ISSN 0021�3640, JETP Letters, 2013, Vol. 98, No. 3, pp. 174–180. © Pleiades Publishing, Ltd., 2013.Original Russian Text © S.V. Karyagin, 2013, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2013, Vol. 98, No. 3, pp. 197–203.

174

Gamma resonance on long�lived isomers (lifetimeτ > 10–4 s) is suppressed by a factor of k = Γtτ, where Γt

is the total linewidth. Usually, Γt � Γdd ~ 104 s–1, whereΓdd is the dipole–dipole broadening. It is suggested [1]that the narrowing of lines to about the natural widthΓ = 1/τ is possible only by artificial (e.g., radio�fre�quency) methods. However, the experiments reportedin [2–5] on the yield of 88.034�keV photons from the109mAg isomer in silver plates can be explained only ifthe total linewidth is close to 1/τ at τ ≅ 57 s, i.e., nar�rower than the dipole–dipole broadening by a factorof ~105. Since no measures for the narrowing of theline were used in [2–5], i.e., strong narrowing (by afactor of ~105) occurred naturally, this effect can becalled NSN. The authors of [4, 5] attributed NSN tothe averaging of hyperfine interactions in the transi�tion time ~τ of the nucleus from 109mAg to 109Ag. At thesame time, a sufficiently efficient process of naturalmodulation of the hyperfine interaction has not yetbeen found (see [4, 5] and references therein). In thiswork, a mechanism of the suppression of inhomoge�neities of the hyperfine interaction that is much moreefficient than the lattice vibrations is proposed. Thismechanism is due to fluctuations of the Fermi contactfield Hc. Let estimate the value Hc = |Hc | for the Agatom with the [Kr]4d105s1 shell, where [Kr] is the kryp�ton�like core [6]. An unpaired 5s electron creates thefollowing field on the nucleus (at the point r = 0) [7]:

(1)

where μ0 = 9.3 × 10–21 G cm3 is the Bohr magneton(G cm3 = erg/G); the polarization vector of the 5s�

electron S = (iσx + jσy + kσz)ψel/2 is the quantumaverage over the spin state specified by the spinor ψel;i, j, and k are the unit vectors along the X, Y, and Zaxes, respectively; σx, σy, and σz are the Pauli matrices;

and is the density of the 5s�electron on thenucleus (independent of the time). In the hydrogen�like atom (see Eq. (16.24) in [8]) with the effective

charge number of the nucleus, the density isroughly estimated by the formula

(2)

where a0 = 0.53 × 10–8 cm is the Bohr radius, n is theprincipal quantum number, Z is the charge number ofthe nucleus, and ζ is the screening factor of thenucleus by inner electrons. For silver, Z = 47 and n = 5.

For ζ ~ 0.02–0.2, ~ 1022–1025 cm–3 and the

reasonable interval Hc(0) ~ 103–106 G are obtained inqualitative agreement with the hyperfine interaction inMössbauer spectra for various nuclei [1, 9, 10].

The 4d105s1 electrons in a metal are in bands. In thiscase, the 5s�“orbit” as a part of an unclosed system isnonstationary and its electron density differs from that

Hc 0( ) 16π/3( )μ0 ψ5s 0( )2S,=

ψel+

ψ5s 0( )2

ψ5s 0( )2

ψns 0( )2 1/π( ) Zζ/ na0( )[ ]

3,∼

ψ5s 0( )2

Explanation for the Effect of Natural Strong Narrowing of Mössbauer Lines on Long�Lived Isomers. Variety of Nuclei and Media with This Effect

S. V. KaryaginGol’danskii Division of the Structure of Matter, Semenov Institute of Chemical Physics, Russian Academy of Sciences,

ul. Kosygina 4, Moscow, 119991 Russiae�mail: akaryagina@gmail.com

Received May 13, 2013; in final form, July 1, 2013

In memory of Yu.A. Izyumov

The effect of natural strong narrowing (NSN) of Mössbauer lines on long�lived isomers has been explained.This effect was doubted for more than thirty years because its mechanism was unknown. The mechanism ofNSN has been revealed. Its threshold condition has been established. The theory of inhomogeneous broad�ening of a line in the case of NSN has been developed. Broadenings owing to magnetic, quadrupole, and2L�pole hyperfine interactions are suppressed almost to zero at NSN. The mechanism of NSN provides awidth about the natural width. The linewidth at NSN is independent of the magnitude and direction of themagnetic field if it is lower than ~100 G. The variety of nuclei and media with NSN or with the collapse ofthe hyperfine interaction (responsible for NSN) is outlined.

DOI: 10.1134/S0021364013160066

JETP LETTERS Vol. 98 No. 3 2013

EXPLANATION FOR THE EFFECT OF NATURAL STRONG NARROWING 175

for the stable 5s�orbit of a free atom. Band electronscolliding with the 5s�“orbit” repopulate it, leading tothe stochastic variation of the field Hc(0, t) in the timet with the frequency of fluctuations νF. The Sommer�feld model [6] for metallic silver gives

(3)

where ρsvavσs is the frequency of collisions of electronswith the core, ρs is the electron number density in the5s�band, vav ~ (3/5)1/2

vF is the average velocity ofthese electrons (see Eq. (2.31) in [6]), vF is the Fermi

velocity, σs ~ π is the cross section, rs is the radius ofthe core, and κ is the efficiency of collisions. Sincesome collisions occur without the repopulation of the5s “orbit,” κ < 1. In nonmetals, κ � 1 because therepopulation of the 5s�“orbit” is suppressed because ofits strong coupling with the lattice. In metals, κ ~ 1.For example, for Cu, Ag, and Au, κ ≅ 1. For Ag, ρs =5.9 × 1022 cm–3, vav ~ 108 cm/s, vF = 1.4 × 108 cm/s, σs ~8 × 10–16 cm2, rs ~ 1.6 × 10–8 cm, and νF ~ 5 × 1015 s–1.A close estimate νF ~ κvav/2rs ~ 3 × 1015 s–1 was givenin [6, Chapter 31, p. 281, note 5]. If the 4d–5sexchange is taken into account when a 4d�electronknocks out a 5s�electron and occupies its place,becoming the 5s�electron, Eq. (3) is changed to thenew estimate

(3a)

where ρd ~ 10ρs is the total electron number density inthe 4d�band. The electron crosses the core in time τs ~2rs/κvav ~ 3 × 10–16/κ s. In the same time interval, itpenetrates into the 5s “orbit,” perturbing it and,hence, changing the magnitude and direction of Hc.Since, up to τsνF ~ 15 collisions with the 5s “orbit” canoccur in this time, Ne = 0, 1, or 2 electrons can simul�taneously get this “orbit.” Since the distribution ofeach of Ne electrons in the 5s “orbit” and between the“orbits” of their “atom” and neighboring “atoms”stochastically depends on the time,

. (4)

Here, C(0, t) = Σaca(0, t) is the contact spin density

vector where ca(0, t) = Sa(0, t). The sub�script a = 1, 2 enumerates the electrons of the 5s�“orbit.” The vector Hc(0, t) jumps both in direction(because the polarizations of the electrons getting the5s�“orbit” are random) and in magnitude (because theincident electrons perturbing the 5s�“orbit” randomlychange the spin density distribution in it and, thereby,its value at the point r = 0).

The unpaired electron of the 5s�“orbit” acts notonly with the field Hc on its nucleus but also with thefields Hs ~ μ0/(2rs)

3 ~ 300 G on neighboring nuclei.Since the fields Hs and Hc fluctuate with the frequen�

νF ρsvavσsκ 5 1015κ s

1–,×∼∼

rs2

νF ρd ρs+( )vavσsκ 5 1016 s

1–,×∼ ∼

Hc 0 t,( ) 16π/3( )μ0C 0 t,( )=

ψ5s a, 0 t,( )2

cies of the same order of magnitude and |Hs | � |Hc | onaverage, the field Hs can be included in Hc. Further�more, fluctuations of the number of electrons in the4d�“orbits” create the fields Hd ~ 300 G with the fre�quencies ~νF on their and neighboring nuclei. Thesefluctuating fields Hd, as well as Hs, can be added to thestochastic field Hc.

Without NSN in high�purity silver [5], themoments of the 107Ag, 109Ag, 109mAg, and 109Cd nucleicreate the dipole fields Hd* on neighbors that areslower than Hc, Hd, and Hs. The magnetic moment

operator has the form [7] = μμN /I, where =

i + j + k ; , , and are the spin projectionoperators on the X, Y, and Z axes, respectively; I is themaximum spin projection; μ is a number; μN = 5.05 ×

10–24 G cm3 is the nuclear magneton; and is the

quantum average of . The maximum at the

distance r = 2rs = 3.2 × 10–8 cm has the form

(5)

where μ = = ψ+ ψ is the quantum average of the operator in the spin state of the nucleus with the

spinor ψ and max = . The highest fields

are created by 109mAg (~0.6 G) and 109Cd(~0.1 G). However, their contributions to the line�width are negligible because of very small concentra�tions (10–11 and 10–5%, respectively). For this reason,the fields Hd* are 99.99999% created by the 107Ag and109Ag nuclei, for which ~ 0.1, < ~0.02 G, and the dipole–dipole contribution to thebroadening is δωdd* ~ Λ , where Λ = /� =

4.75 × 103 s–1 G–1. For 109Ag and 109mAg, δωdd* ~ 10and 400 s–1, respectively. The relative dipole–dipolebroadening kdd* = τδωdd* is 600 for 109Ag and 24000 for109mAg. The external field in [5] was |Hex | ~ 1 G. Sinceits spread is |δHex |max � 1 G, the broadening of the lineowing to δHex can be δω∗ex line ~ 2 × 104 s–1 andk∗ex line ~ 106. (The subscript * marks the estimates dis�regarding NSN.)

Except for the fields |Hc | ~ 105 G and |Hl | = |Hex +Hd* | ~ 1 G, other (ferro�, antiferro�, para�, etc.) mag�netic fields in the experiments reported in [5] wereabsent because silver is diamagnetic at |Hex | ~ 1 G andPauli paramagnetism is manifested in silver only at|Hex | � 104 G.

The Zeeman hyperfine interaction for only one ofthe nuclei of the source is proportional to the scalar

product , where HR = Hc + Hl. Since the angular

μ I I

Ix Iy Iz Ix Iy Iy

I⟨ ⟩

I Hd* max

Hd* max µr( )r/r5µ/r3–=

= μ μN/r3μ 0.153 G,×=

µ⟨ ⟩ µ

µ

µ μ μN

Hd* max

μ Hd* Hd* *

Hd* max* μ μN

μ

IHR

176

JETP LETTERS Vol. 98 No. 3 2013

KARYAGIN

distribution of fluctuations of the field Hc at τav ⇒ ∞ isspherically symmetric and Hl(t) ≅ const, averaging

over τav ⇒ ∞ gives Hcav∞ = (t)dt/T = 0. How�

ever, Hlav∞ ≠ 0 and HRav∞ = Hcav∞ + Hlav∞ = Hlav∞ ≠ 0.At the same time, = 0 because the chaotic

motion of the spin in the fluctuating field Hc at τav ⇒ ∞and |Hc | � |Hl | gives a spherically symmetric distribu�

tion of quantum averages at different times. In this

case, NSN appears because = 0 on allnuclei. At finite τav values, the dynamics of the spin ofthe nuclei under the action of Hc(t) should be takeninto account.

The spin state of the nucleus is described by the

equation i�∂ψ/∂t = –[( )μμNN/I]ψ, where ψ isthe (2I + 1)�component spinor and I is the spin of thenucleus [7]. For simplicity, let the direction of the fieldHR change from one fluctuation to another fluctuationby jumps (random in direction and magnitude), but itsmagnitude and direction remain unchanged in eachindividual fluctuation. The problem appears to besomewhat more difficult if relative changes in the fieldHR during each fluctuation are small, i.e., |HR –HRav |/|HRav | � 1 and the field HR deviates from theunit vector k = HRav/|HRav | at an angle smaller than

π/2. Here, HRav = dt/T is the average value of HR

in the interval (0, T) of the current fluctuation with alength of T ~ 1/νF. Let the Z axis be directed along k.Then, each component ψν of the spinor ψ correspondsto its individual solution [7]

(6)

where Iν are the eigenvalues of the operator , which

correspond to the components ψν: I1 = I, I2 = I – 1, …,I2I + 1 = –I; HR = ; c

ν = b

νexp(–iϕ

ν); and the num�

bers bν are real and normalized: = = 1.

(The superscript of the spinor component ψν is writtenin the quantities c

ν, b

ν, ϕ

ν, and I

ν as a subscript, rather

than as a superscript in order to avoid confusion withthe exponent.) Since µ(t) is an everywhere continuousfunction of t, the vectors µ at the end of the precedingfluctuation coincide with those at the beginning of thecurrent fluctuation:

(7)

Hc0

τ

∫τ ∞⇒

lim

I⟨ ⟩ av∞

I

I⟨ ⟩

IHR⟨ ⟩ av∞

IHR

HR0

T

∫

ψν cν i μμNIν/I�( ) HR td

0

t

∫–exp=

= bν i ϕν μμNIν/I�( ) HR td

0

t

∫+ ,–exp

Iz

HR

Σνbν

2Σν cν

2

µat the end of the last fluctuation

= µat the start of the current fluctuation.

Since µ = μμN/I, where = ψ+ ψ is the quan�

tum average of and the spin vector at the end ofthe preceding fluctuation coincides with that at thebeginning of the current fluctuation. For estimates, ψν =cνexp(–iω

νt), where ω

ν = (μμNI

ν/I�)HRf and HRf =

dt/T is the average magnitude of the field during

the current fluctuation T ~ 1/νF. In this case, HRf =Hcf + Hl, where Hl is independent of t and Hcf =

dt/T. It is noteworthy that |HRav | = ≅

HRf, but |HRav | > HRf. The unit vector u = µ/ =

/ has the projections ux = , uy =

, and uz = on the X, Y, and Z axes,

where = ψ+ ψ, = ψ+ ψ, = ψ+ ψ,

and = ( + + )1/2. Therefore,

+ + = 1. The unit sphere u2 = 1 is calledu�sphere and the end of the vector u on it is calledu�point. The uniform covering of the entire u�sphereby the u�points gives µav = 0. As a result, simple esti�mates of the linewidth appear. The terms “uniform”and “spherically symmetric” are used here disregardingfluctuation�induced symmetry breakings. These break�ings are taken into account in the estimates of broaden�ings by Eq. (14). During the current fluctuation (0, T),the vector u moves about the Z axis over the arc of a cir�cle described by the equation

(8)

where Ω = μμNHRf/I� is the Larmor frequency. Equa�tion (8) is represented in the system of axes � = (X, Y,Z) associated with the average field HRav (see textabove Eq. (6)) and with the initial vector u parallel tothe magnetic moment of the nucleus µ at the end ofthe preceding fluctuation. The Z axis is parallel to thefield HRav and makes the angle θ with the vector u. TheX, Y, and Z axes are mutually orthogonal. The positivedirections of the X, Y, and Z axes correspond to theunit vectors i = [HRavxu]/|[HRavxu]|, j = [kxi], and k =HRav/|[HRav |, respectively. In this case, |[HRavxu | =|sinθ|, ku = cosθ, iu = 0, and ju = sinθ. Each fluctua�tion corresponds to an individual system of axes � =(X, Y, Z) and an individual arc. In this sense, the termsfluctuation, arc, and system of axes � are equivalent.According to Eq. (7), the end of each arc is the begin�ning of the next arc. As a result, a continuous u�chainof random arcs appears on the u�sphere. Since theangle θ is random, its average in the time τav is θav =π/2. Let HRε, Hcε, and Ω

ε be the rms values for HRf, Hcf,

and Ω over all fluctuations on a given nucleus in thetime τav. Let the vector u rotate from the old positionto the new position about the Z axis parallel to the new

I⟨ ⟩ I⟨ ⟩ I

I I⟨ ⟩

HR0

T

∫

Hc0

T

∫ HR t/Td0

T

∫µ

I⟨ ⟩ I⟨ ⟩ Ix⟨ ⟩/ I⟨ ⟩

Iy⟨ ⟩/ I⟨ ⟩ Iz⟨ ⟩/ I⟨ ⟩

Ix⟨ ⟩ Ix Iy⟨ ⟩ Iy Iz⟨ ⟩ Iz

I⟨ ⟩ Ix⟨ ⟩2

Iy⟨ ⟩2

Iz⟨ ⟩2

ux2 uy

2 uz2

ux θ Ωt( ), uycossin θ Ωt( ),sinsin= =

uz θ,cos=

JETP LETTERS Vol. 98 No. 3 2013

EXPLANATION FOR THE EFFECT OF NATURAL STRONG NARROWING 177

field HR at θ = θav = π/2, i.e., ux = cos(Ωεt), uy =

sin(Ωεt), uz = 0. In this case, during the fluctuation

interval 1/νF, the vector u makes a step to a new u�point circumscribing the arc δϕ

ε ~ Ω

ε/νF. The U�point

rotates over the arc δϕ about the axis parallel to thefield HRf = Hcf + Hl. At Hcε � |Hl |, the fields HRf1 andHRf2 in the fluctuations f1 and f2 hardly correlate witheach other because

(9)

in view of < 1 G, Hcε ~ 5 × 104 G (see table), and

( /Hcε)2 ~ 10–9. Thus, the directions of the field HRf

in different fluctuations hardly correlate with eachother. Consequently, the u�point makes uncorrelatedrandom walks over the u�sphere. Such a walking of theu�point can approximately be described by the 2D dif�

fusion coefficient Du = aw/2 = /2νF, where a =δϕ

ε= Ω

ε/νF is the rms angular step and w = Ω

ε is the

rms angular velocity in the step. The maximum dis�placement of the u�point in the time τav � 1/νF over ageodesic line constitutes the angle Θu = (2Dumτav)

1/2.In order to avoid a complicated description of multi�sheet walks of the u�point over the u�sphere, thesewalks are projected on the plane. Then, the entire u�chain of the length Lu ~ τavΩε

> 109 rad forms a coil

disk with the radius Θum with the area Sm = π =

π2τavDu = πτav /νF = πτav(HcεΛ/I)2/νF. As a result,

the u�sphere having an area of 4π rad2 is � times cov�ered by the u�coil, where � = Sm/4π =τav(HcεΛ/I)2/4νF. It can be thought that the u�sphere isuniformly filled with u�points disregarding fluctua�tions if the u�coil covers the u�sphere with no less thanone layer, i.e., � > 1:

(10)

Under condition (10), the filling of the u�sphere by u�points is spherically symmetric disregarding fluctua�tions whose contribution was taken into account in the

HRf1HRf2( )/ HRf1 HRf2( )[ ]av Hl /Hcε( )2 � 1,∼

Hl

Hl

Ωε

2

Θum2

Ωε

2

� Sm/4π τavΩε

2/4νF τav HcεΛ/I( )

2/4νF= = =

= τav μμNHcε/I�( )2/4νF 1.>

estimate of broadening by Eq. (14). The disregard offluctuations is equivalent to the case τav ⇒ ∞ underthreshold condition (10).

Condition (10) separates the spherically symmetricdistribution of u�points over the entire sphere from anasymmetric distribution when u�points occupy only apart of the u�sphere. If condition (10) is satisfied at alarge number of u�points τavνF, inequality (10) is infi�nitely enhanced in the limit τavνF ⇒ ∞ and the infi�nite�layer distribution of u�points on the u�spherebecomes exactly spherically symmetric. Then, theaverage magnitude of the vector µ over u�points isexactly zero,

(11)

Here, the subscript 0 corresponds to the infinite num�ber of u�points. When condition (10) is not satisfied,the spherical symmetry disappears and large regionswithout u�points appear. Thus, criterion (10) guaran�tees the symmetric filling of the entire u�sphere withu�points at which µav0 = 0.

The averaging of the quantum average of the mag�

netic hyperfine interaction � = –HR /I over τav

gives

where

(12)

Here, is factorized because averaging over τav isperformed for the same nucleus at the absolute value

remaining constant on the entire u�sphere, i.e., = const. Since the field Hl is almost constant,

(Hlu)av = |Hl |(cosθ')av, where θ' is the angle between Hl

and u. For 109Ag (see table), the u�sphere is uniformlyfilled because � > 1 when u and –u are equiprobableand, therefore, the expected value is

(13)

μav0 μ⟨ ⟩ av0 0.= =

IμμN

�⟨ ⟩ av HR IμμN/I⟨ ⟩( )av– HR µ⟨ ⟩( )av–= =

= Hc Hl+( ) µ u( )av �c⟨ ⟩ av– �l⟨ ⟩ av,+

�c⟨ ⟩ av µ Hcu( )av; �l⟨ ⟩ av– µ �lu( )av.–= =

µ

µ

µ

�l⟨ ⟩ av0 µ Hl θ 'cos( )av0– 0,= =

Inhomogeneous linewidth at NSN for the parameters νF = 5 × 1016 Hz, = 4.8 × 104 G, τav = τ = 60 s, and kmon line ~ 10

I μΛ,

s–1/G ω1ε, s–1 Ω

ε, s–1 δϕ

ε,

radDu,

rad2/sLu/2π Θum/π �

109Ag 1/2 0.13 6.18 × 102 2.96 × 107 0.59 × 108 1.18 × 10–9 0.035 5.6 × 108 0.65 1.04109mAg 7/2 ~4.0 1.90 × 104 9.1 × 108 2.6 × 108 5.21 × 10–9 0.66 2.4 × 109 2.85 20.2

|⟨ ⟩/I |crΓc, s

–1 kc Γ', s–1 k' kdd* kdd1 kdd2 kqu

109Ag 1 0.0099 0.59 1.5 × 10–7 0.9 × 10–5 6 × 102 2 × 10–7 6 × 10–17 0109mAg ≅6/7 0.26 15.6 3.8 × 10–6 2.3 × 10–4 2.4 × 104 7.2 × 10–6 2.4 × 10–15 �1

Hcε

I

178

JETP LETTERS Vol. 98 No. 3 2013

KARYAGIN

where the subscript av0 means the average over theinfinite number of probes. Thus, the expected contri�bution to the hyperfine interaction from �ll vanishes.However, deviations of (cosθ')av from (cosθ')av0 = 0 arepossible for various nuclei. The standard deviation of(cosθ')av from (cosθ')av0 can be estimated using aknown trick [11] of the theory of errors. Let A be theaveraged quantity; Aav0 be the expected average for A,i.e., average for A at the infinite number of probes; Aav

be the average value for a certain sample; Nav be thenumber of values (probes) of the quantity A in thesample; and [A – Aav0]ε be the expected rms value forA – Aav0. Then, on the set of all samples, the standarddeviation of averages Aav from the expected averageAav0 is

(14)

where (Nav – 1)1/2 ≅ is taken at large Nav values.Since (cosθ')av0 = 0, Eq. (14) gives

(15)

where [cosθ']ε = =

(1/3)1/2. Thus, = 0; i.e., the contribution fromHl to the NSN disappears and the inhomogeneousbroadening from this contribution is

(16)

because = μμN/I. The subscript cr stands foraveraging over the nuclei of the crystal. In particular,

= 1 for I = 1/2 and ≅ 6/7 for I = 7/2.The estimates are valid under threshold conditions (10).

In terms of the h�vector h = Hc/|Hc |, Eq. (12) is

represented in the form = – . Theend of the h�vector (the h�point) moves on the h�sphere = 1 by the jumps δϕc ~ π/2. The number ofh�points that can be located in one layer at an averagedistance of ~π/2 from each other on the h�sphere isNc ~ 4π/(π/2)2 = 16/π ~ 5. In the time τav = τ ≅ 60 s,the sphere is covered by τavνF/Nc ~ 1018–1019 such lay�ers, creating a uniformly dense covering of the sphereby h�points. Then, the expected average is =

– = 0 because the vectors h and –h areequiprobable for the same vector u. This resultobtained from the symmetry reasons corresponds tothe averaging of averages ≠ 0 over the infinite

ε Aav Aav0–( ) A Aav0–[ ]ε/ Nav 1–( )

1/2=

≅ A Aav0–[ ]ε/Nav

1/2,

Nav1/2

ε θ 'cos( )av θ 'cos( )av0–[ ] θ'cos[ ]ε/ τavνF( )1/2

≅

= 1/ 3τavνF( )1/2

,

ϕ θ ' θ ' θ '/4πdsincos2

0

π

∫d0

2π

∫( )1/2

�l⟨ ⟩ av0

Γ ' µ /�( ) Hl[ ]ε

θ 'cos[ ]ε/ τavνF( )1/2=

= I⟨ ⟩/I crHlε/ 3τavνF( )1/2

, k ' Γ'τ,=

µ I⟨ ⟩

I⟨ ⟩/I cr I⟨ ⟩/I cr

�c⟨ ⟩ av µ Hc hu( )av

h

�c⟨ ⟩ av0

µ Hc hu( )av0

�c⟨ ⟩ av

number of samples (the number of points in the sam�ple is Nav = τavμF). Each nucleus of the sample is char�

acterized by the individual sample and average over τav. Usually, try reasons corresponds to the aver�

aging of averages ≠ . According to

Eq. (14), the standard deviation for is obtainedin the form

(17)

Suggesting that |Hc | and cosθcu are uncorrelated witheach other, we have

(18)

where θcu is the angle between Hc and u and [cosθcu]ε =

[cosθ']ε = (1/3)1/2. Thus, /� = 0; i.e., the

contribution from Hc in the averaged hyperfine inter�action disappears and the inhomogeneous broadeningfrom this contribution in view of Eq. (14) is given bythe expression

(19)

where the rms value [cosθcu]ε = (1/3)1/2 is obtained

under the assumption that all h values are equiproba�ble at a given u value, ω1ε = HRεΛ ≅ HcεΛ ≅ [Hc]εΛ, andkc is the relative broadening (in units of the naturalwidth 1/τ).

At νF = 5 × 1016 Hz (see Eq. (3a)), threshold con�dition (10) is satisfied for 109Ag in the limit if =

4.8 × 104 G. Then, kc ~ 0.6 and 15.6 for 109Ag and109mAg, respectively (see table), which gives the broad�ening of the line kc line ~ 15.6. At NSN, the broadeningk' in Eqs. (16) from magnetic inhomogeneities issmaller than kc by a factor of k/kc = Hlε/Hcε ~ 10–4 andthe dipole–dipole broadening Γdd ~ �ddavkdd* is sup�pressed by a factor of τavνF ~ 1019, because it is propor�tional to the product of two nuclear dipole momentseach suppressed by a factor of (τavνF)1/2 at the NSN.For comparison with the experiment, it is importantthat the hyperfine interaction in the field HR = Hc + Hl

at � Hl is averaged over τav to zero: =

+ = 0 (see Eq. (13) and text aboveEq. (17)). This means that the components of thehyperfine interaction of the γ�transition collapse intoa singlet. As a result, the absorption of a resonance γ�ray photon is independent of the angle between theexternal magnetic field Hex and the wave vector of theγ�ray photon. It is worth noting that the dependence ofthe resonance cross section on this angle predicted

�c⟨ ⟩ av

�c⟨ ⟩ av �c⟨ ⟩ av0

�c⟨ ⟩ av

ε �c⟨ ⟩ av �c⟨ ⟩ av0–( ) µ Hcu[ ]ε/ τavνF( )

1/2.≅

Hcu[ ]ε

Hc θcucos[ ]ε

=

≅ Hc[ ]ε

θcucos[ ]ε

Hcε

1/3( )1/2

,≅

µ Hcu( )av0

Γc ε Hcu( )av[ ] μ cr/�=

= ω1ε I⟨ ⟩ cr/I / 3τavνF( )1/2

; kc Γcτ,=

Hcε

Hcε

�⟨ ⟩ av0

�c⟨ ⟩ av0 �l⟨ ⟩ av0

JETP LETTERS Vol. 98 No. 3 2013

EXPLANATION FOR THE EFFECT OF NATURAL STRONG NARROWING 179

in [12] was at first glance confirmed in the experimentsreported in [5].

Fluctuations of Hc result in the collapse of the hyper�fine interaction of any multipolarity. The quadrupolehyperfine interaction averaged over electrons has theform [7, 14] �qu = ΣiΣk[IiIk + IkIi – 2I(I +1)δik/3]Bik�, where the subscripts i, k = 1, 2, 3 specifythe x, y, and z axes, respectively; δik is the Kroneckerdelta; Ii and Ik are the spin projection operators on theaxes i and k, respectively; and Bik are the componentsof the tensor (see Eq. (120.6) in [7]). In the prin�cipal axes X*, Y*, and Z* of the tensor with the trace

Bx*x* + By*y* + Bz*z* = 0, �qu = 2�[( – )Bx*x* +

( – )By*y*] and the quantum average is =

ψ+�quψ = 2�[( – )Bx*x* + ( –

)By*y*], where the spinor ψ is specified in the cur�rent fluctuation axes X, Y, and Z, whereas the matrices

, , and are defined in the principal axes X*,

Y*, and Z* of the tensor . The quantities =

ψ+ ψ, = ψ+ ψ, and = ψ+ ψ varyowing to the fluctuations of the direction of the fieldHc. If the operator U transforms ψ from the X, Y, andZ axes to the X*, Y*, and Z* axes, the averages at the

spinor are = ψ+U+ Uψ, = ψ+U+ Uψ,

and = ψ+U+ Uψ.

The fluctuations of Hc and systems � = (X, Y, Z)adequate to them are enumerated by the subscript r =1, 2, …, R: �1 = (X1, Y1, Z1), �2 = (X2, Y2, Z2), …, �r =(Xr, Yr, Zr), …, �R = (XR, YR, ZR). Here, R = rmax =Σr1 ~ τavνF is the maximum r value. Attributing thespinors ψr and matrices Ur, which transform the spinorψr from the system �r = (Xr, Yr, Zr) to the system �* =(X*, Y*, Z*), we obtain the total set of spinors (ψ1, …,ψr, …, ψR) and matrices (U1, U2, …, Ur, …, UR). Then,

the average of over all fluctuations of Hc is

= 2�Σr[( – )Bx*x* + ( –

)By*y*]/R, where = Urψr,

= Urψr, and = Urψr.Under condition (10), the orientation distribution ofthe systems of axes � is spherically symmetric; i.e., allorientations are equiprobable. Furthermore, all spinstates ψ are equiprobable and do not correlate with theorientations of the systems �. All three axes X*, Y*,and Z* in this spherically symmetric mixture of thesystems � and spin states ψ are physically equivalent.

Consequently, at rmax = ∞, Σr = Σr =

Σr , whence = 0, where means

B

Ix*2 Iz*

2

Iy*2 Iz*

2 �qu⟨ ⟩

Ix*2

⟨ ⟩ Iz*2

⟨ ⟩ Iy*2

⟨ ⟩

Iz*2

⟨ ⟩

Ix*2 Iy*

2 Iz*2

B Ix*2

⟨ ⟩

Ix*2 Iy*

2⟨ ⟩ Iy*

2 Iz*2

⟨ ⟩ Iz*2

Ix*2

⟨ ⟩ Ix*2 Iy*

2⟨ ⟩ Iy*

2

Iz*2

⟨ ⟩ Iz*2

�qu⟨ ⟩

�qu⟨ ⟩ av Ix*2

⟨ ⟩ r Iz*2

⟨ ⟩ r Iy*2

⟨ ⟩ r

Iz*2

⟨ ⟩ r Ix*2

⟨ ⟩ r ψr+Ur

+Ix*2

Iy*2

⟨ ⟩ r ψr+Ur

+Iy*2 Iz*

2⟨ ⟩ r ψr

+Ur+Iz*

2

Ix*2

⟨ ⟩ r Iy*2

⟨ ⟩ r

Iz*2

⟨ ⟩ �qu⟨ ⟩ av0 …⟨ ⟩ av0

averaging in the limit rmax = ∞. (The complete deriva�tion was given in [13].) Then (see Eq. (14)), the qua�drupole broadening at the NSN is

(20)

where /� = [Σr( )2/R]1/2/� < 107 s–1 forthe quadrupole interaction in [5].

The Hamiltonian of the 2L�pole hyperfine interac�tion (L > 1) is similarly constructed [14] from theproducts of L operators Ii* by the principal axes. In thiscase, analysis is more complicated, but reveals a simi�lar result [13] of the collapse of the 2L�pole hyperfineinteraction averaged over τav at R = ∞ with the sup�pression of broadening by a factor of (τavνF)1/2.

Thus, at NSN, contributions to broadening are(see table) kmon line ~ 10 from the monopole shift [13],kc line ~ 15.6 from the Fermi field, kqu � 1 from the qua�drupole NSN, k' ~ 2 × 10–4 from the inhomogeneousNSN in the external field, and kdd 2 line = δωdd 2 lineτ ~2.4 × 10–15 from the dipole–dipole hyperfine interac�tion if NSN occurs for both nuclei or kdd 1 line =δωdd 1 lineτ ~ 3 × 10–8 if NSN acts only on one nucleus.Comparison with the experiment was performed in[13]. The stages of the mechanism of NSN are (i) fluc�tuations of the Fermi field, (ii) random walks of thedirection of the spin, (iii) uniformity of the distribu�tion of spin direction over τav when threshold (10) isachieved, (iv) collapse of the hyperfine interaction,and (v) NSN. It is not excluded [13] that τav = pτ,where p > 1. In this case, Hcε ~ 4.8 × 104 G/p1/2 andkc line = 15.6/p. In particular, if p = 2, then Hcε = 3.4 Gand kc line = 7.8. According to Eq. (10), τav = pτ >4νF/(μμNHcε/I�)2. We apply this inequality to an arbi�trary nucleus and apply a similar equality p × 60 s =20 × 1016 s–1/(0.13μN × 4.8 × 104 G/0.5p1/2�)2 to the109Ag nucleus. Then, τ > 60 s (νF/5p × 1016 s–1)(0.13/μ)2 (4.8 × 104 G/Hcε)

2 (I/0.5)2. Since it is notexcluded that νF = 5 × 1015 s–1, I = 1/2, = 1.3, andHcε = 4.8 × 106 G, the lower bound for τ is τmin =60 s(0.1/p)(0.1)2(0.01)2 ~ 0.6 × 10–5 s/p; i.e., it is in theregion of short�lived isomers.

The upper bound of τ is obtained from the condi�tion kc < kcm ~ 10 in the form τ < τmax ~ 106 s ~ 15d [13].These estimates are very rough. For this reason, theregion of the possibility of NSN will be studied notonly in τ but also in other parameters (resonance crosssection, etc.).

The 107Ag and 109Ag isotopes ensure the conditionsfor NSN. At the same time, NSN is possible on a widercircle of nuclei because the collapse of the hyperfineinteraction owing to the fluctuations of Hc is possiblefor many atoms with the outer s1 or s0�shell (the s0�

Γqu �qu⟨ ⟩ε/�( ) τavνF( )

1/2 10 2– s

1–;<=

kqu Γquτav 1,<=

�qu⟨ ⟩ε

�qu⟨ ⟩ r

μ

180

JETP LETTERS Vol. 98 No. 3 2013

KARYAGIN

shell has electron affinity) and with filled inner shells,as in Ag. The effect of NSN belongs to a large group ofeffects of the collapse of the hyperfine structure possi�ble in many types of spectroscopy. The collapse of thehyperfine structure is probable in the chemical ele�ments H 1s1, Li 1s2s1, Na[Ne]3s1, K[Ar]4s1,Cu[Ar]3d104s1, Rb[Kr]5s1, Pd[Kr]3d105s0,Ag[Kr]3d105s1, Cs[Xe]6s1, Pt[Xe]4f145d106s0,Au[Xe]4f145d106s1, and Fr[Rn]7s1, including all theirisotopes. This also refers to 103Rh, which was discussedin [13]. The collapse of hyperfine structure is alsoprobable in the case of uncompleted shells inside anunstable s�shell: Cr[Ar]3d54s1, Nb[Kr]4d45s1,Mo[Kr]4d55s1, Ru[Kr]4d75s1, and Rh[Kr]4d85s1. Forcollapse, atoms with the unstable s�shell should beintroduced into a metallic matrix with a weak internalfield Hin � Hc on them. On atoms of the matrix lattice,Hin can be high (Hin ~ Hc). For this reason, any metalscan in principle be used as the matrix lattice to observethe effects of collapse (e.g., NSN). Collapse effects donot appear when atoms with the s' or s0�shell are intro�duced into an insulator. In particular, collapse andNSN are impossible in AgF, AgCl, AgBr, AgI, andAgAt.

To conclude, the mechanism of NSN is in princi�ple possible on nuclei with the lifetimes τ ~ 10–5–106 sin a wide circle of metallic media involving atoms withthe s1 or s0 outer shell.

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Translated by R. Tyapaev