A.Ya. Ender et al. / Physics Reports 328 (2000) 1}72
COLLECTIVE DIODE DYNAMICS: AN ANALYTICAL APPROACH
A.Ya. ENDER , Heidrun KOLINSKY, V.I. KUZNETSOV , H. SCHAMEL Iowe Physico-Technical Institute, St. Petersburg 194021, Russia Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Physics Reports 328 (2000) 1}72
Collective diode dynamics: an analytical approach A.Ya. Ender *, Heidrun Kolinsky, V.I. Kuznetsov , H. Schamel Iowe Physico-Technical Institute, St. Petersburg 194021, Russia Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany Received June 1999; editor: D.L. Mills Contents 1. Introduction and review 1.1. Introduction 1.2. Review on plasma diode theories 1.3. Experiments and applications of the Bursian and Pierce diode instabilities 2. Notation and basic equations 3. Equilibrium solutions without and with re#ection 4. Criteria for existence and stability of equilibria without re#ection in the Lagrangian representation 4.1. The Lagrangian formulation 4.2. Equivalence between Eulerian and Lagrangian representation 4.3. On the boundary of solutions without re#ection 4.4. Linear dispersion relation 5. The transformation of states 5.1. De"nition and transformation of states without re#ection
4 4 6 15 18 20
24 24 26 28 29 30 30
5.2. Transformation of states with re#ection 5.3. Some properties of the transformation 6. Classi"cation of potential distributions (PD): (g, e)-diagrams and I}< characteristics 6.1. Generalized Pierce diode 6.2. Nonneutral diode 6.3. (g, e)-diagrams and I}< characteristics 7. Stability of equilibrium solutions 7.1. Aperiodical instability boundaries for nonre#ective solutions 7.2. Aperiodical instability boundaries for solutions with partial re#ection 7.3. Aperiodical instability boundaries for solutions with total re#ection 7.4. Dispersion curves and boundaries of oscillatory instability 8. Summary and conclusions Acknowledgements References
32 33 36 36 40 42 44 45 52 55 57 67 70 70
Abstract An analytical study of the plasma states in nonneutral plasma diodes and of their stability is presented for an arbitrary neutralization parameter c, including the Pierce (c"1) and the Bursian (c"0) diode as special cases. Physically such a study is of interest, e.g. in the transport problem of an electron beam in spatially bounded electronic devices. Similarity transformations are obtained which connect equilibrium solutions of di!erent c's. This implies that by simple transformations one can infer from equilibria of the generalized
* Corresponding author. 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 9 2 - 7
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Pierce diode to equilibria of nonneutral diodes. The regimes with partial and total re#ection of electrons are studied in detail for the "rst time. A classi"cation of nonuniform solutions for these regions as well as for the regime without re#ection is presented. The equivalence between the Eulerian and the Lagrangian formulation of the diode dynamics is proved, and both, the aperiodical and oscillatory eigenmodes of the generalized Pierce diode are examined. New bifurcation points in the branches of dispersion relations are discovered. 2000 Elsevier Science B.V. All rights reserved. PACS: 41.75.!i; 85.30.Fg; 52.35.Qz; 85.45.!w; 84.47.#w Keywords: Plasma diode dynamics; Beam plasma device; Generalized Pierce diode; Nonneutral diode; Re#ection by virtual cathodes; Space charge limited current
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1. Introduction and review 1.1. Introduction This article reports on recent theoretical progress in the understanding of diode dynamics as a collective phenomenon. Many systems in physical science and technology, in which charge carriers are transported between two electrodes, are known to exhibit a diodic behavior, the most celebrated diodic event being the phenomenon of `space-charge-limited #owa. This #ow limitation, which in most situations severely restricts the operation conditions, indicates a bifurcation of states, i.e. a transition between two distinct stable states. An understanding of these bifurcation processes is hence of crucial importance for all systems in which a space-charge-induced #ow limitation takes place. Consequently, most of the present work is devoted to bifurcation studies including investigations on the existence of diode states and on their stability properties. Diode dynamics essentially a!ects physical and technical processes such as that found in Q-machines [1,2], thermionic converters [3}5], microwave generators [6,7], electronic switches [8], low-pressure discharges and processing [9], accelerators [10], inertial con"nement devices [11,12], xerographic technologies [13], semiconductor devices [14}16], etc., to mention some of them. Common to these devices is the fact that the particle transport can be modeled in good approximation by a one-dimensional, ballistic and monoenergetic #ow. Despite the simplicity of this model } "rst ideas and descriptions date back a long time [17}19] } it gained new interest in recent years mainly due to its potential applicability and its richness in the phenomena associated with. Focusing on the dominant collective behavior of the lighter charged species (mostly electrons), we describe the #uid-like dynamics on the fast time scale by continuity and momentum equation supplemented by Poisson's equation. At "rst glance, an investigation of this type may look simple and straightforward; however, as shown later, the dynamics shows up with several subtleties and pitfalls, some of them being described for the "rst time. The nonlinearity and time dependency of these plasma equations, the possibility of particle re#ection due to the occurrence of an internal potential minimum and the boundary conditions render the problem highly nontrivial. It is our concern in this theoretical report, to present a complete picture of the dc states of diodes under these conditions, to evaluate almost completely their linear stability behavior and to o!er for the "rst time a transformation between dc states, which yields a substantial simpli"cation of the analysis. Also, a review of previous works on diode dynamics is given. Fig. 1 shows schematically the underlying diode model. A beam of electrons of given density n> and velocity v enters the diode region at the emitter (donor) electrode at x"0. The latter is kept at zero potential. If no re#ection occurs, all electrons leave the diode at the collector, at x"¸. A bias voltage is applied across the diode region. Electrons which are re#ected by the internal space charge potential return to the emitter electrode and are totally absorbed. It is furthermore assumed that the diode region is occupied uniformly by in"nitely massive ions of constant density n . We allow any value of n . This implies that charge neutralization (n "n>) is only one of the many options discussed. The ions are, hence, treated as immobile, a dynamical situation often found to be valid in lowest approximation as our focus is on the fast electron processes. (A practical example where ions do not participate in the dynamics and hence can be treated by a constant
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Fig. 1. Schematic view of the electron dynamics in the diode. Some of the electrons, all enter the diode region at x"0, are re#ected by a virtual cathode. Ions contribute by their constant density n only.
density is their perpendicular injection. If their velocity is su$ciently large they will leave the diode region essentially unchanged, i.e. without having experienced a change in their density distribution both in longitudinal and transversal direction. Hence, fresh ions with a homogeneous distribution will provide the necessary background.) The adjective ballistic implies that the #ow is assumed to be collisionless, i.e. that the electron mean free path well exceeds the diode length ¸, a situation often met. Such a regime of a plasma diode, often referred to as the Knudsen regime, can be found, for example, in Q-machines [1,2], where the su$ciently dilute and hot plasma is provided by the surface ionization of a neutral particle beam that hits a hot plate (emitter). A comparable situation is given in high-temperature thermionic energy converters [3}5]. Cold electron beam evaporation diode (or triode) systems, in which the thermal spread of the beam is negligible and where ballistic transport prevails, are found in micro- and nanoelectronic semiconductor "lm capacitors as well [15,16,20]. Field emitter arrays of the Spindt cathode type [21}23] constitute another important class of applications. Typically, these devices experience a current self-quenching due to space charge e!ects and show current}voltage characteristics with portions of a negative di!erential resistivity (see later). A monoenergetic electron #ow can also be generated in triodes where a highly accelerating voltage is applied between the thermocathode and the grid. The space between the grid and the collector constitutes a further experimental realization of the model. A current limitation is observed whenever one of the following quantities exceeds the critical value: the particle density, the current density and the gap distance. In this situation, an aperiodic instability sets in that destabilizes the diode. As a result, a virtual cathode arises internally which re#ects electrons partially or totally, depending on the parameter regime involved. In the case of charge neutrality, n>"n , this instability is known as the Pierce instability and the diode under short-circuited conditions as the Pierce diode [19]. On the other hand, if at all ions are absent and hence if only electrons are present, the diode is called Bursian diode [18]. One reason, as mentioned before, why such an idealized con"guration exhibits such a complex dynamical pattern lies in the boundary conditions. The #ow characteristics for "xed boundary conditions are generally di!erent at injection and at exit. In contrast to periodic systems the #ow at
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injection has thus no information about the actual status of the diode. To meet the "xed boundary conditions in case of a time-dependent behavior, such as that treated by a linear stability analysis of the homogeneous state under short-circuited conditions, a Fourier analysis with complex frequencies and wavenumbers is required which has rather intricate consequences [24]. Textbooks on plasma physics and related topics, on the other hand, almost exclusively deal with unbounded or periodic plasmas } three of the few exceptions are the monographs of Llewellyn [25], Birdsall and Bridges [26] and Nezlin [27] } and are hence not suited to describe such situations theoretically. In these more simple cases either the frequency or the wavenumber can be chosen real, depending on the situation. Hence, bounded plasmas are something else possessing their own characteristic dynamics. Furthermore, the ballistic nature of the #ow suggests to bring in also the Lagrangian picture, in which the dynamics is described in a reference system comoving with a #uid element. In certain situations this description is preferred to an Eulerian description, as it allows pragmatically a solution in contrast to the latter which sometimes gets lost in analytically unsolvable equations. An example is the linear stability analysis with an arbitrary degree of neutralization and a dc bias. We hence pursue in this report both descriptions, the Eulerian and the Lagrangian. In addition, it is also helpful to consider a Dirichlet boundary value problem, namely given the potential at the electrodes, and contrast it to a Cauchy boundary value problem, where the normalized potential g and the normalized electric "eld e are prescribed at the emitter. As we shall show, the use of both concepts, the Lagrangian method, promoted in Bayreuth and the so-called (g, e)-method, developed in St. Petersburg, allow us to achieve results that have not been obtained before. With the (g, e)-method we are able to investigate completely the aperiodic instability also in situations of dc virtual cathodes, i.e. when particle re#ections occur. On the other hand, the Lagrangian description allows to solve the linear stability problem in the case of nonre#ective equilibria, encompassing oscillatory instabilities, too. These methods are hence in some sense complementary. Although a number of works have emerged in the last decades, many of them having been referred to in the adjacent review of plasma diode theories and experiments, a systematic study is lacking. In addition, the region of existence of solutions in parameter space and the interrelations between di!erent solutions have not yet been analyzed. This paper intends to "ll this gap. Before we enter into the detailed mathematical analysis we report and summarize previous works on diode dynamics. 1.2. Review on plasma diode theories 1.2.1. Bursian diode (pure electron diode) In this review, we summarize the highlights in the analytic description of plasma diodes. First we treat pure electron diodes since from these all the later developments originate. The very "rst paper on the theory of bounded collisionless plasmas was written by Child [17]. He made the "rst analytic description of the static electric potential in a vacuum diode in which an ion #ow leaves the emitter with zero velocity and an accelerating negative voltage, ! ,
(1)
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the normalized interelectrode distance (Pierce parameter) d" : u> ¸/v ,: ¸/j , " and the normalized bias between collector and emitter
(2)
e/e m) is the plasma frequency corresponding to n>, and e, m are the (positive) elementary charge and the electron mass, respectively and j is the formal Debye length " j " : v u>\. " The set of equations governing the diode dynamics depends on whether electron re#ection takes place or not. If no electron re#ection takes place, our basic system of equations becomes R n#R (nv)"0 , (4a) O D R v#vR v"!e , (4b) O D !R e"n!c , (4c) D which are the cold electron #uid equations coupled with Poisson's equation. In Eqs. (4a)}(4c) the normalized coordinates q"u> t, f"z/j " and #uid quantities
(5)
n"n /n>, v"v /v are used, and the normalized electric potential and electric "eld are given by
(6)
g"e /mv , e"eEj /mv , " where e"!R g holds. D The system (4) is supplemented by the Dirichlet boundary conditions
(7)
n(0, q)"1, v(0, q)"1 ,
(8a)
g(0, q)"0 ,
(8b)
g(d, q)" (0, q)"r ,
(10)
v (0, q)"1,
(11)
v > (0, q)"1 ,
where r is the re#ection parameter which may generally depend on time. In Ref. [40] the two regimes are called type-I and type-II #ow and the splitting parameter l of that paper is related to the present one by l"(1!r)/(1#r). Note that (4) (resp. (9)) contain as special cases the Bursian diode (pure electron diode), c"0, and the generalized Pierce diode, c"1.
3. Equilibrium solutions without and with re6ection The simplest equilibrium is the one where the incoming beam is completely transmitted (r"0). Dropping the time dependence, from Eqs. (4a), (4b) and (8a), (8b) for the density we easily "nd n"(1#2g)\ ,
(12)
and Poisson's equation (4c) becomes g(f)"(1#2g)\!c .
(13)
Multiplying (13) by g(f), we get by integration g(f)!e "2[(1#2g)!cg!1] ,
(14)
where e represents the electric "eld at the emitter. By a second integration we can obtain g(f) and by applying the boundary condition (8c) we can establish a relation between the electric "eld at the emitter e and the collector potential c\[1!Z(g)], f4f
,
">c\+[Z(g)#2r>\]![Z(g )#2r>\],, f'f ,
where we introduced the functions >" : [ce #(1#r!c)] , Z(g) " : [c(1#2g)!(1#r)]>\ .
(17a) (17b)
(18) (19)
In the regime without re#ection, r"0, the value of the potential minimum has to be greater than !1/2, and Eqs. (17a), (17b) equals Eq. (14). If we now use the conditions g"! and g"0 for the minimum point then we obtain for the boundaries of this regime "e "((2!c) . (19a) In the regime with partial re#ection, 0(r(1, we will, according to Ref. [36], distinguish between two types of potential distributions (PD): a PD of the "rst kind, where the re#ection point lies inside the diode region, and a PD of the second kind, where the re#ection point is at the collector. In the regime of partial re#ection of the "rst kind the potential of the re#ection point is equal to !1/2 and the electric "eld dg/df vanishes in this point. With these values we obtain from (17a), (18) and (19) a relation between the re#ection coe$cient r and the emitter electric "eld strength e : "e ""[2(1#r)!c] . (20) The boundaries of the regime of partial re#ection are given by substituting r"0 and r"1, respectively, (2!c)4"e "4(4!c) . (21) In the considered region the typical PDs for large enough diode lengths d are of wavy type with alternating potential minima and maxima. These extremum points can be directly derived from (17a), (17b) by setting g"0. For the regime without re#ection (r"0) we obtain g
(22a) "!#+1![ce #(1!c)],/2c , g "!#+1#[ce #(1!c)],/2c . (22b)
For the regime with re#ection the maximum potential nearest to the emitter for e (0 is found to be g
"!#+1#r#>,/2c . (23)
Note that, whereas the electric "eld strength in the point of re#ection always vanishes in the case of partial re#ection of the "rst kind, it generally di!ers from zero in the cases of re#ection of the second kind or of total re#ection. The expression for the PD is found analytically by integration of
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Eqs. (17a). To the left of the "rst extremum we obtain sgn(e ) +>[(1!Z(g))!(1!Z(0))]#(1#r)[arcsin Z(0)!arcsin Z(g)], . f" c Extremum points in the PD are given by Z"$1. For e (0 the "rst maximum is at e (1#r) p #arcsin[(1#r!c)/>] ! , f "
c c 2
(24)
(25)
and the "rst minimum is at
(1#r) 3p e f " #arcsin[(1#r!c)/>] ! .
c c 2 and the PD between f and f reads
f"f
#c\>(1!Z(g))#(1#r)c\
(26)
p !arcsin Z(g) . 2
For e '0, the location of the "rst minimum is found from (24) and becomes (1#r) p e f " !arcsin[(1#r!c)/>] ! .
c 2 c
(27)
(28)
The PD solution beyond the "rst minimum and the next maximum can be obtained by integration of (17b):
f #c\[p #arcsin ZI (g)![ce #(1!c)][1!ZI (g)]], r"0 , f" r'0 , f #c\(1!r)[p #arcsin ZI (g)![1!ZI (g)]],
where ZI (g) is given by
ZI (g)"
[c(1#2g)!1][ce #(1!c)]\, r"0 , [c(1#2g)!(1!r)]/(1!r), r'0 .
(29)
(30)
From (29) we can immediately extract the spatial period of the PD beyond the "rst minimum which is given by 2pc\(1!r) .
(31)
If all electrons are re#ected, r"1, the electric "eld strength in the re#ection point becomes according to (17)}(19) eH"(e #c!4) , (32) which, as a rule, di!ers from zero. There are no electrons at all to the right of this point and the PD is a parabola g(f)"!!eH(f!fH)!c(f!fH)/2 .
(33)
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These formulas allow us to carry out a complete analysis of the PD in a diode with varying external parameters. Figs. 2a}d show how the PD changes when e is varied for c"1 (the generalized Pierce diode). Figs. 2a and b represent solutions without re#ection, Figs. 2c and d with re#ections. In Figs. 2a and c e '0 and in Figs. 2b and d e (0 hold. In the regime without re#ection, as it follows from (22a), (22b), "e "(1 holds. In this case the potential vanishes at the points f"2mp with m"1, 2, 3,2 independent of e . In this regime the maximum potential grows with increasing "e " and the minimum diminishes until !1/2. In the regime with partial re#ection, Figs. 2c and d, g stays for increasing "e " equal to !1/2 and g to the right of the re#ection point decreases
until !1/2. As a result the wave amplitude in the wavy region "rst increases with increasing "e " and then, when the transition to the regime with re#ection occurs, it begins to decrease and at "e ""3"1.732 it vanishes. In contrast to the regime without re#ection, where the period is independent of "e ", the period in the regime with re#ection is proportional to 1!r and decreases with increasing "e " and "nally disappears at "e ""3. Therefore, in the latter regime, as a rule, for given diode length d and collector potential < several PD solutions exist.
Fig. 2. (a) The electric potential g of the generalized Pierce diode (c"1) as a function of space f for several values of the emitter electric "eld e . Its stability behavior is denoted by di!erent drawings: solid (dashed) lines represent aperiodically stable (unstable) solutions whereas dotted lines mark marginally stable solutions. A transition of the stability is in addition emphasized by an open circle. Here the situation of no re#ection and e '0 is drawn. (b) Same as Fig. 2a, except for e (0. (c) Same as Fig. 2a, except that solutions with re#ection are shown for e '0. (d) Same as Fig. 2c, except for e (0.
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4. Criteria for existence and stability of equilibria without re6ection in the Lagrangian representation 4.1. The Lagrangian formulation The problem, we dealt with in Section 3, namely the establishment of equilibrium solutions, can be alternatively solved in Lagrangian formulation. Restricting the analysis to equilibria without re#ection we will derive an existence criterium and show, that part of the solution can be simpli"ed by a scaling argument. Furthermore, a linear dispersion relation will be derived which solves the linear stability problem of these types of equilibria. Following essentially Ref. [40] and the papers cited therein we "rst transform the Eulerian system (4) into a Lagrangian system. De"ning the stream function q (f, q) by R q "!n, R q "nv , (34) D O we get by a f-integration of the "rst equation
D
n(f, q) df . (35) For "xed q we obtain by inversion of (35) f(q , q). It represents the position of a #uid element at time q which was emitted at time q . Switching to Lagrangian quantities [40] and integrating Poisson's equation with respect to q we "nd (36) f$ (q , q)!(q!q )#cf(q , q)"!e (q) , where dot stands for R . In the steady state the electric "eld on the emitter is time independent. In O generality e will depend on time. The dynamics represented by (36) is controlled by two constraints: q (f, q)"q!
(i) the transit condition f(q , q #¹)"d , and
(37)
(ii) the potential condition
O\2
dq f$ (q , q)f(q , q)"< ,
(38) O where dash stands for d . ¹ is the transit time, the #uid element needs to reach the collector. From O Eqs. (36)}(38) we see how the external control parameters c, d and < enter into the description; d and < are introduced by the boundary conditions whereas c appears in the dynamical equation (36) only. It is worth noting that the Lagrangian formulation is superior to the Eulerian formulation in situations where two or more particle streams with "nite velocity are involved [24]. In the present paper with only one #uid (if we disregard re#ections) both methods work equally well. In the case of equilibrium without re#ection we solve (36) by the ansatz q" : q!q , s(q) " : f(q , q) ,
(39)
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and obtain s(q)#cs(q)"q!e . The solution for s(0)"0 and s(0)"1 is given by (c!1) q e sin(cq) . s(q)" # [cos(cq)!1]# c c c
(40)
(41)
In addition, we have to ful"ll the two constraints (37),(38) which become s(¹)"d ,
(42)
s(¹)"(1#2!=(+cos F(+[1!Z(g)]![1!Z(0)], The value of a/K then becomes with K from (56): a cos F(Z de"ned by (18) and (19) (see also (55)). The essential step in "nding the transformation is to preserve the value of > when c changes from unity to an other value: >(c, e (c)),>(1, i) , where for simplicity we introduce the shorthand notation
(70)
i" : e (1) . (71) Eq. (70) uniquely de"nes e (c), where the parametric c dependence is explicitly pointed out. There are some restrictions on the choice of possible i's, but this will be discussed later. We, however, remind that in the regime without re#ection "i"41 (see (19a)).
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With e (c), given by (70), the new PD(c) is uniquely determined and could be found by (24). However, we now show that the PD(c) follows directly from PD(1) by a shift and a scaling transformation of the coordinates g and f. This transformation will be denoted by g,g(1)Pg(c) ,
(72a)
f,f(1)Pf(c) .
(72b)
The "rst one will be determined by preserving the value of = =(c; g(c)),=(1; g(1)) .
(73)
From conditions (70) and (73) it follows immediately that Z(c; g(c)),Z(1; g(1)) .
(74)
The second transformation is found by the ansatz f(c)"c\[f(1)#S(c)] ,
(75)
where the shift S(c) is needed for the compensation of the g-independent terms in (24), i.e. the terms associated with Z(c; 0). The concrete expressions are then easily found. In the present case r"0, we get from (70) and (18) e (c)"c\[i!(1!c)] . We see that the rhs must be nonnegative which leads to the inequality "i"5"1!c" .
(76)
(77)
This means that for given c only the subset of PD(1) with e (1),i satisfying (77) can be transformed into a new PD(c). If the equality sign in (77) holds, we have e (c)"0, which means that e (c) cannot change sign in the region given by (77). If we also take into account that e "i, we obtain that sgn e "sgn i. From (73) (resp. (74)) and (19) we "nd g(c)#"c\[g(1)#] . (78) In (78) only the parametric c-dependence and not the f-dependence is considered. Substituting (75) into (24) we obtain f(1)#S(c)"sgn e (c)+>(1, i)[(1!Z(1; g(1)))!(1!Z(c; 0))] # [arcsin Z(c; 0)!arcsin Z(1; g(1))], .
(79)
On the other hand, by setting c"1 in (24) we also have f(1)"sgn i+>(1, i)[(1!Z(1; g(1)))!(1!Z(1; 0))] # [arcsin Z(1; 0)!arcsin Z(1; g(1))], .
(80)
Subtracting (80) from (79) we obtain S(c)"sgn i+>(1, i)[(1!Z(1; 0))!(1!Z(c; 0))]#arcsin Z(c; 0)!arcsin Z(1; 0), . (81)
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Furthermore, we have >(1, i)""i",
Z(c; 0)"(c!1)/"i",
Z(1; 0)"0 ,
(82)
and from (81) we get for the shift,
(c!1) c!1 S(c)"sgn i "i" 1! ! #arcsin . i "i"
(83)
If we introduce y"arcsin((c!1)/i), the rhs of (83) can be written as i(1!"cos y")#y, from which it follows that in the present case of "i"41 the sign of S(c) equals the sign of (c!1)/i. Hence, the direction of the shift in f-space is determined by sgn((c!1)/i). sgn S(c)"sgn((c!1)/i) .
(83a)
We start with an arbitrary reference point in the (f, g)-plane on the PD(1), given by (f(1), g(1)). The new reference point (f(c), g(c)) is then given by (75) together with (83) and (78). Hence both, abscissa and ordinate of the new reference point are obtained by a shift and a rescaling. In the latter case, the shift of the potential is found from (78) as (1!c). (By the way, also e (c) follows from a similar type of transformation, as seen from (76).) If we select as an initial reference point the point of re#ection (g(1)"!1/2) then according to (78) the new reference point will again be a point of re#ection (g(c)"!1/2). 5.2. Transformation of states with reyection Now, we lift the situation of PDs without re#ection and include the ones with partial re#ection. In this case the starting equation (24) describes the solution to the left of the "rst minimum, which is the re#ection point. To obtain the appropriate transformation we can proceed as before which means that (70)}(74) are still valid. We merely have to use expressions (18), (19) and (24) with rO0. In principle, r could be a transformed quantity, too. However, we in addition demand that it has invariance properties, too: r(c)"r(1),r ,
(84)
and show its validity. From (70) and (18) we have instead of (76) e (c)"c\[i#r!(1#r!c)] , (85) and from (73) and (19) we obtain (78) again. Eq. (85) represents >"const. If we remain within the class of PDs with partial re#ection of the "rst kind, which means that e (c) is given by (20), then for > we have >(c, e (c))"[1#r(c)] , (86) from which it follows that the constancy of > implies the constancy of r. For this class of PDs, (84) is hence not an additional demand but comes out automatically. However, for the class of PDs of second kind, both conditions >"const. and r"const. are independently needed.
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With rO0 the ansatz (75) can still be made, but the shift will now depend on r. We "nd analogous to the case r"0 S(c)"sgn i+>(1, i)[(1!Z(1; 0))!(1!Z(c; 0))] # (1#r)[arcsin Z(c; 0)!arcsin Z(1; 0)], ,
(87)
extending (81) with respect to rO0. The other quantities are easily obtained. We "nd >(1, i)"(i#r) ,
(88)
Z(c; 0)"[c!(1#r)]/>(1, i) ,
(89a)
Z(1; 0)"!r/>(1, i) .
(89b)
For PDs with partial re#ection of the "rst kind, where i"1#2r
(89c)
which follows from (20), we then get >(1, i)"1#r and the shift becomes S(c)"sgn i(1#r)+(1!Z(1; 0))!(1!Z(c; 0)) # arcsin Z(c; 0)!arcsin Z(1; 0), ,
(90)
where Z(c; 0)"[c!(1#r)]/(1#r) and Z(1; 0)"!r/(1#r). We expect that the transformation obtained so far for the region to the left of the re#ection point can be applied also to the region to the right of it. To show this, we have to look at the second parts of Eqs. (29) and (30) (namely that for r'0). The two conditions (78) and (84) yield ZI (c; g(c)),ZI (1; g(1)) and therefore we "nd c[f(c)!f (c)]"f(1)!f (1) ,
which means that the transformation formula (75) together with the shift remains unchanged. A similar statement can also be made for the case of total re#ection, as seen from formula (33) as well as from the formulas for the regime with partial re#ection with r"1 as obtained in Section 5.2. 5.3. Some properties of the transformation In the case of partial re#ection of the "rst kind, while c(2, the parameter r varies from 0 to 1 and i is determined by r from Eq. (89c). For c'2 we obtain from the positiveness of the rhs of (85) with the substitution of (89c), c/2!14r41 ,
(90a)
where the second inequality is obvious. For c"4 a zone of partial re#ection of the "rst kind vanishes. The allowed range of r is depicted in Fig. 5 by the nonhatched region. In terms of i, de"ned by (71), the inequalities for r (90a) transform to max+1, (c!1),4"i"(3 .
(91)
Fig. 6 shows the allowed range in the ("i", c) plane. In the vertically hatched region there is no correspondence between the solutions for cO1 and c"1, respectively. On the boundary of this region e (c)"0 holds.
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Fig. 5. The region of allowed values of the re#ection coe$cient r and the charge-nonneutrality parameter c, represented by the nonhatched area. Fig. 6. The region of allowed values of "i" (where i is the emitter electric "eld for c"1) and of c, represented by the nonhatched area.
The transformation of coordinates (75) and (78) can be supplemented by the transformation of the transit time. This transformation can be found by the ansatz ¹(c)"c\[¹(1)#S ] . (92a) 2A The formula for the shift S is easily obtained as follows. From (57a) we obtain for cO1 and 2A c"1, respectively, in the notation of Section 4.2, ¹(c)c!f(c)c"!sgn i>(1; i)+[1!Z(c; g(c))]![1!Z(c; 0)], , ¹(1)!f(1)"!sgn i>(1; i)+[1!Z(1; g(1))]![1!Z(1; 0)], . Subtracting the second from the "rst equation and taking into account (74), Z(1; 0)"0 and the expression for S(c) given by (81), we obtain
S "sgn i arcsin Z(c; 0)"arcsin 2A For S
c!1 . i
(92b)
we may also write 2A S "S(c)#sgn i[i!(1!c)]!i , (92c) 2A where S(c) is given by (83). To summarize this section, we have shown that the equilibrium states of the nonneutralized beam plasme diode can be obtained by a linear transformation of the states of the generalized Pierce diode (c"1; < arbitrary). The transformation is given by (75) and (78) in all cases. For di!erent situations only the shift has to be adapted since it generally depends on c and i, as seen from (81), (83), (87) and (90). Due to the linearity of the transformation extrema are mapped onto itself and each subregion, in which the potential is monotonic, has its counterpart after transformation. Each peculiarity of the
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35
solution is hence preserved. Furthermore, a re#ection point keeps its value and remains a re#ection point after transformation. An example of the transformation is given in Figs. 7a}f, where it is shown how a certain piece of the source PD of the generalized Pierce diode (c"1; Figs. 7a, c and e) is mapped into the corresponding PD of the nonneutralized diode for c"0.8 and c"1.2,
Fig. 7. (a) The source PD of the generalized Pierce diode (c"1) as a function of space with i"0.2 which is a boundary of the transformation in the nonre#ective regime. (b) The PDs after the transformation of the source PD for two values of c. (c) Same as Fig. 7a, except for i"0.5 lying now inside the allowed range. (d) Same as Fig. 7b. (e) The source PD in the regime with partial re#ection of the "rst kind, r"0.3 and i"!1.265. (f ) Same as Fig. 7b.
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respectively (Figs. 7b, d and f ). The transformed PDs agree with the curves calculated from Eqs. (24), (27)}(30). The beginning and the end of each piece of a PD are marked by a thick point. Note that in the case of a positive shift S(c) we need a piece of the source PD from a nonphysical region, !S(c)4f40, in order to obtain the piece of a PD of a diode with cO1 in the physical region located between the emitter and the left thick point. Figs. 7a and b demonstrate a mapping at the boundary of transformation (see Fig. 6) for the nonre#ective regime. As it follows from (77), the value of the boundary emitter "eld strength i"0.2 is the same for both c values and is transformed to e (c)"0. According to (83a) a shift of the origin turns into a positive position at c"1.2 and into a negative one at c"0.8. The topological similarity of the PDs is conserved for all cases. Figs. 7c and d represent a mapping again for the nonre#ective regime but inside the allowed range, i"0.5. In the case of a partial re#ection of the "rst kind, Figs. 7e and f, the value of the potential minimum is conserved being equal to !1/2.
6. Classi5cation of potential distributions (PD): (g, e)-diagrams and I+V characteristics In the previous section we have shown that for the construction of all possible potential distributions (PD) for a diode with arbitrary c, it is su$cient to know the PDs of the Pierce diode (c"1) for all values of the emitter electric "eld i,e (1). All the PDs with arbitrary c can be obtained from these solutions by a simple transformation of the coordinates, given by (75), (76) and (78). Therefore, the investigation of the Pierce diode for all values of i is a central issue. In Section 6.1 we solve this problem for the generalized Pierce diode in which both i and \+ce [>#=(!=((e ) (e #c)>(e ) e #c Here >(e ) is de"ned by (55a). We shall mark this boundary line as A . Fig. 18a shows an example of such a line for c"1. The line A intersects the line e "0 in the point d"pc\, where the collector potential < turns out to be 2(1!c)c\. As expected, at c"1 (see Fig. 18a) the abscissa of this point coincides with the threshold of the aperiodical instability for the classical Pierce diode [19] (at e "0, (e ) >(e )
if e '0 ,
if e (0 .
(120)
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Fig. 21. An (g, e)-diagram for c"1 and d"4.5p.
Using (18) with r"1 for >(e ) and (32) we can calculate the derivative d(e ) We shall show that for e '0 the derivative d; in addition unlike the classical Pierce diode the collector potential < is varied. The three main results we have achieved are: 1. a complete classi"cation of the equilibria, 2. a transformation, which allows to deduce nonneutral equilibria (cO1) from generalized Pierce diode equilibria (c"1), 3. an extended, almost complete, stability scenario. The classi"cation distinguishes solutions without and with electron re#ection and furthermore discriminates between solutions in which the potential minimum (`virtual cathodea), re#ecting a fraction of electrons back to the emitter, is located inside the gap (partial re#ection of "rst kind). In this case, the re#ection ratio r, is related to the transmitted current I by I"1!r. It varies between 0 and 1. For completeness, also the case of total re#ection has to be considered. The transformation is of self-similar type and casts a generalized Pierce diode solution onto a nonneutral equilibrium solution. Hence, it is su$cient to know the former one in order to obtain a solution for a nonneutral diode with an arbitrary c. The transformation makes use of a Cauchy boundary value problem with given injection conditions. Therefore, e (c) is found by the given i"e (1); then, d(c) is found by d(1) and "nally islands. The distinction can be caused, for instance, by di!erent heating of the electron gas. For
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simplicity, we assume the electron temperature to be equal in all the islands. Then the function U (E, ¹) can be taken as a Fermi function with an e!ective temperature h (expressed in energy GG> units), which depends on the power pumped into the island:
f (E)" exp
\ E#u . #1 h
(7)
Substituting (6) and (7) into Eq. (2) gives I"A
E#(*;/2) 1#exp ! e \
\
1#exp
E#u h
\
dE
(8)
with A"4pme*;/(2p ). To obtain an explicit dependence of the conduction current on voltage, temperature and other parameters, it is necessary to "nd the relationship between the electron temperature, the lattice temperature and the power fed into the "lm by passing current. The peculiarities of the dissipation of the energy of electrons in small particles will be treated at length in Sections 5 and 6. Here we note only that the motion of electrons is ballistic in the particles whose size is smaller than the electron mean free path in the bulk metal. For this reason electrons lose their energy in collisions with defects and with the island as a whole when they are re#ected from the potential barrier. The re#ection is almost elastic and the transferred momentum equals *p"2p, where p is the electron momentum. Thus the transferred energy is *E"Em/M, where m and E are respectively the mass and energy of electron and M is the mass of the atom (or the island). It should be noted that the factor m/M, which determines the part of the energy lost by electron in a collision, remains somewhat uncertain. However, this fact need not concern us, since the factor enters into a phenomenological coe$cient which may be determined from experiment. The number of collisions per unit time equals 1/q"v/a, where v is the electron velocity and a is the island dimension. Hence the power transferred from an electron to the lattice is *P"*E/q&((m/M)(E/a) .
(9)
The total power transferred by electrons to the whole "lm can be obtained by averaging (9) over the nonequilibrium component of the electron energy distribution and multiplying the result by the volume of the islands. Under steady-state conditions, this power must be equal to the power fed into the "lm P"I;, where ; is the applied voltage. Hence the following expression can be obtained for the electron temperature: (10) h "k¹ "(h #aHI; . Here h is the lattice temperature in energy units and aH is a coe$cient which is independent of the "eld and temperature and determines the e$ciency of electron heating. In the case e 'h , the current is carried mainly by electrons tunneling near the Fermi level, because the barrier is rather steep and its transparency depends on energy less sharply than the electron energy distribution function. Then the expression for the current becomes I"Ae exp(!u/e )+1#(p/6)(h /e ), exp(*;/2e ) ,
(11)
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where u is the work function. In the opposite case e (h , when the transparency depends on energy stronger than the distribution function, the current is carried predominantly by electrons tunneling near the top of the barrier and we obtain I"Ah exp(!u/e )+1#(p/6)(h /e ), exp(*;/2e ) . (12) Thus, the maximum contribution to the tunneling current belongs to the group of electrons which are located at some distance from the Fermi energy, depending on the shape of the barrier (characterized by the parameter e ) and electron temperature h . If h is low, the tunneling occurs just above the Fermi level. As h increases, the maximum of the tunneling #ux is shifted to the top of the barrier. Some contribution to the conduction current can also originate from overbarrier electrons, but their part seems to be comparatively small (since the emission current is found to be much smaller than the conduction current, see Section 3.3). All these factors cause the deviation from the Ohm law. In work [64], we proposed a method for determination of the model parameters e and aH, and hence of the electron temperature, from the dependences of the conduction current on the applied voltage ("eld) and temperature. The typical electron temperature corresponding to beginning of the deviation of conduction current}voltage characteristics from linearity was found to be about 0.15 eV (+2;10 K) (Fig. 3.1). This model not only explains the variation of the conduction current versus electric "eld, temperature and work function, but also allows one to understand the main regularities of electron emission from the island "lms, which will be discussed in the following sections. To conclude, a few additional comments are in order. Proposing the model of electron heating described above, we were guided by the following considerations. The deviation of the conduction current from Ohm's law and the occurrence of the electron and light emission (discussed in detail below) set in at approximately the same voltage applied to the "lm. This suggests that these phenomena should have a common cause. Such a cause may be the generation of hot electrons. Taking into consideration the weak temperature dependence of the conduction current in the Ohmic region, we have assumed the tunnelling conduction mechanism. In the case of the thermionic mechanism, the conduction current would be strongly dependent on temperature, which is not observed. The next comment is connected with the work function of the islands. We have assumed that the work function as well as the Fermi energy are the same in all the islands. However, this assumption can be not valid for very small islands where the e!ects of the Coulomb blockade can be essential. Generally, such e!ects should not be pronounced in the conductivity of two-dimensional ensembles of islands at room temperature. It is known (see e.g. [71]) that two major prerequisites must exist for the observation of the Coulomb blockade: (1) one should provide the condition e/2CIX X , U!(r)"$ V 2mu Rx
(37)
where the function u!(x) has the following form:
de\ OV, x40,
x50, e $ u5; , V ge\AV, x50, e $ u4; . V In (38) we have used the notations u!(x)" ge AV,
(38)
2m (e $ u) ,
V
(39)
2m "e $ u!;" .
V
(40)
q"
c"
Since we are considering the radiation, it is natural to assume that e ' u. The unknown V coe$cients d, g, g in Eq. (38) can be determined from the continuity condition for the function U!(r) and its derivative at x"0 [122]. In doing so we can "nd an explicit expression for the V function U!(r) and hence calculate the required probabilities of the inelastic transitions. Indeed, V taking into account (37) and (38) and substituting function (30) into the expression for the current density
Rt RtH e
tH !t , I " V i2m Rx Rx
(41)
one can calculate the elastic and inelastic (i.e. connected with u!(x)) component of the current density. Relating then the inelastic component to the incident #ux (I" k /m), we obtain the V V probability of the inelastic transitions
IL 2e q D!(e , u)" V " A "d" . V V I c
k V V After the determination of d and its substitution into (42), we arrive at the result
(e (e $ u), V V 2 2eA (eV (eV # u)C(eV # u), V D!(e , u)" V mu c
(e (e ! u)C(e ), V V V (e (e $ u)C(e $ u), V V V
e (;, V e 4;, V e 5;, V e 5;, V
e $ u4; V e # u5; V e ! u4; V e $ u5; V
(42)
, , , .
(43)
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The function C(e ) has here the form V ; C(e )" . V ((e #(e !;) V V
(44)
In (43), the case e 4;, e $ u4; describes the usual bremsstrahlung for the `#a sign and V V the inverse e!ect for the `!a sign. The case e 4;, e # u'; corresponds to the surface V V photoe!ect and the case e 5;, e ! u(; to the inverse surface photoe!ect. Finally, the V V situation e 5;, e $ u5; represents a usual inelastic scattering at the potential barrier. V V Before addressing the problem of the spectral density of the radiation, let us brie#y turn to the inelastic tunneling between the islands. Instead of Eq. (16), one has in this case
x40 ,
0,
;(x)" ;, 04x(a , 0, x'a .
(45)
The solution procedure remains the same with the di!erence that, instead of Eqs. (33) and (34), one uses the functions corresponding to potential (45) and the functions u!(x) from (38) must be replaced by
x40 ,
de\ OV,
u!(x)" f eAV#f e\AV, 04x4a , ge OV, x5a .
(46)
The details of the calculations can be found in Appendix A. In addition, the procedure of joining the functions and their derivatives should be carried out not only at x"0, but also at x"a . In this case the probability of the inelastic scattering with absorption (the sign `#a) and radiation (the sign `!a) of photons is given by
2e q D!( e , u)" A "g" . V c V k V
(47)
Here
"g""4
; k V +ci (ch ca !ch i a )#(i qsh ca !ck sh i a ), V
u G(i , k )G(c, q) V
(48)
and sh x, ch x are respectively hyperbolic sine and cosine. We also have introduced the notation G(i k )"(i !k)shi a #4i k chi a . V V V
(49)
The above method of calculation, within a uni"ed approach, of all processes connected with the inelastic electron re#ection from the barrier and the inelastic tunneling was "rst used to this end in [122].
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The radiative transitions considered up to this point correspond to transitions induced by the external "eld. To obtain the probabilities of spontaneous transitions which we need, we are using the following arti"cial expedient [168]. Let us choose such a normalization of the vector potential of the electromagnetic "eld that the "eld quantization volume < will contain in the average N photons characterized by the energy u and a given wave vector and polarization. Under such conditions one obtains
2p N . (50) < u To calculate the probability of spontaneous transitions, it is necessary to substitute (50) into (42) (or (43)) and (47) and then to set N "1. In this way we shall obtain the probabilities of spontaneous transitions into the "eld state with a given polarization, frequency u and wave vector q . To "nd the full probability of radiation in a unit frequency interval and a solid angle dX, the probabilities obtained above should be multiplied by the density of the "nal state of the "eld equal to A"c
< < q dq do(u)" dX" u dX . (2pc) (2p )du
(51)
The total probability of spontaneous radiation in inelastic electron re#ection from the barrier or the inelastic tunneling is given by (52) = "[D\(e , u)] ) do(u) . V , The spectral density of radiation of all electrons into a solid angle dX from the surface area S equals
2 uS dE(u, X)" dp v = (e , u) dp dp f (e)[1!f (e! u)] V V V W X (2p ) V
CYS \ \ 2mh Su[exp( u/h )!1]\ ) de =(e , u)Z(e u) . (53) " V V V (2p ) CV Y S In this expression, f (e) is the Fermi function with an e!ective electron temperature h "k ¹ and Z(e , u) is V 1#exp((k# u!e )/h ) V , (54) Z(e , u)"ln V 1#exp((k!e )/h ) V where k is the Fermi energy. Having the explicit expressions for the probability of inelastic electron re#ection from the barrier (43) and of the inelastic tunneling (47), it is easy to calculate, using formula (53), the spectral density of radiation in any speci"c case of barrier parameters, frequency range etc. Unfortunately, a simple analytical expression for the general case does not exist. It can be obtained only in various limiting cases. For instance, in the frequency interval h ( u(u, k , the total spectral density of radiation in all directions is
2e 2 E(u)"S
ue\ SF k(k# u)! ( u) . 3(2c ) 3
(55)
(56)
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In the case speci"ed by condition (55) the main contribution to the radiation stems from bremsstrahlung of hot electrons occurring in their collision with the surface barrier. It is instructive to compare the above quantum mechanical treatment of the bremsstrahlung with its classical description. The total (to all directions) power of bremsstrahlung from a charge moving with an acceleration v is known to be dE 2 e " v . dt 3 c
(57)
Hence the total energy radiated during the time of the acceleration is
dE 2 e . (58) dt" du v e\ SR dt dt 3p c \ \ Since the change of the velocity of an electron in its collision with a step-like barrier occurs jumpwise, the value ut in the exponent of formula (58) can be neglected, which gives a simple expression for the spectral density of radiation: E"
2 e "*v" . E(u)" 3p c
(59)
Here *v is the change in the velocity of an electron in its re#ection from the barrier (*v"2v ). Thus V we arrive at the result 16 e 8 e v " e . E(u)" 3p c V 3p mc V
(60)
In the quantum mechanical treatment, expressions (52), (43) and (50) give for the contribution of an electron to the spectral density of the bremsstrahlung 2e (e (e ! u) dX V V .
u= "(cos h) m pc
(61)
Here h is the angle between the vector of the electric "eld in the radiated wave and the normal to the surface. The integration of (61) over all angles under the condition e < u gives the expression V that is exactly coincident with (60). In closing this section let us address brie#y the radiation generated in the inelastic tunneling. In the case i a '1 , (62) relations (47) and (52) predict that the probability of the tunneling accompanied by the radiation of a quantum u equals 4e
(e (e ! u)i e\? G cos h dX . =" pmc; V V
(63)
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Hence, the spectral density of this radiation into a solid angle dX is
2eh u 3 cos hdX de (e (e ! u)eI\CV F ) i e\? G . dE(u, X)+S (64) V V V pm c;
S It should be kept in mind that the integrand in (64) is not quite accurate at both the lower and the upper limit of integration. The inaccuracy at the lower limit is due to the necessity of taking into account Pauli's exclusion principle for the states with e (k. The inaccuracy at the upper limit is V caused by the fact that condition (62) is not obeyed at e P;. However, these inaccuracies are V partially compensated by other physical factors and for this reason cannot signi"cantly in#uence the value of integral (64). Indeed, the inaccuracy of the approximation at the lower integration limit is partially counterbalanced by the very low tunneling probability (because of the inequality a i IJ X . u, s" , , J IJ \ Substitution of (68) in (66) yields the following equation for s: 1 K Rs #u(k)s"! e\ IJ (R , 8p o¸ Rt
(68)
(69)
where u(k)"s(k #k). , J The electron energy losses due to the generation of lattice vibrations are given by [164]
Rs Ru de "K
d(r!r(t))dr"K dk (k #k) e\ IJ (R , , , J Rt Rt dt IJ R "0 . u Rt X X!*
(70)
(71)
Here 2¸ is the size of the system along the z-axis. The solution of Eq. (69) can be written as
R K dt e\ IJ (RY sin[u(k)(t!t)] . s"! 8po¸u(k)
(72)
Substituting this solution into (70), we get
de R K "! dk (k #k) dt cos [u(k)(t!t)]e IJ (R\(RY . , , J dt 8po¸
(73)
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Assuming now that the electron oscillating motion is described by (t) and taking it in the simplest form
(t)"¸ sin
v t , ¸
(74)
we can easily estimate, within the context of (73), how the "nite size of the system in#uences the electron}lattice energy exchange. First of all, we shall show that for ¸PR, (73) yields the known equation for hot-electron energy losses in an in"nite medium [164]. Note that for ¸PR, tPR, we have
R
p dt cos [u(k)(t!t)]cos [k ( (t)! (t))]" +d[k v#u(k)]#d[k v!u(k)], . J J J 2
When deriving the latter equation, it is important to keep a proper order in proceeding to the limits ("rst ¸PR, then tPR). As a result, we obtain from (73) (for ¸PR, tPR):
K de "! dk ku(k)d[(kv)!u(k)] . 8po¸ dt
(75)
The result, given in Ref. [164], follows from (75) immediately. Now let us consider a "nite system. We employ the following expansion: e\ IJ (R"e\ IJ * TR*" J (k ¸)e\ LTR , L J L\ where J is the Bessel function. This expansion makes in possible to reduce (73) to the form L J (k ¸) de K L J "! dk (k #k) , , J n v/¸$u(k) dt 8po¸ IJ L >\ v ; sin n t!k (t) # sin[$u(k)t#k (t)] . (76) J J ¸
Hence, we see that de/dt is a rapidly oscillating function for tPR. If v/¸'max u(k),u (u " " is the Debye frequency), i.e. in the absence of resonances, then the mean value of this oscillating function tends to zero since the averaging procedure implies the operation 1/tR dtde/dt for tPR. Thus, we have shown that the electron motion under consideration is not accompanied by the energy losses associated with the Cherenkov generation of acoustic lattice vibrations, though the latter mechanism of energy dissipation is dominant in bulk metals. In the case of an in"nite metal, the expression for the total losses (due to all hot electrons) can be derived by multiplying (76) by the number of electrons with energies exceeding the Fermi energy (only such electrons can generate lattice vibrations). As a result, this expression can be reduced (see Ref. [164]) to the form a(¹ !¹) where p ms a" nl (77) 6 ¹ (n is the electron concentration).
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For small metal particles, whose characteristic size ¸ satis"es the inequality v /¸'u $ "
(78)
(where v is electron velocity on the Fermi surface), these bulk losses have been shown to vanish. $ 6.3. Surface vibrations of small particles Since we will be discussing metal particles with dimensions on the order of or less than the mean free path, we can use the following model to calculate the surface energy exchange. The electron gas is in a spherical potential well of radius R (if thermal vibrations are ignored) and height < . This model was used in Ref. [170] to study optical absorption in island metal "lms. As we have already mentioned, the reason for the energy exchange is an interaction of the electron with thermal vibrations of the surface. These vibrations can be classi"ed somewhat crudely as either shape vibrations (so-called capillary vibrations), in the course of which the volume does not change, or surface vibrations, which are accompanied by a change in density (acoustic vibrations). A theory for the surface vibrations of a spherical particle is set forth in detail (for the case of vibrations of the surface of an atomic nucleus) in Ref. [171]. We begin our analysis with the capillary vibrations. We expand the radius of the vibrating surface in spherical harmonics > (h, u): HI R(h, u)"R
1# a > (h, u) . HI HI HI
(79)
The Hamiltonian of the capillary vibrations can then be written in the following form, in accordance with Ref. [171]:
1 1 "p " HI #D u "a " . H" +D "a "#C "a ",, H HI H HI H H HI 2 2 D H HI HI
(80)
Here p "D aH is a generalized momentum, and u "(C D ) is the frequency of the HI H HI H H H capillary vibrations. The constants D and C depend on the island dimensions in di!erent ways. H H According to Ref. [171], they are given by D "MnR /j, H
C "p R (j!1)(j#2) . H
(81)
Here M is the mass of the atom, n is the density, and p is the surface energy. It can be seen from (81) that the frequency of the shape vibrations depends strongly on the radius of the metal island, R :
C (j!1)j(j#2) " p . u " H H D MnR H
(82)
For the discussion below we will take a quantum-mechanical approach in which p and a HI HI are replaced by corresponding operators, which are related to the operators of creation (b> ) and HI
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annihilation (b ) of surface phonons by HI p( "i(D u /2)(b !b> ) , (83) HI H H HI HI
(b #b> ) . (84) a( " HI HI HI 2D u H H After this replacement, the Hamiltonian of the capillary vibrations takes the standard form:
1 (85) HK " u b> b # . H HI HI 2 HI To "nd the electron}phonon energy exchange, we need an explicit expression for the corresponding Hamiltonian. According to the model adopted above, the potential energy of an electron in a metal island is =(r)"< *(r!R(h, u)) , where 1, x'0 , *(x)" 0, x(0 .
(86)
(87)
Using expansion (79) for R(h, u), we "nd from (86) =(r)+< *(r!R )#d(r!R )< R a > (h, u) . (88) HI HI HI The second term in (88) describes the energy of the electron}phonon interaction associated with the surface vibrations. Writing this term in the second-quantization representation [using (84)], we "nd
2(b #b> )a> a . (89) HK "< R 1tH "d(r!R )>H (h, u)"t JYLYKY HI HI JLK JYLYKY JLK HI 2D u H H The operators a> and a in (89) create and annihilate an electron in the corresponding state. JLK JYLYKY The meaning of the subscripts on these operators becomes clear when we recall that the electron wave function in a `spherical potential square wella is 1 t (r)" R (r)> (h, u) . (90) JLK JK C J JL Here C is a normalization factor, and the radial wave function is JL j (k r), for r(R , R (r)" J JL (91) J h(iK r) for r'R . J JL The quantity j (x) in (91) is the spherical Bessel function, and h(x) is the spherical Hankel J J function. In addition,
2m e , k " JL
JL
2m (< !e ) K " . JL JL
(92)
Here m is the mass of an electron, and e is the energy of the electronic levels in a spherical JL square potential well. These conditions are found from the condition for the joining of the electron
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wave function and its derivative at the point r"R In view of the rapid decay of the electron wave function inside the barrier, we write k in the following form, as in Ref. [170]: JL k "k #*k , (93) JL JL JL where k are the roots of the equation JL j (k R )"0 . (94) J JL Eq. (94) corresponds to the case of an in"nitely deep potential well. Assuming that *k is small in JL comparison with k (this point is easily checked), we "nd the following result from the condition for JL the joining of the wave function and its derivative at the point R : *k "!k /R K . (95) JL JL JL Here we have used the asymptotic expression
ip 1 exp !Kr! (l#1) , r'R . R (r)+ JL 2 iKr
(96)
Since we will be interested below in the electron levels near (and above) the Fermi energy, we can use the method of Ref. [170], "nding approximate solutions of (94) through the use of the asymptotic representation of the spherical Bessel function: p (2n#l) . (97) k " JL 2R Now, in accordance with (93), (95), and (97), we have an explicit expression for k . Consequently, JL the electron wave functions in (90) and (91) have been determined completely. Using them, we can put the Hamiltonian for the electron}photon interaction, (89), in the form
2
e e p p JL JYLY H "< du dh sin h> ) (h, u) D u (< !e )(< !e ) JL H H JL JYLY ;> (h, u)(b #b> )a> a . JYLY HI HI JLK JYLYKY
(98)
6.4. Surface electron}phonon energy exchange Now that we have explicit expressions for the electron and photon spectra and also for the Hamiltonian of the electron}photon interaction, we can move to the problem of determining the electron}photon exchange. This exchange can be taken into account systematically by a kineticequation approach. For brevity, we will be using the notation l"+l, n, m,,
q"+j, k, .
(99)
The change per unit time in the distribution of electrons among states caused by the scattering of electrons by phonons is then given by Rf JJ ,If " = +[(N #1) f (1!f )!N f (1!f )]d[e !e # u ] JJ JJYO O JJ JYJY O JYJY JJ JY J O Rt JYO #[N f (1!f )!(N #1) f (1!f )]d[e !e ! u ], . O JJ JYJY O JYJY JJ JY J O
(100)
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Here f "1a>a 2 is the electron distribution function, and N "1b>b 2 the phonon distribuJJ J J O O O tion function. As usual, the angle brackets mean an average over the statistical operator. Furthermore, in our case we have e "e , u "u , i.e., the spectrum is degenerate. It is simple matter to J JL O H derive an explicit expression for the transition probabilities, by "rst writing the interaction Hamiltonian (98) in the compact form HK " C [b #b>]a>a . JJYO O O J JY JJYO
(101)
Then 2p = " "C " . JJYO
JJYO
(102)
The energy transferred from the electrons to the phonons per unit time is RE R " e f " e If . J JJ Rt Rt J JJ
(103)
We note that the electron distribution in a metal island, f , depends on only the electron energy: JJ F "f (e ). JJ J Treating the phonon system as a heat reservoir (with respect to the electron subsystem), we take the phonon distribution function N to be Planckian with a temperature ¹. Expanding collision O integral (100) in a series in the small quantity u (i.e., actually expanding in the ratio of the phonon O energy to the Fermi energy), we "nd the following result for expression (103):
f (e )[1!f (e )] RE J JY d(e !e ) . + = N ( u ) (104) JJYO O O JY J k ¹ Rt JJYO We can now write an explicit expression for the electron distribution function. Because of the intense electron}electron interaction, the power acquired by the electron subsystem from the external source becomes distributed among many electrons rapidly. As a result, a Fermi distribution with some e!ective electron temperature ¹ , is established: e !e \ $ #1 , (105) f (e )" exp T T k ¹ where e is the Fermi energy. Substituting (105) into (104), we "nd $ ¹ RE " !1 = N ( u )(e !e )d(e !e ) . (106) JJYO O O JY J J $ ¹ Rt JJYO To pursue the calculations we need to use the explicit expression for = which follows from JJYO (102) and from a comparison of (98) and (101):
4p = " JJYO (< !e )(< !e ) M JL M JYLY
p p du dh sin h> (h, u)> (h, u)> (h, u) . JL JYLY HI
(107)
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Substituting (107) into (106) we "nd that an explicit dependence of the integrand on the indices characterizing the electron states remains only in the spherical harmonic, because of the presence of the function d(e !e ),d(e !e ) in the integral. We can thus sum over the electron indices (96). J $ JL $ In doing so, we make use of the orthogonality of the spherical harmonics: J d(u!u)d(h!h) . (108) > (h, u)> (h, u)" JK JK sin h J K\J In our case the summation over l is bounded by the condition e (e . This circumstance does JL $ not introduce any signi"cant error, however, since the maximum value of l is large, l &10. This
estimate of l follows from the relation
pl 1
e " . (109) $ 2R 2m As a result of these calculations, we "nd from (106)
RE ¹ R m e < u N $ H H . + !1 (110) D Rt ¹ 2p
u H HI Here u "< !e is the work function of the metal, and N "N(u ) is the Planckian $ H H distribution function of the capillary vibrations. We are left with the task of evaluating the phonon sum in (110): H H u N(u ) k ¹ H 2j#1 H + H . (111) D
D H H H \H H In (111) we have recognized that the energy corresponding to the Debye frequency of the capillary vibrations is considerably smaller than k ¹ (at room temperature). It follows from (82) in this case that
p u "u + j . " H
MnR M As a result of these calculations we "nd
(112)
RE 4pR 3 v m u < k (¹ !¹) $ n " . (113) + Rt 3 u 16p R p In the literature, the power transferred from the electrons to the phonons is customarily written in the form
RE 4pR a(¹ !¹) . " Rt 3
(114)
Here we have assumed that the particle is a sphere in our case. The constant a, which is a measure of the rate of the electron-phonon energy exchange, is given in our case by
3 v m u < a" . k $n " 16p R u p M
(115)
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Let us evalute this quantity for a gold particle (a sphere) with the following parameter values: n"6;10 cm\, v "10 cm/s, R "10\ cm, p "10 erg/cm, u "3;10 s\ [165], and $ " (< /u )"5. We "nd a"2;10 erg/(cm s deg). The value found for a is two orders of magnitude lower than the corresponding value in bulk metals. An experiment carried out to determine a in small particles has con"rmed this estimate [172]. A reduction in the intensity of electron}phonon interaction in small particles has also been found experimentally in another recent work [173]. In this work, a somewhat di!erent system has been studied than in our case (a dielectric core surrounded by an ultrathin Au shell). As we mentioned earlier, in addition to the shape vibrations (the capillary vibrations) of the particles there are surface vibrations which do involve a change in density (acoustic vibrations). The dispersion relation for these phonons is u "k s , (116) LH LH where s is the sound velocity, and the wave vector k is determined by the roots of the equation LH j (k R )"0 . (117) H LH The interaction of the electrons with these vibrations can be dealt with by an approach like that taken above. As a result we "nd the following expression for the value of a determined by the surface acoustic vibrations:
1 nv m u < . (118) k $ " a+ u 16p R o s Here u is the Debye frequency of the acoustic vibrations, and o is the density of the material. " An estimate of a from (52) for the same gold particles as discussed above yields a value an order of magnitude smaller than the result in (49). Consequently, the interaction with capillary waves is predominant for these particles. We would simply like to point out that the idea of classifying the vibrations as either capillary or acoustic is valid only if u and u are substantially di!erent. This " " condition is satis"ed in the case under consideration here. 6.5. Derivation of the equation describing the sound generation by hot electrons The Hamiltonian of the interaction of an electron, residing in point r, with atoms of the lattice can be written as R H " + SR, .
(A.3)
This relationship allows Eq. (A.1) to be represented as 2e AU!(r)e . (A.4)
>\ Equations for functions U! can be obtained by substituting (A.4) into (A.2) and equating the G terms that contain the same components A. In particular, the equation for U! is G G t(r, t)"i
2m R D# (e$ u!; ) U!" t(r) . V V
Rx
(A.5)
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Consider now in more detail the case of a rectangular barrier along x having a width a and height ; (see Eq. (45)). With this barrier, one obtains t(r)"t(x)e IW W>IX X
e IV V#R e\ IV V, x40 , ,e IW W>IX X B eG V#B e\G V, 04x4a , x5a . C e I V,
(A.6)
The coe$cients R , B , B and C are determined from conditions of joining the function (A.6) and its derivatives in the points x"0 and x"a . The procedure of the joining gives a system of algebraic equations which determine the unknown coe$cients: 1#R "B #B , ik (1!R )"i (B !B ) , V B eG ?#B e\G ? "C e IV ? , i (B eG ? !B e\G ? )"ik C e IV ? . V
(A.7)
Considering the structure of function (A.6), the dependence of U (r) on the variables y and z can V be separated into an isolated factor: U (r)"u(x)e IW W>IX X . V
(A.8)
The substitution of (A.8) into Eq. (A.5) gives du 2m # (e $ -)u(x)"ik (e IV V!R e\ IV V), x40 , V
V dx du 2m # (e $ -!;)u(x)"i (B eG V!B e\G V), 04x4a , dx
V
(A.9)
du 2m # (e $ -)u(x)"ik C e IV V, x5a . dx
V It can be easily checked by immediate substitution into (A.9) that its solution can be represented as
Rt(x) u(x)"$ #u!(x) . 2mu Rx
(A.10)
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Here the shape of u!(x) is given by formula (46). The "rst term in (A.10) is a partial solution of the nonuniform system of equations (A.9) and u!(x) gives a solution of the uniform system. The conditions of joining the function (A.10) and its derivative in the points x"0 and x"a give a system of algebraic equations which determine the coe$cients d, f , f , and g: d"f #f , f eA? #f e\A? "ge O? ,
(A.11) ; (c#iq) f #(!c#iq) f "G (B #B ) ,
u ; (c!iq)eA? f !(c#iq)e\A? f "G e IV ? C .
u The probability of inelastic tunneling is determined by the coe$cient g and that of the inelastic re#ection from the barrier by the coe$cient d. Let us illustrate this statement for the case of tunneling. As can be seen from Eqs. (A.1), (A.4), (A.6),(A.8) and (A.10), the electron wave function at xPR is given by
2e A t(r, t)"C e> IV e IWW>IXXe\ C R#i c V
$ C ik e IV V#ge OV 2mu V
;e IW W>IX Xe\ C! SR .
(A.12)
E It should be recalled that the summands U!(r) and U!(r) do not contribute to the inelastic W X current if the dispersion of the electromagnetic wave is not considered. By substitution of function (A.12) into expression (41) for the current I we obtain V
e k e 2eA 1 k
k V V $ V #1 "C " . I " V "C "# q"g"# V m m mc 4 mu mu
(A.13)
The cross-contribution of the "rst and second terms in (A.12) to the current vanishes when averaged over the wave period. The "rst summand in (A.13) determines the probability of the elastic tunneling. In the braces, the "rst term accounts for the probability of the inelastic tunneling while the second one gives a correction to the probability of the elastic tunneling due to the presence of the wave "eld. The intelastically re#ected current can be calculated similarly to (A.13). In this case, "d" appears instead of "g" in the expression for the current. The coe$cients g and d are found from (A.11) with allowance for (A.7). The expression for "g" is given by (48). For "d", we obtain
; k V +[ci !ci ch ca ch i a #qk sh ca sh i a ] V
u G(i , k )G(c, q) V # [ck ch ca sh i a #(i q ch i a sh ca )], . (A.14) V The calculation method presented above can easily be generalized for the case of an asymmetric barrier. "d""4
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Appendix B As suggested in Section 7, the mutual e!ect of metal islands on the local "eld strength in a given island can be formally accounted for by replacing the depolarization factor ¸ with ¸ !*¸ . H H H Below, we will "nd the explicit form of *¸ for a particular model. H It should be noted that the problem of the exact determination of the local "eld presents considerable di$culties even in the simplest case of two identical spherical particles (see e.g. [229,230]). Usually, an approximation is applied in which the electrostatic potential induced by an island is expanded as a power series in multipoles. As a rule one restricts the consideration to the lowest multipoles (most often, the dipoles). Such an approximation is not very appropriate for an island "lm in which the distance between the islands is of the same order as their size. In what follows we describe a method of the calculation of the local "eld which does not apply the multipole expansion [231]. Consider a linear periodic chain consisting of identical metal islands. The mutual e!ect of the islands on each other will be maximum when the external electric "eld E is parallel to the chain axis. We assume just such an orientation of the "eld. Since both the size of the islands and spacings between them are in our case much smaller than the length of the incident wave, the problem of the determination of the resulting "eld reduces to the solution of the Laplace equation with appropriate boundary conditions: *u (r)"0 , u>(r)" "u\(r)" , (B.1) Ru> Ru\ " . e Rn Rn The signs (#) and (!) correspond here to the limiting values of the function u inside and outside the surface S, respectively, and n is the outward normal to it. The dielectric susceptibility of the medium is taken equal to 1, and that of the islands to e. The solution of Eq. (B.1) can be represented as
ds o (r) . (B.2) u (r)"!Er# "r!r!ak" I\ Here o (r) is the surface charge density which, with the account for the assumed periodicity of the chain, obeys the following integral equation [232]:
cos h 1 1!e 1!e I ds o (r) "! En . (B.3) o (r)# 2p"r!r!ak" 2p 1#e 1#e I\ h is the angle between the vectors r!r!ak and n , where n is the outward normal to the I PY PY surface in the point r and a is a vector connecting the centers of two adjacent islands. The direction of the vector a to one or another side along the chain plays no role, since the summation is carried out over all k's.
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Eq. (B.3) has an exact solution for the case of a sungle island, i.e. for k"0 (or aPR): 3 e!1 o" En . 4p e#2
(B.4)
The substitution of (B.4) into (B.2) with retaining only k"0 gives the known result for a dielectric sphere in a uniform external electric "eld. It will be recalled once again that at frequencies much lower than the plasma frequency, a conducting metal island behaves as a dielectric. In a linear chain of identical spherical islands, the surface charge density o (r) depends only on the angle h between the "eld E and the radius vector r in the point on the sphere surface. Therefore o can be expanded in the general case into a series o " C P (cos h) . K K K
(B.5)
As it is clear from the symmetry considerations, Eq. (B.5) contains only odd Legendre's polynomials. By substitution of (B.5) into (B.3), multiplying the result by P (cos h) sin h and then by L integration over all h's, one obtains a system of algebraic equations which determine the unknown coe$cients C : K
1 4 (n#m)! R L>K> C kL>K> K (2m#1)(2n#1) n!m! a I K 1 # n(2n#1)
1 E 1#e # C "! d . 2n#1 L 6p L 1!e
(B.6)
Here R is the island radius; besides, the indices m and n pass over odd numbers only. The sums over k's in (B.6) converge rapidly to unity. For example, 1/k+1.2; 1/k+1.04. I I It can be easily found from Eq. (B.6) that when a"3R, i.e. when the gap between the islands equals the island radius, the coe$cient C and all the subsequent ones are much smaller than C . The value of C can be comparable to that of C only in the case if the islands almost touch each other. For a known charge distribution, the electrostatic potential can be found with the aid of Eq. (B.2). However, our problem is substantially simpli"ed by the circumstance that it is su$cient to determine only the component of the local "eld inside the island normal to its surface. If the surface charge density is known, this component of the local "elds is easily calculated from the boundary conditions: (E !E )n "4po (r) , * P E n "eE n . P * P
(B.7)
Here E is the "eld at the outward side of the island. It follows from (B.7) that E n "4po /(e!1) . * P
(B.8)
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The normal component of the local "eld will have its maximum value, equal to the full local "eld, when its direction coincides with that of the external "eld. It is just in this direction that the surface charge density induced by the "eld is maximum. This direction corresponds to h"0 in expansion (B.5). In such a case o" " C . (B.9) F K By orienting n in Eq. (B.8) along the external "eld (i.e. taking h"0), we obtain 4p 4p E " C . (B.10) o" " * e!1 F e!1 K K As noted above, the coe$cient C is much larger than C and all the following coe$cients even in the case when the gap between the adjacent islands is equal to the island radius. At larger gaps this tendency is all the more pronounced. Thus, one may retain in sum (B.10) only the coe$cient C which is determined by Eq. (B.6). As a result. one obtains from Eq. (B.10) the local "eld inside a metal island comprising a part of the periodic linear chain of identical islands:
1 4 R 1 \ . (B.11) E "E 1#(e!1) ! * 3 3 a k I As the local "eld E is rewritten in a standard shape E "E+1#(e!1)[¸"*¸],\ , (B.12) * its comparison with (B.11) gives ¸"1/3 for a spherical island. The parameter D¸ for this situation is *¸+(R/a) . (B.13) In a similar way one can consider a periodic chain of ellipsoidal islands. Suppose the islands are identical ellipsoids of revolution with their major axes oriented along the chain. Taking into account the explicit form of the right-hand side of (B.3), it is convenient to present the charge distribution over the ellipsoid surface as (R R ) , , C P (cos h) . (B.14) o" K K (R cos h#R sin h) , , K Here h is the angle between the radius vector to a point at the surface and the major axis of the ellipsoid which coincides with the direction of the external "elds; R and R are the major and , , minor semiaxes of the ellipsoid. The coe$cients C can also be found from a system of algebraic K equations, similar to (B.6), which can be obtained in the same way as described above. In the same approximation as that applied to derive Eq. (B.11), we "nd for the chain of ellipsoidal particles:
R 1 4 E "E 1#(e!1) ¸ ! (1!e ) , , 3 N a * k I
\
,
(B.15)
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where ¸ is the depolarization factor, determined by formula (147), and e "1!R /R . The , N , , value a stands as before for the distance between the centers of the islands. For *¸ we obtain the , following expression from (B.15): (B.16) *¸ +(1!e)(R /a) . N , , If the gaps between the islands are very small, the values of the local "elds can easily be determined more exactly by using (B.10) and retaining not only C , but also C (and possibly C ). References [1] P.G. Borziak, O.G. Sarbej, R.D. Fedorovich, Phys. Stat. Sol. 8 (1965) 55. *** [2] L.I. Andreeva, A.A. Benditsky, L.V. Viduta, A.B. Granovskii, Yu.A. Kulyupin, M.A. Makedontsev, G.I. Rukman, B.M. Stepanov, R.D. Fedorovich, M.A. Shoitov, A.I. Yuzhin, Fizika Tverdogo Tela 26 (1984) 1519 (in Russian). [3] A.A. Benditskii, L.V. Veduta, Yu.A. Kulyupin, A.P. Ostranitsa, P.M. Tomchuk, R.D. Fedorovich, V.A. Yakovlev, Izvestiya Akademii Nauk SSSR, Ser. Fiz. 50 (1986) 1634 (in Russian). [4] A.A. Benditskii, L.V. Veduta, V.I. Konov, S.M. Pimenov, A.M. Prokhorov, P.M. Tomchuk, R.D. Fedorovich, N.I. Chapliev, V.A. Yakovlev, Poverkhnost Fiz. Khim. Mekh. No. 10 (1988) 48 (in Russian). *** [5] D.A. Ganichev, V.S. Dokuchaev, S.A. Fridrikhov, Pisma v ZhTF 8 (1975) 386 (in Russian). [6] P.G. Borziak, Yu.A. Kulyupin, Elektronnye Processy v Ostrovkovykh Metallicheskikh Plenkakh (Electron Processes in Island Metal Films), Naukova Dumka, Kiev, 1980 (in Russian). *** [7] S.A. Nepijko, Fizicheskiye Svoistva Malykh Metallicheskikh Chastits (Physical Properties of Small Metal Particles), Naukova Dumka, Kiev, 1985 (in Russian). ** [8] H. Pagnia, N. Sotnik, Phys. Stat. Sol. (a) 108 (1988) 11. ** [9] R.D. Fedorovich, A.G. Naumovets, P.M. Tomchuk, Prog. Surf. Sci. 42 (1993) 189. * [10] L.I. Maissel, R. Gland (Eds.), Handbook of Thin Film Technology, McGraw-Hill, New York, 1970. [11] L.I. Maissel, in: Physics of Thin Films, Vol. 3, Academic Press, New York, 1966, p. 61. [12] K.R. Lawless, in: G. Hass, R. Thun (Eds.), Physics of Thin Films, Vol. 4, Academic Press, New York, 1967. [13] E. Yamaguchi, K. Sakai, I. Nomura, T. Ono, M. Yamanobe, N. Abe, T. Hara, K. Hatanaka, Y. Osada, H. Yamamoto, T. Nakagiri, J. Soc. Inform. Display 5 (1997) 345. ** [14] E. Bauer, H. Poppa, Thin Sol. Films 12 (1972) 167. ** [15] K.L. Chopra, Thin Film Phenomena, McGraw-Hill, New York, 1969. [16] V.M. Ievlev, L.I. Trusov, V.A. Kholmiansky, Strukturnye prevraschenia v tonkikh plenkakh (Structure Transformations in Thin Films), Metallurgia, Moscow, 1982 (in Russian). [17] L.I. Trusov, V.A. Kholmiansky, Ostrovkovye metallicheskiye plenki (Island Metal Films), Metallurgia, Moscow, 1973 (in Russian). * [18] A. Barna, P. Barna, J. Pocza, Vacuum 17 (1967) 219. [19] G. Honjo, K. Takeyanagi, K. Yagi, K. Kobayashi, Jpn. J. Appl. Phys. 2 (1974) 539. [20] S.A. Nepijko, Mikroelektronika 5 (1976) 86 (in Russian). [21] A. Barna, P. Barna, R. Fedorovich, H. Sugawara, D. Radnoczi, Thin Solid Films 36 (1976) 75. [22] D. Leonard, M. Krishnamurthy, C.M. Reaves, S.P. Denbaars, P.M. Petro!, Appl. Phys. Lett. 63 (1993) 3203. [23] J.M. Moison, F. Houzay, F. Barthe, L. Leprince, F. Andre, O. Vatel, Appl. Phys. Lett. 64 (1994) 196. [24] P. Tognini, L.C. Andreani, M. Geddo, A. Stella, P. Cheyssac, R. Kofman, A. Miglori, Phys. Rev. B 53 (1996) 6992. * [25] R. Kern, H. Niehus, A. Schatz, P. Zeppenfeld, J. George, G. Comsa, Phys. Rev. Lett. 67 (1991) 855. ** [26] E. Sondergard, R. Kofman, P. Cheyssac, A. Stella, Surf. Sci. 364 (1996) 467. [27] M. Zinke-Allmang, in: M. Tringides (Eds.), Surface Di!usion: Atomistic and Collective Processes, Plenum Press, New York, 1997, p. 389. * [28] G.R. Carlow, R.J. Barel, M. Zinke-Allmang, Phys. Rev. B 56 (1997) 12519. [29] G. Rosenfeld, M. Esser, K. Morgenstern, G. Comsa, Mat. Res. Soc. Symp. Proc. 528 (1998) 111. [30] G. Rosenfeld, K. Morgenstern, I. Beckmann, W. Wulfhekel, E. Laegsgaard, F. Besenbacher, G. Comsa, Surf. Sci. 401 (1998) 402}404.
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[199] H. Araki, T. Hanawa, Thin Solid Films 158 (1988) 207. [200] A.S. Sukharier, S.V. Zagrebneva, E.N. Petrov, V.M. Suchilin, V.M. Trusakov, Bulletin of Inventions of USSR, No 8, Certi"cate No 296179 (1971). [201] A.S. Sukharier, S.V. Zagrebneva, V.A. Osipov, E.N. Petrov, V.M. Trusakov, B.L. Serebrjakov, Bulletin of Inventions of USSR, No 25, Certi"cate No 349044 (1972). [202] R.D. Fedorovich, A.G. Naumovets, P.M. Tomchuk, 9th Intern. Conf. on Vacuum Microelectronics Digest, St Petersburg, 1996, p. 179. [203] S.S. Ivanets, N.G. Nakhodkin, A.I. Novoselskaja, R.D. Fedorovich, Bulletin of Inventions of USSR, Certi"cate No 1271278 (1985). [204] G.A. Vorobyov, V.V. E"mov, L.A. Troyan, S. Lubsanov, Abstr. 5th Symp. on nonheated cathodes, Tomsk, 1985, p. 240. [205] C.A. Spindt, I. Brodie, L. Humphrey, E.R. Westerberg, J. Appl. Phys. 47 (1976) 5248. * [206] P.R. Schwoebel, I. Brodie, J. Vac. Sci. Technol. B 13 (1995) 1391. [207] R.D. Fedorovich, A.G. Naumovets, P.M. Tomchuk, Condensed Matter Physics, 7 (1996) 5. * [208] L.V. Viduta, R.D. Fedorovich, Abstracts of the 16th All-Union Conference on Emission Elecronics, Makhachkala, 1976, p. 40 (in Russian). [209] M.I. Elinson, A.G. Zhdan, G.A. Kudintseva, M.Ye. Chugunova, Radiotekhnika i Electronika 10 (1965) 1500 (in Russian). ** [210] V.V. Nikulov, G.A. Kudintseva, M.I. Elinson, L.A. Kosulnikova, Radiotekhnika i Electronika 17 (1972) 1471 (in Russian). [211] G.A. Kudintseva, M.I. Elinson, in: M.I. Elinson (Ed.), Nenakalivayemye Katody (Nonheated Cathodes), Sovietskoye Radio, Moscow, 1974, p. 29 (in Russian). ** [212] K. Sakai, I. Nomura, E. Yamaguchi, M. Yamanobe, S. Ikeda, T. Hara, K. Hatanaka, Y. Osada, H. Yamamoto, T. Nakagiri, Proc. 16th Int. Display Res. Conf. (Euro Display 96) (1996) 569. [213] A. Asai, M. Okuda, S. Matsutani, K. Shinjo, N. Nakamura, K. Hatanaka, Y. Osada, T. Nakagiri, Society of Information Display, Int. Symp. Digest Tech. Papers (1997) 127. [214] A. Benditskii, D. Danko, R. Fedorovich, S. Nepijko, L. Viduta, Int. J. Electron. 77 (1994) 985. [215] B.M. Singer, Patent USA No 3919555 (1975). [216] R.L. Parker, A. Krinsky, J. Appl. Phys. 14 (1972) 2700. [217] J.F. Morris, Thin Solid Films 11 (1972) 259. [218] L.V. Viduta, A.P. Ostranitsa, R.D. Fedorovich, S. Chumak, Ultradispersnye chastitsy i ikh ansambli (Ultrdispersed Particles and Their Ensembles), Naukova Dumka, Kiev, 1982, p. 110 (in Russian). [219] Yu.A. Kulyupin, K.N. Pilipchak, R.D. Fedorovich, Bulletin of Inventions of USSR, No 43, Certi"cate No 1193843 (1985). [220] Yu.A. Kulyupin, K.N. Pilipchak, R.D. Fedorovich, Bulletin of Inventions of USSR, No 40, Certi"cate No 1279433 (1986). [221] Yu.A. Kulyupin, K.N. Pilipchak, in: Dispergirovannye Metallicheskiye Plenki (Dispersed Metal Films), Institut Fiziki AN Ukr SSR, Kiev, 1972, p. 238 (in Russian). [222] Jagdeep Shah (Ed.), Hot Carriers in Semiconductor Nanostructures: Physics and Applications, Academic, San Diego, 1992. ** [223] J.W. Gadzuk, L.J. Richter, S.A. Buntin, D.S. King, R.R. Cavanagh, Surf. Sci. 235 (1990) 317. * [224] R.R. Cavanagh, D.S. King, J.C. Stephenson, T.F. Heinz, J. Phys. Chem. 97 (1993) 786. [225] M. Brandbyge, P. Hedegard, T.F. Heinz, J.A. Misewich, D.M. Newns, Phys. Rev. B 52 (1995) 6042. [226] R.G. Sharpe, St.J. Dixon-Warren, P.J. Durston, R.E. Palmer, Chem. Phys. Lett. 234 (1995) 354. [227] J.W. Gadzuk, Surf. Sci. 342 (1995) 345. [228] J.W. Gadzuk, J. Vac. Sci. Technol. A 15 (1997) 1520. * [229] A. Goyette, N. Navon, Phys. Rev. B 13 (1976) 4320. [230] R. Ruppin, J. Phys. Soc. Japan 58 (1989) 1125. [231] E.D. Belotskii, P.M. Tomchuk, Int. J. Electronics 73 (1992) 915. [232] D.Ya. Petrina, Zhurn. Vychislitelnoi matematiki i mat. "ziki 24 (1984) 709.
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BARYON RESONANCE EXTRACTION FROM nN DATA USING A UNITARY MULTICHANNEL MODEL
T.P. VRANA , S.A. DYTMAN , T.-S.H. LEE Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 328 (2000) 181}236
Baryon resonance extraction from nN data using a unitary multichannel model T.P. Vrana , S.A. Dytman *, T.-S.H. Lee Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Received 12 July 1999; editor: G.E. Brown Contents 1. Introduction 2. The CMB unitary multi-channel model 2.1. Representation of three-body "nal states 2.2. Model details 2.3. Resonance parameter extraction 2.4. Relationship to other models 3. Database 3.1. nN elastic data 3.2 nNPnnN data 3.3. Description of the nNPgN analysis 4. Illustrative examples } P and S partial waves 4.1. Single-channel}single-resonance case 4.2. Two-channel}two-resonance case
184 188 188 188 191 194 195 196 196 197 197 198 200
4.3. Cusp structure in the S partial wave 4.4. Model dependence in analysis of the S partial wave 4.5. Elastic data dependence 4.6. The non-resonant amplitude 5. Results and discussion 5.1. Details of "tting 5.2. General results 5.3. Detailed discussion } D , D , and S partial waves 5.4. Observables 6. Conclusions Acknowledgements References
203 204 208 209 209 210 214 223 231 233 235 235
Abstract A unitary multi-channel approach, "rst developed by the Carnegie-Mellon Berkeley group, is applied to extract the pole positions, masses, and partial decay widths of nucleon resonances from the partial wave amplitudes for the transitions from nN to eight possible "nal baryon}meson states. Results of single energy analyses of the VPI group using the most current database are used in this analysis. A proper treatment of threshold e!ects and channel coupling within the unitarity constraint is shown to be crucial in extracting resonant parameters, especially for the resonance states, such as S (1535), which have decay thresholds very
* Corresponding author. Tel.: 412-624-9244; fax: 412-624-9163. Present Address: Fisher Scienti"c Corp., Pittsburgh, PA. E-mail address:
[email protected] (S.A. Dytman) 0370-1573/99/$ - see front matter 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 0 8 - 8
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close to the resonance pole position. The extracted masses and partial decay widths of baryon resonances up to about 2 GeV mass are listed and compared with the results from previous analyses. In many cases, the new results agree with previous analyses. However, some signi"cant di!erences, in particular for the resonances that are weakly excited in nN reactions, are found. 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 13.30.!a; 13.75.Gx; 13.30.Eg; 14.20.!c
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1. Introduction The interest in the study of baryon resonances began many years ago and led to the important discovery of SU(3) symmetry. Although many states were discovered, the quality and scope of the data limited the analyses. Interest has grown signi"cantly in the last few years because of the prospects for new data of high quality at facilities such as the Thomas Je!erson National Accelerator Facility, the Bonn synchrotron, and Brookhaven National Laboratory. These are excited states of the nucleon with excitation energy of about 0.3}1 GeV. Since most of these states were discovered in amplitude analyses of nNPnN scattering data, they are labeled by the approximate mass and the nN quantum numbers: the relative orbital angular momentum ¸, the total isospin ¹, and the total angular momentum J. For example, the D (1520) resonance has a mass of about 1520 MeV, isospin ¹", total angular momentum J", and decays into a ¸"2 nN state. Information about the various baryon resonances is evaluated and tabulated in biennial publications of the Particle Data Group [1]. In their 1998 publication, they list about 20NH (¹") and 20D (¹") states. Some states are well established in the data and most analyses agree on their properties. On the other hand, some very large discrepancies exist between di!erent analyses. For example, the extracted full widths for the S (1535) state (A 4H state of PDG [1]) are 66 MeV [2], 120$20 MeV [3], 151$27 MeV [4], 151}198 MeV [5], and 270$50 MeV [6]. This state has a number of striking properties } unusual strong decay width to gN [1] and an unusually #at transition form factor [7] } that have made it a very interesting state to understand. Although there are a number of strongly excited states (rated 4H or 3H by PDG), there are many weakly excited states whose properties are poorly determined with the existing data. The di!erences between various amplitude analyses originate for various reasons, including handling of data, method of parameterizing non-resonant background, and handling of the many NH decay channels. We emphasize the fact that the baryon resonances always decay into a baryon and one or more of various mesons; the existing analyses di!er from each other signi"cantly in the methods used to describe this intrinsically multi-channel problem. Thus, model dependence in extraction of resonance properties makes it di$cult to test predictions of theoretical models with existing data. With the new experimental facilities, the situation will soon be greatly improved when more exclusive data for di!erent "nal states, such as gN, nD, and oN, become available. One of the major goals of the new experiments is to obtain data for a large number of reactions. This greatly increases the probability of seeing new resonances, e.g. those that couple weakly to the nN channel, but puts signi"cant demands on models to interpret the data consistently. The objective of this work is to revive the multi-channel analysis of partial wave amplitudes into resonance parameters of the Carnegie-Mellon Berkeley (CMB) group. Although this is meant to be a signi"cant step toward a full analysis of all contributing reactions, the equally important problem of extracting partial wave amplitudes from the observables is not considered in this paper. The baryon resonances (called NH states from now on in this paper) extracted from amplitude analyses are thought to be predominantly composed of 3 valence quarks because of the SU(3) symmetry seen in the spectrum of hadrons of low total angular momentum. This notion has been the basis for developing various quark models, ranging from the well-studied Constituent Quark Models [8}11], Chiral Bag Models [12,13], NJL models [14], Soliton models [15], to the most recent Chiral Constituent Quark Models [16]. All of these hadron models are motivated by QCD
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and are constructed phenomenologically to describe the hadron spectrum in terms of suitable parameterizations of quark con"nement mechanisms and quark}quark interactions. With some additional assumptions on the decay mechanisms, these models can use the constructed wave functions to predict the decay widths of the NH states. For example, the one-body form of quark currents is used [10,17] in most of the constituent quark models in predicting the NHPcN decay width. A P model is assumed in Ref. [18] for creation of q q pairs in the calculation of the decay of an NH into nucleons and mesons. The comparison of these predictions with the decay width data, such as that listed by PDG, is clearly a more detailed test of hadron models. In recent years, lattice QCD calculations have been extended to predict masses of the low-lying baryons [19]. Although the lattice QCD calculations are far superior to the empirical quark models, they are much more di$cult to apply to the decay widths. In any case, precise data of resonance parameters such as masses and partial decay widths for various "nal states are needed to distinguish QCD-inspired hadron models and, ultimately, understand the non-perturbative aspects of QCD. The determination of resonance parameters is almost always a two-step process. First, phase shift analyses (elastic data) and isobar analyses (inelastic data) are used to separate the cross section and polarization observable data into partial wave amplitudes, often in the form of ¹ matrices. With a large data set, this determination has less model dependence than the extraction of resonance properties in the second step. If non-resonant e!ects are small, resonances will show up as counterclockwise rotations in the complex energy plane (Argand diagram) as the energy dependence of these ¹ matrices is plotted. Partial wave amplitudes have been determined by various groups for elastic scattering and much less often for inelastic scattering. Older elastic scattering analyses of Carnegie-Mellon Berkeley (CMB) [20] and KarlsruK he Helsinki (KH) [3,21] stressed the importance of theoretical constraints such as dispersion relations to ensure the uniqueness of the "t. The data situation has improved signi"cantly since then; the more recent work of the Virginia Polytechnic Institute and State University (commonly called VPI) group is the most visible recent e!ort [2]. They have regularly updated the data base and attempted to cull out older less viable results. They determine single-energy (also called energy-independent) ¹ matrices for partial waves up to ¸"6 and satisfy a smaller set of dispersion relations than the earlier work. All these analyses are available in the VPI repository [2]. We use the VPI single-energy elastic partial wave amplitudes from the 1995 analysis (SM95) in this work. At the same time, we recognize the point made by HoK hler [22] that this analysis is not as well constrained as the older analyses. We use the single-energy solutions rather than the more debatable smoothed solutions. The second step is to extract the resonance parameters from the partial wave amplitudes. In a simple picture, each transition amplitude in a nN reaction would be parameterized as the product of the excitation strength of the incident channel to a given resonance and the decay strength of the resonance into the allowed "nal states with a resonance propagator for the intermediate state. However, this s-channel resonant mechanism is far from complete since the u-channel and t-channel mechanisms, as implied by crossing symmetry or meson-exchange mechanisms, are known to be important. Thus, the parameterization of the amplitude in each partial wave must contain a resonant part and a non-resonant part (called the background term in most of the literature). Furthermore, the threshold e!ects associated with each decay channel (nN, gN, cN, nD, oN, uN, nNH(1440) and others) must be treated correctly within the multi-channel
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unitarity condition. Thus, resonance extraction requires a signi"cant calculational e!ort and many articles have presented various ways to proceed in practice. The PDG mostly bases its recommended values for baryons on a few works that study the full resonance region (1170 MeV( =(2200 MeV). These include older work by the CMB group [6] and the KH group [3], and more recent work by the Kent State University (KSU) group [4], and the VPI group [2]. A very recent work of Feuster and Mosel [5] "ts data for =(1900 MeV. All of these e!orts use the data of nN reactions. All maintain unitarity, though the methods employed are quite di!erent. This is reasonable since there is more than one way to implement the unitarity condition. However, these analyses di!er signi"cantly from each other in handling the multichannel character of the nN reactions. The CMB and KSU groups use a formalism that allows for many channels, while the KH and VPI groups focus on the nN elastic channel. KH accounts for all inelasticity in absorption parameters and VPI uses a dummy channel to account for all inelasticity. Feuster and Mosel [5] "t elastic scattering and inelastic scattering cross-section data with asymptotic two-body "nal states directly, and account for all the remaining components of the total inelastic cross section with a dummy inelastic channel. For most strongly excited, isolated states, these "ve analyses tend to agree within the assigned errors. However, as mentioned above, signi"cant di!erences between them exist in many cases. This paper will revive the CMB approach [6] and apply it to extract baryon resonance parameters from partial wave amplitudes of nN reactions with a large variety of "nal states. This approach emphasizes the analytic properties of scattering amplitudes in the complex energy plane that are consistent with the dispersion-relation approach and potential scattering theory. Other methods of handling multi-channel unitarity are through the K-matrix approximation [2,5,23] and the KSU model [4]. Although the transition amplitude is parameterized in a form similar to these models, there are distinct and important di!erences. In particular, the threshold for each of eight possible channels is treated correctly with two- and three-body unitarity requirements imposed. Thus, resonances can be found as poles in the ¹-matrix, a feature missing in most K-matrix models. Obviously, the CMB approach is most suitable for extracting resonances that are close to inelastic channel thresholds. Although the KSU approach by Manley and Saleski and more recent K matrix models [5] also account for the structure due to channel openings, their multi-channel parameterization is completely di!erent from the CMB model in realizing the unitarity condition. In the CMB model, dispersion relations are used to guarantee analyticity in the amplitudes. The KSU parameterization is an extension of a K-matrix formulation. It does not allow the analytic continuation into the complex energy plane, which is required to "nd the resonance pole. Feuster and Mosel present poles from a speed plot analysis, stating that technical problems associated with their partial wave decomposition prevent direct determination of poles from their K matrix analysis. There is the additional problem that resonances are associated with poles in the ¹ matrix rather than the K matrix. In the CMB model, the non-resonant t-channel and u-channel mechanisms are simulated by including the transitions to sub-threshold one-particle states with masses below the nN threshold and with the possibility of producing either an attractive or a repulsive potential. Existing models treat these e!ects in a variety of ways. The empirical method used here is rather di!erent from the polynomial parameterization often used in previous amplitude analyses. The consequence of this di!erence is very signi"cant in practice since the intrinsic `structurea of the non-resonant term due to the channel opening is built in correctly in the CMB approach, but can be easily missed if the
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`smoothnessa of the background term is the only criterion. Modern K matrix methods [5] "t coupling strengths in various diagrams. These have the advantage of "tting fewer parameters than the more empirical approaches and have the correct partial wave decomposition of the diagrams include. The disadvantages include ignoring all o!-shell intermediate state scattering and the inability to describe resonances of total angular momentum larger than . The CMB approach was published in 1979. Although inelastic data were used in the analysis, the elastic nN scattering data were emphasized. In this work, we follow closely their theoretical approach, making only small changes in the parameterization of the amplitudes. However, we make signi"cant changes in the data set used. We include all nNPgN data, some of which was unavailable then. (They depended on a separate analysis of backward n\p elastic scattering and other data [24] to model the eta cusp. This analysis would give different results with the present data set.) We also directly "t nNPnnN inelastic data, represented by the amplitudes determined in the isobar model "t of Manley et al. [25]. By including 30% more data, the Manley et al. inelastic amplitudes are more accurate than those used in the original CMB analysis. Furthermore, we will use the most recent VPI energy-independent amplitude [2] as the nN elastic scattering input. This amplitude is signi"cantly di!erent from the one used by the CMB group and is also more accurate than the KH amplitude in the S channel, as discussed in Ref. [26]. These di!erences in the input data make our results for some cases (in particular the S (1535) state) signi"cantly di!erent from the CMB values listed in Particle Data Table. Batinic et al. [27] have applied the CMB model to perform an analysis with only 3 channels: nN, gN, and a dummy channel meant to represent the complex set of nnN channels. Their focus is on the dynamics associated with the gN channel. They "t the KH80 [3] energy-independent amplitudes of the nN elastic data and the nNPgN data. The gNPgN amplitudes are the predictions of the model. The Brown et al. data [28] is the largest body of nNPgN data in the energy region close to threshold, but it is felt to have signi"cant systematic errors [27,29]. Batinic et al. give it a weight 5 times smaller than the other data to deemphasize it. We will use the same nNPgN data with the same weights in our analysis. This is an important part of the determination of the parameters associated with the S (1535) resonance. The present work is the "rst step in an ongoing program to develop a model appropriate for the new generation of data coming from Je!erson Lab and other labs. Although we anticipate further development of the model, the goal of this work is to apply the CMB model in a form very similar to its original implementation. We closely follow the procedures of Ref. [6], but use it to determine the baryon spectrum and decay branching fractions from the modern data. Future publications will address interesting but nontrivial issues such as how to determine the `besta baryon spectrum and how to di!erentiate between resonance poles, bound state or virtual poles, and poles that have changed Riemann sheet by moving across an inelastic threshold cut to the sheet most directly reached from the physical data. In Section 2, we present a detailed account of the CMB multi-channel unitary model. To illustrate the main features of the model, we discuss in Section 3 the examples of an isolated resonance and the two-resonance}two-channel situation. The full results are presented and discussed in Section 4. A summary and outlook are given in Section 5. A more complete discussion of the methods and results is given in Ref. [30].
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2. The CMB unitary multi-channel model 2.1. Representation of 3-body xnal states The "rst step of a multi-channel analysis to determine NH parameters is to extract from the nNPnnN data a set of partial-wave amplitudes for the transitions from a nN state to various quasi-two-particle channels in which one of the two particles is either a nN or a nn resonant state: nD, oN, nNH(1440) and a very broad (nn) N channel. For convenience, we will use the (2 notation p to label the S-wave, isoscalar nn state. This procedure was introduced by the CarnegieMellon Berkeley (CMB) group in the 1970s. A similar procedure was used later by the Virginia Polytechnic Institute and State University (VPI) and Kent State University (KSU) groups [25] to obtain partial wave amplitudes from all of the nNPnnN data available in 1984. These amplitudes, called VPI}KSU amplitudes, will be used in this work. A more precise analysis should start with the original data of nNPnnN reactions. This event data has been stored by the VPI group. The raw nNPnnN data was not received from them in time for the present analysis. 2.2. Model details To extract resonance parameters from VPI}KSU amplitudes, it is necessary to employ a multichannel formulation of the nN reaction. This is accomplished in the CMB approach by assuming that the transition amplitudes of the nN reaction can be written in the center-of-mass (c.m.) frame as , (1) ¹ " f (s)(o (s)c G (s)c (o (s) f (s) , @ @ ?@ ? ? ?G GH H@ GH where s is the total center-of-mass energy squared, indices a, b denote the asymptotic channels which can be either a stable two-particle state (nN, gN, KK,2) or a quasi-two-particle state (nD, oN,2). These asymptotic channels are coupled to a set of intermediate states (resonances) denoted by indices i, j. The scattering matrix de"ned by Eq. (1) is related to the S-matrix via S"1#2i¹ with SRS"1. Hence we have the following unitarity condition: + Im(¹ )" ¹H ¹ (2) ?@ ?A A@ A The crucial step of the CMB model is to choose a parameterization of various quantities in Eq. (1) such that Eq. (2) is satis"ed. Furthermore, the resulting analytic structure of the scattering amplitudes is consistent with the well-developed dispersion relations for nN elastic scattering and multi-channel potential scattering theories. This is accomplished by using the following prescription. First, it is assumed that the ith resonance to be found is identi"ed with the bare particle, i, with a bare mass squared, s . The strength constant c and form factor f (s) in Eq. (1) de"ne the decay G ?G ? of the ith resonance into an asymptotic channel a. The form factor is de"ned by
p J? ? , f (s)" ? Q #(p#Q ?
(3)
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Fig. 1. Schematic diagram for the Dyson equation iteration.
where Q and Q are empirical constants de"ning how quickly a channel, with orbital angular momentum l , opens up. p is the center of mass momentum for channel a. In this analysis, we set ? ? Q and Q equal to the pion mass. These are the same values as were used by CMB; they chose this parameterization and these values as a result of a study of the best way to model the left-hand cut. The right-hand or unitarity cut is set through the channel propagators, Im (s)"f o (s). The ? ? ? phase space factor o in Eq. (1) is de"ned by ? (4) o "p /(s ? ? for a stable two-particle state. For quasi-two-particle channels, o clearly must be de"ned consis? tently with the phase space factors used in de"ning the propagation of this resonant two-body state during the collision, as required by the unitary condition Eq. (2). In the CMB model, this is achieved by assuming that the only interaction during collisions is a vertex interaction which converts intermediate states into asymptotic states. Then the propagator G in Eq. (1) can be GH graphically depicted in Fig. 1 and is de"ned by the Dyson equation: G "G #G R G . (5) GH GH GJ JI IH Eq. (5) is an iterative equation. A sum over repeated indices is implicit. G, G , and R all vary with s and are N;N matrices, where N is the number of intermediate resonant and non-resonant states considered. Each bare intermediate state has a propagator, G , de"ned by G (s)"d e /(s!s ) , (6) GH GH G G where e "#1 for states that correspond to the resonances that will be "t. To simulate the G t-channel and u-channel mechanisms, two subthreshold bare states with a mass s below the nN G threshold are introduced. These subthreshold states will simulate an attractive background potential for e "!1, and a repulsive potential for e "#1. In principle, this prescription can G G simulate any t- and u-channel mechanisms if a su$ciently large number of bare states are included. This is well known in potential scattering theory. The actual number of the subthreshold states needed is not intrinsically known. We use two subthreshold states and one state at very high energy for every partial wave. In the original work, CMB [6] found negligible di!erences between "ts using a (smaller) number of subthreshold states and "ts using actual potentials that simulate the left-hand cut. They are allowed to couple to nN and gN asymptotic states. The self-energy, R , describes the dressing of bare particles by the coupling with two-particle GH channels, as depicted in Fig. 2. It therefore must depend on the strengths (c 's) and form factors G? ( f 's), and is assumed to take the following form: ? + R (s)" c U (s)c (7) GH AG A AH A
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Fig. 2. Schematic diagram for the self-energy (R) iteration. Each resonance is allowed to couple to all open channels.
For the contributions from stable two-particle channels, we have U (s)" (s) A A
(8)
with
1 dsf H(s)g (s; s) f (s) . (9)
(s)" A A A A p A Q Here s is the threshold for channel c, and g (s) is the propagator of the two-particle channel c. A A The task now is to choose g (s; s) such that the unitarity condition Eq. (2) is satis"ed and the A desired analytic structure of the amplitude can be generated. It can be shown that the (s) of the A CMB model is obtained by assuming that
o (s) o (s) 1 A !P A , (10) g (s; s)" A s!s p s!s#ie where the density of states o (s) has been de"ned in Eq. (4), and P means taking the principle-value A part of the propagator. Substituting Eq. (10) into Eq. (9), we then obtain the following dispersion relation for the auxiliary function : A Im( (s))"f (s)o (s) , A A A (11) Im (s) s!s A ds . Re( (s))"Re( (s ))# A A p Q (s!s)(s!s ) We see that Eq. (11) is a subtracted dispersion-relation which has a form similar to what has been established in many studies of nN scattering. In the complex-s plane, one can choose the subtraction point s such that the resulting scattering amplitude has a pole on the left-hand side and a branch cut from s "(m #m ) to #R. For each channel, we choose the value at L, L , threshold for s , and we set the subtraction constant, Re( (s )), such that Re( (s)) is 0 at threshold. A This arbitrary choice has the primary e!ect of shifting the value of the bare mass squared, s , for G each resonance in a partial wave, but does not a!ect the physical mass of the `dresseda resonance. Here we assume that this dynamical assumption is valid for all stable two-particle channels like gN, and KK. For a quasi-two-particle channel c, the function U (s) in Eq. (7) must account for the mass A distribution of one of the two particles which is itself a resonance state. To be speci"c, let m be the
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mass of the stable particle in the channel c and m and m be the masses of the two daughter particles from the decay of the resonant subsystem into a channel r. Then the form assumed by the CMB model can be more explicitly written as
U (s)" A
(Q\K
K >K
ds p(s ) (s ) P P A P
(12)
where the mass distribution of the quasi-particle was taken to be c Im( (s))/n P P p(s)" . (M!s)#cIm( (s)) P P P
(13)
Here (s) is again de"ned by the subtracted dispersion relation Eq. (11) for the appropriate P resonant subsystem. Eq. (13) has the commonly used Breit}Wigner resonance form. The coupling strength c is related to the width C of the considered resonance state by C "[cIm( (M))]/M . P P P P P P In this work r is either a nN state or a nn state. From the empirical values of the widths for D and o, mass distribution functions for the nD and oN channels can be "xed. For the other quasi-twoparticle channels, pN and nNH(1440), the width C is "xed at a standard value in the "t [4]. The P above formalism for the quasi-two-body channels provides a unitarity cut along the real s-axis from the three-body nnN threshold to in"nity. Furthermore, it makes the resulting scattering amplitudes satisfy the unitarity condition Eq. (2). This is how the three-body unitarity is implemented in CMB model. The Dyson equation, Eq. (5), is algebraic and can be solved by inverting an N;N matrix. Schematically, we have G (s),[H\(s)] GH GH
(14)
with the matrix element of H de"ned by H (s)"[(s!s )/e ]d !R (s) . GH G G GH GH
(15)
Now all of the ingredients needed for calculating the ¹-matrix elements of Eq. (1) are in place. The variable parameters (i.e. couplings c and poles s ) are then adjusted to "t the VPI}KSU partial GA G wave ¹-matrix elements. 2.3. Resonance parameter extraction Following Ref. [6] a resonance position is identi"ed with a pole of the scattering ¹ matrix in the complex energy plane. This can only be done for models that can be evaluated for complex values of s, i.e. models that have the correct analyticity structure. In the CMB model, the determinant of the H matrix de"ned by Eq. (15) equals zero at the pole position, s"s , in the complex s-plane. Only the poles located close to the real axis are interpreted as resonances. This procedure involves an analytic continuation of (s) into the complex s-plane for Im(s)(0. Clearly, the analyticity, A de"ned by the dispersion relation, Eq. (11), plays a `dynamicala role in "nding the resonance parameters. This is one of the main di!erences between our approach and the KSU approach.
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To proceed, we need to evaluate Eqs. (11) and (12) for complex s. Each ¹-matrix element has a branch cut beginning at the elastic and each inelastic threshold. The branch cut can have a strong e!ect on the amplitude, in extreme cases producing a cusp, e.g. in the nNPnN S partial wave. Above each threshold, the amplitude is multi-valued. This is traditionally described by a Riemann sheet structure [31]. The amplitude is continuous when analytically continuing to the appropriate new sheet as the value of s crosses the branch cut, but discontinuous when staying on the same sheet. The function, (s) de"ned by Eqs. (9) and(10) is the channel propagator for the `"rst sheeta A of the complex s-plane, labeled . At s5s , it has a discontinuity in its imaginary part A' A determined by unitarity as s crosses the real axis. The resonance pole is on the `second sheeta in which (s)" (s); it has the same discontinuity A A'' as the "rst sheet except for the opposite sign,
(s#ie)! (s!ie)" (s!ie)! (s#ie) for s's . (16) A'' A'' A' A' It is (s) which is used in the search of the resonance pole positions. Using the same strategy as A'' CMB [6], we search the nNPnN ¹ matrix for poles on the sheet most directly reached from the physical region. As discussed by Cutkosky and Wang [32] and elsewhere in the literature, each resonance has additional poles on other Riemann sheets associated with each inelastic threshold. However, the pole closest to the physical region is most closely associated with the physical characteristics of the resonance. The formalism presented here can be extended to search for poles on other sheets and try to distinguish between resonance poles and poles that arise from bound states of composite particles. A resonance pole is found by searching for a zero in the determinant of the H matrix de"ned in Eq. (15). Once a pole is found, H(s) is diagonalized at the pole position s . This is done to eliminate resonance}resonance interference e!ects when multiple resonances are present in a partial wave. By using the resulting eigenfunctions s , the ¹-matrix in the vicinity of the pole can be G written as [6] (B (Im( )g g (Im( )(B B !d ?C C C D D D@ . ?@ # (17) ¹ " ?@ ?@ D(s) 2i CD The g describe the coupling of the resonance to channel c, as de"ned in Eq. (19). This form can be A shown to be equivalent to the full CMB model. Eq. (17) is a general form for a Breit}Wigner resonance shifted by non-resonant (background) reaction mechanisms. To determine the resonance properties, we look at the ¹ matrix in the vicinity of the pole with a simplifying assumption. The form of the background part, B , is assumed to be smooth in the immediate vicinity of a pole. This ?@ allows us to use the denominator, D(s), to de"ne the Breit}Wigner resonance parameters at the pole, ignoring the background. The denominator of the above expression is then matched to a relativistic Breit}Wigner form. The full denominator can be written as D(s)"r!s!v y , A A A where
, y ""g "" c s . GA G A A G
(18)
(19)
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The real constants, r and v, are de"ned by equating D(s )"0. The resonance mass comes from the real part of the denominator and the width comes from the imaginary part. The general form of the Breit}Wigner denominator for a multi-channel situation is "(= !s)!i= C (20) A A where = is the mass of the resonance and C is the decay width of the resonance to asymptotic A state c. Thus, shifts in the resonance mass can be identi"ed with the real parts of the denominator and the imaginary part can be identi"ed with a sum over the partial decay widths of the resonance to various channels, c. To obtain numerical values, we use a linear approximation for the real part of the summation in Eq. (18). Using the above de"nitions, the following qualitative statements about the relationship between the resonance parameters and the model parameters can be made. The resonance width is approximately equal to twice the imaginary part of the denominator at resonance. Because the self-energy term in the denominator has strong energy dependence, the physical pole position is shifted from the bare pole by an amount that depends of the value of the coupling parameter (c) and both the value and shape of Re (s) in the vicinity of the pole. The mass of the resonance is further shifted from the physical pole because the mass is determined on the real axis of the complex energy plane. Shifts from the real part of the bare pole to the mass can be positive or negative; the size of the shift depends on many factors and can be quite large (more information can be found in Section 5). Eq. (17) has an energy-dependent pole shift due to Re (s). We wish to remove this energy dependence to make a connection between the full ¹-matrix in Eq. (17) and a standard relativistic Breit}Wigner shape. Making the assumption that the `reala part of the term y is linear close A A A to the resonance pole: D
5
v Re y +a#bs , (21) A A A the ¹-matrix can be re-written in the form of a relativistic Breit}Wigner resonance, without an energy-dependent pole shift: (B Im g (B Im g D@ D D ?C C C (1#b (1#b B !d ?@ # . (22) ¹ " ?@ ?@ ( P\? )!s!iv y Im /(1#b) 2i A A A CD >@ In Eq. (22), the quantitative expressions for the resonance parameters can then be identi"ed in terms only of quantities evaluated at the resonance mass: Re D(M )"0 de"nes resonance mass PM , Im D(M ) de"nes resonance width , C" M Re D(M ) y Im
A C de"nes branching fraction into channel c . C" A A y Im
? ? ?
(23)
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In the above equations, D(s) is the derivative of D with respect to s, Re D(s)"!(1#b). This term accounts for the fact that there is an energy dependence to Re (s), which shifts the pole position. This formulation is identical to the generalized Breit}Wigner form that is the basis of most "tting and theoretical models. However, we only use this form in the immediate vicinity of the pole to determine the resonance parameters. The data are still "t with the full model and the pole position is then determined from that "t. This type of prescription for obtaining the mass, width and partial widths of a resonance is model dependent. It is important to realize that all de"nitions of baryon resonance parameters are model dependent. For a highly elastic and isolated resonance, the model is not very important. However, very few resonances "t this description and we are trying to present a formulation that minimizes the model dependence. 2.4. Relationship to other models The CMB model has a number of features that are not included in commonly used models. Here, we present a discussion of formulations similar to standard models by making approximations to the full CMB model. These simpler formulas will be used in Section 4.4 to show the corresponding model dependence in the extracted resonance properties. The full CMB model contains a dispersion relation which guarantees the analyticity. The imaginary part of the channel propagator function (Eq. (11)) is the relativistic phase space function and the real part is then calculated from a dispersion integral. Models which are not analytic [4] include only the phase space, so we set Re (s)"0 for all s to simulate them. The full CMB model uses a Dyson equation to allow for conversion into open intermediate states (resonant or non-resonant) and open asymptotic channels. The bare propagator (Eq. (6)) is `dresseda by all the open intermediate and asymptotic states. The K-matrix formulation [2,4,5,23] uses the bare propagator of the CMB model in place of the dressed propagator as a K-matrix rather than a ¹-matrix and identi"es the resonance properties with its parameters. A non-analytic unitary multichannel K-matrix using a relativistic Breit}Wigner form [23] can then be constructed for the contribution of resonance R to the reaction between initial state i and "nal state j: (Im (s) c c (Im (s) 0G 0H H , G K0 (s)" GH s !s 0
(24)
where Im (s) is the product of the form factor and phase space for channel i as de"ned in Eq. (11). G M and C are the mass and partial width for decay to channel i of resonance R; the total width is 0 0G C . These three physical quantities are de"ned by 0 M "(s , 0 0 Im (s )(c ) G 0 0G , C " 0G M 0 C " C . 0 0G G
(25)
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For the non-relativistic Breit}Wigner case, (Im (s) c c (Im (s) 0G 0H H , G K0 (s)" GH M != 0 where C is de"ned by 0
(26)
C "2 Im (s )(c ) . (27) 0 G 0 0G G In either case, the corresponding ¹-matrix can then be found through the standard de"nition: ¹"K*(I!iK)\ .
(28)
Since both K and ¹ are matrices, there is no simple closed expression for ¹ corresponding to Eqs. (24) and (26) except in the single-channel case. Non-resonant K-matrices must also be de"ned. To maintain unitarity, the full K-matrix is obtained by adding all the resonant and non-resonant K-matrices (e.g. [38]). The corresponding ¹-matrix includes e!ects of resonance interference and coupling to non-resonant processes, but only on-shell. Various theoretical schemes have been built on the K-matrix method. At low energies, the characteristics of the D (P (1232)) and the S (1535) have been determined with an e!ective Lagrangian method [38]. More complete formulations have been developed [5,33]. In these models, the mesons and baryons are each fundamental particles. Although the number of parameters is reduced from what is required for the more empirical models, there are still a number of ambiguities in the construction of the Lagrangian and in the proper development of a multiresonance, multi-channel model.
3. Database We devote a separate section to the database because it plays a critical role in the results we obtain. There are many possible asymptotic states that can couple to each resonance. There are also a number of resonances which couple weakly to the nN channel so that the state is only seen in the inelastic data. Just as it is important to include various inelastic channels with proper threshold e!ects in the theoretical model, it is also important to include as much of the relevant data as possible with appropriate error bars. Although it is best to use the original data, various partial wave analyses producing ¹ matrix representations of the data are available. This kind of analysis can be done with much less model dependence than is found in the analysis used for determination of baryon resonance properties. Nevertheless, these analyses make choices in the data used in the "ts and their absolute normalizations since not all data sets are consistent with each other; these choices can add error beyond what was in the original data, thus adding uncertainty to the "t. On the other hand, "tting partial wave amplitudes allows a simpler "tting strategy } separate "ts can be made for each partial wave and less computer time is required. This is the same procedure chosen by KSU [4]. We choose to "t the single-energy partial wave amplitudes of the VPI group for elastic scattering [2] and the isobar
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model "ts of KSU}VPI for inelastic pion production [25]. We also make a separate partial wave analysis of the nNPgN data. 3.1. nN elastic data Data in this channel is easiest to measure. Therefore, the data are more complete and of higher quality than in the other channels. In many resonance parameter analyses, these data have a dominant role in the results. This takes advantage of using the best data. However, the inelastic data must be included to "nd resonances that would not be seen in nN elastic scattering. The Constituent Quark Model [8,18] predicts the existence of a large number of states (roughly a number equal to the number of states seen to date) and "nds that many of them couple weakly to the nN channel. There have been numerous analyses of the elastic data. Since complete experimental results are not yet available, theoretical constraints must be employed to get unambiguous "ts. Older analyses of CMB [20] and KH [3] had strong reliance on dispersion relations to generate unique "ts. KH80 used "xed-t and "xed-h dispersion relations at many angles. The real parts of these results were later compared with partial-wave "xed-t dispersion relation predictions by Koch [21]. Although KH80 results are quite noisy close to threshold in the high ¸ partial waves, there is good qualitative agreement with the additional constraints. CMB80 uses hyperbolic constraints in the Mandelstam variables. Although the older works have the best theoretical underpinnings, we use the latest nN elastic partial wave amplitudes of VPI [2]. The VPI analysis uses a signi"cantly larger data set than was available for the earlier CMB80 and KH80 elastic analyses. Most of the new points are at =(1600 MeV, but there are also many new results at higher energies [2]. Thus, a number of older data points with signi"cantly larger estimated errors could be dropped from the "t, resulting in an improved "t. These more recent results are consistent with "xed-t dispersion relations at h"0 and low =. Although improvements in the VPI analysis are expected, we use this somewhat debatable [22] approach in this work since this provides a way to include all of the data published since the older work. Desire for an update of the 1980s work has been expressed at conferences for several years, but no such work is in progress. 3.2. nNPnnN data For the best available representation of inelastic data, the choices are much more limited than for elastic data. Since there are no model-independent methods for the 3-body "nal states, isobar models are employed. Since no recent interpretation of the nNPnnN data is presently available, we use the quasi-two-body channel decomposition of Manley et al. [25]. That work "t an isobar model to the nNPnnN data, isolating the contributions of nD, oN, pN (with p representing the nn strength in an isoscalar s-wave), and nNH(1440) channels. Although the pN channel will absorb some of the non-resonant nnN strength, these choices ignore some of the non-resonant nnN strength and states such as nNH(1520) that might be expected to share strength. They used a data sample of about 241,000 events spread over 1320}1910 MeV in =; statistical accuracy is much poorer than for the elastic channel and there is no data at the highest energies where resonances are found. All data published after this analysis were very close to threshold. It will be clear from our analysis that these data need augmentation and that the isobar analysis should be repeated.
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As with the choice in elastic data amplitude, this choice is a compromise between including all the existing data in a simple form and using a more proper analysis. An analysis the nN inelastic data without the assumption of the isobar model would be extremely di$cult and unjusti"ed with the quality of the present data set. In the CMB analysis [6,32], the nNPnnN data were weighted by a factor of smaller than the results of Manley et al. [25] because they felt the errors were understated. We have a similar attitude and weight the inelastic data by a factor of . Otherwise, the "ts to the higher quality elastic data are degraded. 3.3. Description of the nNPgN analysis The most signi"cant inelastic channel at low = is the gN state; data involving this channel is thought to be crucial in analyzing S (1535) because the s-wave nNPgN cross section is large and rapidly changing close to the resonance mass. Unfortunately, the data in this channel is both limited and of uncertain quality [29]. The data set with the most points close to threshold was published by Brown et al. [28]. Clajus and Nefkens [29] argue that these data have unknown errors in the assigned values of =, making them unusable. As mentioned earlier, Batinic et al. [27] put a very small weight on these data points in their partial wave "t to the nNPgN data. To produce partial wave amplitudes for this reaction, we reproduce the Batinic et al. "t to the nNPgN data, e!ectively leaving out the controversial Brown et al. [28] data set. Since there is very little data available for the reaction nNPgN, we use a simpli"ed version of the full CMB model to simultaneously "t the nN elastic ¹ matrices and the nNPgN data to provide a parameterization for each of the partial waves with ¸44 contributing to the g production cross section. The partial wave analysis followed the procedures used by Ref. [27], but used the newest VPI nN elastic partial wave amplitudes [2] instead of the older KarlsruK he Helsinki (KH80) amplitudes. The channels used in the analysis are nN, gN, and a dummy channel consisting of a "ctitious meson with mass chosen so that the dummy channel opens at about the energy where nN inelasticity due to channels such as nnN start. The mass of that "ctitious meson changes from partial wave to partial wave. The values used in the analysis are given in Ref. [27]. The partial waves used are the I" partial waves through G . All partial wave parameters were varied simultaneously. More details on this procedure given in this section can be found in the Refs. [27,30]. For this process, the S (1535) resonance makes up most of the total cross section. Therefore, the S partial wave is the most accurately extracted. The other partial wave amplitudes are smaller and less accurately determined. The results of this "t are ¹-matrices that best model the partial wave data. The best "t is shown as an error band in the "gures of the results section. For the "nal "ts, 40 data points between threshold and 2.3 GeV were used for each partial wave. This insures that these data provide the appropriate contribution to the total chisquare. 4. Illustrative examples ⴚ P33 and S11 partial waves The purpose of this section is to introduce features of the model through the examination of speci"c partial waves. The P and S partial waves are chosen for this purpose. The P (1232) or
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D is the best example of an isolated and elastic, i.e. simple, resonance; its characteristics are largely well-established. On the other hand, the S partial wave has some of the most interesting structure of all the partial waves contributing to nN scattering. Because of this structure, extra care must be taken when extracting resonance parameters for this partial wave. Some of the interesting features exhibited in this partial wave are listed below. 1. There are 2 PDG 4* resonances (S (1535) and S (1650)) which overlap signi"cantly. 2. The S (1535) has strong coupling to both the nN and gN channels and is very near the gN threshold (+1487 MeV). This produces a strong cusp in the nN elastic S ¹-matrix element. 3. There are 7 decay channels that have measurable coupling to the S states, each of which has di!erent phase space which can cause structure in all of the other channels via unitarity. All of the above features are adequately handled in the CMB Model. This section will help in understanding the unitarity, analyticity, and other properties of the model, which are especially important in the S partial wave. A discussion of the cusp associated with the S partial wave will also be given. Furthermore, this section will present a systematic study of the model dependence of resonance parameters extraction by examining results obtained when leaving out various features of the full CMB model. 4.1. Single-channel}single-resonance case The equations from Section 2 give expressions for a reaction ¹-matrix element for any number of open scattering channels or asymptotic states (nN, gN, etc.) and any number of intermediate states (S (1535), P (1232), etc.). The ¹-matrix elements for one asymptotic state and one intermediate state are much simpler. The resulting equations can then be applied with good success to the P partial wave near the D(1232) resonance. Despite the complexity of the CMB formulation, the ¹-matrices for an isolated, single-channel resonance have a form similar to that found in the Breit}Wigner shapes commonly used. The one asymptotic state-one intermediate state case requires two parameters (using D to label the P (1232) intermediate state and nN as the asymptotic state), one coupling cD and one bare L, pole energy, sD . Since there is only one intermediate state the matrix equations are reduced to scalar equations and the math is simple. There is only one-channel propagator which is determined L, analytically with a contour integral. (There is a functional form for the channel propagator only in the S-wave case. The results for the nN channel in an S-wave are shown in Fig. 3.) There is also one self-energy term according to Eq. (9): RD D (s)"cD (s) . L, L, Since the function varies with energy, R does also. The H matrix is then de"ned by
(29)
H"(sD !s)!RD D "(sD !s)!cD . (30) L, L, The G matrix (the dressed propagator) is then calculated from H as de"ned by Eqs. (14) and (15) in Section 2: 1 G" , (sD !s)!cD
L, L,
(31)
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Fig. 3. (s) (channel propagator) distributions for the nN and gN channels in S partial wave. These functions are used in the two-channel}two-resonance test case and in the "nal "ts.
¹ is de"ned using Eq. (1) and the above equations: L,L, cD Im
L, L, ¹ " . L,L, (sD !s)!cD
L, L,
(32)
The results of a model calculation for the ¹ matrix is shown in Fig. 4 with a solid dot at the physical resonance mass. The equations produce the characteristic shape of a resonance with a maximum in the imaginary part and a zero in the real part at the resonance mass. A similar signal should be seen for strong resonances, but there will be shifts in the real data when underlying backgrounds of varying smoothness are included. Eq. (32) has the usual form of a relativistic Breit}Wigner resonance as expected, but there are some important new features. The real part of the pole is shifted from the bare pole energy (sD ) by cRe (s) and the pole gains an imaginary part, cIm (s) due to couplings of the resonance to the asymptotic state as it propagates. In this model, these shifts are energy dependent and come from both unitarity and analyticity requirements. The analyticity condition, Eq. (11) then allows an analytical continuation of the above expressions to the complex s plane where the physical pole position, mass, and width of the resonance must then be determined by a search discussed in Section 2. In other contexts, unitarity requirements are satis"ed through inclusion of "nal state interactions. These methods add terms to the denominator similar to what comes from the CMB model. However, these models are not necessarily analytic. Near the resonance peak, Eq. (32) can be expressed in terms of a mass and width identical to a generalized Breit}Wigner shape. Thus, the features of the CMB model can be absorbed into e!ective constants for the case of an isolated resonance.
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Fig. 4. One-resonance}one-channel test case ¹ matrix (real and imaginary parts) using parameters appropriate for the P (1232) resonance. Solid dots are placed at the energy of the resonance mass.
4.2. Two-channel}two-resonance case The equations get rapidly more complicated as the number of resonances and open channels in a partial wave increases. The two-channel (or asymptotic states) two-resonance (or intermediate states) case is still instructive. The relevant equations are given below for the ¸"0 isospin (S ) partial wave. In reality, this partial wave has two strong channels and two strong resonances. The results shown below use realistic parameters for these states, but are not meant to be an accurate representation of data because non-resonant processes and other channels are ignored. For clarity, channel labels nN and gN and resonance labels of 1535 (referring to S (1535)) and 1650 (referring to S (1650)) are used. Parameters that must be determined include four coupling strengths (c ,c ,c , and c ) and two bare poles (s and s ). L, E, L, E, The terms in the R self-energy matrix are constructed from Eq. (7): R "c
#c
, L, L, E, E, R "c c
#c c
, L, L, L, E, E, E, R "c
#c
. L, L, E, E, The R matrix is symmetric by construction so it takes on the simple form
R
R
. (34) Fig. 5 shows the R (s) function. It is a weighted sum of and , shown in Fig. 3, and L, E, thus carries the threshold behavior of both the gN and nN channels. The other elements have R(s)"
R
(33)
R
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Fig. 5. Self energy function, R for the test case where only nN and gN channels are included.
similar qualitative behavior, but varying weights of because the two resonances couple to the two asymptotic channels with di!erent strength. The H matrix becomes
s !s!R !R . H(s)"(s !s)!R(s)" G !R s !s!R The dressed propagator, the G matrix, has a simple form since it is a 2;2 matrix:
1 s !s!R G"H\" "H" R where "H" is the determinant of H:
R , s !s!R
(35)
(36)
"H""(s !s!R )(s !s!R )!R . (37) The ¹ matrix elements are then de"ned in Section 2 in Eq. (1). Explicitly, the nN elastic ¹-matrix element is c Im
L, L, R !s!R ! s !s!R c Im
L, L, # R s !s!R ! s !s!R 2c c R Im
L, L, L, # . (s !s!R )(s !s!R )!R
¹ " L,L, s
(38)
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T.P. Vrana et al. / Physics Reports 328 (2000) 181}236 Table 1 Model parameters for a test calculation in the S partial wave. These are the bare mass, the physical mass and width, and the dressed pole location for the two resonances included in the calculation. No background contributions are included State
Bare mass (MeV)
Physical mass (MeV)
Full-width (MeV)
Physical pole (GeV)
S (1535) S (1650)
1500 1650
1518 1680
83 200
1.507}0.032i 1.676}0.101i
Table 2 Model parameters for test calculation in the S partial wave. These are the coupling parameters and branching fractions (b.f.) between the two asymptotic channels and the two resonances included in the calculation. No background contributions are included State
cnN (GeV)
S (1535) S (1650)
0.5 0.9
cgN (GeV)
0.85 !0.04
nN b.f. (%)
gN b.f. (%)
42 96
58 4
Note that all the functions used in this expression depend on s. The above expression is the sum of two resonances with an interference term. (If there is only one resonance, this formula simpli"es to the same as Eq. (32).) The terms that look like single resonances are more complicated than in the previous example. The self-energy terms have the necessary analytic cuts from the gN as well as the nN channels and also make the amplitude unitary. There are analogous ¹-matrix elements for the processes: nNPgN and gNPgN. In this case, two poles must be found in the complex energy plane. Since the two resonances have signi"cant interference, the non-diagonal elements of the G matrix are large. This means the second resonance has a contribution to the propagator of the "rst resonance at the pole of the "rst resonance. Referring to Eq. (38), there are shifts to the mass and width of each resonance due to the presence of the other in addition to those due to the asymptotic channel couplings which were not seen in the one-channel}one-resonance case. Here, the H matrix (the inverse of G or the `denominatora matrix) must then be diagonalized at each pole to isolate the contributions from each individual resonance. The nN elastic and nNPgN ¹ matrices in this partial wave corresponding to the above equation for representative parameters (see Tables 1 and 2 for a complete listing of the relevant values) are shown in Fig. 6. The eta threshold has a signi"cant e!ect on the observables. Therefore, the nN elastic ¹-matrix element peaks at the gN threshold rather than at the peak of the S (1535) resonance. The physical masses are shown as solid dots which are near the peak of the cross section and the imaginary part of ¹ for the higher state, but above those positions for the lower state. For all channels, the peak of the ¹-matrix is shifted by a few dozen MeV from the bare mass by the
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Fig. 6. ¹ matrices for two-channel}two-resonance test case. The real and imaginary parts are shown for the nNPnN and nNPgN reactions. Solid dots are placed at the values of the resonance masses obtained from these ¹ matrices.
self-energy terms. (The S (1535) resonance shape is also modi"ed by the presence of a strong inelastic threshold.) There is also signi"cant interference between the two resonances. Both cross-section bumps are unusually narrow compared to the physical widths because of interference e!ects. We "nd that the S (1535) properties are signi"cantly altered by the interference with S (1650), similar to the "ndings of Sauermann et al. [33] in a K-matrix photoproduction calculation. 4.3. Cusp structure in the S partial wave There are two main causes of the strong cusp structure observed in nN elastic scattering di!erential cross section, as well as the S partial wave amplitudes. The "rst is that the S (1535) resonance couples strongly to both the nN and gN channels and the gN channel threshold (+1487 MeV) is just below the 1535 pole. The second is that the orbital angular momentum is zero (nN S-wave); thus, the cross section in this partial wave increases linearly with momentum. Therefore, through analyticity and unitarity, a cusp structure appears in the nN elastic channel (Fig. 7). Fig. 5 shows the self-energy term labeled R for the two-channel}two-resonance model of Section 4.2. This self-energy has the appropriate analytic phase space factors and therefore shows a cusp. Since R enters directly into the ¹-matrix elements, the cusp shows up there as well. Fig. 8 shows the ¹-matrix elements for elastic scattering and o production. Although the cusp structure is apparent in the ¹ matrices for all channels other than that of eta production, it is most evident in the nN elastic channel because most of the decay width of the S (1535) is split roughly equally between the nN and gN channels. All other inelastic channel openings have the potential of
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Fig. 7. Best "t calculation for S partial wave at the gN threshold. ¹ matrices for nNPnN and nNPo N are shown.
creating a cusp, but this is by far the strongest case. At present, the nN data is not of high enough quality to see any other cusp structures. 4.4. Model dependence in analysis of the S
partial wave
The variation between this and other models can best be seen in an analysis of the data in the nN S channel, as discussed in previous sections. Because of the peculiarities of this channel, data for both nNPnN and nNPgN reactions would be required for a high quality determination of the S (1535) parameters and data for nNPnnN would be required for a good determination of properties for the higher energy states. Presently, the inelastic data is of much less quality than for the elastic channel. To test model dependence in the resonance parameters, we have analyzed all available and subsets of the data for the S channel with the full CMB model and various approximations that simulate the models employed by various other groups. Although the "nal "ts were of similar quality for all cases, there can be large di!erences in the extracted resonance parameters. In Tables 4 and 5, masses, widths and branching fractions are given for two of the three S resonances used in this analysis for a number of di!erent model types using various data sets to constrain the "ts. The four columns on the left of the table describe features of the model used and what data was used in the "t. The "ve columns on the right give the results of the "t for the resonance parameters } mass, width, and branching fractions. The Unitarity column labels how unitarity was imposed in the "ts. K-matrix means that unitarity was imposed by commonly used K-matrix methods of Moorhouse et al. [23]. For the results shown in the tables, we recreate the K-matrix "t of Ref. [23] with the appropriate phase space factors ( (s)) from our model as discussed in Section 2.4. Dyson equation means that unitarity was imposed using the Dyson-equation approach of the CMB model. Channels can interact any
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Fig. 8. ¹-matrices for S partial wave for elastic scattering and "nal states of o N, pN, and nNH(1440) calculated with the full model using "nal "t parameters. The dotted line is calculated with only resonance couplings turned on and the dashed line is calculated with only non-resonant couplings enabled. The full calculation, which cannot be a sum of the dotted and dashed lines because the resonant and non-resonant diagrams interfere, is shown as a solid line.
number of times in various forms before "nally decaying because the Dyson equation is iterative. The column labeled Disp Rel refers to whether or not the dispersion relation was used to make the phase space factors in the self-energy terms analytic functions or not. If the dispersion relation is evaluated, the phase space factors and hence the self-energies are analytic functions of the square of
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Table 3 Channels included in this analysis. Since isospin symmetry is assumed to simplify the analysis, all charge states of a particle have the same mass. We choose channels very similar to those used by KSU with identical nomenclature. All channels used in "ts have "xed spin, orbital, and total angular momentum. In the text, the oN channel is denoted by (o N)l with s equal to twice the total spin of the oN system and l giving the orbital angular momentum Q in spectroscopic notation. The orbital angular momentum of the nD is also ambiguous, so it is also given as a subscript Channel
Baryon mass
nN gN oN pN uN nD nN* KK
939 939 939 939 939 1232 1440 1116
Baryon width
Meson mass
Meson width
139 549 770 800 782 139 139 498
115 200
153 800
Table 4 Model dependence for the S (1535). See text for details Unitarity
Disp. Rel.
Res. Type
Channels in "t
Mass (MeV)
Width (MeV)
nN (%)
gN (%)
nnN (%)
K-Matrix K-Matrix K-Matrix K-Matrix K-Matrix K-Matrix Dyson eq. Dyson eq. Dyson eq.
NO NO NO NO NO NO YES YES YES
NRBW NRBW NRBW RBW RBW RBW RBW RBW RBW
nN nN, gN ALL nN nN, gN ALL nN nN, gN ALL
1518 1532 1535 1514 1533 1534 1531 1526 1542
87 108 126 84 110 125 72 114 112
43 45 42 35 44 42 16 36 35
6 39 44 0 40 43 62 41 51
51 16 14 65 16 15 22 23 14
the CM energy, s; otherwise they are not. When using the Dyson equation and not evaluating the dispersion relation (in the "ts, we set Re "0.0), all intermediate interactions in the scattering process occur on-shell. Note that this does not a!ect unitarity. If unitarity is achieved through K-matrix methods, the mass and widths are direct parameters in the "t. The column Res Type describes the form of resonance used. NRBW refers to a non-relativistic Breit}Wigner shape and RBW means a relativistic Breit}Wigner shape was used. For the NRBW case, a resonance has a c/(w !w!ic) form and the c couplings have units of (energy. For the RBW case, a resonance has a c/(s !s!ic) form; the c couplings have units of energy. Finally, the column labeled Channels in xt describes the types of data used in the "t. For the nN case, only the VPI nN elastic S ¹-matrix elements were used in the "t. For the nN, gN cases both the VPI elastic data as well as constraints from a partial wave analysis of nNPgN done by this
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Table 5 Model dependence for the S (1650). See text for details Unitarity
Disp. Rel.
Res. Type
Channels in "t
Mass (MeV)
Width (MeV)
nN (%)
gN (%)
nnN (%)
K-Matrix K-Matrix K-Matrix K-Matrix K-Matrix K-Matrix Dyson eq. Dyson eq. Dyson eq.
NO NO NO NO NO NO YES YES YES
NRBW NRBW NRBW RBW RBW RBW RBW RBW RBW
nN nN, gN ALL nN nN, gN ALL nN nN, gN ALL
1645 1689 1694 1682 1692 1690 1692 1676 1689
233 225 259 161 233 229 138 104 202
35 67 72 78 75 65 65 54 74
49 31 16 5 15 25 35 45 6
16 2 12 17 10 10 0 1 20
group were used. For cases labeled All, nNP+nN, gN, nnN, are all included in the "t. The nnN channel is composed of quasi-two-body channels (o N) , (o N) , (nD) , pN, and nNH(1440). 1 " " Partial wave quasi-two-body ¹-matrix elements of Manley et al. [25] are used for these channels. The last row in each table contains results for our full model. All channels are used in the "t, unitarity comes from use of the Dyson equation, analytic phase space factors are used, and the bare resonance has a RBW dependence. Signi"cant model dependence is seen along with sensitivity to the data used. The lower state (S (1535)) has signi"cant model dependence as noted above while S (1650) tends to have poorer quality data, i.e. missing data or data with error bars that are too small. We do not show any results for the third resonance (S (2090)) because the data quality dominates the "tting, causing wide variation in "t results, e.g. masses vary between 1509 and 2028 MeV. The data quality for this partial wave is discussed in detail in Section 5.3. We note that without the dispersion relation, the ¹ matrices for the K-matrix model and the model using the Dyson equation for the resonance propagator are equivalent. Therefore, only the K-matrix results without the dispersion relation are given in the table. In general, the choice of relativistic vs. non-relativistic shape for the bare Breit}Wigner resonance does not have a strong in#uence for an isolated resonance such as the 1535 MeV state. However, the 1650 MeV state has a weaker signal (in part because of poorer data quality) and the two shapes can produce larger di!erences. Even there, the agreement in the case where all data is used (line 3 for NRB= vs. line 6 for RB=) is very good. More important di!erences are found when comparing K-matrix vs. Dyson equation results with the dispersion relation included (e.g. line 6 vs. line 9). The former is close to the model employed by Manley and Saleski [4]. These two models have di!erences of about 10% in the total width and up to 50% in the branching fractions. The most important deviation from the full result comes from the use of a truncated data set. For the 1535 MeV state, ignoring the interference with the gN "nal state causes the model to "t the Breit}Wigner shape to the cusp at the gN threshold. The VPI work [2] has a very small width for the 1535 MeV state; although the gN channel is mocked up, none of the actual data is used. For
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even the full model, leaving out the nnN "nal state data (such as was done by Batinic et al. [27]) produces 20% deviations in the branching fractions. We reproduce the updated results of the Batinic et al. paper [27]. Both the mass and the width of the S (1535) tend to increase as more data channels are added into the "t. It is interesting to note that the small widths (i.e. (100 MeV) are in situations where only the nN elastic data is used in the "t. Also, the branching fraction of gN tends to be smaller than that of nN unless the analyticity is taken into account in the phase space factor. In other words, when the cusp is handled appropriately, the gN channel becomes the dominant decay mode. The Manley}Saleski [4] analysis, which does not address the analyticity issue, is unable to match the cusp well and concludes that nN is the dominant decay mode. The S (1650) and the third S resonance have more random shifts in their resonance parameters. At energies in the region of the excitation of the third S state, only elastic and g production data exist and the signal is not strong. Therefore, the "t parameters for the third S state are largely determined by "tting non-statistical #uctuations of the data. The "tted masses vary widely from case to case and we feel the results with the di!erent models do not give information about features of the models. 4.5. Elastic data dependence A major question is the balance between model dependence and data dependence. The previous section showed the model dependence using various subsets of the data used in this analysis. In this section, we present results for "ts of the S channel using two di!erent sets of elastic partial wave amplitudes. In Table 6, we compare our standard results with the results of a "t using the combination of CMB80 and KH80 data sets. This "t uses the same elastic data as Manley and Saleski [4] except for the additional nNPgN data. The di!erences seen in the table are somewhat similar (in total width) to but somewhat smaller (in branching fraction) than those presented in the previous section, verifying that the major di!erence between the present work and Manley and Saleski is likely due to model dependence. The changes in the results are larger for the second state than for the lowest state, but the di!erences are signi"cant for both states. The di!erences are much larger for the third S , but are not shown because of the weak evidence for this state. Table 6 S Resonance parameters from "ts to di!erent elastic data sets. All results use the nNPgN and nNPnnN partial wave amplitudes as in the full analysis. Comparisons are made for results using the VPI and a mixture of the CMB80 and KH80 elastic partial wave amplitudes (as was done in Ref. [4]). The "rst two lines in the table give results for S (1535) and the last two lines give results for S (1650). Although all channels were used in the "t, only the total nnN branching fraction is given Elastic data set
Mass (MeV)
Width (MeV)
nN b.f. (%)
gN b.f. (%)
nnN b.f. (%)
VPI CMB/KH80 VPI CMB/KH80
1542 1535 1689 1691
112 137 202 222
35 35 74 58
51 53 6 15
14 12 20 27
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209
The balance between the branching fraction into nN and gN for the lower state are very similar in the two cases, but the total width found using the older data is 22% larger. This gives more evidence for the "ndings of the previous section that di!erences in branching fractions are a primary result of the di!erence between this model and that used by Manley and Saleski. 4.6. The non-resonant amplitude A major problem in any extraction of resonance parameters is in the careful separation of resonant and non-resonant mechanisms. Ideally, an independent calculation would be used for the non-resonant diagrams. Here, we choose a more objective strategy and use a smooth background that can then be `dresseda to contain the correct threshold behavior. To do this, we add two subthreshold resonances (one repulsive and one attractive) and one very high-energy resonance in each partial wave as &bare' propagators. These propagators are then dressed identically to the true resonances. More details are given in Section 2.2. The separation into resonant and non-resonant components is shown in Fig. 8 for various reactions in the S partial wave. We show the magnitude of the ¹-matrix for four di!erent "nal states calculated with "nal "t parameters. The three lines shown correspond to including only non-resonant couplings, only resonant couplings, and all couplings. Since the non-resonant and resonant processes are intermixed in the Dyson equation, there is no way to sum them to get the full result. While some reactions are dominated by resonant processes (e.g. elastic scattering), others are dominated by the non-resonant processes (e.g. nNPo N). The resonance excitation must be sampled through a variety of channels to provide the full picture. At very high energies (=&1.9 GeV), the lack of data allows the non-resonant processes to dominate. Resonance extraction at these high energies is very unclear with the data presently available.
5. Results and discussion We have applied the CMB model to the database presented in Section 3 } nN single-energy elastic ¹ matrices of VPI [2], the inelastic ¹ matrices of Manley et al. [25], and our own partial wave analysis of the nNPgN data. The nNPnnN raw data was not available in time for the present analysis. A reanalysis of the nNPnnN data is in progress [34] by our group; a more complete analysis can be presented when that work is "nished. The analysis presented here contains features of CMB [6] and KSU [4] since we use the formalism of the former and a data set similar to that used by the latter. However, the present analysis goes beyond any previously published. We present general results and a detailed discussion of the D , D , and S partial waves. D is an excellent example of an isolated resonance, but has strong inelastic couplings. The D and S partial waves each have a strong state (well understood for the former and poorly understood for the latter) along with less well-understood states; the interpretation of most of these states are sensitive to the features of the present model. We will compare our results to previous work with an emphasis on works that treat many channels. We also compare to the composite results of PDG [1].
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5.1. Details of xtting Although the nNPnnN data set used here is identical to that used by KSU, they do not include the nNPgN data and use older elastic analyses. KSU includes an gN channel in the S partial wave "t, using the requirements of unitarity to "x its coupling strength to the resonances. For the elastic channel, they use CMB and KH80 elastic amplitudes simultaneously. For the inelastic channels involving two pions, we de"ne channels identical to those chosen in Ref. [4]. They are listed in Table 3. Each channel has a speci"c orbital and spin angular momentum; the nomenclature used is given in the table. These nNPnnN channels are almost identical to those chosen in Ref. [25]. The only di!erence is in the choice of the mass and width of the "ctitious isoscalar-spin 0 meson, taken as 1 GeV for each parameter in Ref. [25]; like Ref. [4], we choose a mass and width of 800 MeV. We agree with their conclusion that the results are not sensitive to this choice. The number of states sought in each partial wave was the same as used by KSU [4]. We also included the same number of open channels in each partial as KSU because of the choices in the amplitudes "tted in Ref. [25]. If no amplitudes are provided by Ref. [25] for a partial wave, e.g. for F , an appropriate dummy channel is used to absorb the #ux. Table 8 labels rows of this type as &Flux'. The number of parameters depends on the number of resonances to be "t and the number of open channels included in the "t. For example, the S partial wave has three resonances and we "t values for each bare pole and its couplings to as many as 8 channels. There are also two subthreshold and one high energy `statesa used to simulate background. The masses of these states are constrained to be far away from any of the actual resonances. Three parameters, a pole and coupling strengths to nN and gN are "t for each subthreshold background pole. The high-energy state is allowed to couple to all open channels. Thus, there are 38 parameters "t in this partial wave. For the D partial wave, only one resonance (with coupling to four channels) and three background poles were "t, a total of 15 real parameters. In the elastic channel, the data quality is reasonable at values of = from threshold to about 2.0 GeV. At values of = larger than roughly 1.8 GeV, no inelastic data is available for the nnN "nal state. The inelastic data has signi"cant #uctuations. If the data are to be represented by a smooth function (assumed in all analyses), the error bars are underrepresented. In fact, the Manley et al. paper [25] states that only diagonal errors were included in their output. Correlations should be signi"cant in the analysis and can only add to the uncertainties. Although the elastic data were able to be "t well, the inelastic data were not. Since the elastic data is of much higher quality than the inelastic data, the inelastic error bars were weighted by a factor of 2 lower than the elastic error bars in order to ensure a reasonable "t to the elastic data. (The original Cutkosky et al. paper [6,32] used a factor of 3 to weight the inelastic data.) For elastic data, values of s/datapoint were 1.7 and 1.6 for the S and D waves, respectively. For the inelastic amplitudes, s/datapoint values were 9.2 and 22.2. The s values are given for the partial wave amplitude values without the extra weighting factor. (We do not quote s per degree of freedom because the parameters are shared between the elastic and inelastic "ts.) Although we get qualitatively better "ts to the elastic data than Manley and Saleski [4] in most cases, similar quality "ts are found for the inelastic amplitudes. (No values of s are given by KSU.) However, the shapes of our inelastic ¹ matrices are qualitatively di!erent in many cases. Speci"c partial waves will be discussed in Section 5.3 (Fig. 9).
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211
Fig. 9. ¹ matrices according to the best "t for the D partial wave. The VPI partial wave amplitudes are shown by data points with error bars. The KSU "t of Manley and Saleski is shown as dashed lines and the "t of this work is shown as solid lines.
For a complicated multi-parameter "t, errors are di$cult to determine because correlations can be signi"cant. The partial wave data we use as input quotes only diagonal errors. We include error estimates in all extracted quantities due to propagation of errors quoted in the partial wave data. In addition, we add contributions determined from additional "ts where the background parameterization is varied. To allow a full error analysis, separate "ts were made to data over a limited energy range to isolate single resonances. A 2;2 K matrix was used to model 2 channels at a time and a number of "ts were made for each resonance to determine errors on each of the extracted quantities. In one "t, the errors on the resonance mass and width and the error on the largest branching ratio were
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Fig. 9. (Continued.)
determined. With nN as one channel and the channel with the largest remaining branching fraction (often nD) for the resonance under consideration as the second channel, the resonance mass, width, and branching fraction together with a simple parameterization of the non-resonant amplitude were "t to the data. Errors for the branching fraction to the channels with less coupling strength were determined in "ts where the mass and width of the resonance were "xed at the "nal "t value of this work and only the branching fraction and background were "t. For the K matrix in these cases, the channel for which the branching fraction was being determined was one K-matrix channel and the nN channel or the channel with the largest branching fraction was chosen as the second K-matrix channel. In all cases, two "ts were done with di!erent simple assumptions for the background dependence, either #at vs. linear or linear vs. quadratic. The "rst type was most often
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213
Fig. 9. (Continued.)
used. Two components to the error for each parameter under study were determined from each pair of "ts, a relative error for each "t parameter from the "t with the largest s and the absolute di!erence between the two determinations of the "t parameter. The "rst component is representative of the statistical error in the data points; the second error is due to systematic e!ects between the di!erent background parameterizations. When two reasonable "ts were obtained, the components were added in quadrature. For prominent states, the "ts were easily made. However, most states required many trials to "nd "ts of the appropriate quality. Background shapes, channel choices, = ranges, and weighting of the inelastic channel in the s determination were all varied until two good "ts were obtained. All error bars quoted in the tables discussed below were determined from these "ts.
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5.2. General results A list of resonances found in this analysis is given in Tables 7 and 8 and compared to the results of KSU [4], those of CMB [6] and the latest recommended values given by PDG [1]. For various states, many other results exist. We make direct comparisons only with the previous results that use multi-channel models and provide error bars for their determinations. We show masses and widths in Table 7 and branching fractions to various inelastic channels in Table 8. These are the results of the analysis discussed in Section 2.3. Poles corresponding to most of the states found by KSU are found in this analysis, although the properties can be di!erent; states for which there is no evidence in the present analysis are P (1900), F (2000), and D (1940), all of which are 1H states according to PDG [1]. Although we obtain "t results for many weak states, a detailed study of the validity of the "t for weak states was not attempted. Discussion in this section will be limited to PDG 3H and 4H resonances. In Table 9, we show the results for pole positions of this analysis. The pole positions have less model dependence than the parameters in Tables 7 and 8. The analysis presented here is not identical to any previous analysis. The data base used is similar to KSU, but the formalism has signi"cant di!erences. The formalism is identical to CMB, but the data base used is quite di!erent. CMB used the best representation of data at the time } their own elastic amplitude analysis [20] and the inelastic quasi-two-body amplitudes of the SLAC-Berkeley [35] and Imperial College [36]. Batinic et al. [27] use a truncated version of the formalism used here and still another data set. Strong isolated resonances that have a strong elastic coupling are "t well with all models and results for the resonance parameters, such as the D (1675) and F (1680) masses and widths, tend to have close agreement between previous results and the new results. PDG gives a range of 15 MeV for both masses and a range of 40 and 20 MeV for the D and F full-widths and our values are within these intervals. For the elastic branching fraction, PDG suggests a range of 10%; our results are inside the range for F and just outside it for D . The bene"t of the multichannel analysis is readily apparent for states with a very small elastic branching fraction. For example, D (1700) and P (1710) (both PDG 3H resonances) are not seen in the VPI elastic analysis [2] because there are no strong signs of the resonance in the elastic ¹ matrix. However, there is a strong resonance signal in the nNPnD ¹ matrix in each case (Fig. 10). For the cases where KSU di!ers signi"cantly from the consensus of previous results, the S (1620) mass and the S (1650) elasticity, the new results tend to agree with the older values. To test for the dependence on the elastic data used in the "t, we re-did "ts with the same elastic data used by KSU and found results qualitatively similar to our "nal analysis. An unusual feature of the CMB analysis is the large value for the S (1535) width, about 40% larger than any other analysis. This analysis obtains a width in closer agreement with KSU and KH than CMB. We have been unable to reproduce the large width. We note that CMB based their "t on a subsidiary analysis of Bhandari and Chao [37] with a compatible model. Although the nNPgN data has changed little since then, the elastic data we use is a global "t to all data rather than the single data set they used. The VPI SM95 single energy solution is systematically larger and somewhat #atter than the data used in that 1977 work. The most confusing aspect is that Bhandari and Chao quote a full-width of 139$33 MeV, much more compatible with the present results than with the full CMB results.
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215
Table 7 Results for masses, full widths, and elastic fractions for all resonances found in this analysis. All resonances found in the KSU analysis were searched for, but not all were found. No attempt was made to "nd new resonances because then data quality is not good enough for a new search. See text for more details Resonance
Mass (MeV)
Width (MeV)
Elasticity (%)
Reference
S (1535) ****
1542(3) 1534(7) 1520}1555 1550(40) 1689(12) 1659(9) 1640}1680 1650(30) 1822(43) 1928(59) +2090 2180(80) 1479(80) 1462(10) 1430}1470 1440(30) 1699(65) 1717(28) 1680}1740 1700(50) 2084(93) 1885(30) +2100 2125(75) 1716(112) 1717(31) 1650}1750 1700(50) 1518(3) 1524(4) 1515}1530 1525(10) 1736(33) 1737(44) 1650}1750 1675(25) 2003(18) 1804(55) +2080 2060(80) 1685(4) 1676(2) 1670}1685 1675(10)
112(19) 151(27) 100}250 240(80) 202(40) 173(12) 145}190 150(40) 248(185) 414(157)
35(8) 51(5) 35}55 50(10) 74(2) 89(7) 55}90 65(10) 17(3) 10(10)
350(100) 490(120) 391(34) 250}450 340(70) 143(100) 478(226) 50}250 90(30) 1077(643) 113(44)
18(8) 72(5) 69(3) 60}70 68(4) 27(13) 9(4) 10}20 20(4) 2(5) 15(6)
260(100) 121(39) 383(179) 100}200 125(70) 124(4) 124(8) 110}135 120(15) 175(133) 249(218) 50}150 90(40) 1070(858) 447(185)
12(3) 5(5) 13(5) 10}20 10(4) 63(2) 59(3) 50}60 58(3) 4(2) 1(2) 5}15 11(5) 13(3) 23(3)
300(100) 131(10) 159(7) 140}180 160(20)
14(7) 35(1) 47(2) 40}50 38(5)
Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB
S (1650) ****
S (2090) *
P (1440) ****
P (1710) ***
P (2100) *
P (1720) ****
D (1520) ****
D (1700) ***
D (2080) **
D (1675) ****
(¹able continued on next page)
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T.P. Vrana et al. / Physics Reports 328 (2000) 181}236 Table 7 (Continued) Resonance
Mass (MeV)
Width (MeV)
Elasticity (%)
Reference
F (1680) ****
1679(3) 1684(4) 1675}1690 1680(10) 2311(16) 2086(28) +1990 1970(50) 2168(18) 2127(9) 2100}2200 2200(70) 1617(15) 1672(7) 1615}1675 1620(20) 1802(87) 1920(24) 1850}1950 1890(50) 1721(61) 1744(36) +1750 1995(12) 1882(10) 1870}1920 1910(40) 1234(5) 1231(1) 1230}1234 1232(3) 1687(44) 1706(10) 1550}1700 1600(50) 1889(100) 2014(16) 1900}1970 1920(80) 1732(23) 1762(44) 1670}1770 1710(30) 1932(100) 1956(22) 1920}1970 1940(30)
128(9) 139(8) 120}140 120(10) 205(72) 535(117)
69(2) 70(3) 60}70 62(5) 22(11) 6(2)
350(120) 453(101) 547(48) 350}550 500(150) 143(42) 154(37) 120}180 140(20) 48(45) 263(39) 140}240 170(50) 70(50) 299(118)
6(2) 20(4) 22(1) 10}20 12(6) 45(5) 9(2) 20}30 25(3) 33(10) 41(4) 10}30 10(3) 6(9) 8(3)
713(465) 239(25) 190}270 225(50) 112(18) 118(4) 115}125 120(5) 493(75) 430(73) 250}450 300(100) 123(53) 152(55) 150}300 300(100) 119(70) 599(248) 200}400 280(80) 316(237) 526(142) 250}450 320(60)
29(21) 23(8) 15}30 19(3) 100(1) 100(0) 98}100 100(0) 28(5) 12(2) 10}25 18(4) 5(4) 2(2) 5}20 20(5) 5(1) 14(6) 10}20 12(3) 9(8) 18(2) 5}20 14(4)
Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB
F (1990) **
G (2190) ****
S (1620) ****
S (1900) ***
P (1750) * P (1910) ****
P (1232) ****
P (1600) ***
P (1920) ***
D (1700) ****
D (1930) ***
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217
Table 7 (Continued) Resonance
Mass (MeV)
Width (MeV)
Elasticity (%)
Reference
D (2350) *
2459(100) 2171(18) +2350 2400(125) 1724(61) 1752(32) 1873(77) 1881(18) 1870}1920 1910(30) 1936(4.5) 1945(2) 1940}1960 1950(15)
480(360) 264(51)
7(14) 2(0)
400(150) 138(68) 251(93) 461(111) 327(51) 280}440 400(100) 245(12) 300(7) 290}350 340(50)
20(10) 0(1) 2(1) 9(1) 12(3) 5}15 8(3) 44(1) 38(1) 35}40 39(4)
Pitt-ANL KSU PDG CMB Pitt-ANL KSU Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB
F (1752) * F (1905) ****
F (1950) ****
There has been signi"cant interest and controversy in the properties of states in the P partial wave. In large part this is because the data in this partial wave has always been poor. In the present "t, s/datapoint for this partial wave is 3.8 for the elastic channel and ranges from 8.5 to 17.6 for the inelastic channels. With the low quality of the existing inelastic data, no e!ort was made to determine the correct number of P states. We "nd three P states with properties somewhat di!erent than those previously obtained, although all of our values are within the suggested ranges of PDG. In our results, P (1440) has a mass that just overlaps the PDG window and one of the largest widths obtained. In our "t, this state has a signi"cant contribution from non-resonant interactions; that together with the low quality of the present data produces a large estimated error for the values given. KSU results for this state show a somewhat smaller width and a much smaller error bar. We can only comment that the KSU model handles background quite di!erently than the present model and they used older nN elastic data. For the P (1710), we "nd mass and width values well within the large PDG ranges; however, KSU has a very large width with a large estimated error. Since this state sits on the tail of the P (1440) and does not have a strong signal in any channel, it is clear that the properties of the 2 lowest P states are closely coupled and multichannel analyses are very appropriate. A later analysis of Cutkosky and Wang [32] of this partial wave using the CMB model compared results obtained with the VPI (SM89) and CMB80 partial wave analyses. They "nd qualitatitively similar results (large width for the Roper and small width for the 1700 MeV state) to those obtained here. Evidence for the highest energy P state in the present data is very poor because only elastic data exists in the appropriate energy range. Other unusual cases found in this new analysis include the P (1910) and S (1900) states. The P (1910) is found at signi"cantly higher mass and has signi"cantly larger width than previous determinations. Since this state is at high mass, the inelastic data is very important in determining its properties. However, there is almost no existing inelastic data in this partial wave other than a few points in the nNPnNH(1440) reaction. PDG has given this state a 4H rating, but with the present data this rating should be downgraded. Although VPI does not "nd the S (1900) state, it is very prominent in KSU and PDG. We "nd the mass at 1802 MeV and a width of 48 MeV (with
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T.P. Vrana et al. / Physics Reports 328 (2000) 181}236 Table 8 Results for decay branching ratios of all resonances found in this analysis. Fractions are expressed as a percentage of the full width found in Table 7 Resonance
Channel
Pitt-ANL
KSU
PDG
S (1535)
nN gN o N (o N) " (n*) " (pN) . nNH(1440) nN gN o N (o N) " (n*) " (pN) . nNH(1440) nN gN o N (o N) " (n*) " (pN) . nNH(1440) nN gN o N (n*) . (pN) 1 K" nN gN o N (n*) . (pN) 1 K" nN gN o N (n*) . (pN) 1 K" nN gN o N Flux nN gN
35(4) 51(5) 2(1) 0(1) 1(1) 2(1) 10(9) 74(2) 6(1) 1(1) 13(3) 2(1) 1(1) 3(1) 17(7) 41(4) 36(1) 1(1) 1(1) 2(1) 2(1) 72(2) 0(1) 0(1) 16(1) 12(1) 0(1) 27(4) 6(1) 17(1) 39(8) 1(1) 10(10) 2(1) 61(61) 4(1) 2(1) 10(1) 21(20) 5(5) 4(1) 91(1) 0(1) 63(1) 0(1)
51(5) 43(6) 2(1) 1(1) 0(0) 1(1) 2(2) 89(7) 3(5) 0(0) 3(2) 2(1) 2(2) 1(1) 10(10) 0(3) 49(22) 0(1) 6(14) 4(10) 30(22) 69(3)
35}55 30}55 0}4
22(3) 9(2)
20}30 5}10
9(4)
10}20
3(7) 49(10) 2(4) 37(10) 15(6)
5}25 15}40 10}40 5}25
S (1650)
S (2090)
P (1440)
P (1710)
P (2100)
P (1720)
D (1520)
0}1 0}3 0}7 55}90 3}10 4}14 3}7 0}4 0}5
60}70 0}8
27(79) 24(18) 32(71) 2(6) 13(5)
10}20
87(5)
70}85
59(3)
50}60
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Table 8 (Continued) Resonance
D (1700)
D (2080)
D (1675)
F (1680)
F (1990) G (2190)
S (1620)
S (1900)
P (1750)
Channel
Pitt-ANL
KSU
PDG
(o N) 1 (n*) " (n*) 1 (pN) . nN gN (o N) 1 (n*) " (n*) 1 (pN) . nN gN (o N) 1 (n*) " (n*) 1 (pN) . nN gN o N (o N) " (n*) " nN gN (o N) $ (o N) . (n*) $ (n*) . (pN) " nN gN Flux nN gN (o N) " (u N) " nN o N (o N) " (n*) " nNH(1440) nN o N (o N) " (n*) " nNH(1440) nN nNH(1440) Flux
9(1) 11(2) 15(2) 1(1) 4(1) 0(1) 7(1) 79(56) 11(1) 0(1) 13(2) 0(2) 6(6) 17(10) 40(10) 24(24) 35(2) 0(1) 0(1) 1(1) 63(2) 69(1) 0(1) 3(1) 5(1) 1(1) 14(3) 9(1) 22(3) 0(1) 77(77) 20(1) 0(1) 29(28) 51(51) 45(1) 14(3) 2(1) 39(2) 0(1) 33(6) 30(2) 5(1) 28(1) 4(1) 6(6) 83(1) 11(11)
21(4) 15(4) 5(3)
15}25 5}12 10}14 0}8 5}15
1(2) 13(17) 80(19) 5(10) 2(4) 23(3) 26(14) 21(14) 3(7) 27(12) 47(2)
0}35
40}50
0(0) 0(0) 53(2) 70(3)
1}3
2(1) 5(3) 1(1) 10(3) 12(3) 6(2) 94(2)
1}5 0}12 0}2 6}14 5}20
22(1)
10}20
29(6) 49(7) 9(2) 25(6) 4(3) 62(6)
20}30 7}25
41(4) 5(7) 33(10) 16(8) 6(9) 8(3) 28(9) 64(9)
50}60 60}70
30}60 10}30
(¹able continued on next page)
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T.P. Vrana et al. / Physics Reports 328 (2000) 181}236 Table 8 (Continued) Resonance
Channel
Pitt-ANL
KSU
PDG
P (1910)
nN nNH(1440) Flux nN (n*) . nNH(1440) nN (n*) . nNH(1440) nN (n*) . nNH(1440) nN (o N) 1 (n*) " (n*) 1 nN K" Flux nN K" Flux nN (o N) . (n*) $ (n*) . nN (o N) . (n*) $ (n*) . nN (o N) $ (n*) $ Flux
29(29) 56(7) 15(15) 100(1) 0(1) 0(1) 28(5) 59(10) 13(4?1) 5(61?) 41(2.9) 53(8.2) 5(1.6) 1(1) 4(1) 90(1.7) 9(8) 91(11)
23(8) 67(10) 10(1) 100(0) 0(0) 0(0) 12(2) 67(5) 20(4) 2(2) 83(26) 15(24) 14(6) 8(4) 4(3) 74(7) 18(2)
15}30
P (1232) P (1600) P (1920) D (1700)
D (1930) D (2350) F (1752)
F (1905)
F (1950)
7(14) 93(15) 0(1) 60(60) 40(1) 0(1) 9(2.2) 24(1) 44(1) 23(1) 44(1) 36(1) 20(20)
98}100
10}25 40}70 10}35 5}20
10}20 5}20 1}7 25}50 5}20
82(2) 2(0) 98(0) 2(1) 22(14) 48(16) 28(18) 12(3) 86(3) 0(1) 1(3) 38(1) 43(1) 18(3)
5}15 0}60 0}25 35}40 0}10 20}30
a large estimated error) while KSU "nds 1950 MeV and 263 MeV. This signi"cant di!erence is largely due to the elastic data sets used. There is a strong bump at about 1900 MeV in the elastic ¹ matrices used by KSU which has vanished in the VPI partial wave amplitudes. For the branching fractions presented in Table 8, the only recent result is KSU, which is presumably weighted heavily in the PDG listings (also shown in the table). We divide the oN and nD channels into the appropriate spin channels since the component angular momenta can sometimes have more than one value. For D , the oN spin can be with orbital angular momentum 2 or with orbital angular momentum 0 or 2. Since only the spin orbital angular momentum 0 case was found to be important in the nNPnnN isobar analysis [25], this is the only
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221
Table 9 Pole positions. The complex energy of the pole for each state is given along with the physical mass Resonance
Res mass (MeV)
Pole position (MeV)
S (1535) S (1650) S (2090) P (1440) P (1710) P (2100) P (1720) D (1520) D (1700) D (2080) D (1675) F (1680) F (1990) G (2190) S (1620) S (1900) P (1750) P (1910) P (1232) P (1600) P (1920) D (1700) D (1930) D (2350) F (1752) F (1905) F (1950)
1545 1693 1822 1479 1699 2083 1716 1520 1729 2002 1687 1679 2311 2168 1633 1798 1721 1995 1234 1687 1889 1732 1932 2459 1724 1873 1936
1525}51i 1663}120i 1795}110i 1383}158i 1679}66i 1810}311i 1692}47i 1504}56i 1704}78i 1824}307i 1674}60i 1667}61i 2301}101i 2107}190i 1607}74i 1795}29i 1714}34i 1880}248i 1217}48i 1599}156i 1880}60i 1726}59i 1883}125i 2427}229i 1697}56i 1793}151i 1910}115i
oN channel we include for this partial wave. The nD channel can couple with orbital angular momentum of 0 or 2 in this partial wave and both possibilities are included in the "ts. As with KSU, uncertain "ts due to underestimated error bars and/or missing data make interpretation di$cult in some cases. We quote values for this analysis and give estimated errors for each quantity. Since we use the same inelastic ¹ matrices as KSU for input to the "t, the results should be qualitatively similar. Based on our study in Section 4.4, we feel there is roughly 20% di!erence between the KSU and Pitt-ANL results due to model dependence in the most sensitive quantities. The poor "ts make this true less often than might be expected. For F (1680), the elastic branching fraction is &70% in both analyses and the largest inelastic channels are nD P-wave and (nn) N in Q both; agreement is within errors for the largest values. For D (1700), the elastic branching fraction is small and the inelastic strength in concentrated in the nD s-wave, so we are in close agreement with KSU. However, S (1620) is a strong state where agreement is not good. There are 2 strong
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T.P. Vrana et al. / Physics Reports 328 (2000) 181}236
Fig. 10. ¹ matrices for D
partial wave. The same labeling is used as in Fig. 9.
inelastic channels, o N and nD D-wave. Although there is agreement in the strength of o N, the elastic strength is much smaller for KSU and the nD branching fraction is of course larger. The full-widths seen are only 15% di!erent and the estimated errors overlap. The s per datapoint for the S partial wave in the elastic channel is 2.1 and ranges from 4.3 to 17.2 for the inelastic channels. The calculation of Capstick and Roberts uses a relativized quark model to calculate resonance masses and a P model for creation of qq pairs [18]. They "t the two quark level coupling parameters to the nN decay amplitudes of the non-strange resonances rated by PDG as 2H or better and then predict the remaining decay amplitudes for various meson#baryon "nal states (including many "nal states for which there are no data). Although their primary purpose was to
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223
Fig. 10. (Continued.)
look for signatures of `missinga quark model resonances, we note that the qualitative agreement with our analysis is satisfactory. There are notable successes such as D (1520) where quantitative agreement comes with all signi"cant decay channels and notable failures such as S (1535) where both nN and gN decay widths (and the full-width) are overestimated by a factor of 4. 5.3. Detailed discussion } D , D , and S partial waves Presented in this section are detailed results for three representative partial waves. It includes "gures of all the channel ¹ matrices and a discussion of the results. These partial waves have
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Fig. 10. (Continued.)
reasonable quality data and contain both prominent and less prominent states. Numerical data can be found in Tables 4, 5, 8, and 7. The ¹ matrices are found in Fig. 9 for the D partial wave, 10 for D , and 11 for S . In each we show the relevant phase shift amplitudes [2,25] along with the "t of Manley and Saleski [4] (dashed lines) and our "nal "t (solid line). Only data up to ="2.15 GeV were used in the "ts because the data at higher values of = are of diminished quality. Since all the data in each partial wave were simultaneously "t with the requirements of unitary, not all channel data shown in the "gures is "t equally well. In general, the "ts to the elastic data are good (s/datapoint &2) and the "ts to the inelastic data are poor (s/datapoint'10) for both the KSU "ts and the present results, as discussed above.
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225
Fig. 10. (Continued.)
The D partial wave contains a strong (PDG [1] 4H), isolated resonance for which the results of the one resonance example (see Section 4.1) apply fairly well. In this case, both the elastic and nNPnD (Dwave) inelastic channels are prominent in the D partial wave. Since neither threshold is nearby and there is no interfering resonance, the ¹ matrix in each of these channels has the characteristic shape of an isolated resonance with a peak in the elastic and inelastic cross section near the resonance mass. The non-resonant amplitude is a smooth function. The KSU and present "ts are similar in the ability to match features in the data in the two strong channels near the cross section peak at =K1680 MeV. The two analyses di!er in their description of the high energy data (=51.8 GeV), but the data are very sparse there. Neither model "ts the smaller o production channels well. The o N channel shows that the model expectations (based on the requirements of unitarity and the data in the other channels) do not match the data in this channel and a poor "t results. Essentially all analyses for the D (1680) resonance give similar values for the mass, full-width, and elastic fraction (see Table 7). Even though VPI [2] accounts for the inelasticity by using a dummy channel, the "t parameters for this state are similar to those obtained in the present "t. PDG [1] gives quite small error bars for the mass and width of this state, re#ecting the unanimity of the "tting results for this state. It is encouraging that the complicated features of the present model are not important for the simple case presented in this partial wave. The D partial wave contains a 4H state at 1520 MeV (PDG standard value, the actual mass is slightly di!erent), a 3H state at 1700 MeV, and a 2H state at 2080 MeV. Since the lowest two resonances have signi"cant overlap in energy, the features of the CMB model are important for this partial wave. The lowest state is highly elastic, but also shows up prominently in (o N) 1 and (nD) "nal states. Note that the peak in the imaginary ¹ matrix is inverted. As already 1 mentioned, the second state is barely seen in the elastic channel, thus is not found in the VPI [2] analysis. Both KSU and the present analysis "nd most of the decay strength in the nnN "nal states with less than 10% of the decay strength to the elastic channel. KSU di!ers with this model in the
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Fig. 11. ¹ matrices for S partial wave. The same labeling is used as in Fig. 9.
distribution of the inelastic strength. While KSU "nds most of the strength in oN, this model "nds it in nD. The KSU model "ts available inelastic data better. Feuster and Mosel [5] "nd a systematically lower width than other models for the lowest state, but their width for the 2nd state is unusually large. Evidence for the third state is very weak in this analysis; there is only elastic data at =51.9 GeV and there is very little evidence of structure. It is best "t with a very large width. The three resonances have varying strength in the di!erent channels. As a result, the "tted curve also changes signi"cantly from channel to channel. In the nD channels, the shapes are complicated because the 1700 MeV state is important. The imaginary part of the ¹ matrix has a dip at the 1520 MeV state and a peak at the 1700 state. Nevertheless, this model and the KSU model "t the features well and the "tted parameters for the higher energy state agree within stated errors.
T.P. Vrana et al. / Physics Reports 328 (2000) 181}236
227
Fig. 11. (Continued.)
The S partial wave was already discussed in Section 4 with regard to the signi"cant model dependence seen. As a result of this model dependence, the S (1535) full-width has been quoted as 66}250 MeV based on di!erent analyses of similar data (almost all "ts are based prominently on the elastic nN channel). The relatively small width of the S (1535) in this analysis (112$30 MeV) is determined in large part by a signi"cant overlap of the S (1535) with the S (1650). This overlap causes a large interference e!ect. Although a similar e!ect was seen in one description of the cpPgp data that uses a formalism similar to the Pitt-ANL model [33], many models based on a ¹-matrix formalism (e.g. [38}40]) have a formalism where it is di$cult to account for more than one resonance in a partial wave.
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Fig. 11. (Continued.)
The multichannel K-matrix analysis of Feuster and Mosel [5] gets values of the full-width in the range of 151}198 MeV for di!erent form factors. However, their model does not handle the gN cusp as well because it lacks the analytic phase space, does not model the nnN inelasticity well because a zero width particle is used, and does not include o!-shell intermediate state scattering e!ects because of the K-matrix approximation. Although 8 channels are included in the "t, many of them turn out to have small coupling to S (1535) (in agreement with PDG [1]). The two major channel couplings for S (1535) are nN and gN. Therefore, another determining factor in the full-width is the total cross section for the
T.P. Vrana et al. / Physics Reports 328 (2000) 181}236
229
Fig. 11. (Continued.)
n\pPgn in Fig. 11 and the corresponding ¹ matrix in Fig. 11b. The cross section of Fig. 12 is made up of nNPgN ¹-matrix amplitudes for S through G , however its main component is S . Although the shape of the cusp (see Fig. 7) does carry information about the gN channel, the overall data quality isn't satisfactory. We await higher quality data for a more conclusive determination of this amplitude. The branching fraction of the S (1535) to the nN channel is at the low end of the PDG range. A surprising feature of the S (1535) has been its unusually large decay width to gN. We "nd strong evidence for a value at the high end of the PDG range. We believe our result because our model fully accounts for the threshold enhancement (cusp e!ect) due to the gN channel opening and the
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T.P. Vrana et al. / Physics Reports 328 (2000) 181}236
Fig. 12. Total cross section for the n\pPgn reaction. The data was taken from sources in Ref. [27], leaving out the low energy Brown et al. data. The solid line shows the total cross section calculated with the "nal "t result of this work using all partial waves. The contribution from the S partial wave would be very similar to the full result.
interference of the two overlapping resonances. Since no other model has such a complete formulation, they can have model-dependent systematic errors as a result. The properties of S (1650) are largely determined through the prominent bump in the elastic channel. The elastic ¹ matrix has a peak in the imaginary part at about the right energy and the real part is decreasing at that energy; a strong resonance signal. The strongest inelastic signal comes in the o N channel. The decay width to gN is small in this analysis, in agreement with previous analyses. The striking di!erence between branching fractions to gN has been another unusual feature of these states. The existence of the third S resonance is very weakly supported by the existing data. In our analysis, this state is most often determined by the need to "t non-statistical features in the partial wave amplitudes. Further data will be required to sort out the question of whether or not it exists. The role of the inelastic gN channel is very important in the determination of resonance parameters for the S (1535) state, as shown in some detail in Section 4. Unfortunately, the existing data for nNPgp is sparse and of uncertain quality [29] (see Fig. 12). Recently published data [41] for cpPgN has been interpreted as evidence for a very large width (K200 MeV). We do not include the Krusche et al. data in this analysis. However, very simple resonance models [41,42] were used to determine the large width. Since the large body of nN data was not "tted with their model, possible inconsistencies exist in their parameterization. Although the simple model has validity for the gN coupling to S (1535), we feel there will still be signi"cant e!ects from interference with S (1650) and coupled channel e!ects (e.g. cNPS (1650)PnNP S (1535)PgN).
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231
Fig. 13. Pitt-ANL model calculation using "nal fit compared with the total cross section for the process n\pPnn. The "t was to ¹-matrix amplitudes derived in part from the data shown here. Data shown is from the VPI SAID database.
5.4. Observables ¹-matrix amplitudes were "t using the Pitt-ANL model. We do not "t directly to data; rather, we "t the partial wave amplitudes. It is then interesting to compare the actual experimental data to our calculated observables to verify that nothing has been lost by not "tting to the actual data points. The f and g spin non-#ip and spin #ip amplitudes are de"ned in terms of the Pitt-ANL ¹-matrix elements for nN elastic scattering by [43] 1 f (h)" [(l#1)¹l #l¹l ]Pl (cos h) , > \ ql 1 g(h)" (¹l !¹l )Pl (cos h) , (39) > \ ql where Pl and Pl are the Legendre polynomials and their "rst derivative with respect to cos h, respectively. The ¹l are the amplitudes for a particular nN scattering charge channel, where the ! l$ refers to the total spin J"l$, and q is the center of mass momentum of the "nal state pion. The ¹l are de"ned by Eq. (1) for the nNPnN channel. ! The di!erential cross section (dp/dX) and polarization P are: dp "" f (h)"#"sin h g(h)" , dX dp P"2Im [ f (h)gH(h)] sin h . dX
(40)
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Fig. 14. Pitt-ANL model calculation using "nal fit parameters compared to the di!erential cross section for the process n>pPn>p. Data from the VPI SAID database is shown.
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233
Fig. 15. Pitt-ANL model calculation using "nal fit parameters of the polarization, P, for the process n\pPn\p. Data from the VPI SAID database is shown.
Fig. 13 shows a plot of the total cross section vs. the Pitt-ANL calculation for the process n\pPnn. Fig. 14 shows plots of the di!erential cross section, dp/dX, at a number of di!erent energies for the process n>pPn>p. Fig. 15 shows plots of the nucleon recoil polarization, P, for a number of di!erent energies for the process n\pPn\p. In each case, the calculation using "nal "t parameters is in good agreement with the data. For the P data, the error bars are sometimes large. For the plot in Figs. 13}15 the data is taken from VPI's SAID database. It is a large collection of scattering data for a number of di!erent reaction channels, including nN elastic and charge exchange reactions. Based on our studies, we can conclude that our resonance parameters are very close to what would have been obtained if the "t had been made to the original data.
6. Conclusions We have presented a new study of excitation of baryon resonances through pion-nucleon interactions. An expanded version of the Carnegie-Mellon Berkeley (CMB) was used in this work.
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We added the gN channel to the calculation and used the latest data sets. The model satis"es many of the desirable theoretical constraints } two- and three-body unitarity, time reversal invariance, and analyticity. Because the model is based on a separable interaction, it is linked to a Hamiltonian approach, unlike many of the models previously used [2,4]. The model has a key feature of allowing large numbers of asymptotic channels and more than one resonance per partial wave. Since analyticity is satis"ed and all inelastic thresholds are included, structure in the physical observables due to non-resonant e!ects are carefully treated. Resonances can then be found as poles in the complex s plane. As in the original CMB model, non-resonant processes are included as very wide resonances at very high or very low energies which are then subject to the same threshold e!ects as the resonances. This empirical approach is well established in potential scattering theory, but could be replaced later by theoretical calculations based on meson exchange models, e.g. [44]. Rather than deal with the complexities of a direct "t to the cross section and polarization data, we use the partial wave amplitudes of the VPI and KSU groups [2,25] and our own "t to the nNPgN data. A re-evaluation of the nNPnnN data is in progress by our group. The features of this model are expected to be very important in the S partial wave where there are two overlapping resonances at 1535 and 1650 MeV and a strong threshold opening. The lower state pole and the gN channel opening are close together and the quantum mechanical interference of the two states is found to be signi"cant. We show how the interpretation of these states has very strong model dependence and how relaxing the features of the present model can skew the "tting results for the width and branching fractions to nN and gN. We get a new result for the branching fraction to gN at the high end of what has been previously published for S (1535), making its characteristics even more unusual. The general "tting results are presented in Tables 7 and 8. We searched for the same states seen in the previous analysis of Manley and Saleski [4]. This simpli"cation was made because the data quality was not high enough for a valid search for weakly excited states. Although values of s/datapoint are roughly 2 for elastic "ts, they are 10 or more for "ts to the nNPnnN ¹ matrix data. This is perhaps due to the lack of a treatment of correlations in the error analysis of the pion production data. Modern experimental techniques could greatly improve the inelastic data set. We strongly encourage new measurements of the inelastic channels. A signi"cant e!ort was made to determine error bars re#ecting both the estimated errors in the ¹ matrices used in the "ts and the di!ering choices of background energy dependence. We "nd strong and isolated states (e.g. P (1232) and D (1675)) with very similar parameters as previous analyses [2}4,6]. States with strong model dependence such as S (1535) or with signi"cant changes in the data set such as S (1900) and D (1700) get quite di!erent results. We "nd a full width of S (1535) at the low end of the range of previous values, especially with respect to the interpretations of recent cpPgp data [41]. The results presented here do not include the photoproduction data; rather, the present results are dependent on a fairly weak data set for n\pPgn. Nevertheless, the large widths obtained in analyses highly dependent on the Mainz eta photoproduction data are only loosely coupled to the large data set used in this study. We are in the process of further extending the model to include photoproduction and electroproduction reactions. The Born terms are included for production of pions and etas and the resonance spectrum is being re"t. Results of the present paper will then be subject to change.
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Acknowledgements We are grateful to R. Arndt, R. Workman, and M. Manley for sharing their partial wave data and their analysis results with us. We have also bene"ted greatly from the programming and database work of D. Ciarletta, J. DeMartino, J. Greenwald, D. Kokales, M. Mihalcin, and K. Bordonaro at the University of Pittsburgh. Special thanks go to C. Tanase for investigating the analytic structure of this model in detail. References [1] Particle Data Group, Phys. Rev. D 54 (1996) 1. [2] Richard A. Arndt, Igor I. Strakovsky, Ron L. Workman, Marcello M. Pavan, Phys. Rev. C 52 (1995) 2120. [3] G. HoK hler, F. Kaiser, R. Koch, E. Pietarinen, Handbook of Pion}Nucleon Scattering, [Physics Data No. 12-1 (1979)]. [4] D.M. Manley, E.M. Saleski, Phys. Rev. D 45 (1992) 4002. [5] T. Feuster, U. Mosel, Phys. Rev. C 58 (1998) 457. [6] R.E. Cutkosky, C.P. Forsyth, R.E. Hendrick, R.L. Kelly, Phys. Rev. D 20 (1979) 2839; R.E. Cutkosky, C.P. Forsyth, J.B. Babcock, R.L. Kelly, R.E. Hendrick, in: N. Isgur (Ed.), Baryons 1980, Proceedings of the IV International Conference on Baryon Resonances, University of Toronto, 1980, p. 19. [7] F. Foster, G. Hughes, Rep. Prog. Phys. 46 (1983) 1445. [8] N. Isgur, G. Karl, Phys. Rev. D 18 (1978) 4187; D 19 (1979) 2653. [9] S. Capstick, N. Isgur, Phys. Rev. D 34 (1986) 2809, S. Capstick, B.D. Keister, Phys. Rev. D 51 (1995) 3598. [10] R. Bijker, F. Iachello, A. Leviatan, Ann. Phys. (N.Y.) 236 (1994) 69. [11] F. Cardarelli, E. Pace, G. Salme, S. Simula, Phys. Lett. B 357 (1995) 267. [12] G.E. Brown, M. Rho, Phys. Lett. B 82 (1979) 177; G.E. Brown, M. Rho, V. Vento, Phys. Lett. B 94 (1979) 383. [13] S. Theberg, A.W. Thomas, G.A. Miller, Phys. Rev. D 22 (1980) 2838; 24 (1981) 216; D.H. Liu, A.W. Thomas, A.G. Williams, Phys. Rev. C 55 (1997) 3108. [14] Chr. V. Christov et al., Prog, Part. Nucl. Phys. 37 (1996) 91; R. Alkofer, H. Reinhart, H. Weigel, Phys. Rep. 265 (1996) 139. [15] See review by L. Wilets, Non-Topological Solitons, World Scienti"c, Singapore, 1989. [16] Zhenping Li, Phys. Rev. D 50 (1994) 5639; A. Glozman, D.O. Riska, Phys. Rep. 268 (1996) 263; P.-N. Shen, Y.-B. Dong, Z.-Y. Zhang, Y.-W. Yu, T.-S.H. Lee, Phys. Rev. C 55 (1997) 2024. [17] S. Capstick, Phys. Rev. D 46 (1992) 2864. [18] S. Capstick, W. Roberts, Phys. Rev. D 47 (1993) 1994; Phys. Rev. D 49 (1994) 4570. [19] F. Butler, H. Chen, J. Sexton, A. Vaccarino, D. Weingarten, Phys. Rev. Lett. 70 (1993) 2849. [20] R.E. Cutkosky, R.E. Hendrick, J.W. Alcock, Y.A. Chao, R.G. Lipes, J.C. Sandusky, R.L. Kelly, Phys. Rev. D 20 (1979) 2804; R. Kelly, R.E. Cutkosky, Phys. Rev. D 20 (1979) 2782. [21] R. Koch, Z. Phys. C 29 (1985) 597; Nucl. Phys. A 448 (1986) 707; R. Koch, E. Pietarinen, Nucl. Phys. A 336 (1980) 331. [22] G. HoK hler, nN Newsletter, G. HoK hler, W. Kluge, B.M.K. Nefkens (Eds.), 13 (1997) 320. [23] R.G. Moorhouse, H. Oberlack, A.H. Rosenfeld, Phys. Rev. D 9 (1979) 1. [24] R. Bhandari, Y.-A. Chao, Phys. Rev. D 15 (1977) 192. [25] D.M. Manley, R.A. Arndt, Y. Goradia, V.L. Teplitz, Phys. Rev. D 30 (1984) 904. [26] G. HoK hler, nN Newsletter, G. HoK hler, W. Kluge, B.M.K. Nefkens (Eds.), 9 (1993) 1. [27] M. Batinic, I. Slaus, A. Svarc, B.M.K. Nefkens, Phys. Rev. C 51 (1995) 2310. [28] R.M. Brown et al., Nucl. Phys. B 153 (1979) 89. [29] R. Clajus, B.M.K. Nefkens, n N Newsletter, G. HoK hler, W. Kluge, B.M.K. Nefkens (Eds.), 7 (1991) 76. [30] T.P. Vrana, University of Pittsburgh Ph.D. Thesis, unpublished. [31] William R. Frazer, Archibald W. Henry, Phys. Rev. 134 (1964) B1307. [32] R.E. Cutkosky, S. Wang, Phys. Rev. D 42 (1990) 235.
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PHASES OF DENSE MATTER IN NEUTRON STARS
Henning HEISELBERG , Morten HJORTH-JENSEN NORDITA, Blegdamsvej 17, DK-2100 K~benhavn }, Denmark Department of Physics, University of Oslo, N-0316 Oslo, Norway
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Phases of dense matter in neutron stars Henning Heiselberg *, Morten Hjorth-Jensen NORDITA, Blegdamsvej 17, DK-2100 K~benhavn }, Denmark Department of Physics, University of Oslo, N-0316 Oslo, Norway Received August 1999; editor: M. Kamionkowski Contents 1. Introduction 1.1. The past, present and future of neutron stars 1.2. Physics of neutron stars 2. Phases of dense matter 2.1. Prerequisites and de"nitions 2.2. Nucleonic degrees of freedom 2.3. A causal parametrization of the nuclear matter EoS 2.4. Hyperonic matter 2.5. Kaon condensation 2.6. Pion condensation 2.7. Super#uidity in baryonic matter 2.8. Quark matter 3. Thermodynamics of multi-component phase transitions 3.1. Maxwell construction for one-component systems 3.2. Two-component systems in a mixed phase 4. Structure of neutron stars 4.1. Screening lengths 4.2. Surface and Coulomb energies of the mixed phase 4.3. Is the mixed phase energetically favored?
240 240 241 243 245 248 265 269 275 276 277 281 283 283 285 285 286 287 289
4.4. Melting temperatures 4.5. Funny phases 4.6. Summary of neutron star structures 5. Observational consequences for neutron stars 5.1. Masses from radio pulsars, X-ray binaries and QPO's 5.2. TOV and Hartle's equations 5.3. Neutron star properties from various equations of state 5.4. Maximum masses 5.5. Phase transitions in rotating neutron stars 5.6. Core quakes and glitches 5.7. Backbending and giant glitches 5.8. Cooling and temperature measurements 5.9. Supernovae 5.10. Gamma-ray bursters 6. Conclusions 6.1. Many-body approaches to the equation of state 6.2. Phase transitions and sti!ness of EoS from masses of neutron stars Acknowledgements References
*Corresponding author. E-mail addresses:
[email protected] (H. Heiselberg),
[email protected] (M. Hjorth-Jensen) 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 1 0 - 6
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Abstract Recent equations of state for dense nuclear matter are discussed with possible phase transitions arising in neutron stars such as pion, kaon and hyperon condensation, super#uidity and quark matter. Speci"cally, we treat the nuclear to quark matter phase transition, the possible mixed phase and its structure. A number of numerical calculations of rotating neutron stars with and without phase transitions are given and compared to observed masses, radii, temperatures and glitches. 2000 Elsevier Science B.V. All rights reserved. PACS: 12.38.Mh; 21.30.!x; 21.65.#f; 26.60.#c; 97.60.Gb; 97.60.Jd Keywords: Neutron star properties; Phase transitions; Equation of state for dense neutron star matter
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1. Introduction 1.1. The past, present and future of neutron stars The discovery of the neutron by Chadwick in 1932 prompted Landau to predict the existence of neutron stars. The birth of such stars in supernovae explosions was suggested by Baade and Zwicky in 1934. First theoretical neutron star calculations were performed by Tolman, Oppenheimer and Volko! in 1939 and Wheeler around 1960. Bell and Hewish were the "rst to discover a neutron star in 1967 as a radio pulsar. The discovery of the rapidly rotating Crab pulsar in the remnant of the Crab supernova observed by the Chinese in 1054 A.D. con"rmed the link to supernovae. Radio pulsars are rapidly rotating with periods in the range 0.033 s4P44.0 s. They are believed to be powered by rotational energy loss and are rapidly spinning down with period derivatives of order PQ &10\}10\. Their high magnetic "eld B leads to dipole magnetic braking radiation proportional to the magnetic "eld squared. One estimates magnetic "elds of the order of B&10}10 G. The total number of pulsars discovered so far has just exceeded 1000 before the turn of the millenium and the number is increasing rapidly. A distinct subclass of radio pulsars are millisecond pulsars with periods between 1.56 ms4 P4100 ms. The period derivatives are very small corresponding to very small magnetic "elds B&10}10 G. They are believed to be recycled pulsars, i.e. old pulsars with low magnetic "elds that have been spun up by accretion preserving their low magnetic "eld and therefore only slowly spinning down. About 20 } almost half of the millisecond pulsars } are found in binaries where the companion is either a white dwarf or a neutron star. Six double neutron stars are known so far including the Hulse}Taylor PSR 1913#16. The "rst binary pulsar was found by Hulse and Taylor in 1973 and by measuring the general relativistic corrections to Newtonian gravity one could determine all parameters in the binary system as both masses, orbital periods and period derivatives, orbital distances and inclination. Parameters are overdetermined and thus provide a test of general relativity. Inward spiralling or orbital decay is an additional test of general relativity to an unprecedented accuracy. The binary neutron stars all have masses in the narrow interval 1.3}1.5M , which may either be due to the creation process or that heavier neutron stars > are unstable. With X-ray detectors on board satellites since 1971 almost two hundred X-ray pulsars and bursters have been found of which the orbital period has been determined for about sixty. The X-ray pulsars and bursters are believed to be accreting neutrons stars from high (M910M ) and > low mass (M:1.2M ) companions, respectively. The X-ray pulses are most probably due to > strong accretion on the magnetic poles emitting X-ray (as northern lights) with orbital frequency. The X-ray bursts are due to slow accretion spreading all over the neutron star surface before igniting in a thermonuclear #ash. The resulting (irregular) bursts have periods depending on accretion rates rather than orbital periods. Recently, bursters and pulsars have been linked by observations of X-ray pulsations in bursts from several low mass X-ray bursters [1]. The pulsations are with spin frequency 300}400 Hz and increase by a few Hz only during the burst. The small increase is expected from cooling after a thermonuclear explosion, which leads to smaller size and moment of inertia and, conserving angular momentum, to larger frequency. The radiation from X-ray bursters is not blackbody and therefore only upper limits on temperatures can be extracted from observed luminosities in most cases. Masses are less accurately measured than for binary
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pulsars. We mention recent mass determinations for the X-ray pulsar Vela X-1: M"(1.9$0.1)M , and the burster Cygnus X-2: M"(1.8$0.2)M , which will be discussed > > later. A subclass of six anomalous X-ray pulsars are slowly rotating but rapidly spinning down indicating that they are young with enormous magnetic "elds, B&10 G, and thus named Magnetars [3]. Recently, quasi-periodic oscillations (QPO) have been found in 12 low mass X-ray binaries. The QPO's set strict limits on masses and radii of neutrons stars, but if the periodic oscillations arrive from the innermost stable orbit [4], it implies de"nite neutron star masses up to MK2.3M . > Non-rotating and non-accreting neutron stars are virtually undetectable. With the Hubble space telescope one single thermally radiating neutron star has been found [5]. Its distance is only 160 pc from Earth and its surface temperature is ¹K60 eV. From its luminosity one deduces a radius of the neutron star R414 km. In our galaxy astrophysicists expect a large abundance &10 of neutron stars. At least as many supernova explosions have occurred since Big Bang which are responsible for all heavier elements present in the Universe today. The scarcity of neutron stars in the solar neighborhood may be due to a high initial velocity (asymmetric `kicka) during their birth in supernovae. Recently, many neutron stars have been found far away from their supernova remnants. Future gravitational microlensing observation may determine the population of such `invisiblea neutron stars as dark matter objects in the galactic halo. From the view of physicists (and mass extinctionists) supernova explosions are unfortunately rare in our and neighboring galaxies. The predicted rate is 1}3 per century in our galaxy but the most recent one was 1987A in LMC. With luck we may observe one in the near future which produces a rapidly rotating pulsar. Light curves and neutrino counts will test supernova and neutron star models. The rapid spin down may be exploited to test the structure and possible phase transitions in the cores of neutron stars [6}8]. The recent discovery of afterglow in gamma ray bursters (GRB) allows determination of the very high redshifts (z51) and thus the enormous distance and energy output E&10 ergs in GRB if isotropically emitted. Very recently evidence for beaming or jets has been found [2] corresponding to `onlya E&10 ergs. Candidates for such violent events neutron star mergers or a special class of type Ic supernova (hypernovae) where cores collapse to black holes. The latter is con"rmed by recent observations of a bright supernova coinciding with GRB 980326. The marvelous discoveries made in the past few decades will continue as numerous earth-based and satellite experiments are running at present and more will be launched. History tells us that the future will bring great surprises and discoveries in this "eld.
1.2. Physics of neutron stars The physics of compact objects like neutron stars o!ers an intriguing interplay between nuclear processes and astrophysical observables. Neutron stars exhibit conditions far from those encountered on earth; typically, expected densities o of a neutron star interior are of the order of 10 or more times the density o +4;10 g/cm at &neutron drip', the density at which nuclei begin to dissolve and merge together. Thus, the determination of an equation of state (EoS) for dense matter is essential to calculations of neutron star properties. The EoS determines properties such as the mass range, the mass}radius relationship, the crust thickness and the cooling rate. The same EoS
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Fig. 1. Possible structure of a neutron star.
is also crucial in calculating the energy released in a supernova explosion. Clearly, the relevant degrees of freedom will not be the same in the crust region of a neutron star, where the density is much smaller than the saturation density of nuclear matter, and in the center of the star, where density is so high that models based solely on interacting nucleons are questionable. These features are pictorially displayed in Fig. 1. Neutron star models including various so-called realistic equations of state result in the following general picture of the interior of a neutron star. The surface region, with typical densities o(10 g/cm, is a region in which temperatures and magnetic "elds may a!ect the equation of state. The outer crust for 10 g/cm(o(4;10 g/cm is a solid region where a Coulomb lattice of heavy nuclei coexist in b-equilibrium with a relativistic degenerate electron gas. The inner crust for 4;10 g/cm(o(2;10 g/cm consists of a lattice of neutron-rich nuclei together with a super#uid neutron gas and an electron gas. The neutron liquid for 2;10 g/cm(o(;10 g/cm contains mainly super#uid neutrons with a smaller concentration of superconducting protons and normal electrons [9]. At higher densities, typically 2}3 times nuclear matter saturation density, interesting phase transitions from a phase with just nucleonic degrees of freedom to quark matter may take place [10]. Furthermore, one may have a mixed phase of quark and nuclear matter [6,11], kaon [12] or pion condensates [13,14], hyperonic matter [6,15}20], strong magnetic "elds in young stars [21,22], etc. The "rst aim of this work is therefore to attempt at a review of various approaches to the equation of state for dense neutron star matter relevant for stars which have achieved thermal equilibrium. Various approaches to the EoS and phases which may occur in a neutron star are discussed in Section 2 while an overview of the thermodynamical properties of the mixed phase and possible phases in neutron stars are presented in Sections 3 and 4. Our second aim is to discuss the relation between the EoS and various neutron star observables when a phase transition in the interior of the star occurs. Astronomical observations leading to global neutron star parameters such as the total mass, radius, or moment of inertia, are important since they are sensitive to microscopic model calculations. The mass, together with the moment of inertia, are also the gross structural parameters of a neutron star which are most accessible to observation. It is the mass which controls the gravitational interaction of the star with other systems such as a binary companion. The moment of inertia controls the energy stored in rotation
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and thereby the energy available to the pulsar emission mechanism. Determining the possible ranges of neutron star is not only important in constraining the EoS, but has important theoretical consequences for the observational prediction of black holes in the universe. Examples are the galactic black hole candidates Cyg X-1 [23] and LMC X-3 [24], which are massive X-ray binaries. Their masses (0.25M and 2.3M ) are, however, smaller than for some low-mass X-ray binaries > > like A0620-00 [25] and V404 Cyg [26], which make better black hole candidates with mass functions in excess of three solar masses. There is a maximum mass a non-rotating neutron star can have. There is however no upper limit on the mass a black hole can have. If, therefore, one can "nd a dense, highly compact object and can argue that its rotation is slow, and can deduce that its mass is greater than the allowed maximum mass allowed to non-rotating (or slowly rotating) neutron stars, then one has a candidate for a black hole. Since neutron stars are objects of highly compressed matter, this means that the geometry of space}time is changed considerably from that of a #at space. Stellar models must therefore be based on Einstein's theory of general relativity. Based on several of the theoretical equations of state and possible phases of matter discussed in Sections 2}4, various properties of non-rotating and rotating neutron stars are presented in Section 5. The relevant equations needed for the study of the structure of a neutron star are summarized in this section as well, for both non-rotating and rotating stellar structures. There we also discuss the observational implications when phase transitions occur in the interior of the star. In addition to studies of the mass}radius relationship and the moment of inertia, we also extract analytical properties of quantities like the braking index and the rate of slowdown near the critical angular velocity where the pressure inside the star just exceeds that needed to make a phase transition. The observational properties for "rst- and second-order phase transitions are also discussed. Other properties like glitches and cooling of stars are also discussed in Section 5. Summary and perspectives are given in Section 6. Finally, we mention several excellent and recent review articles covering various aspects of neutron stars properties in the literature addressing the interesting physics of the neutron star crust [27], the nuclear equation of state [14], hot neutron star matter [28] in connection with protoneutron stars, and cooling calculations [29]. However, as previously mentioned, our aim will be to focus on the connection between the various possible phases of dense neutron star matter in chemical equilibrium and the implications of "rst- and second-order phase transitions for various observables.
2. Phases of dense matter Several theoretical approaches to the EoS for the interior of a neutron star have been considered. Over the past two decades many authors [10] have considered the existence of quark matter in neutron stars. Assuming a "rst-order phase transition one has, depending on the equation of states, found either complete strange quark matter stars or neutron stars with a core of quark matter surrounded by a mantle of nuclear matter and a crust on top. Recently, the possibility of a mixed phase of quark and nuclear matter was considered [6] and found to be energetically favorable. Including surface and Coulomb energies this mixed phase was still found to be favored for reasonable bulk and interface properties [11]. The structure of the mixed phase of quark matter
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embedded in nuclear matter with a uniform background of electrons was studied and resembles that in the neutron drip region in the crust. Starting from the outside, the crust consists of the outer layer, which is a dense solid of neutron-rich nuclei, and the inner layer in which neutrons have dripped and form a neutron gas coexisting with the nuclei. The structure of the latter mixed phase has recently been calculated in detail [27] and is found to exhibit rod-, plate- and bubble-like structures. At nuclear saturation density o K2.8;10 g/cm there is only one phase of uniform nuclear matter consisting of mainly neutrons, a small fraction of protons and the same amount of electrons to achieve charge neutrality. A mixed phase of quark matter (QM) and nuclear matter (NM) appears already around a few times nuclear saturation density } lower than the phase transition in hybrid stars. In the beginning only few droplets of quark matter appear but at higher densities their number increase and they merge into: QM rods, QM plates, NM rods, NM bubbles, and "nally pure QM at very high densities if the neutron stars have not become unstable towards gravitational collapse. In this section we review various attempts at describing the above possible phases of dense neutron star matter. For the part of the neutron star that can be described in terms of nucleonic degrees of freedom only, i.e. b-stable matter with protons, neutrons and electrons (and also muons), we will try to shed light on recent advances within the framework of various many-body approaches. This review is presented in Section 2.2. For the more exotic states of matter such as hyperonic degrees of freedom we will point to recent studies of hyperonic matter in terms of more microscopic models in Section 2.4. The problem however with e.g. hyperonic degrees of freedom is that knowledge of the hyperon}nucleon or hyperon}hyperon interactions has not yet reached the level of sophistication encountered in the nucleon}nucleon sector. Mean "elds methods have however been much favoured in studies of hyperonic matter. A discussion of pion and kaon condensation will also be presented in the two subsequent subsections. Super#uidity is addressed in Section 2.7. In general, we will avoid a discussion of non-relativistic and relativistic mean "elds methods of relevance for neutron matter studies, mainly since such aspects have been covered in depth in the literature, see e.g. Refs. [17,18,30,31]. Moreover, as pointed out by Akmal et al. [14], albeit exhibiting valuable tutorial features, the main problem with relativistic mean "eld methods is that they rely on the approximation kr;1, with k the inverse Compton wavelength of the meson and r the interparticle spacing. For nuclear and neutron matter densities ranging from saturation density to "ve times saturation density, kr is in the range 1.4 to 0.8 for the pion and 7.8 to 4.7 for vector mesons. Clearly, these values are far from being small. The relativistic mean "eld approximation can however be based on e!ective values for the coupling constants, taking thereby into account correlation e!ects. These coupling constants have however a density dependence and a more microscopic theory is needed to calculate them. Our knowledge of quark matter is however limited, and we will resort to phenomenological models in Section 2.8 in our description of this phase of matter. Typical models are the so-called Bag model [32] or the Color-Dielectric model [33]. However, before proceeding with the above more speci"c aspects of neutron star matter, we need to introduce some general properties and features which will enter our description of dense matter. These are introduced in the "rst subsection. The reader should also note that we will omit a discussion of the properties of matter in the crust of the star since this is covered in depth by the review of Ravenhall and Pethick [27]. Morever, for neutron stars with masses +1.4M or greater, >
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the mass fraction contained in the crust of the star is less than about 2%. We will therefore in our "nal EoS employ results from earlier works [34,35] for matter at densities 40.05 fm\. 2.1. Prerequisites and dexnitions At densities of 0.1 fm\ and greater, we will in this work require properties of charge neutral uniform matter to be made of mainly neutrons, protons, electrons and muons in beta equilibrium, although the presence of other baryons will be discussed as well. In this section we will merely focus on distinct phases of matter, such as pure baryonic matter or quark matter. The composition of matter is then determined by the requirements of chemical and electrical equilibrium. Furthermore, we will also consider matter at temperatures much lower than the typical Fermi energies. The equilibrium conditions are governed by the weak processes (normally referred to as the processes for b-equilibrium) b Pb #l#l , J
b #lPb #l , J
(1)
where b and b refer to e.g. the baryons being a neutron and a proton, respectively, l is either an electron or a muon and l and l their respective anti-neutrinos and neutrinos. Muons typically J J appear at a density close to nuclear matter saturation density, the latter being n +0.16$0.02 fm\ , with a corresponding binding energy E for symmetric nuclear matter (SNM) at saturation density of E "B/A"!15.6$0.2 MeV . In this work the energy per baryon E will always be in units of MeV, while the energy density e will be in units of MeV fm\ and the number density n in units of fm\. The pressure P is de"ned through the relation P"n RE/Rn"n Re/Rn!e ,
(2)
with dimension MeV fm\. Similarly, the chemical potential for particle species i is given by k "(Re/Rn ) , G G
(3)
with dimension MeV. In our calculations of properties of neutron star matter in b-equilibrium, we will need to calculate the energy per baryon E for e.g. several proton fractions x , which N corresponds to the ratio of protons as compared to the total nucleon number (Z/A), de"ned as x "n /n , N N In this work we will also set G"c" "1, where G is the gravitational constant. We will often loosely just use density in our discussions.
(4)
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where n"n #n , the total baryonic density if neutrons and protons are the only baryons present. N L In that case, the total Fermi momentum k and the Fermi momenta k , k for protons and $ $N $L neutrons are related to the total nucleon density n by n"(2/3p)k "x n#(1!x )n"(1/3p)k #(1/3p)k . (5) $ N N $N $L The energy per baryon will thus be labelled as E(n, x ). E(n, 0) will then refer to the energy N per baryon for pure neutron matter (PNM) while E(n, ) is the corresponding value for SNM. Furthermore, in this work, subscripts n, p, e, k will always refer to neutrons, protons, electrons and muons, respectively. Since the mean free path of a neutrino in a neutron star is bigger than the typical radius of such a star (&10 km), we will assume throughout that neutrinos escape freely from the neutron star, see e.g. the work of Prakash et al. in Ref. [28] for a discussion on trapped neutrinos. Eq. (1) yields then the following conditions for matter in b equilibrium with e.g. nucleonic degrees freedom only k "k #k , L N C
(6)
and n "n , (7) N C where k and n refer to the chemical potential and number density in fm\ of particle species i. If G G muons are present as well, we need to modify the equation for charge conservation, Eq. (7), to read n "n #n , N C I and require that k "k . With more particles present, the equations read C I (n>G #n>G )" (n\G #n\G ) , J @ J @ G G
(8)
and k "b k #q k , (9) L G G G J where b is the baryon number, q the lepton charge and the superscripts ($) on number densities G G n represent particles with positive or negative charge. To give an example, it is possible to have baryonic matter with hyperons like K and R\> and isobars D\>>> as well in addition to the nucleonic degrees of freedom. In this case the chemical equilibrium condition of Eq. (9) becomes, excluding muons, kR\ "kD\ "k #k , L C kK "kR "kD "k , L kR> "kD> "k "k !k , N L C kD>> "k !2k . L C
(10)
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A transition from hadronic to quark matter is expected at high densities. The high-density quark matter phase in the interior of neutron stars is also described by requiring the system to be locally neutral n !n !n !n "0 , C S B Q
(11)
where n are the densities of the u, d and s quarks and of the electrons (eventually muons as SBQC well), respectively. Moreover, the system must be in b-equilibrium, i.e. the chemical potentials have to satisfy the following equations: k "k #k , B S C
(12)
k "k #k . Q S C
(13)
and
Eqs. (11)}(13) have to be solved self-consistently together with e.g. the "eld equations for quarks at a "xed density n"n #n #n . In this section we will mainly deal with distinct phases of matter, S B Q the additional constraints coming from the existence of a mixed phase of hadrons and quarks and the related thermodynamics will be discussed in Section 3. An important ingredient in the discussion of the EoS and the criteria for matter in b-equilibrium is the so-called symmetry energy S(n), de"ned as the di!erence in energy for symmetric nuclear matter and pure neutron matter S(n)"E(n, x "0)!E(n, x ") . N N
(14)
If we expand the energy per baryon in the case of nucleonic degrees of freedom only in the proton concentration x about the value of the energy for SNM (x "), we obtain, N N E(n, x )"E(n, x ")# (dE/dx)(n)(x !)#2 , N N N N
(15)
where the term dE/dx is to be associated with the symmetry energy S(n) in the empirical mass N formula. If we assume that higher-order derivatives in the above expansion are small (we will see examples of this in the next subsection), then through the conditions for b-equilbrium of Eqs. (6) and (7) and Eq. (3) we can de"ne the proton fraction by the symmetry energy as
c(3pnx )"4S(n)(1!2x ) , N N
(16)
where the electron chemical potential is given by k " ck , i.e. ultrarelativistic electrons are C $ assumed. Thus, the symmetry energy is of paramount importance for studies of neutron star matter in b-equilibrium. One can extract information about the value of the symmetry energy at saturation density n from systematic studies of the masses of atomic nuclei. However, these results are limited to densities around n and for proton fractions close to . Typical values for S(n) at n are in the range 27}38 MeV. For densities greater than n it is more di$cult to get a reliable information on the symmetry energy, and thereby the related proton fraction. We will shed more light on this topic in the next subsection.
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Finally, another property of interest in the discussion of the various equations of state is the incompressibility modulus K at non-zero pressure K"9 RP/Rn .
(17)
The sound speed v depends as well on the density of the nuclear medium through the relation
v dP dP dn K " " " . dn de c de 9(m c#E#P/n) L
(18)
It is important to keep track of the dependence on density of v since a superluminal behavior can occur at higher densities for most non-relativistic EoS. Superluminal behavior would not occur with a fully relativistic theory, and it is necessary to gauge the magnitude of the e!ect it introduces at the higher densities. This will be discussed at the end of this section. The adiabatic constant C can also be extracted from the EoS by C"(n/P) RP/Rn .
(19)
2.2. Nucleonic degrees of freedom A major part of the densities inside neutron stars can be well represented by nucleonic degrees of freedom only, namely the inner part of the crust to the outer part of the core, i.e. densities ranging from 0.5 to 2}3 times nuclear matter saturation density. There is a wealth of experimental and theoretical data, see e.g. Ref. [36] for an overview, which lend support to the assumption that nucleons do not loose their individuality in dense matter, i.e. that properties of the nucleon at such densities are rather close to those of free nucleons. The above density range would correspond to internucleon distances of the order of &1 fm. At such interparticle distances there is little overlap between the various nucleons and we may therefore assume that they still behave as individual nucleons and that one can absorb the e!ects of overlap into the two nucleon interaction. The latter, when embedded in a nuclear medium, is also di!erent from the free nucleon}nucleon interaction. In the medium there are interaction mechanisms which are obviously absent in vacuum. As an example, the one-pion exchange potential is modi"ed in nuclear matter due to `softeninga of pion degrees of freedom in matter. In order to illustrate how the nucleon}nucleon interaction is renormalized in a nuclear medium, we will start with the simplest possible many-body approach, namely the so-called Brueckner} Hartree}Fock (BHF) approach. This is done since the Lippmann}Schwinger equation used to construct the scattering matrix ¹, which in turn relates to the phase shifts, is rather similar to the G-matrix which enters the BHF approach. The di!erence resides in the introduction of a Pauliblocking operator in order to prevent scattering to intermediate particle states prohibited by the Pauli principle. In addition, the single-particle energies of the interacting particle are no longer given by kinetic energies only. However, several of the features seen at the level of the scattering matrix, pertain to the G-matrix as well. Therefore, if one employs di!erent nucleon}nucleon interactions in the calculation of the energy per baryon in pure neutron matter with the BHF G-matrix, eventual di!erences can be retraced at the level of the ¹-matrix. We will illustrate these
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aspects in the next subsection. More complicated many-body terms and relativistic e!ects will be discussed in Sections 2.2.2 and 2.2.3. 2.2.1. From the NN interaction to the nuclear G-matrix The NN interactions we will employ here are the recent models of the Nijmegen group [37], the Argonne < potential [38] and the charge-dependent Bonn interaction (CD}Bonn [39]). In 1993, the Nijmegen group presented a phase-shift analysis of all proton}proton and neutron}proton data below 350 MeV with a s per datum of 0.99 for 4301 data entries. The above potentials have all been constructed based on these data. The CD}Bonn interaction has a s per datum of 1.03 and the same is true for the Nijm-I, Nijm-II and Reid93 potential versions of the Nijmegen group [37]. The new Argonne potential < [38] has a s per datum of 1.09. Although all these potentials predict almost identical phase shifts, their mathematical structure is quite di!erent. The Argonne potential, the Nijm-II and the Reid93 potentials are non-relativistic potential models de"ned in terms of local potential functions, which are attached to various (non-relativistic) operators of the spin, isospin and/or angular momentum operators of the interacting pair of nucleons. Such approaches to the NN interaction have traditionally been quite popular since they are numerically easy to use in con"guration space calculations. The Nijm-I model is similar to the Nijm-II model, but it includes also a p term, see Eq. (13) of Ref. [37], which may be interpreted as a non-local contribution to the central force. The CD}Bonn potential is based on the relativistic meson-exchange model of Ref. [40] which is non-local and cannot be described correctly in terms of local potential functions. For a given NN interaction