UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Depakeing of NMR spectra Sternin, Edward 1982

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1982_A6_7 S75.pdf [ 2.42MB ]
Metadata
JSON: 831-1.0085015.json
JSON-LD: 831-1.0085015-ld.json
RDF/XML (Pretty): 831-1.0085015-rdf.xml
RDF/JSON: 831-1.0085015-rdf.json
Turtle: 831-1.0085015-turtle.txt
N-Triples: 831-1.0085015-rdf-ntriples.txt
Original Record: 831-1.0085015-source.json
Full Text
831-1.0085015-fulltext.txt
Citation
831-1.0085015.ris

Full Text

DEPAKEING OF NMR SPECTRA by EDWARD STERNIN B . S c , The U n i v e r s i t y of B r i t i s h Columbia, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as conforming to the r e g u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September 1982 © Edward S t e r n i n , 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of r n y ^ 1 c ^  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Oc+. / S ~ , DE-6 (3/81) ABSTRACT NMR s p e c t r a of many systems governed by a x i a l l y symmetric second rank tensor i n t e r a c t i o n s e x h i b i t P 2(cos©) dependence on the angle 9 between the symmetry a x i s and the e x t e r n a l l y a p p l i e d magnetic f i e l d . For the s o - c a l l e d "powder samples" c o n s i s t i n g of many randomly o r i e n t e d domains the r e s u l t i n g spectrum i s the s u p e r p o s i t i o n of c o n t r i b u t i o n s from each such domain. T h i s study d e a l s with a numerical technique e n a b l i n g one to ob t a i n the l i n e s h a p e s of such i n d i v i d u a l c o n t r i b u t i o n s r e s p o n s i b l e f o r the given powder spectrum. The e l e c t r i c d i p o l a r i n t e r a c t i o n between two spin 1/2 n u c l e i produces a c h a r a c t e r i s t i c powder lineshape c a l l e d "Pake doublet", a f t e r G.E.Pake, hence the name "de-pake-ing". An i t e r a t i v e a l g o r i t h m capable of d e a l i n g with s p e c t r a produced by a v a r i e t y of systems i s developed and checked by a p p l y i n g i t to a wide range of simulated NMR s p e c t r a . A set of c h a r a c t e r i s t i c " s i g n a t u r e s " a s s o c i a t e d with d i f f e r e n t experimental s i t u a t i o n s i s e s t a b l i s h e d and the l i m i t s of the a p p l i c a b i l i t y of the technique are determined. i i i TABLE OF CONTENTS I . I n t r o d u c t i o n . 1 I I . B a s i c t h e o r y . ' 4 I I I . D e p a k e i n g : d e t a i l s of t h e a l g o r i t h m . 16 IV. D e p a k e i n g : a p p l i c a t i o n t o NMR s p e c t r a 22 V. F i n a l r e m a r k s . 39 REFERENCES 41 APPENDIX 42 i v LIST OF FIGURES F i g . 1. Normalized simulated l i n e s h a p e s produced by domains of d i f f e r e n t o r i e n t a t i o n . 6 F i g . 2. R e l a t i o n s h i p between o r i e n t e d and powder s p e c t r a . 9 F i g . 3. Simple case of depakeing. 13 F i g . 4. An i l l u s t r a t i o n of i t e r a t i v e depakeing. 22 F i g . 5. R e s o l v i n g two s p e c t r a l l i n e s by depakeing. 25 F i g . 6. Extreme case of an e x c e s s i v e l y l a r g e o r i e n t a t i o n independent broadening. 27 F i g . 7. E f f e c t of a moderately l a r g e o r i e n t a t i o n independent broadening.- 28 F i g . 8. E f f e c t of a small o r i e n t a t i o n independent broadening. 30 V F i g . 9. Effect of linewidth being comparable to the ACS. 31 Fig.10. Effect of zero ACS ( "super-lorentzian" 1ineshape ). 33 Fig.11. Effect of an error in the location of the zero of the f i r s t moment. 35 Fig.12. Depakeing of a two component spectrum. 37 v i ACKNOWLEDGEMENT It i s a great pleasure to thank my supervisor, Prof. Myer Bloom, for the help and guidance he has given me throughout th i s wor k. I am indebted to Dr. Alexander MacKay for his patience in teaching me the art of NMR experiments and for many valuable discussions. G.W. Johnson, M.A. Potts and many members of the Computing Centre had to deal with a l o t of my computational demands; their assistance is gr a t e f u l l y acknowledged. I am grateful to my parents, Maya and Vulf Sternin, for their constant encouragement and support. 1 I. INTRODUCTION An a x i a l l y symmetric quadrupolar i n t e r a c t i o n f o r a s p i n 1 or a d i p o l a r i n t e r a c t i o n between two s p i n 1/2 n u c l e i g i v e s r i s e to a nuclear magnetic resonance (NMR) spectrum which i s a doublet with i t s s p l i t t i n g p r o p o r t i o n a l to 3cos 2 9-1 , where 9 i s the angle between the e x t e r n a l l y a p p l i e d magnetic f i e l d and the symmetry a x i s of the e l e c t r i c f i e l d g r a d i e n t or the vector j o i n i n g the two s p i n 1/2 n u c l e i , r e s p e c t i v e l y . The two l i n e s are symmetric about the p o i n t at which the s i n g l e resonance would have been expected i f no s p l i t t i n g o c c u r r e d . When i n s t e a d of being a s i n g l e c r y s t a l the system i s a c r y s t a l l i n e powder with a l a r g e number of randomly o r i e n t e d domains, the s u p e r p o s i t i o n of the doublets a r i s i n g from each of these domains produces a c h a r a c t e r i s t i c lineshape dominated by a p a i r of peaks a s s o c i a t e d with © = J T/2 , the most probable value of 9 . T h i s l i n e s h a p e i s c a l l e d a "Pake doublet", a f t e r G.E.Pake who was the f i r s t to study such systems in d e t a i l [1,2]. T h i s t h e s i s d e a l s with a c e r t a i n numerical technique which enables one to o b t a i n the d i s t r i b u t i o n of i n t e n s i t y in an o r i e n t e d ( s i n g l e c r y s t a l ) spectrum which i s r e s p o n s i b l e for the observed Pake doublet, hence the name "de-Pake-ing" f i r s t i ntroduced by Bloom e t . a l . [ 3 ] . The i n i t i a l a p p l i c a t i o n of depakeing was to quadrupolar 2H NMR s p e c t r a . A s i m i l a r experimental s i t u a t i o n of r e s t r i c t e d 2 molecular motion i s also encountered in many other cases, p a r t i c u l a r l y in 3 1 P spectroscopy of various biochemical systems, the membrane phospholipids being the prime case in mind. This is because the phosphate phosphorus of phospholipids in natural abundance provides a well-defined i n t r i n s i c probe for motion and structure. As was shown by various authors ( for a review see Refs. 4-6 ), 3 1 P NMR spectra of both model and b i o l o g i c a l membranes are sensitive to the occurence of g e l - t o - l i q u i d - c r y s t a l l i n e hydrocarbon chain t r a n s i t i o n s , but the interpretation of such spectra i s often made d i f f i c u l t by the presence of phosphorus-proton dipolar interactions which cause broadening of the spectra. The usual way to avoid these d i f f i c u l t i e s is to apply to the sample a continuous radio frequency f i e l d at the resonance frequency of the protons, e f f e c t i v e l y saturating the proton resonance and thus decoupling the phosphorus nuclei from the neighbouring protons. In this way the phosphorus spectra become sharper and their features easier to i d e n t i f y . Obviously, the information about the dipolar interactions themselves is completely l o s t . Depakeing, on the other hand, provides one with an alternative way of determining the parameters of the phosphorus spectrum, l i k e the anisotropy of the chemical s h i f t , and, at the same time, preserves the information about the interactions responsible for the broadening. However, the direct application to 3 1 P NMR of the formulae derived in [3] for the 2H NMR case was impossible mainly because the oriented spectrum was assumed to be composed of doublets symmetric around the Larmor frequency o 0 and so i t 3 was s u f f i c i e n t t o d e a l w i t h o n l y o n e - h a l f o f t h e o r i e n t e d s p e c t r u m . Thus t h e r e was a need f o r c e r t a i n e x t e n s i o n and r e f i n e m e n t of t h e t e c h n i q u e w h i c h prompted t h i s s t u d y . W i t h o u t s u g g e s t i n g i n d i s c r i m i n a t e use of d e p a k e i n g , i t can be s t a t e d t h a t t h e n u m e r i c a l a l g o r i t h m d e r i v e d h e r e i s of v e r y g e n e r a l n a t u r e and can be a p p l i e d t o a l l s y s t e m s s a t i s f y i n g a c e r t a i n g e n e r a l a s s u m p t i o n a b o u t t h e symmetry of m o l e c u l a r m o t i o n . M o r e o v e r , i t can be e x p e c t e d t h a t t h i s a l g o r i t h m w i l l y i e l d m e a n i n g f u l r e s u l t s even i n some e x p e r i m e n t a l s i t u a t i o n s w h i c h o n l y p a r t i a l l y s a t i s f y t h e a s s u m p t i o n of t h e model, as i s d e m o n s t r a t e d by some s i m u l a t e d s p e c t r a i n p a r t IV of t h i s s t u d y . Some p r e l i m i n a r y r e s u l t s o b t a i n e d i n t h i s l a b o r a t o r y and e l s e w h e r e [7] show t h e extreme u s e f u l n e s s of d e p a k e i n g i n t h e r e l i a b l e i n t e r p r e t a t i o n of 3 1 P and 1 9 F NMR s p e c t r a . 4 I I . BASIC THEORY. F o l l o w i n g t h e t r e a t m e n t of [ 8 ] , we s t a r t by c o n s i d e r i n g t h e e f f e c t t h e s p i n d ependent i n t e r a c t i o n s w h i c h t r a n s f o r m as s e c o n d rank t e n s o r s have on t h e Zeeman e n e r g y l e v e l s of a s p i n s y s t e m : where H, i s t h e f i r s t o r d e r p e r t u r b a t i o n H a m i l t o n i a n , t h e A* a r e th e s p i n o p e r a t o r s of t h e s y s t e m and t h e F < v ( t ) a r e f u n c t i o n s of th e s p a t i a l c o o r d i n a t e s , w h i c h a r e f u n c t i o n s of t i m e . Examples of s u c h s e c o n d rank t e n s o r i n t e r a c t i o n s a r e e l e c t r i c q u a d r u p o l a r , m a g n e t i c d i p o l a r and a n i s o t r o p i c c h e m i c a l s h i f t i n t e r a c t i o n s . I f t h e i n t e r a c t i o n r e s p o n s i b l e f o r t h e l i n e s h a p e i s a x i a l l y s y m m etric or i f r a p i d a x i a l l y s y m metric m o l e c u l a r m o t i o n i s p r e s e n t , t h e s e c u l a r p a r t of t h e H a m i l t o n i a n ( i . e . t h e te r m w i t h q=0 ) i s , t o f i r s t o r d e r i n p e r t u r b a t i o n t h e o r y , where 9 i s t h e a n g l e between t h e e x t e r n a l l y a p p l i e d m a g n e t i c f i e l d and t h e m o l e c u l a r a x i s of symmetry. We s h a l l r e f e r t o t h e NMR s p e c t r u m f o r a g i v e n v a l u e of 9 as t h e " o r i e n t e d s p e c t r u m " . In t h e c a s e of a s y s t e m composed of many randomly o r i e n t e d (2.1 ) H, = E A»F*(t) , (2.2) H(9) = H ( 0 ) P 2 ( c o s 9 ) = H ( 0 ) ( 3 c o s 2 9 - 1 ) / 2 5 domains, t h e r e s u l t i n g s p e c t r u m i s t h e s u p e r p o s i t i o n o f many d i f f e r e n t s p e c t r a a r i s i n g from e a c h s u c h domain. T h e s e s p e c t r a w i l l be r e f e r r e d t o as "powder s p e c t r a " o r "powder p a t t e r n s " . W i t h t h e above a s s u m p t i o n s t h e o r i e n t e d s p e c t r u m of a s p i n 1/2 n u c l e u s ( e . g . 3 1 P ) i s a s i n g l e l i n e s h i f t e d by u f l P 2 ( c o s 9 ) from t h e Larmor f r e q u e n c y u 0 . Here t h e f r e q u e n c y u f l i s d e t e r m i n e d by t h e a n i s o t r o p y of t h e c h e m i c a l s h i f t t e n s o r . The s p e c t r u m i s d i s t r i b u t e d o v e r t h e ra n g e o f f r e q u e n c i e s between -1/2O 4 and u„, as may be seen i n F i g . 1 . Here and i n a l l o t h e r f i g u r e s z e r o of t h e f r e q u e n c y s c a l e c o r r e s p o n d s t o t h e Larmor f r e q u e n c y u 0 . F o r r e a s o n s of c l a r i t y we s h a l l f i r s t f o l l o w t h e t r e a t m e n t o f [3] and make a few s i m p l i f y i n g a s s u m p t i o n s (we s h a l l examine t h e s e a s s u m p t i o n s and t h e ways t o g e n e r a l i z e our t r e a t m e n t i n p a r t I I I of t h i s s t u d y ) . L e t us assume t h a t t h e o r i e n t e d s p e c t r u m c o r r e s p o n d i n g t o 9 = 0 i s g i v e n by a n o r m a l i z e d f u n c t i o n F e - 0 ( o ) , where u i s f r e q u e n c y e x p r e s s e d r e l a t i v e t o o 0 as a m u l t i p l e of some c o n s t a n t , so t h a t t h e s p e c t r u m i s w e l l w i t h i n t h e i n t e r v a l [ 0 , 1 ] . We s h a l l n e g l e c t a l l i n t e n s i t y o u t s i d e t h i s i n t e r v a l , i . e . , (2.3) F e = 0 U ) = 0 , o<0, u>1 . Then t h e n o r m a l i z a t i o n i s (2.4) o 1. e=50° 0=90° A 0=30° A A e=o° JV I 0 -1 C J Fig.1 N o r m a l i z e d s i m u l a t e d l i n e s h a p e s p r o d u c e d by domains o f d i f f e r e n t o r i e n t a t i o n . S c a l i n g i s a c c o r d i n g t o P 2(cos9). L i n e becomes i n f i n i t e l y narrow a t t h e "magic a n g l e " 9=54.6° 7 Let us examine what happens to F e(u) as 0 i s varied. Eq. (2.2) predicts that as 9 changes, the shape of the spectral l i n e does not change, but i t i s scaled according to (2.5) u = u (6 ) = x(3cos 29-1)/2 = xP 2(cos0) , where x corresponds to 9 = 0 : x = u(0). This i s i l l u s t r a t e d in Fig.1. Note that since integral intensity must remain constant, - t - o o (2.6) [ F e ( o ) d o = 1, V 9 , - OB where the symbol V means "for any value of", the following expression holds: (2.7) F e(o) = F e(xP 2(cos9)) = F 0 ( x ) / P 2 ( C O S 9 ) . Also the powder spectrum G(u) i s the superposition of contributions from a l l possible angles 9 : TT/2 (2.8) G(o) = | d9F eU)p(0) , 0 where p(e) = sin 9 is the s o l i d angle weighting factor, designating what part of a randomly oriented mixture of domains has orientation between 9 and 9+d9. Let us change the integration in Eq.(2.8) to the frequency 8 domain : p(9 ) d e = p(o)do , or ( 2 . 9 ) pU) = p(9) (do/ d e ) - 1 = [ 3 x ( 2 o + x ) ] - 1 / 2 , where E q . ( 2 . 5 ) was u s e d . A n o t h e r c o n s e q u e n c e o f E q . ( 2 . 5 ) i s t h a t t h e a l l o w e d v a l u e s o f u a r e between -x/2 and x. Thus we have t o i n t r o d u c e t h e f u n c t i o n l ( u , x ) : (2. 1 0 ) I ( u , x ) = 1, -x/2<u<x 0, o t h e r w i s e C o m b i n i n g t h e r e s u l t s o f E q . ( 2 . 5 ) , ( 2 . 7 ) , ( 2 . 9 ) a n d (2 . 1 0 ) we c a n r e w r i t e E q . ( 2 . 8 ) a s : x ( 2 . 1 1 ) G U ) = J d u F e U ) p ( o ) -x/2 1 = J d x F 0 ( x ) [ 3 x ( 2 o + x ) ] - ' / 2 l U , x ) , 0 w h i c h i s e x a c t l y t h e E q . (2) o f [ 3 ] . F i g . 2 i l l u s t r a t e s t h e r e l a t i o n s h i p b e t w e e n a g a u s s i a n F 0 ( u ) and c o r r e s p o n d i n g G ( u ) . I n h e r e n t t o most o f t h e e x i s t i n g ways o f o b t a i n i n g NMR s p e c t r a i s t h e f a c t t h a t t h e s p e c t r u m i s m e a s u r e d a t a d i s c r e t e s e t o f p o i n t s . Thus we o n l y know G ( o ) a t w = ±n/N, where n 0,1,...N a nd so t h e r e g i o n -1<o<1 i s s u b d i v i d e d i n t o 2N e q u a l l y 9 Fig.2 R e l a t i o n s h i p between an o r i e n t e d spectrum F 0 ( o ) and i t s corresponding powder p a t t e r n G(o). Notice the c h a r a c t e r i s t i c high edge and low shoulder of the powder p a t t e r n . 10 s p a c e d i n t e r v a l s . L e t us d e n o t e our measured powder p a t t e r n by g( ± n ) = G ( ± n / N ) . We s h a l l r e p r e s e n t F 0 ( x ) by a h i s t o g r a m f ( 2 k ) w i t h t h e w i d t h of e a c h h i s t o g r a m i n t e r v a l b e i n g 2/N, i . e . , where k = 1,2...<N/2. How w i l l t h i s d i s c r e t i z a t i o n of G and F a f f e c t E q . ( 2 . 1 l ) ? I m m e d i a t e l y i t s h o u l d be n o t e d t h a t due t o t h e n a t u r e o f t h e f u n c t i o n I ( u , x ) a d i s t i n c t i o n has t o be made between c a s e s where k>n and k=n. Note t h a t o n l y one h a l f of t h e h i s t o g r a m i n t e r v a l ( ( 2 k - 1 ) / N , ( 2 k + 1 ) / N ] , where k=n, i s a l l o w e d by l ( u , x ) ; i . e . t h e i n t e g r a t i o n i n t h i s i n t e r v a l i s a c t u a l l y o v e r t h e i n t e r v a l ( 2n/N,(2n+1)/N]. W i t h t h i s i n mind E q . ( 2 . 1 1 ) can be r e w r i t t e n a s , ( f o r n e g a t i v e u = -n/N) (2.12) F 0 ( x ) = f ( 2 k ) , (2k-1)/N<x<(2k+1)/N N/2 (2.13) g ( - n ) = I f ( 2 k ) k>n (2k+1)/N f d x [ 3 x ( x - 2 n / N ) ] - 1 / 2 ( 2 k - 1 ) / N + f ( 2 n ) (2n+1)/N J d x [ 3 x ( x - 2 n / N ) ] - 1 / 2 2n/N P e r f o r m i n g t h e i n t e g r a t i o n y i e l d s : 11 N/2 f(2k) (2.14) g(-n) = E In k>n 3 1 / 2 [ (2k+1 )( 2 k - 2 n + 1 ) ] 1 ' 2 + 2k-n+1 [(2k-1)(2k-2n-1)] 1/ 2+2k-n-1 f (2n) + In 31/2 (2n+1) 1/ 2+n+1 S i m i l a r l y , f o r p o s i t i v e o = n/N : N/2 f(2k) (2.15) g(n) = I In 2k>n 3 1 / 2 [(2k+1)(2k+2n+1)] l / 2+2k+n+1 '[ (2k-1 ) (2k + 2n-1 ) ] 1 / 2 + 2k+n-1 f (2n) + In 31/2 [(n+1)(3n+1)] 1 / 2+2n+1 (3 1' 2+2)n The above expressions (2.14) and (2.15) w i l l now allow us to s y s t e m a t i c a l l y c a l c u l a t e the values of f(2k) ( o r i e n t e d spectrum) from the e x p e r i m e n t a l l y known values of g(n) (powder spectrum). Let us see how t h i s can be done. Consider an a n i s o t r o p i c chemical s h i f t spectrum with i t s edge at negative f r e q u e n c i e s . It should look something l i k e F i g . 2 . I t can be seen that at s u f f i c i e n t l y l a r g e values of n, g(-n) disappears i n t o n o i s e . Thus i f we choose n, such that i t i s w e l l i n s i d e the noise r e g i o n , we can assume that (2.16) f(2k) =0, V k>n,. 12 Then we can use expression (2.14) to calculate f(2n,). It i s obvious that the sum in that expression does not have any non-zero terms because of (2.16). Thus we can rewrite (2.14) as : f(2n,) = g(- n i)3 1/ 2(ln|[(2n,+1) 1 / 2+n,+1]/n,|)- 1 . Now we can repeat the procedure for f(2n,-2). Note that now there is one non-zero term in the sum, namely, the one a r i s i n g from f(2n,) : g(-n,+1) = f(2n 1)3- 1 / 2ln|{[3(2n,+l)] 1 / 2+n 1+2}/{(2n l-1) 1/ 2+n 1}| + f ( 2 n , - 2 ) 3 1 / 2 l n | [ ( 2 n i - 1 ) 1 / 2 + n , ] / ( n , - 1 ) | . The above expression can again be inverted to determine f(2n!~2) from the known value of g(-n,+l) and the already determined value of f(2n,). This process can be continued u n t i l the entire oriented spectrum is obtained. An example of the use of the above procedure i s shown in Fig.3. The simulated oriented spectrum in Fig.3a is a single gaussian l i n e . The powder pattern in Fig.3b has some random noise added to i t to make the situation more r e a l i s t i c . The signal-to-noise r a t i o (S/N) is deliberately kept low ( approximately 50 ) for although modern NMR techniques can produce much better results i t is important to see what depakeing can do in a more d i f f i c u l t case of a system yi e l d i n g limited signal-to-noise r a t i o . The depaked spectrum in Fig.3c 13 F i g . 3 Simple case of depakeing. a) simulated o r i e n t e d spectrum: a s i n g l e gaussiari l i n e ; b) corresponding powder p a t t e r n with random noise added. S i g n a l - t o - n o i s e (S/N) i s approximately 50. R a t i o of o r i e n t a t i o n independent to o r i e n t a t i o n dependent broadening R=0.01; c) r e s u l t of depakeing of (b). 1 4 agrees very well with the i n i t i a l oriented spectrum. The S/N ra t i o i s much smaller than for the powder spectrum, which is consistent with the fact that the algorithm derives the information on the oriented spectrum from the small fraction of the c r y s t a l l i n e domains oriented around 0=90° ("edge"). Also the noise increases as the distance from u 0 and thus the numerical c o e f f i c i e n t s in (2.14) and (2.15) increase making the contributions from the random noise at the edges, of the powder spectrum much greater than the contributions from the random noise of the same intensity in the middle of i t . The choice of n, close to the relevant part of the powder pattern could, in addition to savings in time and expence of computation, also make an improvement in the appearance of the depaked spectrum by eliminating wings of the oriented spectrum where noise i s most evident, but i t would only be a visua l improvement since the noise elsewhere in the depaked spectrum is largely independent of the choice of n,. The following remark should be made about the depakeing procedure as described so far. Notice that only one of the formulae (2.14) and (2.15) i s s u f f i c i e n t to determine f(2k), i. e . a l l the relevant information is contained in only one-half of the powder pattern. Obviously i t is advantageous to use the "edge" side since the signal-to-noise r a t i o i s considerably higher there. However, the expressions (2.14) and (2.15) are not t o t a l l y symmetrical so i t is necessary to determine beforehand which side is the "edge" and which one is the "shoulder". In the case of Fig.3 this i s t r i v i a l enough. 15 However, i n a d i f f e r e n t e x p e r i m e n t a l s i t u a t i o n t h i s m i g h t not be so e a s y . F o r example, e a c h of t h e o r i e n t e d s p e c t r a , c o n s t i t u t i n g t h e powder p a t t e r n , o f t e n has an o r i e n t a t i o n i n d e p e n d e n t component, t a k i n g i t s o r i g i n i n t h e o r i e n t a t i o n i n d e p e n d e n t p a r t of t h e H a m i l t o n i a n . When R, t h e r a t i o of t h e o r i e n t a t i o n i n d e p e n d e n t t o t h e o r i e n t a t i o n d e p e n d e n t b r o a d e n i n g , i s s m a l l , as i n f a c t i s t h e c a s e i n F i g . 3 where o r i e n t a t i o n i n d e p e n d e n t b r o a d e n i n g a c c o u n t s f o r 1% o f t h e t o t a l l i n e w i d t h , t h e asymmetry of t h e powder p a t t e r n i s l a r g e and i t s edge i s e a s i l y i d e n t i f i a b l e . I f , however, t h e s y s t e m has two components w i t h a n i s o t r o p i e s of o p p o s i t e s i g n ( as i s t h e c a s e f o r t h e 3 1 P s p e c t r a of membrane sy s t e m s w i t h m i x t u r e s of h e x a g o n a l and b i l a y e r p h a s e s ) , or i f t h e o r i e n t e d s p e c t r u m i s not a s i n g l e l i n e but a d o u b l e t l e a d i n g t o t h e symmetric powder p a t t e r n ( w h i c h c an be i d e n t i f i e d w i t h [ G ( o ) + G ( - u ) ] / 2 i n o u r n o t a t i o n ) , as i s t h e c a s e f o r t h e q u a d r u p o l a r i n t e r a c t i o n s , t h e n t h e s p e c t r u m w i l l have b o t h "edges" and " s h o u l d e r s " on b o t h s i d e s of t h e s p e c t r u m . In t h e s e c a s e s , and a l s o when t h e w i d t h of t h e s i n g l e l i n e becomes l a r g e , t h e a s s u m p t i o n t h a t F 0 ( x ) i s c o n t a i n e d w i t h i n t h e i n t e r v a l [0,1] has t o be r e c o n s i d e r e d . 16 I I I . DEPAKEING DETAILS OF THE ALGORITHM. What happens t o our d e p a k e i n g f o r m u l a e , i f t h e a s s u m p t i o n t h a t F 0 ( x ) i s c o n t a i n e d w e l l w i t h i n t h e i n t e r v a l [0,1] i s n o t made t o b e g i n w i t h ? An o b v i o u s way t o d e a l w i t h s u c h a c a s e i s t o c o n s i d e r t h e F 0 ( x ) as c o n s i s t i n g of two f u n c t i o n s , e a c h c o n t a i n e d w i t h i n i n t e r v a l s [-1,0] and [0,1] and t o t r e a t e a c h of them s e p a r a t e l y . O b v i o u s l y , t h e r e w i l l now be f o u r terms i n t h e e x p r e s s i o n f o r G ( u ) , two from e a c h s i d e . A f t e r some t r i v i a l c o n s i d e r a t i o n s d e a l i n g m o s t l y w i t h s i g n s and l i m i t s of i n t e g r a t i o n t h e f o r m u l a f o r n e g a t i v e o = -n/N, a n a l o g o u s t o t h e e x p r e s s i o n ( 2 . 1 3 ) , w i l l t a k e t h e f o l l o w i n g form : (3.1) g ( - n ) E f ( 2 k ) A [ ( 2 k - 1 ) / N , ( 2 k + 1 ) / N ] k>n + E k<-n/2 f ( 2 k ) A [ ( 2 k - 1 ) / N ) , ( 2 k + 1 ) / N ] + f ( 2 n ) A [ 2 n / N , ( 2 n + 1 ) / N ] + f ( - n ) A [ - n / N , ( - n - 1 ) / N ] where A [ a , b ] b J d x [ 3 x ( x + 2 n / N ) ] - 1 / 2 , a and t h u s t h e d e p a k e i n g e x p r e s s i o n s w i l l become : 1 7 (3.2) g(-n) = I 3 - 1 / 2 f ( 2 k ) l n 2k>2n [ (2k+1 ) (2k-2n+1)3 1 / 2 + 2k-n+1 [ (2k- 1 ) ( 2 k - 2 n - 1 ) ] 1 / 2 + 2k-n-1 + Z 3 " 1 / 2 f ( 2 k ) l n 2k<-n [(2k+1)(2k-2n+1)] 1 / 2+2k-n+1 [(2k-1)(2k-2n-1)] 1/ 2+2k-n-1 + 3 " 1 / 2 f ( 2 n ) l n (2n+1) 1/ 2+n+1 + 3 " 1 ^ 2 f ( - n ) l n ( 3 1 / 2 - 2 ) n [(n+1)(3n+1)] 1/ 2-2n-1 A f t e r r e p e a t i n g the same f o r p o s i t i v e o = + n/N, we get (3.3) g(n) = I 3 " 1 / 2 f ( 2 k ) l n 2k<-2n [(2k+1)(2k+2n+1)] 1 / 2+2k+n+1 [(2k-1)(2k+2n-1)] 1 / 2+2k+n-1 + 1 3 " , / 2 f ( 2 k ) l n 2k>n [ (2k+1 )(2k + 2 n + 1 ) ] l / 2 + 2k+n+1 [(2k-1)(2k + 2 n - 1 ) ] 1 / 2 + 2k+n-1 + 3 " 1 / 2 f ( 2 n ) l n -n (2n+1) l / 2-n-1 + 3 " 1 / 2 f ( n ) l n [(n+1)(3n+1)] 1/ 2+2n+1 ( 3 1 / 2 + 2 ) n 18 The above two formulae are to be used in a l l subsequent cal c u l a t i o n s . It should be pointed out that these expressions d i f f e r s l i g h t l y from the ones presented in [3]. F i r s t l y , there are two extra terms appearing here because the assumption about F 0(x) being confined to the positive values of x has been removed. Secondly, the histogram intervals are here defined as ((2k-1)/N,(2k+1)/N] as opposed to (2k/N,2(k+1)/N] in [3], which results in di f f e r e n t integration l i m i t s . However, i f these changes are made, the equations in [3] become i d e n t i c a l to the formulae (3.2) and (3.3). Having developed the general formalism of our method, we should now pay some attention to the c a l c u l a t i o n a l strategy best suited for the occasion. An obvious way of using formulae (3.2) and (3.3) i s to apply them simultaneously. Thus as we start from the noise region, the f i r s t systematic intensity we encounter as the running index ±n decreases is interpreted as a "shoulder" intensity coming from the oriented spectrum on the opposite side. As the index reaches one-half of i t s sta r t i n g value, the intensity not accounted for as "shoulder" intensity is now assigned to the "edge" of the powder pattern of the anisotropy of the same sign. Altogether, when the running index reaches zero from both sides, the data w i l l only have been used once, i . e . the procedure is a straightforward one-shot calculat ion. Such an approach, successful when applied to simulated spectra, proved to be inadequate when depakeing of real spectra was attempted [9]. This can be explained in the following way. 19 As one sees from the shape of the powder pattern, the "shoulder" intensity is always smaller than the "edge" intensity. This means that when the f i r s t non-zero intensity i s encountered and is treated as "shoulder" intensity, the appropriate contributions to the powder pattern at d i f f e r e n t frequencies are obtained by multiplying this intensity by sometimes very large numerical factors, as determined by (2.9). Only the intensity remaining after subtracting a l l "shoulder" contributions i s assigned to the appropriate "edge". On the other hand, the algorithm does not distinguish between the noise and the "true" intensity and treats both in the same manner. Thus, i f noise is present in the spectrum, i t w i l l be treated as the true "shoulder" and w i l l make disproportionately large contributions at the frequencies of associated "edge", masking the information we are a f t e r . It has to be mentioned that in addition, the signal-to-noise ratios are much smaller for the shoulder side, so that even a minimal amount of noise can have serious adverse e f f e c t s . To avoid these d i f f i c u l t i e s an i t e r a t i v e algorithm was proposed in [3] for use with symmetric spectra. A similar i t e r a t i v e approach was used in this study with, in general, asymmetric spectra. Here "asymmetric" refers to the lineshape of the powder pattern and is not related to the so-called "asymmetry parameter", associated with deviations from a x i a l symmetry of dipolar e l e c t r i c or magnetic f i e l d s [10]. To describe t h i s i t e r a t i v e algorithm l e t us f i r s t r e c a l l that during simultaneous cal c u l a t i o n of both sides of the 20 oriented spectrum, the intensity in a channel of the powder pattern i s divided into two parts, one being interpreted as a "shoulder" side of the pattern corresponding to the oriented spectrum of the anisotropy of the same sign and another being interpreted as an "edge" side of the pattern corresponding to the oriented spectrum of the anisotropy of the opposite sign. In the i t e r a t i v e procedure, on the other hand, we f i r s t obtain an i n i t i a l estimate ( f i r s t i t e r ation) by treating a l l of the intensity of the powder pattern on one side as "edge" intensity which means using only the f i r s t and the t h i r d terms in the expressions (3.2) and (3.3). This produces a spurious contribution in the calculated side of the oriented spectrum, coming from the "shoulder" i n t e n s i t i e s of the other side of the oriented spectrum being interpreted as "edge" i n t e n s i t i e s of the same side. In the second i t e r a t i o n a l l terms in (3.2) and (3.3) are used to calculate t h i s other side of the oriented spectrum. Obviously, since the f i r s t i t e r a t i o n overestimates the oriented spectrum as i t treats a l l the intensity as an "edge" when only a part of i t i s actually an "edge", the second i t e r a t i o n w i l l underestimate the other side of the oriented spectrum by subtracting too large a contribution. If the process is now continued, the same is true of a l l odd and even i t e r a t i o n s . Since the spurious contribution is much smaller than the true F 0(x) i t i s systematically eliminated by the i t e r a t i o n process. In most cases three to five iterations were found to be s u f f i c i e n t for convergence. It should be noted that such an i t e r a t i v e approach 21 i n t r o d u c e s a c e r t a i n asymmetry i n t o t h e p r o b l e m : how does one c h o o s e t h e s i d e t o be u s e d as t h e i n i t i a l e s t i m a t e ? I n t h e c a s e of an o b v i o u s asymmetry, one s h o u l d , no d o u b t , use t h e "edge" s i d e , but what i f t h e s p e c t r u m i s s u f f i c i e n t l y b r o a d e n e d t o mask o f f t h e s i g n of t h e asymmetry? A c o n s i d e r a b l e amount of t i m e was s p e n t i n t h i s s t u d y on c h e c k i n g t h e s t a b i l i t y of t h e a l g o r i t h m a g a i n s t s u c h e r r o r s of judgement and i t s c o n v e r g e n c e was f o u n d t o be e x c e l l e n t . F i g . 4 c o n t a i n s an i l l u s t r a t i o n of t h e use o f t h e i t e r a t i v e d e p a k e i n g as a p p l i e d t o a s i m u l a t e d powder p a t t e r n . As may be s e e n , even when t h e f i r s t i t e r a t i o n i s made from t h e wrong ( i . e . " s h o u l d e r " ) s i d e of t h e powder p a t t e r n t h e i t e r a t i v e p r o c e s s q u i c k l y c o n v e r g e s t o t h e same c o r r e c t o r i e n t e d s p e c t r u m . To summarize, we have d e v e l o p e d an a l g o r i t h m w h i c h a l l o w s one t o o b t a i n t h e o r i e n t e d NMR s p e c t r u m from t h e powder s p e c t r u m of t h e s y s t e m w i t h l o c a l a x i a l symmetry. The p r o c e d u r e not o n l y d e t e r m i n e s t h e a n i s o t r o p y of t h e s y s t e m , ( e . g . a n i s o t r o p y of c h e m i c a l s h i f t ) but a l s o g i v e s t h e l i n e s h a p e of t h e o r i e n t e d s p e c t r u m , t h u s p r o v i d i n g t h e p r e v i o u s l y u n a v a i l a b l e i n f o r m a t i o n a b o u t i n t e r a c t i o n s r e s p o n s i b l e f o r t h e b r o a d e n i n g . 22 -10 0 10 -10 0 10 Fig.4 An i l l u s t r a t i o n of i t e r a t i v e depakeing. a) i n i t i a l simulated o r i e n t e d spectrum; b) powder pa t t e r n corresponding to ( a ) . Random noise added with S/N = 70; c) f i r s t i t e r a t i o n , t r e a t i n g r i g h t side of (b) as "edge"; d) same f o r the l e f t s i d e of (b); e) and f) f o u r t h and f i f t h i t e r a t i o n s converge to ( a ) , independently of the side of the f i r s t i t e r a t i o n . 23 IV. DEPAKEING : APPLICATION TO NMR SPECTRA Having developed the formalism of depakeing, we now i n v e s t i g a t e the l i m i t a t i o n s of the method and i t s u s e f u l n e s s i n d i f f e r e n t experimental s i t u a t i o n s . In t h i s chapter we deal with simulated s p e c t r a only. T h i s way we have t o t a l c o n t r o l over the l i n e s h a p e that i s being depaked, and so the unambiguous i n t e r p r e t a t i o n of the r e s u l t s of our c a l c u l a t i o n s i s p o s s i b l e . By s i m u l a t i n g d i f f e r e n t experimental s i t u a t i o n s we l e a r n to recognize t h e i r c h a r a c t e r i s t i c " s i g n a t u r e s " i n the depakeing, which i s a p r e r e q u i s i t e to the i n t e r p r e t a t i o n of r e a l experimental data. A l l the c a l c u l a t i o n s presented here were made on the f a c i l i t i e s of the Computing Centre of the U n i v e r s i t y of B r i t i s h Columbia, which i n c l u d e d an AMDAHL IV computer and l i b r a r i e s and device supports provided by the Michigan Terminal System. A h i g h l y t e r m i n a l - i n t e r a c t i v e depakeing program was developed over the course of t h i s study. The FORTRAN l i s t i n g of i t s c u r r e n t v e r s i o n can be found i n the APPENDIX. Let us now c o n s i d e r a few aspects of the NMR s p e c t r a that c o u l d make t h e i r d i r e c t i n t e r p r e t a t i o n d i f f i c u l t and see how the depakeing can be of help. One t h i n g that the depakeing can do very s u c c e s s f u l l y i s to r e s o l v e two neighbouring s p e c t r a l l i n e s . An i l l u s t r a t i o n of t h i s i s given in F i g . 5 . Here the powder p a t t e r n a s s o c i a t e d with 24 an o r i e n t e d s p e c t r u m c o n s i s t i n g of two c l o s e l y s p a c e d l i n e s does not e x h i b i t any p r o m i n e n t f e a t u r e s t h a t would e n a b l e one t o i n t e r p r e t i t as a t w o - l i n e s p e c t r u m . O n l y a s l i g h t bend i n t h e "edge" s i d e o f t h e s p e c t r u m s u g g e s t s t h e e x i s t e n c e of t h e s e two l i n e s . A f t e r d e p a k e i n g , however, t h e two l i n e s a r e c l e a r l y r e s o l v e d and, moreover, t h e i r r e l a t i v e w i d t h s and i n t e n s i t i e s a r e d e t e r m i n e d c o r r e c t l y , a s one can see by c o m p a r i n g them t o t h e i n i t i a l F 0 ( x ) u s e d i n t h e s i m u l a t i o n . S i m i l a r r e s u l t i s o b t a i n e d i n t h e c a s e o f a r e a l s p e c t r u m , as was shown i n [ 3 ] . T h e r e t h e f a c t t h a t t h e d e p a k e i n g g i v e s c o r r e c t i n t e g r a l i n t e n s i t i e s o f t h e l i n e s was u s e d t o a s s i g n t h o s e l i n e s t o d i f f e r e n t p o s i t i o n s i n t h e c h a i n of a p e r d e u t e r a t e d membrane l i p i d m o l e c u l e . A n o t h e r e x p e r i m e n t a l d i f f i c u l t y w h i c h was b r i e f l y m e n t i o n e d e a r l i e r i s t h e p r e s e n c e o f an o r i e n t a t i o n i n d e p e n d e n t component i n e a c h of t h e c o n t r i b u t i o n s from d i f f e r e n t o r i e n t a t i o n s w h i c h c o n s t i t u t e t h e powder p a t t e r n . Such an o r i e n t a t i o n i n d e p e n d e n t b r o a d e n i n g c o n t r i b u t e s g r e a t l y t o t h e t o t a l w i d t h of t h e s p e c t r u m and as t h i s c o n t r i b u t i o n i n c r e a s e s i t may o b s c u r e t h e "edge" and s h i f t t h e maximum i n t e n s i t y i n t h e powder p a t t e r n t o w a r d s t h e Larmor f r e q u e n c y u 0 . In t h e l i m i t o f l a r g e R, i . e . when t h i s b r o a d e n i n g i s much g r e a t e r t h a n t h e o r i e n t a t i o n d e p e n d e n t b r o a d e n i n g d e t e r m i n e d by t h e a x i a l l y s ymmetric i n t e r a c t i o n s , t h e powder p a t t e r n becomes a b r o a d l i n e a l m o s t s y m m e t r i c a r o u n d o 0 . Here i t must be made c l e a r t h a t t h i s e x p e r i m e n t a l s i t u a t i o n does not c o n f o r m t o t h e a s s u m p t i o n s t h a t were made i n t h e v e r y b e g i n n i n g , namely, t h a t e i t h e r t h e 25 a | 1 V i i i i i i i i i !_J[ c 1 1 - 1 0 0 1 0 Fig.5 R e s o l v i n g two s p e c t r a l l i n e s by depakeing. a) i n i t i a l o r i e n t e d spectrum; b) powder p a t t e r n with random noise added, S/N=70 ; c) depaked spectrum. 26 interactions are a x i a l l y symmetric or that there i s a rapid a x i a l l y symmetric molecular motion in the system. Thus i t i s not surprising that an attempt to depake such a l i n e does not produce any meaningful results, as i s apparent from Fig.6, where R=10. However, when the orientation independent broadening i s not so overwhelmingly large compared to the orientation dependent broadening so that the assumptions of the model are s a t i s f i e d only p a r t i a l l y , depakeing s t i l l proves to be useful. An example of such a case where the r a t i o of orientation independent and orientation dependent broadenings R=0.4 is presented in Fig.7. Note that the direc t determination of the anisotropy of chemical s h i f t (ACS) from the powder pattern in Fig.7b i s p r a c t i c a l i y impossible. The depaked spectrum in Fig.7c, however, provides such information ! Furthermore, the lineshape of the depaked spectrum contains the information about the orientation independent broadening i t s e l f . By comparing Fig.7a and Fig.7c one can see that the o r i g i n a l lineshape i s closely reproduced by the depaked spectrum everywhere except within a certain interval around Larmor frequency u 0. In this p a r t i c u l a r case the width of t h i s i n t e r v a l i s almost as large as the ACS, so that the lineshape i s seriously affected and there is a certain ambiguity in the interpretation of the depaked spectrum. This i s a l i m i t i n g case of the a p p l i c a b i l i t y of depakeing. I f , however, the r a t i o R is made smaller and thus the width of the interval determined by the orientation independent broadening i s reduced and i t does not obscure most of the lineshape i t becomes easy to estimate the width of this 27 1 0 0 1 0 F i g . 6 Extreme case of an e x c e s s i v e l y l a r g e o r i e n t a t i o n independent broadening. a) i n i t i a l o r i e n t e d spectrum, not i n c l u d i n g the o r i e n t a t i o n independent broadening; b) powder p a t t e r n assuming very l a r g e (R=10) o r i e n t a t i o n independent broadening; c) r e s u l t of depakeing of ( b ) . - 1 0 0 1 0 Fig.7 E f f e c t of a moderately l a r g e o r i e n t a t i o n independent broadening. a) i n i t i a l o r i e n t e d spectrum, not i n c l u d i n g the o r i e n t a t i o n independent broadening; b) powder p a t t e r n assuming R=0.4 ; c) r e s u l t of depakeing of ( b ) . 29 o r i e n t a t i o n i n d e p e n d e n t b r o a d e n i n g i n a d d i t i o n t o t h e d e t e r m i n a t i o n of t h e p r e c i s e l i n e s h a p e of t h e o r i e n t e d s p e c t r u m . Such a c a s e when R=0.1 and t h e l i n e s h a p e of t h e powder p a t t e r n a p p r o a c h e s what i s commonly c a l l e d i n t h e l i t e r a t u r e a " s u p e r - l o r e n t z i a n " shape [ 1 1 ] i s i l l u s t r a t e d i n F i g . 8 . More p r e c i s e d e t e r m i n a t i o n of t h e w i d t h of t h e o r i e n t a t i o n i n d e p e n d e n t component i s hampered by t h e d i f f i c u l t i e s of f i n d i n g t h e e x a c t w i d t h of t h i s i n t e r v a l i n the* d e p a k e d s p e c t r u m , but a p a r t f r o m t h a t a s i m p l e " r u l e of thumb" a p p e a r s t o h o l d : a s s u m i n g g a u s s i a n l i n e s h a p e f o r t h e o r i e n t a t i o n i n d e p e n d e n t b r o a d e n i n g i t s w i d t h p a r a m e t e r a i s a p p r o x i m a t e l y o n e - h a l f of t h e w i d t h of t h i s i n t e r v a l . A n o t h e r p r o b l e m t h a t i s o f t e n e n c o u n t e r e d e s p e c i a l l y where s o p h i s t i c a t e d NMR i n s t r u m e n t s a r e not e a s i l y a v a i l a b l e i s t h e p o s s i b i l i t y t h a t t h e l i n e w i d t h of t h e o r i e n t e d s p e c t r u m i s c o m p a r a b l e t o t h e ACS. One way t o overcome t h i s p r o b l e m i s t o use h i g h e r f i e l d NMR but i f s u c h a s o l u t i o n i s not p o s s i b l e d e p a k e i n g can p r o v i d e an e a s i l y . a v a i l a b l e a l t e r n a t i v e . F i g . 8 has a l r e a d y p r o v i d e d us w i t h a good example of an o r i e n t e d s p e c t r u m w h i c h e x t e n d s on b o t h s i d e s of u 0 . A n o t h e r example i s g i v e n i n F i g . 9 . Here t h e w i d t h of t h e o r i e n t e d l i n e i s t w i c e t h e ACS and t h e r a t i o R i s assumed s m a l l ( R=0.1 ). Even t h e extreme c a s e when t h e ACS i s z e r o and t h e l i n e s h a p e o f t h e powder s p e c t r u m i s " s u p e r - l o r e n t z i a n " can be h a n d l e d s u c c e s f u l l y by d e p a k e i n g . An example i s p r e s e n t e d i n F i g . 1 0 . Here ACS=0 and t h e r a t i o R= 0 . 0 2 . As e x p e c t e d , t h e l i n e s h a p e of t h e o r i e n t e d s p e c t r u m i s r e p r o d u c e d p r e c i s e l y by d e p a k e i n g and s i n c e - 1 0 0 .10 Fig.8 E f f e c t of a small o r i e n t a t i o n independent broaden i n g . a) i n i t i a l o r i e n t e d spectrum, not i n c l u d i n g the o r i e n t a t i o n independent broadening; b) powder p a t t e r n assuming R=0.1 ; c) r e s u l t of depakeing of (b). 31 0 1 0 F i g . 9 E f f e c t of l i n e w i d t h being comparable to the ACS a) i n i t i a l o r i e n t e d spectrum. The l i n e w i d t h , not i n c l u d i n g the o r i e n t a t i o n independent broadening, i s twice as l a r g e as the ACS; b) powder p a t t e r n assuming R=0.1; c) r e s u l t of depakeing of (b). 32 the o r i e n t a t i o n independent component c o n s t i t u t e s only 2% of the t o t a l l i n e w i d t h the d i s t o r t i o n e f f e c t s are c o n f i n e d to a very small i n t e r v a l around o 0 . T h i s can be of great h e l p i n the i n t e r p r e t a t i o n of some 1H NMR s p e c t r a , l i k e c o n c e n t r a t e d aqueous s o l u t i o n s of soap. A d e t a i l e d d i s c u s s i o n of such logarithmmic l i n e s h a p e s was given in [11]. I f , however, the d i f f i c u l t y of the ACS being zero or of the same order of magnitude as the o r i e n t a t i o n dependent broadening i s compounded by the presence of a very l a r g e o r i e n t a t i o n independent broadening the unambiguous i n t e r p r e t a t i o n of the depaked spectrum becomes imposs i b l e . Another kind of d i f f i c u l t y that a user of depakeing should be aware of can be i l l u s t r a t e d with the help of Fig.10. The powder p a t t e r n in Fig.10b was obtained by s i m u l a t i o n i n which 1000 data p o i n t s were used. On the other hand, the width parameter e of the o r i e n t a t i o n independent broadening which determines the lower l i m i t of the l i n e w i d t h of i n d i v i d u a l l i n e s of which the powder p a t t e r n i s composed corresponds to only 2 p o i n t s ( s i n c e in t h i s case R=0.02 and the o r i e n t a t i o n dependent broadening i s one-tenth of the t o t a l width of the spectrum ). Thus i t becomes e s s e n t i a l to determine p r e c i s e l y the zero of the f i r s t moment of the spectrum which d e f i n e s the l o c a t i o n of the Larmor frequency u 0 . Even a small mistake here w i l l have a s e r i o u s e f f e c t on the depaked spectrum s i n c e the few c e n t r a l data p o i n t s most i n c o r r e c t l y t r e a t e d c o n t a i n a l a r g e p o r t i o n of the o v e r a l l i n t e n s i t y of the spectrum. T h i s i s i l l u s t r a t e d i n Fig.11 which d i f f e r s from Fig.10 i n only that the -10 0 10 Fig.10 E f f e c t of zero ACS ( " s u p e r - l o r e n t z i a n " l i n e s h a p e " ). a) i n i t i a l o r i e n t e d spectrum. ACS=0; b) " s u p e r - l o r e n t z i a n " powder p a t t e r n assuming R=0.02; c) r e s u l t of depakeing of ( b ) . 34 depaked spectrum was obtained a f t e r the l o c a t i o n of the zero of the f i r s t moment was a r t i f i c i a l l y s h i f t e d to the l e f t by one p o i n t . To ensure that t h i s type of e r r o r does not occur d u r i n g the depakeing of r e a l experimental s p e c t r a where the exact l o c a t i o n of o 0 i s not known beforehand and one has to r e l y completely on the c a l c u l a t i o n of the zero of the f i r s t moment great care should be e x e r c i s e d i n performing such c a l c u l a t i o n s . The depakeing program presented i n the APPENDIX c o n t a i n s a number of f e a t u r e s devoted to t h i s problem and s i m i l a r measures should be taken whenever depakeing i s attempted. From the experimental poi n t of view t h i s may be i n t e r p r e t e d as a requirement that the peaks of a powder spectrum should be d e f i n e d by a s u f f i c i e n t l y l a r g e number of data p o i n t s . In l e s s extreme cases t h i s d i f f i c u l t y does not a r i s e and a minor e r r o r in the d e t e r m i n a t i o n of the zero of the f i r s t moment w i l l not have any s e r i o u s e f f e c t on the r e s u l t s of depakeing. F i n a l l y , we should i n v e s t i g a t e what happens i f the spectrum i s a c t u a l l y two superimposed s p e c t r a with a n i s o t r o p i c s of opposite s i g n . T h i s c o u l d be the case i f the system c o n s i s t s of a mixture of two phases. Here i t should be e x p l i c i t l y noted that there i s a r e s t r i c t i o n on both c o n t r i b u t i n g components to have the same i s o t r o p i c chemical s h i f t , f o r only then are the r e s u l t s of depakeing meaningful. For many of the known membrane systems t h i s requirement i s s a t i s f i e d , so t h i s i s not a very s e r i o u s r e s t r i c t i o n ; however, one should keep i t i n mind when a p p l y i n g depakeing to systems where such an assumption can not be made a p r i o r i . F i g . 1 2 c o n t a i n s an example of depakeing of -10 0 10 Fig.1 1 E f f e c t of an e r r o r i n the l o c a t i o n of the zero of the f i r s t moment. a) i n i t i a l o r i e n t e d spectrum. ACS=0; b) " s u p e r - l o r e n t z i a n " powder p a t t e r n assuming R=0.02; c) r e s u l t of depakeing of (b) using i n c o r r e c t value for the l o c a t i o n of the zero of the f i r s t moment. 36 a two component spectrum. It should be noted that depakeing has a tremendous potential in cases l i k e t h i s . It i s a well established practice in the biochemical l i t e r a t u r e to interpret 3 1 P NMR spectra as belonging to lamellar or hexagonal phases of the phospholipids according to their lineshape, the main i d e n t i f i e r s being the sign and the magnitude of the ACS [ 4 ] . In some regimes there is often a p o s s i b i l i t y of coexistence of the two phases. In every such case the interpretation of the spectrum has been limited to a rather q u a l i t a t i v e discussion accompanied by some simulation of the lineshape at best. In almost a l l of these cases proton decoupling was necessary to obtain some degree of precision in the determination of the ACS. This, of course, results in a loss of information about the dipolar interactions in the system. Depakeing, on the other hand, can resolve a l l of these problems : the true two-phase spectra are easily i d e n t i f i e d ; the ACS of the two phases is determined precisely; the integral intensity of each l i n e provides a measure for the r e l a t i v e amount of each phase present in the system; in addition, the depaked spectrum contains the information about the dipolar interactions in each phase. One should be cautioned here that a high quality powder spectrum is absolutely e s s e n t i a l . When only the q u a l i t a t i v e information is sought one tends to disregard various lineshape d i s t o r t i o n e f f e cts due to the f i n i t e pulse length [ 1 2 ] , the dead time of the amplifier etc., but depakeing of such distorted spectra inevitably produces a r t i f a c t s of various kinds since the lineshape i s i t s primary source of information. - 1 0 0 1 0 F i g . 1 2 Depakeing of a two component spectrum. a) i n i t i a l o r i e n t e d spectrum; b) powder p a t t e r n ; c) r e s u l t of depakeing of ( b ) . 38 To summarize, we have by now l e a r n e d enough a b o u t d e p a k e i n g and t h e way i t works under d i f f e r e n t c o n d i t i o n s t o be a b l e t o i n t e r p r e t t h e r e s u l t s w i t h a c e r t a i n d e g r e e of c o n f i d e n c e . In g e n e r a l , we e x p e c t d e p a k e i n g t o produce- t h e t r u e l i n e s h a p e of t h e o r i e n t e d s p e c t r u m when t h e s y s t e m s a t i s f i e s t h e a s s u m p t i o n of a x i a l symmetry. When t h i s a s s u m p t i o n i s s a t i s f i e d o n l y p a r t i a l l y we c a n e x p e c t c e r t a i n d i s t o r t i o n e f f e c t s w h i c h we c a n r e c o g n i z e by t h e i r c h a r a c t e r i s t i c " s i g n a t u r e s " i n t h e depaked s p e c t r u m . 39 V. FINAL REMARKS. In conclusion, the following points should be reiterated. F i r s t l y , depakeing is a numerical procedure of, es s e n t i a l l y , deconvolution of a composite integral function and so, following [3], the author does not claim to have circumvented the inevitable uncertainty associated with the method. On the other hand, depakeing is an extremely useful and easy to apply technique which produces results in some instances superior to previously available methods. As an example, in [7] the main body of research consisted of an extensive set of d i f f i c u l t experiments involving 1 9 F NMR of samples oriented between glass plates. Thus i t was possible to make comparison of the oriented spectra obtained later by depakeing of powder spectra of the same systems to the ones obtained d i r e c t l y . The agreement was found to be excellent, with the differences between the two type of spectra being, i t appears, completely accountable for by dif f e r e n t water content of the oriented and the powder samples used ( private communication with the authors of Ref. 7 ). Secondly, the need to have undistorted experimental spectra cannot be overemphasized. It is always advisable to exercise caution with respect to such experimental factors as the f i n i t e length of the RF ( radio frequency ) pulses, dead time of the amplifier, inhomogeneities of various kind d i s t o r t i o n s introduced by the indiscriminate use of f i l t e r s , excessive phase 40 c o r r e c t i o n s , e t c . I t i s impossible even to l i s t a l l the sources of l i n e s h a p e d i s t o r t i o n that have to be d e a l t with i f depakeing i s to be s u c c e s s f u l l y a p p l i e d s i n c e the l i n e s h a p e i s i t s primary source of i n f o r m a t i o n . I t i s t h e r e f o r e very encouraging that s e v e r a l methods of d e a l i n g with some of these experimental d i f f i c u l t i e s have been developed r e c e n t l y [12,13]. T h i s should i n c r e a s e the range of systems whose sp e c t r a can be i n t e r p r e t e d with the a i d of depakeing. 41 REFERENCES 1. G.E.Pake, J. of Chem. Phys. J_6, 327 (1948). 2. G.E.Pake, "Nuclear Magnetic Resonance", S o l i d State Physics v.2, Seitz and Turnbull, eds. New York, Academic Press Inc., (1956). 3. M.Bloom, J.H.Davis and A.L.MacKay, Chem. Phys. Lett. 80, 198 (1981). 4. P.R.Cullis and B.De K r u i j f f , Biochim. Biophys. Acta 507, 207 (1978). 5. J.Seelig, Biochim. Biophys. Acta 515, 105 (1978). 6. M.A.Hemminga and P.R.Cullis, J. of Magn. Resonance 47, 307 (1982). 7. S.R.Dowd, M.Englesberg, V.Simplaceanu and C.Ho, Xth I n t l . Conference on Magnetic Resonance in B i o l o g i c a l Systems, Stanford, C a l i f o r n i a , USA, Aug.29 - Sept.3, 1982. 8. A.Abragam The Princip l e s of Nuclear Magnetism. Oxford Univ. Press, London, (1961). 9. Prof. M.Bloom, private communication. 10. C.P.SIichter, Princi p l e s of Magnetic Resonance, Harper&Row, New York, (1963). 11. M.Bloom, E.E.Burnell, S.B.W.Roeder and M.I.Valic, J. of Chem. Phys., 66, 3012 (1977). 12. M.Bloom, J.H.Davis and M.I.Valic, Canadian J. of Phys, 58, 1510 (1980). 13. M.Ranee and R.A.Byrd, in preparation. 42 APPENDIX What f o l l o w s i s the FORTRAN l i s t i n g of the depakeing program developed in the course of t h i s study. I t should be noted that the program in i t s present form i s h i g h l y i n t e r a c t i v e and thus r e q u i r e s an a p p r o p r i a t e t e r m i n a l to run i t . In a d d i t i o n , the program makes use of a number of standard l i b r a r y s u broutines provided by the Michigan Terminal System (MTS) which i s adopted at present by the Computing Centre of the U n i v e r s i t y of B r i t i s h Columbia. Thus the d i r e c t use of t h i s program i s r e s t r i c t e d to MTS support. However, a l l these subroutines deal with i n t e r a c t i v e input/output and not with a c t u a l c a l c u l a t i o n s of the depaked s p e c t r a , so that an experienced programmer should have no d i f f i c u l t y adapting the program to the computing system a v a i l a b l e to him. Subroutine ASKIF has been k i n d l y made a v a i l a b l e for p u b l i c use by i t s author, Dr. Jess Brewer of the Department of P h y s i c s , U n i v e r s i t y of B r i t i s h Columbia. Comments were used throughout the program to make i t s s t r u c t u r e e a s i l y understandable but the f o l l o w i n g remarks may be of h e l p to the p o t e n t i a l user. The program r e q u i r e s the user to s p e c i f y c e r t a i n parameters which determine what part i s to be c o n s i d e r e d as the b a s e l i n e and what part - as the spectrum to be depaked. The b a s e l i n e c o n s i s t s of the two i n t e r v a l s : zero to BL ( B a s e l i n e L e f t ) and 43 BR ( B a s e l i n e Right ) to NP ( Number of P o i n t s i n the spectrum ). Thus the i n t e r v a l from BL to BR i s assumed to c o n t a i n a l l the r e l e v a n t d ata. B a s e l i n e c o r r e c t i o n i s performed by averaging over the i n t e n s i t y of the s p e c t r a l p o i n t s i n the b a s e l i n e and s u b t r a c t i n g the r e s u l t i n g number from the i n t e n s i t y of every s p e c t r a l p o i n t . C o n s i d e r a b l e a t t e n t i o n i s p a i d to the de t e r m i n a t i o n of the zero of the f i r s t moment of the spectrum. I n i t i a l l y , the f i r s t moment about the l e f t s i d e of the spectrum i s c a l c u l a t e d over the p o i n t s between BL and BR. Then the c a l c u l a t i o n of the f i r s t moment about t h i s p o i n t i s performed r e p e a t e d l y , with the i n t e r v a l of i n t e g r a t i o n extending symmetrically on both s i d e s of t h i s p o i n t from zero u n t i l i t i n c l u d e s both BL and BR, and the r e s u l t i s p l o t t e d . If BL and BR were chosen so that no systematic i n t e n s i t y was missed t h i s p l o t should l e v e l o f f at the l e v e l determined by the i n i t i a l ( BL to BR ) c a l c u l a t i o n of the f i r s t moment. If there i s no such l e v e l l i n g o f f BL should be decreased and/or BR should be in c r e a s e d to i n c l u d e a l l r e l e v a n t i n t e n s i t y . If the f l a t t a i l i s too long i t i s a d v i s a b l e to choose BL and BR c l o s e r to the region of i n t e r e s t ; t h i s reduces computation time without s a c r i f i c i n g a n y t hing. I t should be noted that the computation time i s approximately p r o p o r t i o n a l to the square of the number of p o i n t s being c a l c u l a t e d . The program p r o v i d e s the p o s s i b i l i t y to adjust a l l the r e l e v a n t parameters r e p e a t e d l y u n t i l the optimum i s reached before beginning the depakeing i t s e l f . The program performs depakeing one i t e r a t i o n at a time and 44 always displays the two most current i t e r a t i o n s . There are two places in the program where some form of d i g i t a l f i l t e r i n g is possible. F i r s t l y , the input data can be averaged into bins of arb i t r a r y s i z e . This i s recommended for i n i t i a l t r i a l s only when an approximate result i s needed quickly. The method of averaging i s deliberately kept crude to emphasize i t s preliminary character. Secondly, more refined binomial smoothing over no more than ±5 points can be applied to the result of depakeing. This is intended for improving, i f necessary, the appearance of the depaked spectrum by reducing the random noise. Such " f i l t e r i n g " should be used with great care to ensure that no systematic intensity is affected by i t . The program allows many attempts to be performed without losing the "raw" data. In fact, by setting the length of the interval for this binomial smoothing to zero the data can be recovered at any moment. A l l the p l o t t i n g in this program is intended for interactive use only and thus lacks the desired d e t a i l s and refinement. The f i n a l plots including a l l the figures of this thesis were produced using a set of p l o t t i n g routines written and supported by Mr. Geoff Johnson of the Department of Physics of the University of B r i t i s h Columbia. The output of the program is currently in terms of (X,Y) coordinate pairs and could be e a s i l y adapted to conform to the format of the graphic support available elsewhere. 45 C DEPAKE : depaking an NMR spectrum. This smart program C w i l l ask you for everything i t needs. C To compile : ( @ MTS ) C $RUN *FTN SCARDS=DEPAKE.SRC SPUNCH=DEPAKE.OBJ C The recommended way of running this program i s : C , $RUN DEPAKE.OBJ+*IG 2=infile 3=outfile 9=plotfile C C by : Edward Sternin Latest update 01.IX.82 C C INPUT (powder) on 2 C OUTPUT on 9 (plot) & 3 (oriented spectrum X,Y) C C Names of things used by the program C C NP= # OF PTS (CHANNELS) IN THE SPECTRUM C SW= SPECTRAL WIDTH, KHZ C N0= point, about which the f i r s t moment is zero C BR= BASELINE RIGHT C BL= BASELINE LEFT C I = CHANNEL INDEX, COUNTED FROM NO C P(l) / R(I) are i n t e n s i t i e s of powder C and oriented spectra C FAST= size of the averaging bin for the C crude primary ca l c u l a t i o n C HEX=1 corresponds to "edge on the l e f t " (HEXagonal) C configuration C HEX=-1 - to the bilayer configuration. C C PART 1 : PARAMETERS C IMPLICIT REAL(A-H,0~Z) REAL*8 AREA,AR,P,C2 INTEGER BL,BR,BLSAVE,BRSAVE,FAST,HEX,BINO,ATT LOGICAL ASKIF COMMON SW,NO,NP,NITER DIMENSION P(10000),R(10000),Y(12000),X(10000) DIMENSION PF(10000),IX(12000) DIMENSION XM(10000),WM1(10000),Y1(10000),X2(2),Z1(2) DIMENSION XSAVE(10000),YSAVE(10000) C C Default values C NP=4096 SWSAVE=50 BRSAVE=2200 BLSAVE=1800 C C C Determination of basic parameters C 46 C C C CALL FREAD(-2,'ENDFILE',1) DO 103 1=1,10001 103 CALL FREAD(2,'R:',P(I),&151) 151 NPSAVE=I-1 101 BL=BLSAVE BR=BRSAVE NP=NPSAVE SW=SWSAVE WRITE(6,160)BL,BR,NP,SW 160 FORMAT(/' x ' / x7 x x' / X x ' / I x x x x x x l x x x x x l x x x x x x l ' / 0 BL BR NP'// 'BL = ',16,' BR = ',16,' NP = ',16,/ 'SW = spectral width = ',G10.4,' kHz'/) IF(ASKIF(6,37,'Are you s a t i s f i e d with these values ?')) 1 GO TO 105 102 WRITE(6,161) 161 FORMAT('Enter BL,BR,NP,SW') CALL FREAD(6, '31 : ' ,BLSAVE,BRSAVE,NPSAVE, 'R:' ,SWSAVE) To sketch or not to sketch... Bin averaging. 105 BL=BLSAVE BR=BRSAVE NP=NPSAVE SW=SWSAVE DO 154 1=1,NP 154 PF(I)=0 FAST=1 IF(.NOT.ASKIF(6,33,'Do you want a crude ca l c u l a t i o n ?*)) 1 GO TO 104 WRITE(6,163) 163 FORMAT('Enter # of pts to be averaged into one bin.'/ 1 ' ( i t ' s good to have NP = a multiple of t h i s #)') CALL FREAD(6,'I:',FAST) 104 CONTINUE L=0 D6=SW/NP PMAX=0 J8=NP-FAST+1 DO 111 1=1,J8,FAST L=L+1 J9=I+FAST-1 DO 131 J=I,J9 PF(L)=PF(L)+P(J) 131 CONTINUE PF(L)=PF(L)/FAST 47 IF(PMAX.LT.PF(L))PMAX=PF(L) X(L)=(I+FAST/2)*D6 111 CONTINUE C3=NP/FAST C4=BL/FAST C5=BR/FAST NP=AINT(C3) BL=AINT(C4) BR=AINT(C5) C C B a s e l i n e c o r r e c t i o n C B1 =0 D6=SW/NP DO 112 I=1,BL B1=B1+PF(I) 112 CONTINUE DO 113 I=BR,NP B1=B1+PF(I) 113 CONTINUE B1=B1/(NP-BR+BL+1) DO 114 I=1,NP PF(I)=PF(I)-B1 114 CONTINUE C C Determine zero of the f i r s t moment C AR1 = 0 AR2 = 0 DO 115 I=BL,BR AR1=AR1+PF(I) AR2=AR2+I*PF(I) 115 CONTINUE N0=AINT(AR2/AR1+.5) C C Primary p l o t t i n g C X2(1)=N0*D6 X2(2)=X2(1) Z1(1)=-0.1*PMAX Z1 (2 ) = 1 .05*PMAX CALL ALSIZE(10.0,7.0) IF(.NOT.ASKIF(6,24,'Plot of the input data ?')) 1 GO TO 108 CALL ALAXIS('FREQUENCY (KHZ)',15,'INTENSITY',9) CALL ALSCAL(0.0,SW,-0.1*PMAX,1.05*PMAX) CALL ALGRAF(X,PF,L,0) CALL ALDASH(0.1,0.1,0.1,0.1) CALL ALGRAF(X2,Z1,-2,0) 108 CONTINUE 48 C C CALCULATION OF THE MOMENTS C WM11=0 WM12=PF(N0) WM2 = 0 WM3 = 0 WM4 = 0 KM=BR-N0 IF(KM.LT.(NO-BL))KM=N0~BL DO 141 K=1,KM WM13=PF(N0-K)-PF(N0+K) WM14=PF(N0-K)+PF(N0+K) WM12=WM12+WM14 WM11=WM11+K*WM13 WM2=WM2+K*K*WM14 WM3=WM3+K*K*K*WM13 WM4=WM4+K**4*WM14 WM11=WM11+K*WM13 WM12=WM12+WM14 WM1(K)=WM11/WM12 XM(K)=K*1.0 141 CONTINUE CALL ALSCAL(0.0,0.0,0.0,0.0) CALL ALAXIS('DISTANCE FROM NO',16,'FIRST MOMENT minus 1 NO',21) CALL ALGRAF(XM,WM1,KM,0) CALL PLCTRL('SEND',0) WM2=WM2/WM12 WM3=WM3/WM12 WM4=WM4/WM12 WRITE(6,168)WM2,WM3,WM4 168 FORMAT('Moments a r e : M2 = ',G16.8,' M3 = ',G16.8, . 1 ' M4 = ',G16.8) WRITE(6,166)BL,N0,BR,NP,SW 166 FORMAT('BL = ',G10.4,' NO = ',G10.4,' BR = ',G10.4, 1 ' NP = ',G10.4,' SW = ',G10.4) I F ( A S K I F ( 6 , 3 8 , ' D o a l l t h e s e p a r a m e t e r s seem OK, S i r ? ' ) ) 1 GO TO 200 GO TO 101 C C PART 2 : D e P a k e i n g C C F i r s t t i m e a r o u n d we know t h a t t h e c o n t r i b u t i o n C from t h e o t h e r s i d e i s z e r o ( s i n c e we h a v e n ' t C been t h e r e y e t ). C 200 NMAX=2*AINT(MAX0(BR-N0,N0-BL)/2.) C =even # C 49 210 HEX=1 IF(ASKIF(6,41,'Expecting a bilayer (edge on the right) ?' 1))HEX=-1 IPLOT=0 YMIN=1.E10 YMAX=-1 .El 0 DO 201 1=1,NP 201 R(I)=0. C DO 213 1=1,NMAX N=NMAX-I+1 NN=2*N NSGND=HEX*N SUM=0. IF(I.EQ.1)GO TO 218 N21=NN+2 N22=2*NMAX JFIN=N22-N21+1 IF(HEX.EQ.1)GO TO 217 N21=-2*NMAX 217 DO 212 J=1,JFIN,2 M=N21 - 1+J 212 SUM=SUM+R(N0+M)*RAT1(M,-NSGND) 218 NOR=N0+2*NSGND ORIENT=(1.7 3 2050808*PF(N0-NSGND)-SUM)/RAT2(-NSGND) R(NOR)=ORIENT C IPLOT=IPLOT+1 Y(lPLOT)=ORIENT IX(lPLOT)=NOR IF(YMIN.GT.ORIENT) YMIN=ORIENT IF(YMAX.LT.ORIENT) YMAX=ORIENT C 213 CONTINUE C NITER= 0 NS = 0 DO 2 19 1 = 1,1 PLOT 219 X(I)=D6*IX(I) 220 CALL DEPLOT(X,Y,IPLOT,NS,YMIN,YMAX) NITER=NITER+1 WRITE(6,229)NITER 229 FORMAT('Number of it e r a t i o n s completed = ',18) IF(.NOT.ASKIF(6,18,'Do you want more ?'))GO TO 306 NPSAV=IPLOT DO 224 1 = 1 ,I PLOT XSAVE(I)=X(I) 224 YSAVE(I)=Y(I) C C Iterative de-Pake-ing 50 C IPLOT=0 IHEX=HEX*(-1)**NITER DO 231 1=1,NMAX N=NMAX+1-I NSGND=IHEX*N SUM1=0. SUM2=0. IF(I.EQ.1)GO TO 238 N3l=2*AINT(N/2.)+2 N32=-2*NMAX N33=-2*N-2 IF(IHEX.EQ.-1)GO TO 237 N31=2*N+2 N33=-N-1 C note that the la s t to contribute is (-N-2) C +(-n) 237 J1FIN=2*NMAX-N31+1 J2FIN=N33-N32+1 DO 232 J=1,J1FIN,2 M=N31-1+J 232 SUM1=SUM1+R(N0+M)*RAT1(M,-NSGND) DO 233 J=1,J2FIN,2 M=N32-1+J 233 SUM2=SUM2+R(N0+M)*RAT1(M,-NSGND) 238 NOR=N0+2*NSGND ORIENT=1.7 32050808*PF(N0-NSGND)-SUM1-SUM2 ORIENT=ORIENT-R(NO-NSGND)*RAT3(-NSGND) ORIENT=ORIENT/RAT2(NSGND) R(NOR)=ORIENT C IPLOT=IPLOT+1 Y(lPLOT)=ORIENT IX(lPLOT)=NOR 231 CONTINUE C C Rearrange for pl o t t i n g purposes C DO 240 J=1,4096 240 Y1(J)=-.1234E56 DO 241 1 = 1 ,1 PLOT J=IX(I) X(J)=J*D6 241 Y1(J)=Y(I) NPLOT=0 DO 242 J=1,4096 Y2=Y1(J) IF(Y2.EQ.-.1234E56)GO TO 242 IF(YMIN.GT.Y2)YMIN=Y2 IF(YMAX.LT.Y2)YMAX=Y2 51 NPL0T=NPL0T+1 Y(NPL0T)=Y2 X(NPLOT)=x(J) 242 CONTINUE IPLOT=NPLOT C GO TO 220 C C End of de-Pake-ing C C C PART 3 : Decision making C 306 CONTINUE IF(.NOT.ASKIF(6,42,'Want to do any more of your 1 manipulations?'))GO TO 303 IF(ASKIF(6,55,'Want to recalculate with d i f f e r e n t 1 averaging bin size ?'))GO TO 105 IF(ASKIF(6,37,'Want to al t e r some basic parameters ?')) 1 GO TO 101 IF(ASKIF(6,45,'Want to start dePakeing from the other 1 side ?'))GO TO 210 303 IF(.NOT.ASKIF(6,35,'Output X,Y depaked data f i l e 1 on 3 ?')) GO TO 302 C C Normalize and output on 3 ready for p l o t t i n g elsewhere C (XBAT:GRAPH.O) C DO 313 1=1,NPSAV X(lPLOT+I)=XSAVE(I) 313 Y(lPLOT+I)=YSAVE(I) NNORM=IPLOT+NPSAV YNORM=0. DO 314 1=1,NNORM 314 YNORM=YNORM+Y(I ) YNORM=YNORM/FLOAT(NNORM) DO 315 1=1,NNORM Y(I)=Y(I)/YNORM 315 WRITE(3,361)X(I),Y(I) 361 FORMAT(2G14.6) 302 IF(.NOT.ASKIF(6,15,'Ready to quit ?'))GO TO 306 C CALL PLOTND WRITE(6,360) 360 FORMAT('Well... See ya.') STOP END C FUNCTION RATI(M,N) MP=M+ 1 52 MM=M-1 RAT=SQRT(MP*(MP+2.*N))+MP+N RAT=RAT/(SQRT(MM*(MM+ 2.*N))+MM+N) RAT 1=ALOG(ABS(RAT)) RETURN END • C FUNCTION RAT2(N) RAT=SQRT(2.*IABS(N)+1.)+N+1. IF(N)421,422,422 422 RAT2=ALOG(ABS(RAT/N)) RETURN 421 RAT2=ALOG(ABS(N/(RAT-2))) RETURN END C FUNCTION RAT3(N) NABS=IABS(N) RAT=SQRT((NABS+1 .)*(3.*NABS+1 . ))+2.*N+1 . IF(N)431,432,432 4 32 RAT3=ALOG(ABS(RAT/3.732050808/NABS)) RETURN 431 RAT3=ALOG(ABS(-.267949l92*NABS/(RAT-2.))) RETURN END C C SUBROUTINE DEPLOT(X,Y,I PLOT,NS,YMIN,YMAX) COMMON SW,NO,NP,NITER LOGICAL ASKIF INTEGER ATT,BlNO DIMENSION X(40 96),Y(4096),Y1(4096),X2(2),X3(2) DIMENSION Z2(2),Z3(2),ATT(36) DIMENSION XPSAVEU096) , YPSAVE ( 4096 ) DATA ATT/1,0,0,0,0,0,2,1,0,0,0,0,6,4,1,0,0,0, 1 20,15,6,1,0,0,70,56,28,8,1,0,252,210,120,45,10,1/ DO 51 1 = 1,1 PLOT 51 Y1(I)=Y(I) CALL ALSIZE(10.0,7.0) IF(YMIN.GT.-0.1 *YMAX)YMIN=-0.1*YMAX X3(1 ) = 0.0 X3(2)=1.0*SW Z3(1 ) = 0.0 Z3(2)=0.0 X2(1)=N0*SW/NP X2(2)=X2(1) Z2(1)=YMIN Z2(2)=1.05*YMAX 52 CALL ALAXIS('FREQUENCY (KHZ)',15,'INTENSITY',9) CALL ALSCAL(0.0,SW,YMIN,1.05*YMAX) 53 CALL ALDASH(0.1,0.1,0.1,0.1) CALL ALGRAF(X2,Z2,2,0) CALL ALGRAF(X3,Z3,-2,0) CALL ALDASH(0.0,0.0,0.0,0.0) CALL ALGRAF(X,Y1 ,-1 PLOT,NS) IF(NITER.GE.1)CALL ALGRAF(XPSAVE,YPSAVE,-NPSAV,NS) CALL PLCTRL('SEND',0) IF(ASKIF(6,22,'Is this s a t i s f a c t o r y ?')) GO TO 50 IF(.NOT.ASKIF(6,27,'Want to smooth the output ?*))GO TO 50 58 WRITE(6,59) 59 FORMAT('Enter correlation 1/2 length for smoothing'/ 1 'in channel numbers, not greater than 5 : ') CALL FREAD(6,'I:',BINO) J52=BINO*2+1 DO 521 1 = 1 ,1 PLOT Y1(I)=0 NATT=0 DO 522 J=1,J52 N=I-BINO+J-1 IF(N.LE.O) GO TO 522 IF(N.GT.IPLOT) GO TO 522 J53=ATT(6*BINO+IABS(BINO-J+1)+1) Y1(I)=Y1(I)+Y(N)*J53 NATT=NATT+J53 522 CONTINUE 521 Y1(I)=Y1(I)/NATT GO TO 52 C 50 DO 524 1 = 1,1 PLOT XPSAVE(I)=X(I) YPSAVE(I)=Y1(I) 524 Y(I)=Y1(I) NPSAV=IPLOT RETURN END C C C..ASKIF... C C...Short Logical Function to get .TRUE, or .FALSE, result as C a Y or N answer to the NCH-character Question in STRING. C A l l I/O on LUN. (No question i f LUN < 0!) C...Used to avoid the usual C "Enter 1 for . . . and 0 for ..." nonsense in FORTRAN. C Also eliminates the need for FORMAT statement when C a short question is being asked. C Unfortunately you must count characters in STRING. C FUNCTION ASKIF (LUN, NCH, STRING) 54 LOGICAL ASKIF, EQUC LOGICAL*1 STRING(80) LOGICAL*1 CHAR, YESU, YESL, NOU, NOL, BLNK C DATA YESU/'Y'/, YESL/'y'/, NOU/'N'/, NOL/'n'/, BLNK/' '/ C c < C ASKIF = .FALSE. ITTY = IABS(LUN) IF (LUN .LT. 0) GO TO 1000 IF (NCH .LE. 0) GO TO 1000 WRITE (LUN,123) (STRING(I),I=1,NCH) 123 FORMAT ('&',80A1) C 1000 CHAR = BLNK READ (ITTY,1111) CHAR 1111 FORMAT (A1) C IF (EQUC(CHAR,YESU)) GO TO 2000 IF (EQUC(CHAR,YESL)) GO TO 2000 C...Blank Line w i l l be interpreted as "No." IF (EQUC(CHAR,BLNK)) RETURN IF (EQUC(CHAR,NOU)) RETURN IF (EQUC(CHAR,NOL)) RETURN C C...If noninteractive (LUN < 0) default "No" for C unrecognizable character C IF (LUN .LE. 0) RETURN WRITE (ITTY,2222) 2222 FORMAT ('&Please answer Y or N: ') GO TO 1000 C 2000 ASKIF = .TRUE. RETURN C END 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0085015/manifest

Comment

Related Items