Dislocation plays a crucial role in controlling the strength and plasticity of bulk materials. However, determining the densities of geometrically necessary dislocations (GNDs) and statistically stored dislocations (SSDs) is one of the classical problems in material research for several decades. Here, we proposed a new approach based on indentation size effect (ISE) and strengthening theories. This approach was performed on a laser powder bed fused (L-PBF) Hastelloy X (HX), and the results were verified by the Hough-based EBSD and modified Williamson–Hall (m-WH) methods. Furthermore, to better understand the new approach and essential mechanisms, an in-depth investigation of the microstructure was conducted. The distribution of dislocations shows a clear grain orientation-dependent: low density in large <101> preferentially orientated grains while high density in fine <001> orientated grains. The increment of strengthening in L-PBF HX is attributed to a huge amount of edge-GNDs. Planar slip is the main operative deformation mechanism during indentation tests, and the slip step patterns depend mostly on grain orientations and stacking fault energy. This study provides quantitative results of GND and SSD density for L-PBF HX, which constructs a firm basis for future quantitative work on other metals with different crystal structures.

To improve the understanding of high temperature mechanical behaviours of LPBF Ni-based superalloys, this work investigates the influence of an elongated grain structure and characteristic crystallographic texture on the anisotropic tensile behaviours in LPBF Hastelloy X (HX) at 700 °C. Two types of loading conditions have been examined to analyse the anisotropy related to the building direction (BD), including the vertical loading (loading direction//BD) and the horizontal loading (loading direction ⊥ BD). To probe the short-term creep behaviours, slow strain rate tensile testing (SSRT) has been applied to address the strain rate dependent inelastic strain accumulation. In-situ time-of-flight neutron diffraction upon loading was performed to track the anisotropic lattice strain evolution in the elastic region and the texture evolution in the plastic region. Combined with the post microstructure and fracture analysis, the anisotropic mechanical behaviours are well correlated with the different microstructural responses between vertical and horizontal loading and the different strain rates. A better creep performance is expected in the vertical direction with the consideration of the better ductility and the higher level of texture evolution.

Additive manufacturing (AM), also known as 3D printing, is a general concept of building a three-dimensional object layer-by-layer. AM breaks through the manufacturing limitations in conventionally subtractive manufacturing, leading to a great design freedom of components with complex geometries. The potential of integrating AM into existing manufacturing process with additional functionality raises interest in various fields, such as aerospace, automotive and medical applications. To ensure robust AM applications, this PhD project has carried out investigations on the mechanical behaviours of AM components with respect to the characteristic microstructure and the geometrical effects. The investigated materials include Hastelloy X (HX, a solid-solution strengthened Ni-based superalloy) and stainless steel 316L (SS 316L, a solid-solution strengthened austenitic stainless steel) manufactured by laser powder bed fusion (LPBF). The high temperature tensile behaviours, short-term creep resistance and low cycle fatigue performance have been examined. The aim of this thesis is to conduct a fundamental studies that can be applied to different material grades with single phase face-centred cubic (FCC) crystallographic structure. 

LPBF HX shows a great potential for the burner tip repair application in gas turbines. Due to the complex geometry of the burner and the requirement of high temperature mechanical performance, the tensile properties have been systematically examined. Multiple testing variables have been applied, including the specimen geometry, the elevated temperature, the strain rate and the loading direction (LD). Combined with the prior and post microstructural analysis, the deformation and fracture mechanisms have been investigated. For the thin-walled specimens, a clear texture transition is found when it comes to the thinner specimen, and it leads to the lower yield strength as a result. In addition, as the high surface roughness of the LPBF as-built specimen can cause inaccuracy of the yield strength determination due to the overestimated loading cross section, especially for the thin-walled specimen, a calibration method based on the crystallographic texture results has been proposed. Meanwhile, anisotropic tensile behaviours are observed at all the testing conditions due to the elongated grain structure and the characteristic texture along the building direction (BD). At elevated temperatures, the grain boundary embrittlement takes place at 700 °C that leads to the ductility loss in the horizontal loading (LD ⊥ BD). Slow strain rate tensile testing (SSRT) has been performed to probe the short-term creep resistance at 700 °C, since it is a useful tool to address the strain rate dependent in elastic strain accumulation. Surprisingly, the ductility of the vertical loading (LD // BD) remains at a high level not only at 700 °C but also at SSRT condition, and the high ductility results from the evident texture evolution and crystallographic orientation dependent deformation twinning. The good ductility of the vertical loading indicates a better creep performance compared to the horizontal loading. In-situ and ex-situ neutron diffraction measurements upon loading have also been applied for a full-length investigation on the anisotropic tensile behaviours. 

Thin-wall effects on strain-controlled low cycle fatigue (LCF) behaviours of LPBF SS 316L have been investigated by using the tubular fatigue specimens with different wall thicknesses. The comparison between the machined and as-built surface conditions have been drawn. The fully reversed LCF tests were successfully performed without the buckling problem in thin-walled structures owing to the tubular geometry. The surface roughness and the distinct microstructure at the surface region lead to the inferior fatigue strain-life, especially with the low applied strain range. The combined effects have been quantified by estimating the fatigue notch factor, K_f . The LCF tests have also been performed on the regular cylindrical specimens and compared to the wrought SS 316L. A comparable fatigue strain-life is found between the LPBF and the wrought SS 316L. Yet, the secondary hardening caused by strain-induced martensitic phase transformation is only observed in the wrought SS 316L, while continuous cyclic softening is shown in the LPBF SS 316L. In addition, as high level of residual stress (RS) is commonly found in the as-built specimen, the effect of stress relief heat treatment (600 °C /4 hours) on the LCF behaviours has been examined. A great reduction of RS is found after the heat treatment, and higher responding stresses are shown in the stress relieved specimen, which indicates a better fatigue stress-life. 

In summary, the deformation and fracture mechanisms of LPBF HX and SS 316L under different loading conditions have been systematically investigated. Via increasing deeper knowledge of LPBF material behaviours, the LPBF applications can be expanded to a greater extent. 

Additiv tillverkning (AM), även kallat 3D-printning, är ett allmänt koncept där man bygger ett tredimensionellt objekt lager för lager. AM bryter igenom tillverkningsbegränsningar i konventionell subtraktiv tillverkning, vilket leder till stor designfrihet för komponenter med komplex geometri. Potentialen med att integrera AM i befintliga tillverkningsprocesser med ytterligare funktionalitet väcker intresse inom flera olika områden, såsom flyg-, fordons- och medicinska tillämpningar. För att säkerställa robusta AM-tillämpningar har det i detta doktorandprojekt genomförts undersökningar av de mekaniska egenskaperna hos AM-komponenter med avseende på den karakteristiska mikrostrukturen och de geometriska effekterna som uppstår vid tillverkningen. De undersökta materialen inkluderar Hastelloy X (HX, en lösningshärdad Ni-baserad superlegering) och rostfritt stål 316L (SS 316L, ett lös-ningshärdat austenitiskt rostfritt stål) tillverkat med laser-pulverbäddsomsmältning (LPBF). Dragprovning vid förhöjd temperatur, korttidskrypmotstånd och lågcykelutmattning har studerats. Syftet med denna avhandling är att genomföra grundläggande studier som kan tillämpas på olika materialtyper med en yt-centrerad kubisk (FCC) kristallstruktur.

LPBF av HX visar en stor potential för reparation av brännarmunstycken i gasturbiner. På grund av brännarmunstyckenas komplexa geometri och kravet på mekanisk prestanda vid hög temperatur har dragegenskaperna systematiskt undersökts. Flera testvariabler har undersökts, inklusive provets geometri, den förhöjda temperaturen, töjningshastigheten och belastningsriktningen (LD). I kom-bination med mikrostrukturanalys före och efter testerna har deformations- och brottmekanismerna undersökts. Med avseende på väggtjocklek finner man en tydlig förändring i kristallografisk textur när man går mot tunnare provstavar vilket leder till lägre sträckgräns. Eftersom den höga yt-ojämnheten hos LPBF-material kan orsaka en felaktig bestämning av sträckgränsen, särskilt för tunnväggiga strukturer, har en kalibreringsmetod baserad på texturresultaten föreslagits. Dock så ob-serveras anisotropa dragprovsegenskaper vid alla testförhållanden på grund av den utsträckta kornstrukturen och den karakteristiska kristallografiska texturen längs byggriktningen (BD). Vid förhöjda temperaturer sker en korngränsförsprödning vid 700 °C vilket leder till en duktilitetsförlust i den horisontell belastningsriktning (LD ⊥ BD). Slow strain rate tensile testing (SSRT) har utförts för att undersöka det korttidskrypmotståndet vid 700 °C eftersom det är ett användbart verktyg för att hantera den töjningshastighetsberoende oelastiska töjningsackumuleringen vid förhöjda temperaturer. Överraskande nog förblir duktiliteten hög för den vertikala belastningen (LD //BD), inte bara vid dragprov vid 700 °C utan även vid den långsammare SSRT-provningen. Den höga duktiliteten är ett resultat av den utpräglade texturutvecklingen och en kristallografiskt orienteringsberoende tvilling-bildning. Den höga duktiliteten för den vertikala belastningsriktningen indikerar ett bättre krypmotstånd jämfört med den horisontella belastningsriktningen. In-situ och ex-situ neutrondiffraktionsmätning vid belastning har också använts för en mer djupgående undersökning av de anisotropa egenskaperna.

Inverkan av väggtjocklek vid töjningskontrollerad lågcykelutmattning (LCF) har undersökts för LPBF SS 316L genom att använda rörformiga utmattningsprovstavar med olika väggtjocklekar samt genom en jämförelse mellan bearbetade och icke-bearbetade ytförhållanden. De i drag/tryck-last symmetriska LCF-testerna kunde genomföras framgångsrikt utan bucklingsproblem tack vare den rörformiga geometrin. Ytojämnheten och den distinkta mikrostrukturen i de ytnära områdena ledde till sämre utmattningslivslängd, speciellt vid låga applicerade töjningsomfång, och de kombinerade effekterna har kvantifierats genom att uppskatta anvisningsfaktorn, K_f . LCF-tester har även utförts på konventionella cylindriska provstavar samt så har en jämförelse gjorts med konventionellt (smidd) SS 316L. En jämförbar livslängd i termer av töjningskontrollerad utmattning finns mellan LPBF och kon-ventionell SS 316L. Dock observerades ett sekundärt hårdnande för konventionell SS 316L som orsakas av töjningsinducerad martensitisk fastransformation, medan LPBF SS 316L uppvisade ett kontinuerligt cyklisk mjuknande. Dessutom, efter-som en hög nivå av restspänningar (RS) vanligtvis förekommer i LPBF-material, har effekten av avspänningsglödgning (600 °C /4timmar) på LCF-beteendet undersökts. En stor minskning av RS noterades efter värmebehandlingen och de avspänningsglödgade proverna hade en högre spänningsrespons, vilket indikerar bättre utmattningsegenskaper i termer av spänningsomfång mot livslängd.

Sammanfattningsvis har deformations- och brottmekanismerna för LPBF HX och SS 316L under olika belastningsförhållanden systematiskt undersökts. En djupare kunskap om materialbeteenden hos LPBF-legeringar kan leda till att tillämpningsområdet för LPBF-tekniken breddas.  

To ensure the robust design freedom of metallic additive manufacturing, the fatigue properties and the dimensional limitation of as-built components by laser powder bed fusion (PBF-LB) are investigated. Fully reversed and strain-controlled fatigue tests were carried out on tubular specimens with different wall thicknesses, 1 mm and 2 mm, for the purpose of studying the thin-wall effect without having risk of buckling problem during compression. Two wrought conditions are also enclosed as a comparison, which are the cold worked (CW) and solution annealed condition (SA). In the as-built PBF-LB tubular specimens, deformed microstructure and deformation twins are discovered close to the surface region, together with a higher roughness of the inner surface due to the heat accumulation. The surface roughness is evaluated as micro-notches, and a higher fatigue notch factor, K-f, at lower applied strain range is revealed. The factors influencing K-f include, the non-conductive inclusions serving as crack initiation sites at the surface region, and the deformation twins formed by the local stress concentration. The strain-life of PBF-LB samples is comparable with the wrought samples. However, the fatigue strength of the responding mid-life stress shows greater difference and is in the following order, CW wrought &gt; PBF-LB &gt; SA wrought. Secondary cyclic hardening owing to deformation induced martensitic transformation is found in both of the wrought samples. Yet, only cyclic softening exhibits in the PBF-LB samples, which is the result of the suppressed martensitic transformation and the dislocation unpinning from the cell boundaries.

This study investigates the anisotropic mechanical and microstructural behavior of the laser powder bed fusionLaser powder bed fusion (LPBF) manufactured Ni-based superalloy Hastelloy X (HX) by using slow strain rate (10⁻⁵ and 10⁻⁶s⁻¹) tensile testing (SSRT) at 700 °C. LPBF HX typically exhibits an elongated grain structure along the building direction (BD) and the texture analysis from the combination of neutron diffractionNeutron diffraction and EBSD discloses a major texture component <011> and a minor texture component <001> along BD, and a texture component <001> in the other two sample directions perpendicular to BD. Two types of tests have been performed, the horizontal tests where the loading direction (LD) is applied perpendicular to BD, and the vertical tests where LD is applied parallel to BD. The vertical tests exhibit lower strength but better ductility, which is explained by the texture effect and the elongated grain structure. A comparison of the mechanical behavior to the wrought HX shows that LPBF HX has better yield strength due to the high dislocation density as proved by TEM images. Creep voids are observed at grain boundaries in SSRT for both directions and are responsible for the poor ductility of the horizontal tests. The vertical ductility in SSRT maintains the same level as the reference tensile test at the strain rate of 10⁻³s⁻¹, due to the extra deformation capacity contributed by the discovered deformation twinningDeformation twinning and lattice rotation. The deformation twinningDeformation twinning, which is only observed in the vertical tests and has not been found in the conventionally manufactured HX, is beneficial to maintain the ductility but does not strengthen the material.

Additive manufacturing (AM), also known as 3D printing, is a concept and method of a manufacturing process that builds a three-dimensional object layer-by-layer. Opposite to the conventional subtractive manufacturing, it conquers various limitations on component design freedom and raises interest in various fields, including aerospace, automotive and medical applications. This thesis studies the mechanical behavior of thin-walled component manufactured by a common AM technique, laser powder bed fusion (LPBF). The studied material is Hastelloy X, which is a Ni-based superalloy, and it is in connection to a component repair application in gas turbines. The influence of microstructure on the deformation mechanisms at elevated temperatures is systematically investigated. This study aims for a fundamental and universal study that can apply to different material grades with FCC crystallographic structure.

It is common to find elongated grain and subgrain structure caused by the directional laser energy input in the LPBF process, which is related to the different printing parameters and brands of equipment. This thesis will start with the study of scan rotation effect on stainless steel 316L in an EOS M290 equipment. The statistic texture analysis by using neutron diffraction reveals a clear transition when different level of scan rotation is applied. Scan rotation of 67° is a standard printing parameter with intention to lower anisotropy, yet, the elongated grain and cell structure is still found in the as-built microstructure. Therefore, the anisotropic mechanical behavior study is carried out on the sample printed with scan rotation of 67° in this thesis.

Thin-walled effects in LPBF are investigated by studying a group of plate-like HX specimens, with different nominal thicknesses from 4mm down to 1mm, and a reference group of rod-like sample with a diameter of 18mm. A texture similar to Goss texture is found in rod-like sample, and it becomes <011>//BD fiber texture in the 4mm specimen, then it turns to be <001> fiber texture along the transverse direction (TD) in the 1mm specimen. Tensile tests with the strain rate of 10⁻³ s⁻¹ have been applied to the plate-like specimens from room temperature up to 700 ℃. A degradation of strength is shown when the sample becomes thinner, which is assumed to be due to the overestimated load bearing cross-section since the as-built surface is rough. A cross-section calibration method is proposed by reducing the surface roughness, and a selection of proper roughness parameters is demonstrated with the consideration of the calculated Taylor’s factor and the residual stress. The large thermal gradient during the LPBF process induces high dislocation density and strengthens the material, hence, the LPBF HX exhibits better yield strength than conventionally manufactured, wrought HX, but the work hardening capacity and ductility are sacrificed at the same time.

Two types of loading condition reveal the anisotropic mechanical behavior, where the vertical and horizontal tests refer to the loading direction being on the BD and TD respectively. The vertical tests exhibit lower strength but better ductility that is related to the larger lattice rotation observed from the samples with different deformation level. Meanwhile, the elongated grain structure and grain boundary embrittlement are responsible for the low horizontal ductility. A ductile to brittle transition is traced at 700 ℃, so a further study with two different slow strain rates, 10⁻⁵ s⁻¹ and 10⁻⁶ s⁻¹, are carried out at 700 ℃. Creep damage is shown in the slow strain rates testing. Deformation twinning is found only in the vertical tests where it forms mostly in the twin favorable <111> oriented grain along the LD. The large lattice rotation and the deformation twinning make the vertical ductility remain high level under the slow strain rates. The slow strain rate tensile testing lightens the understanding of creep behavior in LPBF Ni-based superalloys.

In summary, this thesis uncovers the tensile behavior of LPBF HX with different variations, including geometry-dependence, temperature-dependence, crystallographic texture-dependence and strain rate-dependence. The generated knowledge will be beneficial to the future study of different mechanical behavior such as fatigue and creep, and it will also enable a more robust design for LPBF applications.

Additiv tillverkning, eller 3D-utskrifter, är tillverkningsmetoder där man skapar ett tredimensionellt objekt genom att tillföra material lager for lager. Till skillnad från konventionella avverkande tillverkningsmetoder elimineras många geometriska begränsningar vilket ger större designfrihet och metoderna har därför väckt stort intresse inom en rad olika områden, inklusive flyg-, fordons- och medicinska tillämpningar. I denna avhandling studeras mekaniska egenskaper hos tunnväggiga komponenter tillverkade med en vanligt förekommande laserbaserad pulverbädds-teknik, laser powder bed fusion (LPBF). Det studerade materialet är Hastelloy X, en Ni-baserad superlegering som är vanligt förekommande for både nytillverkning och reparation av komponenter för gasturbiner. Inverkan av mikrostruktur på deformationsmekanismerna vid förhöjda temperaturer undersöks systematiskt. Detta arbete syftar till att ge grundläggande och generisk kunskap som kan tillämpas på olika materialtyper med en kubiskt tätpackad (FCC) kristallstruktur.

Det är vanligt att man hittar en utdragen kornstruktur orsakad av den riktade tillförseln av laserenergi i LPBF-processen, vilket kan relateras till olika processparametrar och kan variera mellan utrustningar frän olika leverantörer. Denna avhandling inleds med studien av effekten av scanningsstrategi vid tillverkning av rostfritt stål 316L i en EOS M290-utrustning. En statistisk texturanalys med hjälp av neutrondiffraktion påvisar en tydlig övergång mellan olika mikrostrukturer när olika scanningsstrategier tillämpas. En scanningsrotation på 67 mellan varje lager är en typisk standardinställning med avsikt att sanka anisotropin i materialet, dock finns den utdragna kornstrukturen oftast kvar. I denna avhandling studeras därför de anisotropa egenskaperna hos material tillverkade med 67 scanningsrotation.

Effekten av tunnväggiga strukturer i LPBF undersöks genom att studera en uppsättning platta HX-prover, med olika nominella tjocklekar från 4 mm ner till 1 mm, samt en referensgrupp med cylindriska prov med en diameter på 18 mm. Kristallografisk textur som liknar den av Goss-typ återfinns i de cylindriska proverna vilket gradvis övergår från en fibertextur med <011> i byggriktningen for 4mm-proven till en fibertextur med <001> i tvärriktningen for 1mm-proven. Dragprovning med en töjningshastighet på 10⁻³ s⁻¹ har utförts på de platta provstavarna från rumstemperatur upp till 700 ℃. En sänkning av styrkan uppvisas när proven blir tunnare, vilket kan antas bero på att det lastbarande tvärsnittet överskattas på grund av den grova ytan. En metod för tvärsnittskalibrering föreslås genom att kompensera for ytråheten, och valet av lämplig ytfinhetsparameter motiveras med hänsyn till den beräknade Taylor-faktorn och förekomsten av restspänningar. Den stora termiska gradienten som uppstår for LPBF-processen inducerar en hög dislokationstäthet vilket höjer materialets styrka och följaktligen uppvisar LPBF HX högre sträckgräns an konventionellt tillverkad, smidda HX, men förmågan till deformationshårdnande samt duktiliteten i materialet sänks samtidigt.

Tester utförda i två olika belastningsriktningar, vertikalt respektive horisontellt mot byggriktningen, demonstrerar det anisotropiska mekaniska beteendet. De vertikala testerna uppvisar lägre hållfasthet men bättre duktilitet vilket kan relateras till en större benägenhet for kristallstukturen att rotera när deformationsgraden ökar. Samtidigt är den utdragna kronstukturen ansvarig for den lägre duktiliteten for de horisontella proverna. En övergång från ett duktilt till ett mer sprött beteende noterades vid 700 ℃, och därför initierades ytterligare en studie där tester med två lägre töjningshastigheter, 10⁻⁵ s⁻¹ och 10⁻⁶ s⁻¹, utfördes vid 700 ℃. Det kan noteras att krypskador återfinns i tester med en långsam deformationshastighet och deformationstvillingar uppstår endast i de vertikala provstavarna där det främst bildas tvillingar i korn orienterade med <111> riktningen längs belastningsriktningen. Den stora förmågan till rotation i kristallstrukturen och deformationstvillingarna bidrar till att den vertikala duktiliteten förblir hög även i testerna med en låg deformationshastighet. Testerna med en långsam draghastighet bidrar därför till en bättre förståelse av krypbeteendet i LPBF Nibaserade superlegeringar.

Sammanfattningsvis så bidrar denna avhandling till bättre förståelse av de mekaniska egenskaperna hos LPBF HX i olika utföranden och förhållanden, inklusive geometriberoende, temperaturberoende, deformationshastighetsberoende samt inverkan av kristallografisk textur. Den genererade kunskapen kommer att vara till stor nytta vid fortsatta studier av olika mekaniska egenskaper som utmattning och kryp, samt bidrar till att möjliggöra en mer robust design for LPBF-tillämpningar.  

Laser powder bed fusion (LPBF) of Ni-based superalloys shows great potential for high temperature applications, for example, as a burner repair application for gas turbines where the thin-walled structure is important. It motivates this work to investigate the evolution of microstructure and the anisotropic mechanical behavior when plate-like specimens are built with a thickness from 4 mm down to 1 mm. By performing texture analysis using neutron diffraction, a clear transition in fiber texture from <011> to <001> is indicated when the specimen becomes thinner. The residual stress shows no thickness dependence, and at the subsurface the residual stress reaches the same level as the yield strength. Due to the rough as-built surface, a roughness compensation method for mechanical properties of thin-walled structures is outlined and demonstrated. Tensile tests from room temperature up to 700 °C have been carried out. Anisotropic mechanical behavior is found at all temperatures, which is strongly related to the anisotropic texture evolution. Stronger texture evolution and grain rotations are discovered when the tensile loading is applied along the building direction. The mechanical behavior has been compared to a wrought material, where the high dislocation density and the subgrain structure of the LPBF material result in a higher yield strength. Combining the statistical texture analysis by neutron diffraction with mechanical testing, EBSD grain orientation mapping and the investigation of dislocation structures using transmission electron microscopy, this work illustrates the significance of texture for the thin-wall effect and anisotropic mechanical behavior of LPBF materials.

Introduction/Purpose

The study of thin-walled structure improves the design freedom of additive manufacturing. Understanding the relation between anisotropic mechanical properties and microstructure gives better control on manufacturing process and challenges the limit of the practical application. As one of the key Ni-based superalloy in aerospace industries, thin-walled Hastelloy X will be beneficial to the light weight application and build a more sustainable environment.

Methods

Thin-walled structures with different thickness form 1mm to 4mm was built by Selective Laser Melting process from EOS M290 machine, and the used powder was EOS NickelAlloy HX. The microstructure and crystallographic orientation have been studied by SEM and EBSD. Tensile tests with directions parallel and perpendicular to building direction (BD) have been carried out at elevated temperature from 400˚C to 700˚C.

Results

The elongated grains have been observed partly parallel and partly 45˚ tilted to the BD from back scattered SEM images, and the contouring region shows smaller grain size . Along the BD, the major preferred orientation is <101> and the minor is <001>. The tensile test result indicates higher strength but lower elongation in the direction perpendicular to BD, and also a big elongation drop between 600˚C and 700˚C. EBSD result from highly deformed area shows different texture evolution mechanism between two different tensile directions.

Conclusions

The local thermal gradient created by the scanning strategy guides the grain growing direction, which is <001>, and <101> turns to be the preferred orientation along BD. The elongated grains are the main reason for the anisotropic mechanical properties. When the tensile direction is parallel to BD, the orientation evolution fits the theory and indicates lattice rotation.