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On the equivalence of mass-change shift and second-order doppler shift in the Mössbauer effect
Volume 30A, number 3
PHYSICS LETTERS
ON T H E E Q U I V A L E N C E SECOND-ORDER DOPPLER
6October 1969
OF MASS-CHANGE SHIFT S H I F T IN T H E M O S S B A U E R
AND EFFECT
M. G. CLARK and A . J . STONE
Department of Theoretical Chemistry, University Chemical Laboratory, Cambridge, UK Received 25 August 1969
Contrary to Dehn~s recent suggestion, there is no reason to revise the view that the two formulations of the thermal red shift of a M'Sssbauer resonance are equivalent, and are to be interpreted as a relativistic time-dilatation effect. Dehn [1] r e c e n t l y questioned the g e n e r a l a c c e p t a n c e of the e q u i v a l e n c e of s e c o n d - o r d e r D o p p l e r shift [2] and m a s s - c h a n g e shift [3] e x p l a n a t i o n s f o r the t h e r m a l r e d shift of M S s s b a u e r g a m m a r a y s . By use of e s s e n t i a l l y c l a s s i c a l a r g u m e n t s b a s e d on the r e l a t i v i s t i c D o p p le r shift f o r m u l a [4] he concluded that an a n i s o t r o p i c s e c o n d - o r d e r shift ( a r i s i n g , in fact, f r o m the nonr e l a t i v i s t i c p a r t of the D o p p l e r shift f o r m u l a ) had b e e n o v e r l o o k e d p r e v i o u s l y , and that the s e c o n d o r d e r D o p p l e r and m a s s - c h a n g e shifts w e r e additive. T h i s note points out that the a n i s o t r o p i c s e c o n d - o r d e r t e r m v a n i s h e s in a quantum t r e a t m e n t , and that the two f o r m u l a t i o n s a r e s i m p l y d i f f e r e n t d e s c r i p t i o n s of the s a m e effect. Dehn t r e a t e d the D o p p l e r e f f e c t in k i n e m a t i cal t e r m s , but it can a l s o be d e r i v e d f r o m the d e t a i l e d d y n a m i c s of a s y s t e m . C o n s i d e r a solid at r e s t in the l a b o r a t o r y f r a m e of r e f e r e n c e (any m a c r o s c o p i c m o t i o n can be t r e a t e d s e p a r a t e l y ) . To c a l c u l a t e the e n e r g y of a M S s s b a u e r g a m m a r a y e m i t t e d (or a b s o r b e d ) by the jth nucleus we m u s t c a l c u l a t e the change in the H a m i lt o n ia n
H = ~i (mic2+p~/2mi) + V ,
(1)
w h e r e c is the speed of light in vacuo, m i and Pi a r e the p a r t i c l e m a s s e s and m o m e n t a , and V inc l u d e s all o t h e r e n e r g y t e r m s .
5H = 5mjc 2 + ~. ~ (PiaSPia/mi) + ~
(2)
- p 2 6 m j / 2 m 2 + ~V, w h e r e , in the second t e r m , a runs o v e r the t h r e e C a r t e s i a n c o m p o n e n t s and the sum o v e r i a r i s e s f r o m the fact that the r e c o i l m o m e n t u m is s h a r e d by the whole solid [5]. 6V g i v e s r i s e to no e f f e c t s r e l e v a n t to our d i s c u s s i o n , and will be neglected. The o b s e r v e d e n e r g y E of the g a m m a r a y is (for emission) 144
E = -<~
:
_~mjc 2 _ <~ ~ Pia@ia i
a
mi
> + < _:_2_ > ~ m j ,
2m 2
(3)
w h e r e - 6mqc2 = E o is the p r o p e r e n e r g y of the t r a n s i t i o n . The second and t h i r d t e r m s in eq. (3) d e s c r i b e the D o p p l er shift of the e m i t t e d r a y to second o r d e r ; the second is the n o n - r e l a t i v i s t i c shift and the t h i r d is the r e l a t i v i s t i c c o r r e c t i o n . T h i s l a t t e r t e r m was noted by both Pound and Rebka [2] and J o s e p h s o n [3], and d e a r l y i d e n t i fied as the r e l a t i v i s t i c t i m e - d i l a t a t i o n effect [3,6, 7] c a u s e d by the n u c l e a r motion. The second t e r m i n c l u d e s the new a n i s o t r o p i c t e r m p r o p o s e d by Dehn. J
.--,'¢-,PiaOPia (/z~az~
~i
~i ~a /Piol6Pia ):
"
'
~//
)-
(4)
= ~ ~ ((Pi~) <6Pia)/mi ' s i n c e Pia and 6pi a a r e u n c o r r e l a t e d , t h e r e being many v i b r a t i o n s of the nuclei during the l i f t e t i m e of a M~issbauer l e v e l . H o w e v e r , (/~io~ = 0 f o r all i, a ; thus the n o n - r e l a t i v i s t i c D o p p l er shift v a n i s h e s and D e h n ' s t e r m does not o c c u r . D e h n ' s t e r m can only o c c u r if the su m o v e r i in eq. (2) r e d u c e s to a single t e r m i = j , i.e., ff all the r e coil m o m e n t u m is taken up by the e m i t t i n g p a r t i c l e alone. In conclusion, t h e r e is no r e a s o n to r e v i s e the e x i s t i n g f o r m u l a t i o n s [2,3] of the t h e r m a l r e d shift, or to doubt that they p r o v i d e equivalent d e s c r i p t i o n s of the s a m e effect. M.G.C. g r a t e f u l l y acknowledges the f i n a n c i a l
Volume 30A, n u m b e r 3
PHYSICS
LETTERS
s u p p o r t p r o v i d e d b y h i s F e l l o w s h i p of St. J o h n ' s College, Cambridge.
4 (1960) 274. 3. B.D. Josephson, Phys. Rev. L e t t e r s 4 (1960) 341. 4. C. M~ller, The Theory of Relativity (CLarendon P r e s s , Oxford, 1952) p. 62. 5. H. F auenfelder, The MSssbauer effect (W.A. Benjamin, Inc., New York, 1963)pp. 20-21. 6. Reference 4, pp. 48-51, 247 and 258-263. 7. C.W. Sherwin, Phys. Rev. 120 (1960) 17.
R~f~'c~ces 1. J. T. Dehn, Phys. L e t t e r s 29A (1969) 132. 2. R.V. Pound and G. A. Eebka J r . , Phys. Rev. L e t t e r s
OF
EFFECT OF METASTABLE
RECOIL ATOMS
J . C. P E A R L ,
6 O c t o b e r 1969
ON THE VELOCITY DISTRIBUTION PRODUCED BY ELECTRON IMPACT
*
D. P. D O N N E L L Y a n d J . C. Z O R N
Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan 48104, USA Received 3 September 1969
We have shown both e x p e r i m e n t a l l y and by a s i m p l e k i n e m a t i c a r g u m e n t that the velocity distribution in a b e a m of m e t a s t a b l e atoms produced by e l e c t r o n b o m b a r d m e n t of ground s t a t e atoms deviates s t r o n g l y from the velocity d i s t r i b u t i o n of the ground s t a t e atoms.
T h e v e l o c i t y d i s t r i b u t i o n of m e t a s t a b l e a t o m s o f t e n e n t e r s e x p l i c i t l y i n t o t h e a n a l y s i s of r e s o n a n c e [1] a n d c o l l i s i o n [2,3] e x p e r i m e n t s d o n e w i t h a t o m i c b e a m s . H o w e v e r , t h e r e h a v e b e e n f e w d i r e c t m e a s u r e m e n t s a n d no c o r r e c t t h e o r e t i c a l d e s c r i p t i o n of t h e v e l o c i t y d i s t r i b u t i o n i n a b e a m of m e t a s t a b l e * R e s e a r c h supported in p a r t by the National Aeronautics and Space Administration.
3!:t
~
~
e • 77.6"
0
•
70.4"
.C
~-0. o
100
ZOO
300
4oo
soo
TIME OF FLIGHT (~lSec) Fig. 1. T i m e - o f - f l i g h t s p e c t r a for helium. Points r e p r e s e n t e x p e r i m e n t a l data. Curves a a r e t h e o r e t i c a l accounting for the angular d i v e r g e n c e of the incoming, ground s t a t e b e a m , a s s u m i n g E - E * = 0.75 eV. Curve b is t h e o r e t i c a l if r e c o i l effects a r e neglected. 145
PHYSICS LETTERS
ON T H E E Q U I V A L E N C E SECOND-ORDER DOPPLER
6October 1969
OF MASS-CHANGE SHIFT S H I F T IN T H E M O S S B A U E R
AND EFFECT
M. G. CLARK and A . J . STONE
Department of Theoretical Chemistry, University Chemical Laboratory, Cambridge, UK Received 25 August 1969
Contrary to Dehn~s recent suggestion, there is no reason to revise the view that the two formulations of the thermal red shift of a M'Sssbauer resonance are equivalent, and are to be interpreted as a relativistic time-dilatation effect. Dehn [1] r e c e n t l y questioned the g e n e r a l a c c e p t a n c e of the e q u i v a l e n c e of s e c o n d - o r d e r D o p p l e r shift [2] and m a s s - c h a n g e shift [3] e x p l a n a t i o n s f o r the t h e r m a l r e d shift of M S s s b a u e r g a m m a r a y s . By use of e s s e n t i a l l y c l a s s i c a l a r g u m e n t s b a s e d on the r e l a t i v i s t i c D o p p le r shift f o r m u l a [4] he concluded that an a n i s o t r o p i c s e c o n d - o r d e r shift ( a r i s i n g , in fact, f r o m the nonr e l a t i v i s t i c p a r t of the D o p p l e r shift f o r m u l a ) had b e e n o v e r l o o k e d p r e v i o u s l y , and that the s e c o n d o r d e r D o p p l e r and m a s s - c h a n g e shifts w e r e additive. T h i s note points out that the a n i s o t r o p i c s e c o n d - o r d e r t e r m v a n i s h e s in a quantum t r e a t m e n t , and that the two f o r m u l a t i o n s a r e s i m p l y d i f f e r e n t d e s c r i p t i o n s of the s a m e effect. Dehn t r e a t e d the D o p p l e r e f f e c t in k i n e m a t i cal t e r m s , but it can a l s o be d e r i v e d f r o m the d e t a i l e d d y n a m i c s of a s y s t e m . C o n s i d e r a solid at r e s t in the l a b o r a t o r y f r a m e of r e f e r e n c e (any m a c r o s c o p i c m o t i o n can be t r e a t e d s e p a r a t e l y ) . To c a l c u l a t e the e n e r g y of a M S s s b a u e r g a m m a r a y e m i t t e d (or a b s o r b e d ) by the jth nucleus we m u s t c a l c u l a t e the change in the H a m i lt o n ia n
H = ~i (mic2+p~/2mi) + V ,
(1)
w h e r e c is the speed of light in vacuo, m i and Pi a r e the p a r t i c l e m a s s e s and m o m e n t a , and V inc l u d e s all o t h e r e n e r g y t e r m s .
5H = 5mjc 2 + ~. ~ (PiaSPia/mi) + ~
(2)
- p 2 6 m j / 2 m 2 + ~V, w h e r e , in the second t e r m , a runs o v e r the t h r e e C a r t e s i a n c o m p o n e n t s and the sum o v e r i a r i s e s f r o m the fact that the r e c o i l m o m e n t u m is s h a r e d by the whole solid [5]. 6V g i v e s r i s e to no e f f e c t s r e l e v a n t to our d i s c u s s i o n , and will be neglected. The o b s e r v e d e n e r g y E of the g a m m a r a y is (for emission) 144
E = -<~
:
_~mjc 2 _ <~ ~ Pia@ia i
a
mi
> + < _:_2_ > ~ m j ,
2m 2
(3)
w h e r e - 6mqc2 = E o is the p r o p e r e n e r g y of the t r a n s i t i o n . The second and t h i r d t e r m s in eq. (3) d e s c r i b e the D o p p l er shift of the e m i t t e d r a y to second o r d e r ; the second is the n o n - r e l a t i v i s t i c shift and the t h i r d is the r e l a t i v i s t i c c o r r e c t i o n . T h i s l a t t e r t e r m was noted by both Pound and Rebka [2] and J o s e p h s o n [3], and d e a r l y i d e n t i fied as the r e l a t i v i s t i c t i m e - d i l a t a t i o n effect [3,6, 7] c a u s e d by the n u c l e a r motion. The second t e r m i n c l u d e s the new a n i s o t r o p i c t e r m p r o p o s e d by Dehn. J
.--,'¢-,PiaOPia (/z~az~
~i
~i ~a /Piol6Pia ):
"
'
~//
)-
(4)
= ~ ~ ((Pi~) <6Pia)/mi ' s i n c e Pia and 6pi a a r e u n c o r r e l a t e d , t h e r e being many v i b r a t i o n s of the nuclei during the l i f t e t i m e of a M~issbauer l e v e l . H o w e v e r , (/~io~ = 0 f o r all i, a ; thus the n o n - r e l a t i v i s t i c D o p p l er shift v a n i s h e s and D e h n ' s t e r m does not o c c u r . D e h n ' s t e r m can only o c c u r if the su m o v e r i in eq. (2) r e d u c e s to a single t e r m i = j , i.e., ff all the r e coil m o m e n t u m is taken up by the e m i t t i n g p a r t i c l e alone. In conclusion, t h e r e is no r e a s o n to r e v i s e the e x i s t i n g f o r m u l a t i o n s [2,3] of the t h e r m a l r e d shift, or to doubt that they p r o v i d e equivalent d e s c r i p t i o n s of the s a m e effect. M.G.C. g r a t e f u l l y acknowledges the f i n a n c i a l
Volume 30A, n u m b e r 3
PHYSICS
LETTERS
s u p p o r t p r o v i d e d b y h i s F e l l o w s h i p of St. J o h n ' s College, Cambridge.
4 (1960) 274. 3. B.D. Josephson, Phys. Rev. L e t t e r s 4 (1960) 341. 4. C. M~ller, The Theory of Relativity (CLarendon P r e s s , Oxford, 1952) p. 62. 5. H. F auenfelder, The MSssbauer effect (W.A. Benjamin, Inc., New York, 1963)pp. 20-21. 6. Reference 4, pp. 48-51, 247 and 258-263. 7. C.W. Sherwin, Phys. Rev. 120 (1960) 17.
R~f~'c~ces 1. J. T. Dehn, Phys. L e t t e r s 29A (1969) 132. 2. R.V. Pound and G. A. Eebka J r . , Phys. Rev. L e t t e r s
OF
EFFECT OF METASTABLE
RECOIL ATOMS
J . C. P E A R L ,
6 O c t o b e r 1969
ON THE VELOCITY DISTRIBUTION PRODUCED BY ELECTRON IMPACT
*
D. P. D O N N E L L Y a n d J . C. Z O R N
Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan 48104, USA Received 3 September 1969
We have shown both e x p e r i m e n t a l l y and by a s i m p l e k i n e m a t i c a r g u m e n t that the velocity distribution in a b e a m of m e t a s t a b l e atoms produced by e l e c t r o n b o m b a r d m e n t of ground s t a t e atoms deviates s t r o n g l y from the velocity d i s t r i b u t i o n of the ground s t a t e atoms.
T h e v e l o c i t y d i s t r i b u t i o n of m e t a s t a b l e a t o m s o f t e n e n t e r s e x p l i c i t l y i n t o t h e a n a l y s i s of r e s o n a n c e [1] a n d c o l l i s i o n [2,3] e x p e r i m e n t s d o n e w i t h a t o m i c b e a m s . H o w e v e r , t h e r e h a v e b e e n f e w d i r e c t m e a s u r e m e n t s a n d no c o r r e c t t h e o r e t i c a l d e s c r i p t i o n of t h e v e l o c i t y d i s t r i b u t i o n i n a b e a m of m e t a s t a b l e * R e s e a r c h supported in p a r t by the National Aeronautics and Space Administration.
3!:t
~
~
e • 77.6"
0
•
70.4"
.C
~-0. o
100
ZOO
300
4oo
soo
TIME OF FLIGHT (~lSec) Fig. 1. T i m e - o f - f l i g h t s p e c t r a for helium. Points r e p r e s e n t e x p e r i m e n t a l data. Curves a a r e t h e o r e t i c a l accounting for the angular d i v e r g e n c e of the incoming, ground s t a t e b e a m , a s s u m i n g E - E * = 0.75 eV. Curve b is t h e o r e t i c a l if r e c o i l effects a r e neglected. 145