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Giant resonances in hot nuclei
Nuclear Physics A482 (1988) 3c- 12c North-Holland, Amsterdam
3c
GIANT RESONANCES IN HOT NUCLEI
David M. BRINK Department of Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK.
The spectrum of gamma-radiation emitted by a highly excited nucleus can be calculated in two ways. In the first method the transition probability for gamma emission is related to the photon absorption cross-section by detailed balance. The second method relies on the fact that an excited hot nucleus has thermal fluctuations. In particular it has a fluctuating dipole moment which produces thermal radiation. The two methods are closely related and in both cases the spectrum of the radiation emitted is dominated by the giant dipole resonance. The equivalence of the detailed balance and thermal radiation theories can be demonstrated explicitly for a coupled oscillator model of the giant resonance. 1. INTRODUCTION The nuclear photo effect was studied extensively in the early 1950's. It was known that the cross-section for the absorption of photons by nuclei had a broad resonance and that the position and width of the resonance varied slowly with mass number and was not strongly dependent on details of nuclear structure. The theory of the photo effect had been discussed by Goldhaber and Teller 1 and by Steinwedel and Jensen 2 on the basis of collective models. According to these models the particular normal modes reached by the absorption of electric dipole radiation are those in which there is a mass motion of the neutrons against the protons. These vibrations are strongly damped due to interaction with other modes of oscillation. If this giant dipole collective mode can be described in terms of a damped simple harmonic oscillator the dependence of the cross-section on energy can be calculated classically and the shape of the photo absorption cross-section is given by (rE)2
(1)
a ( E ) o¢ ( E 2 _ E2o) 2 + ( r E ) 2 .
In my thesis s in 1955 I was interested in obtaining estimates of the partial widths of neutron resonances due to electric dipole emission. This could be done by comparing the emission with the inverse process, i.e. photo absorption. In order to do this I assumed that the energy dependence of the photo effect was independent of the detailed structure of the initial state so that the cross-section for absorption of a photon of energy E would still have the energy dependence given by eq(1) if it was possible to perform the photo effect on an arbitrary excited state.
0375-9474/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
4c
D.M. Brink~Giant resonances in hot nuclei
Nuclear states with high excitation energies and large angular m o m e n t a can be produced in heavy ion reactions. These states decay by emitting neutrons, protons and heavier fragments. They also emit gamma-rays. Properties of such 'hot' nuclei can be studied by measuring the spectra of high energy gamma-rays and the g a m m a spectrum can be calculated by the methods of ref. 3. The first experiments showing the emission of high energy gamma-rays from a highly excited nucleus were made by a group at Berkeley 4. Subsequently the effects of the giant dipole resonance on g a m m a emission from highly excited nuclei have been studied at many other laboratories and recent results will be presented at this conference. There is an alternative theory for calculating the spectrum of gamma-rays emitted by an excited hot nucleus. A hot nucleus has thermal fluctuations. In particular it has a fluctuating dipole moment which produces thermal electromagnetic radiation. The frequency spectrum of the dipole moment and the associated radiation is influenced by the existence of collective modes and is dominated by the giant dipole resonance in which there is a mass motion of the neutrons against the protons. In this talk the thermal radiation theory will be related to the more conventional theory based on detailed balance. There is a simple model for the giant resonance in which the damping of the collective motion is due to a coupling with other modes of excitation. These modes are represented by harmonic oscillators and the coupling is assumed to be linear. If the whole system is excited to a temperature T then the thermal fluctuations of the dipole moment, as well as the emission and absorption of radiation, can be calculated. In this model the equivalence of the detailed balance and thermal radiation theories can be demonstrated explicitly. 2. D E T A I L E D B A L A N C E The probabilities for emission and absorption of radiation between two excited nuclear states are related by detailed balance. The average partial width per unit energy range for emission of a photon with energy E from an initial excited state with excitation energy Ea is given by dr -
"~
-
Z 2 , ~ ~ ~ p(Eb) , , , 2nab8 t ~ , ~bJ o-~--E-~" (rnc} ~'t aj
(2)
Here Eb = E~ - E is the energy of the final state, p(E,,) a n d p(Eb) are the densities of states in the neighbourhood of the initial and final nuclear states and nabs(E, Eb) is the cross section for absorbing a photon energy range E by the nucleus in an excited state Eb. The total transition rate for the emission of radiation by the state E a is obtained by integrating the expression (2) over all photon energies. Eq(2) for the transition rate is simplified by assuming that the absorption cross-section nabs(E, Eb) by a nucleus in an excited state Eb is independent of Eb and is the same as the
absorption cross-section from the ground state. Is this assumption correct ? On the basis of the collective model of the giant dipole resonance it is reasonable to assume that the mass motion of the neutron against the proton should not be affected much if some of the
D.M. Brink~Giant resonances in hot nuclei
5c
nucleons are in excited states. The position and width of the resonance could, however, depend on the energy of the Eb of the initial state and in rapidly rotating nuclei they may also depend on the spins of the states involved. Such changes could be quite marked if the nucleus undergoes some phase transition at high excitation energy or high spin. It is even possible that the assumption that aabs(E) is independent of Eb is not valid at all. In light nuclei the shape of the gamma absorption cross-section has much more structure than predicted by the collective model formula (1) and it is quite possible that the irregularities in the energy variation of aaba should depend on the absorbing state b. Eq(2) can be simplified further by using approximate expressions for the density of states factors. The statistical theory of level densities relates the density of states ratio in eq(2) to the entropy S (E) of the nucleus at the excitation energies of the initial and final states
p(Eb)/p(Ea) ~ exp(S(Eb) -- S(Ea)).
(3)
If the excitation energy is sufficiently high then
S(Eb) - S(Ea) ~ (Eb -- E a ) / T
(4)
where T is a nuclear temperature. Then the density of states ratio is a Boltzmann factor
p(Eb)/p(Ea) ~-, e x p ( - E / T )
(5)
where E = Ea - Eb is the gamma ray energy. Using eq(5) the expression (2) for the partial width for gamma emission simplifies to dF -
E2 (rhc)2a,~ba(Z)e-E/T
(6)
and for a given nucleus depends only on the gamma-ray energy and the nuclear temperature. 3. GAMMA EMISSION AS THERMAL RADIATION The gamma spectrum (6) was obtained by using the principle of detailed balance and by making a statistical approximation to the level density ratio. The appearance of temperature in eq(6) suggests an alternative picture of the emission process. An excited nucleus is a hot nucleus and a hot nucleus will emit thermal radiation. The aim of the next part of this talk will be to show that the spectrum of thermal radiation produced by a hot nucleus satisfies eq(6). We consider a nucleus in an enclosure in equilibrium with thermal radiation at a temperature T. Let a and b be two nuclear states with energies E~ > Eb and occupation probabilities Pa and Pb. The condition for thermal equilibrium is
(Aba + Bbaf(V))Pa = Babf(V)pb
(7)
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D.M. Brink/Giant resonances in hot nuclei
where Aba and Bba are the Einstein coefficients for spontaneous and stimulated emission and
Bah
is the absorption coefficient for dipole transitions between the states a and b. The
function f ( v ) --
8rv2hv 1 e3 (ehvl T - 1)
(8)
is the energy per unit volume per unit frequency range of the black body radiation at energy hi,, = Ea - Eb. The usual argument based on thermal equilibrium gives the Einstein relations
8~rv2 hv2 Aba --
e3
(9)
Bab
(lO)
Bb~ = B~b
which contain the principle of detailed balance. It is useful to write the absorption probability per unit time as B<,bf(v) = aabI(E)
(11)
(clh~,)f(v)
(12)
where E = h v and I ( E ) --
is the photon flux per unit energy range integrated over all solid angles. The absorption coeffÉcient aab averaged over states a with energy near Eb + E is proportional to the g a m m a absorption cross section (13)
aab = aabs(E, Eb)lP(Ea)
where p(Za) is the density of states of the nucleus at Ea. Using e q s ( l l ) , (12) and (13) the expression (9) for the coefficient of spontaneous emission can be written as 1 E2 Ab~ -- h (h~e) 2 a~bs(Z, Eb)/p(E~).
(14)
Summing over final states b in an energy interval d E around Eb gives eq(2). The partial width per unit energy range for emitting g a m m a rays with energy E s u m m e d over the probability distribution p~ of the initial state a is
dr i i
d---E =
p ( E a ) d E : p ( E b ) d E b ~ ( E + Eb -- Ea)AbaPa.
(15)
Assuming a Boltzmann distribution for the occupation probability p= =
(e-Eolr)lZ
(16)
and using expression (14) for the coefficient of spontaneous emission eq(15) reduces to dr
E2 -e -E/T -- (~rhe) 2
< aabs(E) > T
(17)
D.M. Brink~Giant resonances in hot nuclei
7c
where
< aabs(E) >T = J aabs(E, Eb)Pbp(Eb)dEb
(18)
is the average gamma absorption cross-section for a target nucleus excited to temperature T. Eq(18) has the same form as eq(6) but there is a difference in the interpretation. Eq(6) gives the transition rate for gamma-ray emission averaged over initial states in a narrow energy range around E~ and uses the approximation (5) for the density of states ratio. It corresponds to using a microcanonical ensemble in statistical mechanics. Eq(17) is derived for the canonical distribution (16) of occupation probabilities and requires no assumption about the density of states ratio. 4. A COUPLED OSCILLATOR MODEL The giant dipole resonance in a nucleus is damped due to coupling with microscopic degrees of freedom, and this damping gives a width to the resonance. In classical mechanics damping can be represented by a friction term in the equations of motion, but friction cannot be incorporated directly into a quantum mechanical theory. We need an appropriate Hamiltonian for the collective variable coupled to the other degrees of freedom in order to work out the effects of the damping in a consistent way. Caldeira and Leggett s have argued that the details of the microscopic degrees of freedom are not important and that it is sufficient for most purposes to represent the internal degrees of freedom by a set of simple harmonic oscillators coupled to the collective coordinate by a coupling which is linear in the harmonic oscillator coordinates. Following this line of thought we consider a schematic Hamiltonian for the nucleus of the following form s
H=
2MOql + M122q2 +
2rniOx~ + miw~ Xi+miw~ ] j. i
Here q is the dipole coordinate and 12 is the unperturbed frequency of the dipole mode. The xi are the coordinates of the oscillators representing the microscopic degrees of freedom and wi is the frequency of the coordinate xi. The function fi(q) measures the strength of the coupling between q and xi and the form of the coupling is chosen so that it does not produce a big shift in the collective frequency 12. More specifically, the coupling potential has two terms
V, = ~ xiS~(q)
(20)
i
v~ = 1/2 ~ , :~(q)2 l'niw/2"
(21)
By itself V1 would produce a substantial shift in the collective frequency. Most of this shift is cancelled by the counterterm V2 so that 12 is near the final resonance frequency
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D.M. Brink~Giant resonances in hot nuclei
including the effects of coupling. In general fi(q) might be a non-linear function of the collective coordinate q. The case where fi(q) is a linear function of q is especially simple,
fi(q) = Ciq
(22)
where Ci are constants, because then the coupled problem at finite temperature can be solved exactly. We assume that the nuclear Hamiltonian can be represented by eq(19) and that the coupling is linear in q (eq 22). Then the potential energy in H is a quadratic function of the q and xi and the Hamiltonian can be diagonalized by introducing normal coordinates ~a with frequency wa. The new Hamiltonian is 02
i/2(-9
= Z
+ w~)
(23)
cl
and the dipole coordinate q is a linear combination of the normal coordinates, q ----Z
Q~"
(24)
c~
Most quantities of interest can be expressed in terms of a dipole strength function defined by
S(.,) : ~
IQ~l~6Cw - ~ ) .
(25)
c~
We suppose that the nucelus has a temperature T. It will emit a photon of frequency ¢#~ if the oscillator quantum number of the coordinate ~c, changes from n~ to na - 1. The partial width for emitting a photon with frequency w~ is
r . = G(,:.)IQ.I 2 ~
I < n~l~-In- - 1 > I~p(,,~)
(26)
rta=l
where p(n~) is the probability of finding the normal mode ~ in the excited state n~. This is given by a Boltzmann distribution
P(~) =
exp(-n~hw~/T)
z.
Z . = [1 - exp(-hco,~/T)] -1.
(27)
(28)
In eq(26) G(wa) is a phase space factor depending on the photon frequency. The matrix element I < n-I¢-Jn-
- 1 > 12 = hn./(2w.).
(29)
The sum in eq(26) can be evaluated with the result
r . = hG(~.) 2we,
IQ.I 2
(e ~'¢o:/T -- 1)"
(30)
D.M. Brink~Giant resonances in hot nuclei
9c
The width for emitting photons in the frequency range (w, w + dw) is (dr/dw)d~ where
dr
t~G(~) S(~)
-~ =
2w
(31)
(earl T - l )
and the shape of the spectral distribution is determined by the strength function S(w). The cross-section for photon absorption can be calculated in a similar way with the result .2c ~
S(w)
< aaba(E) >T = ~ w 3 G ( w ) ( 1 _-~-h~/T)" Equations (31) and
(32) satisfy the
(32)
reciprocity relation (17). One interesting consequence
of eq(32) is that the absorption cross-section for a nucleus depends on its temperature. The temperature dependence, however, is unimportant if the photon energy is large compared with T
E = hw > > T.
(33)
5.CALCULATION OF THE STRENGTH FUNCTION The strength function S(w) can be obtained from the Hamiltonian (19) by making the transformation to normal coordinates explicitly. It can be found more easily by calculating the classical response function. We add a term qF(t) to the Hamiltonian (19) and solve the classical equation of motion by a Fourier transform method. Defining the Fourier transform of f(t) by
](w) =
/?
ei~t f(t)dt;
Imw > 0
and assuming that all coordinates and momenta are zero at t -- 0 the equations for the Fourier transforms of the normal coordinates are
G(~)(~
- ~ ) = Q~P(~).
Hence
~(~) = r . o . ~ ( ~ ) = ~(~)P(~)
(34)
where the response function a(w) is given by ~ = ~(~) = ~ ( ~ [Q~[ : ~2)
/o °° (~,--z_S(~')d~' ~2--).
(35)
Using the relation 1
x - iE = P
+ i~r6(r)
if E > 0 and E --* O, we get Imct(w) = -~wS(w).
(36)
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D.M. Brink/Giant resonances in hot nuclei
The Fourier transforms of the coupled equations for q and xi are
(Cik, i + miw~ ]
M(I22 - w2)q(w) + E
(37)
i
m~(~
~ ) ~ ( ~ ) + c~4(~) = 0.
-
(3s)
Eliminating the ~i(w) we obtain C2
or the response function a(w) is given by
1
- M f l 2+K(w)
,~(~)
(39)
where K(w) = - w 2 M - w 2 E rniw~ (wC~ _ w2) ' i
(40)
Leggett uses a function J(w) defined by C?
(41)
J(w) = 2 -'('. rniwi so that g(w) = -w2(M+
2 ~oo
j(wl)dwt . w7-~_----~2)).
~.~
(42)
The strength function S(w) can be obtained from a(w) by using eq(36). The result depends on the form of the coupling function J(w). Caldeira and Leggett s consider a case where the distribution of the frequencies and coupling strengths Ci is such that
J(w) ~ rlw
(43)
with a high frequency cut-off. The constant 77 is a friction constant. With this form of
J(w) K(w) ~ -ir/w and 1
"Tw
Ima(w) = M (122 - w2)2 + (-~0~)~ with "7 = rl/M. Hence 2
,-/w 2
s @ ) = 7rM ( n 2 - ~2)2 + , ? z 2 "
(44)
D.M, Brink~Giant resonances in hot nuclei
1lc
This has exactly the form assumed in eq(1). The above discussion uses the formalism developed by Caldeira and Leggettb.The theory of a quantum oscillator in a black body radiation field has been discussed recently by Ford, Lewis and O'Connell6. It is related to earlier work on the quantum Langevin equation by Ford, Kac and Mazur 7. 6. CONCLUSIONS In this talk the relation between the emission spectrum and the absorption cross-section of gamma-rays by nuclei has been obtained by two different methods. The first is based on a detailed balance argument, while the second views gamma emission by an excited nucleus as a form of thermal radiation analogous to black body radiation. There are some slight differences between the results obtained by the two methods. The first difference is that eq(2) corresponds to using a microcanonical distribution for the initial occupation probabilities, while eq(17) is derived using a canonical distribution. The Boltzmann factor in eq(6) has its origin in the approximation (5) for the density of states ratio. The second method, leading to eq(17), makes no assumptions about the energy dependence of the density of states. The second difference is in the meaning of nabs in eqs(6) and (17). The factor Crabs(E) in eq(6) is the absorption cross-section for photons of energy E and is assumed to be the same for absorption from the ground state or from any excited state. The factor < nabs(E) >T is the thermal average of the absorption cross-section for photons of energy E. In general it depends on temperature but it would be independent of temperature if the absorption cross-section were the same for all initial states. The coupled oscillator model studied in sections 4 and 5 gives a nice illustration of the second method as both the emission spectrum and the absorption cross-section can be calculated and the relation (17) can be verified explicitly. It turns out that the thermal average (18) gamma absorption cross-section given in eq(32) depends on temperature. Thus the assumption that the absorption cross-section is independent of the initial state cannot be exactly correct. However, the temperature dependence is not important if the nuclear temperature T is much less than the energy hWD of the giant dipole resonance. This condition is always satisfied in present experiments. This weak dependence on temperature is probably a special feature of the model studied here. A more realistic model with nonlinear coupling could give a much stronger variation of < a(E) >T with temperature. REFERENCES 1) M. Goldhaber and E. Teller, Phys.Rev. '/4 (1948) 1046. 2) H. Steinwedel and J.H.D. Jensen, Z.Naturforsehung 5A (1950) 413. 3) D.M. Brink, Thesis, Oxford (1955). J.O. Newton, B.Herskind, R.M. Diamond, E.L. Dines, J.E. Draper, K.H. Lindenberger, ~ .Schuck, S. Shih and F. Stephens, Phys.Rev.Lett. 46 (1981) 1383. 45)4A.O. Caldeira and A.J. Leggett, Ann.Phys. 149 (1983) 374; Ann. Phys. 153 (1984) 6) G.W. Ford, J.T. Lewis and R.F. O'Connell, Phys.Rev.Lett. 55 (1985) 2273. 7) G.W. Ford, M. Kac and P. Mazur, J.Math.Phys. 6 (1965) 504.
3c
GIANT RESONANCES IN HOT NUCLEI
David M. BRINK Department of Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK.
The spectrum of gamma-radiation emitted by a highly excited nucleus can be calculated in two ways. In the first method the transition probability for gamma emission is related to the photon absorption cross-section by detailed balance. The second method relies on the fact that an excited hot nucleus has thermal fluctuations. In particular it has a fluctuating dipole moment which produces thermal radiation. The two methods are closely related and in both cases the spectrum of the radiation emitted is dominated by the giant dipole resonance. The equivalence of the detailed balance and thermal radiation theories can be demonstrated explicitly for a coupled oscillator model of the giant resonance. 1. INTRODUCTION The nuclear photo effect was studied extensively in the early 1950's. It was known that the cross-section for the absorption of photons by nuclei had a broad resonance and that the position and width of the resonance varied slowly with mass number and was not strongly dependent on details of nuclear structure. The theory of the photo effect had been discussed by Goldhaber and Teller 1 and by Steinwedel and Jensen 2 on the basis of collective models. According to these models the particular normal modes reached by the absorption of electric dipole radiation are those in which there is a mass motion of the neutrons against the protons. These vibrations are strongly damped due to interaction with other modes of oscillation. If this giant dipole collective mode can be described in terms of a damped simple harmonic oscillator the dependence of the cross-section on energy can be calculated classically and the shape of the photo absorption cross-section is given by (rE)2
(1)
a ( E ) o¢ ( E 2 _ E2o) 2 + ( r E ) 2 .
In my thesis s in 1955 I was interested in obtaining estimates of the partial widths of neutron resonances due to electric dipole emission. This could be done by comparing the emission with the inverse process, i.e. photo absorption. In order to do this I assumed that the energy dependence of the photo effect was independent of the detailed structure of the initial state so that the cross-section for absorption of a photon of energy E would still have the energy dependence given by eq(1) if it was possible to perform the photo effect on an arbitrary excited state.
0375-9474/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
4c
D.M. Brink~Giant resonances in hot nuclei
Nuclear states with high excitation energies and large angular m o m e n t a can be produced in heavy ion reactions. These states decay by emitting neutrons, protons and heavier fragments. They also emit gamma-rays. Properties of such 'hot' nuclei can be studied by measuring the spectra of high energy gamma-rays and the g a m m a spectrum can be calculated by the methods of ref. 3. The first experiments showing the emission of high energy gamma-rays from a highly excited nucleus were made by a group at Berkeley 4. Subsequently the effects of the giant dipole resonance on g a m m a emission from highly excited nuclei have been studied at many other laboratories and recent results will be presented at this conference. There is an alternative theory for calculating the spectrum of gamma-rays emitted by an excited hot nucleus. A hot nucleus has thermal fluctuations. In particular it has a fluctuating dipole moment which produces thermal electromagnetic radiation. The frequency spectrum of the dipole moment and the associated radiation is influenced by the existence of collective modes and is dominated by the giant dipole resonance in which there is a mass motion of the neutrons against the protons. In this talk the thermal radiation theory will be related to the more conventional theory based on detailed balance. There is a simple model for the giant resonance in which the damping of the collective motion is due to a coupling with other modes of excitation. These modes are represented by harmonic oscillators and the coupling is assumed to be linear. If the whole system is excited to a temperature T then the thermal fluctuations of the dipole moment, as well as the emission and absorption of radiation, can be calculated. In this model the equivalence of the detailed balance and thermal radiation theories can be demonstrated explicitly. 2. D E T A I L E D B A L A N C E The probabilities for emission and absorption of radiation between two excited nuclear states are related by detailed balance. The average partial width per unit energy range for emission of a photon with energy E from an initial excited state with excitation energy Ea is given by dr -
"~
-
Z 2 , ~ ~ ~ p(Eb) , , , 2nab8 t ~ , ~bJ o-~--E-~" (rnc} ~'t aj
(2)
Here Eb = E~ - E is the energy of the final state, p(E,,) a n d p(Eb) are the densities of states in the neighbourhood of the initial and final nuclear states and nabs(E, Eb) is the cross section for absorbing a photon energy range E by the nucleus in an excited state Eb. The total transition rate for the emission of radiation by the state E a is obtained by integrating the expression (2) over all photon energies. Eq(2) for the transition rate is simplified by assuming that the absorption cross-section nabs(E, Eb) by a nucleus in an excited state Eb is independent of Eb and is the same as the
absorption cross-section from the ground state. Is this assumption correct ? On the basis of the collective model of the giant dipole resonance it is reasonable to assume that the mass motion of the neutron against the proton should not be affected much if some of the
D.M. Brink~Giant resonances in hot nuclei
5c
nucleons are in excited states. The position and width of the resonance could, however, depend on the energy of the Eb of the initial state and in rapidly rotating nuclei they may also depend on the spins of the states involved. Such changes could be quite marked if the nucleus undergoes some phase transition at high excitation energy or high spin. It is even possible that the assumption that aabs(E) is independent of Eb is not valid at all. In light nuclei the shape of the gamma absorption cross-section has much more structure than predicted by the collective model formula (1) and it is quite possible that the irregularities in the energy variation of aaba should depend on the absorbing state b. Eq(2) can be simplified further by using approximate expressions for the density of states factors. The statistical theory of level densities relates the density of states ratio in eq(2) to the entropy S (E) of the nucleus at the excitation energies of the initial and final states
p(Eb)/p(Ea) ~ exp(S(Eb) -- S(Ea)).
(3)
If the excitation energy is sufficiently high then
S(Eb) - S(Ea) ~ (Eb -- E a ) / T
(4)
where T is a nuclear temperature. Then the density of states ratio is a Boltzmann factor
p(Eb)/p(Ea) ~-, e x p ( - E / T )
(5)
where E = Ea - Eb is the gamma ray energy. Using eq(5) the expression (2) for the partial width for gamma emission simplifies to dF -
E2 (rhc)2a,~ba(Z)e-E/T
(6)
and for a given nucleus depends only on the gamma-ray energy and the nuclear temperature. 3. GAMMA EMISSION AS THERMAL RADIATION The gamma spectrum (6) was obtained by using the principle of detailed balance and by making a statistical approximation to the level density ratio. The appearance of temperature in eq(6) suggests an alternative picture of the emission process. An excited nucleus is a hot nucleus and a hot nucleus will emit thermal radiation. The aim of the next part of this talk will be to show that the spectrum of thermal radiation produced by a hot nucleus satisfies eq(6). We consider a nucleus in an enclosure in equilibrium with thermal radiation at a temperature T. Let a and b be two nuclear states with energies E~ > Eb and occupation probabilities Pa and Pb. The condition for thermal equilibrium is
(Aba + Bbaf(V))Pa = Babf(V)pb
(7)
6c
D.M. Brink/Giant resonances in hot nuclei
where Aba and Bba are the Einstein coefficients for spontaneous and stimulated emission and
Bah
is the absorption coefficient for dipole transitions between the states a and b. The
function f ( v ) --
8rv2hv 1 e3 (ehvl T - 1)
(8)
is the energy per unit volume per unit frequency range of the black body radiation at energy hi,, = Ea - Eb. The usual argument based on thermal equilibrium gives the Einstein relations
8~rv2 hv2 Aba --
e3
(9)
Bab
(lO)
Bb~ = B~b
which contain the principle of detailed balance. It is useful to write the absorption probability per unit time as B<,bf(v) = aabI(E)
(11)
(clh~,)f(v)
(12)
where E = h v and I ( E ) --
is the photon flux per unit energy range integrated over all solid angles. The absorption coeffÉcient aab averaged over states a with energy near Eb + E is proportional to the g a m m a absorption cross section (13)
aab = aabs(E, Eb)lP(Ea)
where p(Za) is the density of states of the nucleus at Ea. Using e q s ( l l ) , (12) and (13) the expression (9) for the coefficient of spontaneous emission can be written as 1 E2 Ab~ -- h (h~e) 2 a~bs(Z, Eb)/p(E~).
(14)
Summing over final states b in an energy interval d E around Eb gives eq(2). The partial width per unit energy range for emitting g a m m a rays with energy E s u m m e d over the probability distribution p~ of the initial state a is
dr i i
d---E =
p ( E a ) d E : p ( E b ) d E b ~ ( E + Eb -- Ea)AbaPa.
(15)
Assuming a Boltzmann distribution for the occupation probability p= =
(e-Eolr)lZ
(16)
and using expression (14) for the coefficient of spontaneous emission eq(15) reduces to dr
E2 -e -E/T -- (~rhe) 2
< aabs(E) > T
(17)
D.M. Brink~Giant resonances in hot nuclei
7c
where
< aabs(E) >T = J aabs(E, Eb)Pbp(Eb)dEb
(18)
is the average gamma absorption cross-section for a target nucleus excited to temperature T. Eq(18) has the same form as eq(6) but there is a difference in the interpretation. Eq(6) gives the transition rate for gamma-ray emission averaged over initial states in a narrow energy range around E~ and uses the approximation (5) for the density of states ratio. It corresponds to using a microcanonical ensemble in statistical mechanics. Eq(17) is derived for the canonical distribution (16) of occupation probabilities and requires no assumption about the density of states ratio. 4. A COUPLED OSCILLATOR MODEL The giant dipole resonance in a nucleus is damped due to coupling with microscopic degrees of freedom, and this damping gives a width to the resonance. In classical mechanics damping can be represented by a friction term in the equations of motion, but friction cannot be incorporated directly into a quantum mechanical theory. We need an appropriate Hamiltonian for the collective variable coupled to the other degrees of freedom in order to work out the effects of the damping in a consistent way. Caldeira and Leggett s have argued that the details of the microscopic degrees of freedom are not important and that it is sufficient for most purposes to represent the internal degrees of freedom by a set of simple harmonic oscillators coupled to the collective coordinate by a coupling which is linear in the harmonic oscillator coordinates. Following this line of thought we consider a schematic Hamiltonian for the nucleus of the following form s
H=
2MOql + M122q2 +
2rniOx~ + miw~ Xi+miw~ ] j. i
Here q is the dipole coordinate and 12 is the unperturbed frequency of the dipole mode. The xi are the coordinates of the oscillators representing the microscopic degrees of freedom and wi is the frequency of the coordinate xi. The function fi(q) measures the strength of the coupling between q and xi and the form of the coupling is chosen so that it does not produce a big shift in the collective frequency 12. More specifically, the coupling potential has two terms
V, = ~ xiS~(q)
(20)
i
v~ = 1/2 ~ , :~(q)2 l'niw/2"
(21)
By itself V1 would produce a substantial shift in the collective frequency. Most of this shift is cancelled by the counterterm V2 so that 12 is near the final resonance frequency
8c
D.M. Brink~Giant resonances in hot nuclei
including the effects of coupling. In general fi(q) might be a non-linear function of the collective coordinate q. The case where fi(q) is a linear function of q is especially simple,
fi(q) = Ciq
(22)
where Ci are constants, because then the coupled problem at finite temperature can be solved exactly. We assume that the nuclear Hamiltonian can be represented by eq(19) and that the coupling is linear in q (eq 22). Then the potential energy in H is a quadratic function of the q and xi and the Hamiltonian can be diagonalized by introducing normal coordinates ~a with frequency wa. The new Hamiltonian is 02
i/2(-9
= Z
+ w~)
(23)
cl
and the dipole coordinate q is a linear combination of the normal coordinates, q ----Z
Q~"
(24)
c~
Most quantities of interest can be expressed in terms of a dipole strength function defined by
S(.,) : ~
IQ~l~6Cw - ~ ) .
(25)
c~
We suppose that the nucelus has a temperature T. It will emit a photon of frequency ¢#~ if the oscillator quantum number of the coordinate ~c, changes from n~ to na - 1. The partial width for emitting a photon with frequency w~ is
r . = G(,:.)IQ.I 2 ~
I < n~l~-In- - 1 > I~p(,,~)
(26)
rta=l
where p(n~) is the probability of finding the normal mode ~ in the excited state n~. This is given by a Boltzmann distribution
P(~) =
exp(-n~hw~/T)
z.
Z . = [1 - exp(-hco,~/T)] -1.
(27)
(28)
In eq(26) G(wa) is a phase space factor depending on the photon frequency. The matrix element I < n-I¢-Jn-
- 1 > 12 = hn./(2w.).
(29)
The sum in eq(26) can be evaluated with the result
r . = hG(~.) 2we,
IQ.I 2
(e ~'¢o:/T -- 1)"
(30)
D.M. Brink~Giant resonances in hot nuclei
9c
The width for emitting photons in the frequency range (w, w + dw) is (dr/dw)d~ where
dr
t~G(~) S(~)
-~ =
2w
(31)
(earl T - l )
and the shape of the spectral distribution is determined by the strength function S(w). The cross-section for photon absorption can be calculated in a similar way with the result .2c ~
S(w)
< aaba(E) >T = ~ w 3 G ( w ) ( 1 _-~-h~/T)" Equations (31) and
(32) satisfy the
(32)
reciprocity relation (17). One interesting consequence
of eq(32) is that the absorption cross-section for a nucleus depends on its temperature. The temperature dependence, however, is unimportant if the photon energy is large compared with T
E = hw > > T.
(33)
5.CALCULATION OF THE STRENGTH FUNCTION The strength function S(w) can be obtained from the Hamiltonian (19) by making the transformation to normal coordinates explicitly. It can be found more easily by calculating the classical response function. We add a term qF(t) to the Hamiltonian (19) and solve the classical equation of motion by a Fourier transform method. Defining the Fourier transform of f(t) by
](w) =
/?
ei~t f(t)dt;
Imw > 0
and assuming that all coordinates and momenta are zero at t -- 0 the equations for the Fourier transforms of the normal coordinates are
G(~)(~
- ~ ) = Q~P(~).
Hence
~(~) = r . o . ~ ( ~ ) = ~(~)P(~)
(34)
where the response function a(w) is given by ~ = ~(~) = ~ ( ~ [Q~[ : ~2)
/o °° (~,--z_S(~')d~' ~2--).
(35)
Using the relation 1
x - iE = P
+ i~r6(r)
if E > 0 and E --* O, we get Imct(w) = -~wS(w).
(36)
10c
D.M. Brink/Giant resonances in hot nuclei
The Fourier transforms of the coupled equations for q and xi are
(Cik, i + miw~ ]
M(I22 - w2)q(w) + E
(37)
i
m~(~
~ ) ~ ( ~ ) + c~4(~) = 0.
-
(3s)
Eliminating the ~i(w) we obtain C2
or the response function a(w) is given by
1
- M f l 2+K(w)
,~(~)
(39)
where K(w) = - w 2 M - w 2 E rniw~ (wC~ _ w2) ' i
(40)
Leggett uses a function J(w) defined by C?
(41)
J(w) = 2 -'('. rniwi so that g(w) = -w2(M+
2 ~oo
j(wl)dwt . w7-~_----~2)).
~.~
(42)
The strength function S(w) can be obtained from a(w) by using eq(36). The result depends on the form of the coupling function J(w). Caldeira and Leggett s consider a case where the distribution of the frequencies and coupling strengths Ci is such that
J(w) ~ rlw
(43)
with a high frequency cut-off. The constant 77 is a friction constant. With this form of
J(w) K(w) ~ -ir/w and 1
"Tw
Ima(w) = M (122 - w2)2 + (-~0~)~ with "7 = rl/M. Hence 2
,-/w 2
s @ ) = 7rM ( n 2 - ~2)2 + , ? z 2 "
(44)
D.M, Brink~Giant resonances in hot nuclei
1lc
This has exactly the form assumed in eq(1). The above discussion uses the formalism developed by Caldeira and Leggettb.The theory of a quantum oscillator in a black body radiation field has been discussed recently by Ford, Lewis and O'Connell6. It is related to earlier work on the quantum Langevin equation by Ford, Kac and Mazur 7. 6. CONCLUSIONS In this talk the relation between the emission spectrum and the absorption cross-section of gamma-rays by nuclei has been obtained by two different methods. The first is based on a detailed balance argument, while the second views gamma emission by an excited nucleus as a form of thermal radiation analogous to black body radiation. There are some slight differences between the results obtained by the two methods. The first difference is that eq(2) corresponds to using a microcanonical distribution for the initial occupation probabilities, while eq(17) is derived using a canonical distribution. The Boltzmann factor in eq(6) has its origin in the approximation (5) for the density of states ratio. The second method, leading to eq(17), makes no assumptions about the energy dependence of the density of states. The second difference is in the meaning of nabs in eqs(6) and (17). The factor Crabs(E) in eq(6) is the absorption cross-section for photons of energy E and is assumed to be the same for absorption from the ground state or from any excited state. The factor < nabs(E) >T is the thermal average of the absorption cross-section for photons of energy E. In general it depends on temperature but it would be independent of temperature if the absorption cross-section were the same for all initial states. The coupled oscillator model studied in sections 4 and 5 gives a nice illustration of the second method as both the emission spectrum and the absorption cross-section can be calculated and the relation (17) can be verified explicitly. It turns out that the thermal average (18) gamma absorption cross-section given in eq(32) depends on temperature. Thus the assumption that the absorption cross-section is independent of the initial state cannot be exactly correct. However, the temperature dependence is not important if the nuclear temperature T is much less than the energy hWD of the giant dipole resonance. This condition is always satisfied in present experiments. This weak dependence on temperature is probably a special feature of the model studied here. A more realistic model with nonlinear coupling could give a much stronger variation of < a(E) >T with temperature. REFERENCES 1) M. Goldhaber and E. Teller, Phys.Rev. '/4 (1948) 1046. 2) H. Steinwedel and J.H.D. Jensen, Z.Naturforsehung 5A (1950) 413. 3) D.M. Brink, Thesis, Oxford (1955). J.O. Newton, B.Herskind, R.M. Diamond, E.L. Dines, J.E. Draper, K.H. Lindenberger, ~ .Schuck, S. Shih and F. Stephens, Phys.Rev.Lett. 46 (1981) 1383. 45)4A.O. Caldeira and A.J. Leggett, Ann.Phys. 149 (1983) 374; Ann. Phys. 153 (1984) 6) G.W. Ford, J.T. Lewis and R.F. O'Connell, Phys.Rev.Lett. 55 (1985) 2273. 7) G.W. Ford, M. Kac and P. Mazur, J.Math.Phys. 6 (1965) 504.