domain: shs scco

text:
Ioannis Karakis

Dr Matthew P Pase
email: matthewpase@gmail.com

Alexa Beiser

Sarah L Booth

Paul F Jacques

Gail Rogers

Charles Decarli

Ramachandran S Vasan

Thomas J Wang

Jayandra J Himali

Cedric Annweiler

Dr Sudha Seshadri
email: suseshad@bu.edu

Association of Serum Vitamin D with the Risk of Incident Dementia and Subclinical Indices of Brain Aging: The Framingham Heart Study

Keywords: Alzheimer's disease, brain, dementia, diet, lifestyle, magnetic resonance imaging, neuropsychology, nutritional status, risk factors, vitamin D

Background-Identifying nutrition-and lifestyle-based risk factors for cognitive impairment and dementia may aid future primary prevention efforts.

Objective-We aimed to examine the association of serum vitamin D levels with incident allcause dementia, clinically characterized Alzheimer's disease (AD), MRI markers of brain aging, and neuropsychological function.

INTRODUCTION

As the population continues to age rapidly, it is important to pinpoint modifiable risk factors for dementia, including factors pertaining to lifestyle and diet. Beyond its function in maintaining bone health, vitamin D is increasingly recognized as an important multipurpose steroidal hormone [START_REF] Cherniack | Hypovitaminosis D in the elderly: From bone to brain[END_REF]. Vitamin D deficiency has been associated with a number of neurological illnesses including multiple sclerosis [START_REF] Munger | Serum 25-hydroxyvitamin D levels and risk of multiple sclerosis[END_REF], Parkinson's disease [START_REF] Knekt | Serum vitamin D and the risk of Parkinson disease[END_REF], cerebrovascular disease [START_REF] Buell | 25-Hydroxyvitamin D, dementia, and cerebrovascular pathology in elders receiving home services[END_REF], and dementia [START_REF] Balion | Vitamin D, cognition, and dementia: A systematic review and meta-analysis[END_REF][START_REF] Annweiler | Low serum vitamin D concentrations in Alzheimer's disease: A systematic review and meta-analysis[END_REF].

Meta-analysis of studies comparing patients with and without dementia has indicated that those with Alzheimer's disease (AD) have lower levels of vitamin D [START_REF] Balion | Vitamin D, cognition, and dementia: A systematic review and meta-analysis[END_REF][START_REF] Annweiler | Low serum vitamin D concentrations in Alzheimer's disease: A systematic review and meta-analysis[END_REF]. Similarly, metaanalysis of cross-sectional studies has identified that low levels of vitamin D are associated with poorer global cognitive performance [START_REF] Balion | Vitamin D, cognition, and dementia: A systematic review and meta-analysis[END_REF] and executive function [START_REF] Annweiler | Metaanalysis of memory and executive dysfunctions in relation to vitamin D[END_REF]. However, drawing conclusions from case-control and cross-sectional studies is difficult because individuals with cognitive impairment may have pre-existing risk factors for vitamin D deficiency, including advanced age, comorbid conditions, poor mobilization leading to limiting sun exposure and suboptimal nutritional status [START_REF] Buell | 25-Hydroxyvitamin D, dementia, and cerebrovascular pathology in elders receiving home services[END_REF].

Community-based prospective cohort studies have the ability to relate vitamin D levels to the future risk of incident dementia, yet few studies have been reported. Our objective was to explore the prospective association of serum 25(OH)D concentrations with incident allcause dementia and clinically characterized AD in the large, community-based, Framingham Heart Study. We also examined the associations of serum 25(OH)D concentrations with neuropsychological function and MRI markers of subclinical brain aging. We hypothesized that low vitamin D status would be associated with evidence of subclinical brain injury on MRI, impairment in executive function, and an increased risk of dementia.

MATERIALS AND METHODS

Study sample

The sample was comprised of participants from the Original and Offspring Framingham Heart Study cohorts. The Original cohort first began in 1948, with participants examined approximately every two years. The Offspring cohort was initiated in 1971, comprised of the offspring of the Original cohort and their spouses. The design and selection criteria of the Framingham Heart Study have been described elsewhere [START_REF] Kannel | An investigation of coronary heart disease in families. The Framingham offspring study[END_REF][START_REF] Dawber | Epidemiologi-cal approaches to heart disease: The framingham study[END_REF]. Vitamin D levels were measured during examination cycle 20 (1986-1990) for the Original cohort and examination cycles 6 (1995-1998) and 7 (1998-2001) for the Offspring cohort. Of the 5209 participants initially enrolled into the Original cohort, 1369 attended the 20th examination cycle. Of the 5124 participants enrolled into the Offspring cohort, 2058 attended examination cycles 6 or 7, during the window in which vitamin D was measured (which straddled the two exams). The participant flow diagram is shown in Fig. 1. The final sample for analysis of incident dementia (n= 1663) comprised participants from both the Original and Offspring study cohorts and was limited to participants aged 60 years or older at the time vitamin D levels were measured. Analysis of neuropsychological (n = 1291) and MRI (n = 1139) outcomes was limited to participants enrolled in the Offspring study cohort, but age restrictions were not imposed. All participants were free from dementia at baseline and almost all participants enrolled in this study were Caucasian. All protocols were approved by the Boston University Medical Center and Tufts Medical Center Institutional Review Boards, and participants provided written informed consent.

Serum vitamin D status

Serum samples were obtained in the morning after an overnight fast and frozen at -70ΔC for no more than 3 years. At examination 20 for the Original cohort, serum 25(OH)D was determined by a competitive protein-binding assay, as previously described [START_REF] Rienstra | Vitamin D status is not related to development of atrial fibrillation in the community[END_REF]. Intra-and inter-assay coefficients of variation were 7% and 10%, respectively. At Offspring examinations 6 and 7, serum 25(OH)D was determined by radioimmunoassay (DiaSorin, Stillwater, MN), as previously described [START_REF] Wang | Vitamin D deficiency and risk of cardiovascular disease[END_REF]. The limit of detection for 25(OH)D was 1.5 ng/mL, and no samples had concentrations below the limit of detection. All samples were run in duplicate and the values averaged. Total (intra-assay and inter-assay) coefficients of variation for control values of 14.4 and 54.7 ng/mL were 8.5% and 13.2%, respectively. Vitamin D deficiency was conservatively defined a priori as serum 25(OH)D concentrations below the consensual cut-off value of 10 ng/mL [START_REF] Holick | Vitamin D deficiency[END_REF].

Incident dementia

We examined the 9-year risk of incident dementia and clinically characterized AD, beginning from the time vitamin D was measured. Dementia was diagnosed according to the criteria of the Diagnostic and Statistical Manual of Mental Disorders, 4th Edition (DSM-IV), and AD based on the criteria of the National Institute of Neurological and Communicative Disorders and Stroke and the Alzheimer's Disease and Related Disorders Association for definite, probable, or possible AD [START_REF] Mckhann | Clinical diagnosis of Alzheimer's disease: Report of the NINCDS-ADRDA Work Group under the auspices of Department of Health and Human Services Task Force on Alzheimer's Disease[END_REF]. Diagnosis was made by the dementia review committee comprising at least one neurologist and one neuropsychologist.

Participants have been monitored for the development of all-cause dementia and AD since 1974 using previously described surveillance techniques [START_REF] Wolf | Relationofobesityto cognitive function: Importance of central obesity and synergistic influence of concomitant hypertension. The Framingham Heart Study[END_REF][START_REF] Seshadri | Plasma homocysteine as a risk factor for dementia and Alzheimer's disease[END_REF]. Briefly, participants with suspected dementia or cognitive impairment were either self-referred or referred to us by their primary care provider, family members, or other health care professionals. We also administered the Mini Mental State Examination (MMSE) at each examination cycle in order to detect cognitive impairment. If suspected for cognitive impairment, participants underwent neuropsychological assessment by a neuropsychologist who referred suspected cases of dementia to a neurologist. Participants flagged for potential dementia by a neurologist were reviewed by our dementia review committee. Before making a diagnosis, the committee were required to consider all available evidence including neurologic and neuropsychological assessments, neuroimaging, medical records, and, if available, interviews with family or autopsy data. The review committee determined both type of dementia and date of onset.

Neuropsychological measures

We examined a battery of neuropsychological tests measuring different aspects of cognitive function. Neuropsychological testing was performed an average of 1.9 years after the quantification of vitamin D levels. The individual tests included Trail Making A and B, Logical Memory delayed, Visual Reproductions delayed, Similarities, and the Hooper Visual Organization Test. Trial Making A and B are speeded tasks in which participants have to link a series of scrambled numbers in ascending order (A) and alternate between ascending numbers and letters (B). Outcome scores were calculated as time to complete tasks B -A, with the variable transformed such that higher scores indicated superior task switching (executive control) and processing speed. The test of Similarities from the Wechsler Adult Intelligence Scale involves asking participants to indicate in which way two items are alike. Higher scores reflect better abstraction, reasoning, verbal comprehension, and categorization. Logical Memory delayed from the Wechsler Memory Scale (WMS) is a task of verbal memory with higher scores indicating better access encoding, storage, and retrieval of verbal memory. Visual reproductions delayed from the WMS is primarily a test of visual memory with higher scores reflecting better access encoding, storage and retrieval of visual memory. Finally, the Hooper Visual Organization Test measures visual analysis and synthesis with higher scores indicating better visuo-perceptual skills. Both Trail Making B-A and the Hooper Visual Organization test were natural log transformed to normalize their skewed distributions. All neuropsychological tests have been used previously and have adequate reliability and construct validity [START_REF] Strauss | A Compendium of Neuropsychological Tests: Administration, Norms, and Commentary[END_REF][START_REF] Au | New norms for a new generation: Cognitive performance in the framingham offspring cohort[END_REF]. More details of the neuropsychological protocol in the Framingham Heart Study Offspring cohort can be seen elsewhere [START_REF] Au | New norms for a new generation: Cognitive performance in the framingham offspring cohort[END_REF].

Structural brain measures

MRI was performed an average of 1.8 years after the quantification of vitamin D levels. MRI techniques used in the Framingham Heart Study Offspring cohort have been described previously [START_REF] Seshadri | Association of plasma total homocysteine levels with subclinical brain injury: Cerebral volumes, white matter hyperintensity, and silent brain infarcts at volumetric magnetic resonance imaging in the Framingham Offspring Study[END_REF][START_REF] Jeerakathil | Stroke risk profile predicts white matter hyperintensity volume: The Framingham study[END_REF]. A Siemens Magnetom 1-T field strength MRI machine was used. The images were analyzed by operators blind to serum 25(OH)D concentrations. The following MRI markers of subclinical brain aging were measured: total brain volume (TBV), hippocampal volume (HPV), white matter hyperintensity volume (WMHV), and presence of cover brain infarcts (CBIs). Brain volume was determined by manual outlining of the intracranial vault to determine the total cranial volume and by subsequent mathematical modeling to determine total brain parenchymal volume. We computed TBV as the ratio of total brain parenchymal volume to total cranial volume, thus correcting for differences in head size [START_REF] Seshadri | Association of plasma total homocysteine levels with subclinical brain injury: Cerebral volumes, white matter hyperintensity, and silent brain infarcts at volumetric magnetic resonance imaging in the Framingham Offspring Study[END_REF]. HPV was estimated using operator-defined, manually traced boundaries to define the region of interest. WMHV was determined according to previously published methods [START_REF] Jeerakathil | Stroke risk profile predicts white matter hyperintensity volume: The Framingham study[END_REF] and participants were categorized as having extensive WMHs if the log-WMHV was more than 1 SD above the age-adjusted mean in the cohort. The presence of CBIs was determined manually in accordance with STRIVE criteria [START_REF] Wardlaw | Neuroimaging standards for research into small vessel disease and its contribution to ageing and neurodegeneration[END_REF]. Such methods have been published previously, demonstrating adequate inter-rater and intra-rater reliabilities [START_REF] Seshadri | Association of plasma total homocysteine levels with subclinical brain injury: Cerebral volumes, white matter hyperintensity, and silent brain infarcts at volumetric magnetic resonance imaging in the Framingham Offspring Study[END_REF][START_REF] Decarli | Measures of brain morphology and infarction in the framingham heart study: Establishing what is normal[END_REF][START_REF] Decarli | Method for quantification of brain, ventricular, and subarachnoid CSF volumes from MR images[END_REF].

Covariates

A physician-administered medical history and examination were performed at the sixth and seventh Offspring examinations [START_REF] Kannel | An investigation of coronary heart disease in families. The Framingham offspring study[END_REF]. Vascular risk factors (systolic blood pressure, smoking, diabetes mellitus, and prevalent cardiovascular disease) were defined as in the Framingham Stroke Risk Profile [START_REF] Chobanian | Seventh report of the Joint National Committee on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure[END_REF]. BMI was defined as weight (in kilograms) divided by the square of height (in meters) [START_REF] Wolf | Relationofobesityto cognitive function: Importance of central obesity and synergistic influence of concomitant hypertension. The Framingham Heart Study[END_REF]. Homocysteine levels were determined as previously described [START_REF] Seshadri | Plasma homocysteine as a risk factor for dementia and Alzheimer's disease[END_REF], and log transformed. Information regarding vitamin D-containing supplement use was obtained with a detailed food-frequency questionnaire [START_REF] Rimm | Reproducibility and validity of an expanded self-administered semi quantitative food frequency questionnaire among male health professionals[END_REF]. Educational achievement was studied as a 4-class variable (no high school degree; successful completion of 12 years of education leading to high school degree but no college; some college; college degree).

Statistical analysis

The 25(OH)D measures within each cohort were converted to Z scores, since different assays were used in the two cohorts [START_REF] Rienstra | Vitamin D status is not related to development of atrial fibrillation in the community[END_REF]. Serum vitamin D levels were examined in 4 ways: 1) as a standardized continuous variable; 2) as a dichotomous variable based on the a priori deficiency cut-off value of <10 ng/mL [START_REF] Holick | Vitamin D deficiency[END_REF]; 3) as a dichotomous variable based on the 20th percentile of the 25(OH)D value of each cohort; and 4) as a dichotomous variable based on the 80th percentile of the 25(OH)D value of each cohort. The 20th and 80th percentile cutoffs of vitamin D levels were <19.0 and ≥40.0 ng/mL for the Original cohort dementia sample; <13.6 and ≥25.5 ng/mL for the Offspring cohort dementia sample; and <13.6 and ≥25.4 ng/mL for the combined neuropsychology and MRI Offspring cohort sample.

The association between baseline serum vitamin D and the risk of developing all cause dementia and AD was investigated using Cox proportional hazards regression. We examined the association between baseline serum vitamin D and the neuropsychological and MRI outcomes using multivariable linear and logistic regression for continuous and dichotomous measures, respectively. The effects of potential confounding variables were dealt with in 3 models. Model 1 adjusted for age and gender (and time from vitamin D measurement to cognitive testing/MRI acquisition for those outcomes, plus attained educational level for cognitive outcomes). Model 2 included additional adjustments for vascular risk factors (smoking, hypertension, diabetes, prevalent cardiovascular disease and homocysteine). Model 3 included further adjustment for BMI and self-reported vitamin supplement use. Analyses were performed with SAS software, version 9.2 (SAS Institute, Cary, NC). A 2sided probability value of <0.05 was considered statistically significant.

Sensitivity analysis

We performed a sensitivity analysis using the current Institute of Medicine guidelines of vitamin D deficiency (<12 ng/mL), insufficiency (12 to <20 ng/mL), and adequacy (20 to less than 50 ng/mL) [START_REF]Committee to Review Dietary Reference Intakes for Vitamin D and Calcium[END_REF]. Cox proportional hazards regression were used to separately compare the risk of incident dementia and AD for those with deficient and insufficient vitamin D levels, relative to those with adequate vitamin D levels.

RESULTS

The sample characteristics are summarized in Table 1. The mean age of the dementia cohort was older, given that cohort entry was restricted to participants 60 years of age or older at baseline. Mean 25(OH)D concentrations were 25.1±11.4 ng/mL for the dementia cohort and 19.8 ± 7.4 ng/mL for the cognitive/MRI outcome cohort, although not directly comparable given the aforementioned differences in their determination. Using the predetermined cutoff value of <10 ng/mL, 96 (6%) participants in the dementia cohort, and 104 (8%) in the cognitive outcomes cohort were vitamin D deficient.

Vitamin D and the incidence of dementia

Over a surveillance period of 9-years, 267 participants (16.1%) developed dementia of any type, out of which 208 (12.5%) fulfilled criteria for AD. 25(OH)D levels were not associated with dementia or clinically characterized AD in any of our adjusted models (Table 2).

Vitamin D and neuropsychological function

A statistically significant association was found between low serum vitamin D levels (examined as continuous, dichotomized as vitamin D deficient, or at the cohort-specific 20th percentile level) and poorer Trail Making B-A scores (Table 4). Low levels of vitamin D were also associated with poorer performance on the Hooper Visual Reproductions test. The strength of such associations was generally attenuated by the additional adjustment for BMI and vitamin D supplement use. Vitamin D levels were not associated with performance on any of the other neuropsychological tests.

Vitamin D and markers of subclinical brain aging on MRI

Participants who were vitamin D deficient were found to have smaller HPVs after adjustment for age, gender, and vascular risk factors (Table 3). Vitamin D was not associated with HPV when modeled as a continuous variable or according to cohort specific percentiles. There were no statistically significant associations between vitamin D and the other MRI variables.

Sensitivity analysis

Additional adjustment for geographic location and season did not alter the results (data not shown). Vitamin D was not associated with incident dementia or clinically apparent AD when defining thresholds of deficiency and insufficiency as per the Institute of Medicine guidelines (Supplementary Table 1).

DISCUSSION

This large community-based prospective cohort study examined the associations of vitamin D levels with incident dementia, MRI markers of brain aging and neuropsychological function. The primary finding was that low serum 25(OH)D concentrations were associated with domain-specific cognitive performance including executive function, processing speed and visuo-perceptual skills. 25(OH)D deficiency was also associated with lower HPV. There were no associations observed between vitamin D levels and incident dementia or clinically characterized AD.

To our knowledge, the Cardiovascular Health Study is the only other large prospective cohort study to report on the association between serum vitamin D concentrations and the risk of all-cause dementia and AD [START_REF] Littlejohns | Vitamin D and the risk of dementia and Alzheimer disease[END_REF]. Unlike the present study, the Cardiovascular Health Study found that vitamin D deficiency was predictive of incident dementia, including AD. Associations with dementia were evident when comparing those with adequate 25(OH)D levels (≥20 ng/ml) to those who were deficient (<10 ng/ml; the same threshold used in the current study) and to those who were deficient according to a more lenient threshold (≥10 to <20 ng/ml). Similarly, a study of 10,186 Danish individuals reported that reduced vitamin D levels were associated with dementia as determined by review of medical records [START_REF] Afzal | Reduced 25-hydroxyvitamin D and risk of Alzheimer's disease and vascular dementia[END_REF]. A study of 498 women found that higher dietary intake of vitamin D was associated with a lower 7-year risk of AD [START_REF] Annweiler | Higher vitamin D dietary intake is associated with lower risk of alzheimer's disease: A 7-year follow-up[END_REF]. However, in line with our study, a reanalysis of 40 women with available serum showed that, unlike vitamin D dietary intakes, serum 25(OH)D concentrations were not associated with the onset of AD after 7 years [START_REF] Annweiler | Serum vitamin D deficiency as a predictor of incident non-Alzheimer dementias: A 7-year longitudinal study[END_REF].

Although we did not find associations between vitamin D and incident dementia, we did find that low vitamin D concentrations were associated with poorer executive function, processing speed and visuo-perceptual skills, but not verbal or visual memory. Associations between vitamin D and similar aspects of cognitive function have been observed previously in some cross-sectional [START_REF] Buell | Vitamin D is associated with cognitive function in elders receiving home health services[END_REF][START_REF] Menant | Relationships between serum vitamin D levels, neuromuscular and neuropsychological function and falls in older men and women[END_REF] and prospective studies [START_REF] Llewellyn | Vitamin D and risk of cognitive decline in elderly persons[END_REF]. Consistent with other studies, we did not find an association between vitamin D and episodic memory [START_REF] Buell | Vitamin D is associated with cognitive function in elders receiving home health services[END_REF][START_REF] Jorde | Neuropsychological function in relation to serum parathyroid hormone and serum 25-hydroxyvitamin D levels. The Tromso study[END_REF][START_REF] Mcgrath | No association between serum 25hydroxyvitamin D3 level and performance on psychometric tests in NHANES III[END_REF][START_REF] Tolppanen | The association of serum ionized calcium and vitamin D with adult cognitive performance[END_REF]. The neuropsychological findings of the present study are also consistent with a recent metaanalysis, which found low vitamin D concentrations to be associated with poorer executive function and processing speed, while associations with episodic memory were reported to be uncertain [START_REF] Annweiler | Metaanalysis of memory and executive dysfunctions in relation to vitamin D[END_REF].

Our study found that vitamin D deficiency was associated with smaller HPVs. This association was not found to be linear, suggesting a specific threshold effect. Although hippocampal atrophy is a hallmark of early AD [START_REF] Risacher | Baseline MRI predictors of conversion from MCI to probable AD in the ADNI cohort[END_REF][START_REF] Bateman | Clinical and biomarker changes in dominantly inherited Alzheimer's disease[END_REF], hippocampal atrophy is also associated with vascular pathology [START_REF] Laakso | Hippocampal volumes in Alzheimer's disease, Parkinson's disease with and without dementia, and in vascular dementia: An MRI study[END_REF]. As vitamin D deficiency has been associated with the risk of cardiovascular disease in the Framingham Heart Study Offspring cohort [START_REF] Wang | Vitamin D deficiency and risk of cardiovascular disease[END_REF], the observed association between vitamin D deficiency and HPV may be mediated through vascular pathways. The associations identified between vitamin D deficiency and executive function, processing speed and visuo-perceptual skills are also consistent with an underlying vascular mechanism. In support of this notion, other studies have identified associations between vitamin D deficiency and the development of atherosclerosis [START_REF] Kassi | Role of vitamin D in atherosclerosis[END_REF][START_REF] Juonala | Childhood 25-OH vitamin D levels and carotid intima-media thickness in adulthood: The cardiovascular risk in young Finns study[END_REF], perhaps due to the effects of vitamin D on endothelial dysfunction, vascular smooth muscle cell proliferation and migration, as well as calcification [START_REF] Kassi | Role of vitamin D in atherosclerosis[END_REF].

There are numerous other neurobiological mechanisms that may explain the links between vitamin D, neuropsychological function, and HPV identified in the current study. Vitamin D receptors, and the enzyme essential for the production of vitamin D in its active form, are ubiquitous in the nervous system in regions of the brain associated with cognition, behavior and neurological disease, including the hippocampus [START_REF] Kalueff | Neurosteroid hormone vitamin D and its utility in clinical nutrition[END_REF][START_REF] Mccann | Isthere convincing biological or behavioral evidence linking vitamin D deficiency to brain dysfunction?[END_REF][START_REF] Eyles | Distribution of the vitamin D receptor and 1 alpha-hydroxylase in human brain[END_REF]. Vitamin D receptors are also expressed in immune cells, with vitamin D able to modify innate and adaptive immune responses, lessen inflammatory cytokine production and boost anti-inflammatory defense [START_REF] Cherniack | Hypovitaminosis D in the elderly: From bone to brain[END_REF][START_REF] Buell | Vitamin D and neurocognitive dysfunction: Preventing "D"ecline?[END_REF][START_REF] Aranow | Vitamin D and the immune system[END_REF]. These mechanisms are of interest given that neuroinflammation and immune system actions may contribute to AD [START_REF] Heppner | Immune attack: The role of inflammation in Alzheimer disease[END_REF]. Vitamin D also has a role in buffering antioxidant defenses [START_REF] Cherniack | Hypovitaminosis D in the elderly: From bone to brain[END_REF][START_REF] Buell | Vitamin D and neurocognitive dysfunction: Preventing "D"ecline?[END_REF], stimulating neurotrophin release [START_REF] Kiraly | Vitamin D as a neuroactive substance: Review[END_REF], and downregulating calcium channel expression [START_REF] Brewer | Vitamin D hormone confers neuro-protection in parallel with down regulation of L-type calcium channel expression in hippocampal neurons[END_REF][START_REF] Dursun | A novel perspective for Alzheimer's disease: Vitamin D receptor suppression by amyloid-beta and preventing the amyloid-beta induced alterations by vitamin D in cortical neurons[END_REF]. Vitamin D may attenuate amyloid-β (A β) accumulation by stimulating the phagocytosis of the Aβ peptide [START_REF] Masoumi | 1 alpha, 25dihydroxyvitamin D3 interacts with curcuminoids to stimulate amyloid-beta clearance by macrophages of Alzheimer's disease patients[END_REF] and by enhancing brain-to-blood Aβ efflux transport at the blood-brain barrier [START_REF] Ito | 1alpha,25-Dihydroxyvitamin D3 enhances cerebral clearance of human amyloid-beta peptide(1-40) from mouse brain across the blood-brain barrier[END_REF], with a decreased number of amyloid plaques as a result [START_REF] Yu | Vitamin D3enriched diet correlates with a decrease of amyloid plaques in the brain of AbetaPP transgenic mice[END_REF]. Conversely, vitamin D receptor knock-out mice manifest behavioral abnormalities [START_REF] Kalueff | Neurosteroid hormone vitamin D and its utility in clinical nutrition[END_REF], and reduced concentrations of the vitamin D hormone receptor mRNA have been directly linked with AD [START_REF] Sutherland | Reduction of vitamin D hormone receptor mRNA levels in Alzheimer as compared to Huntington hippocampus: Correlation with calbindin-28k mRNA levels[END_REF].

Vitamin D deficiency is prevalent in the US and other Western countries, particularly amongst non-Caucasians [START_REF] Forrest | Prevalence and correlates of vitamin D deficiency in US adults[END_REF][START_REF] Daly | Prevalence of vitamin D deficiency and its determinants in Australian adults aged 25 years and older: A national, population-based study[END_REF]. Our results suggest that maintaining adequate levels of vitamin D may potentially prevent the accumulation of vascular brain injury, protecting against hippocampal atrophy and executive dysfunction. As vitamin D deficiency can be corrected through supplementation, diet (i.e. oily fish), and increased sun exposure, there is a need to ascertain whether correcting vitamin D deficiency can help mitigate the risk of vascular brain injury and cognitive decline. Should a causal link exist between low vitamin D status and accelerated brain aging, routine screening and treatment for vitamin D deficiency, particularly in high risk populations, may help lower the burden on cognitive impairment.

This study has several strengths including that the sample was relatively large and community-based. Furthermore, the prospective follow-up for dementia allowed us to assess the temporal relationship between vitamin D concentrations and the risk of developing dementia over a 9-year follow-up period. Our study also has limitations. Relatively few participants in our study were found to be deficient in vitamin D. We thus modeled vitamin D deficiency, not only according to a cut-off associated with deficiency, but also according to cohort specific percentiles. As we did not evaluate the concentrations of parathyroid hormone, we cannot determine whether the associations between vitamin D concentrations and neuropsychological function are due to the low vitamin D or to secondary hyperparathyroidism [START_REF] Buell | 25-Hydroxyvitamin D, dementia, and cerebrovascular pathology in elders receiving home services[END_REF], although repletion with vitamin D would be indicated in both cases. The fact that our sample was largely white Caucasian limits the generalizability of our findings to other ethnic groups. Finally, our data is observational and despite several adjustments, one cannot exclude possible residual confounding.

In conclusion, identifying nutritional and lifestyle-based factors that can prevent or ameliorate cognitive decline may have substantial benefits for the individual and noteworthy financial and public health ramifications for society. The current study adds support to a growing body of literature implicating vitamin D deficiency in the development of neuropsychological impairment and subclinical brain injury. Correcting vitamin D deficiency through diet, lifestyle change or supplementation may potentially help prevent brain injury and cognitive decline. However, we did not find an association between levels of vitamin D and the risk of dementia or clinically characterized AD in the Framingham Heart Study. The contribution of vitamin D to dementia thus remains uncertain and requires future investigation. Elucidating why vitamin D appears to be linked to dementia in some cohorts but not others will provide important information for future clinical trials aimed at developing dietary based interventions to protect against cognitive decline and dementia. Selection of study participants. MRI, magnetic resonance imaging.

Fig. 1 .

 1 Fig. 1.

Table 1

 1 Characteristics of the dementia, cognitive and imaging analysis cohorts

	Variable	Dementia cohort	Cognitive and MRI
		(n = 1663)	cohort (n= 1291)
	Clinical and demographic
	Age, years	72.4 ± 6.7	59.5± 9.1
	Women, n (%)	973 (59)	703 (55)
	Education, n (%)
	No high school degree	308 (19)	37 (3)
	High school degree	598 (37)	380 (29)
	Some college	369 (23)	401 (31)
	College graduate	345 (21)	473 (37)
	25(OH)D, ng/mL	25.1± 11.4	19.8± 7.4
	25(OH)D <10 ng/mL, n (%)	96 (6)	104 (8)
	Vitamin D supplement use, n (%)	538 (34)	571 (48)
	Body mass index, kg/m 2	27.3 ± 4.7	27.9± 5.3
	Smoking, n (%)	173 (10)	172 (13)
	Diabetes, n (%)	214 (13)	120 (9)
	Hypertension, n (%)	1123 (68)	477 (37)
	History of CVD, n (%)	404 (24)	98 (8)
	Homocysteine *	10 (8,13)	8 (7,10)
	Cognitive measures
	Mean time to cognitive assessment		1.88 (1.31)
	Logical Memory Delayed, score	-	10.55± 3.54
	Visual Reproductions Delayed, score	-	8.19 ±3.37
	Trail making, B -A score *	-	0.67 (0.47,1.00)
	Similarities, score	-	16.75± 3.49
	Hooper Visual Reproductions, score *	-	25.5 (23.5,27.0)
	MRI measures
	Mean time to MRI assessment		1.85 (1.30)
	Total brain volume, %	-	79.78± 3.29
	Hippocampal volume, %	-	0.34 ±0.06
	WMHV, % *	-	0.04 (0.02,0.08)
	Extensive WHMV, n (%)	-	147 (12.9%)
	Covert brain infarcts, n (%)	-	133 (11.7%)
	Mean ±SE presented, unless specified otherwise. CVD, cardiovascular disease; WMHV, white matter hyperintensitiy volume.

Table 2

 2 Cox adjusted regressions depicting the association between 25(OH)D levels and the risk of all-cause dementia and clinically characterized Alzheimer's disease

	Model	All-cause dementia	Alzheimer Disease
		HR (95% CI)	p	HR (95% CI)	p
	25(OH)D as standardized continuous variable
	1	0.96 (0.85-1.09)	0.512	1.03 (0.90-1.19) 0.660
	2	1.02 (0.89-1.16)	0.777	1.10 (0.95-1.27) 0.211
	3	1.01 (0.88-1.16)	0.926	1.10 (0.94-1.29) 0.223
	25(OH) D <10 ng/mL
	1	1.32 (0.76-2.28)	0.328	1.17 (0.61-2.23) 0.642
	2	1.06 (0.57-1.98)	0.847	0.97 (0.47-2.00) 0.930
	3	1.01 (0.51-2.00)	0.983	0.84 (0.37-1.94) 0.689
	25(OH)D < the cohort-specific 20th percentile
	1	0.96 (0.71-1.29)	0.763	0.79 (0.56-1.13) 0.203
	2	0.86 (0.63-1.17)	0.332	0.70 (0.49-1.02) 0.064
	3	0.87 (0.63-1.21)	0.412	0.71 (0.48-1.04) 0.080
	25(OH)D > the cohort-specific 80th percentile
	1	0.89 (0.64-1.23)	0.475	1.10 (0.77-1.55) 0.608
	2	0.98 (0.70-1.37)	0.910	1.19 (0.83-1.71) 0.353
	3	0.93 (0.66-1.33)	0.696	1.14 (0.78-1.66) 0.503
	Hazards ratios for incident all-cause dementia and Alzheimer's disease are shown per cohort-specific 1-SD increment in 25(OH)D levels, for the
	presence of a 25(OH)D concentration <10 ng/mL, for<cohort-specific 20th percentile, and for> the cohort-specific 80th percentile. Model 1
	includes adjustment for age and gender. Model 2 includes additional adjustment for vascular risk factors (smoking, hypertension, diabetes,
	prevalent cardiovascular disease, and homocysteine). Model 3 includes additional adjustment for BMI and vitamin D supplement use.

Table 3

 3 Multivariable adjusted models depicting the association between 25(OH)D levels and the MRI variables HPV, hippocampal volume; SBI, silent cerebral infarcts; TBV, total brain volume; WMHV, white matter hyperintensity volume. Regression coefficients represent the change in mean of each standardized MRI variable per cohort-specific 1-SD increment in 25(OH)D levels; or by the presence of a 25(OH)D concentration <10 ng/mL; < cohort-specific 20th percentile; or > the cohort-specific 80th percentile. Model 1 adjusts for age, gender, and time from vitamin D measurement to MRI. Model 2 adjusts for the addition of vascular risk factors (smoking, hypertension, diabetes, prevalent cardiovascular disease, and homocysteine). Model 3 adjusts for the addition of BMI and vitamin D supplement use.

	Model TBV HPV WMHV WMH (extensive) SCI	β ±SE p β±SE p β ±SE p OR (95% CI) p OR (95% CI) p	25(OH)D as standardized continuous variable	1 0.14 ± 0.08 0.075 0.003 ±0.002 0.103 -0.03± 0.03 0.212 1.00 (0.84-1.18) 0.973 0.85 (0.71-1.04) 0.108	2 0.09 ± 0.08 0.268 0.003 ±0.002 0.114 -0.03± 0.03 0.212 1.00 (0.84-1.20) 0.965 0.90 (0.74-1.09) 0.291	3 -0.002± 0.09 0.984 0.002 ±0.002 0.299 -0.03± 0.03 0.295 1.02 (0.85-1.23) 0.815 0.91 (0.74-1.12) 0.374	25(OH) D <10 ng/mL	1 -0.34 ±0.29 0.249 -0.01 ±0.01 0.033 -0.004± 0.09 0.964 1.00 (0.53-1.89) 0.995 1.34 (0.73-2.48) 0.349	-0.03± 0.10 0.778 0.96 (0.50-1.84) 0.896 1.12 (0.60-2.11) 0.721 2 -0.16 ±0.29 0.591 -0.01 ±0.01 0.034	-0.05± 0.10 0.621 0.90 (0.45-1.81) 0.773 1.18 (0.61-2.27) 0.629 3 0.04 ± 0.31 0.889 -0.01 ±0.01 0.058	25(OH)D < the cohort-specific 20th percentile	0.01 ±0.06 0.869 0.88 (0.56-1.39) 0.585 1.03 (0.65-1.63) 0.908 1 -0.36 ±0.20 0.073 -0.01 ±0.004 0.111	0.01 ±0.07 0.829 0.89 (0.56-1.41) 0.619 0.92 (0.57-1.46) 0.717 2 -0.23 ±0.20 0.248 -0.01 ±0.004 0.112	-0.01± 0.07 0.890 0.88 (0.53-1.43) 0.595 0.89 (0.54-1.47) 0.652 3 -0.09 ±0.22 0.687 -0.005 ±0.005 0.296	25(OH)D > the cohort-specific 80th percentile	-0.07± 0.06 0.277 0.91 (0.59-1.41) 0.676 0.87 (0.53-1.41) 0.557 1 0.28 ± 0.20 0.163 0.002 ±0.004 0.633	-0.08± 0.07 0.207 0.88 (0.56-1.38) 0.577 0.92 (0.56-1.50) 0.736 2 0.18 ± 0.20 0.362 0.002 ±0.004 0.713	-0.08± 0.07 0.226 0.94 (0.59-1.50) 0.798 0.97 (0.59-1.61) 0.904 3 0.05 ± 0.21 0.822 -0.000 ±0.004 0.921	Bold highlight denotes p< 0.05;

J Alzheimers Dis. Author manuscript; available in PMC 2016 June 17.

* Median (25th percentile, 75th percentile) reported. J Alzheimers Dis. Author manuscript; available in PMC 2016June 17.

Acknowledgments

This work was supported by the Framingham Heart Study's National Heart, Lung, and Blood Institute contract (N01-HC-25195) and by grants from the National Institute of Neurological Disorders and Stroke (R01 NS17950) and from the National Institute on Aging (R01 AG016495, AG008122, AG033193, AG031287). Dr Pase is funded by an Australian National Health and Medical Research Council (NHMRC) Early Career Fellowship (APP1089698).

Supplementary Material

Refer to Web version on PubMed Central for supplementary material.

Author Manuscript

Karakis et al.

Page 17

Table 4

Multivariable adjusted models depicting the association between 25(OH)D levels and neuropsychological function


Félicien Schopfer

François Mallet

Dominique Mailly

Christophe Texier

Gilles Montambaux

Christopher Bäuerle

Laurent Saminadayar

Dimensional crossover in quantum networks: from macroscopic to mesoscopic Physics

Keywords: numbers: 73.23.-b, 03.65.Bz, 75.20.Hr, 72.70.+m, 73.20.Fz

We report on magnetoconductance measurements of metallic networks of various sizes ranging from 10 to 10 6 plaquettes, with anisotropic aspect ratio. Both Altshuler-Aronov-Spivak (AAS) h/2e periodic oscillations and Aharonov-Bohm (AB) h/e periodic oscillations are observed for all networks. For large samples, the amplitude of both oscillations results from the incoherent superposition of contributions of phase coherent regions. When the transverse size becomes smaller than the phase coherent length L φ , one enters a new regime which is phase coherent (mesoscopic) along one direction and macroscopic along the other, leading to a new size dependence of the quantum oscillations.

Quantum interference effects lie at the heart of mesoscopic physics. It is well known that they govern both thermodynamic as well as electronic transport properties of quantum conductors. One of the most spectacular manifestations of such quantum interferences is the Aharonov-Bohm effect [START_REF] Aharonov | [END_REF] in a mesoscopic ring whose perimeter is of the order of the phase coherence length L φ : when applying a magnetic flux through the ring, the conductance oscillates with a periodicity φ 0 = h/e, the flux quantum, h being the Planck constant and e the charge of the electron [2]. Such a magnetoconductance oscillation is a direct consequence of the coupling of the electron charge to the vector potential, and is thus the most direct evidence of the quantum nature of the conduction in mesoscopic systems [3].

An important point is the understanding of how such quantum effects disappear when going from mesoscopic to macroscopic conductors. If one considers lines of N mesoscopic metallic rings, the Aharonov-Bohm (AB) conductance oscillations δG AB /G vanishes to zero as 1/ √ N . This has been beautifully demonstrated by studying lines of silver rings with N varying from 1 to 30 [4].

On the other hand, there exist magnetoconductance oscillations which do survive such an ensemble averaging, since they are due to interferences between time reversed trajectories. These oscillations ∆G AAS , known as Altshuler-Aronov-Spivak (AAS) oscillations, have a period φ 0 /2 [5,6]. The robustness of these oscillations towards ensemble averaging, as opposed to the AB oscillations was also experimentally demonstrated in ref. [4]. The relative amplitude ∆G AAS /G was found to be independent of N . This robustness has also been demonstrated in large two-dimensional metallic networks of dif-ferent topologies [7,8]. It must be stressed that all these experiments have been carried out in a regime where the phase coherence length L φ is much smaller than the system size. In this context, one deals with the simple case of an ensemble averaging consisting in a summation of uncorrelated contributions from phase coherent regions.

A crucial question is to know what happens to the ensemble averaging when the system size decreases and becomes of the order or smaller than the phase coherence length L φ . In this Letter, we report on the size dependence of the amplitudes of both Aharonov-Bohm and Altshuler-Aronov-Spivak magnetoconductance oscillations in silver networks of anisotropic aspect ratio. We show that the amplitude of both AB and AAS oscillations exhibit an unexpected dependence with N when the smallest dimension of the network becomes smaller than the phase coherence length: in this case, the network can be considered as a fully coherent object (mesoscopic) in one direction, whereas macroscopic in the other.

Samples are fabricated on a silicon substrate using electron beam lithography on polymethyl-methacrylate resist. Silver is deposited from a 99.9999 % purity source using an electron gun evaporator and lift-off technique without any additional adherence layer. All samples have been evaporated in a single run to ensure that the sample characteristics (elastic mean free path l e and phase coherence length L φ ) are similar. Two different topologies have been studied in this work: the square lattice and the so-called T 3 lattice [9]. The wires forming the networks are 60 nm wide, 50 nm thick and 640 nm (690 nm) long for the square (T 3 ) lattice. The size of the plaquettes (square or diamond) is chosen such that the magnetic field corresponding to one flux quantum φ per plaque-tte is B = 100 G. All networks with number of plaquettes N varying from 10 to 10 6 have the same aspect ratio L x /L y = 10 (see figure 1). As a consequence, their resistances are similar and of the order of 100 Ω. Measurements have been performed at 400 mK; this allows to stay in the linear regime with a relatively high current (≈ 4 nA) and optimizes the signal to noise ratio without heating the electrons. At this temperature, the phase coherence length, determined from standard weak localization measurements on a 120 nm wide wire fabricated on the same wafer, is about L φ ≃ 6 µm, the diffusion constant D ≃ 105 cm 2 s -1 and the thermal length L T = hD/k B T ≃ 0.45 µm [10]. In figure 2 we show typical data for the magnetoresistance of a square network with 3000 plaquettes. At low field (figure 2a), oscillations with a period B = 50 G, corresponding to φ 0 /2 per plaquette are identified as the AAS oscillations. At fields typically higher than the field which suppresses weak localization, we observe a different type of oscillations. These oscillations have a periodicity of B = 100 G, corresponding to φ 0 ; these are AB oscillations. In order to emphasize the different periodicity of these magnetoconductance oscillations, we display their Fourier spectra in figures 2c (low field) and 2d (high field): in the high field regime, the main peak clearly appears at 0.01 G -1 , whereas in the low field regime, it appears at 0.02 G -1 . To our knowledge, this is the first time that both AAS and AB oscillations are observed on such large samples.

We now concentrate on the variation of the amplitude of the AB as well as AAS oscillations versus the number of plaquettes N . To measure the AB oscillations we sweep the magnetic field from 7000 G to 13000 G, whereas for the AAS oscillations we cover a field range of ±1200 G. To extract precisely the amplitude of the AB oscillations, we take the Fourier transform over 20 periods after sub- traction of a smooth background to remove low frequency fluctuations. We also measure the background noise by repeating the measurement exactly in the same conditions but at fixed magnetic field, and taking again the Fourier transform. The amplitude of the AB signal is then obtained from the Fourier spectrum after subtraction of the background spectrum integrated over the same frequency range, in a similar way used for persistent current measurements [11]. This procedure is only necessary for very large networks (typically larger than 10 5 plaquettes) since for smaller networks the noise is negligible. For the determination of the AAS amplitude such a procedure is not necessary, as the background noise is always negligible. However, the second harmonic (φ 0 /2) of the AB oscillations has the same frequency as the first harmonic (φ 0 /2) of the AAS oscillations. For small networks (typically N ≤ 100) this contribution cannot be neglected. In order to extract the AAS signal, we therefore determine first the amplitude of the second harmonic of the AB oscillations at high field and then subtract this amplitude from the first harmonic of the oscillations measured at low field [START_REF]) between the AAS oscillations and the second harmonic of the AB oscillations[END_REF].

In figure 3 we display the amplitude of magnetoconductance oscillations (AAS and AB) extracted from the Fourier spectra as a function of the number N of plaquettes. For large networks (N > ∼ 300), the amplitude of the AB oscillations clearly decreases as 1/ √ N , whereas the amplitude of the AAS oscillations are independent on the number of plaquettes as naively expected. More surprising is the behavior observed for small networks: when they contain typically less than N ≃ 300 plaquettes, the amplitude of the AB oscillations varies faster than 1/ √ N . At the same time the AAS amplitude now de- pends on N (figure 3). In the following, we will show that this new behavior results from a dimensional crossover when the transverse size of the network becomes smaller than the phase coherence length: one then enters a new regime where the transport properties are effectively onedimensional on the two-dimensional network.

Let us first recall general considerations on the magnetoconductivity oscillations: on one hand, the AAS oscillations are the Fourier harmonics of the weak localization correction ∆σ(B) to the average conductivity [5]. On the other hand, the amplitude of the AB oscillations can be obtained from the correlation function of the conductivity δσ(B)δσ(B ′ ) . The expressions of these two quantities are indeed related [14,15,16,[START_REF] Akkermans | Physique mésoscopique des électrons et des photons[END_REF]. In the limit where the thermal length L T is smaller than L φ , this yields

δσ 2 AB = e 2 h 4πL 2 3 Vol ∆σ AAS (1)
where δσ AB and ∆σ AAS are respectively the first harmonics of the AB and AAS oscillations and Vol the volume of the sample. In this formula, the temperature dependence originates from the conductivity correlation function which probes a finite energy scale of width k B T [3,[START_REF] Akkermans | Physique mésoscopique des électrons et des photons[END_REF]. The key feature is the proportionality between δσ 2 AB and ∆σ AAS . Indeed, both quantities can be written in terms of the coherent part of the return probability to the origin for a diffusive particle. Consequently both must probe in the same way the influence of the geometry [START_REF] Akkermans | Physique mésoscopique des électrons et des photons[END_REF].

We consider a network of dimensions L x × L y (see figure 1). It is important to keep in mind that experiments presented here are performed on several networks of different sizes, but of constant aspect ratio L x /L y = 10. The length and width of the networks thus scale with the number of plaquettes N as

L x ∝ L y ∝ √ N .
The dimensionless conductance g = G/(2e 2 /h) of the network is then related to the conductivity by Ohm's law g ∝ σL y /L x . Combined with eq. ( 1) and given that Vol ∝ L x L y , this yields for the amplitudes of the conductance oscillations ∆g AAS and δg AB :

δg 2 AB = 2πL 2 T 3L 2 x ∆g AAS (2)
which is the key relation from which we now discuss our results, bearing in mind that temperature, and thus L T and L φ are fixed parameters.

Let us first consider large networks with both dimensions larger than the phase coherence length: L x , L y ≫ L φ . Since interfering time reversed trajectories extend over a typical size L φ , they do not feel the boundaries of the system and therefore ∆σ is size independent. Therefore the AAS amplitude varies as ∆g AAS ∝ L y /L x and since this ratio is constant, this amplitude is independent on N :

∆g AAS ∝ N 0 (3)
In this regime we also see from eq. ( 2) that δg 2 AB ∝ L y /L 3

x , which leads to:

δg AB ∝ N -1/2 (4)
This is exactly what is observed for large networks: when the number of plaquettes is larger than ≃ 300, electrons diffuse on what they feel as a two-dimensional network.

For smaller networks, the transverse dimension L y eventually becomes smaller than the phase coherence length: we enter a regime where the network becomes transversally coherent whereas it remains longitudinally incoherent: L y ≪ L φ ≪ L x . In this case, we have the usual quasi-1D scaling ∆σ AAS ∝ L φ /L y . Therefore we find ∆g AAS ∝ 1/L x and δg 2 AB ∝ 1/L 3 x , which leads to:

∆g AAS ∝ N -1/2 (5) δg AB ∝ N -3/4 (6)
This is precisely what is observed for small networks on figure 3.

It remains now to check whether the position of the crossover observed on figure 3 agrees with our estimate of the phase coherence length. The crossover occurs for a size N ≃ 300 corresponding to L y ≃ 3.8 µm. This length has to be compared with the coherence length L φ ≃ 6 µm measured at T = 400 mK. This comparison, which cannot be more than qualitative, supports our analysis.

To summarize, the dimensional crossover observed for the scaling of the AB oscillations corresponds to the different scaling L y /L 3

x → L φ /L 3 x of the variance of the conductance fluctuations, with N ∝ L x L y . At this point, it is useful to compare these dependences with the case of a 1D chain where the number N of rings scales linearly with the length L x of the chain, so that δg 2 AB ∝ 1/L 3 x ∝ 1/N 3 and ∆g AAS ∝ 1/N . Since the conductance g scales as 1/N , this yields for the relative fluctuations ∆g AAS /g ∝ N 0 and ∆g AB /g ∝ 1/ √ N as was observed experimentally [4].

An interesting way of checking our analysis comes from equation ( 2): we can see that the ratio δg 2 AB /∆g AAS ∝ L 2 T /L 2 x is proportional to 1/N and more importantly is independent on L φ . This fundamental relation between δg 2 AB and ∆g AAS is clearly shown on figure 4, where we have plotted the ratio δg 2 AB /∆g AAS as a function of the number of plaquettes N : one sees that it follows perfectly the predicted 1/N behavior, with no dimensional crossover. This is a definitive check of our interpretation of the experimental data in terms of dimensional crossover.

In conclusion, we have measured both Aharonov-Bohm φ 0 periodic oscillations and Altshuler-Aronov-Spivak φ 0 /2 periodic oscillations in metallic networks containing 10 to 10 6 plaquettes. Ensemble averaging can lead to different size dependences for small and large networks. The crossover takes place when the width of the network is of the order of the phase coherence length; this behavior does correspond to a dimensional crossover between effectively one-and two-dimensional networks. In this new one-dimensional regime we observed, we have shown that the amplitude of the AB oscillations varies as N -3/4 and the AAS oscillations as N -1/2 , a behavior which has never been observed up to now. Moreover, we have been able to probe experimentally the fundamental relation between AB and AAS magnetoconductance oscillations due to their common physical origin.

FIG. 1 :

 1 FIG. 1: Scanning Electron Micrograph of several samples of various sizes; the two contacts are visible for the small sample.

FIG. 2 :

 2 FIG. 2: (color online). Magnetoresistance of a square network containing 3000 plaquettes: a) low field data; b) high field data; c), d) Fourier amplitudes of a), b) respectively.

FIG. 3 :

 3 FIG.3:(color online). AAS amplitude ∆gAAS and AB amplitude δgAB as a function of the number N of plaquettes for different networks of various sizes Lx × Ly ∝ N , for two topologies, square (squares) and T3 (diamonds)[START_REF]Note that the amplitudes of these magnetoconductance oscillations depend also[END_REF].

FIG. 4 :

 4 FIG. 4: (color online). ∆gAAS/(δg AB ) 2 as a function of the number of plaquettes N for square networks (squares) and diamonds networks (diamonds).

We are indebted to the Quantronics group for the use of its evaporator and silver source. It is our pleasure to acknowledge H. Bouchiat, B. Douçot, L. P. Lévy and J. Vidal for fruitful discussions. This work has been supported by the French Ministry of Science, grants # 02 2 0222 and # NN/02 2 0112, and the European Commission FP6 NMP-3 project 505457-1 "Ultra-1D".


Vinícius Pascotto Gastaldo
email: vinicius.gastaldo@ufms.br

Mala N Rao

Alexey Bosak

Matteo D'astuto

Andrea Prodi

Marine Verseils

Yannick Klein

Christophe Bellin

Luigi Paolasini

J A Adilson

De Oliveira

Vinícius Pascotto Gastaldo

Adilson J A De Oliveira

Edmondo Gilioli

Samrath Lal Chaplot

Andrea Gauzzi
email: andrea.gauzzi@sorbonne-universite.fr

Direct observation of Jahn-Teller critical dynamics at a charge-order Verwey transition

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The interplay of electron and lattice dynamics beyond the adiabatic approximation is at the forefront of research in the study of many-body systems, such as chemical reactions [1], ionization processes [2] and polaron transport [3]. Favorable conditions to investigate the non-adiabatic regime are usually found in mixed-valence systems, since charge fluctuations drive a strong electron-lattice coupling. One fascinating situation occurs when these fluctuations stabilize a static order of valence electrons. In a seminal paper [4], Verwey proposed that this situation is indeed realized as a static Fe 2+ /Fe 3+ charge order (CO) in the archetypal mixed-valence system magnetite Fe 3 O 4 , which would explain a pronounced structural distortion concomitant to a jump of the electrical resistivity at T V = 120 K, known as Verwey transition. Verwey's picture has been subsequently questioned because a simple ionic description does not account for the complex structural changes occurring at T V [5][6][7][8][9][10][11] and an alternative picture of polarons involving the dynamics of 3 Fe sites (trimerons) has been proposed [12,13]. Subsequent studies of the critical dynamics at the transition have given evidence of electronic [14] but no phonon [13,15] mode softening, which leaves open the question of the cooperative role of the electronic and lattice degrees of freedom on the transition.

A similar phenomenology is found in other mixed-valence systems like chemically doped manganites and nickelates with perovskite-like structure [16][17][18]. Also in these systems, a static CO picture seems to be oversimplified due to disorder and electronic inhomogeneities inherent to chemically substituted compounds and alternative scenarios of dynamic charge fluctuations like Zener-polarons [19] have been proposed. In order to elucidate the above controversial points, here we investigate the Jahn-Teller (JT) dynamics governing the interplay of charge, orbital and spin orderings in mixed-valence transition-metal compounds, as discussed in the seminal papers by Goodenough [20] and by Wollan and Koehler [21] and extensively studied later [22][23][24]. The challenge is that a reliable investigation of this dynamics is hindered in most mixed-valence systems by the coexistence of charge-ordered and -disordered phases [24], incommensurate structural modulations like stripe phases [25] and electronic phase separation [26].

In the present work, we show that the above difficulty is overcome in the mixed-valence compound (NaMn 3 )Mn 4 O 12 (NaMnO) [27]. Similar to magnetite and manganites, at low temperatures, T CO =175 K, NaMnO exhibits a Mn 3+ /Mn 4+ CO transition of the Verwey type consisting of a cubic Im 3 to monoclinic I2/m structural transition accompanied by a large jump of the electrical resistivity [28]. The unique characteristics of the quadruple perovskite structure (AA 3 )B 4 O 12 allow to obtain an equal proportion of Mn 3+ and Mn 4+ ions in the octahedral B-sites without chemical substitutions. In addition, the peculiar four-fold coordination of the Mn 3+ ions in the A sites avoids the formation of oxygen defects. It turns out that (NaMn 3 )Mn 4 O 12 exhibits an almost full charge order of the B-site Mn 3+ and Mn 4+ ions concomitant to a zigzag ordering of the e g 3d z 2 orbitals of the Mn 3+ ions and leading to a comparatively simple structural modulation with commensurate propagation vector q CO = 1 2 , 1 2 , 0 [29]. The above favorable conditions prompt us to probe the lattice dynamics governing the CO Verwey transition in NaMnO.

We first searched for structural anomalies at T CO by diffuse x-ray scattering (DS), a suitable probe of static short-range structural correlations precursor of a structural transition. As reported on magnetite [30], these correlations produce anomalous DS intensities different from ordinary thermal diffuse scattering. Second, we investigated dynamic anomalies by inelastic x-ray scattering (IXS) at T CO . Owing to the low-Z of Na, the scattered intensities mainly arise from the Mn and O atoms playing the dominant role in the lattice dynamics, a further favorable condition for both experiments.

II. EXPERIMENTAL DETAILS AND LATTICE DYNAMICAL CALCULATIONS

NaMnO single crystals were synthesized by high-pressure synthesis using a multi-anvil apparatus, as described elsewhere [31]. Prior to the DS and IXS experiments, two untwinned single crystals selected from the same batch were mechanically made into the shape of a needle and etched down to 50 µm diameter with HCl in order to remove the damaged surface layer. The crystals were subsequently glued on a capillary and oriented with the [001] direction of the cubic lattice perpendicular to the scattering plane. For the DS measurements, the samples were submitted to shutterless exposure with wavelength λ=0.715 Å at the ID28 beamline of the European Synchrotron Radiation Facility (ESRF). Scattered intensities were measured using a PILATUS 2M detector, as described elsewhere [32]. The rotation axis was set perpendicular to the beam and parallel to its polarization. The resolution was 0.1 • per image and the exposure time was 0.6 s for each image. The data were analyzed using the CrysAlis software package and reconstructed using ESRF home-made software. Laue symmetry was applied to the data. The IXS experiment was performed at the ID28 beamline using a (999) Si monochromator line of wavelength λ=0.6968 Å, corresponding to an energy resolution of 3.0 meV. Constant-Q energy scans were collected in transmission mode with Q ⊥ [001].

For both experiments, the sample temperature was controlled using a liquid-nitrogen cryostream apparatus.

Lattice dynamical calculations for the cubic Im 3 phase of NaMnO were carried out using a shell model with pair-wise interionic interaction potential that includes short-range and long-range Coulomb terms. The parameters of the potential satisfy the conditions of static and dynamic equilibrium. The calculations, carried out using the current version of the DISPR software [33], include total energy calculations of the crystal structure, the phonon dispersion relation and the IXS cross section for each phonon mode. We computed the onephonon IXS cross section for a given momentum Q and energy ω transfer using the expression:

S(Q, ω) = A q,j 2ω(q, j) n [ω(q, j)] + 1 2 ± 1 2 • • |F j (Q)| 2 δ (Q -G ∓ q) δ [ω ± ω(q, j)] (1)
where A is a proportionality constant and the sum is ex-tended over all phonon modes of wave vector q, energy ω and branch j, n [ω(q, j)] = exp ω(q,j)

k B T -1 -1
is the Bose-Einstein distribution function for each mode, the sign ± indicates anti-Stokes and Stokes processes, respectively, G = Q ∓ q is a reciprocal lattice vector and F j (Q) is the dynamical structure factor:

F j (Q) = N k=1 f k (Q) Q • e(q, j, k) √ m k e iG•R k e -W k (Q) (2)
where the sum is extended over all N atoms in the unit cell, f k , m k , e(q, j, k), R k and W k are the form factor, mass, normalized eigenvector, vector position and Debye-Waller factor of k-th atom and q and j are the phonon wave vector and branch, as above.

III. DIFFUSE X-RAY SCATTERING RESULTS

By approaching gradually T CO from room temperature, we expected to observe anomalous features in the DS data at the propagation vector q CO of the CO phase, which would reflect the incipient CO transition. Surprisingly, we instead found anomalous features at completely different wave vec-

tors Q = (h ± 1 3 , k ± 1 3 , 0).
As seen in Figs. 1, 2a-b and 3ab, these features appear well above T CO and become clearly visible already at ≈ 230 K; they consist of X-or bar-shaped clouds visible especially around the most intense (4m, 0, 0) and (2m + 1, 2n + 1, 0) peaks, respectively, and equivalent peaks of the cubic Im 3 phase (m, n integers). This observation is accompanied by a pronounced peak broadening along the [110] and [1 10] directions. The DS intensity profiles along these directions evolve continuously upon cooling down from room temperature. As seen in Fig. 3c, the tail of the Bragg peak is progressively enhanced until well-defined satellite peaks appear at Q = (h± 1 3 , k± 1 3 , 0) in the vicinity of T CO . The second-order satellites at Q = (h ± 2 3 , k ± 2 3 , 0) are also visible. This result indicates an incipient structural modulation with propagation vector q DS = ( 13 , 1 3 , 0). Upon further cooling, the intensity of the DS anomalies progressively increases until they abruptly disappear at T CO (see Fig. 1), concomitant to the appearance of the satellite peaks of the CO structural modulation at q CO . This indicates that the instability at q DS is suppressed by the CO instability at q CO .

IV. INELASTIC X-RAY SCATTERING RESULTS

In order to unveil the dynamics of this latent instability, we measured IXS spectra at Q = G 400 + q = (4 ± ξ, 0, 0) and Q = G 440 + q = (4 ± ξ, 4 ± ξ, 0), i.e. along the [010] (∆) and [110] (Σ) directions crossing the X-and bar-shaped DS anomalous features of Fig. 3a. We focused on the low-energy ±30 meV range, which enabled us to investigate the quasielastic (QE) peak and the longitudinal and mixed longitudinaltransverse acoustic phonons. Data were taken at 177 K, i.e. just above T CO , and at 207, 237 and 297 K. T CO was precisely determined to be 175 K by cooling the sample until the satellite peaks of the CO phase appeared and by reducing the cooling rate until the thermal hysteresis disappeared. Representative spectra are shown in Fig. 4. The mode energies were determined by fitting the experimental phonon peaks using pseudo-Voigt functions. The Stokes and anti-Stokes lines of the acoustic phonons were fitted simultaneously by imposing the constraint set by the Bose factor for the intensities. Elastic lines were analyzed using a sum of Gaussian and Lorentzian line shapes to account for the contributions of the experimental resolution and of quasi-elastic scattering, respectively.

The main result is an enhanced intensity of the QE peak upon cooling down to T CO , similar to the case of incommensurate orbital ordering transition in the related compound (CaMn 3 )Mn 4 O 12 [34]. Interestingly, in the present case, the QE intensity progressively decreases by moving away from the above (hk0) peaks, except for a sizable enhancement at Q = G hk0 + q DS (see Fig. 4f), consistent with the behavior of the DS intensity profile of Fig. 3c. Upon cooling, no anomaly is seen in the mode dispersion at q DS (see Fig. 4e), which is again consistent with the DS result that the instability at q DS is suppressed at T CO .

In the search for anomalous modes at the verge of the CO transition, in a second IXS experiment we measured the spectra along the same ∆ and Σ directions as before in an extended energy range up to 90 meV, which enabled us to probe all modes. To look for anomalies, ideally one should follow the changes in the modes across T CO . In practice, this is hindered by the formation of twinned domains in the monoclinic CO phase below T CO . Still, the thermodynamic fluctuations of the CO order parameter near the transition can be sufficiently strong to produce measurable structural anomalies. It is expected that, slightly above T CO , these fluctuations produce a measurable rounding in the temperature-dependence of the critical softening of the q CO phonon in the cubic phase. This expectation is confirmed by the spectra taken along [ 110] (Σ) (see Fig. 5). Upon cooling, note a pronounced 6% softening of an optical mode around 35 meV at the critical wave vector Q * = G4 00 -q CO = (-4.5, -0.5, 0). The critical behavior of the softening of this q CO phonon is evident by analyzing its dispersion along Σ. Additional Q points collected by 8 extra analyzers located in the scattering plane or slightly above give further evidence of this softening. These additional points enable us to reconstruct the dispersion curve along an arc in the (hk0) plane crossing G4 00 , as indicated in Figs. 3a,b and6a-c. In conclusion, Fig. 6 clearly shows a progressive mode softening as the critical wave vector Q * is approached while, away from Q * , the mode displays a conventional hardening upon lowering temperature. The magnitude of the softening is noteworthy, considering that it occurs in the fluctuation region of the cubic phase and that, in the absence of anomalies, a mode hardening should instead occur.

The Q-dependence of the QE peak in Fig. 6b further indicates that the softening is correlated with incipient shortrange structural correlations at T CO . Knowing that this peak is well described by a Lorentzian function [35], we focus on the changes in the peak intensity and width observed at T CO . The energy dependence of the peak carries significant information on the lattice dynamics because the peak is broader than the instrumental resolution. We exclude that the broadening is due to the contribution of low-energy acoustic phonons, for these phonons are located at higher energies. Our analysis of the peak data as a function of Q shows a significant peak broadening at Q * (see Fig. 6b), which corroborates the picture of critical behavior of the lattice dynamics at q CO .

V. PHONON CALCULATIONS RESULTS

We should now try to single out the driving force of the lattice softening by identifying the symmetry of the soft phonon. In the high-temperature cubic Im 3 phase, the small point group is cubic T h at Γ and monoclinic C 2h at q CO = N, where the softening is observed, so the symmetry of the measured phonons at q CO is either A g or B u . In NaMnO, there are 8 modes of the former symmetry and 26 of the latter. We then carried out lattice dynamical calculations on the hightemperature cubic Im 3 phase. The room temperature structural data used as input for the calculations are taken from [27]. In Table I below we report the calculated energies and intensities of the 34 A g or B u modes, symmetric with respect to σ z , at the wave vector Q * measured in the IXS experiment. We do not consider the remaining 26 B g or A u modes, for they are antisymmetric with respect to σ z and therefore not measured in the present IXS geometry. In Table II, we give the eigenvector of the relevant A g mode of calculated energy 35.3 meV, attributed to the soft mode observed by IXS.

The reliability of the calculations is seen in Fig. 7, where one notes the agreement between experimental and calculated dispersions along the representative [ 110] direction within the experimental resolution. Hereafter, we focus on the above 34 A g and B u modes at the Q * wave vector of interest. In Fig. 5, we note the agreement between experimental and calculated positions and intensities of these modes at Q * (see Table I for a full list of the calculated values). Thus, we are confident that the shell model employed in the calculations is reliable to identify the polarization of the soft mode.

Within the energy resolution of the IXS experiment, ∆E=3.0 meV, three of the calculated modes, two with B u symmetry and one with A g symmetry, have energies of 35 ± 1.5 meV at Q * (see Table I). However, only the polarization of the A g mode, of calculated energy 35.3 meV, matches the displacement of the Mn and O atoms governing the structural modulation in the CO phase previously reported [29]. This is understood using the straightforward symmetry argument that A g is the totally symmetric representation of the C 2h small group of the modulated structure. The match between distortion pattern of the phonon and structural modulation is evident in Fig. 8. Note, in particular, the JT-like elongation of half of the MnO 6 octahedra in the pristine cubic cell along the apical [100] or [010] direction, leading to a zig-zag pattern along the [110] direction of the q CO propagation vector. This leads to II for all atoms. Dark and light blue colors denote the two distinct 4e and 4f B-sites in the Ag symmetry (see Table II). Right: Schematic structural modulation described by the propagation vector qCO in the monoclinic C2/m symmetry, concomitant to the Mn 3+ /Mn 4+ charge ordering at TCO (adapted from [29]). The unit cell of the parent Im 3 structure and the supercell are indicated by thick solid black and red lines, respectively. Thin broken and solid black lines represent the B-site MnO6 octahedra in the parent and distorted structures, respectively. Note the match between the pattern of the atomic displacements of the Ag mode and the structural modulation, in particular the zig-zag displacement of the Mn 4+ ions (black arrows) and the Jahn-Teller elongation of the octahedra of the Mn 3+ ions (orange arrows) leading to a zig-zag orientation of the 3d z 2 orbitals of these ions.

the formation of the √ 2 × √ 2 supercell previously reported in the monoclinic CO phase [29]. The observed JT distortion of these octahedra is as expected for the JT-active Mn 3+ ions, consistent with the scenario of Mn 3+ /Mn 4+ order previously reported. The remaining octahedra, occupied by the Mn 4+ ions in the CO phase, exhibit a modest distortion with negligible volume change, for these ions are not JT active. They instead exhibit an alternated displacement along the [110] direction to accommodate the elongation of the Mn 3+ octahedra. In summary, the CO modulated structure arises from a critical slowing down of the Jahn-Teller 35.3 meV A g phonon.

VI. CONCLUSIONS

In conclusion, we directly observed the critical lattice dynamics at the CO Verwey transition in the mixed-valence compound NaMnO by means of DS and IXS. This critical dynamics consists of a phonon softening at the propagation vector q CO of the CO modulated structure. Remarkably, the softening is sizable already in the thermodynamic fluctuation region of the cubic phase slightly above T CO . Lattice dynamical calculations enabled us to attribute the softening to a A g optical mode coupled to the JT dynamics of the MnO 6 octahedra in presence of an incipient Mn 3+ /Mn 4+ charge disproportionation. The match between calculated polarization of the soft phonon and experimental distortion pattern of the CO phase shows that the JT instability is the microscopic mechanism of the transition. The data also unveil a competing structural instability with a distinct commensurate propagation vector appearing at temperatures much higher than T CO and suppressed by the CO instability. Further studies using experimental probes of the short-range crystal structure, such as extended X-ray absorption fine-structure (EXAFS), may further elucidate the long-standing controversy on the scenarios of static charge ordering vs. dynamical charge fluctuations governing the physics of mixed-valence transition-metal oxides including magnetite, manganites and superconducting cuprates.

Table I. Calculated energy and intensity of the 34 Ag or Bu phonon modes measured in the IXS experiment at the critical wave vector qCO = ( 1 2 1 2 0) of the charge-order Verwey transition. The calculated energy and intensity values are plotted graphically as vertical bars in Figure 5. In bold the Ag mode attributed to the soft mode observed in the IXS experiment (see Figures 5 and6).

Energy [meV]

Calculated intensity [arb. units] Symmetry 7.9 3.89 ×10 Table II. Calculated atomic displacements, ux, uy and uz, along the x-, y-, and z-directions for the 35.3 meV Ag phonon. Atomic site label, Wyckoff position and symmetry are given for both parent Im 3 and distorted C/2m structures. Atomic coordinates are given in reduced lattice units. The structural data of the Im 3 structure are taken from [27], with unit cell parameter a = 7.3036 Å. The atomic coordinates of the C/2m structure are given by choosing a unit cell with unique axis z and origin at the centre, corresponding to the I112/m space group [START_REF] Aroyo | International Tables for Crystallography[END_REF].

Atomic site

Figure 1 . 1 3 1 2

 111 Figure 1. Temperature evolution of the DS intensity profile along the Σ direction crossing the (440) peak. Blue arrows indicate the position of the features at qDS = ( 1 3

Figure 2 .

 2 Figure 2. Detailed view of the reciprocal space maps of DS intensities near the (400) and (040) peaks at the CO transition temperature, TCO=175 K (a), and at room temperature (b).

Figure 3 .

 3 Figure 3. Color online. (a,b): reciprocal space DS intensity maps near the ( 400) peak at TCO=175 K and at room temperature (290 K). At TCO, note the anomalous X-shaped features at Q = G hk0 ± qDS = (h ± 1 3 , k ± 1 3 , 0). Circles indicate the Q points measured in the IXS experiment by using the main analyzer (full circles) and the extra analyzers (open circles) (see also text and Fig. 6). The broken red line indicates the direction of the intensity profiles shown in panel (c). (c): DS intensity profiles at TCO=175 K and 290 K (red and blue lines) and ratio of these two profiles (black line) along the Σ direction crossing the ( 400) peak, where the DS anomalous features are most visible. Blue arrows mark the wave vectors h ± 1 3 , k ± 1 3 , 0 and h ± 2 3 , k ± 2 3 , 0 where the anomalous DS intensities are observed. As in the DS data, quasi-elastic scattering is absent at room temperature. The asterisk indicates the polycrystal contribution to the diffraction pattern, visible as rings in panels (a,b).

Figure 4 .

 4 Figure 4. Color online. (a-d): Temperature evolution of the lowenergy IXS spectra at G440 + qCO = (3.5, 4.5, 0) down to 177 K, i.e. just above TCO. Note the increasing intensity of the elastic peak with decreasing T and the Stokes and anti-Stokes acoustic modes.Solid lines are a fit of these 3 peaks, as explained in text. (e): Dispersion of the acoustic mode along Σ at 297 and 177 K. Note a conventional mode hardening upon cooling, with no anomalies either at qDS or at qCO. (f): temperature dependence of the elastic intensity at various Q-points along Σ. The intensity increases upon cooling and decreases by moving away from G440, except at qDS, consistent with the X-shaped feature of the DS intensity in Fig.3a.

Figure 5 .

 5 Figure 5. Color online. IXS spectra taken at 295 K (red points) and TCO=175 K (blue points) at the CO wave vector Q * = (-4.5, -0.5, 0). Vertical bars indicate the calculated energy and intensity of the Ag and Bu modes (see TableI). The Ag mode calculated at 35.3 meV, indicated by a thick blue bar, is attributed to the soft mode highlighted in the black frame and in the inset. The polarization of this Ag mode is given in TableII. Inset: detail of the softening region. Solid red and blue lines are a fit of the mode at 295 and 175 K, respectively. The vertical broken lines and the arrow indicate the ≈2 meV energy shift of the mode. Broken lines are a fit of the two spectra including the contribution of all modes.

Figure 6 .

 6 Figure 6. Color online. (a): IXS dispersion of the soft mode of Fig. 5 taken at 290 K and TCO=175 K along the [ 110] (Σ) direction and along the arc indicated in panel (c). Full and open symbols are points measured by the main and extra analyzers, respectively (see text). Lines are a guide to the eye. Note a pronounced softening at TCO by approaching the wave vector Q * = G400-qCO. (b): Q-dependence of the width of the quasi-elastic peak at TCO. Note the peak broadening at Q * . Full and open symbols are as in (a) and (c). (c): Location in the (hk0) plane of all measured Q-points in reciprocal lattice units [r.l.u.]. Broken red and gray lines indicate the Brillouin zone boundary of the cubic unit cell and of the monoclinic supercell, respectively. (d-f): detail of the spectra at Q * = (-4.5, -0.5, 0), Q2 = (-4.64, -0.48, 0) and Q1 = (-4.78, -0.47, 0). Broken lines are a fit of the full spectrum to which all modes contribute. Arrows indicate peak position.

Figure 7 .

 7 Figure 7. Experimental (points) and calculated (lines) phonon dispersion along the [ 110] (Σ) direction in reciprocal lattice units [r.l.u.]. The experimental points have been measured by IXS at 290 K (see main text). The red line indicates the mode attributed to the soft mode observed experimentally at the zone boundary N = (-1 2 , 1 2 , 0) = qCO marked by a red arrow. At this point the mode symmetry is Ag. See in Table I the list of calculated mode energies and intensities at N and in Table II the calculated polarization of the soft Ag mode.

Figure 8 .

 8 Figure 8. Color online. Left: calculated polarization of the Ag 35.3 meV mode at qCO = ( 1 2

  TableI). The Ag mode calculated at 35.3 meV, indicated by a thick blue bar, is attributed to the soft mode highlighted in the black frame and in the inset. The polarization of this Ag mode is given in TableII. Inset: detail of the softening region. Solid red and blue lines are a fit of the mode at 295 and 175 K, respectively. The vertical broken lines and the arrow indicate the ≈2 meV energy shift of the mode. Broken lines are a fit of the two spectra including the contribution of all modes.

  Atomic coordinates Atomic displacements [arb. units]

	Parent Im 3 structure	Distorted C/2m structure	x	y	z	ux	uy	uz
	Na 2a m 3	Na	2a 2/m	0	0	0	0	0	0
			MnA'1 2b 2/m	0	0	1/2	0	0	0
	MnA' 6b mmm	MnA'2 2c 2/m	1/2	0	0	-0.53	0.39	0
			MnA'3 2d 2/m	0	1/2	0	-0.03	0.65	0
	MnB 8c	3	MnB1 4e MnB2 4f	1 1	1/4 1/4	1/4 1/4	1/4 3/4	0.32 0	-0.34 0	-0.19 0
			O1	4i	m	0.3132	0.1828	0	0.31	0	0
	O 24g m	O2 O3	4i 8j	m 1	0.6868 0.1828	0.1828 0	0 0.3132	0.22 0.11	0.65 -0.21	0 0.06
			O4	8j	1	0	0.3132	0.1828 -0.05 -0.19	-0.09

ACKNOWLEDGMENTS

The authors gratefully acknowledge the ESRF for the beamtime allocated to experiments HC1405 and HC2499 and financial support provided by the FAPESP (projects 2015/21206-1, 2013/27097-4 and 2013/07296-2), CAPES (Finance Code 001, project CAPES-COFECUB 88887.130195/2017-01), CNPq (projects 426965/2018-3 and 311462/2017-0) and the Indian National Science Academy for financial support of an INSA Senior Scientist position to S.L.C. VII. APPENDIX


C Detlefs

F Duc

Z A Kazeĭ

J Vanacken

P Frings

W Bras

J E Lorenzo

Z A Kazeȋ

P C Canfield

G L J A Rikken

Direct observation of the high magnetic field effect on the Jahn-Teller state in TbVO 4

Keywords: numbers: 75, 25, +z,75, 10, -b

HAL is a

1 Phase transitions induced by quadrupolar interactions have recently received much attention, mostly due to their influence on the magnetic properties of the sample, e.g. in the giant-magneto-resistive effect of manganite compounds [START_REF] Dagotto | Nanoscale Phase Separation and Colossal Magnetoresistance: The Physics of Manganites and Re[END_REF] and the magneto-electric effect in multi-ferroic materials [2,3]. The cooperative Jahn-Teller (JT) effect arises from the same quadrupolar interactions [4], and may indeed be interpreted as ferro-quadrupolar order. In JT compounds, the balance between magnetic and quadrupolar effects can be tuned by varying the strength of an externally applied magnetic field. To date, however, only a small amount of experimental evidence has been reported [5], as the required magnetic fields tend to be rather large.

In this letter we present the first direct experimental evidence for the modification of the JT effect by a external magnetic field. This effect was predicted by both qualitative [6,7] and quantitative [8,9] theories, but previous experiments [5,9,10,11,12] observed only indirect signatures of this behavior.

Terbium ortho-vanadate, TbVO 4 , along with DyVO 4 , is a textbook example for a JT transition [4] induced by quadrupolar interactions between the Tb 4f moments, mediated through phonons. At high temperatures, TbVO 4 crystallizes in the tetragonal zircon structure with space group I4 1 /amd [13] The JT transition has been studied extensively, both experimentally and theoretically.

The early studies, however, focused on the sample's properties in the absence of applied magnetic fields [4]. The effect of large external magnetic fields on TbVO 4 was studied only recently. Quantitative mean-field calculations [8] found that the JT distortion is suppressed when fields above ≈ 29 T are applied along the c-axis of the sample, in good agreement with susceptibility measurements [9]. Diffraction experiments have so far not been carried out, as the required combination of high magnetic fields and X-ray diffraction equipment has only recently become available [14,15,16].

The experiments were performed on the DUBBLE CRG beamline, BM26B, at the ESRF [17], using the experimental configuration described in [16]. Flux-grown single crystals of TbVO 4 [18,19] were ground into a fine powder and embedded in low molecular weight polyvinylpyrolidone in order to prevent movement of the powder grains due to magnetic forces, while at the same time improving the thermal contact. The sample was then mounted in a pulsed magnet and cryostat assembly [16].

The data shown here were taken by accumulating ≈ 45 magnetic field pulses per spectrum.

For each field pulse a mechanical shutter exposed the image plate detector for 4.9 ms centered around the maximum field. Representative powder diffraction spectra are shown in Fig. 1.

The JT transition manifests itself as splitting of some, but not all, powder lines (see table I). For small distortions, ǫ δ ≪ 1, the orthorhombic lattice parameters can be approximated as

a o ≈ (1 + 1 2 ǫ δ )ā and b o ≈ (1 -1 2 ǫ δ )ā, where ā = 1 2 (a o + b o ) ≈ √ 2a t .
To first order in ǫ δ , the splitting of a pair of Bragg reflections (H, K, L) o and (K, H, L) o is then given by In samples with large magneto-crystalline (MC) anisotropy, such as TbVO 4 , the phase diagram depends not only on the magnitude of an applied magnetic field, but also on its orientation relative to the crystal symmetry axes. The grains in a powder sample are aligned randomly, so that for a macroscopic measurement, e.g. magnetic susceptibility, the direction of the field has to be averaged over the full solid angle of 4π.

δ(2θ) ≈ λ 2 ā2 K 2 -H 2 sin(2θ) ǫ δ = tan(θ) K 2 -H 2 H 2 + K 2 + ā2 2c 2 L 2 ǫ δ . (1)
In a diffraction experiment, however, the orientation is constrained: Any grain contributing to the diffraction signal at the angle 2θ must be oriented such that the Bragg planes of the corresponding reflection (H, K, L) form the angle θ with the incident beam. The powder averaging must then be taken only over the azimuthal angle, ψ, i.e. rotation about the scattering vector Q = (H, K, L).

In our experiment the external field, µ 0 H, was aligned along the incident beam axis; the angle between the field and the Bragg planes is therefore the Bragg angle, θ.

For systems with substantial MC anisotropy any simulation of the powder diffraction spectrum must thus be carried out peak by peak: For each Bragg reflection the magnetic field direction and the corresponding influence on the phase diagram and the lattice parameters must be calculated as function of ψ, yielding, in this case the distortion ǫ δ (ψ), and the corresponding scattering angle, 2θ calc (HKL, ǫ δ (ψ)). The resulting diffraction patterns are then averaged over the azimuthal angle, taking into account the instrumental resolution.

In order to quantitatively describe the effect outlined above for TbVO 4 we have performed comprehensive mean field calculations of the magnetoelastic distortion of TbVO 4 as function of the strength and direction of the externally applied magnetic field. Our calculations closely follow those of [8] and [9] and will be published in full detail elsewhere. In particular, we have used crystal field parameter set 2 of [8], and a quadrupolar constant G δ = 130 mK.

The order parameter (OP), ǫ δ , is proportional to the expectation value of the quadrupole operator [START_REF][8, 9] we have defined the quadrupole operator P xy in the high-temperature, tetragonal coordinate system[END_REF],

Q xy = P xy , ǫ δ = B δ C δ 0 Q xy , (2)
where P xy = 1 2 (J x J y + J y J x ). The coefficient B δ /C δ 0 = 13.9 • 10 -4 was determined from the zero-field, low temperature value of the spontaneous deformation, ǫ δ .

We first consider the influence of the applied magnetic field on the quadrupole moment Q xy . Two effects must be taken into account: Changes in the magnitude of the distortion will result in variations of the splitting between the pairs of peaks, whereas (for a given grain) the preferential population of domains with positive or negative distortion will lead to a magnetically induced texture that manifests itself in a shift of spectral weight from one partner to the other.

A magnetic field along the c axis leads to a non-vanishing expectation value of J z in competition with the OP. A magnetic field along a o or b o , on the other hand, will induce a magnetization J x ± J y along the corresponding axis. Above the JT transition a magnetostrictive orthorhombic distortion of δ-symmetry is induced. In the ordered state, T < T Q , however, one has to distinguish between a field applied along a o and b o : Along the two axes, the susceptibility will be of slightly different value, thus breaking the degeneracy of ±ǫ δ .

Crossing the phase boundary in the presence of such a magnetic field will then lead to the preferential population of one of these domains. Fields along the [110] o directions should lead to a distortion with γ-symmetry (B 1g -type), as observed in DyVO 4 [20,21]. Finally, if the external field is applied along an arbitrary direction, then both δ and γ strains are induced simultaneously, lowering the symmetry of the crystal to monoclinic. Fortunately, the γ magnetostrictive distortion in TbVO 4 is two orders of magnitude smaller than the δ distortion and can therefore be neglected.

Typical theoretical results for the (311) diffraction peak at T = 7.5 K and 39 K are shown in Fig. 2. The (311) partner of the (311)/(131) pair corresponds to a positive quadrupole moment, Q xy > 0 (see eq. 1).

At T = 39 K > T Q there is no spontaneous quadrupole moment at H = 0. For azimuthal angles ψ below ≈ 22 • a positive quadrupole moment is induced, whereas for larger azimuthal angles the induced quadrupole moment is negative, i.e. corresponding to the (131) partner.

We thus expect the magnetic field to induce a magnetoelastic splitting with unequal domain populations of ≈ 33% Q xy > 0 (contributing to the (311) diffraction peak) and ≈ 67% Left: T = 7.5 K. Right: T = 39 K. At T = 39 K and 30 T, the dotted curve (in red) corresponds to the calculations including the magnetocaloric effect, and the green line to the calculations with the coefficient B δ /C δ 0 reduced by 25%.

subject to pinning on defects, grain boundaries, surfaces, and elastic interactions between twinned domains [13].

The calculated dependences Q xy (H) for fixed T and different azimuthal angles ψ allow us to simulate the X-ray powder diffraction data. For the (202)/(022) pair the values were averaged over the interval 0 • < ψ < 180 • since the two-fold symmetry persists. For the (311)/(131) pair the averaging was performed over 0 • < ψ < 360 • . The parameters of the resolution function were fitted to the experimental data at T > T Q and H = 0. Different parameter sets were used for the (311)/(131) and ( 202)/(022) pairs, and the intensity parameter had to be adjusted to match the data at low temperature and high field. At T > T Q , calculations for the (202)/(022) pair adequately reproduce the almost symmetric splitting of the line by the magnetic field observed in the data (Fig. 3). The calculated magnitude of the splitting, however, is ≈ 25% larger than the experimental one (Fig. 4).

The origin of this discrepancy remains unclear. In particular, it does not seem to be related to the value of the interaction parameters used in the calculations since the value of the spontaneous splitting at H = 0 is correctly described (Fig. 4). Similar deviations between theory and experiment are observed at the (311)/(131) pair. Calculations describe the shape of the (311) line in different fields: the intensity ratio for two components of the split peak is about 1:2. But again the calculated value of the splitting is larger than the experimental one by about 25% (see Fig. 3). The OP may be decreased by sample heating, e.g. due to the magneto-caloric effect. A reasonable temperature increase of about 8-10 K, however, cannot fully explain the observed discrepancy.

The experimental observations at T > T Q and H = 30 T are well described by our calculations when the coefficient B δ /C δ 0 is reduced by 25%; this value, however, is too small for the spontaneous deformation. This may be interpreted as the variation of the quadrupolar constant G δ under high magnetic field, as was observed for the quadrupolar constant G γ of DyVO 4 and considered within an improved "compressible Ising model" [21,22]. However for TbVO 4 no dependence of G δ was observed when the OP increases as the temperature decreases, and its low field behavior is described adequately by the mean field theory [4].

For the quadrupole ordered phase the results of the numerical calculations for the (202) and (311) peaks are compared with the experimental data in Fig. 3. For both peaks the averaging were performed for two domains (ǫ δ and -ǫ δ ) for 0 < ψ < 360 • , in steps of ∆ψ = 5 • . For the (202)/(022) pair the calculations do not reveal a noticeable decrease of the splitting under external field of 30 T, while for the (311)/(131) pair a small shift is expected for the low-angle component.

We examined, and refuted, several possible explanations for the discrepancy between the theoretical and experimental results. Firstly we ascertained that various weak pair interactions (bilinear, quadrupole of γ-, α-and ǫ-symmetry) which were omitted in our calculations do not change the results noticeably. Then we estimated the effect of the sample heating due to the magnetocaloric effect in our experiment. In the adiabatic regime, the temperature change is large enough (about 25 K at 7.5 K) for the magnetic field orientation close to the easy magnetization axis. Additional studies are necessary to elucidate the influence of the magnetocaloric effect in our experiments.

In summary, we have directly observed for the first time the effect of magnetic fields on the Jahn-Teller distortion of TbVO 4 , and compared the experimental data to a detailed mean field theory. Due to the polycrystalline nature of the sample and its strong MC anisotropy the calculated spectra had to be averaged over all possible orientations of the powder grains relative to the applied magnetic field. The applied magnetic field was found to influence both the magnitude of the OP, as observed in the splitting of the (311)/(131) and (202)/(022) pairs of Bragg peaks, and the relative domain populations, reflected in the intensity ratio between the partners of a pair. Our theory is in qualitative agreement with the experimental results, even though small quantitative discrepancies persist. This new technique was found to be a powerful tool to study JT systems, and allows us to observe new aspects of magnetic behavior of the classical and well studied JT compound TbVO 4 . These high magnetic field data may be used both to revise known theoretical models and to develop new, improved ones.

The authors acknowledge the NWO/FWO Vlaanderen and ESRF for granting the beam-

  , with lattice parameters a t = b t = 7.1831(3) Å and c = 6.3310(4) Å. Upon lowering the temperature through T Q ≈ 33 K, it undergoes a cooperative JT transition: The crystal spontaneously distorts along the [110] direction (B 2g strain, δ-symmetry distortion) to the orthorhombic space group F ddd with lattice parameters a o = 10.239(2) Å, b o = 10.029(2) Å, and c = 6.3154(13) Å at 22 K. The distortion is surprisingly large, reaching ǫ δ = 2(a o -b o )/(a o + b o ) = 2% at 22 K [13], and increasing further toward lower temperatures.

  Our experimental configuration, with photon energy E = 21 keV (λ = 0.59 Å), allowed us to observe scattering angles up to 2θ = 12.8 • , covering the reflections, in the orthorhombic notation, (111), (220), (311), (131), (202), and (022). Within this range, only the (311)/(131) and the (202)/(022) pairs are sensitive to the JT distortion.

FIG. 1 :

 1 FIG. 1: X-ray powder diffraction spectra of TbVO 4 . The inset shows the (311)/(131) and (202)/(022) pairs of reflections that are sensitive to the orthorhombic distortion, ǫ δ .

FIG. 2 :

 2 FIG. 2: Calculated dependence of the quadrupole moment Q xy on the external magnetic field for different azimuthal angles in the interval (0-100 • ) at the diffraction peak (311). Left: T = 7.5 K, Right: T = 39 K.

QFIG. 3 :

 3 FIG.3: Comparison between calculated and measured spectra for different temperatures and fields.

FIG. 4 :

 4 FIG. 4: Average order parameter as function of applied magnetic field for selected temperatures. Continuous lines: Experimental data. Dashed lines: Theory

TABLE I :

 I Bragg reflections of TbVO 4 , scattering angle at E = 21 keV, and the splitting due to the JT-effect. Values are based in the lattice parameters given in Ref. 13 for T = 22 K.

	(HKL) o	2θ	δ(2θ)/ǫ δ
		[deg]	[deg/%]
	(111)	7.14	0
	(220)	9.45	0
	(311)/(131) 11.78/11.94 0.076
	(202)/(022) 12.61/12.69 0.036
	(400)/(040) 13.24/13.52 0.134
	(222)	14.32	0
	(331)	15.19	0
	(113)	16.81	0

time for these experiments, and thank the staff of DUBBLE CRG and the ESRF for help in setting up these experiments. We thank M. Nardone, J. Billette, and A. Zitouni for their excellent work on the pulsed field coil and cryostat. Ames Laboratory is supported by the Department of Energy, Office of Science under Contract No. W-7405-Eng-82. Z. A. K. acknowledges financial support from the RFBR (project No. 07-02-01043). Part of this research was funded by the ANR (ANR-05-BLAN-0238) and EuroMagNet (EU contract 506239).