspects of crystal nucleation and growth during the progressive solidification of Hannuoba alkaline basalt.

Based on covariant relations among CSD variables, we can further discuss the characteristics of magma systems and various ascent and flow dynamic processes and reveal the significance of dynamics. The slope and intercept of the CSD regression line are negatively correlated (Figure 8a). The correlation between slope and intercept is well-known and is related to the closure of the system [66]. The slope of the CSD regression line and the maximum particle size determine the survival time of an open or closed system. At the same time, the giant crystal often more easily enters or leaves the system and is usually independent of CSD changes, so the relationship between the maximum particle size and the slope and intercept can reveal many crucial magmatic consolidation dynamics. The slope and maximum particle size are positively correlated (Figure 8b), showing that the nucleation rate and growth rate are stable and that titanomagnetites are formed under the stable and open system. The relationship between the intercept and maximum particle size is shown as negative correlations (Figure 8c), representing the characteristics of a closed system [45,62,63,66]. Overall, the titanomagnetite in these samples probably nucleates and grows in a stable open or closed system and is fractionally influenced by the environment.

Most titanomagnetite in most Hannuoba samples is represented by a straight line in the classic CSD diagram of ln (population density) versus size (S-CSD) (Figure 5), indicating a simple nucleation and growth process [47]. These simple straight CSD lines imply a relatively stable cooling rate, which was experimentally confirmed [43].

S-CSD lines represent the process of crystal nucleation and growth in a stable system without magma mixing [45]. This implies that titanomagnetite crystallizes in a simple and stable environment during the consolidation of Hannuoba alkaline basalt.

L-CSD lines show concave downward CSDs as tiny crystals, which may be due to a decrease in nucleation density during the final stage of groundmass crystallization or the possible Ostwald ripening process [48,49,62,72,74,77,88,89]. Another possible explanation is that the minor fine-grained crystals that are smaller than 0.01 mm are omitted. Some of them are too difficult to quantify during CSD analysis, resulting in the loss of minor fine-grained crystals. Under such conditions, these L-CSD lines can be seen as approximate S-CSD lines. In experiments (e.g., [34]), regardless of how the dwelling time changes (the cooling process ends; the temperature drops to the target temperature and keeps constant for some time), the chemical composition of titanomagnetite remains stable under a fixed cooling rate. However, textural information can be reflected in the CSD diagram: a longer dwelling time results in a larger slope of the CSD lines and a smaller intercept, resulting in the loss of small crystals and the formation of L-CSD. Titanomagnetites with different sizes in the L-CSD samples (for ZHT09 and 17XCNG01 as examples in Figure 9) have homogeneous chemical characteristics, suggesting that the titanomagnetites in these samples undergo a fixed cooling process with some dwelling time.

F-CSD lines might result from two-stage crystallization, crystal aggregation or magma mixing [47,62,63]. If the situation above exists, there will be two different crystal groups of titanomagnetite in the F-CSD sample (for ZHT13 as an example in Figure 9), and the chemical composition of titanomagnetite in different crystal groups should be different because physical and chemical conditions must change in the two crystallization processes, which will influence the cation substitution of titanomagnetite. However, the chemical composition in ZHT13 is almost fixed, and it seems that there are not two titanomagnetite crystal groups (Figure 9). Hannuoba samples are xenolith-bearing alkaline basalts without crustal magma chamber processes [53,54], so magma mixing is also less likely. Another assumption is that the CSD curve may also be fractal at a fixed cooling rate. It has been found in experimental petrology [4,43]. It means that titanomagnetite might undergo one fixed cooling rate. So, the F-CSD curves may not reflect two-stage crystallization or magma mixing.

Combined with CSD theory, experimental petrology and the geological background of Hannuoba alkaline basalt, these titanomagnetites most likely experienced only one cooling process, and magma mixing and aggregation are unlikely.

5.2. A New Parameter: Estimation for Apparent Cooling Rate

The cooling rate can be determined when the degree of undercooling and the growth time of a mineral are already known. However, the degree of undercooling and the growth time are difficult to independently determine for natural samples and can only be determined via experimental petrology [4,6,43]. To explore the cooling rate of natural samples, we can determine the apparent cooling rate using the CSD curve of titanomagnetite [4,43,45,46,47,66,83]. For a series of comagmatic samples, the shallower the slope is, the slower the cooling rate and the larger the crystal size, as indicated by CSD theory [46,47] and observed in experimental petrology [4,43]. However, the potential coarsening process will influence the maximum particle size, CSD slope, intercept and other texture parameters, and these influences will affect the estimation of the cooling rate in natural samples. We need to identify a new parameter to represent the apparent cooling rate in natural samples.

Magnetite and spinel are common spinel minerals in basic and ultrabasic igneous rocks. Their crystallization habits are similar, they are an equiaxed crystal system, symmetric m3m, and often octahedral {111}. Therefore, in the experimental petrology of Giuliani et al. [43], the cooling rate experiment of spinel can be compared with the crystallization law of titanomagnetite, which has certain reference value. As is shown in Figure 10a–d, in the experimental study performed by Giuliani et al. [43], the cooling rate was negatively correlated with the spinel CSD slope, maximum particle size and the ratio of the maximum particle size to the area abundance and positively correlated with the CSD intercept. These parameters can all describe the apparent cooling rate in natural samples to some extent. However, in a constant cooling rate (80 °C/min) experiment [34], the dwelling time was negatively correlated with the titanomagnetite CSD intercept and positively correlated with the maximum particle size and CSD slope. The ratio of maximum particle size to area abundance in titanomagnetite remains stable at a fixed cooling rate, regardless of how the dwelling time changes (Figure 10e–h). The possible Ostwald ripening process greatly influences the maximum particle size. Continuous crystal growth will happen when the dwelling time lasts, and the titanomagnetite content also changes.

The CSD slope, intercept and maximum particle size (L_max) exhibit a strong trend with dwelling time and cooling rate. However, the ratio of maximum particle size to area abundance seems to not be affected by dwelling time but only changes with cooling rate. From 80 °C/min to 0.01 °C/min, the ratio of the maximum particle size to the titanomagnetite area abundance (L_max/timt content) ranges from 1.02 to 20.38, including the average number under a fixed cooling rate in Pontesilli et al. [34]. Due to the different physical and chemical conditions (e.g., quenching temperature, starting temperature, pressure, oxygen fugacity, and starting melt composition), the average number under a fixed cooling rate in the experiment in Pontesilli et al. [34] is not very appropriate but also near to the fitting line (black solid point in Figure 10c,d). However, in the same experiment, the ratio of the maximum particle size to the area abundance can describe the cooling rate (black hollow point in Figure 10c,d), and this parament also makes sense in Hannuoba samples. In Hannuoba alkaline basalt samples, the L_max/timt content ranges from 9.22 to 48.98, implying various apparent cooling rates.

5.3. Cation Redistribution Behavior with the Apparent Cooling Rate in Titanomagnetite in Hannuoba Samples

The cooling rate is one factor that influences the redistribution of cations in the titanomagnetite solid solution. Melt composition is also a key factor that influences the redistribution of cations in the titanomagnetite solid solution. Under equilibrium conditions, the main factor that changes crystal composition is the coexisting melt composition. When the melt cools and deviates from equilibrium conditions, the composition of the melt near the titanomagnetite surface changes, resulting in a change in the chemical components of the titanomagnetite; this is reflected in the partition coefficients of key major elements. Therefore, it is important to consider the partition coefficient of major elements in titanomagnetite.

For the Hannuoba basalt, as the L_max/timt content decreases (increasing the apparent cooling rate), titanomagnetite crystals become progressively enriched in Fe³⁺ and depleted in Mg²⁺, Al³⁺ and Ti⁴⁺. Fe²⁺ shows a large fluctuation range, which is relatively unrelated to the cooling rate. The contents of other cations, such as Cr³⁺ and Ca²⁺, are low and have little relationship with the cooling rate. Thus, only Fe³⁺ and Ti⁴⁺ variations are consistent with the experimental petrology results [6,7]. Here, we use the partition coefficient to describe the proportion of incompatible major elements between titanomagnetite and the melt. The partition coefficients of Mg, Al and Ti are strongly and negatively related to the L_max/timt content and are better than ion (apfu). Similarly, only DTi (the partition coefficient of Ti between titanomagnetite and the melt) is consistent with the experimental results and is more related to the apparent cooling rate. Therefore, in the natural samples, the contents of Ti⁴⁺, DTi and Fe³⁺ in titanomagnetite have strong relationships with the apparent cooling rate (Figure 11).

The main end members of titanomagnetite (Fe²⁺_(1+x)Fe³⁺_(2−2x)Ti⁴⁺_xO₄) (0 < x < 1) are ulvöspinel (Fe₂TiO₄) and magnetite (FeFe₂O₄) [20,33]. In the case of an increasing cooling rate, Fe²⁺ and Ti⁴⁺, from the ulvöspinel end member, gradually decreases [6,7]. Fe³⁺ replaces Ti⁴⁺, so Fe³⁺ and Fe²⁺, which are derived from the magnetite endmember, increase. These replacements explain why Fe²⁺ exhibits a large fluctuation but is irrelevant to the cooling rate.

The composition of the residual melt dynamically changes during the solidification process as the minerals successively crystallize. The variations in the crystallinity of the melt affect the residual melt composition and the chemical composition of crystals in the groundmass, including titanomagnetite. In this study, the crystallinity of melt is reflected in the degree of crystallization of clinopyroxene, plagioclase and other minerals. In Hannuoba basalt, clinopyroxene and plagioclase are mainly groundmass crystals. When the apparent cooling rate increases, the content of groundmass clinopyroxene increases, and most Mg²⁺ enters it, resulting in decreased Mg content in titanomagnetite (Figure 11 and Figure 12). Small titanomagnetite mostly crystallizes later than large titanomagnetite crystals. Although several small-crystal sections may be large crystals because of the cross-section effect (the plane randomly cut through the crystal does not always pass through the center of the crystal, and the size of the major axis of the section can vary arbitrarily from zero to the longest axis of its triad) [90,91] and intersection probability problem (the cross-section is always easier to cut through larger crystals, and smaller ones are less likely to be cut through) [91]. The slight positive correlations between Mg²⁺, Al³⁺ and the long axis of titanomagnetite for sample ZHT09 (Figure 9) suggest that small titanomagnetite (<50 μm) compositions are significantly affected by the near solidus residual melt composition. Therefore, the different crystallinities affect cationic substitution in the titanomagnetite solid solution.

For other cations, the correlations between the contents of Ca²⁺, Ni²⁺, and V³⁺ and the cooling rate are minimal.

Therefore, the cooling rate does control the partition coefficient in titanomagnetite, but there are significant differences in crystallinity during the crystallization process under kinetically controlled conditions. This phenomenon implies that the cooling rate influences the partition coefficients of elements under kinetically controlled conditions to some extent.

5.4. Calibration of a New Titanomagnetite Geospeedometer

The equation for the geospeedometer established by Mollo et al. [6] should be re-examined because, in this study, the content changes in Mg and Al in titanomagnetite are mainly controlled not by the cooling rate but by the possible melt composition and crystallinity of the melt. Therefore, the geospeedometer that applies to MgO and Al₂O₃ might be unsuitable in this study. It is possible that Equation (2) is not applicable because of different systems, but as described below, Mollo et al.’s [6] equation is indeed affected by the melt composition and degree of crystallization. The cooling rate equation

$$\begin{gathered} \mathrm{CR} (°\mathrm{C}/\min )=17.1333\left(\pm 0.2221\right)\times \exp (-1.5211\left(\pm 0.0452\right)\times \\ \frac{{\mathrm{TiO}}_2}{\mathrm{MgO}+{\mathrm{Al}}_2{\mathrm{O}}_3})(\mathrm{wt}\%) \end{gathered}$$

(2)

is used to calculate the cooling rates of the Hannuoba samples, ranging from 0.0001 °C/min to 0.06 °C/min. In other words, the cooling rates calculated by Equation (2) are close to 0, and within the SEE of Equation (2) (0.507 °C/min), which is not applicable and accurate in this study because the L_max/timt content greatly differs (9.22 to 48.98), implying variations in the cooling rate. The results should have a range of variation and should not be close to 0. Importantly, the negative correlation between TiO₂/(Al₂O₃ + MgO) and L_max/timt content (Figure 13b) is inconsistent with the kinetically controlled crystallization trend. This means that when the cooling rate decreases, the L_max/timt content and TiO₂/(Al₂O₃ + MgO) should both increase. However, this is not shown in Figure 13b. Using the titanomagnetite data (Figure 13d) and experiments of this study (Figure 13c) [6], the ratio of the partition coefficient of Ti between liquid and titanomagnetite to Fe²⁺ (DTi/Fe²⁺: and DTi is Ti⁴⁺ in titanomagnetite to TiO₂ in the melt) in titanomagnetite is consistent with the trend of equilibrium (Figure 13d). This means that when the cooling rate decreases, the L_max/timt content increases. The DTi/Fe²⁺ should also increase, as shown in Figure 14d, indicating that DTi/Fe²⁺ is an important parameter of titanomagnetite composition reflecting cooling rate. For titanomagnetite in Hannuoba alkaline basalt, the contents of Mg²⁺ and Al³⁺ cannot sensitively reflect the cooling rate because of changes in the composition and crystallinity of the coexisting melt. In natural dike samples [5], the changes in melt composition were not clear, so Mg²⁺ and Al³⁺ were slightly influenced by the coexisting melt. Therefore, using this geospeedometer [6] to calculate the cooling rate may ignore the influence of melt composition. DTi/Fe²⁺ is expected to be consistent with natural and experimental samples to measure the cooling rate. Under kinetically controlled conditions, the crystallization of clinopyroxene, plagioclase and other minerals results in variations in the residual melt composition, which is reflected in the crystallinity of the melt. Different pressure conditions, quenching temperatures and cooling rates lead to different crystallinities in the melt. The temperature and pressure of Hannuoba alkaline basalt are 1070–1146 ± 28 °C and −1–6.7 ± 1.4 kbar, which is close to experiment petrology (see Appendix A for a detailed overview).

The experiment [6] provides texture and chemical compositional data that can help us to build a calibration dataset. The dataset is provided in Supplementary Materials Table S2. The regression equation is obtained by multiple linear regression using parameters consistent with the equilibrium trend in both natural samples and experimental petrology, as well as other possible parameters (such as initial temperature; final temperature; pressure; clinopyroxene volume content; glass volume content; Ti⁴⁺, Al³⁺, Fe³⁺, Fe²⁺, Mn²⁺, Mg²⁺, and Mn²⁺ in titanomagnetite; DTi, DAl, DMg, DMn, and DTi/Fe²⁺, etc. Finally, the regression is as follows:

$$\mathrm{CR}\left(°\mathrm{C}/\min \right)=-2.255\left(\pm 0.051\right)\times \frac{\mathrm{DTi}}{\mathrm{timt} {\mathrm{Fe}}^{2+}}+0.032\left(\pm 0.010\right)\times \mathrm{Cpx} \mathrm{content}+18.739\left(\pm 0.386\right)$$

(3)

We compared Equation (2) with Equation (3), which shows that the accuracy of the equation has been improved, especially at a slow cooling rate. At a slow cooling rate, the points are more gathered in Equation (3) than in Equation (2) (Figure 14).

Pressure has little effect on the geospeedometer, according to Equation (3). Using the geospeedometer developed in this study, we can more accurately estimate the cooling rate of melt solidification.

Equation (2) in Mollo et al., [6] and Equation (3) in this study are used to estimate the cooling rate of alkaline basalt from Hannuoba (Figure 15). The cooling rates of the Hannuoba alkaline basalt range from 0.7 to 7.0 (±0.5) °C/min, as determined using the new geospeedometer (Equation (3)) proposed in this study. For Equation (2), the Hannuoba samples mostly nucleated and grew in a very low and constant cooling rate environment with a change in L_max/timt content. For Equation (3), a low cooling rate with a L_max/timt content represents equilibrium conditions, and rapid cooling results in a small L_max/timt content, showing kinetically controlled conditions. As seen when comparing Equations (2) and (3), the calculated cooling rate is more accurate when the cooling rate is slow (<3 °C/min) (Figure 14), and the influence of crystallinity on Equation (3) is eliminated to a greater extent, demonstrating that the geospeedometer can reflect the real conditions in nature. Our samples show that rapid cooling is rare in continental basalt, suggesting that prediction accuracy is quite important under a slower cooling rate (<10 °C/min) and that the geospeedometer in this study is more suitable for a lower cooling rate.

Equation (3) is used to predict other samples, and the results are listed in Supplementary Materials Table S3. This geospeedometer is found to have a large error under some conditions. It cannot accurately predict the cooling rate from titanomagnetite when the rock is solidified, except when under natural conditions in the laboratory [34]. It cannot be used to predict the cooling rate in intermediate rock, acidic rock or Fe-rich basaltic melts [4,42,87]. Ilmenite and chromite cannot be used in this geospeedometer [42]. If the melt condition, oxygen fugacity and pressure are relatively unchanged, Equation (2) can accurately predict the cooling rate of natural samples [5,6,7].

6. Conclusions

In a quantitative textural analysis of igneous rocks, we found that the ratio of the maximum particle size to the area abundance of titanomagnetite was utilized to obtain information about the apparent cooling rate in Hannuoba alkaline basalt in this study. The disequilibrium crystallization process is reflected in the solid solution substitution of alkaline basalt groundmass titanomagnetite. DTi/Fe²⁺ (the ratio of the partition coefficient of Ti between liquid and titanomagnetite to Fe²⁺ in titianomagnetite) and Cpx volume content have strong correlations with the cooling rate in this study. The composition and crystallinity of the coexisting melt compensate for the effect of the cooling rate on Mg²⁺ and Al³⁺. Taking the partition coefficient of Ti between the liquid and titanomagnetite, Fe²⁺ in titanomagnetite and crystallinity into account, a new titanomagnetite geospeedometer is calibrated. The new geospeedometer can better predict slow cooling rates [5,6,87], which are common in natural samples. The cooling rates of the Hannuoba alkaline basalt are estimated to range from 0.7 to 7.0 (±0.5) °C/min, which might also cover the ranges of cooling rates for many other continental alkaline basalts.

Our study suggests that the relationship between crystal size distribution and mineral chemistry is a promising tool for exploring the effects of disequilibrium crystallization on cation redistribution in rock-forming minerals. In addition, researchers prefer using common rock-making minerals in igneous rock, such as clinopyroxene, olivine and plagioclase, to calculate temperature, pressure, water content, oxygen fugacity and other information [1,24,92,93,94,95,96,97] to explore the source lithology and solidification history of igneous rocks. However, kinetically controlled crystallization processes increase the uncertainties of such calculations since equilibrium conditions are assumed in these models. Therefore, the influence of the cooling rate should be fully considered when calculating this information. The new titanomagnetite geospeedometer can better measure the cooling rate of basalt and may help evaluate mineral thermobarometers [8,9,10,11,12,13,14,15,16,17,17,18,18,19,19,20,21,22,23,24,25,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100] and better recognize thermal remanence magnetization and ancient magnetic fields [7,15,16].