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Time reversal symmetry in low energy electron diffraction
Volume 31A, number 4
PHYSICS
e l e c t r o n s r e l e a s e d by photoexcitation will r e c o m b i n e with the s e l f - t r a p p e d holes emitting the 339 n m l u m i n e s c e n c e . On the other hand, the glow peaks a r o u n d 415 n m and the e m i s s i o n band peaking at 415 n m may be the r e s u l t of r e c o m b i nation of t h e r m a l l y excited holes and the c~ptured e l e c t r o n s . Since the s a m p l e which was b e t a - r a y e d and then completely r e m o v e d f r o m the beta ray s o u r c e s t i l l e m i t t e d the a f t e r - g l o w peaking at 415 nm, in the m e a s u r e m e n t s of the a b s o r p t i o n band peaking at 690 n m , a r e d f i l t e r was i n s e r t e d between the s a m p l e and the detector. The r e l a t i o n between the 228 n m band and the 690 n m band could not be examined a c c o r d i n g l y .
o
o ~n O
. . . .
I
. . . .
i
600
. . . .
I
. . . .
,
800 nm WAVE
23 February 1970
LETTERS
LENGTH
The a u t o r wishes to thank Mr. I. Morita for his helpful d i s c u s s i o n s and e n c o u r a g e m e n t .
Fig. 2. Absorption spectrum induced by beta rays in quenched CsI. by quenching or may be r e c o m b i n e d with s e l f t r a p p e d holes r e s u l t i n g in the e m i s s i o n of 339 n m l u m i n e s c e n c e . The a b s o r p t i o n band peaking at 690 n m is t e n t a t i v e l y a s c r i b e d to e l e c t r o n capt u r e d at a l a t t i c e defect, though the model of this l a t t i c e defects has not yet been known. The
References 1. T.Towyama, I. Morita and M.Ishiguro, J. Phys. Soc. Jap. 25 (1968) 1133. 2. S. Masunaga, I. Morita and M.Ishiguro, J. Phys. Soc. Jap. 21 (1966) 638.
* * * * *
TIME
REVERSAL
SYMMETRY
IN LOW
ENERGY
ELECTRON
DIFFRACTION
D. P. WOODRUFF and B..W. HOLLAND
School of Physics, University of Warwick, Coventry, UK Received 31 October 1969
Time reversal invariance leads to symmetries in low energy electron diffraction results over and above those arising from the crystal structure. Experimental results from a copper (111) surface confirm these predictions.
While t i m e r e v e r s a l s y m m e t r y i s exploited in m a n y a r e a s of p h y s i c s , t h e r e i s l i t t l e evidence that its i m p o r t a n c e i s fully r e c o g n i s e d in the field of Low E n e r g y E l e c t r o n Diffraction (LEED). In this l e t t e r we wish to d e m o n s t r a t e that t i m e r e v e r s a l i n v a r i a n c e gives r i s e to s y m m e t r i e s in LEED data, over and above those a r i s i n g f r o m the c r y s t a l s t r u c t u r e . Pogany and T u r n e r [1] have a l r e a d y given such a d e m o n s t r a t i o n for e l e c t r o n m i c r o s c o p y . We f i r s t point out that the validity of the r e c i p r o c i t y t h e o r e m i s not dependent on the c o n v e r g e n c e of the Born s e r i e s a s a s s u m e d by Pogany and T u r n e r .
The T - m a t r i x [2] i s so defined that the c r o s s s e c t i o n for e l a s t i c s c a t t e r i n g (even when i n e l a s t i c s c a t t e r i n g can occur) f r o m an ingoing state l a b e l l e d by the wave v e c t o r k (we n e g l e c t spin), to outgoing state l a b e l l e d k ' i s -~k' = (rrb/21rk)21 (k' ITIk~l 2. T h e r e is the o p e r a t o r obtained by taking the expectation value of the t r a n s i t i o n o p e r a t o r for the whole s y s t e m , with r e s p e c t to the state of the c r y s t a l . F u r t h e r , it follows d i r e c t l y f r o m t i m e r e v e r s a l i n v a r i a n c e that (-klT I-k' ~ = (k'lT Jk~. Hence the i n t e n s i t i e s o b s e r v e d in LEED m u s t satisfy the r e c i p r o c i t y
~
207
Volume 31A, number 4
PHYSICS LETTERS
23 February 1970
IlO)
50
IOO
150 200 250 ELECTRONENERGY!¢V)
ioo
J 350
300
theorem
:
Ik_k,
Experiments were performed on the (111) face of copper and initially the intensity plot (intensity versus electron energy curve) for the specular (00) b e a m was studied for angles of incidence between 6 ° and 25 ° at lO intervals on either side of the surface normal. The plane of incidence for all these m e a s u r e m e n t s cuts the surface along the (115) or (112) d i r e c t i o n . The angle of i n c i d e n c e O was defined ( a r b i t r a r i l y ) as positive when the s u r f a c e component of the i n c i d e n t b e a m d i r e c t e d t o w a r d s the c r y s t a l was along the (11~) d i r e c t i o n with the outward s u r f a c e n o r m a l of the c r y s t a l defined at (111) [3]. The r e s u l t s of this i n v e s t i g a t i o n show that t h e r e is excellent c o r r e lation between i n t e n s i t y plots taken at the s a m e angle of i n c i d e n c e on either side of the s u r f a c e n o r m a l for peak positions and i n t e n s i t i e s , and for all angles on i n c i d e n c e studied. E x a m p l e s of this c o r r e l a t i o n a r e given in fig. 1. This s y m m e t r y is not to be expected on the b a s i s of c r y s t a l l i n e s t r u c t u r e but follows d i r e c t l y f r o m (1). The i n t e n s i t y plots w e r e also i n v e s t i g a t e d for two n o n - s p e c u l a r b e a m s . The b e a m s studied w e r e those n e a r e s t to the s p e c u l a r b e a m which e m e r g e with the s u r f a c e component of t h e i r wave v e c t o r in the d i r e c t i o n s (11~) (denoted h e r e as the (10) beam) and (~i'2) r e f e r r e d to as the (I0) beam). In such c a s e the e x p e r i m e n t c o n s i s t e d of d e t e r m i n i n g the i n t e n s i t y plot of the n o n - s p e c u l a r
208
300
b e a m for n o r m a l incidence and c o m p a r i n g this with an i n t e n s i t y plot of the s a m e beam when e m e r g i n g n o r m a l to the s u r f a c e ; this c o r r e s p o n d s to r e v e r s i n g the ingoing and outgoing wave v e c t o r s of the f i r s t e x p e r i m e n t . In view of the e x p e r i m e n t a l difficulties a s s o ciated with the second e x p e r i m e n t the c o r r e l a t i o n of both peak positions a n d i n t e n s i t i e s is quite r e m a r k a b l e a s may be seen in fig. 2. It is i n t e r e s t i n g to note that t i m e r e v e r s a l i n v a r i a n q e when c o m b i n e d with c r y s t a l s y m m e t r i e s can also give r i s e to f u r t h e r s y m m e t r i e s in the diffraction pattern. In the case studied, for exa m p l e , the 3-fold c r y s t a l axis combined with t i m e r e v e r s a l s y m m e t r y gives r i s e to a 6-fold r o t a t i o n axis for the (00) beam. This should be o b s e r v a b l e in R e n n i n g e r plots (of i n t e n s i t y v e r s u s a z i m u t h angle for c o n s t a n t energy and angle of incidence). Unfortunately, such e x p e r i m e n t s a p p e a r to have been p e r f o r m e d only for the (110) s u r f a c e of tungsten [4] in which no e x t r a s y m m e t r y would be observed.
(1)
*
260
Fig. 2. Intensity plots for the (10) and (10) beams. The lower curves are for normal incidence, the upper ones for the outgoing beam normal to the surface. The arrow indicates the shift in zero intensity between the two curves.
curve). The arrow indicates the shift in zero intensity between the two curves.
_k
3o0 ~6d
(TO)
EL ECTRON E NE~GY levi
Fig. i. Intensity plot of (00) beam for angles of incidence. 0, of +13½° (upper curve) and -13½° (lower
I_k,
200
I
*
Re f~r~'nc¢s 1. A. P. Pogany and P. S. Turner, Acta Cryst. A24 (1968) 103. 2. P. Roman, Advanced quantum theory (AddisonWesley Co. Ltd., 1965). 3. D. P. Woodruff and M. P. Seah, Phys. Stat. Solidi, to be published. 4. A. Gervais, R.M. Stern and M. Menes, Acta Cryst. A24 (1968) 191. $
*
PHYSICS
e l e c t r o n s r e l e a s e d by photoexcitation will r e c o m b i n e with the s e l f - t r a p p e d holes emitting the 339 n m l u m i n e s c e n c e . On the other hand, the glow peaks a r o u n d 415 n m and the e m i s s i o n band peaking at 415 n m may be the r e s u l t of r e c o m b i nation of t h e r m a l l y excited holes and the c~ptured e l e c t r o n s . Since the s a m p l e which was b e t a - r a y e d and then completely r e m o v e d f r o m the beta ray s o u r c e s t i l l e m i t t e d the a f t e r - g l o w peaking at 415 nm, in the m e a s u r e m e n t s of the a b s o r p t i o n band peaking at 690 n m , a r e d f i l t e r was i n s e r t e d between the s a m p l e and the detector. The r e l a t i o n between the 228 n m band and the 690 n m band could not be examined a c c o r d i n g l y .
o
o ~n O
. . . .
I
. . . .
i
600
. . . .
I
. . . .
,
800 nm WAVE
23 February 1970
LETTERS
LENGTH
The a u t o r wishes to thank Mr. I. Morita for his helpful d i s c u s s i o n s and e n c o u r a g e m e n t .
Fig. 2. Absorption spectrum induced by beta rays in quenched CsI. by quenching or may be r e c o m b i n e d with s e l f t r a p p e d holes r e s u l t i n g in the e m i s s i o n of 339 n m l u m i n e s c e n c e . The a b s o r p t i o n band peaking at 690 n m is t e n t a t i v e l y a s c r i b e d to e l e c t r o n capt u r e d at a l a t t i c e defect, though the model of this l a t t i c e defects has not yet been known. The
References 1. T.Towyama, I. Morita and M.Ishiguro, J. Phys. Soc. Jap. 25 (1968) 1133. 2. S. Masunaga, I. Morita and M.Ishiguro, J. Phys. Soc. Jap. 21 (1966) 638.
* * * * *
TIME
REVERSAL
SYMMETRY
IN LOW
ENERGY
ELECTRON
DIFFRACTION
D. P. WOODRUFF and B..W. HOLLAND
School of Physics, University of Warwick, Coventry, UK Received 31 October 1969
Time reversal invariance leads to symmetries in low energy electron diffraction results over and above those arising from the crystal structure. Experimental results from a copper (111) surface confirm these predictions.
While t i m e r e v e r s a l s y m m e t r y i s exploited in m a n y a r e a s of p h y s i c s , t h e r e i s l i t t l e evidence that its i m p o r t a n c e i s fully r e c o g n i s e d in the field of Low E n e r g y E l e c t r o n Diffraction (LEED). In this l e t t e r we wish to d e m o n s t r a t e that t i m e r e v e r s a l i n v a r i a n c e gives r i s e to s y m m e t r i e s in LEED data, over and above those a r i s i n g f r o m the c r y s t a l s t r u c t u r e . Pogany and T u r n e r [1] have a l r e a d y given such a d e m o n s t r a t i o n for e l e c t r o n m i c r o s c o p y . We f i r s t point out that the validity of the r e c i p r o c i t y t h e o r e m i s not dependent on the c o n v e r g e n c e of the Born s e r i e s a s a s s u m e d by Pogany and T u r n e r .
The T - m a t r i x [2] i s so defined that the c r o s s s e c t i o n for e l a s t i c s c a t t e r i n g (even when i n e l a s t i c s c a t t e r i n g can occur) f r o m an ingoing state l a b e l l e d by the wave v e c t o r k (we n e g l e c t spin), to outgoing state l a b e l l e d k ' i s -~k' = (rrb/21rk)21 (k' ITIk~l 2. T h e r e is the o p e r a t o r obtained by taking the expectation value of the t r a n s i t i o n o p e r a t o r for the whole s y s t e m , with r e s p e c t to the state of the c r y s t a l . F u r t h e r , it follows d i r e c t l y f r o m t i m e r e v e r s a l i n v a r i a n c e that (-klT I-k' ~ = (k'lT Jk~. Hence the i n t e n s i t i e s o b s e r v e d in LEED m u s t satisfy the r e c i p r o c i t y
~
207
Volume 31A, number 4
PHYSICS LETTERS
23 February 1970
IlO)
50
IOO
150 200 250 ELECTRONENERGY!¢V)
ioo
J 350
300
theorem
:
Ik_k,
Experiments were performed on the (111) face of copper and initially the intensity plot (intensity versus electron energy curve) for the specular (00) b e a m was studied for angles of incidence between 6 ° and 25 ° at lO intervals on either side of the surface normal. The plane of incidence for all these m e a s u r e m e n t s cuts the surface along the (115) or (112) d i r e c t i o n . The angle of i n c i d e n c e O was defined ( a r b i t r a r i l y ) as positive when the s u r f a c e component of the i n c i d e n t b e a m d i r e c t e d t o w a r d s the c r y s t a l was along the (11~) d i r e c t i o n with the outward s u r f a c e n o r m a l of the c r y s t a l defined at (111) [3]. The r e s u l t s of this i n v e s t i g a t i o n show that t h e r e is excellent c o r r e lation between i n t e n s i t y plots taken at the s a m e angle of i n c i d e n c e on either side of the s u r f a c e n o r m a l for peak positions and i n t e n s i t i e s , and for all angles on i n c i d e n c e studied. E x a m p l e s of this c o r r e l a t i o n a r e given in fig. 1. This s y m m e t r y is not to be expected on the b a s i s of c r y s t a l l i n e s t r u c t u r e but follows d i r e c t l y f r o m (1). The i n t e n s i t y plots w e r e also i n v e s t i g a t e d for two n o n - s p e c u l a r b e a m s . The b e a m s studied w e r e those n e a r e s t to the s p e c u l a r b e a m which e m e r g e with the s u r f a c e component of t h e i r wave v e c t o r in the d i r e c t i o n s (11~) (denoted h e r e as the (10) beam) and (~i'2) r e f e r r e d to as the (I0) beam). In such c a s e the e x p e r i m e n t c o n s i s t e d of d e t e r m i n i n g the i n t e n s i t y plot of the n o n - s p e c u l a r
208
300
b e a m for n o r m a l incidence and c o m p a r i n g this with an i n t e n s i t y plot of the s a m e beam when e m e r g i n g n o r m a l to the s u r f a c e ; this c o r r e s p o n d s to r e v e r s i n g the ingoing and outgoing wave v e c t o r s of the f i r s t e x p e r i m e n t . In view of the e x p e r i m e n t a l difficulties a s s o ciated with the second e x p e r i m e n t the c o r r e l a t i o n of both peak positions a n d i n t e n s i t i e s is quite r e m a r k a b l e a s may be seen in fig. 2. It is i n t e r e s t i n g to note that t i m e r e v e r s a l i n v a r i a n q e when c o m b i n e d with c r y s t a l s y m m e t r i e s can also give r i s e to f u r t h e r s y m m e t r i e s in the diffraction pattern. In the case studied, for exa m p l e , the 3-fold c r y s t a l axis combined with t i m e r e v e r s a l s y m m e t r y gives r i s e to a 6-fold r o t a t i o n axis for the (00) beam. This should be o b s e r v a b l e in R e n n i n g e r plots (of i n t e n s i t y v e r s u s a z i m u t h angle for c o n s t a n t energy and angle of incidence). Unfortunately, such e x p e r i m e n t s a p p e a r to have been p e r f o r m e d only for the (110) s u r f a c e of tungsten [4] in which no e x t r a s y m m e t r y would be observed.
(1)
*
260
Fig. 2. Intensity plots for the (10) and (10) beams. The lower curves are for normal incidence, the upper ones for the outgoing beam normal to the surface. The arrow indicates the shift in zero intensity between the two curves.
curve). The arrow indicates the shift in zero intensity between the two curves.
_k
3o0 ~6d
(TO)
EL ECTRON E NE~GY levi
Fig. i. Intensity plot of (00) beam for angles of incidence. 0, of +13½° (upper curve) and -13½° (lower
I_k,
200
I
*
Re f~r~'nc¢s 1. A. P. Pogany and P. S. Turner, Acta Cryst. A24 (1968) 103. 2. P. Roman, Advanced quantum theory (AddisonWesley Co. Ltd., 1965). 3. D. P. Woodruff and M. P. Seah, Phys. Stat. Solidi, to be published. 4. A. Gervais, R.M. Stern and M. Menes, Acta Cryst. A24 (1968) 191. $
*