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On the possibility of magnon investigations without a direct energy analysis using a pulsed neutron source
NUCLEAR
INSTRUMENTS
AND
METHODS
95
(I97I)
45-48; ©
NORTH-HOLLAND
PUBLISHING
CO.
ON THE POSSIBILITY OF MAGNON INVESTIGATIONS WITHOUT A DIRECT ENERGY ANALYSIS USING A PULSED NEUTRON SOURCE L. D O B R Z Y N S K I *
Joint Institute for Nuclear Research, Dubna, U.S.S.R. Received 5 March 1971 The equation o f the scattering surface is derived from the energy and m o m e n t u m conservation laws for inelastic neutron scattering involving m a g n o n creation or annihilation. F r o m this equation some variants of the diffraction technique arise. Particular attention is paid to the studies made possible by pulsed reactors.
The diffraction technique in the m a g n o n studies by means o f neutron scattering has already been k n o w n for m a n y years1). It was extensively employed in the studies o f all magnetic substances: ferro-, antiferroand ferrimagnets. The purpose o f this paper is to propose a version o f this technique which could be particularly interesting when combined with a pulsed neutron source (PNS). It will be shown that when the m a g n o n dispersion relation is quadratic then the neutron scattering at a given angle is strictly confined to a certain range o f incoming and scattered neutron wavelengths. The implication o f this fact is that different experimental arrangements may be used for the studies o f m a g n o n s with no energy analysis. Let k' and k denote the scattering and initial neutron wavevector, respectively. When neutron scattering involves m a g n o n creation or m a g n o n annihilation then the following m o m e n t u m and energy conservation laws are satisfied: k'-k-2n~
=
q,
(l) where q is the m a g n o n wavevector, E o is the energy gap in the m a g n o n dispersion law and e = + 1, where the upper sign refers to the m a g n o n annihilation and the lower one to the m a g n o n creation. The geometry o f the experiment is shown in fig. 1. Neutrons impinging at a given angle ( 9 0 ° - 0 B ) with respect to the crystal axis are analyzed along the direction described by the chosen scattering angle tp. Let us write =
k; + k'~,
ko = nr/sin 0B, (3) Ik'xl ko
--X,
Ikl
--
Ik;I
~y,
ko
-- Z,
ko
Eo
--~-~0,
k~
the solution of eq. (1) with respect to q gives the equation o f the scattering surface: (l - e~) x 2 + 2 e ~ x y cos tp - 4 ectx sin 0B sin (0 B - ~p) -
- (l + ect) y2 + 4 s ~ y sin E 0 B + ( I - e c 0 z 2 - e[~0+4~sinEOB] = 0.
(4)
This equation describes an ellipsoid centered at z o =0
Xo = 2 e~ sinOB ect cos Oa sin (p - sin (OB - ~o) ~2 sin 2 ~0- 1
(5)
Yo = 2 ec~sin 0 a ec~sin (p cos (0 a - ~o) - sin Oa O~2 sin E q9- 1
k'2_k 2 = s(Eoq-~qE),
k'
the following abbreviations
~
88= 15° ? = 40 °
/
~\
"\\
/gO<8\l\\~ (000) 2~T[ ~,~o
(2)
where k' is perpendicular to the scattering plane. With
Fig. 1. The geometry of the experiment. The scattering surface is drawn for m a g n o n s satisfying the quadratic dispersion relation with ~t = 100, a0 = 0, 0n = 15 °, tp = 40 °, e = 1.
* On leave from the Institute of Nuclear Research, Laboratory II, Swierk post Otwock, Poland.
45
46
L. D O B R Z Y N S K I
if ~2
sin 2 cP-- 1 > 0.
(5a)
The ellipsoid is real when --
1
Some additional restrictions arise from the requirement that x and y must be positive; e.g. in the case of m a g n o n annihilation this requirement leads to the inequality
{4 ~ sin 2 08 [1 - ect sin (2 0 8 - q0 sin ~o] -
l < ~COS2(¢p--2OB)
0~--/3 -
~o(~2 sin2 ~o- 1)} > 0.
(6)
T h e vertical cross section of the ellipsoid (4) is a circle with the radius
R=[4s~sin2Oa[1-ectsin(2Oa-tp)sinq ~] (ect-- 1) (~2 sin2 qo-- 1)
e°~o 3~:. ect-- 1
(7) It can be seen that the radius in eq. (7) varies continuously with q~ and 0a. This observation forms the basis of the first experiment which was p e r f o r m e d by Riste et al.2). In this experiment the angle ~o was kept constant and the vertical cross section of the ellipsoid was studied for different 08. The variation of the radius Rz with respect to 08 leads to values of e. It follows f r o m eq. (6) that for given ~0 > 208 one observes scattering with m a g n o n annihilation (for e = l ) only, while m a g n o n creation is obtained for ~o < 2 0B. F o r example, when c~o = 0 (no energy gap) the condition that scattering with m a g n o n annihilation will occur is e sin (~o - 2 08) sin ~o < 0,
(8)
1 + e sin (¢p - 2 08) sin ~o < 0,
(8a)
1 -
while
(8b)
< ~COS 2 0 a - - 1 .
Since the values of e are normally of the order of 100, the fulfillment of the above condition is not difficult. F o r example, figs. 2 and 3 show the projections of the scattering surfaces in the plane of the experiment for the annihilation and creation cases, respectively. After introducing some further abbreviations:
Ax=x-xo,
Ay=y-yo
(9a)
and
F
1
(9b)
eq. (4) reduces to the simpler form:
(
Ax +
ee c°s ¢PAy
;
2 sin 2 ~p_
l
(8~--1) 2
from
(dy/dx
+
[(Ay)2-(Ayo) 2] +
z 2 = 0,
(10)
which it follows that the tangential points = 0) on the scattering surface are given by: x~,2 = x0 _+ c°s---------2~ ec¢ Ayo,
implies scattering in which a m a g n o n can only be created.
(ll) Yl,z = Yo+-Ayo •
= 233 o¢0= 0
0,=8"
,...~.
¥
0,83
,11
0.,82
///*"
- ,o
o,8~ o.8o
~0
079 0.r8 0/7 i
0.78 0.79 0.8o 0.8f o,82 ~
f
X
Fig. 2. T h e s c a t t e r i n g s u r f a c e (in t h e p l a n e o f t h e e x p e r i m e n t ) for e = 233, ct0 = 0, 0• = 8 ° a n d tp = 20 ° ( m a g n o n a n n i h i l a t i o n case).
t8
2.0
21
~2
X
Fig. 3. T h e s c a t t e r i n g s u r f a c e (in the p l a n e o f t h e e x p e r i m e n t ) for ~ = 233, ~t0 = 0, 0B = 8 ° a n d tp = 8 ° ( m a g n o n c r e a t i o n case).
MAGNON
INVESTIGATIONS
WITHOUT
F r o m these considerations it is seen that the study of magnons without direct energy analysis can be carried out by the following methods: 1. When a steady neutron source is used one may fix the counter and the sample at given positions and change continuously the incoming neutron wavelength. The width of the peak observed will be given by Y2 - Y l . The same can also be done with the inverted geometry where the sample is exposed to a "white b e a m " and the scattered neutron wavelength is scanned. 2. In the case in which PNS is used the situation becomes more complicated. For any point on the scattering surface the total time of flight (in microseconds) is
where LI and L 2 are the neutron flight paths (in meters) measured from the source to the sample and from the sample to the counter, respectively. Therefore the width of the observed peak, ziT, is determined by the condition d T / d x = O (or dT/dy = 0 ) and not dy/dx=O or dx/dy = 0, as is the case with the experiments discussed in method 1" * It can be proved that the solutions of the equation dT/dx = 0 lie between the solutions o f the equations dy/dx = 0 and dx/dy = 0 which can be found analytically.
aT-
AT
f20~
8O0 01z Mo
A DIRECT
ENERGY
47
ANALYSIS
The roots of the equation dT/dx = 0 must be found numerically for any particular experimental arrangement. The advantages of the experiment with PNS may, however, compensate for this difficulty. These are the following. Because x and y are functions of the dispersion parameters e and %, one sees that in order to determine them it is enough to study the variation of A T as a function of L~ and L 2. This may be realized for example by means of varying the distance between the sample and the counter. The geometry of the experiment is then maintained constant. This can facilitate the study of e and eo as functions of different external fields. Information on ~ and % can, in addition, be obtained by studying the magnon width with respect to the scattering angle q~ or 0B. If the resolution of the time-of-flight spectrometer were infinite, then sharp maxima should be observed in the curve of the intensity versus the time-of-flight. These would correspond to the points (Xm,y0 and (Xa,Y2). The necessarily finite nature of the resolution will, however, spread out the singularities in the expression for the neutron cross section. In figs. 4 and 5 the widths of the magnon peaks for different sets of 0B and tp are presented. We note that in a certain region of ~p the peak width increases almost linearly with ~o for the magnon annihilation case. The slope of this line is approximately inversely proportional to the square root of ~. In fig. 6 the calculated shapes of the magnon lines are presented for 0B = 8 ° a n d s = 233, eo = 0,L1 = 13m, L 2 = 2 . 2 m . In the course of the calculations, an infinite resolution of the instrument was assumed. For the same set of parameters, the dependence of the total intensity on I~p-2 0BI is shown. One can clearly see the following: (a) the shape of the magnon peak is
5°I Oa=75°
4Or
-~ .~ ' i
' 4 30
Fig. 4. The dependence o f the magnon line width on the scattering angle for various 0B; Ct = 233; ~t0 = 0, L1 = 13 m, L2 = 2.2 m.
~
dO
45
50
55
Fig. 5. The dependence o f the magnon line width (inarbitrary units) on the scattering angle for various ~; 0B = 15 °, a 0 = 0 , La = L2.
48
L. D O B R Z Y N S K I
~19 = 20"
~0 = t8 °
~/0 =
~0 = 22 °
2.4,"
f i
,
i
tO
20
30
~0
i
L
i
i
i
10
20
30
40
50
0
i
i
,
t
i
fO
20
30
40
50
i
60
i
,
,
i
J
10
20
30
4,0
50
60
Fig. 6. Intensity distributions (in arbitrary units) for ~ = 2 3 3 , a 0 = 0 , 0 B = 8 ', La = 13 m, L 2 = 2 . 2 m and different (0. T h e r i g h t - h a n d side edges o f the s u b s e q u e n t peaks (from the left) correspond to the following times o f flight: 6000/~sec, 6728/tsec 7 4 5 5 # s e c a n d 8190/tsec, in that order; the channel width is 8 sec a n d the n u m b e r o f channels is s h o w n on the horizontal axis
9 8
o4 = 2 3 3 ~o = 0
7 6
5 4 3 2 1
Fig. 7. T h e total intensity dependence on the [(o--2 01~[ calculated for c~ = 233, ~o = 0, 0]~ = 8 °.
rather irregular; so the peak half width, which is normally such a convenient parameter, is not so good an experimental parameter for the purpose of comparison with theory. It is possible to use the first and the second moments of the intensity curve in the analysis of the experimental results; (b) the total intensity drops rapidly with Iq>- 2 0B], which should be expected on the basis of the positions of the scattering surfaces (see
figs. 2 and 3). The part of the ellipsoid which lies close to the y = x line corresponds to neutron scattering on long wavelength magnons. Due to the high population of such magnon states, this part gives the main contribution to the scattered intensity. Because in the case of magnon creation the accessible range of the magnon wave vectors expands more rapidly than in the annihilation case, the sharper drop of intensity in the former case is not unexpected. On the other hand, even for quite large values of lop-2 0al the main contribution to the intensity comes from the long wavelength magnons. This situation can be convenient in the experimental determination of a distortion of the quadratic dispersion law which should be present, e.g. in the case of dipole-dipole interaction. Ending this paper, I would like to give my thanks to Dr. A. Holas for much advice in computational problems and to the staff of the Laboratory II of the Institute of Nuclear Research at Swierk for many interesting and stimulating discussions. References t) R. J. Elliot a n d R. D. Lowde, Proc. Roy. Soc. A230 (1955) 46. 2) T. Riste, K. Blinowski a n d J. Janik, J. Phys. C h e m . Sol. 9 (1959) 153.
INSTRUMENTS
AND
METHODS
95
(I97I)
45-48; ©
NORTH-HOLLAND
PUBLISHING
CO.
ON THE POSSIBILITY OF MAGNON INVESTIGATIONS WITHOUT A DIRECT ENERGY ANALYSIS USING A PULSED NEUTRON SOURCE L. D O B R Z Y N S K I *
Joint Institute for Nuclear Research, Dubna, U.S.S.R. Received 5 March 1971 The equation o f the scattering surface is derived from the energy and m o m e n t u m conservation laws for inelastic neutron scattering involving m a g n o n creation or annihilation. F r o m this equation some variants of the diffraction technique arise. Particular attention is paid to the studies made possible by pulsed reactors.
The diffraction technique in the m a g n o n studies by means o f neutron scattering has already been k n o w n for m a n y years1). It was extensively employed in the studies o f all magnetic substances: ferro-, antiferroand ferrimagnets. The purpose o f this paper is to propose a version o f this technique which could be particularly interesting when combined with a pulsed neutron source (PNS). It will be shown that when the m a g n o n dispersion relation is quadratic then the neutron scattering at a given angle is strictly confined to a certain range o f incoming and scattered neutron wavelengths. The implication o f this fact is that different experimental arrangements may be used for the studies o f m a g n o n s with no energy analysis. Let k' and k denote the scattering and initial neutron wavevector, respectively. When neutron scattering involves m a g n o n creation or m a g n o n annihilation then the following m o m e n t u m and energy conservation laws are satisfied: k'-k-2n~
=
q,
(l) where q is the m a g n o n wavevector, E o is the energy gap in the m a g n o n dispersion law and e = + 1, where the upper sign refers to the m a g n o n annihilation and the lower one to the m a g n o n creation. The geometry o f the experiment is shown in fig. 1. Neutrons impinging at a given angle ( 9 0 ° - 0 B ) with respect to the crystal axis are analyzed along the direction described by the chosen scattering angle tp. Let us write =
k; + k'~,
ko = nr/sin 0B, (3) Ik'xl ko
--X,
Ikl
--
Ik;I
~y,
ko
-- Z,
ko
Eo
--~-~0,
k~
the solution of eq. (1) with respect to q gives the equation o f the scattering surface: (l - e~) x 2 + 2 e ~ x y cos tp - 4 ectx sin 0B sin (0 B - ~p) -
- (l + ect) y2 + 4 s ~ y sin E 0 B + ( I - e c 0 z 2 - e[~0+4~sinEOB] = 0.
(4)
This equation describes an ellipsoid centered at z o =0
Xo = 2 e~ sinOB ect cos Oa sin (p - sin (OB - ~o) ~2 sin 2 ~0- 1
(5)
Yo = 2 ec~sin 0 a ec~sin (p cos (0 a - ~o) - sin Oa O~2 sin E q9- 1
k'2_k 2 = s(Eoq-~qE),
k'
the following abbreviations
~
88= 15° ? = 40 °
/
~\
"\\
/gO<8\l\\~ (000) 2~T[ ~,~o
(2)
where k' is perpendicular to the scattering plane. With
Fig. 1. The geometry of the experiment. The scattering surface is drawn for m a g n o n s satisfying the quadratic dispersion relation with ~t = 100, a0 = 0, 0n = 15 °, tp = 40 °, e = 1.
* On leave from the Institute of Nuclear Research, Laboratory II, Swierk post Otwock, Poland.
45
46
L. D O B R Z Y N S K I
if ~2
sin 2 cP-- 1 > 0.
(5a)
The ellipsoid is real when --
1
Some additional restrictions arise from the requirement that x and y must be positive; e.g. in the case of m a g n o n annihilation this requirement leads to the inequality
{4 ~ sin 2 08 [1 - ect sin (2 0 8 - q0 sin ~o] -
l < ~COS2(¢p--2OB)
0~--/3 -
~o(~2 sin2 ~o- 1)} > 0.
(6)
T h e vertical cross section of the ellipsoid (4) is a circle with the radius
R=[4s~sin2Oa[1-ectsin(2Oa-tp)sinq ~] (ect-- 1) (~2 sin2 qo-- 1)
e°~o 3~:. ect-- 1
(7) It can be seen that the radius in eq. (7) varies continuously with q~ and 0a. This observation forms the basis of the first experiment which was p e r f o r m e d by Riste et al.2). In this experiment the angle ~o was kept constant and the vertical cross section of the ellipsoid was studied for different 08. The variation of the radius Rz with respect to 08 leads to values of e. It follows f r o m eq. (6) that for given ~0 > 208 one observes scattering with m a g n o n annihilation (for e = l ) only, while m a g n o n creation is obtained for ~o < 2 0B. F o r example, when c~o = 0 (no energy gap) the condition that scattering with m a g n o n annihilation will occur is e sin (~o - 2 08) sin ~o < 0,
(8)
1 + e sin (¢p - 2 08) sin ~o < 0,
(8a)
1 -
while
(8b)
< ~COS 2 0 a - - 1 .
Since the values of e are normally of the order of 100, the fulfillment of the above condition is not difficult. F o r example, figs. 2 and 3 show the projections of the scattering surfaces in the plane of the experiment for the annihilation and creation cases, respectively. After introducing some further abbreviations:
Ax=x-xo,
Ay=y-yo
(9a)
and
F
1
(9b)
eq. (4) reduces to the simpler form:
(
Ax +
ee c°s ¢PAy
;
2 sin 2 ~p_
l
(8~--1) 2
from
(dy/dx
+
[(Ay)2-(Ayo) 2] +
z 2 = 0,
(10)
which it follows that the tangential points = 0) on the scattering surface are given by: x~,2 = x0 _+ c°s---------2~ ec¢ Ayo,
implies scattering in which a m a g n o n can only be created.
(ll) Yl,z = Yo+-Ayo •
= 233 o¢0= 0
0,=8"
,...~.
¥
0,83
,11
0.,82
///*"
- ,o
o,8~ o.8o
~0
079 0.r8 0/7 i
0.78 0.79 0.8o 0.8f o,82 ~
f
X
Fig. 2. T h e s c a t t e r i n g s u r f a c e (in t h e p l a n e o f t h e e x p e r i m e n t ) for e = 233, ct0 = 0, 0• = 8 ° a n d tp = 20 ° ( m a g n o n a n n i h i l a t i o n case).
t8
2.0
21
~2
X
Fig. 3. T h e s c a t t e r i n g s u r f a c e (in the p l a n e o f t h e e x p e r i m e n t ) for ~ = 233, ~t0 = 0, 0B = 8 ° a n d tp = 8 ° ( m a g n o n c r e a t i o n case).
MAGNON
INVESTIGATIONS
WITHOUT
F r o m these considerations it is seen that the study of magnons without direct energy analysis can be carried out by the following methods: 1. When a steady neutron source is used one may fix the counter and the sample at given positions and change continuously the incoming neutron wavelength. The width of the peak observed will be given by Y2 - Y l . The same can also be done with the inverted geometry where the sample is exposed to a "white b e a m " and the scattered neutron wavelength is scanned. 2. In the case in which PNS is used the situation becomes more complicated. For any point on the scattering surface the total time of flight (in microseconds) is
where LI and L 2 are the neutron flight paths (in meters) measured from the source to the sample and from the sample to the counter, respectively. Therefore the width of the observed peak, ziT, is determined by the condition d T / d x = O (or dT/dy = 0 ) and not dy/dx=O or dx/dy = 0, as is the case with the experiments discussed in method 1" * It can be proved that the solutions of the equation dT/dx = 0 lie between the solutions o f the equations dy/dx = 0 and dx/dy = 0 which can be found analytically.
aT-
AT
f20~
8O0 01z Mo
A DIRECT
ENERGY
47
ANALYSIS
The roots of the equation dT/dx = 0 must be found numerically for any particular experimental arrangement. The advantages of the experiment with PNS may, however, compensate for this difficulty. These are the following. Because x and y are functions of the dispersion parameters e and %, one sees that in order to determine them it is enough to study the variation of A T as a function of L~ and L 2. This may be realized for example by means of varying the distance between the sample and the counter. The geometry of the experiment is then maintained constant. This can facilitate the study of e and eo as functions of different external fields. Information on ~ and % can, in addition, be obtained by studying the magnon width with respect to the scattering angle q~ or 0B. If the resolution of the time-of-flight spectrometer were infinite, then sharp maxima should be observed in the curve of the intensity versus the time-of-flight. These would correspond to the points (Xm,y0 and (Xa,Y2). The necessarily finite nature of the resolution will, however, spread out the singularities in the expression for the neutron cross section. In figs. 4 and 5 the widths of the magnon peaks for different sets of 0B and tp are presented. We note that in a certain region of ~p the peak width increases almost linearly with ~o for the magnon annihilation case. The slope of this line is approximately inversely proportional to the square root of ~. In fig. 6 the calculated shapes of the magnon lines are presented for 0B = 8 ° a n d s = 233, eo = 0,L1 = 13m, L 2 = 2 . 2 m . In the course of the calculations, an infinite resolution of the instrument was assumed. For the same set of parameters, the dependence of the total intensity on I~p-2 0BI is shown. One can clearly see the following: (a) the shape of the magnon peak is
5°I Oa=75°
4Or
-~ .~ ' i
' 4 30
Fig. 4. The dependence o f the magnon line width on the scattering angle for various 0B; Ct = 233; ~t0 = 0, L1 = 13 m, L2 = 2.2 m.
~
dO
45
50
55
Fig. 5. The dependence o f the magnon line width (inarbitrary units) on the scattering angle for various ~; 0B = 15 °, a 0 = 0 , La = L2.
48
L. D O B R Z Y N S K I
~19 = 20"
~0 = t8 °
~/0 =
~0 = 22 °
2.4,"
f i
,
i
tO
20
30
~0
i
L
i
i
i
10
20
30
40
50
0
i
i
,
t
i
fO
20
30
40
50
i
60
i
,
,
i
J
10
20
30
4,0
50
60
Fig. 6. Intensity distributions (in arbitrary units) for ~ = 2 3 3 , a 0 = 0 , 0 B = 8 ', La = 13 m, L 2 = 2 . 2 m and different (0. T h e r i g h t - h a n d side edges o f the s u b s e q u e n t peaks (from the left) correspond to the following times o f flight: 6000/~sec, 6728/tsec 7 4 5 5 # s e c a n d 8190/tsec, in that order; the channel width is 8 sec a n d the n u m b e r o f channels is s h o w n on the horizontal axis
9 8
o4 = 2 3 3 ~o = 0
7 6
5 4 3 2 1
Fig. 7. T h e total intensity dependence on the [(o--2 01~[ calculated for c~ = 233, ~o = 0, 0]~ = 8 °.
rather irregular; so the peak half width, which is normally such a convenient parameter, is not so good an experimental parameter for the purpose of comparison with theory. It is possible to use the first and the second moments of the intensity curve in the analysis of the experimental results; (b) the total intensity drops rapidly with Iq>- 2 0B], which should be expected on the basis of the positions of the scattering surfaces (see
figs. 2 and 3). The part of the ellipsoid which lies close to the y = x line corresponds to neutron scattering on long wavelength magnons. Due to the high population of such magnon states, this part gives the main contribution to the scattered intensity. Because in the case of magnon creation the accessible range of the magnon wave vectors expands more rapidly than in the annihilation case, the sharper drop of intensity in the former case is not unexpected. On the other hand, even for quite large values of lop-2 0al the main contribution to the intensity comes from the long wavelength magnons. This situation can be convenient in the experimental determination of a distortion of the quadratic dispersion law which should be present, e.g. in the case of dipole-dipole interaction. Ending this paper, I would like to give my thanks to Dr. A. Holas for much advice in computational problems and to the staff of the Laboratory II of the Institute of Nuclear Research at Swierk for many interesting and stimulating discussions. References t) R. J. Elliot a n d R. D. Lowde, Proc. Roy. Soc. A230 (1955) 46. 2) T. Riste, K. Blinowski a n d J. Janik, J. Phys. C h e m . Sol. 9 (1959) 153.