Anharmonic effective potential, thermodynamic parameters, and EX A FS o f hep crystals

V N l ' Journal o f Science, M athem a tics - Physics 25 (2009) 213-219

Anharmonic effective potential, thermodynamic parameters, and EX A FS o f hep crystals

N g u y e n V a n H u n g * , N g o T r o n g H a i, T o n g S y T i e n , L e H a i H u n g Faculty o f Physics, ĩỉa n o i U niversity o f Science, VNU

334 Nguven Trai, Ihanh Xuan Hanoi, Vieinam Received 15 S ep tem b er 2009

A b s t r a c t . A n h a rm o n ic effective potential, effective local force constant, thermal expansion coefficient, three leading cum ulants, an d E X A F S (E x te n d e d X-ray A bso rp tio n Fine Structure) o f hep crystals have been studied. A nalytical expressions for these quantities have b een derived.

N um erical calculations have b een carried out for Z n and Cd. T h ey s h o w a g o o d agreem ent with experim ent results m easured at H A S Y L A B (D E S Y , G e rm an y ) and unnegligible anharm onic ctTects in the considered quantities.

Í. Introdu ction

EX A FS and its param eters are often m easured at low tem peratures and well analysed by the harmonic procedure [1] bccausc Ihe anharm onic contributions to atom ic therm al vibrations can be ncglcctcd. Bui E X A F S m ay provide apparently different inform ation on structure and on other param eters o f the S L ib s t a n c c s at different high tem peratures [2-11,14,15] due to anharm onicity.

T h i^ w o r k is to d e v e l o p m e n t o f a n e w m c t h o i l f o r c a l c u l a t i o n a n d a n a l y s i s o f th e h i g h order anharnionic effective potential, local force constant, three leading cumulants, thermal expansion coefficient, and E X A F S OÍ' hep crystals. Derivation o f analytical expressions for these quantities is based on quantum statistical theory with the anhanmonic correlated Einstein model [9] and M orse

Ị ) o l c n l i a l is used to characterize interaction betw een cach pair o f atoms. N um erical results for Zn and Cd arc found to be in good agreem ent w ith expcrim cnl [16] and show unnegligible anharm onic effects in the considered quantities.

2. F o r m a lis m

According to cum ulant expansion approach the E X A F S oscillation function is given by [11]

X i k ) = F{ k ) . Im

kR

e x p ( 1 )

where F ( k ) is the real atom ic backscatlering a m p l i t u d e , o is the net phase shift, k and A are the wave num ber and the m ean free path o f the photoeleclron, respectively, R = w ith r as the

C o re s p o n d in g author. E-Iiiail: hun g iiv @ v n u .cd u .v n

213

instantaneous bond length betw een absorbing and backscalterm g atom s and ơ ' " ' (n == 1, 2, 3, arc the cum ulants [2].

The total m ean square relative displacem ent (M S R D ) or 2'"^ cuniLilant a ^ Ụ ' ) at a g iv e n tem perature T is given as the sum o f harmonic ơ^(T’) and anharm onic ơ-3(7’) co nlributions 11 1 1

^ L { T ) = ư ^ T ) ^ a ỵ T ) , a ị = 0 { T ị a ^ T ) - , J3(r) = 2 ỵ , : ^ , (2 )

w here Yo is G runcisen param eter, AV/V is the relative volum e change due to therm al expansion, ơ l IS zero-point contribution lo c r ' i r ) .

T he anharm onic effective potential can be expressed as a function o f the d isplacem ent .X = r - along the R® direction, r and Kq being the instantaneous and e q u ilib riu m bondlengths b e tw e e n absorbing and backscattcring atom s, respectively

+kyX^ (3)

where ẲqÌs effective local force constant, and ^3 IS cubic parameter giving the asymmetry due to anharm onicity. (H ere and in the following, the constant contributions arc neglected).

For calculation o f th c im o d y n am ic param eters w e use the further definition y = x — a , a - { r - r ^ ' ) [9, 18], to write Eq. (3) as

w here is an effective local force constant, in principle different from ,

M aking use o f q u an tu m statistical m ethods [13], the physical quan tity is d e leư n in ed b> an averaging procedure using canonical partition function z and statistical density matnx

p

{ y " ' ) = Ỷ ‘ r { p y " ' \ "I = 1, 2 , 3 , . . . { j )

Atomic vibrations are quantized in terms o f phonon, and anharm o n icity is the result o f p h onon- phonon interaction, that is w hy w e express in tenns o f annihilation and creation operators, â and

ã * , respectively

>’ = « o ( â + â ^ ) , «0 = - ^ .

and use the harmonic oscillator state n) as the eigenstate with the eigen valu e = ntìcOị, , ignoring the zero-point energy for convenience, here ũ)^ is coư elated Einstein frequency.

A M orse potential is assu m ed to describe the interatomic interaction, a n d ex p ande d to the third order around its minimum

K(.x) = - 2 e - " ^ ) = o Ị - l + a^.v^ +•■•), (7) w here a describes the width o f the potential and D is the dissociation energy.

In the case o f relative vibrations o f absorber and backscatterer atom s, including the effect o f correlation and taking into account only the nearest neighbor interactions, the effectiv e pair potential is given by

214 N. V ỉiu n ỵ f t a ỉ / V N U J o iin u il o f Science, M aihcm aĩics - P hysics 25 (2 0 0 9 ) 213^219

N. ỉ ' ỉỉuní^ í7 aĩ. / V N U J o u rn a l o f S à c ỉic e , M a íh cn u itics - P hysics 25 (2009) 2 ỉ 3-219 215

ỵ z I' :r-vR“.R, =V{x)+2V +8K ~ J +81" J . (8)

I = a, f >ị *a. b J V V V ^ /

w here the first term on the right c o n c c m s only absorber an d backscalterer atom s, the rem aining sums extend over ihc rem a in in g neighbors, and the second equality is for m on o a to m ic hep crystals.

In acc o rd an ce \Mth Hq. (4), using M orse potential Eq. (7), and A'o = 5 D a “ the effective potential 1-q. (8) IS expressed as

y , A v ) = S D a - f 9 ^ 5 4

1 - — aa ay + — D a I 2 0 , ' 2

V - — D a V »

V 10 J- • •

(9)

w here the local force c o n stan t is given by k ^ ^ = 5 D a - 9

- — a a 10

hcOfr

(10)

where 0 / ,, /I arc the co rre la te d liinstcin temperature and reduced mass, respectively.

For further calculation w c w rite the effective interatom ic potential as the sum o f the harmonic contribution and a p e rtu rb a tio n Ổ V due to the w eak anharm onicity

\ ,

5a — a y

y

^{(1 1)}

Using the above results for correlated atomic vibrations and the p rocedure depicted by Hqs. (5, 6), as well as, the first-order th c rm odynam ic perturbation theory w ith considering the anharm onic com ponent in the potential Hq. (9), w e derived the cum ulants.

The 2 ”'^ cum ulant or m e an square relative displacem ent (M S R D ) is ex pressed as

1 + z

_ 2 ( r \_____ 2 1 + -

Ơ- = Ơ-0 — , ƠQ 1 ^z First and third cu m u la n ts are

W D a -

1 t ^ * ) ư _2 1 _(i t t 1) 9 ư _ 2

Ơ , ơn = - “ (Jn »

20 ® 20 “

( 7 ) - ;

10 3 ( a ' f - 2 ( a ỉ Ỵ \ ^ ^ ( a ỉ Ỵ ,

(12)

(13) (14)

\OODaR

(15)

w hc‘c R is the bond l e n g t h ,ƠQ.ƠQ ^ are zcro-point contributions to and is the conilant value o f ư-Ị at h ig h-tcm perature.

To calculate the total M S R D including anharm onic contribution Eq. (2) an anharm onic factor has hccr derived

and liq. (13) the thermal ex p a n s io n cocfficicnt is resulted as

< . r/ 2 ^{y I ;yl} 0 ^_

^9k

o

I 2

1 4R

, 3 a 2

4R (16)

" h e anham ionic contrib u tio n to the E X A FS phase at a ^iven tem perature is the diffcrencc between the Dial phase and the o n e o f the harm onic EX A FS, and it is given by

2 1 6 N. l ' ỉỉiitĩiị Cĩ (lỉ / VN U J o u rn a l o f Science. M a them aĩics ' P ỉn s ic s 25 (2009) 2 Ĩ Ỉ - 2 Ị 9

<X>ẠT.k)=^

2

k

( 1 7 )

\Ve obtained from I:q. (1), taking into account the above results, the tem perature d e p e n d e n t K- edge EX A FS function including anharm onic effccts as

SĨN }

( r ị ự l ^ ơ ^ i T ) *2H:ỈÁik)

^ sins m{ 2 k R . + a > j ( k ) + ^ ' j k , r ) ) , (18) w here is the square o f the m any body overlap term, N is the atom ic n u m b e r o f cach shell, the rem aining param eters w ere d efined above, the m ean free path Ằ IS d e f i n e d by the im aginary part o f the com plex photoelectron m o m e n tu m p ~ k + i I Ằ , and the sum is o v er all atom ic shells.

3. N u m erica l resu lts and co m p a riso n to exp erim en t

N ow w e apply the above derived expressions to numerical calculations com p ared to experim ent for Zn and Cd m easured at H A S Y L A B (D E S Y , G erm an y ) [16]. M o rse potential param eters o f Zn and Cd have been calculated by generalizing the procedure for cubic crystals [12J lo the one for hep crystals. T hey are com p ared to the E X A FS exj)erimenlal data [16]. Effective local force constants, correlated Einstein frequencies and tem peratures have been calculated using ihcse M orse param eters.

T he results are written in T able 1. T hey are used for calculation o f anharm onic HXA FS and its param eters. The calculated anharm onic effective potentials for Zn and C d are co m p ared to experim ent and to their harm onic c o m p o n en ts (Figure la). T he calculated anharm o n ic factors for Zn and Cd arc shown in Figure lb). T hey agree with the extracted experim ental results [16].

Table 1. Calculated and experimental values o f D, a , , and , Cũị.^, Oị: for Zn, Cd

Bond D(cV) Ơ ( Ả ' )

'b(A ) iy ^ ( x I O ‘^ //z ) 0,:{K )

Zn-Zn, Calc. 0.1698 1.7054 2.7931 39.5616 2.6917 2Ơ5.6101

Zn-Zn, Expt. [161 0 1685 1.7000 2.7650 390105 2.6729 204 1730

Cd-Cd. Calc. 0.1675 1.9069 3.0419 48.7927 2.2798 174.1425

Cd-Cd, Expt. [16] 0.1653 1.9053 3.0550 48.0711 2.2628 172.8499

2 5 ,

> 2

s ”

I

cCL 1

0)>

Zn, Anharmonic 2n Expt 2n. Harmonic Cd. Anharmonic Cd Expt Cd. Harmonic

0 5

- 05 0

X(A)

0 5

0 09

0 ⁰⁸^

£ 0 07

d

B 0 0 6 r

ă

u

Ọ 0 05

I

^{0 04j}

ầ !

I 0 03}

00 2! 0 010

Zn. Present Zn. Expt Cd, Present Cd. Exp(

100 200

a)

300 400

T(K)

b)

500 600 700

Fig. 1. Calculated anharmonic effective potentials and theừ harmonic components (a), and anharmonic factors (b) for Zn, Cd. They are compared to experiment [ 16].

N. V. lỉu ỉĩg ct aĩ. / VN U J o u rn a l o f Science, M a th em a tics - P hysics 25 (2009) 2 Ỉ 3-219 217

Figure 2 illustrates the tem perature dependence o f our calculated r* c u m u lan t (a) describing the net therm al expansion and 2''"^ cum ulant (b) describing D eb y e-W aller factor for Zn and Cd co m pared to experim ent at 77 K and 300 K [16].

0 02

0 0 1 8 0 0 1 6 0 0 1 4

< 0012

2n. Present Zn, Expt Cd. Present C d .E x p t

rg

fNi

0 0 3

0 025

0 02

0 015

0.01

0 005

Zn. Present Zn. Expt Cd. Present C d .E x p t

0 008 0 0 0 6

0 0C4 Ị ‘ '

TOO 200 300 400 500 600 700 100 200 300 400 500 600 700

T(K ) T(K)

(a) (b)

Fig. 2. Calculated temperature dependence o f (a) and 2"*^ (b) cumulants for Zn and Cd compared to experiment at 77 K and 300 K [16].

Figure 3 d em onstrates the tem perature dependence o f our calculated 3'^^ cum ulant and thermal expansion coefficient for Z n and Cd. T hey agree with the m easu re d values at 77 K and 300 K [16]. All three calculated cum ulants o f Zn and C d satisfy iheir fundam ental properties. T h ey contain zero-point contnhiition at low tem perature as quantum effects. At high-tcm peratures the r ‘ and 2"^^ cum ulants are linearly proportional to the tem perature T, but the 3^"^ c u m u lan t to T \ O u r calculated tem perature dcpendcnce o f therm al expansion coefficients for Zn and C d agree w ith experim ental values at 77 K und :o o K. M uicuvci, liicy saiisly Gruciiciscri ilicorem, w here al low icm pcralurcs they behave as and £t hi^h-tcm pcraturcs they approach the constant values as the form o f specific fcal.

X 10 _X 10

fO<

1 -

09 Zn. Present .1

0 9

05 2n, Expt ¹₁

Cd. Present ¹ 0 8

0 ' 06 05

Cd. Expl

2^- 0 7 Í

. a ^/

0 5Ị /

O i H

6 0 4' ;

03 03'

o : 02 '

0 ■ 0 1

0-^ '

0 100

-

0 100 200 300 ^{400 500} 600 700 ₂₀₀

(T’

^(b)

300 400

T(K)

500

Zn. Present I Zn, Expt Cd, Present Cd. Expt

600 700

Fig. Calculated temperature dependence of 3' cumulants (a) and thennal expansion coefficients for Zn and Cd compared to experiment at 77 K and 300 K [16].

218 N. r. iỉỉo ìiĩ eí aỉ. / Ỉ ' NU J o u rn a l o f Science, M a them aỉics - P hysics 25 (2009) 2 Ỉ 3 - 2 Ỉ 9

Figure 4 show s the CX A FS spcclra calculalcd by our ihcorj' at 77 K, 300 K and 500 K (a) nnii their Fourier transform m agn itu d e at 300 K (b) com pared to experim ent [16]. '['he I 'X A l'S arc attenuated and shifted shifted to the right as the tem perature increases. O u r calculated I'ouricr Iransform m agnitude agrees with experim ent [16] and is shifted to the left co m p ared to the harmonic FEFF code results [I]. This is indicative o f the necessity o f including anharm o n ic contributions in Ihc EX A FS data analysis.

k{A-’)

0 18

0) 0 16

•g

■i 014

cỠ>

1 0 1 2 g 0 1 0

oư> 008

cTO

006

2 0 04 o u . 0 02

000

Zn, 30 0K 3A’ < k < 13 5A’

--- — Expt _ --- P re se n t tneory --- F E F F

(a)

Fig. 4. C alculated a n ha rm onic E X A F S at 77 K, 300 K, 500 K (a) and Fourier transform m ag n itu d e at 300 c o m p a re d to experim ent [16] and 10 FHFF result [1] for Zn.

K

4. C on clu sion s

fn this work a new m ethod for calculation and analysis o f anharm onic effective polenlia!, elTcctivc local force constant, three leading cum ulants, and l^XAFS for hep crystals has been explored. I his anharm onic theory contains the harm onic model at low tem peratures and the classical limit at high tem peratures as special eases.

Derived analytical expressions for the considered quanlities satisfy all llicir rundaincntal properties and provide a good agreem ent betw een the calculatcd and experim ental results. Ihis em phasizes the necessity o f including anharm onic contributions in the E X A F S data analysis.

A ck n o w led g em en ts. T he authors thank Prof. R. R. Frahm for useful com m en ts. This work is supported by the basic science research project o f VNIJ Hanoi Ọ G .08.02 and by the rcscarch projcct No. ¹⁰³.^{0 1}.^09.09 o fN A F O S T E D

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