Objective The aim of this study is to investigate the fatigue and hydrothermal degradation behavior among the cubic-containing translucent yttrium oxide stabilized tetragonal zirconia polycrystal (Y-TZP).Materials and Methods Four groups of commercial Y-TZP , containing different amount of cubic crystal, were examined. Artificial aging was accomplished by autoclaving at 122°C under 2 bar pressure for 8 hours. Fatigue simulation parameters were set using an axial 50 N load, a total of 240,000 cycles. Flexural fatigue strength was evaluated.Statistical Analysis Two-way analysis of variance with Tukey's honestly significant difference test was used to determine the difference in fatigue strength values between various type of zirconia materials within control and aging with a significant level of 5%. Weibull analysis was conducted on the fatigue strength data.Results All groups showed the mean flexural fatigue strength had no significant difference in hydrothermally aged specimens compared with specimens without aging (p = 0.154). Group T showed significantly higher flexural fatigue strength followed by group ST, XT, and P. Group XT and P showed no significant difference in strength value (p > 0.05). Group T demonstrated a lower probability of failure than group ST, XT, and P whether with or without aging. Weibull modulus in group T and ST with aging condition was increased and higher than all the experimental groups.Conclusion Cubic-containing zirconia materials  exhibited lower flexural fatigue strength than non-cubic 3Y-TZP zirconia (group T). However, various cubic-containing translucent zirconia was assumed to have high resistance to hydrothermal degradation. Yttrium-stabilized tetragonal zirconia polycrystal (Y-TZP) is widely used in dentistry because of its excellent strength and fracture toughness after machining and sintering compared with other dental ceramics.The t→m transformation can also spontaneously occur over time at low temperature. When the t→m transformation is not triggered by the local stress, this phenomenon is known as aging, hydrothermal degradation, or low-temperature degradation (LTD).2O3stabilizer tends to alter the phase structure from tetragonal to cubic phase resulting in increased translucency and improved hydrothermal aging resistance.The increased patients' esthetic demands for both anterior and posterior teeth are driving the development of highly translucent monolithic zirconia. Translucency is considered as one of the most important factors in controlling esthetics and plays an important role in material selection.However, the clinical recommendations for using translucent zirconia restorations cannot bring clear advantage and there is little information on literature regarding the properties especially in fatigue resistance and the degradation behavior of recently introduced translucent zirconia for monolithic restorations, particularly the cubic-containing zirconia with the varying amount of cubic structure. Therefore, the aim of this study was to investigate the fatigue resistance and hydrothermal degradation behavior among commercial translucent zirconia products containing different amount of cubic crystal  and compare them with conventional zirconia (0% cubic phase). The null hypothesis was that no difference in flexural fatigue strength between hydrothermally aged and nonhydrothermally aged of various cubic-containing translucent zirconia.n = 10/group) and other 10 specimens were kept in the air dry at room temperature (n = 10/group) before being subjected to cyclic loading.Four different commercially available zirconia materials were tested in this study . TwentyAll specimens were cyclic loaded in a chewing simulator  using an axial load of 50 N for 240,000 cycles with a frequency of 1.1 Hz. A stainless steel metal stylus (Ø = 2 mm) was used as an antagonist abrader and the force was transferred to the middle site of the specimen by opposing metal stylus under water at room temperature. After cyclic loading was completed, specimens were subjected to a biaxial flexural strength test according to ISO 6872.σ = [–0.2387P(X–Y)]/b2σis the maximum tensile stress (MPa),Pis the total load to fracture (N),bis the thickness at fracture origin (mm), andXandYare calculated according to:whereX = (1 + ν) ln(r2/r3)2 + [(1–ν)/2)](r2/r3)2Y = (1 + ν)[1 + ln(r1/r3)2] + (1–ν)(r1/r3)2νis Poisson's ratio (ν = 0.30),r1is the radius of the support circle (6.05 mm),r2is the radius of the loaded area (0.8 mm), andr3is the radius of the specimen (7.5 mm). The flexural fatigue strength testing was performed on both specimens with and without hydrothermal aging and compared statistically.whereThe fracture patterns were used in the descriptive fractographic analysis of fractured specimens using scanning electron microscopy  at ×200 and ×500 magnification.n = 10/group). Weibull analysis was conducted on the fatigue strength data for evaluating strength reliability. The description of the Weibull distribution is given byMean flexural fatigue strength and standard deviations were recorded. Assessment of normality distribution and homogeneity of variance were performed using the Shapiro–Wilk test and Levene's test, respectively. The data was normally distributed and homogeneity of variance was equal in each group. Therefore, two-way analysis of variance with Tukey's honestly significant difference test was used to determine the difference in flexural fatigue strength values between various type of zirconia materials in with LTD aging and without LTD aging groups (Pf = 1–exp[–(σ/σ0)m]Pfis the probability of failure,σis the fatigue strength at a givenPf,σ0is the characteristic strength (MPa), andmis the shape parameter (Weibull modulus). The statistical analysis was done using SPSS software, version 24.0  to detect statistically significant differences (α = 0.05).wherep < 0.05), while LTD showed no significant effect on strength degradation in all groups (p = 0.154). The mean flexural fatigue strength showed significantly highest value in group T (815 MPa without aging group and 811 MPa with aging group). Group ST was significantly higher than group XT and P in specimens with and without aging. However, group XT and P showed no significant difference in flexural fatigue strength value regardless of aging condition (p > 0.05) (p = 0.184).The statistical analysis showed a significant difference in flexural fatigue strength among the material groups (> 0.05) . Moreovm) and characteristic strength (σ0) are listed inσ0) was higher for group T than that for group ST, XT, and P, respectively, in both of aging conditions. The probability of failure distribution curve is shown inWeibull parameters including Weibull modulus , moderate (30–50% by volume), and high cubic-containing zirconia (> 50% by volume). From the result of this study, there was a statistically significant difference in flexural fatigue strength among the material groups, therefore the null hypothesis was partially rejected.In vitrocyclic loading condition under humid environment simulates the clinical function of the restoration. The fatigue simulation parameters in this study were set using an axial 50 N load, a total of 240,000 cycles according to previous study.The fatigue resistance is one of the main concerning properties of ceramic materials because it affects the life of Y-TZP restorations. Crack propagation under intermittent loading could lead to catastrophic failure of restorations intraorally.The strength in present results was higher than a previous study which evaluated the flexural fatigue strength of 3Y-TZP, 4Y-PSZ, and 5Y-PSZ on a 3-point bend test for 6,000 cycles. It was reported that the flexural fatigue strength of 3Y-TZP, 4Y-PSZ, and 5Y-PSZ were 640, 304, and 232 MPa, respectively.in vitroaccelerated hydrothermal aging characteristics of material using autoclave condition at 122°C under 2-bar pressure over a period of 8 hours. At this temperature, the nucleation was detected by small surface upheavals which corresponded to the transformation of one or a few grains. Consequently, this condition was used to detect the tetragonal to monoclinic phase transformation. It is possible to assume that 1 hour of autoclave aging in this condition has the same effect as 1 year in clinical services.Autoclave aging was used to simulate the LTD aging condition. This study investigatedm) and characteristic strength (σ0) estimate failure probability expression.-) in water molecules can diffuse into the zirconia lattice then water reacts with ZrO2or Y2O3bonds at crack tips. As a consequence, Zr-OH or Y-OH bonds are formed and the Zr-OH or Y-OH bond brings about the lattice strain on the surface. Strain accumulation creates nucleation sites for phase transformation on the surface; as a result, the metastable tetragonal transformation to stable monoclinic phase can occur with a large volume expansion.Weibull distribution was performed in the present study because the strength properties of ceramic materials are controlled by the number of surface flaws and flaws size distribution. Therefore, it is possible to provide reliability data relating to the fracture of ceramic materials. Weibull parameters including Weibull modulus or shape parameter (σ0) is the strength at a probability of failure of 63.2% for a test specimen and loading protocol. From the current study, the Weibull characteristic strength values were coinciding with the mean flexural fatigue strength. Group ST, XT, and P demonstrated a shift to the left of the distribution curve and indicated a decrease in flexural fatigue strength, in both of aging conditions. Group T presented high Weibull characteristic strength because of the higher amount of tetragonal structure; therefore, transformation toughening phenomenon could take place. As a result, strength for group T was greater than the other groups.The Weibull characteristic strength  exhibited lower flexural fatigue strength than non-cubic 3Y-TZP zirconia (group T). However, hydrothermal aging in autoclave for 8 hours did not cause a statistically significant decrease in the flexural fatigue strength of all experimental groups.Based on the limitations of this


Background: Elexacaftor–tezacaftor–ivacaftor (ETI) is a novel, highly effective CFTR modulator combination proven to enhance lung function and body weight in people with cystic fibrosis (pwCF) carrying a F508del mutation. Recently, we revealed significant reductions in abdominal symptoms (AS) in German, British, and Irish pwCF after 24–26 weeks of ETI using the CFAbd-Score, the first patient-reported outcome measure (PROM) specifically developed and validated for pwCF following FDA guidelines. Notably, many pwCF reported marked changes in their AS during the first days of the new treatment. To capture these immediate effects, we developed the CFAbd-day2day, a CF-specific GI-diary, following FDA and COSMIN guidelines.Aim: To prospectively capture the immediate dynamics of AS using the CFAbd-day2day 14 days before and 14–28 days after ETI initiation. In addition, we aim to provide validation steps of the novel PROM concerning sensitivity to changes.Methods: To develop the CFAbd-day2day, focus groups  were repeatedly consulted. Before and during the new ETI therapy, pwCF prospectively scored AS on a daily basis with the CFAbd-day2day.Results: Altogether, 45 pwCF attended in five CF centers prospectively completed the CFAbd-day2day before (mean ± sd:14 ± 7 days) and after (mean ± sd: 28 ± 23 days) ETI initiation. On the one hand, cumulative scores significantly decreased during the 3–4-week time frame after ETI initiation, compared to 2 weeks prior to therapy. On the other hand, many patients who revealed a relatively stable level of AS before ETI reported changes during the first days of treatment with the highly effective CFTR modulators. Factors like pain and flatulence increased in up to 21% of patients during the first 14 days of therapy, but they improved during days 15–27.Conclusion: The CFAbd-day2day, specifically developed and in the process of validation to prospectively capture GI symptoms in pwCF, provides new substantial insights into the dynamics of AS in pwCF receiving a new treatment with ETI. This novel tool is also helpful in prospectively monitoring patients with specific GI problems. International implementation and further validation steps of the diary are ongoing. For long, abdominal involvement in cystic fibrosis (CF), the most common lethal inherited disease of the Caucasian population, received little attention. Since the availability of pancreatic enzyme supplementation therapy (PERT) in mostly  pancreatIntroduction of highly effective CFTR modulator therapies (HEMTs) in pwCF carrying a rare gating mutation like G551D over a decade ago revealed that correction and potentiation of the defective and/or malfunctioning CFTR channel have marked effects on the health status of pwCF beyond pulmonary function . PatientTo fill this gap, we developed the CFAbd-Score in different steps following FDA guidelines for the development of a PROM , includiRecently, we implemented the CFAbd-Score in an international study with 107 pwCF from Germany and the UK before and at a median of 24 weeks during therapy with elexacaftor–tezacaftor–ivacaftor (ETI). Therein, ETI was found to substantially improve the total CFAbd-Score and its five domains: “pain,” “GERD,” “disorders of bowel movement,” “disorders of appetite,” and “quality of life impairment” . Quite aNevertheless, during the early phases after HEMT introduction, patients reported experiencing many GI symptoms, which may not be adequately represented by a PROM that addresses GI symptoms retrospectively with a 2-week recall period. Furthermore, after HEMT initiation some symptoms could arise more frequently and markedly, but only for quite a short period of time, which may not be adequately recorded with a questionnaire focused on capturing the overall frequency of GI symptoms over the past 14 days.© is suitable for closely recording AS.Consequently, based on the CFAbd-Score, we developed the CFAbd-day2day questionnaire, a prospective diary PROM. This questionnaire is being validated following FDA and COnsensus-based Standards for the selection of health Measurement INstruments (COSMIN) guidelines . AnalogoAs the selection and wording of questions included in the CFAbd-Score have been found to be highly sensitive for capturing and quantifying GI symptoms in pwCF receiving a new ETI therapy , substan© questionnaire.The aim of this study was to prospectively assess changes in abdominal symptoms on a daily basis during a new, highly effective CFTR-modulating ETI therapy in pwCF. Specifically, we aimed to capture GI symptoms for a period of up to 14 days prior to commencing the new therapy, as well as during 14–28 days of the new therapy, using the novel CFAbd-day2day© before and after ETI initiation. Of note, 10 pwCF recruited at the CF centers in Brandenburg an der Havel/Potsdam for this project, which focuses on prospective short term changes in the CFAbd-day2day, were also included in the study previously published in 2022 , Tübingen (n = 15), Gießen (n = 7) and Frankfurt am Main (n = 7) were prospectively considered to complete the CFAbd-day2day in 2022 . Further© questionnaire. The CFAbd-day2day© is a diary version of the CFAbd-Score. Therefore, the CFAbd-day2day© also comprises 28 items grouped into five domains. However, in addition to the modified recall period, some questions have been adapted to prospectively focus on each observational day. The questionnaire also includes an optional comments section for recording changes in dietary habits, enzyme and medication use, and menstrual symptoms. A major intrinsic advantage of the diary character is the ability to record not only symptom frequencies but also their daily intensity. Printed copies of the questionnaire were issued to the participants through the local CF care providers and/or research coordinators. Scoring and analyses of completed, pseudonymized questionnaires were centrally performed at the CF center in Brandenburg an der Havel, Germany.Abdominal symptoms were recorded daily using the CFAbd-day2day© questionnaire were obtained by computing medians of each item over the time frame prior to ETI therapy initiation for each pwCF. Dynamics of symptoms were assessed by computing daily proportions of subjects reporting either improvement or worsening with respect to the aforementioned baseline values for each of the 28 symptoms assessed with the CFAbd-day2day© questionnaire. Furthermore, within-subject averaged absolute daily variation measures were computed for each subject and each questionnaire item to quantify the levels of variation over three time frames: prior to ETI therapy and 2 and 4 weeks after ETI therapy initiation. These measures were computed by averaging the absolute changes experienced by each patient between consecutive days in each item over the aforementioned time frames. Additionally, the maximum deviation with respect to the median in each item was identified for each patient within each time frame. In order to find out the period of time (prior to ETI therapy and 2 and 4 weeks after ETI therapy initiation) at which symptoms reported by the cohort underwent maximum changes, exploratory analyses to compare medians of these measures of variability were conducted using non-parametric Wilcoxon signed-rank tests. Effects of ETI therapy on cumulative CFAbd-day2day responses were assessed by mapping the frequency of events registered by each patient for each item in the CFAbd-day2day© with the following scale: not at all → 0, once → 1, 2–3 times → 2, 4–7 times → 3, more than 7 times → 4, and daily → 5. Afterward, items were grouped into five different domains as defined for the 2-week CFAbd-Score . 45 (90%) of them completed the questionnaire after ETI initiation . Detailed information about the rates of daily answers is provided in ©  reporting improvement was observed for the item “Foul-smelling stools” on day 23 after ETI therapy initiation. Other items for which the proportion of patients reporting improvement reaching almost the 20% level over this time frame were “No appetite” (19%), “Physical activity limitation” (19%), and “Fatty stools” (19%), with the former observed already on day 7 and the latter two from day 20 onward. On the other hand, symptoms for which the proportion of patients did not surpass the 10% level were “Loss of taste,” “Forced feeding,” “Heartburn,” “Nausea,” “Constipation,” “Sadness,” “Waking up at night,” and “Pain during bowel movements.”The proportions of patients reporting worsening in symptoms within the first 4 weeks after ETI treatment initiation  were theComparing maximal absolute deviations from the median over the three time frames revealed that the maximum deviations in all items occurred within the first 2-week period. This was significant for items regarding “No appetite,” “Constipation,” “Fatigue,” “Waking up at night,” and “Abdominal pain duration” . On the © responses revealed a highly significant decrease in the median total score (p = 0.00001). Changes in four domains, namely, “Pain,” “GERD,” “Disorders of bowel movement,” and “Quality of life impairment,” were statistically significant , although medians for “GERD” and “Quality of life impairment” resulted equal for the two time frames  (p < 0.01) was slightly lower at 3–4 weeks (Averaged CFAbd-day2day< 0.001) . On the –4 weeks .©”. The development of the new PROM was based on the CFAbd-Score, the first CF-specific GI PROM designed and validated following FDA guidelines . According to our previous studies with the CFAbd-Score, children report significantly more often on abdominal pain, whereas adults complain significantly more often about gastroesophageal reflux . Accordip < 0.05), similar to the five domains of Pain, GERD, Disorders of bowel movement, Disorders of appetite, and GI-related QoL , who were repeatedly consulted. Consequently, the questionnaire was observed to have high acceptance rate.Diaries are favourable when short term changes are expected. This applies to our setting assessing changes in AS during ETI initiation, which was motivated beforehand by numerous pwCF’s reports on short-term changes in GI symptoms, often commencing hours after receiving the first dosage of HEMT. In the development process of the novel PROM, the 28 items included in the CFAbd-day2day©. They reveal that the PROM is sensitive to detect symptom changes, e.g. those caused by a newly initiated therapy. At present, further steps to validate the CFAbd-day2day© are in progress, including analyses of convergent validity with the CFAbd-Score and AS dynamics in pwCF suffering from CF-related conditions like constipation, DIOS, GERD, CF-related diabetes (CFRD), and liver disease.The results presented in this publication fulfil essential steps in the validation process of the CFAbd-day2day© during a new therapy with the HEMT ETI provides new real-world insights relevant for the CF community and healthcare providers. Furthermore, the symptom diary can be implemented in routine care of pwCF suffering from GI symptoms in order to follow up their dynamics in daily life, as well as the effects of therapeutic interventions. Accordingly, the CF-specific CFAbd-day2day© may facilitate the identification of critical GI complications that may require the individual with CF to consult the attending CF center. Analogous to the CFAbd-Score, the PROM is being translated to other languages and will be implemented in international studies.In summary, implementation of the novel CFAbd-day2day


Many textbook physical effects in crystals are enabled by some specific symmetries. In contrast to such ‘apparent effects’, ‘hidden effect X’ refers to the general condition where the nominal global system symmetry would disallow the effect X, whereas the symmetry of local sectors within the crystal would enable effect X. Known examples include the hidden Rashba and/or hidden Dresselhaus spin polarization that require spin-orbit coupling, but unlike their apparent counterparts are demonstrated to exist in non-magnetic systems even in inversion-symmetric crystals. Here, we discuss hidden spin polarization effect in collinear antiferromagnets without the requirement for spin-orbit coupling (SOC). Symmetry analysis suggests that antiferromagnets hosting such effect can be classified into six types depending on the global vs local symmetry. We identify which of the possible collinear antiferromagnetic compounds will harbor such hidden polarization and validate these symmetry enabling predictions with first-principles density functional calculations for several representative compounds. This will boost the theoretical and experimental efforts in finding new spin-polarized materials. A hidden effect can occur in materials where locally a symmetry is broken, even though global symmetry is preserved. An example is hidden spin-polarization, arising from local inversion symmetry breaking in otherwise globally centro-symmetric materials. Here, Yuan et al uncover a hidden spin-polarization that can occur in antiferromagnets without spin-orbit coupling and identify the key material requirements for this to occur. Another example of effects enabled by the recognized global system symmetry is the removal of spin degeneracy of energy bands due to spin-orbit coupling (SOC) in non-magnetic crystals having broken inversion symmetry (such as the Rashba (R-1)4 and Dresselhaus (D-1)5 effects). When an effect is observed despite the needed enabling symmetry being absent, it is often assumed that the system contains some symmetry-altering imperfections.Many traditional textbook physical effects in crystals are enabled by some specific symmetries, encoded in the crystal space group. Such are the symmetry conditions for the apparent electric polarization which defines various order parameters such as in ferroelectricityIn contrast, the “Hidden Effect X” in materials that are not supported by the nominal enabling symmetry, yet effect X exists locally. The “Hidden effect X” reflects the intrinsic properties of the perfect crystal rather than imperfections that would disappear when the crystal becomes perfect. The understanding of such hidden intrinsic effects is important as it can demystify peculiar observations of phenomena that are unexpected to exist based on the global symmetry of the system.7 and observed18 in centrosymmetric nonmagnetic crystals . Similar form of Hidden effect X are (ii) X = “anisotropic optical circular polarized luminescence” expected only in odd-layered transition-metal dichalcogenides but observed19 also in even-layered crystals. Such effects were originally dismissed as being due to some extrinsic sample imperfection22 but later on were shown to be an intrinsic property pertained to the individual layer23. (iii) X = “spin polarization” induced by SOC in antiferromagnetic systems. The effect is again expected only in non-centrosymmetric crystals (such as BiCoO324) but shown in centrosymmetric crystals (such as CuMnAs and Mn2Au28) where combined symmetry of inversion and time reversal disallows splitting. Here, “centrosymmetric” means the crystal in the non-magnetic state has an inversion. Prominently, the hidden spin polarization in these compounds facilitates the electrical reversal of their antiferromagnetic ordering29. (iv) X = “anomalous Hall effect” induced by SOC expected only in odd-layered ferromagnetic MnBi2Te4 systems but observed in even-layered antiferromagnetic MnBi2Te430 systems via a perturbative applied electric field.Examples of “Hidden Effect X” that is SOC-induced include (i) Rashba or Dresselhaus spin polarization, expected exclusively to occur in non-centrosymmetric crystals, but predicted40; And the hidden effect exists in antiferromagnetic materials where spin-up and spin-down bands are paired. This represents a step further beyond the already known hidden Rashba and hidden Dresselhaus spin-polarization that unavoidably require a sizable contribution from SOC. A careful analysis of the “global (bulk) vs local (sector)” symmetries suggests that antiferromagnets hosting the SOC-independent “hidden” spin polarization effect can be delineated into six types. We scrutinize a vast database of known collinear AFM materials and performed first-principles calculations on several selected candidate compounds assuming zero SOC. We show that such hidden, SOC-independent effects reflect the intrinsic properties of the perfect crystal rather than an effect due to imperfections. The interest in this SOC-independent hidden spin polarization effect stems both from the evolving of the fundamental understanding of general hidden effects in solids, and from the ability to extend the pool of useful materials for potential spintronic applications.Here, we discuss a different form of hidden spin polarization effect see Fig.  whose co38. This involved utilizing first a few individual symmetry operations: U being a spin rotation of the SU(2) group acting on the spin 1/2 space that reverses the spin; T being spatial translation; Θ being time reversal, and I being the spatial inversion. These individual operations are then used for constructing two symmetry products: a SOC-free magnetic symmetry ΘIT, and a spin symmetry UT (where the former product can be simplified to ΘI by proper choice of inversion center). SOC-independent spin splitting39 would occur only when both symmetry products are simultaneously violated. Antiferromagnets with ΘIT symmetry42 will not show such spin splitting. Such symmetry conditions disentangle the SOC-independent splitting from the SOC-induced splitting by considering the symmetry at the zero SOC limit45, where spin and space are fully decoupled.Symmetry is essential to understand the energy bands’ degeneracy of a material. The symmetry conditions for apparent spin splitting or spin polarization was pointed out recently in ref. 39 for magnetic materials. There are three prototypes with no apparent spin splitting effect: (1) AFM compounds that violate UT but preserve ΘIT symmetry referred to as spin splitting prototype 1 (SST-1) antiferromagnets; (2) AFM compounds that preserve both UT and ΘIT symmetry referred to as SST-2 antiferromagnets; (3) AFM compounds that preserve UT but violate ΘIT symmetry referred to as SST-3 antiferromagnets. In addition, there are two prototypes with apparent spin splitting effects: (4) AFM compounds that violate both UT and ΘIT symmetry referred to as SST-4 antiferromagnets; (5) Ferromagnetic (FM) compounds that violate both UT and ΘIT symmetry referred to as SST-5 ferromagnets. The classification defined in bulk crystals39 can be generalized to sectors of a bulk based on the local sector symmetry. Figure Given the symmetry conditions, it is thus possible to classify all different spin splitting prototypesΘIT or UT or both) that disallows apparent spin polarization, one can then classify six hidden spin polarization cases. Following the previous classification of spin splitting prototypes for apparent spin degeneracy and apparent spin splitting39, collinear antiferromagnetic materials with “hidden spin polarization” are those antiferromagnets whose bulk prototype being SST-I  and constitute sector prototype being SST-J . Detailed discussions of the symmetry conditions for hidden spin polarization in collinear AFM are given in Supplementary Information Section “Hidden spin polarization” is expected in collinear antiferromagnets when the bulk has zero net spin polarization, but its constituent sectors allow locally a spin splitting and spin polarization effect. Consider the combination of two possible prototypes constituting sector that gives hidden spin polarization locally but lead to three possible prototypes of the bulk symmetry  are denoted as SST-5 in Fig. UT and ΘIT, thus allows spin splitting without SOC. AFM materials that satisfy the condition are denoted as SST-4 in Fig. Figure We now turn to discuss how the enabling symmetries are applied to individual sectors to give magnetic hidden spin polarization effects in real antiferromagnetic materials.46 and identified a few antiferromagnetic materials of potential candidates for magnetic hidden spin polarization. The identified candidates are: Ca2MnO447, CoSe2O548 and Fe2TeO649, K2CoP2O750 and LiFePO451 whose bulk prototype is SST-1 with sector prototype of SST-4; Sr2IrO452 whose bulk prototype is SST-2 with sector prototype of SST-4; SrCo2V2O853 whose bulk prototype is SST-3 with sector prototype of SST-4; CuMnAs54 and Mn2Au55 whole bulk prototype is SST-1 with sector prototype of SST-5; FeCl2 and CoCl256 whose bulk prototype is SST-2 with sector prototype of SST-5; ErAuGe57 whose bulk prototype is SST-3 with sector prototype of SST-5. These materials form the platform for the exploration of the magnetic hidden spin polarization effects.As a first step, we will try to find real materials that falls into the six categories we defined. This can be done straightforwardly by applying the symmetry conditions to filter out candidate materials in existing antiferromagnetic databases. We conducted such filtering for MAGNDATA database58, we find a list of 37 ferromagnetic 2D materials and 6 antiferromagnetic 2D monolayers that can be used as such building blocks (see Tables The opposite design philosophy (the bottom-to-top approach) is to construct layered bulk antiferromagnets with the hidden effect based on two-dimensional 2D) compounds that belong to SST-4 and SST-5 prototypes. By searching through the database of predicted naturally exfoliate 3D Van der Waals materialsD compoun54, Ca2MnO447 and FeBr256 using PBE + U method59 in the zero SOC limit. The results are presented below. Additional examples with DFT results are presented in Supplementary Information Section The next step is to validate the predicted hidden spin polarization effect in some of these identified real materials. We studied the sector-projected spin textures on certain wavevector planes for three actual antiferromagnetic materials, CuMnAs54 (bulk belonging to SST-1 class with sectors belonging to SST-5 class). The crystal is antiferromagnetically ordered with its magnetic moments collinearly aligned in the (010) direction. The magnetic space group (MSG) of the crystal is Pm’mn (MSG type III). The unit cell consists of two MnAs layers (α-sector and β-sector) that are ferromagnetically ordered , the material has been demonstrated as a useful platform for electrically switching29 the antiferromagnetic magnetization using the hidden spin polarization from the SOC segregated on each sector. Here, we point out a different SOC-independent scenario that might also be contributing to the observed electric switching in this material, i.e., the Zeeman effect within each ferromagnetic MnAs layer creates a local spin split state anchored on the layer. The two MnAs layers are connected by the ΘIT symmetry which restores the spin degeneracy of the bulk and results in a compensated net spin polarization  and Ca3Ru2O760 (bulk belonging to SST-3 with sector belonging to SST-5) in Supplementary Information Section Figure ion Fig. . As showα-sector and β-sector, shown in Fig. α and sector- β(1/√2(|α↑ + |β↑ and 1/√2(|α↓β↓〉) is (+0.5) +(−0.5) = 0 when projected onto sector-α↑〉 and |β↓〉) is 1 when projected onto sector−α and is −1 when projected onto sector.We note that the corresponding hidden spin polarization projected onto 2MnO447 (bulk belonging to SST-1 class with sector belonging to SST-4 class). The crystal is antiferromagnetically ordered with its magnetic moments collinearly aligned in the (001) direction. The MSG of the crystal is I41’/a’cd’ (MSG type III). The unit cell consists of two layers of MnO6 octahedral (α-sector and β-sector) that are antiferromagnetically ordered  and La2NiO462 (bulk belonging to SST-3 with sector belonging to SST-4) in Supplementary Information Section Fig. ion Fig. . However2 (DFT settings for applying the electric field is provided in Methods section). The basic building block of the crystal is the ferromagnetically ordered FeBr2 layer (sector belonging to SST-5 class). The bilayer slab is built by stacking identical FeBr2 layer with alternating magnetic ordering. The two layers are connected by both ΘIT and UT symmetry, the bilayer hexagonal FeBr2 (MSG: PC-3c1) thus belongs to a bulk SST-2 class, featuring a spin degenerate energy band. However, the spin degenerate band structure of the SST-2 class FeBr2  – a transition from SST-2 to SST-4. The spin splitting arises because of the external electric field Ez creates a non-equivalent potential on the sectors and breaks the ΘIT and UT symmetry of the bulk that connects the two layers. DFT calculations for different values of the applied field, inserted in Fig. CB1 is dominantly segregated on the α-sector, while the spin-down (blue) state ΓCB2 is dominantly segregated on the β-sector. Therefore, the hidden effect of two-fold degenerate energy states subspace (when Ez = 0) can be traced back to the individual FeBr2 layers. Because the applied electric field is small, the main characteristic of the observed spin polarization is inherited from the system without electric field. The layer-segregated states shown in Fig. 63 where external electric field lifts the spin degeneracy. We also note that the layer Hall effect in the even-layered MnBi2Te4—in which electrons from the top and bottom layers spontaneously deflect in opposite directions but globally compensate—has been observed with the help of an applied electric field30. These examples not only verify our understanding of the hidden effect being intrinsic to the bulk but also suggest an external electric field as an effective knob for modulating the hidden effect.To demonstrate the symmetry connection between local sectors and the subsequent transition from hidden effect to apparent effect mediated by the breaking of the symmetry connection, we apply in our calculations a perturbative symmetry-breaking external electric field on an antiferromagnetic compound with hidden spin polarization, hexagonal FeBrBr2 Fig.  is lifte38. Still, it is important to note the inclusion of SOC would modify the energy bands in both non-magnetic materials and magnetic materials65: (1) it reduces the degeneracy of certain bands which may cause additional spin splitting. (2) it mixes the spin polarized states of up and down (so spin is no longer a good quantum number), which results in momentum-dependent spin polarization that are not unidirectionally aligned; (3) it opens a gap for the crossing energy bands with opposite spin polarization. In compounds consist of low-Z elements the SOC-induced effect can be neglected.The SOC-independent hidden spin polarization effect persists in the presence of SOC. This is because the effect being inherited from the unusual antiferromagnetic order rather than SOCUT in the spin space group  means there is a spatial translation T that connects the atomic sites with opposite magnetic moments and keeps the crystal structure invariant. By definition, antiferromagnets with primitive lattice translations that reverse the microscopic magnetic moments are known as having black and white Bravais lattice that is classified as MSG type IV; Antiferromagnets without such translation T belongs to MSG type I and type III66. This suggests there is a one-to-one correspondence between the existence or absence of the UT in the spin space group and the MSG being type IV or type I/III.In collinear antiferromagnetic compounds, the existence of 38 or equivalently as “magnetic groups with pseudoscalar electron spin” in ref. 67. Following that, we can prove a chain relation as depicted in Eq.  the existence or not of a ΘT symmetry in the auxiliary group. This is because the ΘU symmetry preserves any collinear magnetic ordering and is a symmetry of any collinear antiferromagnets45. Meanwhile, (b) the existence or not of a ΘT symmetry in the auxiliary MSG (without SOC) corresponds to (c) the existence or not of a ΘT symmetry in the standard MSG (with SOC). Antiferromagnetic materials whose (c) MSG preserve (or violate) ΘT symmetry is classified as (d) MSG type IV (or MSG type I/III)66.(a) The existence or not of a 64—to predict whether the spin splitting effect without SOC will occur. This also allows the use of the tabulated magnetic structure symmetry information provided in material database46 to sort out candidate materials39. For the prediction of the degeneracy of the full bands without SOC, a comprehensive analyze of the spin symmetry group and its irreducible representation is necessary65.The established correspondence relation thus justifies the use of MSG (with SOC)—avoiding the use of the “less familiar” spin symmetryΘIT symmetry, the energy bands are spin degenerate. If the system can be further divided into separate sectors that locally violate ΘIT, then there could exist hidden spin polarization pertaining to the individual sectors. However, one should note that the symmetry condition of having UT for preserving spin degeneracy in noncollinear antiferromagnetic compounds39 is not valid anymore, this is because (1) when the spin arrangement is non-coplanar, the MSG type IV does not guarantee the existence of UT; Moreover, (2) when the spin arrangement is coplanar, MSG type IV guarantees the existence of UT, but the existence of such UT does not always guarantee spin degeneracy. Specifically, when the spin states are not aligned in the same plane of the coplanar plane, the UT symmetry will not reverse the spin states as it works in the collinear magnetic systems. These properties of noncollinear antiferromagnets offer new knobs to tune the hidden versus apparent spin polarization via tilting the local magnetic motifs.While the current paper focuses on the hidden spin polarization in collinear antiferromagnetic compounds, we note that the hidden effect can also exist in noncollinear antiferromagnetic compounds. When a bulk noncollinear antiferromagnetic compound has Analogous to the detection of SOC-induced hidden spin polarization in nonmagnetic compounds  [12], a hidden property can be observed when a probe can resolve the local sectors where the property is not compensated. Specific to hidden spin polarization, the spatial segregation of the spin polarization states allows in principle the detection of the hidden effect in antiferromagnets. Since this effect is intrinsic to the bulk it can be distinguished from the surface effect as the latter sensitively depends on the effective penetration depth of the probing beam [43]. Albeit, for the hidden spin polarization from individual AFM sectors, to detect the AFM spin polarization of the individual sectors, one needs to choose the surface configuration that respects the symmetries of the individual sector that ensure the anti-ferromagnetism of the sector, e.g., mirror plane symmetries perpendicular to the surface plane that connect the spin up and spin down magnetic moments of the AFM sector. Especially, systems with the degenerate states segregated on the different sectors would result in a minimally compensated hidden spin polarization, thus contributing to a robust signal when selectively probing the individual sector, thus being ideal platforms for the detection of the hidden effect.2 discussed in the Results Section), external electric field would break the ΘIT and UT symmetry between the spin split sectors, which then implies a transition from hidden effect to apparent effect. In fact, the electric field applied couples with the electron spin through the magnetoelectric effect68, which is only allowed under specific symmetry conditions41. Additionally, transport properties that are even functions of the sectors can take non-vanishing values in a hidden system. For example, non-reciprocal nonlinear current respond to an applied electric field is recently demonstrated in antiferromagnetic tetragonal CuMnAs69. This serves as a guide in search for systems exhibiting this particular response behavior. Furthermore, the bulk antiferromagnets formed by ferromagnetic layers with alternatively aligned magnetic moments along the direction perpendicular to the ferromagnetic layers (thus hosting hidden spin polarization) could have very different magnetoresistance from the bulk ferromagnets formed by the same ferromagnetic layers but with uniformly aligned magnetic moments. Therefore, switching between the AFM and FM states by external magnetic field could lead to significant change of magnetoresistance, mimicking the tunneling magnetoresistance effect70. These perspectives offer electric and/or magnetic means to control the spin-related properties in antiferromagnets.One of the most desirable features of spin-related phenomena is the possibility of electric and magnetic control. In the case of the hidden spin polarization in AFM, since the unit cell can always be built in terms of two or more sectors, electric field is a practically direct way of inducing and controlling the existence of spin splitting (as well as its magnitude) via modulating the symmetry relationship between the sectors. For example, in the spin degenerate bulk antiferromagnets made of a pair of spin-split antiferromagnetic sectors 75 implemented in the Vienna Ab initio simulation package (VASP). Structural and magnetic configurations are taken from the MAGNDATA database46 derived from experiments. The calculations of SOC-independent spin splitting and spin polarization are done using a non-collinear magnetic setting but without the introduction of spin-orbit coupling . We adopt the GGA + U method76 to account for the on-site Coulomb interactions of localized 3d orbitals involved in the calculations. We used U = 3.9 eV, J = 0 eV on Mn-3d orbits for Ca2MnO4, U = 5.3 eV, J = 0 eV on Fe-3d orbits for insulating FeBr2. These Hubbard U values are derived in ref. 77 using the approach outlined in ref. 78. For CuMnAs, a metal, we used a smaller U value on Mn-3d orbits . We follow the approach proposed by Neugebauer and Scheffler79 to apply a uniform electric field to the bilayer slab in the calculations. This approach treats the artificial periodicity of the slab by adding a planar dipole sheet in the middle of the vacuum region.Electronic structures are calculated using the density functional theory (DFT) methodϕ1〉, |ϕ2〉 onto the atomic orbital basis |ilm〉 and sum over the sites i within sector-η in the primitive unit cell, s = 1,2).We evaluated the hidden spin polarization on sector-Sectors are chosen such that atomic sites within a sector are more closely clustered, while atom pairs associated with different sectors are spatially well separated. This results in weak inter-sector coupling, and consequently physically significant hidden spin polarization effect.Supplementary informationPeer Review File


II is the predominant process by which 30% of annually deposited Hg is emitted back to the atmosphere. Isotopic analyses indicate that above the water table, dark abiotic oxidation decreases peat soil gaseous Hg0 concentrations. Below the water table, supersaturation of gaseous Hg is likely created more by direct photoreduction of rainfall rather than by reduction and release of Hg from the peat soil. Identification and quantification of these light-driven and dark redox processes advance our understanding of the fate of Hg in peatlands, including the potential for mobilization and methylation of HgII.Peatland vegetation takes up mercury (Hg) from the atmosphere, typically contributing to net production and export of neurotoxic methyl-Hg to downstream ecosystems. Chemical reduction processes can slow down methyl-Hg production by releasing Hg from peat back to the atmosphere. The extent of these processes remains, however, unclear. Here we present results from a comprehensive study covering concentrations and isotopic signatures of Hg in an open boreal peatland system to identify post-depositional Hg redox transformation processes. Isotope mass balances suggest photoreduction of Hg II to volatile Hg0 does not play a significant role in mobilizing Hg during peat mass lossMercury isotope signatures in groundwater, soil gas, solid peat, and atmosphere suggest that dark abiotic reduction of peat soil Hg This methyl-Hg can be exported to downstream aquatic systems and subsequently biomagnifies in the food web2. Peatlands receive atmospheric Hg largely through vegetation uptake of gaseous elemental Hg (Hg0)4, the dominant form of Hg in the atmosphere5, including the possibility of surface adsorption of Hg0. Hg0 taken up by vegetation is further oxidized to divalent reactive Hg (HgII) via enzymatic reactions or by the action of reactive oxygen species6. Rainfall input of HgII to peatlands also contributes, but it is smaller in magnitude . HgII from both rainfall and oxidation of Hg0 associated with plant uptake quickly bind to the thiol groups of natural organic matter (NOM)7, potentially forming immobile nanoparticulate β-HgS8. The Hg stored in boreal and subarctic peatlands is globally significant given that peatlands comprise 15 – 30% of terrestrial organic carbon10. Reduction processes may transform some of the deposited HgII to volatile Hg0, which can move vertically within the pore system of the peatland11. Net Hg0 evasion from an open (tree-less) boreal peatland was measured with a micrometeorological method over the course of one year12. Variability in Hg0 evasion rates along a thawing permafrost fen-palsa-bog gradient in the subarctic were related to different amounts of Hg stored in the peat13. While reduction and subsequent evasion might reduce the HgII available for methylation, it also raises questions about peatlands as a long-term Hg sink14, and the suitability of peatlands as archives of earlier Hg deposition16. Given the importance of peatlands in global, regional and local Hg cycles, it is crucial to understand Hg deposition and biogeochemical transformation processes in the peat. The post-depositional processes of Hg related to Hg reduction and oxidation could be resolved by analyzing the abundance and composition of Hg isotopes.The peatlands covering 3% of the Earth’s land surface are hotspots for the production of neurotoxic methyl-mercury (methyl-Hg)18). The fractionation either depends on mass  or is independent of the isotopic mass . Fig. 0 favors light isotopes MDF19), while direct rainfall supply of HgII is not found to cause MDF and MIF. Peat HgII is transformed to gaseous Hg0 through photochemical, biotic and abiotic reduction processes. These processes leave residual HgII enriched in heavier isotopes after losses of Hg0 ((+)MDF22). At the peat surface, UV radiation in sunlight can reduce newly deposited Hg, either when adsorbed onto leaf surfaces23 or when stored in leaf interiors24. Depending on the strength of the HgII bonds to ligands25, photo-reduction of HgII can produce Hg0 with either negative odd-mass MIF (ligands with N/O-Hg bonds26) or positive odd-mass MIF  due to magnetic isotopic effects. In sub-surface peat soils, dark abiotic or biotic reduction of HgII can occur29. Biotic reduction of Hg results in MDF without any significant MIF30, similar to microbial methylation and demethylation of Hg with only significant MDF17. Dark abiotic reduction, as controlled by Nuclear Volume Effect (NVE), results in positive odd-mass MIF in product Hg031. Jiskra et al.14 estimated a 27% loss of Hg in boreal peat soil (a riparian zone soil with tree cover) via dark reduction by NOM, and Yuan et al.32 reported a larger relative Hg loss in a forest ecosystem caused by NOM dark reduction (two thirds) than by microbial reduction (one third). In contrast, dark abiotic oxidation of Hg0 to HgII with positive odd-mass MIF was found based on experimental work with thiol compounds and humic acids18. This was explained by equilibrium fractionation33. Field studies have further demonstrated the quantitative importance of Hg dark abiotic oxidation in arctic tundra soils34. Despite these recent advances in the understanding of how Hg isotopes are fractionated by transformation processes, it still remains unclear how post-depositional processes are affecting the fate of Hg in peatlands.Hg stable isotopes enable us to constrain sources of Hg and its transformation processes because isotopes undergo fractionation during biogeochemical cycling . These investigations are made in two distinct peat microforms, slightly elevated (20 – 30 cm) hummocks and flatter lawns, which differ in the water table level relative to the peat surface and their characteristic vegetation composition. Both of these differences have the potential to affect Hg deposition rates and redox-related mobility processes. This study includes the reports of Hg concentration and isotopes in the peat soil gas of the unsaturated zone above the water table, as well as the Hg isotopes in dissolved gaseous Hg (DGM) of peat groundwater just below the water table.In this study, we identify redox-related Hg post-depositional processes and associated magnitudes in an open (tree-less) boreal peatland system, Degerö Stormyr . Modern Degerö hummock HgAR at 2000–2020CE is 14.4 ± 4.8 μg m−2 yr−1 . This HgAR is, however, more than twice as high as in the Degerö lawn for the same period . Such a significant difference between hummock and lawn sites just 5 m apart on an open peatland can be explained by either a greater Hg sequestration in hummock or a higher post-depositional Hg loss from the lawn. Enhanced Hg sequestration in µg m-2 yr-1 can be achieved by higher peat Hg assimilation in ng g-1 and/or higher peat biomass production  are present in the lawn profile . A higher net peat AR coupled with enhanced net Hg0 assimilation in hummock species are likely to be important reasons for greater HgAR in hummock than in lawn at depths corresponding to the period 2000–2020CE .Surface living vegetation on hummocks has an average Hg concentration of 29 ± 5.5 ng g40, although these considerations alone cannot rule out the possibility of a higher Hg loss from the lawn site . Even though the dominant vegetation species and associated Hg deposition rates are different, Degerö hummock and lawn HgAR profiles show similar stepwise increases from the natural background period  to the second half of the 20th century, where peak fluxes are recorded, followed by a decline  and –1.37 ± 0.17‰  and 0.16 ± 0.07‰  from Northern Hemisphere (NH) remote areas, respectively. Three samples of atmospheric Hg0 at Degerö suggest similar Δ200Hg signatures  to atmospheric Hg0 values reported in the studies mentioned above. According to the current understanding, MIF of even mass Hg isotopes is relatively conservative over the Earth’s surface without being altered during post-deposition transformation processes . We assign Δ200Hg values in atmospheric Hg0  and rainfall HgII  based on the composite records from NH remote areas as the end-members for atmospheric deposition at Degerö. The Δ200Hg in both the hummock and lawn peat profiles averages –0.01 ± 0.05‰  than in the lawn . These results of dominant plant Hg0uptake are in agreement with other studies of peatlands42, as well as different vegetation ecosystems . Δ200Hg becomes slightly positive at 1950–2000CE relative to 1800–1950CE in both hummock  and lawn . A litter decomposition experiment over the course of two-years showed no significant change in the residual HgII Δ199Hg 32, suggesting there would be no significant alteration of Δ199Hg during decomposition of litter and possibly organic matter of peat. The observed more negative Δ199Hg in peat as compared to atmosphere end-members also seems to be in line with the fractionation trajectories of dark abiotic reduction31, as well as photochemical reduction of HgII24, both of which leave more negative Δ199Hg in residual HgII  the associated product Hg0 in the soil gas of the unsaturated zone, i.e., in the peat soil above the peat ground water table, and (ii) in the dissolved gaseous Hg (DGM) of the peat groundwater below the unsaturated zone. In such an assessment, and to further identify Hg post-depositional processes that potentially involved MIF of odd mass, we investigated (i) Hg0 diffusion processes at the atmosphere-peat interface in the unsaturated zone, and (ii) the origin and fate of superficial peat groundwater DGM.Both hummock and lawn Δ0 in the atmosphere average 1.31 ± 0.17 ng m-3 . These concentrations  are consistently below levels observed in the atmosphere. A consistently lower Hg0 concentration in soil gas relative to the atmosphere implies a downward diffusion gradient from the atmosphere into the pore air of the unsaturated zone of the peat soil above the groundwater table. The depth of this unsaturated zone averages 12 ± 5 cm in the lawn 34 and mineral forest soils in North America 59.Over the two year sampling period, the concentrations of Hg0 and peat soil gas Hg0 to deduce the main processes lowering the Hg0 concentration in the peat soil gas34. In this study, soil gas Hg0 shows a lower δ202Hg  and a higher Δ199Hg  than the atmospheric Hg0 above the peatland surface . This similarity of slopes further suggests that dark abiotic oxidation of Hg0 to HgII, coupled with equilibrium isotope exchange, is the dominant process regulating Hg0 concentration in the unsaturated zone of the peat, which is a system that appears to be semi-closed in a long-term perspective.We use Hg isotope fractionation trajectories between atmospheric Hg0 to the gas phase of the unsaturated zone of the peat. By use of Fick’s first law, we calculated the potential vertical diffusion flux of Hg0 from the lower atmosphere into the unsaturated peat layer  in the Degerö peat groundwater 0 – 15 cm below the groundwater surface  of the unsaturated zone of the peat (0.45 ± 0.12 ng m-3), as well as to the atmospheric Hg0 concentration of 1.31 ± 0.17 ng m-3 just above the peatland. The Hg0 saturation level in the groundwater is 1500% and 4300% relative to the atmosphere and the peat soil gas phase, respectively  suggests that there is no net diffusion of DGM to the atmosphere via the air-filled soil pores in the peat. However, at this stage we cannot exclude a potential upward diffusion of DGM along the water-saturated pores that exist side-by-side with air-filled pores in the unsaturated zone of the peat, or through the aerenchymatous tissues of vascular plants. Some of this Hg would diffuse into soil gas to be oxidized by NOM in the unsaturated zone similar to the fate of peat gas Hg0 originating from the atmosphere. The DGM in the groundwater also diffuses downwards along the established concentration gradient. Even if the increase in reduced organic sulfur species below the annual water table  in the sulfidic environment of peatland soils in the region62.During summer the DGM concentrations average 77 pg Lace Fig. . This su200Hg between DGM measured below the ground water table and peat soil gas Hg measured above the same groundwater table . Furthermore, we do not find any evidence of DGM in the peatland groundwater being produced from HgII stored in the peat, giving rise to a current source of Hg0 diffusing through the peat unsaturated zone and back to the atmosphere. However, it cannot be ruled out that DGM evasion to the atmosphere may be of importance during submerged conditions when the water table rises to the peat surface e.g., late fall, winter, and early spring. Such DGM fluxes to the atmosphere have been reported from freshwater lakes65.Another constraint on the origin and fate of DGM in peat groundwater is provided by the composition of Hg isotopes. A significant difference in conservative Δ200Hg even-MIF lies between rainfall and peat soil data , and 52% from rainwater HgII . The high uncertainties in the DGM contribution from these two sources warrant an investigation of other Hg isotope signatures. The product of reduction  is generally enriched in lighter isotopes . At Degerö the DGM δ202Hg is more positive than δ202Hg in solid peat and peat groundwater . Given equal water volumes of rainwater and groundwater, Hg in rainwater could at the most account for half of the quantity of HgII collected in the superficial peat groundwater on the same day. Even though no rainfall DGM isotope signatures have ever been reported, lower δ202Hg and lower Δ199Hg relative to those in rainfall HgII  can last for hours to days during periods of precipitation and downward water flux0 oxidation in the unsaturated zone of peat soil and a non-significant production of DGM from HgII stored in the solid peatland ending up in the peat groundwater suggests that NOM-driven dark reduction is unlikely to be a significant process contributing to the negative peat Δ199Hg observed in peat soil 58, can lead to reduction of HgII, which is generally characterized by a fractionation trajectory with negative Δ199Hg in residual vegetation HgII when complexed by thiol functional groups24. Our observed lower peat Δ199Hg  and 27 ± 20% (3.5 ± 4.2 μg m-2 yr-1), respectively. The size of this proportional loss is also in line with the estimates on Hg0 re-emitted from foliage by photoreduction24.We calculated the HgII complexed by dissolved organic matter from the Degerö catchment area. This loss was estimated to be 1.6 ± 0.2 μg m−2 yr−1 for the period 2009–2014CE12. The Hg export via streamflow mainly originates from the peatland system with a minor contribution from the upland mineral soils 12. Thus, rainfall, snowmelt and peat groundwater are the major contributors of Hg in streamflow37. The 14C dating of the organic matter in streamflow from the Degerö mire shows that this organic matter is less than half a century old73, suggesting that it is Hg deposited during the last 50 years that is possibly mobilized by water flow through the peatland system. This estimated Hg loss in streamflow corresponds to less than half of the losses generated from photoreduction, which is the most important process by which Hg is lost from the open peatland.An additional peat Hg loss pathway is through discharge export of Hg-2 yr-1, Supplementary Fig. n = 6 vs 31% ± 13%, n = 8, P = 0.5, and 5.2 ± 5.6 vs 8.0 ± 3.5 μg m-2 yr-1, n = 6 vs 8, P = 0.6, Supplementary Fig. -2 yr-1) is not due to a higher Hg loss, but rather to less Hg deposition. This emphasizes the importance of vegetation composition and associated primary productivity in the transfer of atmospheric Hg to terrestrial environments40. Our study suggests that photoreductive Hg loss dominates Hg mobility in peatlands, accounting for approximately 30% of the annual Hg deposition  input of atmospheric HgII by wet deposition . The plant uptake of atmospheric Hg0 accounts for the major input, approximately 70%, to the open boreal peatland at Degerö. About 30% of the total Hg input is released back to the atmosphere as Hg0, produced by the photoreduction of HgII at surfaces of the peatland vegetation. The dominance of net plant uptake of atmospheric Hg0 over net rainfall HgII deposition to the vegetated surface of peatlands might be partially due to HgII in precipitation being more readily available for photoreduction and subsequent evasion back to the atmosphere than the HgII associated with plant tissues. The relative importance of HgII photoreduction in rainfall can be explained by the abundance of organic R-O/N functional groups at peat and vegetation surfaces providing a weaker bond to HgII to compete with the reduction process, during the time window required for HgII to re-arrange to stronger bonding RS functional groups70. Once bound to the stronger RS functional groups, HgII is less susceptible to reduction. Our results on detailed post-depositional Hg redox processes provide new information that helps to explain the poorly understood mechanisms behind HgII reduction and Hg0 re-emission5, particularly in open peatland ecosystems 24.Mercury is deposited onto open peatlands by two predominant processes: (i) direct plant uptake of atmospheric Hg0 concentrations in the soil gas relative to the atmosphere. While the downward diffusion flux appears to be negligible as compared to total Hg deposition to the peatland, it clearly rules out upward net diffusion of Hg0 from the peatland air-filled sub-layers to the atmosphere. Below the groundwater table of the peat, the concentration of DGM is supersaturated in relation to both the gas phase of the unsaturated zone of the peat, as well as the atmosphere. The main source of the DGM is likely to be rainwater HgII photoreduction before and/or after deposition to the peatland surface, rather than dark reduction HgII in the peat soil. This means we have not found any reduction mechanisms that would significantly redistribute Hg in gaseous form once it is incorporated into the peat profile and thereby change the peat Hg archive during decomposition processes. There is a small amount of the previously deposited Hg (10%) that is exported downstream together with dissolved organic matter in stream runoff. Peat soil Hg isotopes are not likely to be significantly altered by this stream runoff based on the similarity of Hg isotope composition in boreal forest runoff and soils74. The processes of Hg transformation we unravel here would constrain the mobility of Hg while the peat OM slowly decays below the groundwater table, and thus the possibilities for mobilization of Hg deposited from the atmosphere in earlier decades or centuries. Compared to tree-covered forested ecosystems14, dark reduction of HgII by NOM appear less important in these types of open (tree-less) peatlands where there is more sunlight to promote photoreduction of HgII at the surface. The dominant peat Hg loss caused by photoreduction can likely reduce Hg methylation rates by decreasing the size of this weakly-bond HgII pool which is expected to have a high availability for methylation. Our study highlights that it is the peat surface, instead of peat sub-layers, where main Hg loss occurs. We do, however, suggest that the redox transformation processes during peat decomposition are not significant enough to disqualify peat soil to be a reasonable archive of long-term atmospheric Hg deposition patterns.Mercury concentrations and isotope signatures highlight the oxidation capacity of organic matter in the air-filled pores of the peat above the groundwater table. This lowers the Hg2 catchment area which is 270 m above sea level in the Kulbäcksliden Research Park (64°11’N and 19°33’E). This is located in the municipality of Vindeln municipality, Västerbotten province, Sweden. The climate at Degerö peatland is cold temperate humid, with mean annual values of 523 mm for precipitation and +1.2 °C for temperature (data from 1961 to 199075). Mean temperatures in July and January are +14.7 °C and −12.4 °C, respectively. The vegetation growing season is from May to October (156 ± 15 days based on 2001–200558). The rest of the year is characterized by snow cover, which in general reaches a depth of 0.6 m. Previous work on Hg conducted in this peatland includes the influence of sulphate concentration on peat pore water Hg methylation78, and Hg flux measurements80. The Degerö peatland surface is dominated by lawns with minor occurrence of hummocks.This study was conducted at the Degerö Stormyr, a Sphagnum-dominated minerotrophic open peatland, on areas without tree cover  and two other lawn peat sequences (ca. 350 cm) at the same sites were collected and stored as archive samples. Peat cores were described for some basic information  and then wrapped in plastic film before placement in PVC tubes for transport to the Swedish University of Agricultural Sciences . Cores were frozen and subsequently sliced at roughly 1 cm resolution for the top 50 cm of peat and then at 2 cm resolution for the rest of the core. Each new slice was cleaned with MilliQ water, edges removed and subsampled for further analysis following well-established protocols83. The dimension of the largest subsample of each slice was measured using a Vernier caliper to obtain the volume for calculating the dry bulk density and to estimate the cut loss between each slice. Subsequently, the largest sub samples were dried for geochemical analysis using the freeze-dryers at SLU . In this paper, we focus on the top 1 m to understand the Hg geochemical cycle in the zone where the water table fluctuates and the acrotelm (above peat groundwater table level) transitions into the catotelm (below groundwater table level).One 3 m-long peat sequence (DEG20-PH01A) was collected from the Degerö hummock site in July 2020 . Fourteen hummock and thirteen lawn samples were dated to a post-bomb period, whose ages were calibrated using the NH Zone 1 calibration curve provided by Calibomb software of Queen’s University, Belfast88. The age models for hummock peat profiles (27 dates) and lawn peat profiles (32 dates) were generated from post-bomb calibrated ages and pre-bomb 14C results using the Bacon model  with the ‘rbacon’ package in R software (https://CRAN.R-project.org/package=rbacon)89. Details on the dated material, radiocarbon ages and calibrated ages are shown in Supplementary Table In total 27 plant macrofossil samples from hummock profiles and 32 from lawn profiles, were selected for radiocarbon analyses following established protocols90 for Sphagnum mosses. Volume abundances of all components are expressed as percentages with the exception of Andromeda polifolia seeds, Eriophorum vaginatum spindles, Carex spp. nutlets, Sphagnum spore capsules, Cristatella mucedo statoblasts and macrofossil charcoal fragments, which are presented as the number (n) found in each of the subsamples. Zonation of the macrofossil diagram was made using psimpoll 4.2791, using the optimal splitting by information content option.A total of 25 and 26 fresh samples from the top 1 m hummock and lawn profiles, respectively, were chosen for macrofossil identification. Macrofossil samples were warmed in 10% NaOH and sieved (mesh diameter 180 μm). Macrofossils were identified using a binocular microscope (×10 – ×50) based upon modern type material. Identifications were also made with reference to Michaelis, (2011)92. Briefly, spectra were obtained from 12 freeze-dried samples from hummock profile in fluorescence mode at ambient temperature under high vacuum (10−8 − 10−6 mbar). The storage ring was operated at 2.5 GeV with a ring current of 250 mA. A fixed double-crystal monochromator with Si(111) crystals was used to monochromatize the white beam. Scans were taken at the energy range of 2462–2500 eV with a step size of 0.2 eV. Data averaging, normalization, and Gaussian curve deconvolution were conducted using Athena, WinXAS, and Microsoft Excel  were collected and analyzed at Beamline 4B7A in Beijing Synchrotron Radiation Facilities (BSRF)0 concentration at sampling rates from 0.05 to 0.42 LPM. We did not find a significant difference between these different rates . This indicates that the sampling rates were sufficiently low to avoid drawing in air from above the peat.We sampled peat soil gas and atmosphere in both Degerö hummock and lawn for the measurements of Hg concentration and Hg stable isotopes over two summers from 2020 to 2021 , were buried below the lawn peat surface to serve as groundwater reservoirs. Each reservoir tube was 3 m long and buried with one end at −30 cm below the lawn peat surface, and the other end at -50 cm. At both ends of each PVC reservoir tube, there was a vertical, 1 m-long PVC access tube with 2 cm ID. This extended above the peat surface allowing peat water to be easily pumped from the lower end of each of the 3 m-long reservoir tubes . The collection system consists of an inclined acid-washed Teflon-coated black plate connecting to a clean funnel at the lower edge of the plate. A 500 ml glass bottle was placed at the outlet of the funnel. Three rainfall samples were collected within 1 h during rain events on July 50 concentration in peat groundwater, approximately 2000 L of peat water was collected for pre-concentration from 23rd June to 13th July 2021, enabling four aliquots of DGM isotope measurements . We adapted a rainfall Hg purging method described in Jiskra et al.54 to our peat groundwater DGM extraction system  for NIST 1515, and 458 ± 13  for BCR 482.Freeze-dried peat samples were analyzed for total Hg (THg) concentration on a combustion cold vapor atomic absorption spectrometer  at the Swedish University of Agricultural Science, Uppsala, Sweden. The analytical performance of the DMA-80 was assessed by multiple measurements on reference materials, NIST 1515 (Apple leaves) and BCR 482 (Lichen). Results were not statistically different from the certified values, with Hg concentrations of 42.9 ± 3.9 ng g-2 yr-1, Eq. i was obtained by Hg concentration (ng g-1), density (g cm-3), thickness (cm) and age interval (yr).Hg accumulation rate , three atmosphere, seven peat soil gas and four peat water DGM samples were determined from 20%  inverse aqua regia solutions using cold-vapor multi-collector inductively coupled mass spectrometry . Sample isotopic ratios were corrected for mass bias by sample-standard bracketing using NIST 313394. Results are reported as δ-values in per mil (‰) representing Hg mass dependent fractionation by reference to NIST 3133  and 0.07 ± 0.10‰ , respectively. Hg isotopic signatures in procedural standards are reported for δ202Hg (maximum 2σ = 0.17‰), Δ199Hg (maximum 2σ = 0.13‰), Δ200Hg (maximum 2σ = 0.08‰), Δ201Hg (maximum 2σ = 0.19‰) and Δ204Hg (maximum 2σ = 0.44‰).The quality control of Hg isotope measurements is assessed by analyzing ETH-Fluka and procedural standards (Apple leaves, NIST 1515, 200Hg in NH remote Hg0 and rainfall HgII to quantify the atmospheric Hg deposition pathways to peat (Eqs. 3.We use Δeat Eqs.  and 53.40 and rainfall HgII deposition, respectively.Symbols α and θ represent the proportion of HgFurther information on research design is available in the Supplementary InformationPeer Review FileReporting SummarySource Data


Biological evidence indicewates that the brain atrophy can be involved at the onset of neuropathological pathways of Alzheimer's disease. However, there is lack of formal statistical methods to perform genetic dissection of brain activation phenotypes such as shape and intensity. To this end, we propose a Bayesian hierarchical model which consists of two levels of hierarchy. At level 1, we develop a Bayesian nonparametric level set (BNLS) model for studying the brain activation region shape. At level 2, we construct a regression model to select genetic variants that are strongly associated with the brain activation intensity, where a spike-and-slab prior and a Gaussian prior are chosen for feature selection. We develop efficient posterior computation algorithms based on the Markov chain Monte Carlo (MCMC) method. We demonstrate the advantages of the proposed method via extensive simulation studies and analyses of imaging genetics data in the Alzheimer's disease neuroimaging initiative (ADNI) study. Imaging genetics is an emerging interdisciplinary field with a focus on assessing the impact of genetic variation on brain function and structure. It is a useful tool to uncover the etiologies of complex neuropsychiatric diseases, such as Autism  data, single nucleotide polymorphisms (SNP) data and clinical data in the Alzheimer's Disease Neuroimaging Initiative (ADNI) study. Alzheimer's disease (AD) is one of the most common neurodegenerative disorders that impair mental functioning. It affects approximately eight percent of people who are 65 years of age or older. It has been shown that AD leads to nerve cell death and tissue loss in the brain  study  prior. We term our method as Bayesian nonparametric level set (BNLS) method. BNLS consists of two levels of hierarchy.At level 1, a Bayesian nonparametric level set model is developed for characterizing the shape of consistent brain activation regions across multiple subjects. The level set method has been widely used in image segmentation problems  that are strongly associated with brain activation intensities.p voxels in a brain region ℬ ⊂ ℝ3 and genetic variants of m SNPs from n subjects. Let i index the subject, j index the voxels and k index the SNPs. Denote by yij the observed imaging signal at voxel vj ∈ ℬ. Let Sik be the genetic variant for SNP k.Suppose we collect brain images consisting of yij follow a normal mixture model:At Level 1, we model the brain signal intensity within brain activation regions by assuming x) = 1 if x > 0 and δ(x) = 0 if x ≤ 0. The level set function ϕ(v):ℬ → ℝ determines the brain activation regions. For any voxel v in the brain, if ϕ(v) > 0 implying that δ{ϕ(vj)} = 1, then it is located in a activation region and the brain signal yij has an average activation intensity μi. Otherwise, the voxel is located outside the brain activation regions with a mean intensity zero. The parameter yij across all voxels j for subject i.where δ and (2).v) with mean zero and covariance kernel function, denoted asAt Level 1, to guarantee the robustness and flexibility of modeling the activation regions shape, we assign a Gaussian process prior to the level set function ϕ:ℬ × ℬ → ℝ is a symmetric positive definite kernel function.where κ refers to a point mass measure at a real value v. The parameter w is the prior inclusion probability indicating how likely each feature is to be selected. Pre-defined value ν0 usually select very small so that the “spike”  represents an inverse gamma prior with shape w1 and rate w2.where IG(l(·)} and {λl} are respectively eigen functions and eigenvalues for the kernel function κ that are shared cross all patient samples. Here, L is the number of basis functions and it can be determined according to the proportion of the variation of the GP, denoted as α∈{0, 1}, that can be explained by the basis expansions, i.e., Hϵ[x] → δ[x] as ϵ → 0. Note that its first derivative is To implement posterior computation algorithm, we need to consider model approximations. First, we consider the basis expansion approximation S = (Sik), l, j = ψl(vj). Then our Bayesian hierarchical model with prior specifications can be represented asWrite Γ is a diagonal matrix with  element being where η without imposing sparsity which leads to more efficient posterior computation. The model is represented asWe also consider a conjugate normal prior on For posterior computation, we adopt the Riemann Manifold Metropolis adjusted Langevin algorithm (MMALA)  with brain image intensity levels, we use the null hypothesis that SNP k is uncorrelated with the intensity level inside the activation region (H0:ηk = 0) and the alternative hypothesis that SNP k is not uncorrelated with the intensity level inside the activation region (Ha:ηk≠0). Based on the marginal posterior distribution for ηk, if 0 is included in the posterior 99% credible interval, we assign γk = 1, otherwise γk = 0 where γk is the same indicator variable introduced in SSVS. We approximate the posterior inclusion probability of SNP k: Eγk using the averaged values after burn-ins For the variable selection, we apply an The details of derivations and posterior computation algorithms the Bayesian level set method with spike-and-slab prior and normal prior are provided in the n = 1, d = 2, S = 0 thus no variable selection was involved. For the most complicated simulation setting, we simulated multiple subjects, 3D imaging data and utilized the predictors in real data analysis for selection.We tested the performance for learning activation region shapes and selection influential variables using proposed method starting from the simplest scenario and then gradually extended to the most complicated scenario. For the simplest simulation setting, we simulated a single subject, 2D imaging data and zero predictor matrix, i.e., set 2 (d = 2). We considered three activation region shapes: circles, squares and random shapes. We simulated data by setting σ2 = 1, and the signal intensities μ and the level set function were set as follows:In this simulation study, the objective is to test the Bayesian nonparametric level set method for random shape fitting. We simulated 2D images of size 150 × 150 on a square region Circle shapes: set the signal intensity μ = 1 (weak) and the true level set function v) = exp{−0.5(|v1|+|v2|)}−0.8Square shapes: set the signal intensity μ = 3 (strong) and the true level set function ϕ and draw the true level set from a Gaussian process with mean zero and covariance kernel −3 and run 50,000 iterations with 20,000 burn-ins. The shape estimation results were presented in For the posterior computation, we set ϵ = 1 × 10m = 50 subjects with 3D images simulated for each of them. The 3D image grid was of 20 × 20 × 20  on a square region 2 (d = 3). Like simulation 3.1, we set S = 0 so that there is no variable selection involved. We considered three different shapes of activation region: spheres, diamonds, and random shapes. We set i  and the level set function ϕ(v) were set as follows:We evaluated the proposed method on a total of i~N and the true level set function Sphere shapes: set the signal intensity μi~N and the true level set function ϕ(v) = exp{−0.5(|v1|+|v2|+|v3|)}−0.6Diamond shapes: set the signal intensity μi~N and draw the true level set from a Gaussian process with mean zero and covariance kernel Random signal shapes: set the signal intensity μ−3, α = 0.8 as PCA percent and run 5,000 iteration with 2,000 burn-in. The shape segmentation results were respectively summarized in For the posterior computation, we set ϵ = 1 × 10n = 235 subjects with 3D images simulated for each of them. We only took the first 200 columns (m = 200) from the SNP matrix in real data analysis to form S in the simulations. We randomly selected 5 of them as signal  and the remaining 195  as noise. Like previous simulation studies, we considered three different activation region shapes with different combination of abilities for shape estimation and variable selection quantified by signal-to-noise-ratio SNR(•).In the simulation study, we evaluated the proposed method on the most complicated scenario where there is a total of β) is the signal-to-noise ratio for activation shape estimation and SNR(η) is the signal-to-noise ratio for variable selection., and in simulations, we simulated datasets of different combinations: SNR(β) = 8, 5, 2 and SNR(η) = 8, 5, 2.where SNR for β) = 2 and SNR(η) = 2, the averaged accuracy, sensitivity and specificity for voxels classification are all above 0.94 and for variable selection are all 100%. As SNR(β) increased to 5 and 8, classification performance improves as expected while SNR(η) increased to 5 and 8, variable selection are all 100% accurate. For MSE of η and μ, it decreases in the general trend when SNR(β) increases.The simulation studies indicate our proposed method is accurate for voxels classification and variable selection. For simulations using SSVS, even with the worse scenario when SNR(β) = SNR(η) = 2, the accuracy, sensitivity (true positive rate) and specificity (true negative rate) decrease to 0.893, 0.840, 0.894 and MSE for η and μ increases to 0.081 and 0.141. The proposed method does suffer a decrease performance as expected, but in general, the results are acceptable. We recommend applying the fast algorithm when there is exceedingly large number of candidate SNPs in the study for fast computation purpose.Compared to Bayesian level set method with non-sparse prior for variable selection, the voxels classification is robust but the variable selection generates worse performance. If we compare the scenario when SNR for both brain activation region selection and genetic effects, using estimations from the analysis of ADNI data. The estimated SNR(β)≈0.5, while for η, the SNR varies between 0.5 and 1.0; hence, we consider two scenarios: SNR(η) = 0.5, 1.0. We simulate the activation region through Gaussian Processes (GP), where the spatial correlation is approximated to be 0.9 for neighboring voxels, based on PET image analysis. Given that the underlying assumptions of our Bayesian model differ from those of FVGWAS and FGWAS, the parameter estimations are not directly comparable. Therefore, we focus our comparison on activation region and SNP selection results, omitting the mean squared error (MSE) metrics. The findings are summarized in The comparative analysis indicates that FVGWAS exhibits low power in detecting activation regions but performs exceptionally well in identifying genetic variants, albeit with a slightly inflated false-positive rate. On the other hand, FGWAS, which incorporates spatial smoothness into its estimations, demonstrates high power in detecting activation regions but struggles to control the false-positive rate effectively, resulting in low specificity. In contrast, our proposed Bayesian level set method, augmented with a spike-and-slab prior, outperforms both methods in terms of activation region and genetic variant selection. Importantly, it also effectively controls the false-positive rate.We applied the proposed method on an imaging genetics study to detect genotypes that are associated with imaging phenotypes (both imaging intensities and activation shapes) in application to the Alzheimer's disease. To be specific, the primary goal was to determinate any specific gene markers that are correlated with regional activation levels in the brain, which can serve as potential indicators of disease with different levels of progression.The data was made available by the Alzheimer's Disease Neuroimaging Initiative (ADNI). There are three different groups of subjects: 69 normal subjects (NORM), 117 mild cognitive impairment subjects (MCI), and 49 Alzheimer's disease patients (AD). In total, 235 subjects were included in the study. For genotypes, we selected 614 SNPs that are associated with 34 genes known to be potentially related to AD from the literature. In addition to the genetic data, we also included 5 clinical factors. They are: the subjects age, gender, body weight, neuropsychiatric inventory score (NPISCORE), and functional activity questionnaire score (FAQSCORE). There are 2 missing values in NPISCORE and 4 missing values in FAQSCORE. All the missing values were imputed by the mean values of observed data for the corresponding variables. We standardized each variable so that they had an average value of 0 and a variance of 1. We included 42 brain regions in the analysis. There are 12 regions located in Frontal lobe including ; 8 in Parietal lobe including Parietal_Sup_L, etc.; 6 in Occipital lobe including Occipital_Sup_L, etc. and 16 in Temporal lobe including Temporal_Sup_R, Hippocampus_L, etc. . We studied each of them at three different time points: baseline (bl), month 6 (m6), and month 12 (m12).A and B, βs are different: β(A)≠β(B)), independent intensity levels within subject (μi(A)≠μi(B)) and across subjects (μi(A)≠μj(B)), individual set of influential SNPs (η(A)≠η(B)). Second, we simplified our model by assuming the same level of activation within one anatomical brain region due to the fact that anatomical regions are usually small areas in the brain.We applied the proposed level set image segmentation for activation region fitting, and utilized the spike and slab prior in variable selection as we only included limited number of SNPs in the study. We applied our method to each of the brain anatomical regions. The objective was to learn the brain activation region changes over time, and to select significant SNP biomarkers that were associated with the activation intensities. There were some assumptions in the model in the way we implemented. First, we borrowed the anatomical structure information by assuming separate activation regions , this time, we applied a more stringent rule where only SNPs with posterior inclusion probability larger than 0.5 are selected as highly-influential SNP of interest. We pooled the SNPs selected from at least one anatomical region together. As shown in the Venn diagram , in totaBased on SNP-activation relation, we observe that ADAM10 (SNP rs16940638) is universally associated with the majority of the anatomical brain regions across time: 38 out of 42 at baseline, 33 out of 42 at month 6, and 33 out of 42 at month 12. The SNP has been identified as one of the significant genetic variants using genome-wide association studies (GWAS) and mediation analyses where the objective was to detect SNPs that influence psychiatric and cognitive traits through intermediaries, and would not be detected otherwise  is among the top SNPs to be related with AD gene pathway implicated in Alzheimer's disease  and NEDD9 (SNP rs3734404) in the brain are not documented based on existing literature, but their functionalities can be inferred. DAPK1 (rs10512186) may be related to the late-onset of Alzheimer's disease as it is only selected at month 6 and month 12. Functionally, DAPK1 is a death-associated protein kinase that mediates a number of cellular processes including apoptosis and autophagy  plays a key role in tyrosine-kinase signaling related to cell adhesion  gene is a well-known risk factor due to its influence on blood cholesterol. Although it was included in our analysis, APOE was not found to be significantly associated with brain activation in the FDG-PET data, presumably because APOE variants act in a more global manner, and are not directly linked to the activation level of brain regions in the specific data type.In this work, the genetic data used only involved genotyping data. Although genetic variations can shed some light on the potential association between genes and brain region activation during AD development, it cannot elucidate detailed molecular mechanisms. In future works, we will try to include gene expression and other data types to further study the mechanisms behind the genetic associations.https://adni.loni.usc.edu.Publicly available datasets were analyzed in this study. This data can be found at: JK and TY contributed to conception and design of the study. ZJ implemented the algorithm, performed simulation studies and real data analysis, and wrote the first draft of the manuscript. All authors formulated the statistical model, derived posterior computation algorithm, contributed to manuscript revision, read, and approved the submitted version.


Based on simple electrostaticand harmonic potentialconsiderations,we derive a straightforward expression linking the composition ofa Prussian blue analogue (PBA) to its propensity to undergo collectivestructural distortions. We demonstrate the existence of a thresholdvalue, below which PBAs are undistorted and above which PBAs distortby a degree that is controlled by a geometric tolerance factor. Ouranalysis rationalizes the presence, absence, and magnitude of distortionsin a wide range of PBAs and distinguishes their structural chemistryfrom that of other hybrid perovskites. A value of α = 1 impliesperfect filling, which in turn prevents framework distortion. Decreasingthe value of α generally induces volume-reducing distortionsin the structure to compensate for the poorer geometric fit. Recognizingthe relationship between tolerance factor and structural distortionshas enabled targeted control over perovskite crystal structures, asin A2CrWO6 , and is seen as a crucial design approach foroptimizing functional response in, for example, magnetoresistive14 and photovoltaic15 perovskites.In conventional perovskites, which sharethe same parent ABX17 simple bonding considerationssuggest the picture may be fundamentally different for PBAs in particular.The key distinction is that directional metal–cyanide interactionsfavor a linear linkage geometry,18 in contrastto the tilted geometry generally driven by covalency in conventionalperovskites.19 For example, Mn[Pt(CN)6], which with no A-site cations at all has a tolerance factorwell below 1, nonetheless adopts a cubic crystal structure in whichthe BX3 framework is undistorted.20 Instead it is by filling the A-site  thatdistortions are switched on.21 This behavior contrastswith that of conventional perovskites but is conceptually similarto the activation of tilts in halide hybrid perovskites through hydrogen-bondinginteractions between organic A-site cations and the anionic framework.22 An additional complication that we will addressis that PBAs are highly nonstoichiometric frameworks, and one mightexpect partial site occupancies to play a nontrivial role in the activationor otherwise of collective distortions.Here we address the obvious and important questionof whether ananalogous geometric relationship holds for PBAs. While it is alreadywell established that the tolerance factor approach might be successfullyand straightforwardly extended to molecular perovskite analogues (“hybridperovskites”),3 framework is predominantly electrostatic, andsince the corresponding attraction scales inversely with distance,it is intuitive that distortions reducing the A···Xdistance should be energetically favored  and 1/(rAX – d). For small distortions d ≪ r = rAX, we have an electrostatic stabilizationd2 . Hence for a PBA with formula Axq+B[B′(CN)6]y we can writex and y are the A-site and X-site occupancies, q is thecharge on the A-site cation, and β > 0 is a proportionalityconstant into which we have subsumed the 2/r3 factor of aSo why might fillingthe A-site of PBAs switch on structural distortionsat all? The interaction between an A-site cation and its surroundinganionic BX favored 1a. Howevk is an effectivespring constant that reflects the stiffness of the M–CN–Mlinkages to deformation. So this energy cost also scales as d2 for small distortion magnitudes. Combiningthese two terms, one expects distortions to occur whenever ECoul + Eharm <0; that is,d has now disappeared because both terms scale as d2.Acting against this electrostatic driving forceis the energy costof distorting M–CN–M linkages from their linear ground-state.The leading (harmonic) term in the corresponding energy expansionhas the form xyq values fora large rangeof PBAs of different compositions (details in the Supporting Information). With only a very small number ofexceptions (which we will come to discuss), we observe a partitioninginto undistorted structures at low values of xyq anddistorted structures at high values of xyq . The existence of a universal criticalvalue of k/(2β) for which k and Coulomb prefactors β (which depend on the A···Xdistance) do not vary too much among PBAs. From our data set thiscritical value is about 1.We show in xq+B[B′(CN)6]y is likely to be structurally distorted, based simply on thecorresponding product xyq. Immediately it rationalizeswhy low-vacancy PBA cathode materials switch between distorted andundistorted phases as A-site cations are shuttled in and out duringelectrochemical cycling, for example.23 The reduced A-site occupancy of the oxidized state means there isan insufficient electrostatic driving force to drive distortion, buton reduction the increased A-site occupancy induces long-range symmetrybreaking.24This result provides a simple butpowerful heuristic for estimatingwhether or not a given PBA of arbitrary composition Axyq ≃ 0.8 correspond to low-vacancy PBAs Rb0.90Cu[Co(CN)6]0.92 and Rb0.85Cu[Fe(CN)6]0.95 containing Jahn–Teller-active Cu2+. In the particular case of Rb-containing PBAs, which tendto show columnar Rb order at half-filling,25 the combination of A-site order and cooperative Jahn–Tellerorder from the Cu2+ ions automatically activates an additionaltilt instability through symmetry considerations alone.26 Consequently, the presence of distortions inthese two systems arises from electronic effects rather than froman electrostatic drive to reduce the volume. This point is made clearalso in the small volume strains involved (<4%). We have also madethe assumption that alkali cations are situated on (or near) the 12-coordinateA-site, which is almost universally true29 but is not necessarily the case for some Na-containingPBAs,30 nor is it for the related familyof cyanoelpasolites.31 Mindful of thesevarious caveats, we nonetheless see the value of understanding whichgeneral factors switch on and off collective distortions in differentPBAs.Our derivation is intentionallysimplistic and involves a numberof crude approximations. We have not taken into account electronicdistortion mechanisms, such as the Jahn–Teller effect, whichneeds to be treated separately. Indeed the two outliers at xyq > 1 will minimizeitsenergy by maximizing the value of d; in other words,it should distort as much as the structure allows /Vref.But what determines distortion magnitude? An implicationof theanalysis leading to e allows 1b. This 12 Our data suggest an empiricalrelation δ ≃ 2(1 – α); e.g., a tolerancefactor of 0.9 corresponds to a volume reduction of about 20%.We show in 2Mn[Mn(CN)6] andK1.7Fe[Fe(CN)6]0.9, their similardistortion magnitudes can now be rationalized in terms of their similartolerance factors . Second, becausevolume reduction is cooperative, we expect that the magnitude of distortionshould be largely unaffected by the incorporation of A- or X-sitevacancies 6]y systems  show remarkablylittle variation in distortion magnitude. The small variation in cellvolume is clearly evidenced by the 24 KxMn[Fe(CN)6]y samples in ref (2Mn[Mn(CN)6] originally flagged as the undistortedanomaly in xyq = 2, so one expects a distorted structure,but the tolerance factor is so near unity that the correspondingvalue of δ is only about 6%. We anticipate that the system iscubic because small distortion magnitudes can be accommodated in theflexible PBA structure through dynamic and/or short-range fluctuationsrather than collective symmetry breaking.We make a number of commentsregarding acancies 2b. It iss in ref . By contpΔV term to the lattice enthalpybalance, which behaves as if to reduce the effective spring constant k and so shifts the critical value of xyq to below 1. This effect is consistent with the experimental observationof pressure-driven structural distortions in some PBAs.39 The effect of temperature is formally given by the balance of entropicterms in the free energies, but one simplification is to considerthe effective radii of 24The effect of externalpressure is to include an additional 41 Hence k < 0 and the inequality of 42 formate,43 or thiocyanate44) all favor distorted geometries. Hence the existenceof compositional distortion thresholds is not universal but may nonethelessbe relevant to dicyanometallates,45 borohydrides,46 bifluorides,47 andeven metal–organic frameworks.49So the structural chemistry of PBAs differs conceptually fromthatof other hybrid perovskites as a consequence of the linear ground-stategeometry of the B–X–B linkages. In conventional perovskites,for example, the tilt modes are usually mechanically unstable in theabsence of sufficiently large A-site cations.8) then the combination of high A-site occupancies, low hexacyanometallatevacancy fractions, and a tolerance factor much reduced from unitywill collectively maximize the degree of distortion. Inverting thesecriteria then forms the design strategy for avoiding structural distortions,which may be important for improving the reversible capacity of PBA-basedcathodes.Fromthe perspective of functional PBA design, our results suggesta number of avenues. In cases where structural distortions are favorable(e.g., when targeting hybrid-improper ferroelectrics