A compendium of selected international publications tuyển tập các công bố quốc tế

VIETNAM ATOMIC E N ER G Y INSTITUTE
PHAM DUY HIEN

A COMPENDIUM OF SELECTED
INTERNATIONAL PUBLICATIONS
TUYHN TẬP C Á C CỒNG BỐ Q U Ổ C T Ể

\

[ 7 S C I E N C E AND T E C H N I C S P U B L IS H I N G H O U S E


VIETNAM ATOMIC ENERGY INSTITUTE
PHAM DUY HIEN

A COMPENDIUM OF SELECTED
INTERNATIONAL PUBLICATONS
TUYẾN TẠP CÁC CÔNG BỐ Q UỐ C TÉ

Responsible for Publishing:

Phạm Ngọc Khôi

Editor:

Quỳnh Anh

Cover Design:

Ngọc Tuấn

SCIENCE AND T E C H N IC S PI B U S H IN G HOUSE
70 Tran ílunỊỉ Đao, Hoan Kiem. Ha Noi

I*

Publishing registration number 376- 2014 CXB/13-19/KHKT
Publishing decision number: 20 ODXB-NXISKHKT, 11/03/2014
Ouantitv 100 pcs, SI/-.cm at Tran Cong Joint Stock Company
P rin tin g finished and cnpvriL’ht deposited in 1’’quarter o f 2014


Lời tựa
■

gày 20 3/2014. Lò Phàn ứng hạt nhân Đà Lạt vừa tròn 30 năm vận hành an

N

toàn và khai thác có hiệu quả. Trải qua 30 năm vận hành và phát triển, phải
kè đến sự tham aia đóng góp cúa rất nhiều thế hệ lãnh đạo và các nhà khoa

học. trong đó có m ột người mà tất cà mọi người đều mến mộ và kính trọng, đó chính
là GS Phạm Duy Hiên.
GS Phạm Duy Hién là một trons nhửng nhà khoa học đầu ngành H ạt nhân - người đã
đặt nền móng đầu tiên cho việc hồi sinh Lò Phàn ímg hạt nhân Đ à Lạt - lò phàn ứng
duy nhàt của Việt N am . một còne trinh lịch sừ và là niềm tự hào của ngành hạt nhân
Việt Nam. Từ nhừng ngày đầu với biết bao khó khăn, thử thách ấy, GS Phạm Duy
Hièn - N guyên Phó Viện trưởng Viện Năng lượng N guyên tử Quốc gia, Viện trưởng

Viện N ghiên cửu Hạt nhàn Đà Lạt đă được Đàng, N hà nước giao cho trọng trách lớn
lao - đó là cùng phia Liên Xô thực hiện sử mệnh lịch sứ cùa ngành Hạt nhân Việt Nam
- khôi phục sự hoạt độn2 của Lò Phản ímg hạt nhân Đà Lạt. N gày 20/3/1984, lò phản
ứng được chính thức đưa vào vận hành với công suất danh định 500 kWt, gấp hai lần
so với lò TRIGA M ark-2 trước đây. Với tư cách Viện trướng Viện N ghiên cứu Hạt
nhàn Đà Lạt. giáo sư đã dẫn dẩt Viện tiển lên vừng chẳc. có uy tín trong nước, trong
khu vực và được bạn bè quổc tế thừa nhận. Đối với đội ngũ cán bộ cùa Viện Nghiên
cửu Hạt nhân Đà Lạt. GS Phạm Duy Hiển vừa là người lãnh đạo, người hướng dẫn
khoa học. vừa là nsười bạn. người anh thân tình cùa các thế hệ.

Nhân dịp ký niệm 30 năm khánh thành công trình khôi phục và m ờ rộng Lò Phản ứng
hạt nhân Đà Lạt (20 03 1984-20/03/2014), Viện Năng lượng N guyên tử Việt N am xin
giới thiệu với bạn đọc cuốn "Tuyến tập các công bổ quốc tế" của GS Phạm Duy Hiển.
Đó là những công trình khoa học, chù yếu là những bài báo được đăng tải trên các tạp
chí khoa học thế giới từ nãm 1962 đến nay. Nhìn vào đó chúng ta có thể thấy được sự
nghiêm túc trong khoa học, sự đóng góp to lớn đáng trân trọng của giáo sư cho ngành
Hạt nhân Việt Nam. Kính chúc GS Phạm Duy Hiển có nhiều sức khỏe để tiếp tục cống
hiến cho ngành hạt nhân nước nhà, giúp đào tạo đội ngũ cán bộ khoa học kế cận đảm
đương những trong trách trong ngành hạt nhân Việt Nam.
V iện N ăng lư ợng N guyên tử V iệt Nam



a b o u t t h is c o m p e n d iu m

T h e re sea rc h a rtic le s c o m p ile d in th is b o o k can b e g ro u p e d in to th e v a rio u s d is c ip lin e s and
p u b lic a tio n \ e a r s as fo llo w s:
1 ■ A rticles «1 to # 7 published in the early 1% 0 's , are experim ental w orks on the effect o f
re c o il-fre e e m issio n an d a b so rp tio n o f low e n erg y g a m m a ra y s in c ry sta ls th a t w as
d isco v e re d in 1958 b y a G e rm a n p h y s ic ist R. M o ssb au e r, w h o w o n th e N o b e l P rize in

p h y sics in 1961. T he a n iso tro p y o f th e M o ss b a u e r e ffe c t in /?-S n s in g le c ry stals
d isco v e re d in a rtic le - 3 h a s b een c ite d re g u la rly in th e lite ra tu re w h ile a rtic le # 2 w a s o n e
o f the ten a rtic le s c ite d o n th e M o ss b a u e r e ffe c t in th e S o v ie t E n c y c lo p e d ia o f P h y sic s
p u b lish e d in 1 % ? .

2.

In ill.' co n d itio n s o f w ar and hardship in tile 1960's and I 9 7 0 ’s no experim ental facilities
w e re a v a ila b le fo r re s e a rc h in V ie tn a m and th e o re tic al re se a rc h w a s th e o n ly w a y fo r
s c ie rris ts to k eep p a c e w ilh d e v e lo p m e n ts in tile field. T h is e ffo rt h a d re su lte d in a rtic le s
= s to =13 p u b lis h e d in th e S o v iet JE T F , a lead in g jo u rn al o f p h y sic s at th a t tim e.

.V

The t ext experim ental and theoretical w orks w ere carried out m ain ly in D ubna (1 9 7 2 -7 3 )
on neutron activation 1= 1JI in the IBR-2 pulsed nuclear reactor and on fission isom ers

1
4.

= 15 :o = 2 0 ) in h ea v y ion accelerators.

The . ccelerator and G e detector that w ere installed at the Institute o f P h ysics in 1974
p ro v id e d an o p p o rtu n it} to in v e stig ate for th e first tim e in V ie tn a m th e n u c le a r re a ctio n s
a n d a c t i v a t i o n a i u h s i s V, ith 14 M e V n e u t r o n s ( a r t i c l e s #22

-

# 24), a n d analysis o f ore

sam p les b y th e h ig h re s o lu tio n earn m a sp ec tro sc o p y m e th o d (a rtic le # 2 6 ). A rtic le # 2 7
d e sc rib e s an in te re s tin g fie ld e x p e rim e n t on d e te c tin g th e o il-w a te r c o n ta c t in th e 3 0 0 0 m
deep ?il exploration borehole in the Thai Binh oil field. T he experim ent w as carried out
in 1975 w ith th e Use o f a 14 M eY p u lsed n e u tro n g e n erato r.
V

A rtie es ° 28 to = ; ; d e sc rib e th e a p p lic atio n s o f th e D alat n u c le a r re se a rc h re a c to r th a t
w as c o m m is sio n e d in 19S4. T h e u se o f re a c to r-p ro d u c e d ra d io tra c e rs in h y d ro lo g y and
sed im en to lo g v w as presented in articles =33 and # 3 4 .

6.

The V ietn a m ese scien tists started a new research direction on lon g range transport o f
radio ictiv e air pollutants and environm ental radioactivity m on itoring after the C hernobyl
accident in 1986. The transport o f C hernobyl-derived radionuclides w a s investigated in
article =31. and 25 Years later, the YAEI laboratories su c ce ssfu lly detected radionuclides
transported from the Fukushim a accident (article # 5 2 ). N o te that no international
p u b lic a tio n s on th e se e v e n ts cam e from o th e r S o u th e a st A sian co u n trie s.

7.

V ie tn a m w a s n o t ab le to m e a su re and in v e stig ate ra d io n u c lid e s d e riv e d from the n u c le a r
w eapons tests in the earlv 1 9 6 0 's. A rticles -4 1 to - 4 3 present a su ccessfu l approach to
reconstructing the accum ulated deposition d ensities o f nuclear tests-derived C s-1 3 7 , Sr90. and P u -2 3 9 -2 4 0 o ver the V ietnam land. T ile reconstructed dep osition density o f C s-

o


137 w as favorably com p ared w ith the reference values o f C s-1 37 m easu red in the
experim ents on soil erosion and sedim entation in articles #40 and #53.

8.

To support research on long ran g e tran sp o rt o f air pollutants, the studies on air quality in
urban and rural areas by u sing n u clear and co m plem entary an aly tical techniques w ere
carried out o v er the last tw o decades. T his resulted in articles #36 - # 39 and #44 - #50.
M any o f these articles have enjoyed high citations acco rd in g to G oogle Scholar and
Scopus.

M ost o f articles in this com p en d iu m are the results o f collaborative research w ith m any
V ietnam ese scientists 0 \ er the last four decades. The author w o u ld like to express sincere thanks
to them on the occasion o f the publication o f this book at the initiative o f the V ietnam A tom ic
Energy Institute.

H


Prof. P. D. H IEN ’ PUBLICATIONS IN INTERNATIONAL JOURNALS
No

Pages

P aper

1

Resonance scattering o f gam m a quanta in Te1"5. Soviet Physics JE T P , Vol 15, p 489,

2

Calculation o f the param eters o f experim ental spectra ot resonance absorption o f gamma

quanta in crystals. Soviet Phvsics JETP. 16, 646, (1962).

14

3

A nisotropy o f the M ossbauer effect in p-Sn single crystals. Soviet Physics JE T ? , 16,
559.

20

4
5

11

D ependence o f the y-quanlum resonance absorption spectrum on crystal temperature.
Soviet Physics JE T P . 17. 268 (1963).
Resonance absorption o f y-quantum in barium, strontium and calcium stannates. Soviet
Physics .ỈETF~ 17, 1271. (1 963).

23
26

6

Abrupt chance in probability o f the M ossbauer effect at the phase transition in ferroelectrics. Soviet Physics Letters JETF . (,1963).

31

7

Change o f resonance absorption spectra o f 23 .8 keV uam m a-rays o f S n 119 during phase
transitions o f the B iFeO .-SrtSn M n , ,)O v Soviet Physics Letters JE T P (1964).

33

8

The investigation o f nuclear quadrupole interaction in experim ents on the M ossbauer
effect. Soviet Physics JE TP . 22. 1080. (1965).

36

Q

Evaluation o f the effective resonance integral for resonant neutron absorption in hetero­
geneous reactors. A tom naya Enereiya VÒ127, pp. 3-5, 1969

39

10

Spontaneous em ission o f Y-quanta by a system containing identical nuclei. Soviet Physics
JETP. 3 1 .8 3 ,(1 9 7 0 )

42

11

Influence o f coherence o f resonance absorption o f y - quanta in polycrystalline substanc­
es. Soviet Physics JE TF . 32. 111. {1971).

46

12

Certain problems o f kinematic theory o f bragg scattering o f resonant y-rays

50

13

On the theory o f passatie o f v-quanta through a resonant m edium. Soviet P hvsics JETP,
3 4 .1 9 1 .(1 9 7 2 ).

55

14

Activation analysis w ith thermal and epithermal neutrons o f the Dubna reactor IBR-30.
Radiochem istry \ o \ 15. N 6. 1973.

59

Coulomb excitation O! spontaneous fission isomers by heavy ions .C om m unications o f
15
JIXR P7-7022, (1973)
A ngular đisưibution ot fragm ents o f spontaneously fissionnine isomers. Soviet Nuclear
16

Pin sics. 1 7 .N 3 .4 8 9 .1 1973)

65
76

r

Delayed fission fragm ent angular distributions in some alpha-particle-induced reactions.
Int Coni on Physics a n d C hem istry o f Fission. Rochester, (1974) IA EA-SM -174/15

80

18

Fission-in-flight technique and the slow ing-dow n o f recoil nuclei in gases.
Á Methods, 121. 379 11974).

88

19

D eorientation o f a nucleus and the angular distribution o f g-quanta in the reaction with
hi^h inserted m omentum. Soviet N uclear Physics. 1, 4, (1975) (in Russian).

93

20

On spin assignment for spontaneously fissioning isomers. Soviet Nuclear Physics, 22, 5
1 1975) (in Russian).

100

21

1 -i-MeV neutron acti\ ation analysis o f bauxite (with To N. T. et al.). Zavodskaja Laboratnria (in Russian) 43. 1091. 1977.

109

2"

jwance for fluctuations o f radiation flux in activation analysis A tom naya Energiya,
42 467. n 977).

112

Nucl. Instr.


Determination o f copper using annihilation g-rays . Zcivodskaja
sian) 44, S20. (, 10"S).

Laboratoria (in Rus­
114

24

Determination o f uranium in rocks with disturbed radioactive equilibrium by means of
izamma-spectrometry. Atoinnayu cncixiya. Vol. 50, No. 2, 1981, pp. 146-148

117

25

Neutron emission from "Be nucleus formed under P' and g-irradiation by radioactive
dcca\s. JIXR Preprint. P3-S0-SS0. Five. Int. Conf. cm Neutron Physics, Kiev, (1980).

120

:6

A study of the isomeric ratio for the (n,2n) and (g,n) reactions in 92Mo, 90Zr, 86Sr and
'4Sc . >Vv. Sucl. Phys. Yol 35, 2, 257-263 (1982).

127

27

Application o f the pulsed neutron - neutron melhod for the determination o f oil-water
contacts in a bore hole. Proc. IAEA Consultants 'Meeting on Nuclear Data for Borehole
j n J Bti.'k McJiil .-iNNi/v L'sinc Xuclciir Techniques, Krakow, Poland, November (1983).

134

2S

Determination o f k -factors bv thermal neutron aclivation technique. J. Radiocinal. Nucl.
Client.. Letup's 105. .'51 ( WSo).

139

2l)

X-ra\ fluorescence XRF anal)sis with 'Go excitation source. J. Radioancil. Nucl.
Chcm.. Letups 1 1 8 . : r (N S 'V

144

JO

Determination o f k -factors o f short-lived nuclides (T >= 1 mill) by thermal neutron acti\ at ion technique../.
Xitel. Chem.. Letters 155, 169 (1991).

152

31

3;

The application and development o f ^-standardization method o f neutron activation anal\ sis at Dal.1t research reactor / Rililioaihil. and Nucl. Chơm., 257, 643-647, 2003
Procress and challenges o f nuclear science development in Vietnam - an outlook on the
occasion o f the 10-th anrmersary o f the Dalat Nuclear Research Reactor (1994). Trans­
actions o f the AmcrkiiK X iulcar Socicty: Journal Volume: 70; Journal Issue: Suppl.J;
Contircncc V-th A/c.Tw ;\/.v in nuclear conference, Sydney

159

164

33

Application o f isotope tracer techniques for assessing the seepage o f the hydropower
dam at Tri An. South Vietnam. J. Radioanal. Nucl. (Them., Vol. 206, No. 2 (1996), pp.
295-303

172

-'4

Sediment transport in the Haiphonu Navigation Channel. Radiation Protection Managnicut. JuK Aunusi 20(10

181

35

Variations o f caesium isotope concentrations in air and fallout at Dalat, South Vietnam,
1986-91./ Environ /?.:./■ '.\rivirv. 22, 55 (1994).

190

36

Monitorinc lead in Ml.1']'ended air particulate matter ill Ho Chi Minh City. Atmospheric
Environment 31. I T 3 1 ! wv>“ )

198

T e m p o ra l \ ariation.N OĨ V '.ircc impacts at the receptor as derived from air particulate mon-

3

38
39
40
41
42

f &

itonne (JuUi in 11«' Chi
(1999).

Cit\. Vietnam. Atmospheric Environment, 33, 3133-3142

INAA in comhinati' >n V.
: Ỉ1 c r analytical techniques in the study o f urban aerosol sourc­
e s../ Rudinanul cV V,- .
m Vol. 244, N l, 10.1-107 (2000).
( Minparam e rcccpt' T !".■■
'J. study oi l Si\ PM , and PM, M
I in 1lo chi Minh City Atmo­
spheric i.n\ in/iwh I1 Ỉ 35. v- ! 5. pp 26i)l)-2678, 2001.
A-i'.essmcnt "1 L-niNinn d accretion in catchmcnt areas based on -l0Pb and li7Cs contents
in soil and sediment. <::: ^ " !-, "isotope Production am t Applications in the 21-st Centuf.dit H\ NiL'L'l S 'j
. Vancouver, Canada, September 1999).
Derivation 0 ! ( s cJcp'*'i’:"n ilcnsitv from measurements o f " 7Cs inventories in undisI'lrhcd
o f Ị.IV. !".rn iU’j l RadifJiictivitv, , 62, N3, pp 295-303, 2002).
f rr.ironmciit.ll n u ll'’!, :.
in surlace soils o f Vietnam. ThcScicntiJicWorld Journal
(200-2) 2. 1127-1 m

202
212
217

227
231
241


43

44

45
46

-**240pu. ',0Sr and ,?7Cs inventories in surface soils o f Vietnam. J. o f Environm ental R adio­
activity• 75, 329-337, 2004.
Influence o f m eteorological conditions on P M ,, and PM 25 |0 concentrations during the
monsoon season in Hanoi. Vietnam. Atm ospheric Environm ent, 36, N 21, pp 107-118,
2002
PMF receptor m odelling o f fine and coarse PM 10 in A ir m asses G overning M onsoon C on­
ditions in Hanoi, northern Vietnam. Atm ospheric Environm ent 38. 189-201, 2003.

267

Investigation o f sulfate and nitrate formation on mineral dust particles by receptor m od­
eling 2005. Arm osphcric Environment. Vol. 39. p p 7231-~239, 2005

280

246

255

I Than air quality in the A sian region. Science o f the Total Environm ent 404. 103-112,
2008.
C hance o f air tem perature w ith altitude, atmospheric stability and air pollution. 39-th
Phvsics Olym piad - Hanoi - Vietnam - 2008. Theoretical Problem N o 3. .
ntnu.edu t\v problem s-and-soluiions 2008 IPhO 2008 Theory 3 Problem .pdf

299

49

Regional and local em issions in Red river delta, northern Vietnam. Air O ual Atm os Health
( 2 0 0 ^ : 157-1(17

305

50

Air pollution episodes associated w ith East Asian w inter m onsoons. Science o f the Total
Environment. 409 (201 1) 50(V'-506S

316

51

Atmospheric radionuclides from the Fukushima Dai-ichi nuclear reactor accident observed

in Vietnam. Journal or Emiror.n'.cfUiil Radioactivity'. Vol. I l l , 53-58, 2012.

322

S'1

Natural radioactivity and external dose assessm ent o f surface soils in Vietnam Radiation
Proit'Ctit'H D osimetry 2012. pp 1-10

328

■>"í

Redistributions o f i r ( s and soil com ponents on cultivated hill slopes w ith hedgerows as
conservation m easures. Soil Iiỉhi Tillage Research. V 128, pp. 149-154, 2013

338

54

Concentrations o fN O . s o . , and benzene across Hanoi m easured by passive diffusion sam ­
plers. A fn o sp h eric Environment. 2014. http: ch.doi.org/10.1016/j.atm osenv.2014.01.036

344

55

A com parative study (if research capabilities o f East Asian countries and implications for
Vietnam 2010. H igher E Jucanou Vol. 60. pp 615-625, 2010.

352

56

IAEA publications and papers in the proceedings o f international conferences

363

47

48

289



R E S O N A N C E S C A T T E R IN G O F G A M M A Q U A N T A IN T e125
PHAM ZUY H IEN , V. G. SHAPIRO, and V. s. SHPINEL
Nuclear Physics Institute, M oscow State University
Subm itted to JETP editor O ctober 2, 1961
J. Exptl. Theoret. Phys. (Ư.S.S.R) 42,703-706 (M arch, 1962)
By recording the X rays following internal conversion in the resonance absorption in Te125, we have observed
the M ossbaner effect for the 35.5 keV Y rays o f Te12Sm. For TeOj, we found the values f = 0,12 ± 0.03 at 190oC and f = 0.067 ± 0.008 at room tem perature. The half-life o f the 35.5-keV level was m easured to be (1.4
± 0.2) X 10 * sec, which agrees \sith the result from m easurem ents o f delayed coincidences.

1 .“ *

1. IN TR O D U C T IO N
We have studied the resonance for the 35.5-keV Y rays
em itted by T el25m , w hose decay schem e is shown in

Fig.l.
The usual resonance absorption m ethod cannot
be used in this case since the 35.5-keV transition is
highly converted and the weak 7 line o f this transition
cannot be separated bv the scintillation spectrom eter
from the intense X rays at 27.4 and 31.2-keV, which
cannot be filtered out. We therefore used the proce­
dure of counting X rays following conversion in the
resonance o n the scattering nuclei.
2. D E SC R IPT IO N
EX PERIM ENT

OF

APPARATUS

AND

The Tel25mwas obtained by neutron irradiation
in a reactor of m etallic tellurium enriched to 86%
Tem . The irradiated tellurium was in the form o f the
dioxide, T e 0 2, from which a 20 m g/cm 2 source was
prepared. The scatterers were prepared by precipitat­
ing the dioxide TeO; , enriched to 92% in Te125, onto an
alum inum backing 5 n thick.

oc, • /»(/
M I H -.U At I f
I t Itt a c ---------

FIG.l Decay scheme of Te1”"’
FEU -13 photom ultiplier. The lead collim ator 4, placed
over the crystal, shields the direct radiation from the
source. To reduce scattering from the inner walls of
the collimator, the walls are grooved. Below the scatterer, on the end face o f the photom ultiplier there is
a lead plate 5 thich shields the crystal from radiation
scattered by the walls and o th er parts o f the photomutiplier. The plate itself scatters quite weakly b e­
cause o f the deep cut m ade in it.
The scintillation spectrom eter records the X rays
em itted by the scatterer, together with a small adm ix­
ture o f resonantly scattered Y quanta, on a background
of radiation which is scattered nonresonantly by the
m aterial o f the scatterer, as well as the backing, the
lead, etc. The filler 6, o f 400 |i copper foil, serves to
increase the fraction.

The source 1 (Fig. 2), which is cooled by liquid
nitrogen in a Teflon container, is set in reciprocating
m otion at constant speed by an appropriately shaped
cam. The speed of the source is changed by m eans of
m ulti-step pulleys and a variable speed drive, to which
the m otion is transm itted from an electric m otor via
a worm gear. The scatterer 2 is placed inside the Nal
(Tl) detector crystal 3, which is a hollow cylinder with
i.d.27 m m , wall thickness 4.6 m m , and height 12 mm,
halfway up the cylinder. The crystal is placed on an

£ S f

F IG. 3.
of y quanta in the ra d ia tio n in cid e n t on th e s o n tt e r e r and to re d u c e the bao kg ro u n d . T he s p e c ­
t r o m e te r is i=hi eld ed from lig h t by th e h la e k p a p e r
c a n 7.
P u ls e s fro m the p h o to m u ltip lie r p a s s th ro u g h
an a m p lifie r and a sin g le -c lia n n e l d i s c r im i n a t o r to
two P S -1 0 0 0 counting c ir c u its , one of w hich o p e r ­
a te s w hile th e s o u rc e m oves to w a rd Ihe s c a tte r e d ,
the o th e r w hile it m oves in the o p p o s ite d ire c tio n .
D u rin g th e r e v e r s a l of the lir e e tio n of m o tio n (In
the tra n s itio n s e c tio n s of t " e cam p r o f ile ) , both
cou n tin g c ir c u its a r e sw itc h e d out. C o n tro l of the
o p e ra tio n of the counting c ir c u it s is a c c o m p lis h e d
b y a p a ir o f r e la y s w h ich n r c s e t in o p e r a t io n IIV ÍI

e ffic ie n t o f r e s o n a n c e a b s o rp tio n of y q u a n ta by
nuclei of Ilie s e a t t e r e r , ttK is the K - s h e ll c o n v e r ­
sio n c o e ffic ie n t, a is th e to tal c o n v e rs io n c o e ffi­
c ie n t. U>K is Uic i lu o r e s c e n c e y ie ld of th e X ra y s
(wK = 0 . 8 5 ) ^ '- . Pp i s the a b s o rp tio n c o e ffic ie n t f o r
th e p h o to effec t, 4>(E) is the e n e rg y d is tr ib u tio n of
y quanta e m itte d by Uie s o u rc e w ith o u t r e c o il, and
B' is th e p r o b a b ility o f d e te c tin g an X r a y p ro d u ce d
in the s c a t t e r e r . F u r th e r m o r e ,
li' — )— \ exp (— |JRr/sin ủ ) dữ.

(3)

w h ere HR is th e a b s o rp tio n c o e ffic ie n t o f th e s c a tt e r e r f o r X r a y s , Í2 i s th e so lid a n g le w ith in which
th e X r a y s fro m th e s c a t t e r e r im p in g e on the d e le c to r c r y s t a l. F o r s m a ll v a lu e s of ịiỊ{, we can

re p la c e X in (3) by i ts a v e ra g e v alu e X /2 .
F o r fir\x « 1 (th in s c a t t e r e r ) , the s e co n d te rm
in the fin a l fo rm u la (2) c a n b e n e g le c ted . I n tr o ­
ducing
B ^ / J V W l a + l)

(4)

s lid in g c o n ta c t w hich is a tta c h e d to the s a m e s h a ft
a s the cam . The in s ta b ility >f the r e c o r d in g e q u ip ­
m e n t did not exceed 0.39,.

and I n te g r a tin g d s w ith r e s p e c t to X, w e ob tain

3. COM PUTATIONS O F THE SIZE O F TH E
E F F E C T AND RESULTS OF M EASUREM ENTS

The X in te g r a l in (5) d e s c r ib e s the re s o n a n c e a b ­
s o rp tio n o f y q u a n ta in th e s c a t t e r e r , a n d c o n se ­
q u e n tly ^

We in tro d u c e th e r e la tiv e in te n s ity of th e r e ­
c o rd e d X r a y s r e s u ltin g f r ; in re s o n a n c e a b s o rp tio n
of th e y r a y s (we n e glect the c o n trib u tio n of the
re s o n a n tly s c a tte r e d y q u a n ta ):

s =» / / .

01

w h e re I i s th e n u m b e r of re c o rd e d X r a y s ,
a p p e a r a s a r e s u l t OÍ re s o n a n c e a b s o rp tio n
m in e d fro m the d iffe re n c e in counting r a t e
sp e ed and a t high s p e e d ),
is the n u m b e r
qu a n ta in cid e n t on the s c a : t e r e r .
C o n s id e r an e le m e n ta l-

w hich
(d e te r­
a t z e ro
of y

la y e r d.\ o f th e s t u t ­

t e r e r a t a d is t a n c e X fro.T ’.:.e s u r f a c e f F ig . 3 ) .

T h r r e la tiv e in te n sity of t i e A ra y s producer! in it
a s a r e s u l t o f the resouan- tf a b s o rp tio n of y r a y s
is given by tr.e fo rm u la (tn e f ir s t in te g r a l g iv es
the x - r a y in te n s ity fo r z e ro speed, th e s e co n d for
v e ry high s p e e d s ) :
dS = [ \ e - *** V '
-

\

K

^

,y [L) dL

Up'V 'X. d t \ B i- » '‘

S - D ^ e "e1

v-n'iinty (E)dti\dx.

s = fi/e.x |>( — |ie X )|l — e.xp(

r N<Ĩ0XỊ2)

'’2)].

6

( )

w h e re f and p a r e , re s p e c tiv e ly , the p r o ln b ilitie s
of rric o ille s s e m is s io n a n d a b s o rp tio n of Y I|uanta.
F ro m (0) a n d (I) wc g e l
/{I — exp ( - - !‘.\la0X / 2 )J 0(if,NrJ(lXf2) ] = l/l0B exp ( - JI^Y)

O)
Tho q u a n titie s on th e r ig h t o f (7) c a n b e found fro m
the g e o m e try o f th e e x p e rim e n t and the r e s u l ts of
the m e a s u r e m e n ts . T h e valu es of B f o r the sc a tt e r e r t h i c k n e s s e s u s e d by IIS a r e : 1) X = 3 .5

m g /c m 2, B - 0.23; 2) X = 5.0 M g /c m 2, B = 0.22;

3) X =■ 8 .5 m g /c m 2, B = 0 .2 0 .
To f le te n n in e I„ we m u st co m pute Hie fra c tio n
oi y q u a n ta in the ra d ia tio n in cid e n t on th e s c a tt c r e r . F ro m th e d e c ay s c h c iu e of T c l2sm we find
th a t the r a t i o o f the in te n s ity of th e X r a y s ( 27.4
and 31.2 k e V ) to th e in te n s ity of th e y ra d ia tio n
is 18.5 U>K =■ 15.7.
T a k in g a c c o u n t o f th e d iffe re n c e in a b s o rp tio n
of th e s o u r c e m a te r ia l fo r th e v a rio u s ra d ia tio n s
(th e a b s o r p tio n c o e f f ic ie n t i s 8 c m 2/ g f o r X r a y s

y lij/D (F.)r.E -f

u*‘

— 1 1ỊI A ‘{F.) (Ic] H'dx,
1 (2 )

w h ere Jie is the c o e ffic ie n t ' Í toUil e le c tro n ic a b ­
s o rp tio n of y quanta in th* E'.-a ttr rc-r, ịin is tin* c o ­

m

arid 25.8 c m V g fo r y r a y s ) , we find f o r th e ra tio
the v alue 19.1. The a tte n u a tio n o f the r a d ia tio n by
a c o p p e r f il te r in the e x p e rim e n ta l g e o m e try was
m e a s u re d u s in g a f la t Nul ( T l) c ry s t a l p lac e d a t


FIG. 4. Dependence of recorded radiation intensity on
source velocity V (results for positive and n e g a t i v e velocities

combined).
th e p o s itio n of th e hollo w c y li n d e r c r y s t a l . To
m e a s u r e th e a tte n u a tio n of th e y r a v s w e u s e d th e
c e r i u m X r a y (3 4 .5 k e V ) w h ic h h a s a lm o s t th e
s a m e e n e r g y . I t w a s found th a t th e c o p p e r f i l t e r
t r a n s m i t s 3.2% of th e X r a y s a n d 12.9% o f th e y
q u a n ta . T h u s the r a t i o of th e X r a y a n d y i n te n ­
s i t i e s a f t e r p a s s i n g th ro u g h th e f i l t e r b e c a m c s 4 .8
± 0 .4 . Bv m e a s u r in g th e to ta l n u m b e r o f q u a n ta
in c id e n t on th e f la t c r y s t a l , w e 'f in d th e v a lu e o f I9.
F r o m m e a s u r e m e n ts of th e e f f e c t w ith th e
s o u r c e u n c o o le d a n d c o o le d to liq u id n i tr o g e n t e m ­
p e r a t u r e ( in b o th c a s e s th e s c a t t e r e r w as n o t
c o o le d ) it w a s found t h a t f = ( 1 .7 ± 0 .1 7 ) f .
C o m p u tin g Lhe v a lu e OÍ c = I/I * B e x p ( —n e X )
f o r th e s c a t t e r e r t h ic k n e s s e s X u s e d , a n d u s in g
th e g r a p h o f th e fu n c tio n in s q u a r e b r a c k e t s in (7),
o n e c a n e a s ily fin d th e a p p r o p r i a t e v a lu e s o f f \
T h e c o m p u ta tio n s g a v e :

X. mfja
c. %

3,5
1.3±0,2
\070±0.009

5.0
1 SS±0.25

0 r-03= n.

S.5
2.5±0.4

T h e a v e r a g e w a s f' = 0 .0 6 7 i 0 .0 0 8 ; f = 0.1 2
± 0 .0 3 .
T h e h a lf - w id lh o f th e c u r v e ( F ig . 4 ) , ta k e n f o r
X = 3.5 r u g /c m 2 a n d a s o u r c e t e m p e r a t u r e of
— liK F C , g iv e s f o r llic h a l f - l i f e C1Í th e 3 5 .5 -k e V
le v e l th e v a lu e ( 1 .4 -h 0 .2 ) X 10-9 s e c (ta k in g a c ­
c o u n t of th e f a c t th a t th e o b s e r v e d w id th o f th e lin e
is 2 .1 5 t i m e s th e n a tu r a l w id th , a s a r e s u l t o f s e lf a b s o r p t io n in th e s o u r c e a n d th e f in ite th ic k n e s s o f
tile a b s o r b e r ) , w h ic h a g r e e s w ith th e r e s u l t obtain*
by G r a h a m a n d Bell*-3-} u s in g d e la y e d c o in c id e n c e s :
(1 .5 8 ± 0 .1 5 ) y 10~9 s e c .
M e a s u r e m e n ts u s in g a s c a t t e r e r o f m e ta llic
te l l u r i u m (4 m g /c m 2), w ith th e s a m e e n r ic h m e n t
a s f o r th e T e O j, s h o w e d no s ig n i f ic a n t e ff e c t a t
ro o m t e m p e r a t u r e . T h u s in t h is e x a m p le we a g a in
s e e th a t th e p r e s e n c e o f a lig h t a to m in th e l a t t ic e
i n c r e a s e s th e p r o b a b ility f o r r e c o i l l e s s e m is s io n
o f y q u a n ta . A t h e o r e ti c a l b a s i s f o r th is e ffe c t
h a s b e e n g iv e n by K ag an .
T h e a u th o r s e x p r e s s t h e i r g r a t i tu d e to A. A.
S o ro k in f o r d is c u s s io n s , a n d to s tu d e n t L. A.
B y k o v s k a y a fo r h e lp in th e e x p e r i m e n t .
I.
B e r g s tr o m , in B e ta - a n d G a m m a - S p e c tr o s ­
co p y , e d . K. S ie g b ah n , 1959, p . 5 6 7 .

ZE . C o tto n , J . p h y s . r a d iu m 2 1 , 265 (1960).
3 R . G r a h a m a n d R . B e ll, C a n . J . P h y s . 31, 377
(1953).
4 Y u. M. K agan, J E T P 4 1, 65 9 (1961), S o v ie t
P h y s . J E T P 14, 4 72 (1962).

T r a n s l a te d by M. H a m e rm e s h

111


C A L C U L A T IO N O F P A R A M E T E R S OF E X P E R I M E N T A L S P E C T R A O F R E SO N A N C E

ABSORPTI ON o r G A M M A Q U A N T A I N C R Y S T A L S
2. A. BYKOV and PHAM 7.1 Y HIEN
N u c le ar P h y s ic s I n stitu te , M oscow S tate U n iv e rs ity
Subm itted to J Z T P e d ito r M a rc h 12, 1962
J . E x p tl. T h e o re i. P l u s . (U .S.S.R .) 43. 909-91S (S e p te m b er, 1962)
The p a ra m e te r s of o b s e rv e d s p e c tr a of re s o n a n c e a b s o rp tio n o f y q u an ta in c r y s t a ls a r e
cornDUted fo r sin g le and s p lit lin e s tak in g ac co u n t of s e lf -a b s o r p tio n In th e s o u rc e .
1. INTRODUCTION

r = Ế. ~ h

2. FORMULATION O F T H I PROBLEM
The shape of :he sp e c tru m in e x p e rim e n ts on
reso n a n c e abs orption of y cu&nta u sin g a m oving
s o u rc e is given by the quan;.:y
E

{*.}.

(1)

w here V is the re la tiv e v e io c itv of s o u rc e and ab ­
s o rb e r ; N( V) and N<«) a r e , r e s p e c tiv e ly , the
counting r a t e s of quanta p a fiir.g th ro u g h the r e s o ­
nant a o s o ro e r when tne ve lo c ity of the m otion is V
and when the velocity is su i:.c :e r.tlv la rg e so th at
th e re i s no reso n a n c e a b so r/u .o r..
If E is the e n e rg y of the V quar.tum , ơt E ) the
effective c r o s s se ctio n fo r re s o n a n c e a b so rp tio n ,
w e ! £ I the e n e rg y d istrib u tio n function fo r the
e m issio n s p e c tru m , n orrr.il].7ed to u n it a r e a ,
is the n u m b er of a to m s of t. e p a r tic u la r iso to p e
e n e rg y of the
p e r c m 2 of aijy o rb er, E'J i?
reso n a n c e lev e l, and r IS t e to*.;: wjrith o f the
I c v o i, th en USỈÍI" the 'iimcrif- -jn less v a r ia b le s

m

n

* — ĩĩf~ '

I n e x p e rim e n ts on reso n a n c e a b s o rp tio n of y
quanta in c r y s ta ls , one stue.-.es th e d ep en d en ce of
the a b s o rp tio r on the rciac: ve v e lo c ity o f s o u rc e
and a b s o rb e r . One m u st then som ehow tak e into

account the e ffect of c e rta i" ' f a c to rs ( s e lf- a b s o rp ­
tion in the s o u rc e , a b s o rb e r th ic k n e s s , e tc .) on the
m e a s u re d q u a n titie s. U sual.y in su ch s tu d ie s the
n e c e s s a r y c o rr e c tio n s a r c ro m p u te d on the b a sis
of som e assur'.D tions. O fte:.. noMkever, the lim its
of va lid ity of the a ssu m p tio n s a r c n o t m ade c le a r .
In a’ p a p e r of S hirley, K aplan, and A x e l,- J on the
b a s is of the r e s u lts of a n u m e ric a l co m p u tatio n
m ade by M a rp u lie s and Ehrm an,-*- so m e e m p iric a l
fo rm u la s a re d e rived for the p a ra m e te r s of the
o b se rv e d s p e c tr u m . We s h a ll d e riv e so m e g en e ra l
fo rm u la s fo r com puting Lhe n e c e s s a r y e x p e rim e n ­
tal p a ra m e te r s .

EoVfc

we c a n w rite fo r € ( y ) ( c f . ^ ) ,
r ( í ) = a / \ |] - e x p (—

)l V . (a: + y)dx.

(2)

H ere f is the p r o b a b ility f o r r e c o ille s s em issio n
of y q u an ta and a is th e f ra c tio n of re so n a n c e
quanta In th e e m is s io n s p e c tru m .
F ro m (2) it follow s th a t the a r e a u n d e r the ex ­
p e rim e n ta l a b s o rp tio n c u rv e
s .=

[ z (y) dy

is in d ep en d en t of th e sh a p e o f the e m is s io n lin e
and is d e te rm in e d onlv by the a b s o rp tio n sp e ctru m .
In p a r t ic u la r , Se is in d ep en d en t o f se Jf-a b so rp tio n
in the s o u rc e and p a r a s itic v ib ra tio n s of the equip­
m en t. In th is s e n s e , Sc is a co n v en ien t e x p e ri­
m en ta l p a r a m e te r .
F o r a s in g le a b s o rp tio n lin e , ơ ( x ) Is given by
the fo rm u la
n A 0 ịx) = o f n A/ ( i - x>),

(3)

w h e re Ỉ' is the p ro b a b ility fo r r e c o ille s s a b s o rp ­
tion o f Uie y q u a n tu m ;

v_£y

_

°n - 2/a + 1

r

( I3 and l|j a r e the s p in s o f the ground and e x cited
s ta te s of the n u cleu s: A is the w avelength of the y
q u an tu m , T y is th e r a d ia tio n width o f the lev el ).
We den o te the d im e n s io n le s s q u an tity ơ()f'TiẠ by
Ca .

ư th e a b so rp tio n lin e is s p lit into p co m p o n en ts,
ơ ( x ) h a s the fo rm
nAa {■*) — 2

1X ( F + X ? ’

w h e re Ai is the d isp la c e m e n t o f the i - t h com po­


n e n t r e la tiv e to s o m e fix e d e n e rg y le v e l, f o r e x ­
a m p le the e n e rg y o f th e u n s p lit lin e , in u n its of
r / 2 , CiA * f tC A . w h e r e ổi is th e r e la tiv e in te n ­
s it y of the i - t h c o m p o n e n t.
The fo llo w in g c o m p u ta tio n is b a s e d on th e r e l a ­

We n o te th a t
c h a r a c t e r i z e s th e d is tr ib u tio n
€( y ) j u s t a s w e ll a s d o e s th e u s u a l w idth a t h a lf
m a x im u m 2k ( C a )> w h e re K (C a ) i s d e fin e d by

tio n s (2), (3). a nd (4).

E x p r e s s io n s (7) and i l l ) g iv e th e r e la tio n

3 . SINGLE LIN E S IN EM ISSION AND AB SO R PTIO N

f. 1 x ^ ) 1 = e(0)/2.

(14)

x im (C„)= > iK (C ^ /[l- e ~ c ^ l , ( C Aữ )i.

(15)

We n o te t h a t fo r s m a ll C a i t fo llo w s fro m (6)
A.
S ource w ith o u t s e lf - a b s o r p ti o n . F o r s u c h a
th a t
s o u r c e , We I X ) h a s th e fo rm
£ to) 2ỉ 4 ^ f o r c * - 0 .
(16 )
w t (Jf) = 1 .1 u - **>
(5)
E q u atio n (16) p e r m i ts u s to a s s u m e th a t, w ith in
C o m b in in g (2), (3) a nd (5). w e a r r i v e a t th e fo l­
so m e in te r v a l o f v a r ia tio n o f C a > th e q u a n tity e ( y )
low ing e x p re s s io n f o r € ( > ') :
c a n b e a p p ro x im a te d by a d i s p e r s io n d is tr ib u tio n ,
w h ic h c a n be w r itte n a s
í y) = a /-^ - \
(C/(. .V' 7 — —- 7. ,
(6 )
e (y) = a/ K (CA) X (C„)/{lx (CM))* - r y ') }
(17)
w h e re we have u s e d th e n o ta tio n
if we u se Eq. (11).
y . (X. jr> — . I —
(6 ') In th is c a s e , K { C a ) i s r e la te d s im p ly to /Cịnt:
T he v alu e of of th e m a x im u m a b s o rp tio n € ( 0 )
c a lc u la te d by M ố s s b a u e r a nd W ie d e m a n n ^

I (0) = af II

w as

c , 2' .

\ I (y) dy - a ! " ; .< C A x\d.r .

(8 )

The in te g ra l a p p e a r in g in 'S i car. be co m p u te d e x ­
a c tly icf.
:
ị J 0 {Ca . •*> dx = -K;C a .

(9)

K (X) = \ e ~ Xr‘ \ . : >.2 - / .

(10)

w h e rp 1 .1 ) Is th e B e s s e l functio n o f o r d e r one
and in :a g ln a r y a rg u m e n t. F in a lly we have
5 , = a f iC Ar c *'2 ! /e <( A 2 -

•.
(11)

Since the function K (X ) is c h a r a c te r i s t ic fo r the

p r o c e s s e s w* a r e stu d y in g , we give th e b e h a v io r of
X I > ' in v a rio u s l im itin g c a s e s
ỵ y

- ; f o r >. — 0, K (?. V'h — 2 ị H fo r

' 12)

t(y)dy.

X (C ,) = ------ ---------------—

( 19>

In th e lim itin g c a s e s w e hav e
X (CA) - 2 fo r CA - 0; X (CA) -> 27/ T v * f o r

-

CO.

(2 0 )

T o stu d y th e r e g io n o f v a lid ity o f (19), w e c o m ­
p a re KI C a ) w ith th e r e s u l t s o f th e n u m e r ic a l
c o m p u ta tio n w h ich a r e show n g r a p h ic a lly in ^ 1-.
F o r 0 s C a - - 6, t h e r e is e x a c t a g re e m e n t; fo r
6 < C a - 10. th e r e a r e d e v ia tio n s w hich r e a c h 3%
a t C a * 10.
B. S o u rce w ith s e lf - a b s o r p ti o n . If th e ra d ia tin g

n u c le i a r e d is tr ib u te d u n ifo rm ly th ro u g h o u t the
th ic k n e s s of th e s o u r c e , th e e m is s io n s p e c tr u m h as
th e fo rm
w t (x) = J 0 (CS]X)/N (Cs).

(21)

H e re c s = ơ0fns ; ns is the n u m b e r o f n u c le i of
r e s o n a n tly a b s o rb in g iso to p e p e r c m 2 o f s o u rc e ;
th e n o r m a liz a tio n c o n s ta n t is
N (C.) - ị j , (C,; X) dx

— Of

The q u a n titie s ếí 0 ) and Se a llo w us to c h a r a c ­
t e r i z e the s h a p e of th e a b s o rp tio n s p e c tr u m . F o r
th is p u rp o se w e c o n s id e r the in te g r a l w idth o f th e
d is tr ib u tio n € i y ) :
■'•int (C/i) “ ZJT

(18)

(7)

w h e re Iq ( X I is th e B e s s e l fu n ctio n o f z e ro o r d e r
and im a g in a ry a rg u m e n t.
The o th e r p a r a m e t e r s of th e d is tr ib u tio n e ( y )
a r e c a lc u la te d in th e follo w in g w ay. F o r Se, we
find fro m (6)
Se=

*in» (CA) = ™ ( C A).
C o m b in in g (15) and (18), we g e t th e fo rm u la

(13)

A c c o rd in g to (9), w e h a v e N ( c s ) = ;rK( c s ).
T h e e x p e rim e n ta l s p e c tr u m h a s th e fo rm

e ^ = kỉtịc ) \ J ° (Ca] x) J°

x +

dx ■

(22)

s e is a g a in g iv en by (11). T he m a x im u m a b s o rp tio n
c( 0 ) is g iv en by th e eq u a tio n

ẵẾ


e lO) (23)
U sing the fo rm u la
\ d l , (XVdX — p j , i\) -

1 (*)

(cf. ^8-), we e a s ily get

£ (0) = a- l: — e-_ c -‘ - / 0 vCA2 )] f o r C , - 0 .
In w ork on re s o n a n c e a b s o rp tio n of y q u a n ta in
c r y s t a ls , one d e te r m in e s the p h y s ic a lly im p o r ta n t
p a r a m e te r f ’ fro m th e e x p e rim e n ta lly o b s e rv e d
dependence of 61 0> on c,\. F ro m F ig . 1 we c a n
e a s ily s e e th a t s e lf - a b s o r p tio n in th e s o u rc e m u s t
be tak e n into a c c o u n t. T h e quan tity e( 0 ) is th a t
p a r a m e te r of the o b s e rv e d d is tr ib u tio n w hich is
m o st s e n sitiv e to the shapx of the e m is s io n s p e c ­
tru m . F o r th is r e a s o n , to lie te r m in e V it is p r e ­
f e r a b le in m o st c a s e s to u s e the d e p e n d en c e o f the
in te g r a l a b s o rp tio n on C a - e s p e c ia lly a s the c u rv e
Se< C a ) is m o re c o n v e n ie n t to o b tain e x p e rim e n ­
ta lly (cf. Fig. 2 fo r Ò = OK
A ssu m in g th a t w ith in a p a r t i c u l a r in te r v a l o f
v a ria tio n of
and c<; the o b s e rv e d s p e c tru m
h a s the d is p e r s io n fo rm :
.KI.CẢ * ‘.CA.Ct)
(24)

' a ,T * t£ 7 c3 F + F '

we find fo r the halfw idth K C \ , C s ) th e fo rm u la
K Ca )K(C,)
(25)
X (CA, c .) fro m which we g et re la tio n IS) fo r c s —■ 0.
A c o m p a ris o n of (25) w ith the n u m e ric a l c o m ­
putatio n L1J show s no d e v ia tio n s ex c ee d in g 3%.
F rom th is one can d raw th t follow ing c o n c lu s io n s:

1. O ver a wide ra n g e of v a ria tio n of the p a r a -

FIG. 1. Dependence of maximum absorption e(0) on the
parameters CA and c s .
m e t e r s C a and c s , ( a t l e a s t fo r C a . C s £ 10),
th e e x p e rim e n ta l a b s o rp tio n s p e c tr u m h a s the
d is p e r s io n fo rm (24) w ith th e h a lfw ld th K( Ca » C s )
g iv en by (25). T h u s we s e e th a t a s we"go fro m C a ,
C s — 0 to f in ite v a lu e s o f Ca and C s, only the
w id th o f the o b s e rv e d s p e c tr u m c h a n g e s e s s e n tia lly ,
w h ile th e s h a p e o f th e d is tr ib u tio n €( y ) c h a n g es
n e g lig ib ly .
2.
To an a c c u r a c y s u ffic ie n t f o r th e f u rth e r
c o m p u ta tio n s , th e e q u a lity
r\ J/ . (CA;
t r . X)
.A Jr , t(C,
r .

> .A _ - K 1Ca )K tC,)x(CA, c,)
(26)

4. SP L IT T IN G O F EM ISSION AND ABSORPTION
LIN ES
As a r e s u l t o f in te r a c tio n o f th e e le c t r i c and
m ag n e tic m o m e n ts o f th e n u c le i w ith e x te rn a ] o r
c r y s ta llin e f ie ld s , in m a n y c a s e s th e e m is s io n and
a b s o rp tio n l in e s a r e s p li t in to s e v e r a l co m p o n en ts.

FIG. 2. Dependence of integral absorption on the
parameter Ca for the case of splitting of the absorp­
tion line into two identical components (CịA * C2A
= CA) (A is the difference in energy of the components
in units of r/2). Here s2e - A(y)dy. y « 2vEo/cr.

m


Tt th e o b s e r v e d s p e c t r u m h ; s th o fo rm o f in d iv id ­
u a l, c o m p le te ly s p l i t c o m p o n e n ts (w ith an e n e r g y
d i f f e r e n c e b e tw e e n c o m p o n e n ts w h ic h is » T ) ,
t h e n w e c a n a p p ly th e r e s u l t s o f th e p r e c e d in g s e c ­
t io n to e a c h C om ponent o f th e t r u e a b s o r p t io n (o r
e m i s s io n ) s p e c t r u m . T h e c a s e w h e r e th e s p li tt i n g
i s f‘^ n c o m ^ le c e ,, r e q u i r e s s p e c i a l t r e a t m e n t . We
s h a ll g iv e a g e n e ra l m e th o d f o r co m p u tin g ; th e r e ­
q u i r e d p a r a m e t e r s f o r th e c a s e o f d o u b le t s p littin g
o f o ne o f th e lin e s (th e a b s o r p t io n lin e ) .
S u p p o se th a t

?: ÍT

= c x [] : '•=) - t . ()

Vv - -V:*-.

J j , J 2 a r e th o s e p a r t s o f tile a b s o r p tio n w h ic h r e ­
s u l t fro m t.he “ i n d e p e n d e n t” a c tio n o f th e tw o
c o m p o n e n ts , J 3 is th e p a r t o f th e a b s o r p tio n

c a u s e d by ilie m u tu a l “ o v e r l a p p in g ” o f th e c o m ­
p o n e n ts . J j a n d J j a r e e a s i l y found u s in g (26). It
i s e a s 3' to c a lc u l a t e J* u s in g th e F o u r ie r in te g r a l
r e p r e s e n t a t i o n (of.
I).
W e d e n o te th e F o u r i e r t r a n s f o r m o f Jo (A; X)
by Jo ( *; U'). T h e f u n c tio n s Jo (A; x ) and
■J*(A; Ú.') a r c r e l a te d by th e f o r m u la s
Jo (/•; “) ~ -rq- 'ị J.) (}., (I>) emx d(ù ;

(27)

w h e r e A i s th e e n e r g y d i f f e r e n c e o f th e c o m p o ­
n e n ts in u n its '.vf r / 2 . F o r th e o b s e r v e d d i s t r i b u ­
tio n ểf» (y ) Ith e s u b s c r i p t 2 in d ic a te ? th a t th e lin e
is s p lit' wc have

h (Ã;
y )ii.\,

(2 8 )

T h e q u a n tity 5 c is g iv e n by tn e e x p r e s s io n
f
= of \

c ,
e.tp ; — r-rrr. — —

",
._

•! dx . (29)

w h ich i s e a s i h c a lc v ila te d u s in g 19 a n d ' 2 6 1. We
fin al]V ^ e t th e f o rm u la

s :e- xaf
-

j•

(30)
In the lim itin g c a s e s o f ‘ 's m a l l ” and ‘T a r g e ”
s p li tt i n g w e g e : the e x p e c te -i r e s u l t s :
S i t - x a f X {C,A . C\ A) — s
S ., = TO- : x ,c. ) - 5 ¥ Cm >

A

• 0.

5 .(C ..a ).

X c‘A“' J l (Cj^; oj') j'0 (ClA; U)' — (u)d(i) dm' .

j j‘ (CA\ <I>) j \ (Cs; Oj)

■ - J ■ - J ,) .

\

■ (33)

Jr (CY, w) - x K (CA) exp (— X {CA) !0););

c 5.

-

A ,y fc( c

|^J/Ĩ . ■*! Jo iCĩA-

Jt -

*

(34)

w h e re th e q u a n titie s K ( C a ) a n d K ( C s ) s a ti s f y
th e c o n d itio n
y. {<:,) - X (C ) -

V.

(C

t\).

(35)

IL i s n a tu r a l to a s s u m e t h a t in c la c u la tin g fu n c­
tio n s lik e J 3 i y ) e x p r e s s io n s o f th e ty p e o f (34)
g iv e a s u f f ic ie n tly good a p p r o x i m a t i o n .1^
T h u s w e s h a il a s s u m e th a t
nA‘ {'/:) e '

- c u ClA, CĨA .

(36)

C o m b in in g (32) a n d (3 4 ), w e fin d f o r th e d e s ir e d
q u a n tity an e x p r e s s io n w h ic h , a f t e r c o m p u ta tio n
o f th e i n te g r a l s a p p e a rin g : in i t, t a k e s th e f o rm

.r T ,1 ax ,

.

- n -K (C,,) K (Cs) PXỌ (— X (CAl Cj) jfflj)

S in c e (33) i s v a lid o v e r a w id e r a n g e o f v a lu e s of
th e p a r a m e t e r s , we ecet

J (Ã u)»
-

(3 2 )

y 9 (C»: (0 ) - .1 K ( Q exp (— X ( Q 1» ■),

t

a

To c o m p u te th e o t h e r p a r a m e t e r s , w e T ru st find
th e fu n c tio n * v y ) e x p li c it ly , v.'e w r i te f23) in the

;

Ị J\ (C,; „ ) f

\

F o r m u la (32) r e d u c e s th e p r o b le m o f fin d in g the
a p p r o x im a te e x p r e s s io n fo r J 3 to a c h o ic e o f a
s u it a b le a p p ro x im a tio n f o r th e F o u r i e r tr a n s f o r m
o f J 0 ( A; X ). F o r th is p u r p o s e w e t u r n to (2 6 ).
T a k in g th e F o u r ie r t r a n s f o r m s o f J 0 ( C a ; x ) i
J ữTight, o f (2 6 ), w e g e t th e e q u a tio n

2 a.b

- 2 ’-

a . „) - J L

K (Cọ/).

F o r the c a s e o; e q u a l c o m p o n e n ts ' CjA - C - \ ),
th e d e p e n d e n c e of S jc o n c ' i s sh o w n in F ig .

Ĩ y -

(31)

F o r J 3 we e a s i l y fin d
{C,c.,„

I

\’ J 0 t t; X) e~iut (ix .

dx.

V -ill \C., X f (/) dx
I2 8 #)

‘ As an example of the use of Fourier transforms in physi­
cal applications leading to relations of the convolution type
we mention the problem of the distorting effect of sp c ctro
rnetric apparatus (cf. r i ) .

EM

V . _

.V
l'

'

«A

|:t;X)+itut) - vttV)11f 11»1;I,i I-j.r-,t(V)| -.ựX)-x(|t;I -Ị- y^(.g}|x(X)-f-)t(v)| —2Ayx(^?*il>f

*

U* w

-

* (M r -

A»ỉ (J X 0 .) T X (V)J* r j r - ( > (V) +

-(y -A H

■

M i/nJ In] - |*x (Á1 ~ X íịi)I >. {X} - X (v)J ;•/. (v) f X
w h e re X, u and I' fro m D "V on den o te CjA. CjA
a:ui C s. re s p e c tiv e ly .
Tt i s n a tu ra l to a s su m e the follo w in g “ ad d itio n
la w ” fo r the p a ra m e te r s K ( X):

X rtl - X

- X (À. n».

(38)

w h ere *ci X. ./> is civet) b\ (25). We th en a r r i v e at
the re la tio n
X (Ã) - V Ị*

Ul - X

V' - X (jli. Vi’ .

(39)

final:V we have fo r J j
J> (C . C.J. t .; S. i ' — 4-

[\) K (u> A iv)

•• —i - — - — p ~ - —

“Hw:" : Av|m|W

lx l>.. u 1 • X (u. V - > > vìl. |r''

- .X (À. u)

X VJI. V — X O’. Ã-.:! • :

-■ |y. vx . \

4 - X (u . V '-

(•-. u ) |

(4 0 )

From thi.c we can e a sily ge t the g e n e ra : ex ­
p re s s io n for € ,( } ’ ). B eef.use of the co m p lex ity of
the e x p re s s io n s , we shail r e s t r i c t o u r tre a tm e n t
to the e a s e of c o m p o n e n ts of eq u al in t e n s it y
(> = u = C a • F o r th is ca
*

D - -i-iji-* (X, v ) [2 —
+
•
L
Xs (X, ?.) —;i- \ ‘ 2x (X, v; , J
_ A*' •] — . KJQ'* (X, Â.) 11
1
*f (X, X) f ' i ” Jj •

(42)

w h e re
E_ 4

/(tX)x(X, Jw) í 1
x(>., i-Ll
*»£.*» + A* I T ■ 4x(X,v)J-

T hen the r e s u l ts we w ant c a n be e x p re s s e d as
fo llo w s:
1.
If the in eq u a lity A2 < 2 [D — VD2 — 4 C ] is
s a tis fie d , the o b s e rv e d s p e c tr u m h a s th e fo rm of
a s in g le lin e w ith its m ax im u m a t y = A /2 ; the
in d iv id u al co m p o n e n ts do no t a p p e a r . T h e m ag n i­
tude o f th e m ax im u m a b s o rp tio n is d e te rm in e d bv
th e v alu e of € ,( C a , C a , C s ; &, y ) a t y = Ỏ/2.
T he d ep en d en ce of € 2( C a ; a / 2 ) on C a and A /2
is shown In Fig. 3. F o r l a r g e v a lu e s o f CA and
s m a ll A, the q u a n tity e A /2 i s a nonm onotonic
function o f A. T h is f e a tu re a p p e a r s w hen the ab­
s o rp tio n is l a r g e , and isc o n n e c te d with th e ex iste n ce
of a re g io n o f “ o v e rla p p in g ” o f the c o m p o n en ts.

r .i C i .C ,; V ...

.

!

v ;> V. :>■ V.

V j-J .
ạ i ụ - A j ( í — x U ,X )/2 x (X . V))

Ị>; -V * ;. V . J .V - A )- ’
'
'} •

(41)
On the b a s is of ■41) we var. co m pute the c h a r a c ­
t e r i s t i c s of I he o b se rv e d .-•.f-ctT-jm. F irs t wc d e­
te rm in e the lo cation of the e x tre m a of the a b s o rp ­
tion. The general co n d itier ’o r an e x tre m u m
cf-}/cy = 0 le a d s to an a l s 'b r a i c eq u atio n o f the
fifth d e g re e . The s e a r c h ' r a so lu tio n o f such an
equation is c o n sid e rab ly s im p lifie d b e c au se of the
s y m m e try p ro p e r tie s of * V y ).
We w rite the fo rm u la s fo r tbe q u a n titie s c and
D, w hich wi] be im p o rtsr.i f o r the following:
— X/

1

j /.•

Vj. ] y 1

:

—

FIG. 3. Dependence OÍ absorption at the midpoint of the
observed spectrum on the parameter Cạ when the absoiptioo
line is split into two equal components (CjA = C 2A ■ Ca;

c:s = 0 . 4 ).
The halfw td th o f th e a b so rp tio n line is found fro m
the eq u atio n
c t , c ,;

A, i / 2 + * » )= - 4 -

A, A/2)

th e function « £ ( C a ) is s hown in Wg. 4 .
2. When A2 = 2 (D - v'D2 - 4C J, the f ir s t th re e
d e riv a tiv e s o f * 2 ( y ) v anish a t y = A /2 , and the
a b s o rp tio n s p c c tru m has a “ p la te a u -s h a p e d ”
m ax im u m .
3. When A 2 > 2 [D - y/D2 - 4C Ị, the d is tr ib u ­
tion € 2 ( y ) h a s th re e e x tre m a ( so th a t one s e e s


W e n o te th at, f o r A = 0 th e e x p r e s s i o n f o r th e
m a g n itu d e o f th e m a x im u m a b s o r p t io n d o e s n o t go
o v e r p r e c i s e l y in to th e c o r r e s p o n d in g f o r m u la f o r
s in g le l i n e s ; i t i s e a s y to s e e th a t th e r e a s o n f o r
th is d i s c r e p a n c y i s th e f a c t t h a t th e a p p r o x im a tio n
u s e d h e r e f o r th e F o u r i e r t r a n s f o r m o f J 0 (X , x )
d o e s n o t r e p r o d u c e th e “ h ig h f r e q u e n c y ” p a r t o f
th e s p e c t r u m o f th e f u n c tio n e x a c t l y —t h e h ig h f r e ­
q u e n c ie s a r e o v e r e s t i m a t e d . F r o m th e p r a c t ic a l
p o in t o f v iew th e e r r o r m a d e i s in s i g n if i c a n t , s in c e
th e d i f f e r e n c e b e tw e e n f o r m u la s (23) a n d (45) f o r
2 C a - 1 0 ; C s ^ 10 d o e s n o t e x c e e d 3%.

In c o n c lu s io n w e m e n tio n th e f o llo w in g . If one
FIG. 4. Dependence of halfw:dth
at half maximum oế
the absorption line on the parameter Ca when the absorption
line is split into :wo identical components (CjA “ C ;a = Ca ;
c s = 0.4). The doshed line corresponds to the region of val­
ues of CA where the observed spectrum IS split.
th e s p li tt i n g in th e o r ig i n a l s p e c t r u m >: t h e r e i s a

c o m p u te s f r o m th e e x p e r i m e n t a l d a ta th e q u a n ti­
t i e s f \ A, r a n d a f , t h e n i t i s u s e f u l a s a c h e c k
to m e a s u r e th e v a lu e o f th e a b s o r p t io n a t a d i s ­
ta n c e A / 2 f r o m th e m id p o in t o f th e s p e c t r u m . T h is
q u a n tity s h o u ld be e q u a l to
£c = e ,(C „ , c /t, c , ; A, 0)

m in im u m a t y : A / 2 a n d tw o s y m m e t r i c a l l y
p la c e d m a x im a a t th e p o in ts
= X V - V; (A3 -

2 \D -

7UC, [ K I C J + K M - K I C * ■
Ị 'D- -

4C ; ) '

(43)

KlCf + Ct)K(CA)*iCA + c,, CA)

X5[CA T Cj, CA) -f A-

T h e s e p a r a ti o n o f th e m a x i m a A c b s i s g iv e n by
th e f o r m u la

(4 .

/A)* =- 1 — 2A-* \D -

VD* -

4CI.

(44)

K- (CA) -X (CA, CA) y'(CA. CA) + A- .

We ta k e t h is o p p o r tu n ity to e x p r e s s o u r g r a t i ­
tu d e to P r o f . V . S. S hpinel* f o r h i s c o n tin u e d in ­
t e r e s t and s u p p o rt of o u r w o rk .

We s e e th a t A obs i s a lw a y s l e s s Than X s in c e
c > 0. A s w e s e e f r o m F ig . 5, the d i f f e r e n c e b e ­
tw e e n th e o b s e r v e d a n d th e o r ig i n a l s p li tt i n g is
v e r y l a r g e in t h e c a s e s o f p r a c t i c a l i n t e r e s t . A s
an e x a m p le w e p o in t o u t t h a t e v e n f o r an i n f in ite ly
th in s o u r c e a n d a b s o r b e r th e o b s e r v e d a n d a c tu a l
s p li tt i n g s b e c o m e e q u a l o n ly f o r A > 6.
T h e m a g n itu d e s o f th e m a x im u m and m in im u m
i b s o r p t l o n s a r e g iv e n by th e r e l a t i o n s

tynat — M i 'i ) = e-i (t/a);

t-zrrur. = e2 ( A/2).

(45)

1 S h irle y , K a p la n a n d A x e l, P h y s . R e v . 1 2 3 , 816
(1 9 6 1 ).
2 S. M a r g u l ie s a n d I. E h r m a n , N u c l. I n s t . and.
M e th o d s , 12, 131 (1 9 6 1 ).
3 D e ly a g in , S h p in e l’ a n d B ry u k h a n o v , J E T P 4 1 ,
1347 (1 9 6 1 ), S o v ie t P h y s . J E T P 1 4 , 95 9 (1 9 6 2 ).
4 R. L . M õ s s b a u e r a n d w . H. W ie d e m a n n , z .
P h y s ik 15 9 , 33 (1960).
5 M ax B o r n , O p tik , B e r li n , 19 3 3 .
6 K . M . R y z h ik a n d I . s . G r a d s h te m , T a b lits y
in te g r a l o v , s u m m , r y a d o v i p r o iz v e d e n ii ( T a b le s
o f i n t e g r a l s , s u m s , s e r i e s a n d p r o d u c ts ) 3 r d
e d itio n , G o s te k h iz d a t, 1 9 5 6 .

FIG. 5. Dependence of the ratio
Aobs/A on the parameter Ca when the
absorption line is split into two equal
components
= C2 A * Cạ ,
Cs = 0.4). Aobs is the separation of
the absorption maxima in the observed
spectrum, in units of 172.

7

E . T i tc h m a r s h , I n tr o d u c tio n to th e T h e o r y o f

F o u r i e r I n te g r a l s , O x fo rd , C la re n d o n P r e s s , 19 4 8 .
8 S. G . R a u tia n , U F N 6 6 , 475 (1 9 5 8 ), S o v ie t
P h y s . U s p e k h i 1 , 245 (1 9 5 8 ).

T r a n s l a t e d by M . H a m e rm e s h
159

m


A M SO T HOP) O f THE M O SSBAU ER E F F E C T IN 0 - S n SING LE C R Y S T A L S
N. E. A I.£K > - f . \ ổ l \ l ĩ , P I AM Z!.'v HIKN. V. G. SH A PĨR O , a n d V. a. S H P IN E L '
M u rlea'- P h y s io s I n .'tT jr e , M oscow S tare U n iv e rs ity
S u b m itted 1.0 J t T P e d it o r A p ril 4. iy 6 2
J . E x p tl. T h e o r e i. P h v s . f r .S .S .R .) 43, 7 9 0 -7 9 4 ( S e p te m b e r. 1962)
T he p ro b a b ility f' o 1 r e s o n a n c e a b s o rp tio n ơl 2 3 .8 -k e V y q u a n ta w as m e a s u r e d fo r s in g le
c r y s ta l p l a i t s j f w n .te m e t a ll ic tin cu t a lo n n v a rio u s c ry s t.a ]lo g ra p (iic p la n e s . A la r g e a n is o ­
tro p y A IS ọ b s i\rv e u Ah oh chd MOI i:hange ill g o in g fro m a t e m p e r a t u r e o f
to 77’K. 'J'he
r a tio s of th e f v a lu i1* f o r th e v a rio u s p la te s w e r e f*or. : fím : fóoi : ftjoly = 1 : 0 95 : 0 .7 1 : 0 30
rhe o b s e rv e d d iffe rt'P .c e s in lo c a tio n o f Che a b s o r p tio n m a x im a and th e a s y m m e tr y o f th e
a b s o r p r o n lin e s a r e e x p la in e d by th e q u a d ru p o le s p li tt i n g o f th e e x c ite d s la t e o f th e S n 119
n u c le u s in -he Í - S n 'r y s t a l
1. INTRODUCTION
F r o m q u a lita tiv e arg u m eiii.s o n e m ay e x p e c t th a t
the p ro b ab ility f o r r e c o i li e s s e m is s io n ( o r a b s o rp
tio n ) of V q u a n ta should d e p e n d on th e o rie n ta tio n

of the c r y s ta ilo c r a p h .c axe;- r e l a ti v e to th e d i r e c ­
tion of m o tio n o: Ứ1C V quaT .3. K a g a n o b t a i n e d
e x p re s s io n s Ũ r the a n is o tr- 'p v o f th e M o s sb a u e r
e ffe c t fo r s im p le la ttic e s O' tn e rh o m b ic and
te tr a g o n a l s y s te m s and g a \ f 1 ro u g h e s tim a te of
the a n iso tro p > fo r a w hite i n s in g le c r y s t a l at
T --- 0°K.
The p u rp o se of th e p r e s - n : w o rk w as to in v e s ­
tig a te the exp< c tc d e ffe c t
a S in g le c r y s t a l o f
w hite Lin.
2. D E SC R IPT IO N O F TH E E X P E R IM E N T
T he re s o n a n c e a b s o rp tio n m e a s u r e m e n ts w e re
m ad e a t room t e m p e r a tu r e i:id at liq u id n itro g e n
t e m p e r a tu r e w ith a b s o r b e r ' ;n the fo rm o f p la te s
cu t fro m a s in g le c r y s t a l of n a tu r a l w h ite tin
(/3-Sn) alo n g who (00 1 ), (101). and '100) p la n e s .
A poly c ry s ta l] m e tin foil WdS u se d in c o n tro l
m e a s u r e m e n ts . T he r e l a ti v e th ic k n e s s e s of the
a b s o r b e r s fo r the m e a s u r e m e n ts a t ro o m t e m p e r ­
a tu r e , a s d e to -m in e d by th> . r n o n rc s o n a n t y - ra y
a b s o rp tio n , Wf-re ec u al to . : 1 .0 1 : 0 95 : 0 .9 4 , r e ­
s p e c tiv e ly , w h.le the th ic k r.e s s c f :h e f ir s t fo il, a s
d e te rm in e d by w eighing, w;-,3 57 5 * 0 .8 m g /c m 2.
Ln the m e a s u r e m e n ts at liq u id n itr o g e n t e m p e r a ­
t u r e , the r e l a ti v e th ic k n e s s e s of th e p la te s c u t
a lo n g the (100' and (001) p k n e s and th e p o ly c r y s ta llin e foil w e re r e s p e c tiv e ly 1 : 1.15 :1 .0 8 , and
the th ic k n e s s '.f the f ir s t o: th e s e fo ils w as 20.0
Ị 0.2 m g /c m 2.

T h e s o u r c e o f th e 2 3 .8 -k e V y q u a n ta , which
a r e e m itte d by S n ll3m n u c le i, w a s p r e p a r e d from
lit) d io x id e 5 n 0 2 e n ric h e d to 88% S n 118 and i r r a d i ­
a te d in a r e a c t o r . The s o u r c e th ic k n e s s w as ~ 5
m g / c m 2. T h e s o u r c e w as k e p t a t ro o m t e m p e r a ­
t u r e in a ll th e m e a s u r e m e n ts .
T h e b e a m o f y q u an ta w a s c o llim a te d by a lead
d ia p h ra g m w ith a 6 m m o p e n in g . T he a b s o r b e r s
w e r e p la c c d b e h in d th e d ia p h ra g m in a n a c c u r a te ly
fix ed p o s itio n . R a d ia tio n p a s s i n g th ro u g h th e a b ­
s o r b e r w as d e te c te d by a s c in tilla tio n s p e c t r o m ­
e t e r u s in g a f la t N a l ( T l) c r y s t a l.
T h e m e a s u r e m e n ts w e r e m a d e w ith th e a p p a ­
r a t u s d e s c r i b e d p r e v io u s ly , l2j w h ich g iv e s th e
a b s o r b e r a m o tio n a t c o n s ta n t s p e e d .

3 . R E S U L T S O F M E A SU R E M E N TS AND DIS­
CUSSION
S om e o f th e a b s o rp tio n s p e c t r a ta k e n fo r the
v a r io u s p la te s u n d e r id e n tic a l g e o m e tric a l c o n d i­
tio n s a i a t e m p e r a i u r e o f 77°K a r e sh ew n in the
f ig u r e . As we s e e , th e a b s o rp tio n lin e a p p e a r s at
p o s itiv e v e lo c itie s . The c o u n tin g r a t e a t n e g a tiv e
v e lo c i ti e s c o m e s fro m n o n r e s o n a n t a b s o rp tio n .
T he lo c a tio n o f th e e x p e rim e n ta l p o in ts in th is
r e g io n in d ic a te s th e s ta b ility o f th e re c o r d in g
e q u ip m e n t.
T able I g iv e s th e q u a n titie s c h a r a c te r i z i n g the
e x p e r im e n ta l c u r v e s : vr e s , th e p o s itio n of th e
a b s o rp tio n m a x im u m , and em ax> th e r a tio o f th e

d i f f e r e n c e in c o u n tin g r a t e in th e a b s c n c e o f r e s o ­
n a n c e a b s o r p tio n a n d w ith m ax im u m re s o n a n c e
a b s o r p tio n to th e co u n tin g r a t e w ith o u t r e s o n a n c e
a b s o rp tio n .
The o b s e rv e d h a lfw id th s o f th e lin e s e x c e e d th e


c a lle d t ( A / l i )!. w h ic h d e p e n d o n A a n d on th e e f ­
f e c tiv e t h ic k n e s s C A o f th e a b s o r b e r . In t h is p a p e r
i t w a s sh o w n th a t th e d e te r m i n a ti o n o f th e p ro b n b i li ty f ' by m e a s u r in g th e a r e a o f th e r e s o n a n c e
a b s o r p t io n c u r v c is c o n v e n ie n t b e c a u s e th en s e x p
is in d e p e n d e n t o f th e p e c u l i a r i t i e s in th e e m is s io n
s p e c t r u m w h ich m a y r e s u l t fro m s e lf - a b s o r p ti o n ,
t’r o in s p littin g o f th e e m i s s io n lin e a n d f ro m v i h r a tio n .

Absorption S|«fCtra taken at 7~°K, tor p.ates of various
orientations: a) lOOl). b) (100), c ) poly crystal. The thin
vertical lines sh^'* the positions of the absorption maxima.
T a b le 1
O rie n ta tio n
o( p l j i e

K

1

I

s ." k‘

v ^ m a is tc *

P s ljr c r y lta L
;CCV
(1Ơ1
ClOO)

•n sx

: 5 3 5 - 0 .0 7 I
: 4 0 - 0 .0 7 ị

{

I
I 2 . 5 2 x 0 . 0 7 Iị 1 5 . 5 + o . h

: 5C - 0 . 1 0 !
I
Ị

\

S .4 * 0 .5
5 .5 - 0 .5
M + 0 .5
9 ^ = 0 .5

The s o u r c e c h a r a c t e r i s t i c s w h ich m u s t b e known
in o r d e r to m a k e th e c o m p u ta tio n s a r e Off (w h e r e

f i s th e p r o b a b i l it y f o r r e c o i l l e s s e m is s io n o f y
q u a n ta , a n d a is a p a r a m e t e r d e te r m in in g th e
r e l a t i v e c o n tr ib u tio n o f t h e s e y q u a n ta to th e to ta l
c o u n tin g r a t e ) a n d th e e ff e c tiv e t h ic k n e s s c s of
th e s o u r c e ( c s = n s fr?0, w h e r e n s is th e n u m b e r
o f r e s o n a n tly a b s o r b i n g n u c le i p e r c m 2 o f th e
s o u r c e ) . By m e a s u r in g th e r e s o n a n c e a b s o rp tio n
w ith SnOj a b s o r b e r s o f d if f e r e n t t h ic k n e s s e s
( f r o m 4 .5 m g / c m 2 to 60 m g /c m 2 ) i t w a s found
th a t orf - 0 .2 8 -fc 0 .0 3 , c s
0 .4 .
U sing th e f a m ily o f c u r v e s f o r SeXp a n d em ax
( c f . F ig s . 2 a n d
o f th e p a p e r o f P h a m Zuy H ien
an d B y k o v 3- ) , w e c a n d e te r m i n e A a n d f ' fo r
p o l y c r y s ta l li n e w h ite tin (T a b le n ) . T h e v a lu e s
fo u n d fo r A a g r e e w ith in th e l i m i t s o f e r r o r w ith
th e d a ta o f D e ly a g in e t
if w e ta k e r = 0.31
m m / s e c f o r th e lin e w id th o f th e s o u r c e .
T a b le II

• T » lo c a u o o O' th e a b s o r p t io n » a u m a w a s d « te ra u n e d m ore

»c-* » :T y br add'.'iona! measurements.

^W ithin th e lim i'a o f e x p e r u n c n ta i e r r Of. n o t ĩ u h o f th e »b■o rp tie n M T jir a w a s o b s e r v e d in t h r s p * c tr a ' u n w ith t . n g i e

v a lu e s e x p e c te d on th e b a s i s o f tn e e ff e c tiv e t h ic k ­
n e s s e s of th e a b s o r b e r C a ( c \ = Q ^ f'ơ 9, w h e re

is th e n u m b e r o f r e s o n a n t ly a b s o rb in g n u c le i
p e r c m 2 of a b s o r b e r ) a n d s o j r e e c 5 F o r e x am p le , a t 77®K th e h a lfw id th o f ‘.h e lin e fo r th e
p o ly c r y s ta llin t: fo il is 1 .5 m m 's e c , w h e r e a s it
sh o u ld only be 0 .9 m m / s e c . T he v ib ra tio n le v e l
o c c u r r in g in th e r e l a t i v e m o tio n OĨ s o u r c e a n d a b ­
s o r b e r w as m e a s u r e d , a n d s h o u ld c a u s e a b r o a d ­
e n in g of the lin e by no m o r e th a n 3%. T h e o b ­
s e r v e d b r o a d e n in g m a y b e c a u s e d by q u a d ru p o le
in te r a c t io n . T h is m u s t b e c o n s id e r e d w h en d e t e r ­
m in in g ’-he p r o b a b i l it ie s of r e c o i l l e s s a b s o rp tio n
o f 7 q - a n ta fo r th e d i f f e r e n t ly o r ie n te d p la te s .
In tr e a ti n g Ih e d a ta it is c o n v e n ie n t to u s e th e
m e th o d d e v e lo p e d in a p a p e r by B ykov a n d P h am
Z u y H ien. 3- T o d e te r m i n e th e v a lu e s o f th e
q u a d ru p o le s p li tt i n g A a n d th e p r o b a b ility f , we
u s e th e a n a ly tic f o r m u la s g iv e n b y th em f o r SeXp
( th e a r e a u n d e r th e a b s o r p t io n c u r v e ) and
r»>t; r e l a ti v e rciaxim um a b s o r p tio n ; in
th is is

Tempera- '
Iure, • K 1 A. mm/sec77
293

1 0.5 ±0.1
1 0.4G±0.1

0.32-0.06
0 061 ±0.015

We now p r o c e e d to d i s c u s s th e r e s u l t s f o r th e
s in g le c r y s t a l s . Ir f i r s t a p p ro x im a tio n w e m ay
a s s u m e th a t th e e l e c t r i c f ie ld g r a d i e n t in th e 0 -S n
c r y s t a l h a s a x ia l s y m m e tr y . T h e £ -S n c r y s t a l
b e lo n g s to th e te tr a g o n a l s y s te m , s o t h e r e is e v e ry
r e a s o n to e x p e c t th a t th e s y m m e tr y a x is o f th e field
g r a d ie n t c o in c id e s w ith th e c a x is o f th e c r y s t a l,
i .e ., is p e r p e n d i c u la r to th e (001) p la n e .
A s a r e s u l t o f q u a d ru p o le in te r a c t io n , th e e x ­
c ite d le v e l a t 2 3 .8 keV in S n 119 i s s p li t in to two
s u b le v e ls w ith s p in p r o je c ti o n s *ty2 and ± l/2 on
th e s y m m e tr y a x is . A k i n e m a tic a l c o m p u ta tio n
sh o w s th a t th e p r o b a b i l it ie s o f r e s o n a n c e a b s o r p ­
tio n o f y q u a n ta in th e s e s u b le v e ls a r e d if f e r e n t
in g e n e r a l, a n d d e p e n d o n th e a n g le 0 b e tw e e n th e
s y m m e tr y a x is o f th e c r y s t a l and th e d i r e c ti o n o f
th e in c id e n t y q u a n tu m :
2>+ tt — ( I J - COS2 0 ) .

------ COS4 o ) .

(1 )


w h e re w . f z a r e th e p r o b a b ilitie s o f re s o n a n c e a b s o ro u o n of > q u a n ta in the r e s p e c tiv e s u b le v e ls .
Sincc the d ire c tio n of the in cid e n t y quantum
IS p e rp e n d ic u la r to Ihe plan e of th e a b s o r b e r , fo r
p la te s cut fro m th e s in g le c r y s ta l a lo n g d iffe re n t
p la n e s we got th e follow ing v a lu e s fo r th e r e la tiv e
in te n s itie s a - W± J 2 /w „ ị m of the co m p o n e n ts in

the ab so rp tii-o s p e c tru m :
Orientation of plates

“‘V ;

(1011)

(2)

T hus the s h a p e of th e a b s o rp tio n lin e sh o u ld de pend on th e o rie n ta tio n of the p la te —th e p o s itio n of
the a b s o rp tio n m ax im u m s h ifts to w a rd th e m o re
in te n s e c o m ponent and the lin e b e c o m e s a s y m m e t­
r ic a l. T h e se p e c u lia r itie s should sh o w up m o re
c le a r ly the g r e a t e r A a n c the s m a l le r th e e ff e c ­
tiv e w idth o f the a b s o r b e r . \ t roo m te m p e r a tu r e ,
w ithin th e e x p e rim e n ta l e r r o r s , no a s y m m e tr y of
the lin e s no r s h ift of the re s o n a n c e a b s o rp tio n
m ax im a w e re o b s e rv e d . At 77°K, a s w e s e e fro m
the fig u re , the a b s o rp tio n lin e s fo r th e (001) and
(100) p la te s a c tu a lly a r e a s y m m e tr ic , th e e ffe c t
b e in g la r g e r in the f i r s t c a se . The o b s e rv e d
a s y m m e tr y in d ic a te s [ u sin g the v a lu e s (2)] th a t
the s u b le v e l w ith sp in p ro je c tio n ± Vj h a s h ig h er
e n e rg y than the one with sp in p r o je c tio n ±*/2- It
then follow s th a t the p ro d u ct Qqz z • (w h e re Q is
the o b s e rv e d q u a d ru p o le m o m en t o f th e n u c le u s
and q z z is the c o m ponent of the fie ld g ra d ie n t
a long the s y m m e tr y a x is ) a :list be n eg ativ e.
F o r the le v e ls c o n s id e re c h e re , th e q u a d ru p o le
s p littin g is A = V2eQ qz z . T his q u a n tity h a s a

v a lu e w h ich i s c h a r a c t e r IS li e fo r th e u n it c e l l o f

th e p a r t i c u l a r c r y s t a l. We can th e r e fo r e u s e the
v a lu e of A found w ith the p o ly c ry s ta l fo r the c o m ­
p u tatio n of f' in the p la te s w ith d iffe re n t o r ie n t a ­
tio n s .
U sing the v a lu e s (2) and app ly in g the g e n e ra l
fo rm u la (30) of c3- , we d e te rm in e f ' (T ab le i n ) .
F ro m fo rm u la (30) It follow s that f o r a g iv en v alue
of th e e ữ e c tiv e th ic k n e s s of the a b s o r b e r , the a r e a
u n d e r th e a b s o rp tio n c u rv e depend s w eak ly on the

T a b le HI
Orientation
of plate
(001)
(101)
(100)

'77 "K

W k

0.24+0. OS iV054-1-0.91
0.072-«-0 .1)1
0.36-0.00 O.OTS±O.Ol

re la tiv e in te n s ity o f th e c o m p o n e n ts a n d on the
v alue of A. T h u s p o s s ib le e r r o r s in d e te rm in in g
A and s lig h t d e v ia tio n s o f th e fie ld g r a d ie n t fro m

a x ia l s y m m e tr y h a v e l it tl e e ffe c t o n th e r e s u l ts .
We n o te th a t th e r a t i o s o f th e v a lu e s o f V fo r
a b s o r b e r s o f d i f f e r e n t o r ie n ta tio n s a r e d e te rm in e d
m o re a c c u r a te ly th a n Ih e q u a n titie s th e m s e lv e s ,
s in c e th ey a r e p r a c t ic a l ly in d ep e n d e n t o f th e m e a s ­
u r e m e n ts of o t h e r q u a n titie s (A , Off). T h u s we
have
F’ot T * 77° K: ỈỊQQ /(JQ, / p0|y — i .0.67 0.89
For T - 293° K: tm ■Ú ■fMi f poly * 1:0.95 0.71 0.80
W ithin th e l im it s o f e r r o r , th e r a t i o s do not change
in g oing fro m liq u id n itr o g e n to ro o m te m p e ra tu re .
F o r a d i r e c t c o m p a r is o n o f th e e x p e rim e n ta l
and th e o r e ti c a l r e s u l t s o n e m u s t m a k e a d e ta ile d
c a lc u la tio n o f th e e ff e c t, ta k in g in to a c co u n t the
a c tu a l s t r u c t u r e o f th e tin l a ttic e a n d u s in g a w ide
r a n g e o f te m p e r a t u r e s .
‘ Yu. M. K a g a n , DAN 140, 794 (1961), Soviet
P h y s. D oklady 6 , 881 (1962).
2
P h am Zuy H ie n , S h a p iro , and S h p in e l’, J E T P
42, 703 (1962), S o v ie t P h y s . J E T P 15, 489 (1962).
3G. a . Bykov a n d P h am Zuy H ien, J E T P 43,
909 (1962), t h is is s u e , p . 646.
*
B ry u k h a n o v , D e ly a g in , O p alenko, a n d Sh p in el’,
J E T P 43, 432 (1962), S o v ie t P h y s. J E T P 16, 310
(1963).

T r a n s la te d by M. H a m e rm e s h
134

D E P E N D E N C E OF TH E y-Q U A N T U M R E S O N A N C E A B S O R P T IO N S P E C T R U M ON
C R YST A L TEM PERATU RE
P H A M ZL'Y H1EN and V. s . S H P IN E L ’
N u c le a r P h y s i c s i n s t it u t e , M o sco w S ta te U n i v e r s ity
S u b m in c đ to J E T P e d it o r J u n e 8 , 1962; r e s u b m it t e d O c to b e r 1, 1962
J . Expel. T h e o r e i. P h v s

(V S .s.H i 44. 3 9 3 -3 9 7 ( F e b r u a r y . 1963)

T h e s p e c t r a o f r e s o n a n c e a b s o r p tio n o f 2 3 .8 -k e V V r a y s in S n 0 2 a r e in v e s tig a te d in th e t e m ­
p e r a t u r e r a n g e f r o m 7$ to 6 I5 °K . E x tr a p o la tio n o f th e o b s e r v e d lin e w id th s to z e r o a b s o r b e r
t h ic k n e s s e s l e a d s to v a lu e s w h ic h a r e g r e a t e r th a n th e n a tu r a l w id th a n d w h ic h g ro w w ith i n ­
c r e a s in g t e m p e r a t u r e . T h e e x p e r im e n ta l d a ta sh o w th a t th e b r o a d e n in g i s no t th e r e s u l t o f a
d o u b le t s t r u c t u r e a n d Thai t h e t r u e b ro a d e n e d l i r e h a s an a p p r o x im a te L o r e n lz s h a p e . T h e
p r o b a b i l it ie s f o r a b s c rp -.io n w ith o u t r e c o i l i f ' ) a r e m e a s u r e d in th e i n v e s tig a te d t e m p e r a t u r e
r a n g e . T rie v a lu e s o i f v a ry w e a k ly w ith th e t e m p e r a t u r e . A l in e a r d e p e n d e n c e of th e t e m p e r a : u r e lin e s h if t d u e to a s e c o n d - o r d e r D o p p le r e ff e c t w a s o b s e rv e d a t T £ 300°K. D e v ia ­
tio n s f r o m l i n e a r i t y o c c u r a ; T £ 300°K.
INTRODUCTION*

A s i s w e ll k now n, the p r o b a b i l it y o f r e c o i l l e s s
a b s o rp tio n ( e m i s s i o n ) o f g a m m a q u a n ta in c r y s t a l s
( f ’ ), c a lc u la te s : in th e D e b y e a p p r o x u n a tio n , d e ­
c r e a s e s s h a r p ly w ith i n c r e a s i r .g te i r .p e r a i u r e . It
h a s b e e n a lr e a d y n o te d e a r l i e r th a t :c m a n y c a s e s
o f c o m p lic a te d l a t t i c e s t h e v a lu e of
t u r n s o u t to
d e p e n d lit tl e O’ th e t e i n o e r a i u r e . Ir r e s o n a n c e a b ­
s o r p tio n s p e c t a s in g le l i n e s b r o a d e r th a n th e n a tu ­

r a l widi.il a r e i r e q u e n tl y o b s e r v e d . :~.e w id th b e in g
fo u nd CO e x h ib it a t e m p e r a t u r f d e p e n d e n c e la th e
c a s e of w h ite t n
A c h a n g e in t e m p e r a t u r e
a ls o in flu e n c e s th e p o s it io n of th e r e s o n a n c e lin e
and le a d s to s o - c a l l e d t e m p f - m t j r p s h if ts , d u e to
th e s e c o n d - o r d e r D o p p le r e ff e c t. It i s hoped th a t
a d e ta ile d i n v e s tig a tio n OỈ th e i o r e g o in g t e m p e r a ­
t u r e e ff e c t c a n y ie ld v a lu a b le n fo rr r .a tio n or. th e
d y n a m ic s of th e c r y s t a l l a t i i c r .
In t h e p r e s e n t i n v e s iig a t-o n w e s tu d ie d th e r e s o ­
n a n c e a b s o rp tio n s p e c t r a of 2 ^ .8 -k e V g a m m a r a y s
f"-orr. S n : i ỉ * w ith an a b s o r b e r .n th e f o rm of p o ly c r v s t a l l i n e tin ox id e p o w d e r I S n O j) a t d iff e re n t
te m p c ra iu ro s

6 4 5 'K . T h e o b s e rv e d s p e c t r u m had t h e f o rm o f a
s in g le u n s p lit lin e at a ll t e m p e r a t u r e s . T h e n u m ­
b e r o f p u l s e s c o u n te d a t e a c h o b s e r v a tio n p o in t
r e a c h e d - 10s.
F ig u r e 1 s h o w s th e o b s e r v e d v a lu e s of t h e lin e
h a lf - w id th f o r a b s o r b e r s o f d i f f e r e n t t h ic k n e s s at
th e f o u r t e m p e r a t u r e s i n d ic a te d on th e p lo t. T o
d e te r m i n e th e h a lf - w id th r 0k s w e d r e w th e p e d ­
e s ta l o f th e lin e in s u c h a w ay th a t th e p o in t o f th e
s p e c t r u m lo c a te d a t 7 2 1’o b s a w a y f r o m th e c e n te r
of th e lir.c w a s a t 10 p e r c c n t o f th e lin e h e ig h t. It
i s s e e n f ro m F ig . 1 th a t , w ith in th e l im it s o f th e
e x p e r i m e n t a l e r r o r s , th e e x p e r i m e n t a l p o in ts p e r ­
ta in in g to a s p e c if ie d t e m p e r a t u r e l i e on a s tr a ig h t
lin e . T h e h a lf - w id th s o f th e lin e s e x tr a p o la t e d to

z e r o a b s o r b e r t h ic k n e s s r’e x t r a r e l is te d in c o l ­
u m n 2 of th e ta b l e and e x c c c d th e n a tu r a l lin e w id th ,
th e b r o a d e n in g in c r e a s i n g w ith in c r e a s i n g t e m p e r ­
a tu r e . ( W ith in th e l i m i t s o f th e m e a s u r e m e n t e r ­
r o r s , th e v a lu e s o f l ’ e x t r a r c a lm o s t th e s a m e f o r

m ea su r em en ts and r esu l ts

/ p r e v io u s ly d e s c r i b e d 1- - ' s e tu p w a s u s e d to
p i o t th e a b s o rp tio n s p c c t r a fo r r e s o n a n t f il te r s
o i 'Jíí 'c r e r . ‘. tn i'- k n e s s . Ia lilt e x p e r i m e n t s , th e
s c *ce in th e o rm o f tin o x id e 3 m g -'e m 2 th ic k ,
■A’a:- at -ne t e m p e r a t u r e o f liq u id n itr o g e n , w h ile
t'lf- a b s o r b e r t e m p e r a t u r e r a n g e d b e tw e e n 78 anc

FIG. I. Half-width ol the observed spectrum vs. absorber
thickness fur different temperatures. Dashed line—calculated
for room temperature funder the assumption of Art. 1).

______________________ trw