Study of Dielectric Constant and Loss Tangent of Ammonium Iron Alum

Transcription

1 DOI:0.7598/cst06.35 Chemcal cence Transactons IN: , 5(), 30-3 REEARCH ARTICLE tudy of Delectrc Constant and Loss Tangent of Ammonum Iron Alum ARVIND KUMAR RAWAT and TRILOK CHANDRA UPADHYAY Physcs Department, H.N.B. Garhwal Unversty, rnagar (Garhwal), U.K. 67, Inda Receved 7 eptember 05 / Accepted 5 October 05 Abstract: Wth the help of double-tme temperature dependent Green s functon method and modfyng pseudospn-lattce coupled mode model by addng thrd-and fourth-order phonon anharmonc nteractons terms, epressons for ferroelectrc mode frequency, delectrc constant and loss tangent have been derved. By fttng model values of physcal quanttes n above epressons, temperature dependence of above quanttes have been calculated numercally. Theoretcal results are compared wth epermental results of others, whch show agreement. Keywords: Green s functon, Pseudospn-lattce coupled mode, Anharmonc, Delectrc Introducton Ferroelectrc materals have attracted scentsts due to ther potental applcatons n producton of mall se capactors of hgh capactance, as memory devces for computers, as transducers, and nfrared detectors. Alums are nterestng materals. ome alums show ferroelectrc propertes. Ammonum ron alum (NH )Fe(O ).H O s ferroelectrc below 88K. Below 88K, t s cubc whle above 88K t s paraelectrc. In AFeD alum NH group gves rse to order-dsorder type of mechansm n proton subsystem assocated wth these groups. Ths s assumed to be responsble for the ferroelectrc phase transtons n several alums. Due to order-dsorder character of the ammonum group n AFeD, the H-bonds assocated wth these groups undergo some nd of orderng. It then becomes possble to apply pseudospn model smlar to the case of KDP system, after sutable modfcaton. The proton moton s assocated wth the actve ammonum on. In ths crystal there are very lttle sotope effect on T c and C, the pseudospn moton should be hghly damped wth strong anharmonc phonon nteractons. There has been a consderable nterest n the epermental study of AFeD alum. Derby has carred out crystal growth of AFeD alums. Weber has carred out epermental study of delectrc propertes of AFeD alum. achdeva et al. 3 have done epermental crystal growth studes on AFeD alum.

2 Chem c Trans., 06, 5(), Petrusev has carred out vbratonal spectra of AFeD, elenate and other alums. Gu and L 5 have carred out spectral propertes of AFeD and potassum alum. Gu and Ho 6 have done spectral propertes studes on AFeD alum and ammonum alum. han et al. 7 have studed applcaton of AFeD alum n norganc synthess. Bow et al. 8 have studed epermentally the aduvant acton of AFeD alums. Earler theoretcal studes on AFeD alum has been done by O Relly and Tsang 9. Thereafter Chaudhury et al. 0 have done theoretcal study of AFeD and MAD alums. Chaudhury et al. 0 have appled pseudospn-lattce coupled-mode model. These authors have not consdered thrd order phonon anharmonc nteracton term. Morever, they have decoupled the correlatons at an early stage. As a result some mportant nteractons were dsappeared from ther calculatons. In the present wor, we shall modfy pseudospn-lattce coupled mode model by addng the thrd- and fourth-order phonon anharmonc nteractons terms. By usng double-tme thermal Green s functon method, epressons for ferroelectrc mode frequency, delectrc constant and loss tangent wll be obtaned. Numercal calculatons wll be done to obtan thermal dependence of above quanttes. Theoretcal results wll be compared wth epermental results of others. Theory For AFeD alum, Chaudhury et al 0 have appled pseudospn-lattce coupled mode model. We shall add thrd-and fourth-order phonon anharmonc nteracton terms:- Followng Zubarev 3, we consder the Green s functon Where θ s unt step functon, θ = for t ' > t and = 0 for t ' < t, Ω s proton tunnelng frequency, s spn operator, ω s phonon frequency, A s poston operator, B s momentum operator. Dfferentatng Eq.(b) wth respect to tme t and t ' two tmes and settng ts Fourer transform nto Dyson s equaton form we obtan Green s functon (b) as Where H = V 3 (,, ) A A A V (,, ) Α Α Α Α. 3 G J ( ' ) ( ) ; t t ( t ' t ) ' ' = θ ( t t ) [ ( t) ; ( t )] = G Z 3 G(ω)=G 0 (ω)g 0 (ω) P ~ (ω)g 0 (ω) Z 3 ω ( Α Ω δ = π ω ˆ Γ, ( ) 3 Α Β Β ) (a) (b) (c) Ω ~ Ωˆ = ω, (3) 3 V Α ()

3 3 Chem c Trans., 06, 5(), 30-3 Ω ~ = a b bc, () a = J, (5a) b = Ω, (5b) c = J (5c) and ( ω ) and Γ are obtaned as ~ a b c V ( ) N K a V ωδ ' ω ω ω = ~ ~ ~, (6) ( ) ( ) ( ) ( ~ Ωω Ωω Ωω ω ω ) ω Γ and δ [ ( ) ( )] [ ( ) ( )] ( ω ω ) ( ω ~ πa ~ ~ πv ~ ~ ' Na V Γ(ω) = ~ δω δω Ω ~ δω δω Ω (7) ΩΩ ΩΩ ω ~ ω ) ω Γ We obtan susceptblty χ from Green s functon () smlarly as snce χ = lm πχnµ G (8) 0 N s no of dpoles per unt volume and µ s dpole moment assocated wth O-H O bond. We now that delectrc constant or smply s related to susceptblty χ as = πχ as >>. From Eq. (8), (9) and () we obtan 8πNµ = The tangent loss s obtaned from Eq.(0) tan X { ˆ Ω ( ω Ω ) ΩΓ( ω) } [( ˆ ω Ω Γ )] ( δ) = '' πχ ( ω ˆ ). ( ) (9) (0) ΩΓ = = n () ' From Eqs. (0) and () one observes that delectrc constant and loss tangent depend on modfed soft mode frequency. Hence these depend on tunnelng frequency as well as on anharmonc nteractons terms. Results and Dscusson For AFeD alum, we use the model values of physcal quanttes as Ω = 0.5, T C =88 K, J ' =.6 cm -, V = 6.9 cm -, N = 0.5 0, µ 0 8 cgs =.05, ω =5 cm -, C =5 K, J=83.9 cm -, A 0-7 (ergk - ) =0.05. We calculate the temperature dependence of ferroelectrc mode frequency Ωˆ, and tanδ usng Eq. (3), (0) and () and above model values. The results have been shown n Fgures, and 3 respectvely. Theoretcal results have been compared wth epermental results of Pepnsy et al. for delectrc constant and correlated results for ferroelectrc frequency and tangent loss for AFeD crystal. Our theoretcal results agree wth epermental results of Pepnsy et al.. From Eq. (3) and Fgure, t s observed that ferroelectrc mode frequency decreases as we approach from low temperature sde towards Cure temperature. At Cure temperature frequency becomes nfntesmally small, and ncreases above t. Our fndng s n agreement wth epermental

4 Chem c Trans., 06, 5(), observatons. Our Eq. (0) and Fgure show that delectrc constant frst ncreases as we ncrease temperature from low temperature sde becomng anomalously large at transton temperature. Above T c delectrc constant decreases wth ncreasng temperature. Our fndngs agree wth epermental results of Pepnsy et al. observatons. mlar behavour s predcted by our Eq. () and Fgure 3 for AFed alum. Fgure. Temperature dependence of soft mode frequency Ω (cm - ) of AFeD alum (-our results; epermental results of Pepnsy et al., ) Fgure. Temperature dependence of delectrc constant of AFeD alum ( our results; epermental results of Pepnsy et al., ) Tangent loss Delectrc constant oftmode frequency Fgure 3. Temperature dependence of tangent loss (δ) of AFeD alum (- our results; eperment results of Pepensy et al..)

5 3 Chem c Trans., 06, 5(), 30-3 Concluson As can be understood from the above results, the pseudospn-lattce coupled mode model along wth thrsd-and fourth- order phonon anharmonc nteractons terms eplans the delectrc propertes and ferroelectrc behavour of AFeD alum clearly. The phonon anharmonc nteracton terms sgnfcantly affect the temperature dependence of softmode frequency, delectrc constant and loss tangent n AFeD alum. Our results are better than the results of earler authors, quanttatvely. Acnowledgement The authors are thanful to Emnent Prof B emwal (Former, HOD Physcs) for hs nd blessngs and to hs nterest n the wor. Dr G. M. Kalamse, cence College, Nanded (M..) for encouragement, they are thanful to Prof Laltha redeshmuh (K.U., Warangal), Prof K G ubhadra (K U, Warangal), Prof Kamal ngh (Nagpur Unv & Amrawat Unv) Prof N Neg (H P Unv hmla) and Prof C Bhatt (HOD, & Convener Physcs, HNBGU rnagar (Garhwal)) and Dr M Unyal (HNBGU rnagar (garhwal)) for ther encouragements. References. Derby J J, th Int ummer chool on Crystal Growth, Dalan, Chna, Aug, 00, -7.. Weber, Zets fur Krstall, 00, 9, achdeva A, ngh R, ng P K and Bhattacharya B, Mater Technol., 03, 7(), Petrusev V M, Macedonan J Chem Chem Engg., 05, 3(), Gu, H and L Y, pectrochma Acta Part A, 05, 39, 3-35; DOI:0.06/.saa Gu H and Hao X, 05, 6, 73-76; pectrochma Acta Part A, DOI:0.06/.saa han Yu, et al., General Revew, 005, 7(), Mbow L M, Enno Gregoro De and Ulmer Jeffrey B, Nature Medcne, 0, 7(), 5-6; DOI:0.038/nm Relly O D E and Tsang T, Phys Rev., 967, 57, Chaudhury K R, Nath D, Baneree and Chaudhur B K, Phys Rev B, 98, 6, Zubarev D N, ov Phys Usp., 960, 3(3), 30.. Pepnsy R, Jona and hrane G, Phys Rev., 956, 0, 8-8.