3p fine structure of ferromagnetic Fe and Co from photoemission with linearly polarized light

Pergamon

557-562. 1994

Vol. 90. No. 9. pp.

solid state communications.

Eiscvier Science Ltd Printed in Great Britain 0038-1098(94)$7.00+ .OO 003% 1098(94)EO237-6

3p

FROM

FINE STRUCTURE PHOTOEMISSION

OF FERROMAGNETIC Fe AND Co WITH LINEARLY POLARIZED LIGHT G. Rossi

Laboratorium

and

fiir FestkGrperphysik,

F. Sirotti ETH-Ziirich,

CH-8093,

Switzerland

and Laboratoire

pour I’Utiiisation CNRS-CEA-MEN

du Rayonnement F-91405

Eiectromagnetique,

Orsay, France

Cherepkov* and F. Combet Farnoux Lahoratoire de Photophysique Moiecuiaire.

N.A. CNRS

Universite

de Paris-Sud

F-91405

Orsay, France

G. Panaccione Laboratoire

pour i’lltilisation CNRS-CEA-MEN

du Rayonnement F-91405

(received by E. Molinari

Linear

Magnetic

Dichroism

Eiectromagnetique,

Orsay, France

on 11 February

in the Angular

Distribution

observed in angular resolved photoemission experiments on 3p core levels of the ferromagnetic asymmcstry is strictly polarizrd

radiation.

multipoles interaction.

the polarization

Dy peak-fitting

the J=3/2

the cxp&nentnl

and J= i/2

energies.

experiments

transition

multiplcts,

Circular

photoemission

Absorption

[I] and

Atomic-muitipiet culations

theory

(CD)

in absorption

and atomic-cross

X-ray absorption

on magnetic

magnetic

photoemission

i.e.

to asymmetries

Magnetic

CD

on ferromagnetic

as solid state maniftstations in the Angular

LMDAD

within

in atomic

Distribution

190 000 St. Petersburg,

of pho-

[lo] when an ex-

is observed when linearly

polarizes the

in photocmission

mentum

p-polarized

is built in the experiment

and magnetization

photoelectron

dircctions.[‘i,

light by mo-

91 These con-

ditions allow to fully exploit the intensity of the linearly polarized

synchrotron

radiation

allow high energy resolution,

sources, and therefore

as in standard

core level

photoemission. The effect is very large: magnetic asymmetries of the order of 10%-200/o are measured for Fe 3p and Co 3p from magnetically

strumentation,

of the Linear (LDAD)

photoionization

a proper choice of the polarization.

metals, and of IO-30%

Address: State Academy

surfact- present large

effects.17, 8, 91 Those can be fully

is employed and a chirality

for 4f levels in rare earths. Linear dichroism has also been exploited in X-ray absorption for ferromagnctically

* Permanent

for special

resolved photoemis-

ternal field (in this case the magnetization)

of the order of a few percent

for 2p and 3p levels of transition

angular

core hole states.

site-projected

was also measured, and calculated

ordered solids, and in par-

it has been shown that,

geometries,

dichroism

toelectrons

moments, and spin momcnts.[S] CD in

the same scheme [2]. amounts

materials:

recently

Dichroism

Dichro-

information

for surface crystal field studies.[6]

understood

section cal-

have opened the accrss to a new kind of

antiferromagneticaliy

ticular

magnetic

a a

data.

and/or

sion experiments

ra-

order of solids and surfaces.

applied to Magnetic

and by

of the* 3p core

lineshapes can be derived

photoemission

experimental

and in

ism (MCD) orbital

core level spectra

Quite

121 on core levels of the fer-

from core levels has been interpreted

probe of the ferromagnetic

dichroic

and that spin-selected

metals 131 and of the rare earths

dichroism

to the state

of the hole levels under the exchange

with spin-resolved

(41 have been performcul with polarized synchrotron diation.

magnetic

with circularly

Fe and Co and obtain the m, ordering of the sublevels.

Recent results haveshown the great interc%t of dichro-

romagnetic

is

light

the energy order for the 3p hole-levels of Fe and Co is reversed for

from out analysis and compared

photoemission

The measured

the relevant multipoh~s one can resolve the fno structure

We show that

at X-ray

metals.

Atomic theory shows that the effect is proportional

hole states of ferromagnetic

ism experiments

of the photoelectrons

with linearly polarized

related to the one observed in photoemission

that characterize

calculating

transition

1994)

tal (100)

of Aerospace inRussia

amorphous 557

saturated

single crys-

surfaces as well as from polycrystalline films.

Furthermore

magnetic

and

asymmetries

558

FERROMAGNETIC

of the

order

thicknesses

of 5%

are

The first reports logical

experimental

final

[12].

system,

do not interact

out that

inside

to simplify

the

the

final

one hole spectrum

the photoemission

field which

It

problem,

a

formal

with

momentum

with

the

and

unpolarized

In ferromagnets gular

the (3~)

valence

into

projection

in the final state

ized.

The

core hole statrs

state

multipol~s

throry

soft

is that

thr

magnitude measure

between

coriiponc*nts

jrctions

III,. from

here LAiDAD

3p

core

iron

levels.

CPI~be dt*rivecl

Icvc4

and by comparing

LMI)AD

be attributed

a%yrnmetry,

3p core holes in presence resolution

able

that

i.e.

electron

direction

the

Fe and

mission

Co

that

allows

were

at

Fc( 100)

ergy

was varied

between

test

thr

energy

in figure

de-

the

angular

whose gap wzi

or by a Fe-Ni-R

magnetic

driver

LMDAD

[7]

radiation at LURE

which

crys-

acted

as

be set parallel

by applying

the yoke.

Linear

from the SU7 undulator impinged

wan

vertically

Fe and Co overlayers. could

a Fe( 100) Co

photon

eV and hv=210

depcndcncc

of the

during

t!

surface

was grown

A. The

rn-

eV

LMDAD

to cf-

mcasurcm(*nt

of LMDAD

wan

the

state

bc tlcscribcd

the* problcln

language

in Table

m,

tially

with

111,.

dc*nsity

This

final

matrix,

it is more

sublevel

hole

but

for

convc!nicnt

multipolc~s.[lli]

each magnetic

The

of p-states

state are

1.

photoionization

of a closed

characterized

by a dcfinitc

is described

toionization

total

projcrction

of state

iutc*ractiou

into the componc~nts

by the

with

the

of the:

a given

consiclcration

p,&for

hole state tion

of the

under

to IIse the

hole state

j is split

valiw

can

Due to the exrhangc*

process.

3d subshrll

from

description

by the same

of a one electron

in the state

with

subshell

with

the

value of the projccequation

subshell

the same m,.

The

ay the pho-

which latter

Tab.1:

on the sample

State

multipoles

and np3/*

p&

was iniproblem

for magnetic

statcu.

in the

circuit

mounted

ribbon

direction

around

performed

by a Fe( 100) single

of the sample

to the y-axis

rent to a coil wrapped synchrotron

closed

amorphous

hv=125

photoc-

of IO-20

to the avail-

magnetic

yoke

for polycrystalline

magnetization

or antiparallrl

SuperAco

horseshoe

of 2-10

explanation

a df+finitct

The

spin-

magnetization

layers

on

phoelectro-

to the

evaporation.

tcmperaturc

momentum

given

Inj

mirror

the

core Icvels we will use the pure atomic

of the

the

and

grown

by e-beam

sample

near-normal

Fr buffrr

for thickncssrs

photon

The

multipoles

to recognize

and to draw

were

A

1.

the

300 I\:.

repre-

of the core

spin-resolved

experiments

sketched

by a soft iron

(y-axis)

plane.

epitaxially

200C

The

perpcndirularly

kept

with

of Ill1 can

from

can be compared

en-

lies along

by a cylindrical

analyzer,

onto

interaction.

structure

sublevels

1141, and

Photoemission made

energy

(horizontal)

photocmission

to t hc* m(*a-

sextuplet

intensities

MAC11

B is 50 degrees.

incidence.

was imposed

levels of np,/*

geometry

The

static

of

data.

tal,

were collected

I*‘c

on atonlic

value

of the exchange

of the individual

3p spectra

ba.4

the fine structure

photoemission

spin-resolved

for the structure

the results

of the sextuplet

holc*s of frrromagnrtic spin-dependent

selected

off normal

toelectrons

feet.

ob-

of the experimentally

Th iu . slows t sublevels

cobalt,

a unique

to each sublevel

sents atomic-like

and

multipolrs

31, state

sextuplt:t.

projections

Thorc-

distinguish

by linc*shapc~ analysis.l’J]

wave functions

The

sign and

sp(*ctra,

A sextuplet

th

rived

for

with difft:rc:nt pro-

photoemission

Ijy CalCUlirtiIlg

surcd

WI: to

of the hole state

angle

SO degree

thicknrss

of this

multipolcs.

enal&

fcrromagnctic

Co

tht* tort*

by

angular

LMDAD

by the

state

of LMDAD

to its axis which

For the qualitativr

WC: prcscnt tained

the

consc’quencc.

is d4ncd

of the relevant

fore thr

31’ and

axis. polar-

can he described

One

su blcvels

a

quantization

thr sign and the magnitudc*of

diffc*rcsnt magnrtic

with

for a given experiment

X-rays.

The

The

has a circular

an-

is therefore

also dt+*rminr

of the photoelectrons

slit perpendicular normal.

experiment:

represented,

interaction

sublevels

m, on the magnetic

p-polari&

trance

of the

schematically

sample of total

Geometry

analyzer,

in angular

by the exchange

electrons

(131 which

Fig.1:

Circular

light.[l3]

level observed polarized

re-

for explain-

dichroism

Each

with

effort

the

Those

LhlDAD.

given

distribution

possible

to discuss

core hole state

j is split

(3d)

and

manner

in photoemission

photoemission

is therefore

a new theoretical

in

above

by the magnetic

sample.

of the 3p core level only.[9]

ing and unifying Dichroism

(SO-15&V

electronic

state

sults have stimulated

with atomic

with the remaining

and are not affected

is confined

a phenomeno-

17. 9] in connection

are well in the continuum

threshold),

b

on rare gases [I l] and with

It was pointed

states

90, No.‘9

Y

for monolayer-range

proposed

of the effect

results

Vol.

100) interfaces.[9]

on LMDAD

explanation

theory

measured

of Cr at Cr/Fe(

Fe AND Co

a cur-

polarized source surface

of at

j = l/2

j = 312

sub-

FERROMAGNETIC

Vol. 90, No. 9

PC AND

Co

was considered already in ref.[ 16, I?] We will neglect for simplicity

the spin-orbit

spectrum,

the contribution

small.

interaction

in the continuous

of which is supposed to be

Then for the geometry of the experiment

in lig.1, using eq.(lO) photoelectron

of ref.[l7]

flux ejected in a given direction.

IL’~~D =

(1) =q3i

Ij(M)

shown

for the intensity

- Ij(-M)

of the

we obtain

=

pP,(2j + 1)‘j2 CA, - sin 0 - cos d

where M tion (which

is the direction

of the sample magnetiza-

coincides with the direction

polarization

n in [17]),

u+(w)

of the atomic

is the photoionization

cross section of the nlj subshell. w is the photon energy, and 19 is the angle of the grazing incidence of the light. The parameter

C,:,

is defined as follows

100

120

140

160

180

200

220

240

Photon Energy (eV) where do and dz are the reduced dipole matrix ments for the np-cs and np-cd transitions, and c50and 61 are the corresponding in the approximation section u,l,

phase shifts. Since

adopted by us, neither

nor the parameters

ele-

respectively,

Ci,,

the cross

depend on m,, the

sign and the relative magnitude of LMDAD

for dilTerent

Fig.2:

Upper

panel:

Dependence

of the parame-

ters /3 (fun line, left scale) and C&,(dashed scale) on the photon energy. Lower panel: the calculated experimental

A$,yDAD(left asymmetry,

line, right Variation

of

scale) and of the maximum after background subtraction,

(right scale) as a function of the photon energy.

magnetic sublevels is defined exclusively by the sign and the magniludc

of the state multipoles pr,, (see table I).

Usually in the rxpcrimcnt cross scctiocis is unknown. to consitlc-r the normahA

the absolute valur of the Thcreforc

LMDAD

a ratio of the difference

lo the

it is worthwhile

which is datintrd ay of the rucasurcd

sum

elcrclron intc*nsitic.s

photon

energies from hv=l25

exprrimental +

asymmetry

Idown), where

background tization

I,,

eV. The

- I,jmn) / (I,,

(Id”,,,“) is the peak intensity

subtraction

mcasurcd

in the up (down)

for in-plant

y direction.

after

magne-

These values of

photon energy correspond to final state kinetic energies for the 31, photoclcctrons i.e.

spanning

Ilerc p is the angular

asymmetry

parameter,

and Pz

DAD

asymmetry

of the parameters

p and C,j,,

have

for

A monotonic decrease of the LM-

is observed as the photon energy in-

creases in agreement

is lhc sc*cond Legcndre polynomial.

between 66 eV and 151 eV,

the most surface sensitive conditions

electron spectroscopy.

The calculations

eV to hu=210

is dcfincd hs (I,,

with the calculations.

ence in the magnitude can be connected

The difTer-

between thcwry and experiment

with

the omission of the solid state

been perforrncvl for the ,7p subshell of Fe within an atomic model through the determination of Ihe matrix elements

effects in the theory, with the contribution

of the elasti-

do and cl2 and the corresponding

cally scattered

and with the

we used an averaged configuration imation

phase shifts 60 and 15~: flartree-Fock

for both discrete and continuum

as described

in [18] for photoionization

later on for photoemission proximation theoretical

studies within

and discussed

normal.

energy range and

hau allowed to get a good theory-experiment when the excitation np threshold.

this ap-

used in photoemission

the VUV

agreement

energy does not lie Loo close to the

The present experiments

and 210 eV) fulfill this condition

(hu between 125

which means that we

consider a region where do is monotonously and dz displays a positive maximum beyond a zero value revealing

decreasing

which is expected

the Cooper

minimum

po-

sition. Fig.

incomplete tial angular

in 119, 201. Indtvd,

has been extensively

approx-

wavefunctions

2a shows the dependence of the parameters

/3

linear polarization

Figures 3 and 4 display the experimental tized photoemission

effect makes it possible to fit reliably spectra by sextuplets,

shown previously[9]. tion of simultaneously

is shown in Fig. asymmetry

2b along at selected

fitting

as it has been

procedure

were constrained

spectra.

each component

of the sextuplet

the peak

by the condi-

the unmagnetized

as the LMDAD

as well

The lineshapes and widths of were kept fixed during

the refinement procedure. The fitting parameters, and the results for the peak energies are collected in table 2. Systematically

experimental

In the fitting

energies and amplitudes

broadening

fo r m,=3/2.

difference

the unmagnetized

within

value A$iDAD

unmagnc

Co. The LMDAD

variation

with the maximum

spectra and the LMDAD

for the 3p core levels of Fe( 100) and of polycrystalline

on the photon energy. Both display a smooth The

of the light and the par-

averaging of the detector about the surface

and C,:,

the energy range 110 + 23OcV.

electrons in the experiment

the deeper two states require a lorenzian

almost double with respect to the shallower

four peaks.[9]

The intensities

of the six peaks decrease

with increasing binding energy, as it is suggested by the

FERROMAGNETIC

Vol. 90, No. 9

Fe AND Co general lineshape of the unmagnetized bottom

panels show the experimental

as well bs the calculated LSIDAD

spectrum.

The

differences curves

differences:

the experimental

difference spectrum is reproduced by multiply-

ing the intensity of each sublevel of the 3p sextuplet the corresponding This

implies

state multipoles that

the sublevels deduced

solid state core level spectrum tuplet

states

zation

axis.

character

with m,

open-shell

from the

behave as atomic

projections

sex-

along the magneti-

The solid state effects, namely

of the 3d screening electrons,

of multiplicity

by

of Table 1.

the band

and the effect

of final state configurations

of the two

system. are either all included

in the width

of the sublevels. or lie at final state energies outside the main peak and contribute

negligible intensity

lites are measured for Fe and Co, unlikely Ki 121). The comparison of experimental LXIDAD J=3/2

indicates

and calculated

a reversed order of the J=l/2

and

sublevels.

These

Binding Energy (eV)

(no satelthe case of

results require some comments:

versed level ordering,

a)

the re-

with respect to the Zeeman split-

ting, is due to the exchange coupling of the 3p core hole Fig.3:

Upper panel: fitting of the Fe 3p photoemis-

sion spectrum

from unmagnetiztd

gral background

subtraction).

Fe( 100) (after

In the bottom

shown the mI cornponrnts multiplittl tipolcs of Tablr (continuous

inte-

panel arc

by the, state rnul-

I along with the rt5ulting

curve) and thr rxpc*rirnrntal

convolution

LMDAD

sl~c-

trutn (points).

with

the %I band of Fe or Co (the exchange interaction as a magnetic field acting on the hole

can br rc*prmmtrd

spin only).

This fact was already

core lcvrls whrrc* the spin-orbit the. rxchangr

splitting

arated

and J=1/2

J=R/2

observed for derper

intrraction

is observed within mllltiplcts

mission cxpc*rimcnts with circular 2p [‘I. B] b) ‘h

is large and the well sep-

like in the photoc-

polarized

width of the J=3/2

hc*rr lo tw I. I I f.OS eV and 1.3Gf.05

light on Fe

multiplct

is found

cV rc:spcclivcly for

Ft* 3p and Co :Ip, while- the total width of the sc*xtuplcts art* 2.Gf.l

CV and 3.4f.l

eV. The spin orbit splitting

of the 3p corf’ holes in IJe and Co is known from silicicle data [“L] and is rfvpcctively

1.04f.05

eV and 1.43f.05

eV for I:(, and Co. c) the intensities of the sublevels decrease aq the final state energy decreases: in particular the ratio of the J=3/2 J=3/2,

m,=-3/2

m,=3/2

peak intensity

equal to the ratio between majority

Tab.2:

Fitting

unmagnetized

parameters

and minority

broadening

the various m, components are reported. broadening

was 350meV

The binding

values are accurate

along with the lorentzian

elec-

used in the best fit of

Fe 3p and Co 3p spectra.

energies (whose relative meV)

and the

peak intensity is 1.73 which is roughly

within

30

in eV for

The gaussian

and SOOmeV for Fe and Co re-

spectively. .

64

62

60

58

Fe - 3P

56

Binding Energy (eV) 7?tj Fig.4:

lIpper

mission spectrum gral background

panr*l: fitting

of the Co 3p photoe-

from unmagnrtizrd subtraction).

shown thr m, components multiplird tipolrs

of Tablr

Co (aftt*r

In the bottom

by the state mul-

1 along with thr resulting

(continuous curve) and the experimental trum (points).

intc-

panrl are

convolution

LMDAD

spec-

+3/2 +1/2 -l/2 -312 -l/2 +1/2

B.E.(eL’)

52.2 52.5 52.9 53.3 53.9 54.8

Lor. 0.42 0.42 0.42 0.42 0.85 0.85

co-3p B.E.(eV) 59.6 60.0 60.5 61.2 61.8 63.0

Lor. 0.7 0.7 0.7 0.7 1.0 1.0

.Voi.

FERROMAGNETIC

90. No. 9

Fe AND

co

561

Fe 3p PhoIocmissio~*+ M&WlzmlcaDown/ !

I

t

I .33

Area Ratio

Spin Up - Spin Down Spin Selected lntcnsiks

Soin Uo

Binding Energy (eV) Spin Dawn

Fig.& peaks spin

Spin-sekcted

of figure intensities.

ity (minority) rurvc

-ss

-Jb

-s4

-51

.52

-51

Binding Energy (cV)

Fig.S:

Expcrimc*ntal

and

diamonds

rtqectiv4y)

pilrilIIl~tC!r~of

s&cted

1:~.31) lin4apc:s

up uiagnc*tizatiou

(diamonds).

Table

liueshapes

of the* sextuplets

and

the* scxtuplrt lowor pauela

obtain4

usiug

according

to both

dircctiou

(full

IIIajority

sp(*ctruIu

excikvl

tiou,

etc.).

These

curves

the spiu-rc4ved

(1uli

2. The

rn;~gnctization spiu

obtain4

dircctious

the

fit

can

with

with

data

trons

in the d-band

Ouce

in bee-Ice.

the finch structure

distribution

nority

The

Arca Ratio

minority

kinetic

to

et al.[7]

the J=3/2, and

of the .J=R/2, dc4incd

m,=f3/2

m,=fl/2.

according

is attributed to the

of the

two spin

wave runctions.[24]

possible

rrlevant to

3p sublevel

has mi-

linear

With

The

62

combination

of the

prorrtlure angle.

62

61

60

59

58

of un-

with

63

Binding Energy (eV)

intensity

m,=fl/2

spin-rc.solvcd

65

231 WC

harmonics this

g-.60.

by direct

to the two spin-selrctcd

sphrrical

reproduce

usspcc-

to pure spin states character.

and the J=1/2,

products comes

Fe 3p spectra

measIIremrnts.(l~I,

(-)

along the en-

of the sextuplet

energy

majority

spin states

multiplrts

Spin Up - Spin Down

for Co

levels projcctcd

as it has been confirmed

photoemission

(+)

I. IX

the

compared

one can reconstruct

m, components

highest

approximate

in thr

aud

same analysis

of the spin-selected

spin character,

spin-resolved

of m,

is known,

ing the individual trum.

curve* is shown

of 1.25.

the magnetization ergy

sl’in-asylliclic,try

iu

maguctiza-

of Roth

Thr

The

line)

spt*ctrurri

the*

I shows a ratio

Wum (continuous polycrystallinc*

the. spin

reprc5cnt

bc* directly

exp~~riuIc*utal

total

major-

the difference

panc4.

couq~oncuts

down

The

indicate

with

the

minority

with

spin projectiou

up triangks

triangles

from

and

ancl open

show m,

(d own)

to the* unruagn&izcd

bottom

down

Up

obtained

all majority

spins in I+( 100) along

(dot-dauhrd).

compar4

lincshaprs

5 by selrrting

thr

it brintr-

Fig.7:

Spin-sclectcd

els obtained nority

spin

majority linr)

intensities.

(minority)

is compared

monds)

and

asymmetry

lineshapcs

as in Fig.

the difference curve

is shown

3p core

lev-

majority

and

IJp

triangles

indicate

spins. to thr

of Co

6 selecting (down) Thr

total

Sum (continuous

unmagnrtizcd curve

mi-

spectrum

is reported.

in the bottom

The

panel.

(diaspin-

562

FERROMAGNETIC Fe AND Co

grated 3p spectra as well as spin-resolved LMDAD spectra solely based on the experimental intensities and widths of the sextuplet sublevels and on the calculated atomic LMDAD coefficients. JS shown in figure 5. Figure 6 shows the total majority spin 3p intensity and minority spin 3p intensity for Fe: the energy difference between the center of gravity of the two opposite spin distributions is 0.53 eV. These spin selected lineshapes compare well with the spin-resolved Fe 3p photoemission data by Sinkovic et al.[l4] provided a gaussian broadening of 1.5 eV FWHM is applied to the calculated curves. The broadening of the experimental spin-resolved data is due to the low efficiency of spin-detection, which imposes to severely degrade the energy resolution, with respect to standard core level photoemission spectroscopy. The difference curve (lu, - Idown) and the spin-asymmetry / (&n-up + Ispen-dowm) are alyo (1,pln-up - I,p+down) plotted. The results for Co are shown in fig. 7. The area ratios for the (angle integrated) minority and majority spin 3p spectra are 1.33 for Fe and 1.18 for Co. This result is generally confirmed by spin-resolved photoemission results and has not been explained.

In conclusions, we have measured a large LMDAD effect in 3p core level photoemission of ferromagnetic Fe and Co, and we havegiven an interprrtation of it within a pure atomic model. The calculation of the relrvant state multipolcs, based on atomic wavefunctions, for the

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particular experimental geometry enables one to extract from the experimental photoemission data the fine structure of magnetic core hole atomic-like sublevels. From the energies, widths and intensities of the m, levels fitted to the experimental spectra (unmagnetized and LMDAD), one can reconstruct the total spin-selected spectra, the dichroic spectra, as well as the spin-selected dichroic spectra. The main solid-state correction to the atomic model turns out to be in the relative intensities of the magnetic sextuplet levels and in their individual widths. We have shown that angle resolved core level photoelectron spectroscopy with linearly polarized light yields all the information on the fine structure of magnetic core hole levels, with a major advantage of intensity and (consequently) energy resolution, when compared to photoemission with circular polarized light, or with spin analysis. Extensions of the theory to explain the LMDAD results on 3d valence bands [9] is underway. N.A.C. acknowledges the hospitality of the Laboratoire de Photophysique Moleculaire du CNRS, extended to him during the work on this paper, and the financial support of the Centre National de la Rccherche Scientifiqur. Thanks are due to H.C. Siegmann for support and criticism. This work was supported by the Swiss National Fund under program 24.

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