The Relationship Between the Fractal Dimension and the Evolution of Rock-Forming Minerals Crystallization on the Example of the Northwestern Part of the Lovozero Intrusion

Abstract

This article presents the results of fractal texture analyses of selected minerals (aegirine, eudialyte, orthoclase) in the northwestern part of the Lovozero intrusion. This intrusion is located in northeastern Scandinavia and is a massif made of alkaline rocks. There are rocks such as massive syenites and porphyrtes, as well as iiolites, urtites, and foyaites, accompanied by metasomatic rocks of the contact zone. A box-counting fractal dimension was used to numerically represent the texture of these minerals. In the further part, this coefficient was visualized in the form of maps superimposed on the study area, and some simple arithmetic calculations were performed to highlight the common features of this dimension for the selected rock-forming minerals. In conjunction with the geological interpretation of these results, rock-forming processes in this massif were depicted. This work is preliminary, showing the potential of this calculation method in petrological applications.

1. Introduction

Alkaline rocks, due to their chemical and mineral composition, have typical structures [1,2,3,4]. They crystallize due to the phenitization of magmas as a result of their penetration through layers of supracrustal rocks [5]. During these processes, chemical changes occur that cause changes in the proportions and sizes of minerals. This is accompanied by numerous secondary processes that affect the structure of the rock. It characterizes the way minerals are arranged in rocks and depends on their size, which phase is the majority in the rock, and the time of their crystallization. In the case of alkaline rocks, these are potassium and sodium feldspars as well as plagioclases, accompanying dark minerals represented by alkaline amphiboles and pyroxenes, and accessory minerals. Feldspars usually have a tabular habit, and they are often legible in syenites. Feldspars often form tabular-shaped minerals in cross-section, while amphiboles and pyroxenes form columns and needles. However, in the course of melt evolution and changes after magma processes, the corrosion of some minerals and significant changes in their arrangement occurs. Added to this are after-magma processes causing geochemical transformations of minerals and the crystallization of new phases. All this together floats the character of textures in a given rock and their uniqueness both in terms of heterogeneous intrusion and in terms of different massifs. The authors in this publication present the results of studies of rocks from the NW region of the Lovozero Massif, an intrusion of alkaline rocks located in the Kola Peninsula region. Fractal analysis is a large research tool used in many applied sciences. Applications of fractal analysis in geology are numerous and indicate that this technique will develop. Chen et al. [6] used the fractal dimension factor to determine the nature of grains in the soil. Lehmann et al. [7] dealt with the analysis of soil structure. Martin et al. [8] studied the effective porosity of the soil in the context of water migration. Kincal et al. [9] dealt with the process of rock weathering and the study of secondary porosity using the fractal dimension factor. Zheru et al. [10] dealt with the issues of the fractal geometry of the distribution of elements on the surface of grains. Mohammadi et al. [11] analyzed the issues of changes in gold occurrence parameters for deposit purposes. Zhengli et al. [12] performed similar analyses in the study of gold and copper deposits. Most works of fractal geology focus on the analysis of effective porosity or the assessment of diffusion channels of elements. There are relatively few works dealing with the connection between the shape of minerals, the textures of rocks, and the fractal coefficient. Pioneering analyses in the matter of resolving mineral textures using the fractal analysis method were published at the end of the 20th century by Fowler [13,14], whereas Turcotte [15] described the general work on fractals in geology, and Kruhl [16] focused on the study of igneous rocks. Słaby et al. [17] used fractal analyses to determine the degree of mylonitization of the rock. Voitsekhovsky et al. [18], in turn, analyzed the textural issues of rhythmically layered rock complexes. Huber et al. [19] analyzed rock-forming minerals in basic layered intrusions. Gulbin and Evangulova [20] studied the character of quartz in granites, while Golsanami et al. [21] used fractal studies in the analysis of clay minerals using SEM. In the case of clay minerals, many works have been published since the end of the 20th century [22,23,24,25,26]. In the case of alkaline igneous rocks, there is still a great lack of publications, and the subject of fractal analyses is developing and evolving. Potter et al. [27] analyzed the nature of perovskite in the Afrikanda intrusion. The use of fractal methods for calculating the contour of selected crystals allows their description to be unified and the results to be reproducible [28]. The use of fractal dimension coefficient studies to determine the shape of minerals is still a relatively innovative solution. In rock petrography, the shape of minerals is a very important factor informing about their genesis and secondary processes occurring in the rock [29]. Minerals that crystallize first may have an anhedral shape, while those that crystallize last in free spaces are anhedral [30,31]. Additionally, during crystallization processes, chemical changes occur in the alloy from which the minerals crystallize, which leads to reactions with already crystallized phases, and those that remain may undergo de-mixing processes [32,33]. Moreover, in post-magmatic and metasomatic processes, the systematic removal of some substances may occur, enriching the rocks with others. In the case of alkaline rocks, the original melt could have been ultrabasic (such intrusions and vein rocks are known in the discussed region), and then, as a result of the evolution of the magmas, they were enriched in alkaline components, which influenced the final mineral composition of the discussed rocks [34,35,36]. The crystallization of residual melts and post-magmatic processes led to the occurrence of rare minerals, which often crystallized in an anhedral form, and in the peripheral parts of the melt, where phenitization of previously crystallized rocks occurred, they usually constitute a euhedral form with numerous inclusions. The method of defining the shape of minerals is therefore of great importance in recognizing their genesis. However, today, numerous different terms can be distinguished to describe the shape of minerals, because they were assigned arbitrarily by the observer of the sample [37,38]. Moreover, due to their arbitrary nature, these terms are not precise. They also do not allow for the detection of subtle differences that may appear during the tracking of samples taken in the intrusion. The authors, conducting research using the fractal dimension coefficient, have shown below that it is possible to precisely determine the mineral’s shape coefficient and precisely, repeatably track its changes by defining them quantitatively. This allows us, together with the traditional petrographic description, to determine specific boundaries corresponding to a specific shape that the described mineral takes during its tracking in the rock. This coefficient, together with the basic data, constitutes a significant improvement in the petrographic description of rocks in space, allowing these changes to be tracked in the form of visualization of coefficient change maps. This prompted the authors to study the rocks of the Lovozero Massif and, after recognizing the typical rock-forming minerals illustrated below and determining their shape—to which the fractal dimension coefficient was assigned—to track how it changes in this intrusion based on the collected calculations from rock samples collected in the field. The purpose of this study is to verify the crystallization character factor of minerals from alkaline rocks for further comparison. The example of the Lovozero Massif was chosen deliberately since linear textures dominate in this massif, and they affect the arrangement and size of minerals; the effect of rock phenitization was also found to contribute to the crystallization of skeletal crystals. All these features make it possible to illustrate these changes for further interpretation. The authors believe that this innovative method will allow much more precise determination of different zones of the studied rocks based on changes in the fractal dimension coefficient.

2. Geology of the Lovozero Massif

There are more than a dozen diverse intrusions of alkaline-basic rocks in the NE Fennoscandia region. These intrusions form two separate zones composed of more primitive magmas (such as Africanda and Kovdor) and having a composition contaminated with supracrustal materials (such as Kchibiny, Lovozero, Sokli, and Turij) [3,39,40,41,42,43,44,45,46,47,48,49]. Most of these intrusions are associated with the activity of hot spots in the early Paleozoic plume [50,51,52,53,54,55,56,57,58]. These intrusions are surrounded by Archean and Paleoproterozoic rocks of magmatic and metamorphic character, forming the foundation of the Baltic Shield of the Eastern European Platform [59,60,61,62,63,64,65,66,67,68], and sometimes covered by Pleistocene sediments [69,70]. The formation of these intrusions dates to the period of 465–364 million years ago [48,54,71,72], forming the Kola–Karelian Alkaline Province (Figure 1) [46,73,74]. This province is of particular interest because the formed intrusions represent the largest complex of such rocks in the world, being the subject of exploration, mining, and recultivation [75,76,77].

The Lovozero Massif is one of the largest intrusions of alkaline rocks found in this province (next to Khibiny). It is a central intrusion with an area of 587 km² [55,79] and an age of 370–359 Ma [44,45,67,80]. This intrusion consists of concentrically arranged belts made of various rocks [81,82,83,84]. The outermost one is built of massive syenites and porphyrites, and metasomatic rocks (Figure 2). Inside is present a stratified complex of fojaites and lujavrites, interlacing with each other, creating rhythmic transitions of one rock into another. These rocks formed a layered complex zone dominated by rocks of layered character with distinct linear textures present. Above these rocks there are urtites, jovites, and other rocks containing sometimes interesting, rare mineralization. These rocks also often exhibit a layered character. The third sequence consists of various types of vein formations (melteigites, tinguaites, micro syenites, and veins with natrolite) and pegmatites, as well as extrusive rocks occurring in the intrusion [68,85]. In the Lovozero intrusion, there are many rare minerals such as murmanite, epistolite, narsarsukite, eudialyte, and many others [64,68,76,77,78,86,87]. The rock samples described in this article come from the boundary zone (Flora Mt) and the layered zone (Aluaiv, Karnasurt Mts, see Figure 2), located in the NW part of the discussed intrusion. The presence of these minerals in the Lovozero Massif and other massifs in the NE Scandinavian region also contributes to their intensive exploration and the development of tourism in the region [88,89].

3. Type of Analyzed Rocks

The following rock types were distinguished and studied during the field survey: Massive syenites, lujavrites, foyaites, jovites, and porphyrites.

3.1. Massive Syenites

Massive syenites are exposed in both the Aluaiv and Karnasurt slopes. They are gray–green-colored rocks. Their structure is coarsely crystalline. The texture is compact and disorderly. In some rocks, there is also a pseudo-ophitic, nested texture, emphasized by accumulations of several mineral crystals next to each other. In these syenites, large specimens of microcline and orthoclase twinned in Carlsbad law (twinning of crystals most often along the 010 axis), are visible, sometimes arranged in the form of contacting plates, resembling a pile of bricks. Many of these crystals are interspersed with each other and sometimes also show regeneration (albitization). Usually, these minerals have a tabular shape and are sometimes strongly elongated. Their presence often takes the form of polycrystalline accumulations of various ages in the described rocks. Natrolite is also visible in the vicinity of the feldspars in the form of small aggregates (Figure 3A). Between these crystals, large crystals nepheline and accompanying eudialyte are visible between the other minerals. Nepheline crystals are usually cubic in shape, whereas eudialyte is usually anhedral, filling the spaces between the feldspars and nepheline, which give it a polygonal shape. It is accompanied by apatite in the form of euhedral crystals against a background of femic minerals. Apatite crystals usually approximate a hexagonal shape depending on their orientation and intersection with thin sections. Next to them, large crystals of aegirine are visible in the form of needles interpenetrating the analyzed minerals or large anhedral crystals poikilitic interspersed with nepheline and apatite. Loparite is also visible against the background of these minerals. In addition to aegirine, the rock also contains riebeckite in the form of small crystals and small concentrations of astrophyllite. Small crystals of lamprophyllite are also visible, forming small clusters of anhedral crystals between the eudialyte and aegirine. Ore minerals are represented by lamprophyllite, titanite, ilmenite, and pyrite. Accessory minerals can also be found, such as ussingite, lovozerite, murmanite, and many others [33,91,92]. Enrichment in rare elements is quite common in these rocks. This is most often the case with loparite (Nb, Th, Ce), apatite (Ce), or titanite (Nb, Th, Ce) [93].

3.2. Nepheline-Aegirine Foyaites

These rocks often have a visible green–gray color. In these rocks, the amount of femic minerals is predominant. These are rocks with a coarse-crystalline structure, a compact, disorderly texture, sometimes with visible lineation, emphasized by aegirine crystals. The aegirine minerals in these rocks form large aggregates. Some of them are intergrown with each other in a poikilitic manner. Aegirine often forms crystals in which smaller femic minerals occur in the background, creating poikilitic inclusions (Figure 3B). The potassium feldspar is mainly represented by microcline, usually having a tabular shape, forming a polycrystalline accumulation. Some individuals are overgrown with natrolite, forming palisaded small crystals, and often forming rims emphasizing the shape of the feldspar. Next to the microcline, there are accumulations of nepheline, forming clusters of contacting crystals. These crystals have a square-like character, although, this may undergo certain modifications, especially in polycrystalline aggregates, where several crystals are in contact with each other “by force”. Along with nepheline, there are also single crystals of anhedral apatite. Apatite crystals in the rocks discussed here most often form shapes close to polygons. Alongside these minerals, large crystals of aegirine are visible, forming needle-like and polycrystalline accumulations, separating the leucocratic minerals from each other. Aegirine is accompanied by riebeckite and arfvedsonite. Along with these minerals, titanite crystals are visible with poikilitic in-grown crystals of nepheline and aegirine.

3.3. Aegirine Nepheline Lujavrites

Along with the urtites, aegirine nepheline lujavrites are also visible; green in color with a medium-crystalline structure and linear texture. The microscopic image shows polycrystalline aggregates of microcline, sometimes subjected to the albitization process on the edges of the crystals. They form large irregular crystals, resembling tablets but with an irregular boundary and numerous inclusions of femic minerals. The microscopic image shows nepheline and apatite crystals. Apatite and nepheline crystals in the discussed rocks are visible in an uneven, non-homogeneously crystallized form, close to polygons, also with numerous inclusions of dark minerals. In the vicinity of the microcline, natrolite plaques encrusting these minerals are also visibly surrounded by aegirine in the form of fine-crystalline needles and sheaves, giving a linear character to the rock. Along with these, augite is also visible in the form of small crystals (Figure 3C). These tiny aegirine crystals resemble columns and needles and occur in the form of irregular aggregates and numerous inclusions scattered among other rock-forming minerals. They are accompanied by aenigmatite, forming medium-sized crystals surrounded by aegirine. In addition to these minerals, large crystals of loparite can also be seen in these rocks, usually occurring against a background of aegirine. In these rocks, loparite forms crystals that are repeatedly approximated. Subordinate in these rocks are murmanite and lovozerite, as well as pyrite, chalcopyrite, and zircon.

3.4. Aegirine Eudialyte Lujavrites

These rocks are green in color with red dots with a medium-crystalline, linear texture (Figure 3D). In these rocks, aegirine crystals form numerous needle-like columns of polycrystalline filling the spaces between other minerals occurring in the discussed rocks. Similarly to the above-mentioned lujavrites, they also have numerous inclusions in other minerals. They co-occur with eudialyte, which in these rocks has a distinctly anhedral shape, sometimes forming rounded crystals. Eudialyte appears in the form of small crystals, which sometimes shows zonality. Nepheline and apatite occurring in them also have an anhedral shape similar to polygons, sometimes with an uneven line of the crystal boundary. Nepheline has shown numerous indentations that may resemble the corrosion of this mineral. In the rocks discussed, potassium feldspars occur mainly in the form of small, irregularly shaped crystals, interspersed with numerous inclusions.

3.5. Aegirine Eudialyte Foyaites

There are rocks similar to nepheline-aegirine foyaites, similar to the above, differing in terms of the presence of eudialyte in them. In this rock, aegirine crystals are present, which forms small accumulations “flowing around” individual leucocratic minerals (Figure 3E). These minerals form polycrystalline aggregates composed of small, often deformed, felted crystals, sometimes also called “acmite”. Minerals such as microcline and eudialyte have a lens-like shape and are located inside the zones “flowing” by aegirine. Nepheline forms fine cube-shaped crystals, next to which are visible crystals of microcline, usually also having a rounded shape. Next to these minerals, natrolite is visible, whose crystals surround the above-mentioned phases. Eudialyte often forms zonal crystals. Next to them, loparite and ilmenite are seen as an accessory. Subordinately riebeckite is present.

3.6. Jovites

Jovites are rocks with a light greenish color and a holocrystalline structure; they are porphyritic, compact, and disorderly in texture. In this rock, the microcline forms large tabular crystals resembling an ophitic structure. Small crystals of natrolite are visible next to the microcline. Nepheline, subordinately apatite, is also visible (Figure 3F). In these rocks, co-occurring minerals such as eudialyte and microcline are often seen occurring next to each other, together creating a lenticular shape. Nepheline can co-occur with them, or form separate, much larger cubic crystals. Around these minerals, small needles and columns of aegirine are visible. The accompanying eudialyte crystals have a zonal structure and an anhedral shape. Femic minerals occur in the spaces between the leucocratic ones, although there are fewer of them compared to foyaite. They are mainly formed by aegirine–acmite needles. The rock subordinately shows loparite and titanite.

3.7. Augitic Porphyrites

These are rocks with a gray–green color, a porphyritic, compact, and pseudo-fluid texture, and are sometimes miarolitic. The rock is formed by small crystals of albite and microcline. These minerals usually form accumulations of anhedral small crystals. Aegirine usually forms crystals of a needle and columnar character, sometimes reaching several cm in size, often with numerous inclusions of small leucocratic minerals. Large euhedral crystals of minerals such as eudialyte, epistolite, murmanite, narsarsukite, lorenzenite, and loparite are visible. These minerals have a size reaching up to 2 cm. They are shaped in a skeletal form and have well-crystallized outer walls, while inside, they have a large number of fine minerals, the same as in the vicinity of the phase under discussion. Some aegirine crystals are much larger, reaching several cm in length. They are accompanied by radial clusters of astrophyllite (astrophyllite sun). Eudialyte usually ascends in the form of adhesions of several crystals. Sometimes in these rocks are voids in which these minerals form druse-like accumulations (Figure 4A).

3.8. Vein Rocks

In this massif, there are also vein formations in the form of lamprophyres, and syenite pegmatites built of several dominant phases, e.g., loparite. An example of this is the loparitite from the Aluaiv area. It is a macroscopically black-colored rock with a coarse-crystalline structure and a compact, disorderly texture. The microscopic image shows crystals of aegirine accompanied by astrophyllite and lamprophyllite. These crystals are poikilitic interspersed with large and approximated crystals of loparite, having a zonal structure (Figure 4B). Subordinately in these rocks, the occurrence of microcline can also be found in the form of tabular crystals occurring between the aegirine, and also nepheline and apatite against the aegirine. The microcline is accompanied by natrolite. In the background of aegirine, nepheline crystals have a cubic shape, together with apatite are euhedral minerals. Microcline aggregates usually form polycrystalline small clusters of a diverse shape.

4. Materials and Methods

Surveys were conducted in the western part of the massif on the slopes of Mount Aluaiv, in the northern part in the Karnasurt area where the layered, complex rock sequences were recorded, and also in the “Flora” massif area where porphyrites were found. In total, 242 analyses were performed in 45 minerals and in 16 rock samples. When preparing a sample for testing, it is important to take a sample of fresh rock, i.e., one that has not undergone weathering processes, so that the minerals tested in this rock do not bear traces of decomposition. We carried out a detailed petrographic description of the rocks to ensure the mutual relations and proportions of minerals in terms of quality, quantity, size, and shape. The selection of rock samples with minerals depends on the issue being described; other rock and mineral samples are suitable for selecting crystallization processes as well as other samples for determining secondary processes, such as testing the state of the rock mass, stresses in the rock, and mineral crushing. The main criterion for data validation is the close connection of fractal studies with petrographic analyses. Unaltered rocks in which minerals constitute the rock-forming phase can be tested for their crystallization, where secondary processes occur on a larger scale, and where the reconstruction of the original conditions may be difficult or impossible. In this massif, the authors collected rock samples from surface exposures (cutting away fresh rock) and quarries. It was not possible to collect rocks from drill cores. Our analyses below showed that the fractal dimension coefficient correlates with the distinguished rock types (Figure 2), but there were also some deviations.

From the samples of these rocks collected in the field, thin sections and microphotographs were then taken using a Leica DM2500P polarizing optical microscope in the microscope at the Department of Geology, Soil Science, and Geoinformation at the Maria Curie-Skłodowska University in Lublin. A list of slides and a brief description of them can be found in

Table 1. Fractal dimension factor results were calculated for four minerals (eudialyte—EUD, aegirine—AEG, nepheline—NE, potassium feldspar—KFS) in samples of the northwestern Lovozero Massif. Blank spaces indicate lack of data in a particular case.

Sample	EUD	AEG	NE	KFS	Sample	EUD	AEG	NE	KFS
01aLV21_1	2.73	1.36	2.14	2.05	05LV12	2.65	1.65	2.07	2.13
01aLV21_2	2.27	1.60	2.03	2.34	05LV21_1	2.16	1.62	1.94	1.95
01aLV21_3	2.28	1.83	1.70	1.43	05LV21_2	2.15	1.69	1.76	2.43
01aLV21_4	2.03	2.09	2.63	2.35	05LV21_3	1.44	1.12	1.74	2.04
01aLV21_5		1.08	2.57	2.37	05LV21_4	1.83	1.36	2.00	1.52
01aLV21_7	2.36	1.76	2.64	1.38	05LV21	1.90	1.45	1.86	1.98
01aLV21_6			2.61		06LV21_1	1.41	1.74	2.29
01LV21	2.33	1.62	2.33	1.99	06LV21_2	2.65	1.68	2.25	1.90
01LV12_1	1.81	1.79	1.38	1.99	06LV21_3		1.62		2.10
01LV12_2	2.15	2.21	1.84	2.04	06LV21_4	2.27	1.15		2.74
01LV12_3	1.94	2.37	2.10	1.92	06LV21	2.11	1.55	2.27	2.25
01LV12	1.97	2.13	1.77	1.98	07LV21_1	2.41	1.46	2.10	2.42
01LV21_1	1.97	2.17	2.08	1.53	07LV21_2	2.63	1.57	2.77
01LV21_2	1.80	2.11	1.76	1.90	07LV21_3	2.39	2.71	2.58
01LV21_3	1.87	1.52	1.99	1.80	07LV21_4	2.52	2.00	2.51	2.41
01LV21_4	1.82	2.58	1.34		07LV21	2.49	1.94	2.49	2.42
01LV21	1.86	2.09	1.79	1.74	09LV21_1	2.29	2.66	2.37	2.41
02LV03_1	1.25	1.86	1.46		09LV21_3	1.65	2.46	2.27	2.38
02LV03_2	2.26		1.28	2.09	09LV21_4	2.50	2.14	2.27	2.45
02LV03_3		1.38	1.32	2.56	09LV21	2.15	2.42	2.30	2.41
02LV03	1.55	1.62	1.35	2.32	10LV21_1	2.46	2.00	1.60
02LV21_1	1.45	2.55	2.27	2.63	10LV21_2	1.73	1.41	2.37
02LV21_2	2.28	1.38	1.72	2.48	10LV21_3	1.43	1.66	2.34	2.13
02LV21_3	2.06	1.40	1.86	2.33	10LV21_4	2.41	1.65		2.63
02LV21	1.93	1.77	1.95	2.48	10LV21	2.01	1.68	2.10	2.38
03LV21_1	1.83	1.25	2.34		56LV00_1	2.43	1.38
03LV21_2	2.06	1.79	1.89		56LV00_2	2.01	1.70		1.83
03LV21_3	1.76	1.18	1.96	1.88	56LV00_3	2.49	1.66	1.52	1.64
03LV21	1.88	1.31	2.06	1.88	56LV00	2.31	1.58	1.52	1.74
04aLV03	2.40	1.00			P_Lv21_1 *	1.94	2.60	2.04	1.95
04aLV03	2.40	1.00			P_Lv21_2 *	1.85	2.53	1.93	2.55
05LV03_1	1.76	1.76	1.61		P_LV21 *	1.89	2.57	1.98	2.25
05LV03_2	2.89	1.00			MIN	2.94	2.71	2.77	2.74
05LV03	2.33	1.38	1.61		MAX	0.85	0.95	1.28	1.38
05LV12_1	2.77	2.05			AVERAGE	2.11	1.75	2.01	2.13
05LV12_2	2.23		2.49	2.13	Standart deviation	0.63	0.57	0.81	1.12
05LV12_3	2.94	1.26	1.65

* sample P_LV21 is the loparite vein rocks.

and Figure 5. During the microscope observations, special attention was paid to minerals such as potassium feldspars (microcline), eudialyte, nepheline, and aegirine. The calculation of their shapes was carried out with the help of 3.34.4 QGIS and Ms. Excel 2019 programs (Figure 5) according to the following sequence: after importing the microphotography into the QGIS program, a vector map was created, separating the background (black color) from the area to be further processed (Figure 5B); then, the border of the studied crystals was separated, the area was divided into shapes corresponding to individual minerals (Figure 5D), which were given labels. When preparing a rock sample for testing, its microphotographs were taken, and after the process of changing raster graphics to vector graphics, these minerals were collected for fractal analyses. The process of changing the graphics was done manually (the envelopes of mineral boundaries were made manually). The authors repeated the process of changing the graphics and calculating the fractal dimension on the selected sample. The differences resulting from the possible image resolution in relation to the size of the mineral and the possible error of the human factor in selecting the boundary amounted to 0.45%, which can be considered as an acceptable error of the human factor. In this case, the effect of the original image resolution depends on the size of the crystal. If the mineral being tested is large enough, it can be omitted; if not, it is worth taking a microphotograph with a higher magnification. Specific crystals were then selected for further calculations, taking care that they were not too small and not too large (not protruding beyond the study area), where possible (Figure 5E). After the grid of squares was applied, further calculations were made in this grid (Figure 5F). The resulting counts were processed according to the following Formula (1) [20,21,22,23,24,25,26,27,28]:

$$\mathrm{D}={\displaystyle \frac{\left|{\log }_{\mathrm{x}2}{-\log }_{\mathrm{x}1}\right|}{\left|{\log }_{\mathrm{d}2}{-\log }_{\mathrm{d}1}\right|}}$$

(1)

where x₁ and x₂ correspond to the number of passes of the individual squares x₁ and x₂ of the contour of the shape, with d₁ and d₂ of their respective side lengths. During the fractal analyses, a grid with meshes of 100, 50, 25, and 12 pixels was selected, respectively. In order to illustrate the operation of Formula (1), the authors subjected a circle, square, and rectangle to fractal studies (see Figure 6). The calculation results are presented in this figure. During the analysis of the fractal coefficient, the initial size of the analyzed mineral in relation to the size of the sides is important. The authors tried to select micrographs of the analyzed minerals so that their size exceeded the size of the analyzed boxes and allowed for a precise fractal analysis.

The result obtained as a result of applying the Formula (1) is then averaged by repeating the calculation multiple times with different square sizes. It represents a fractal, box-like dimension that determines the nature of the mineral boundary. When analyzing the results for different shapes, it is worth noting that this dimension determines the ratio of the length (extension) of the boundary to its size. These calculations show that the circle has the smallest box size, the square is larger, and the complex figure has the largest. The fractal coefficient calculated for the crystal boundary does not reflect the entire surface of the crystal, hence the value it can take ranges from 1 (for simple, spherical crystals) to more than 2 (in the case of crystals containing many inclusions and having a highly relaxed boundary). Reasoning further, the fractal dimension is, therefore, a numerical representation corresponding to the shape of the mineral. A corroded mineral with very extensive borders and inclusions will have the highest coefficient. The non-integer values of the fractal dimension result from the properties of the boundaries of these crystals. However, a strict understanding of this coefficient requires the reader to familiarize himself with the professional literature [94,95,96]. From the mineralogical point of view, the range of values of changes in the fractal dimension coefficient is important, which can be assumed as an equivalent of a specific character of the mineral shape. However, it does not depend on the direction in which that shape changes. The relationship of the numerical representation with the shape of the mineral is clear and important. A mineral having a euhedral shape will have its specific numerical range (depending on its representation in the thin sections), but if it corrodes or crystallizes in a space already filled with other minerals, its fractal dimension will be different. These data allow us to represent the extremes-mineral confined—crystallizing in the vicinity of others—minerals having their shape, and minerals undergoing corrosion. Tracing these space changes allows us to plot maps of changes in shape-factor content for a given mineral, which can allow us to determine the change in the degree of influence of various processes. In alkaline magmatic rocks such as syenites, the presence and shape of minerals such as potassium feldspars, feldspars (nepheline), eudialyte, and pyroxene (aegirine) can be crucial to understanding the course of their crystallization. The authors decided to investigate this factor by checking the change, like the four rock-forming minerals in several rock profiles of the Lovozero Massif (the results of these analyses can be found below). To sum up the methodology described above, this study can be presented in the following block diagram:

1.

Sampling and field observation.

2.

Making thin section plates of rocks.

3.

Selecting minerals for testing and taking micrographs.

4.

Petrographic analyses and selection of minerals.

If the amount of minerals is sufficient and the phases prepared for analysis correspond to the assumed processes, proceed to the next stage; otherwise, return to point 1 or 2.

5.

Assessment of the size of minerals for fractal analysis (if the minerals are too small, a micrograph can be taken at point 3 with a different magnification).

6.

Performing fractal analysis.

7.

Further interpretation in combination with petrographic knowledge obtained from observations.

5. Results

Minerals such as aegirine, eudialyte, potassium feldspar (microcline), and nepheline were examined. These minerals were present in all rock samples taken in the massif except for nepheline (which was not present in the porphyrites). Their quantity, determined by planimetric studies, is listed in Table 1.

5.1. Identification of the Shape of the Rock-Forming Minerals in the Studied Rocks

The minerals studied were characterized by varying sizes and formations in these rocks. In massive syenites, aegirine forms large crystals exceeding 1 cm in size. As a rule, these crystals form numerous accumulations, sometimes overgrowing each other’s columns with numerous inclusions of usually nepheline, microcline, and ore minerals. Eudialyte found in the discussed rocks forms anhedral crystals, usually filling the spaces between femic and leucocratic minerals. Its shape is adapted to the voids in which this mineral crystallizes. Nepheline in the studied rocks usually has an anhedral shape and is often found in the vicinity of feldspars or as an intergrowth in aegirine. In the case of nepheline-aegirine foyaites, aegirine forms numerous crystals, sometimes poikilitic overgrowing each other. Accompanying microcline minerals are formed in the form of plaques, but their boundaries are not always equal because they are surrounded by natrolite, which crystallizes on these crystals, contributing to their corrosion. Nepheline sometimes forms polycrystalline accumulations of an anhedral shape. The single crystals of eudialyte that occur are anhedral, having the shape of the rock voids in which this mineral crystallized. The minerals studied in the aegirine and nepheline lujavrites have a similar shape. In these rocks, aegirine forms both coarse-crystalline minerals and aggregates of fine, fused crystals. In the case of aegirine-eudialyte lujavrites, the eudialyte crystals are anhedral in shape, and some are also zonal. In these rocks, eudialyte, although it does not have the properties of shaped minerals, crystallizes in the form of large phases, often having quite developed boundaries. In aegirine-eudialyte foyaites, aegirine crystals usually occur in the form of fine-crystalline aggregates overgrowing each other. Microcline crystals usually have a rounded shape, as does eudialyte. Nepheline, on the other hand, forms finely crystalline accumulations. In jovites, microcline is formed as large plaques, sometimes corroded by natrolite. Nepheline occurs in the form of fine crystals, and aegirine forms columnar crystals with numerous inclusions of leucocratic minerals. In porphyrites, the minerals studied had a euhedral shape, with many feldspar inclusions inside these minerals. Eudialyte often had multiple approximations, occurring in these rocks as adhesions of several minerals. Its crystals reached up to 2 cm in size. Next to it is aegirine, usually forming long needle-shaped crystals, usually reaching up to 1 cm in size or small accumulations of this mineral in pillared form. Accompanying these phases, potassium feldspars occur in the rock background as small crystals, usually tabular in shape, sometimes reaching a larger size reaching up to 0.5 cm. In pegmatites, these minerals are usually very large, reaching up to several cm in size. Particularly in the loparitite rocks, aegirine accompanies these minerals as a crystal crystallizing in their interstices. Sometimes small aggregates of nepheline and crystals of microcline, usually of an anhedral nature, can also be seen next to the loparite.

5.2. Calculated Fractal Dimension Results for Syenite Rock-Forming Minerals

The minerals aegirine, eudialyte, nepheline, and microcline found in the analyzed rocks were measured, and their fractal coefficient is given in Table 1. In the given technique, the estimated error can be defined as less than 1% of the value of the given number.

Aegirine has the lowest fractal dimension factor in samples 03LV21 and 04aLV03 (lujavrites), where it occurs in the form of fine needle-like crystals, highly elongated (Figure 7). The highest fractal dimension factor for aegirine was measured in samples 07LV21, 09LV21 (syenites), and P_LV21 (pegmatite, loparite), in which aegirine occured in the form of large aggregates of crystals with elaborate boundaries and numerous inclusions. In the case of eudialyte, the lowest fractal dimension factor was determined in sample 02LV21 (jovite), where the crystals had rounded shapes, similar to a lens. The highest fractal dimension factor was recorded for sample 05LV12, where eudialyte had jagged boundaries due to crystallization in the space of other minerals. Similarly, a high fractal dimension factor for eudialyte can be determined for sample 56LV00 (porphyrites), where the eudialyte had a euhedral shape but has a lot of inclusions. In the case of nepheline, the lowest fractal dimension factor was determined for sample 02LV03, where it occured as aggregates of several crystals, while the highest fractal dimension factor was found for sample 07LV21, where the occurring nepheline crystals had numerous inclusions of aegirine and other minerals. In the case of microcline, the lowest fractal dimension factor was measured for samples 01aLV21 and 56LV00, where the mineral occured in the form of small tabular crystals. The highest fractal dimension factor for this mineral was determined for samples 02LV21 and 10LV21, where this mineral occured in the form of large tabular crystals corroded by natrolite crystallization, also having numerous mafic mineral inclusions. Examples of minerals with low and high fractal dimension factors are illustrated below. The relationship between high and low aspect ratio is shown in the shape of the mineral (Figure 6). Where the coefficient is low, the mineral has a shape as close as possible to a square (it can be oval, e.g., nepheline, Figure 7E). Some low-index minerals have a euhedral shape (as described above, e.g., aegirine, Figure 7C). The rounded shape may indicate the process of mineral growth that has been arrested (or its rotation, Figure 7A). A mineral crystallizing between other phases, with a large number of inclusions, may have strongly jagged boundaries (Figure 7D). Corrosion leads to very complex boundaries (as shown in the example of eudialyte, Figure 7B).

Analyzing the values of standard deviation for the entire population of minerals examined, presented in the table, it can be stated that potassium feldspar is characterized by relatively the most variable features, the shape of which depends on the order of crystallization and secondary processes (albitization found). Nepheline minerals (most often crystallizing in the discussed rocks as cube minerals) and eudialyte, which usually also have an anhedral shape, have relatively low coefficients. In this interpretation, one must be very careful, because there were only four (6%) out of sixty-nine samples of rocks subjected to phenitization, eighteen (26%) massive rocks and forty-seven (68%) rocks classified as a stratified zone—constituting the majority and influencing the nature of titers in the examined population.

5.3. Distribution of Fractal Dimension Maps of the Crystals and the Interpretation of These Data

The distribution of the fractal dimension factor of the analyzed minerals was obtained by applying data on the geographical location of the sampling points on the map, and showing the average fractal dimension factor, examined for several minerals of a given rock. In this way, a map was created connecting the data obtained from the measurement of points with isolines. Below, the distribution of the fractal coefficient measured for the four minerals discussed above is given. In the case of aegirine (Figure 8), the highest fractal dimension coefficients are found in lujavrite samples in the northwestern part of the massif, where aegirine occurs in significant amounts, forms recrystallized specimens with each other, and often has strongly jagged boundaries.

Lower coefficients can be recorded for jovites (darker field in the western part of the massif), where these crystals form needle-like fine forms, and for porphyrites in the northeastern part of the massif. Nepheline forms several areas of varying fractal dimension coefficients (Figure 9). The highest coefficients were recorded for crystals located in the zone of occurrence of lujavrites and jovites in the northern part of Lovozero and its western part. There, the mineral usually has extensive boundaries and a large number of inclusions. Low coefficients were recorded for pegmatite with loparite, where nepheline forms small cube-shaped crystals, and in the northern part of the massif, where syenites with polycrystalline nepheline aggregates occur.

In the case of eudialyte (Figure 10), the lowest fractal dimension coefficients are found in the northeastern part of the massif.

In these parts where porphyrites occur, these minerals have a euhedral shape, as well as in jovites in the northern part of the massif. In other parts of the massif, their fractal dimension increases and is highest in syenites and lujavrites, where eudialyte crystallizes in the form of crystals occupying voids in the rock between other phases, sometimes also having numerous fusions. In the case of potassium feldspar (microcline, Figure 11), the highest fractal dimension factor can be observed in the northern part of the massif in the region of the occurrence of lujavrites in which the mineral is shaped in the form of large corroded plaques, encrusted with natrolite with numerous inclusions; the lowest fractal dimension factor was recorded for porphyrites in the northeastern parts of the massif, where the mineral occurs in the form of small crystals.

5.4. Interpretation of Fractal Distribution of Crystals in a Discussion of Intrusion Evolution

Aegirine usually has fusions associated with phases of femic minerals such as loparitite. It also co-occurs with apatite and nepheline. Where there is a high fractal dimension, aegirine crystals crystallize in the form of large individuals between these phases. In rocks where the fractal dimension of these minerals is low, they usually occur as adhesions intersecting microcline and plagioclase and forming thin needle-like forms, sometimes repeatedly approximated to form compact polycrystalline aggregates called acmite. Where this mineral crystallizes longer, forming large specimens, its fractal dimension is higher, while where it forms fine aggregates of needle-like pillars, it is much lower. In the case of eudialyte, this mineral shows zonality in many rocks of the massif and forms crystals that fill the space between other phases. In these areas, it is usually characterized by small fractal dimension factors. In other rocks, especially in the zone of phenitization, there is automorphic eudialyte having a large number of inclusions. These crystals sometimes also undergo corrosion, and then their fractal dimension is much higher. In fenitized rocks where skeletal crystals are formed, their fractal dimension is high, while where they crystallize in mineral intersticia, their fractal dimension is lower. The lowest coefficient has those crystals that have adopted a rounded shape and are surrounded by fine needles of aegirine. In the case of nepheline in many rocks, these crystals have a cube-like shape in a cross-section characterized by a relatively low fractal dimension factor. Their coefficient increases significantly when these crystals have extensive walls (they corrode) and numerous inclusions. In the case of microcline (orthoclase), this mineral in the described rocks occurs in the form of plates approximated by Carlsbad’s law, characterized by a relatively low fractal dimension factor. In fenitized rocks, these crystals have a rounded shape, and their fractal dimension is the lowest. The highest coefficient of fractal dimension has those crystals, which are characterized by numerous inclusions of femic minerals.

In interpreting these results in the fields of low and high fractal dimension coefficients for eudialyte, it is possible to indicate the sites of influence of metasomatic processes. In the case of the fractal dimension factor for aegirine, it is possible to indicate the places where this mineral was formed in rocks that crystallized in the deeper parts of the intrusion. Given more time to crystallize, it formed large crystals often with numerous inclusions. Where it crystallized faster, it formed fine aggregates of needle-like crystals. The same is true for nepheline and microcline, where these crystals, having a high fractal dimension factor, indicate the processes of geochemical changes in the rock during their crystallization, and a low factor in rocks where they form a system of single or close-up individuals possibly in the zone of fenitization (rounded shape) [18]. On this basis, combining all the relationships discussed, it is possible to interpret the sites characterized by the occurrence of rocks that must have formed in the deeper parts of the intrusion, and (Figure 12) as a result of the processes of lifting their passages, some geochemical changes contributed to the corrosion and recrystallization of some components; rocks formed in shallower zones where the cooling process was faster and forced linear structures and rapid crystallization of these minerals, as well as in zones associated with the impact of metasomatism where skeletal crystals were formed.

6. Discussion

The present study, combined with a petrographic description, can provide precise data on the textural changes of the minerals in the analyzed rocks. The importance of these data is interesting in that, for each mineral, it is possible to determine the specific values of the limiting intervals in which the mineral occurs in the rock in euhedral form, in anhedral form, and in the case of its corrosion or other processes strongly affecting changes in the crystal boundary (Figure 7). This makes it possible to identify and plot where the crystals crystallized in the rock at a certain stage and also where they corroded. If these intervals are standardized for all the analyzed phases, percent changes can be determined, which, together with the location, can contribute to estimating the degree of influence of various processes occurring in the magma body itself during its crystallization as well as afterward. For example, where potassium feldspar crystals corrode, the map shows fields of increased fractal dimension factor values [97,98,99,100]. Tabular crystals, shaped properly and forming porphyritic textures, are highlighted in blue (see map above, Figure 12). The presented results of the analysis of the fractal dimension coefficient for the rock-forming minerals of alkaline rocks examined in the Lovozero Massif showed some differentiation within the classes of individual rocks (compare the presented graphs in chapter 5 with the geological map on Figure 2). This means that these data can be processed in combination with the petrographic description. Simple graphical presentations of changes in the coefficients can be correlated with the mentioned porphyrites and foyaites and lujavrites of the layered complex. In the fenitization zones where skeletal minerals occurred, there is a tendency for the fractal dimension coefficients to increase, which correlates well with the developed boundaries of these minerals, full of irregularities and inclusions. In the case of the layered rock complex, such a correlation can also be read for the discussed minerals, especially for eudialyte, which, in these rocks often has a shape close to spherical. Further detailed interpretation also indicates many inhomogeneities within the discussed rock classes. This means that the interpretation of these data cannot be subject to simple rules of counting and visualization. In these rocks, these minerals were subject to various secondary processes that were probably related to crystallization processes (interaction of these phases with each other) as well as post-magmatic processes that contributed to the formation of phases such as natrolite or loparite. The visualized data can therefore be the basis for further interpretations regarding the inhomogeneity of the intensity of these processes. This means that in the layered complex, the changes taking place were diverse in the massif space in terms of intensity.

Our team relied on samples taken from the surface (see Section 4—Materials and Methods). In order to detail these data, a larger amount of data is needed. This will allow us to select representative samples in more detail in the future and, on this basis, to create a series of maps regarding changes in coefficients in both typical rocks and those affected by secondary processes. Due to the pioneering nature of the work and the material collected so far, such a distinction cannot be made on this set. However, the presented results show the possibility of correlating the data that are visualized, which is a significant advancement compared to the imprecise descriptions made earlier. These descriptions did not take into account the small nuances that were hidden in the shapes of these minerals, and after their visualization, it can be seen relatively easily.

It is worth mentioning that the interpretation of these data may be of an economic nature. The diversity of the fractal dimension coefficient for these minerals, which we have shown is correlated with the heterogeneity of magmatic and post-magmatic processes, may suggest prospective locations of increased concentration of rare earth elements that crystallized together with the late and residual crystallization phases, which perfectly correlates with the prospective areas of occurrence of these elements studied by traditional methods [101,102,103,104,105]. This allows for a quick and relatively inexpensive way to delineate common areas characterized by features related to the crystallization of specific indicator minerals.

When the characteristic values are known, a specific boundary or set of boundaries can be drawn defining the various stages of the massif’s formation. On this basis (and other indications), the intensity of these processes can be estimated, and the more accurate the sampling and analysis, the better this map will indicate where such changes occur. When these factors are combined with other phases found in these rocks, the intensity of various processes can be determined with high probability. Numerical data can be summed and processed to determine the percent of changes in the rocks. Such spatial distribution of the results makes it possible to determine various interactions, which is very important for determining the crystallization processes of massive magmatic rocks.

7. Conclusions

The results obtained as a result of the research are closely related to the textual characteristics of the minerals in the alkaline rocks under discussion.

• The authors demonstrated the possibility of displaying the shape of the studied rock-forming minerals in the form of a digital indicator, which is measured in a repeatable and precise manner, regardless of the human factor.
• Changes in the fractal dimension coefficient were documented, which can be correlated with a specific type of rock (significant differences for the layered zone and porphyrites).
• Visualization of these results showed the diversity of the fractal dimension coefficient within individual rock types. This allowed the interpretation of areas with different intensity of post-magmatic and metasomatic processes. The correlation of these areas coincides with prospective places of REE mineralization, which was performed in a conventional way.
• Correlation of changes in the fractal dimension coefficient with the results of classical petrographic descriptions allows for the determination of zones characterized by different textural properties of rocks, which can be the basis for further interpretation and application of these data.
• This method is complementary to the classical petrographic description, supplementing it with precise spatial data.

The results of the study make it possible to systematize the studied minerals and determine their characteristics through numerical data. These data can be visualized both in the form of a table and the spatial distribution of these data. Once these data are standardized, one can attempt to determine the intensity of crystallization and secondary processes, as well as their extent in the intrusion in question. In the case of the present samples, zones from deep parts of the intrusion that were affected by secondary processes, rocks that crystallized in shallower parts, and those that were related to the impact of metasomatism were shown. This approach allows a deeper interpretation of data obtained from microscopic observations of rocks in a measurable and reproducible manner.