Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 FRP059

THEORETICAL LIMIT OF ELECTRON BEAM BRIGHTNESSGENERATED FROM ELECTRON GUNS
Masao Kuriki ∗ ,Graduate school of advanced science and engineering, Hiroshima Univerisity
DC BIASED ELECTRON GUN

Abstract
The brightness of the electron beam is important for linear
accelerator and it is fundametally limited by that from electron gun. In this article, we discuss the fundamental limit on
the beam brightness defined as the ratio of the bunch charge
and the 6D emittance from DC and RF guns. The brightness
is limited by the thermal energy of electron, space charge
effect, particle motion in the gun structure, break-down limit,
etc. As the highest brighness according to my considerations, DC gun generates 1013∼16 C/m3 . That from RF gun
is slightly less than that of DC gun, 1013∼15 C/m3 .

INTRODUCTION
In this article, I consider the fundamental limit on the
beam performance from electron guns. The emittance andthe bunch charge are important especially linear accelerator
facility, because the beam performance from the electron
gun determine the accelerator performance. The emittanceand the bunch charge are in the relation of trade-off. The
discussion on the fundamental process which limit the beam
performance is useful to consider a new concept of accelerator facility.
In the following sections, we discuss the possible highest
beam performance from DC and RF guns. DC gun technology is quite conventional and one might consider it is out of
date, but it is not true by several reasons. DC bias gun has alimitation on the acceleration voltage, but it doesn’t mean theperformance is poor. The acceleration by a static field has
a big advantage, because the acceleration voltage is a constant and there is no emittance growth mechanism associatedwith the acceleration. In addition, DC gun can generates
highly spin polarized electron beam, which is strongly demanded by High energy physics, some of material science,
and applications.
RF electron gun can generate a high brightness beam
because the high surface field on the cathode. The peakcurrent is much higher than that of DC gun, but there is
non-linear effects degrading the beam brightness associatedwith the RF acceleration.
In the following section, we discuss the beam brightness
from each gun structure. In this article, we define the beam
brightness as the ratio of the bunch charge 𝑄 to the normalized 6D emittance 𝜀𝑛,6𝐷 as
𝑄
𝐵≡
.
(1)
𝜀𝑛,6𝐷

∗ mkuriki@hiroshima-u.ac.jp

DC biased gun is originally developed as a thermionic gun
which use a thermionic cathode as the electron source. The
DC biased thermionic gun is conventional and widely used.
Some people consider that the DC biased thermionic gun
is out of date, but it is not a correct understanding. SACLA(XFEL in Japan) [1] employs a DC biased thermionic gun(500 kV, pulsed) and the brightness is sufficient to generate X-ray as same as other XFEL facility. The thermionicgun generates a quite long bunch ∼ 1 ns due to the limited
bias voltage comparing to RF field, but this long bunch isdecreased by a multi-stage bunching. If the process is linear,
there is no brightness degradation. The key is controlling thenon-linearity and this is the part which they concern about
the most.
Technical advantage of DC gun is the capability to generate the spin polarized electron beam, which is strongly
demanded from High energy physics. If the spin polarization is required for ALC, negative electron affinity (NEA)
super-lattice (SL) GaAs cathode [2] is the only solution,
which is able to generate 90% spin polarized electron beamwith circularly polarized laser [3]. Although GaAs cathodeis not compatible with RF photo-injector at this moment,
but it is not fundamental limitation.
An unique feature of the DC electron gun is that a perfectly
parallel electron flow from a cathode that completely satisfies
the Poisson equation is made. This is called as space-charge
dominated flow. The emission current density 𝐽 is expressedas [4]𝑉 3/2
𝐽 = 2.33 × 10−6 2 ,(2)𝑑where 𝑉 is gun bias voltage, 𝑑 is the cathode-anode gap
length. The gun bias voltage is limited by vacuum breakdown. The highest voltage ever demonstrated is 500 kV by
the cERL gun at KEK [5].
The vacuum break-down is determined by the highest fieldin the gun structure, and not the voltage. In addition, theaverage acceleration field is lower than the highest peak field
which gives the actual limit. The peak field 𝐸𝑝𝑒𝑎𝑘 should be
proportional to the average field 𝐸 = 𝑉 /𝑑 as 𝐸𝑝𝑒𝑎𝑘 = 𝜅𝐸,
where 𝜅 is determined by the gun geometry. With 𝐸𝑝𝑒𝑎𝑘 , 𝐽
is rewritten as
3/2
𝐸𝑝𝑒𝑎𝑘
𝐽 = 2.33 × 10−6.
(3)
𝜅3/2 √𝑑
The highest current from a DC gun is obtained when 𝐸𝑝𝑒𝑎𝑘is equal to the vacuum break down limit field, 𝐸𝐵𝐷 .
𝐸𝐵𝐷 is determined by material, surface treatment including a local enhancement. Well trained class 1 copper electrode gives 200 MV/m 𝐸𝐵𝐷 [6]. There isn’t many investigation 𝐸𝐵𝐷 for semiconductor. SACLA electron gun employ- 1065 -Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 FRP059

𝐶𝑒𝐵6 and the bias voltage is 500 kV with 50 mm gap, i.e.
the field is 10 MV/m. This gun is operated in pulse with 1 s
duration [7]. cERL gun operates GaAs cathode at 500 kV
with 100 mm gap in CW. 𝐸𝐵𝐷 depends on the pulse lengthand it is proportional to 𝜏−1/6 keeping a same break down
rate for RF case [8]. If the relation is applied to SACLA
electron gun (10 MV/m field at 1 s pulse), we get 32 MV/mwith 1 ns pulse. We use this number as 𝐸𝐵𝐷 for GaAs cathode. Usually, the peak field in a gun structure is observed ina corner of electrode and not on the cathode. Therefore, we
assume 𝜅 = 1 for GaAs case. Other metal which has higher
𝐸𝐵𝐷 than that of GaAs doesn’t cause breakdown even thefield is higher than that on the cathode.
An empirical relation between the breakdown voltage
𝑉𝐵𝐷 and the electrode gap 𝑑 is known as [9],⎧𝑉𝐵𝐷 ∝ 𝑑𝑑 ≤ 1.0 × 10−3{
(4)⎨
{𝑉𝐵𝐷 ∝ 𝑑 0.6 𝑑 > 1.0 × 10−3 .
⎩If 𝑑 is less than 1 mm, 𝑉𝐵𝐷 is proportional to 𝑑, i.e. thebreakdown field is a constant, 200 MV/m for the class 1
copper and 32 MV/m for GaAs. On the other hand, for 𝑑 >1mm, the breakdown voltage is proportional to 𝑑 0.6 and 𝐸𝐵𝐷
is less.According to Eq. (3), smaller 𝑑 gives higher current.

Figure 1: Current density 𝐽 from DC electron gun (A/mm2 )as a function of 𝑑. The gun is operated at 𝐸𝐵𝑅 .
The current density from a DC gun is
3/2
𝐸𝐵𝐷𝑂
𝐸 3/2
𝐽 = 2.33 × 10−6 𝐵𝐷 = 3.68 × 10−8 3/2.
𝜅 𝑑 1.1
𝜅3/2 √𝑑(5)

3/2
𝐽 is proportional to 𝐸𝐵𝐷𝑂
and 𝑑 −1.1 . Figure 1 shows thecurrent density 𝐽 from DC gun as a function of 𝑑 for Cu(black line) and GaAs (red line). The current density is
calculated as the gun is operated at 𝐸𝐵𝐷 . 𝜅 is assumed to be
2 for Cu and 1 for GaAs in these calculations. The current
density is rapidly decreased for large 𝑑.
To extract bunch charge 𝑄 from a cathode with area 𝑆, thebunch length 𝜎𝑧 in RMS is𝑄
𝑄𝜅3/2 𝑑 1.1 𝑐
𝜎𝑧 == 2.72 × 107
,
(6)
3/2
𝐸𝐵𝐷𝑂
𝑆
𝐽𝑆√12

where 𝑐 is speed of light. The bunch length 𝜎𝑧 is defined as
moving with 𝑐.

Because the acceleration is done by the static field, we
account only the thermal contribution as the energy spread.
Space charge makes acceleration and deceleration field, but
it doesn’t contribute to the emittance as long as the field islinear. Because the energy spread is flat over the bunch, theproduct of 𝜎𝑧 and the energy spread by the thermal energy
gives the longitudinal emittance, 𝜀𝑛,𝑧 . It is given as
𝑄𝜅3/2 𝑑 1.1 𝑘𝑇
𝜀𝑛,𝑧 = 2.72 × 107
,
(7)
3/2
𝐸𝐵𝐷𝑂
𝑆 2𝑚𝑐2where 𝑘 is Boltzmann constant, 𝑇 is temperature of cathode.
The thermal energy can be measured as 𝑀𝑇 𝐸 (Mean
Transverse Energy) in the transverse emittance. The normalized transverse emittance 𝜀𝑛,𝑥 is given as

𝑀𝑇 𝐸
,
(8)
𝑚𝑐2
where 𝜎𝑥 is RMS of the transverse beam size. In the ideal
case, 𝑀𝑇 𝐸 depends only on temperature, but it is distorted
by surface roughness, laser wave length (residual energy),
etc. If the thermal energy is isotropic, MTE measured in thetransverse direction is equivalent to that in the longitudinal
direction with a factor of 1/2.
By comparing 𝑀𝑇 𝐸 between bulk GaAs and super-lattice(SL) GaAs, SL GaAs has an advantage, because SL GaAs
crystal forms a mini-band with its width is 35 meV [10].
By assuming a fixed laser wave-length, electron excited by
the laser has an energy and momentum values, but they
are blurred by thermalisation. That is same for bulk and
SL GaAs, but the spread is smaller for SL than that of bulk
GaAs, because the band width of the SL mini-band is smaller.
N. Yamamoto obtained 𝜀𝑡 = 0.15 ± 0.02𝜋𝑚𝑚.𝑚𝑟𝑎𝑑 for SL
GaAs/GaAsP cathode with 1 mm radius and > 760 nm
laser wavelength, which corresponds to 51 meV MTE. This
MTE is in a same order of the thermal energy at 300 K,
𝑘𝑇 ∼ 26 𝑚𝑒𝑉. For copper, the lowest MTE ever observed is
6 meV [11]. In this measurement, the copper is cryogenicallycooled at 30 K. We use MTE 51 meV for GaAs and 6 meVfor copper.
The normalized 6D emittance 𝜀𝑛,6𝐷 is obtained as a product of these emittances without any correlations as,(𝑘𝑇 )2 𝑐 𝑄𝜅3/2 𝑑 1.1
𝜀𝑛,6𝐷 ≡ 𝜀2𝑛,𝑡 𝜀𝑛,𝑧 = 2.72 × 107.
3/2
2(𝑚𝑐2 )2 √12 𝐸𝐵𝐷𝑂 𝜋(9)which is independent from the emission area, 𝑆 and depends
on 𝑄 and 𝑑. Figure 2 shows 𝜀𝑛,6𝐷 as a function of 𝑑 withthe dotted, dashed, and solid lines for 1000 pC, 100 pC,
and 10 pC bunch charge, respectively. The black and redlines show those for copper and GaAs, respectively. It is
proportional to 𝑄 and simply increased with a larger 𝑑.
The beam brightness from DC gun 𝐵𝐷𝐶 is expressed as
the ratio of 𝑄 and 𝜀𝑛,6𝐷 as
3/2
𝑄2(𝑚𝑐2 )2 √12 𝐸𝐵𝐷𝑂 𝜋
𝐵𝐷𝐶 ≡= 3.68 × 10−8
. (10)
𝜀𝑛,6𝐷(𝑘𝑇 )2 𝑐 𝜅3/2 𝑑 1.1
𝜀𝑛,𝑥 = 𝜎𝑥 √It is independent from the bunch charge 𝑄 and the cathodearea 𝑆. It depends on the thermal energy 𝑘𝑇, the breakdown
limit 𝐸𝐵𝐷𝑂 , and the electrode distance 𝑑. Figure 3 shows- 1066 -Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 FRP059

the non-linearity in these processes to suppress the emittance growth. An example is SACLA [1] starting from a
DC biased gun with 1 ns beam, and ending to 10 fs bunchwith the multi-stage bunching. The compensation of the
non-linearity is essential.
Emittance growth by non-linear space charge is also an
issue. In principle, if the space charge is linear in all axes,
the emittance growth doesn’t occur. 3D ellipsoidal bunch is
proposed for the full linearization [12]. An essential problem
is that the solutions in free space (3D ellipsoidal) and in gun
structure (Space charge limited flow) are different. We haveto consider the matching between the different boundary
conditions to obtain the highest brightness from the DC gun.
Figure 2: 𝜀𝑛,6𝐷 is plotted as a function of 𝑑 for 1000, 100,
and 10 pC bunch charge with dotted, dashed, and solid lines,
respectively. The black and red lines are copper and GaAs
cathode, respectively. The cathode radius is 1 mm radius.
the brightness as a function of 𝑑 for Cu (red line) and GaAs
(black line). They are peaked at d=1 mm, 3.8 × 1016 C/m3 for
Cu, and 9.5 × 1013 C/m3 for GaAs. The difference between
copper and GaAs come from the thermal energy 𝑘𝑇 and thebreakdown limit, 𝐸𝐵𝐷𝑂 .

Figure 3: Brightness (C/m3 ) for Cu and GaAs are plotted
as a function of the electrode distance 𝑑 with red and blacklines, respectively. They are peaked at the lowest 𝑑.
As a summary of electron emission from a DC biased gun,the beam emittance can be very small, because the spacecharge limited flow is made by a simple structure. Although
the bias voltage is limited, but the surface field can be large if
a narrow electrode gap is employed. The narrow gap has anadvantage to increase the current density at the space charge
limitation and increase the breakdown limit field, 𝐸𝐵𝐷 . Of
course, there are practical limitations to limit the narrow gap,
but a narrow gap is a possible solution to improve the beam
performance.
If the surface field is limited very low, the bunch lengthfrom gun is huge. In this case, we have to start a low frequency buncher and employ a multi stage bunching sections
to reach a short bunch length such as 10 fs. Although it
doesn’t necessarily make it worse, but we have to care about

RF ELECTRON GUN

A photo-cathode RF gun is a device to generate electron
beam by the photo-electron emission in an RF cavity. The
voltage to extract the beam is higher than that of DC gun,which is capable to generate a high intensity bunch with ashort bunch length. The emittance growth by the non-linear
space charge effect can be suppressed by the linearizationand the quick acceleration.
In the RF electron gun, the bunch length is controlled by
the time width of the laser pulse. However, the space charge
force, especially in near of the cathode, where Lorentz 𝛾
is still low, limits the electron emission. Therefore, when
considering the ultimate short bunch generation from an RF
electron gun, the space charge, not the laser pulse duration,is the main factor that determines it. We consider a deltafunction like laser pulse for the electron emission. The
charge density is computationally infinite, but the amount ofcharge is finite, so the electric fields at the head and tail ofeach bunch can be calculated. In other words, the head andtail of the bunch sees different accelerating fields, resulting
a finite bunch length by velocity modulation. This bunch
length is then used as the shortest bunch length that can begenerated from the RF gun.
Lorentz 𝛾 in a RF gun cavity as a function of the distancefrom cathode 𝑧 is [13]1𝜋𝛾(𝑧) = 1 + 𝛼𝑘𝑧 + {cos 𝜙 − cos( + 𝑘𝑧)} ,(11)22
where 𝑘 is wave number of RF, 𝛼 is defined as
𝑒𝐸0
𝛼≡
,
(12)
2𝑘𝑚𝑐2
where 𝑒 is elementary charge, 𝐸0 is peak field (envelope),
𝑚𝑐2 is the rest mass energy of electron, 𝜙 is RF phase. Asan efficient acceleration, 𝜙 = 𝜋/2 is assumed. The second
cosine term of Eq. (11) shows the variation of 𝛾 in the cellboundary, but we neglect this term since this is not essential
for our purpose. Then, Eq. (11) is rewritten as𝛾(𝑧) ∼ 1 + 𝛼𝑘𝑧.
(13)Time to travel from the cathode to the gun end is
𝐿 𝑑𝑧
𝐿
𝛾
𝜏=∫
=∫
𝑑𝑧,
(14)
0 𝛽𝑐
0
𝑐√𝛾2 − 1
where 𝐿 is the RF gun cavity length, 𝛽 is Lorentz beta, 𝑐 is
speed of light. It can be integrated by replacing 𝑡 = 𝛾2 =(1 + 𝛼𝑘𝑧)2- 1067 -Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 FRP059√(1 + 𝛼𝑘𝐿)2 − 11𝑑𝑡=.
𝑡1 2𝛼𝑘𝑐 √
𝛼𝑘𝑐
𝑡−1

𝜏=∫

𝑡2

(15)

With a typical parameters such as 𝑒𝐸0 = 100𝑀𝑉 /𝑚 and
𝐿 = 0.15𝑚 (1.5 cells in S-band), (1 + 𝛼𝑘𝐿)2 − 1 can beapproximated as ∼ (1 + 𝛼𝑘𝐿)2 , then Eq. (15) becomes
1 + 𝛼𝑘𝐿𝜏∼(16)
𝛼𝑘𝑐
Due to the space charge effect, the acceleration field at
the bunch tail is decreased. We consider a cylindrical bunchwith charge 𝑄 and cross section 𝑆 in electric field.We assume
the electric field is parallel to the cylinder axis. The field at
the bunch head 𝐸0 , and at the bunch tail 𝐸1 is in a relation
𝐸1 = 𝐸0 − 𝑄/(𝜀0 𝑆) according to Gauss’s law, where 𝜀0 is
permittivity of vacuum. The reduction of the accelerating
field by the bunch charge is 𝑄/𝜀0 𝑆. Due to this reduction of
the accelerating field, the bunch tail is delayed as
1𝑄𝑑𝜏 = −.
(17)
𝛼𝑘𝑐 𝜀0 𝑆𝐸0𝑑𝜏 is the delay of the bunch tail from the bunch head. The
bunch length in RMS 𝜎𝑧 caused by the space charge effect
1 1 𝑄
is obtained as
𝜎𝑧 =
,
(18)
√12 𝛼𝑘 𝜀0 𝑆𝐸0
where we assume electron is moving with 𝑐. Figure 4 showswhere Δ𝐸𝑡ℎ and Δ𝐸𝑟𝑓 are contributions by the thermal energy and RF, respectively. Δ𝐸𝑡ℎ is determined by temperature 𝑇 as Δ𝐸𝑡ℎ = 𝑘𝑇 /2.
The energy spread by RF is obtained as follows. Lorentz
𝛾 at the end of the gun cavity is [13]1
𝛾 = 1 + 𝛼 {(𝑛 + ) 𝜋 sin 𝜙 + 𝑐𝑜𝑠𝜙}(20)2where 𝑛 is number of cells and we set 1 for 1.5 cell RF
gun. 𝜙 is RF phase and 𝜋/2 is assumed for the effective
acceleration. Then, we consider the energy variation byphase spread Δ𝜙𝜕𝛾
1
Δ𝜙2Δ𝜙 ∼ −𝛼 (1 + ) 𝜋
,
(21)
𝜕𝜙
2
2
where we took up to the second order. In our case, the bunch
length and the phase variation in a bunch is starting fromzero, and increased up to the final value, 𝜎𝑧 and Δ𝜙. We
assume the phase variation as a function of 𝑧 is linear,𝑧Δ𝜙(𝑧) = 𝑘𝜎𝑧(22)𝐿
where 𝐿 is the cavity length. Δ𝜙2 in Eq. (21) is replacedwith the integral ofΔ𝛾 =𝑘 2 𝜎2𝑧
1 𝐿
𝑧 2.
∫ (𝑘𝜎𝑧 ) 𝑑𝑧 =𝐿 0𝐿3
Equation (21) is rewritten as(23)

𝑘 2 𝜎2𝑧1
Δ𝛾 = 𝛼 (1 + ) 𝜋
.
(24)
2
6
This variation contributes to the longitudinal emittance.
2 =RMS of the energy variation is obtained as 𝑠𝑖𝑔𝑚𝑎𝛾𝛾2̄ − 𝛾̄ 2 . The result is
𝑘 2 𝜎2𝑧1
𝜎𝛾 = 𝛼 (1 + ) 𝜋
.
2
9√5

(25)

The total energy spread is obtained as quadratic sum ofthese components as Eq. (19).
2𝑘 2 𝜎2𝑧 ⎫𝑘𝑇 2 ⎧
1{ 2}
2
𝜎𝐸 = ( ) + ⎨𝑚𝑐 𝛼 (1 + ) 𝜋⎬ ,22

Figure 4: 𝜎𝑧 is plotted as a function of spot size in radius 𝑟
for 1000, 100, and 10 pC bunch charge with dotted, dashed,and solid lines, respectively.
𝜎𝑧 as a function of spot size in radius 𝑟 for 1000, 100, and
10 pC bunch charge with dotted, dashed, and solid lines, respectively. 𝜎𝑧 is proportional to 𝑄 and inversely proportional
to the cathode area, 𝜋𝑟 2 . Because the calculation is based on
the linear approximation for cosine curve, the bunch length
more than 𝜆/4, 25 mm for S-band is not correct. The bunch
splits into two in this case.
The total energy spread of the bunch is determined by thethermal energy of cathode, space charge effect, and energy
variation by RF field. However, the energy spread by the
space charge is linear regarding to the longitudinal position
and it doesn’t contribute to the longitudinal emittance. Then,
we calculate the energy spread as
2 + Δ𝐸 2
Δ𝐸 2 = Δ𝐸𝑡ℎ
𝑟𝑓

(19)

(26)
{
⎩
⎭
9√5 }
The product of 𝜎𝑧 and 𝜎𝐸 gives the longitudinal emittance𝜀𝑛,𝑙𝜎𝜀𝑛,𝑧 = 𝜎𝑧 𝐸2
(27)
𝑚𝑐
2
where 𝜎𝐸 is normalized with 𝑚𝑐 . The thermal energy (orMTE) depends on the material and laser wavelength. The
lowest MTE observed for NEA GaAs and copper are 51 meV[10] and 6 meV [11], respectively. 6 meV MTE was observedwith copper cathode at 30 K.
Transverse emittance of beam from a RF gun has twocomponents as same as in the longitudinal case; thermal
contribution and RF contribution. Time-variant focusing by
RF induces the additional emittance. The first component
is identical to Eq. (8). The second component by RF field
𝜀𝑟𝑓
𝑛,𝑥 is [13]𝛼𝑘𝜎𝑥 𝜎2𝜙
𝜀𝑟𝑓
,
(28)
𝑛,𝑥 =
√2
where 𝜎𝜙 is the bunch size in RF phase. In this estimation by
Kim, the longitudinal bunch size is assumed to be a constant- 1068 -Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 FRP059

in RF gun and the shape is Gaussian. If the bunch starts withzero length and grows up to a bunch length determined by
the space charge effect as same as assumed in Eq. (22), 𝜀𝑟𝑓𝑛,𝑥becomes as22𝛼𝑘𝜎𝑥 𝜎𝜙
𝜀𝑟𝑓
.
(29)
𝑛,𝑥 =
9√5
The transverse emittance is obtained as a sum of these twocomponents. The sum of the energy term of the emittances
is taken quadratically as
2

Figure 6 shows the beam brightness 𝐵 as a function of
the bunch charge 𝑄. The cathode radius is fixed at 5 mm.
𝜀𝑛,𝑧 and 𝜀𝑟𝑓
𝑛,𝑥 is proportional to square of 𝑄, smaller 𝑄 gives
higher brightness. The thermal contribution to 𝜎𝑛,𝑧 and
𝜀𝑛,𝑥 are constant, a saturation at low 𝑄 is observed. The
brightness with 10 pC bunch charge is 4.9 × 1015 C/m3 and6.5 × 1013 C/m3 for copper and GaAs, respectively.

SUMMARY2𝑟𝑓
𝜀2𝑛,𝑥 = (𝜀𝑡ℎ(30)𝑛,𝑥 ) + (𝜀𝑛,𝑥 )𝑡ℎ𝑡ℎ
𝜀𝑛,𝑥 . 𝜀𝑛,𝑥 is dominated by 𝜀𝑛,𝑥 , except for 1000 pC in 𝑟 <0.5𝑚𝑚.
Figure 5 shows 6D emittance 𝜀𝑛,6𝑑 with the same definition as that of DC gun, Eq. (9). 𝜀𝑛,6𝑑 is small for thelarge beam spot size. In the large 𝑟 region where the thermal
contribution is dominant for 𝜀𝑛,𝑧 , 𝜀𝑛,6𝐷 is a constant. The
reason is understood as that the dependence of 𝑟 to 𝜀𝑛,𝑧 and
𝜀𝑛,𝑥 are canceled to each other.

We discussed the highest beam brightness from DC andRF electron guns, by assuming ideal conditions such as the
non-linear space charge is suppressed completely. Beambrightness is measured as the ratio of the bunch charge andthe normalized 6D emittance. Table 1 summarizes the highest beam brightness from these guns in unit of C/m3 . DC
gun generates the highest brightness 3.8×1016 C/m3 with thecryogenically cooled copper cathode, and 9.5 × 1013 C/m3
with GaAs cathode in room temperature. RF gun generates
slightly less brightness as 4.9 × 1015 C/m3 for the copper
cathode, and 6.5 × 1013 C/m3 for GaAs cathode.
Table 1: Summary of the Peak Beam Brightness for DC andRF guns
Gun structureDCRF

Cu cathode3.8 × 1016
4.9 × 1015

GaAs cathode9.5 × 1013
6.5 × 1013

REFERENCES
Figure 5: 𝜀𝑛,6𝑑 is plotted as a function of beam radius 𝑟 for
1000, 100, and 10 pC bunch charge with dotted, dashed, andsolid lines, respectively. The black and red lines those forcopper and GaAs cathode.

[1] K. Togawa et al., Phys. Rev. S. T. Accel. and Beams
12,080706(2009).
[2] T. Nishitani et al., J. Appl. Phys.97,0944907(2005).
[3] X. Jin et al., Appl. Phys. Lett. 105, 203509(2014).
[4] C. D. Child, Phys. Rev. Series I.32(5):pp.492-511 (1911).
[5] R. Nagai et al., Rev. Sci. Instr. 81(3),033304(2010).
[6] S. Kobayashi et al., IEEE Tras. DEI Vol.4. pp.841-847(1997).
[7] K. Togawa et al., Phys. Rev. ST Acc. and Beams
10,020703(2007).
[8] S. Doebert et al., SLAC-PUB-11207(2005).
[9] R. V. Ratham ed., ”High Voltage Vacuum Insulation Basic Concepts and Technological Practice”, Academic Press, London(1995).
[10] N. Yamamoto et al., J. Appl. Phys.102, 024904(2007).
[11] S. Karkare, Photocathode Physics of Photoinjecors (P3) workshop (2018).

Figure 6: The brightness 𝐵 is plotted as a function of 𝑄. The
red and black lines show those for Cu and GaAs, respectively.
The cathode diameter is fixed at 5 mm.

[12] O. J. Luiten et al., Phys. Rev. Lett. 93, 094802(2004).
[13] K-J. Kim, Nucl. Instr. and Meth. A275 201-218(1989).

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